STRUCTURE-PRESERVING ISOGEOMETRIC ANALYSIS FOR COMPUTATIONAL FLUID DYNAMICS BY STEVIE-RAY JANSSEN

Transcription

1 STRUCTURE-PRESERVING ISOGEOMETRIC ANALYSIS FOR COMPUTATIONAL FLUID DYNAMICS BY STEVIE-RAY JANSSEN M.SC. THESIS November 7t, 2016

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3 STRUCTURE-PRESERVING ISOGEOMETRIC ANALYSIS FOR COMPUTATIONAL FLUID DYNAMICS M.SC. THESIS To obtain te degree Master of Science at Delft University of Tecnology, by Stevie-Ray JANSSEN Student of Aerospace Engineering and Applied Matematics, Delft University of Tecnology, Delft, Te Neterlands.

4 Tesis Committee: Aerospace Engineering: Dr. ir. M.I. Gerritsma Prof. dr. S. Hickel Dr. ir. A. Pala Applied Matematics: Dr. M. Möller, Prof. dr. ir. C. Vuik Prof. dr. J.M.A.M. van Neerven TU Delft, supervisor TU Delft TU Eindeoven TU Delft, supervisor TU Delft TU Delft November 7t, 2016 An electronic version of tis dissertation is available at ttp://repository.tudelft.nl/.

5 CONTENTS Preface vii 1 Introduction 1 2 Exterior Calculus Manifolds and Coordinate Spaces Forms and Operations Cains and Cocains Te Codifferential and te Inner Product Poisson Problems Summary of Results Structure Preserving Isogeometric Analysis Te Sobolev-DeRam Complex Basis Functions and Isogeometric Analysis DeRam Conforming Spline Spaces Discretisation of te Poisson Problems Summary of Results Scalar Transport Te Interior Product and te Lie Derivative Transport Problems Discretisation of Transport Problems Summary of Results Euler Equations Te Euler Equations Discretisation of te Euler equations Results, and Analysis Summary of Results Conclusion & Future Directions 85 A DeRam Coomology 87 B Derivation of Adjoint Relations 90 C Generalised Piola Transformations 93 D Well-Posedness of te Poisson Problem 95 E Derivation of Edge Functions 100 F Construction of Bounded Cocain Projections 102 v

6 vi CONTENTS G Derivation of Transport Problems 105 H Construction of Periodic Basis 110 I Derivation of te Euler Equations and Invariants 113 J Proof Discrete Euler Equations Satisfy Conservation Laws 117 K Picard and Newton Linearisation 121 Bibliograpy 125 Index 128

7 PREFACE Man, I went troug tis tesis project like a loose cannon. I finally found myself near te battlefront of knowledge! I was excited to learn, to develop new ideas, and I wanted to do everyting. Call it a young man s ubris. I ve looked at space-time discretisations, te variational multiscale metod, discontinuous Galerkin, Lagrangian and Eulerian mesing tecniques, symplectic time integrators, and, of course, isogeometric analysis and structure-preserving discretisation tecniques. Unsurprisingly, I struggled somewat wit combining tis all into one tesis, wic resulted in te fact tat some material as been removed or moved to appendices to improve bot consistency and readability. I encourage any reader to ceck te appendices as well. In 2013 I attended dr. Gerritsma s course on structure-preserving discretisations for CFD. Te matematics fascinated me even toug I barely understood wat e was talking about pseudo-differential form-coboundary-operator-mumbo-jumbo). I wanted to learn more about tis stuff, and asked im if I could do my tesis project wit im. He, being a nice man and all, accepted my request. He brougt me into contact wit dr. Möller, wo was organising biweekly seminars on isogeometric analysis at tat time. Since tis stuff was also pretty interesting, I ended up combining bot topics into structurepreserving isogeometric analysis for CFD. I would like to tank tem bot for guiding me along te way. I would like to tank some of my predecessors, Hiemstra, dr. Kreeft, Natale, dr. Pala, Rebelo, Tosniwal, old students of dr. Gerritsma, wose teses inspired me and elped me mastering te material. I d like to tank prof. Huges for aving me over at ICES in Austin for a few monts. Tank you for your great lectures on isogeometric analysis. Special tanks to René and Deepes for your guidance and friendsip. I really enjoyed working wit you guys. I d also like to tank te oter guys at ICES for letting me kick teir asses at fußbal. Finally, I d like to tank my girlfriend, Mariel, wo as taken suc good care of me. Stevie-Ray Janssen Rotterdam, November 2016 vii

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9 1 INTRODUCTION Computational Fluid Dynamics CFD), te field of computer simulations of fluid flows, as establised itself as one of te pillars of modern fluid flow researc. Te field as a wide range of teoretical applications wic include te researc of turbulence, tidal dynamics, and blood flows, but also knows important industry applications as te design of aircraft, micro-processors, or submarines. Te Navier-Stokes equations govern many fluid flows. Te underlying geometric structure of tese equations facilitates structures and symmetries. Classical solution metods often neglect tese structures. As a result, tese metods fail to conserve invariant quantities, and are unable to simulate te desired pysical beaviour, [1]. Recently it is sown tat solution tecniques can be constructed tat do preserve te underlying geometry of pysics. Tese tecniques are known as structure-preserving tecniques. "In discretising te incompressible Navier-Stokes one encounters two difficulties, p.265 [2]". Te first difficulty is te discretisation of te non-linear convective term. Te second difficulty used to be te treatment of te saddle-point problem tat describes te relation between pressure and velocity. Structure-preserving metods ave reduced te second difficulty to a triviality. Tese metods aim at preserving fundamental structures of operators and operands found in partial differential equations PDE s). Concepts from differential geometry and algebraic topology are used to analyse tese structures. In tis approac we distinguis between relations tat depend on metric and tose tat are purely topological. Te topological relations can be described using algebraic equations, wic allows exact representation in te discrete setting as argued by Tonti [1]. Satisfying tese topological relations lies at te eart of structure-preserving discretisation tecniques. Te application of differential geometry and algebraic topology as resulted in te development of a wide variety of structure-preserving discretisation tecniques. Rougly speaking, tey can be split in two groups, eac aving a different approac of discretising te Hodge- operator. In te first approac one defines an explicit primal- and dual-grid on wic te action of te Hodge- operator is approximated. Tis approac leads more to a finite-differencing, or finite-volume type metods. Fundamental works include te 1

10 2 1. INTRODUCTION 1 work on mimetic finite difference by Hyman, [3], and te work on discrete exterior calculus by Desbrun et al., [4]. In anoter approac one uses te inner product definition of te Hodge- operator in a variational approac, leading to finite-element type metods. Fundamental works in are te ones on finite element exterior calculus by Arnold et al., [5], conforming mimetic discretisations by Bocev and Hyman, [6], and mimetic spectral element metod Gerritsma et al, [7]. Isogeometric analysis IGA) was pitced by Huges et al. in 2005 to bridge te gap between CAD design and analysis, [8]. In tis approac Non-Uniform Rational B-splines NURBS) are used for bot representation of te pysical geometry and variational analsysis. It was sown tat NURBS posses superior approximation beaviour in analysis over Lagrange polynomials, classically used in finite elements analysis, [9, 10]. Buffa et al. developed a teoretical framework of structure-preserving isogeometric analaysis for elliptic PDE problems, [11]. Variety of successful implementations include te Maxwell equations by Buffa, [12], and te unsteady Navier-Stokes equations by Evans, [13]. Tese works deal wit similar discretisation callenges, e.g. unsteadiness, non-linearity and asymmetric operators, tat are also present in te proposed model problem. Te incompressible Euler equations ave a ric geometric structure as described in [14]. Te Hamiltonian structure of tese was unveiled in te classical work by V. Arnold [15]. An interesting discretisation was introduced by L. Rebolz in wic velocity and vorticity are treated as independent variables [16]. Te structure-preserving discretisation by Pala makes use of tis idea [17]. Te study proposed will contribute to tese developments, and will aim at te furter development and analysis of structurepreserving tecniques for te Euler equations. Te main callenge is to discretise te non-linear convective operator in a structure-preserving manner, i.e. preserving global quantities as kinetic-energy, vorticity, and enstropy. Tis tesis is te result of a researc project on discretisation tecniques for CFD. Tis tesis was written to obtain te degree of Master of Science at te faculties of Aerospace Engineering AE) and Applied Matematics AM) at Delft University of Tecnology. Te project was divided in tree objectives. Te first objective was to study te existing existing structure-preserving IGA framework for elliptic PDE s, capter 2 and capter 3). Te objective for AM was to expand tis teoretical framework to discretise yperbolic PDE s capter 4). Te objective for AE was to construct a structure-preserving IGA discretisation for te incompressible Euler equations, capter 5. All results are combined in tis single tesis document. In line wit te mentioned works we will analyse te PDE s using concepts from differential geometry, exterior calculus, and algebraic topology. Fundamental concepts and teorems from tese fields will be introduced and reviewed in capter 2. We will identify te DeRam complex, wic can also be constructed in a discrete setting, and formulate elliptic PDE problems. In capter 3 we will introduce a variational framework, te foundation of te finite-elements metod, introduce concepts from isogeometric analysis, and construct a framework of structure-preserving isogeometric analysis, using te concepts introduced in capter 2. We will use tese results to discretise various formulations of te scalar Poisson problem. In capter 4 we expand our exterior calculus toolbox, and introduce variational formulations for yperbolic PDE s. We study te scalar transport equation, and discuss various discretisation strategies. Finally, in capter 5 we introduce

11 3 te Euler equations for incompressible fluids, and derive structure-preserving discretisations tat conserve integral invariants over time. Conclusions and future directions will be discussed in capter 6 1

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13 2 EXTERIOR CALCULUS Te laws of pysics are generally analysed using te language of vector calculus. Following te works mentioned in te introduction, we will analyse pysical relations using differential geometry, exterior calculus, and algebraic topology. Te first advantage of tis approac is tat analysis is performed on arbitrary geometries, wic are known as manifolds. Results tat will be introduced do not depend on some local description of pysics. Te second and main advantage of tis approac, is tat it enables a clear distinction between relations tat do depend on te description of local geometry, i.e. tat depend on metric, and tose tat are purely topological, i.e. tat are independent of metric. In te discrete setting local geometry is approximated on some mes, wic induces an approximation error for metric-dependent relations. Topological relations can be described using algebraic equations, wic allows exact representation in te discrete setting. Satisfying tese topological relations lies at te eart of structure-preserving discretisations tecniques. In tis capter we will introduce fundamental concepts from te fields, and aim to relate tese concepts to numerical analysis. In section 2.1 we introduce manifolds and coordinate systems on wic we define primal and dual vector spaces. Dual vectors, known as differential forms, are used to describe pysical quantities. Basic operations as te wedge product and te exterior derivative d on differential forms are presented in section 2.2. Te exterior derivative is a generalisation of te grad, curl, and div operations from vector calculus. Relations between differential forms tat are governed by te exterior derivative are topological, and can be satisfied exactly by discrete spaces using cains and cocains. In section 2.3 we define tese cains and cocains from te field of algebraic topology. We sow tat differentiation is exact in te discrete setting. Metric-dependent relations are introduced in te next section, section 2.4. Here we define an inner product troug te Hodge- operator. In te final section, section 2.5, we return to te pysics, and formulate PDE s on arbitrary geometries using te formalisms introduced in tis capter. 5

14 6 2. EXTERIOR CALCULUS 2.1. MANIFOLDS AND COORDINATE SPACES 2 In tis section we will introduce manifolds and coordinate bases. A manifold is a topological description of some geometry. One needs to introduce a coordinate system to measure metrics, i.e. positions, distances, or angles on a specific manifold. For example, a disk in R 2 is a manifold, one can measure metric using te Cartesian or te equivalent polar coordinate system. Derivatives of te coordinate system s mapping form a basis. Tis basis is used to describe quantities known as vectors. Tere exists a dual basis wic is used to describe quantities known as differential forms. Differential forms are coordinate invariant containers of quantities tat are associated wit points, lines, surfaces, or volumes. We will use tese differential forms as te variables in our coordinate invariant description of PDE s. A n-dimensional manifold M is a topological space wic is locally omeomorpic to Euclidean space R n. We define carts U α to be te open subsets tat cover te manifold M, i.e. M = α A U α. 2.1) For eac cart U α, tere exists a bijective coordinate map ϕ M,α : U α R m, wit m n. Te coordinate map ϕ M,α associates coordinates to eac point p U α, p = x 1, x 2,, x m ) = ϕ 1 M,α ˆx 1, ˆx 2,, ˆx m ), 2.2) wit ˆx 1, ˆx 2,, ˆx m ) ϕ M,α U α ). Te pair U α,ϕ M,α ) defines a local coordinate system. We require tat local coordinate systems are compatible in regions were carts overlap. Hence, on te overlap of carts U α U β we demand existence of a diffeomorpism, wic is te map, ϕ M,β ϕ 1 M,α : ϕ M,αU α U β ) ϕ M,β U α U β ). 2.3) A diffeomorpism is essentially a transformation from one coordinate system to anoter. We illustrate tese concepts in Figure 2.1. Te collection of all pairs A = {U α,ϕ M,α )} α A is known as te atlas of M. We formally define a manifold using te atlas as follows.

15 2.1. MANIFOLDS AND COORDINATE SPACES 7 R n M U α ϕ M,α 2 R m ϕ M,β ϕ 1 M,α U β ϕ M,β Figure 2.1: Differentiable manifold M wit carts U α, and U β. U α U β. Points in U α are described by te map ϕ M,α : U α R n, similarly for U β. On U α U β tere exists diffeomorpism ϕ M,β ϕ 1 M,α. Definition 2.1 Differentiable manifold, Sec. 1.2 [18])). A n-dimensional k-differentiable manifold is a topological space for wic tere exists an atlas A = {U α,ϕ M,α )}, wose coordinate maps are k-times differentiable, i.e. ϕ M,α C k. A smoot manifold is a manifold wit ϕ M,α C. Te continuity constraints are required to perform analysis on manifolds. Eac point p in M is eiter an interior point or a boundary point. We call p = x 1, x 2,, x n ) M an interior point if tere exists an open ball around p tat is fully contained in M, i.e. ɛ > 0 s.t. {x M : dx, p) < ɛ} M, were dx, p) is te distance measured in some metric. A point p is a boundary point if it is not an interior point. Te set of boundary points of M denoted by M. We can use a coordinate system to locally define a linear vector space. First consider a point p on te local coordinate system U, ϕ) on n-dimensional differentiable manifold M. Consider a curve γs) = x 1 s), x 2 s),, x n s)) troug p, wit parametric coordinate s ɛ,ɛ), suc tat γ0) = p, as sown in Figure 2.2. Te derivative wit respect to parametrisation parameter s, known as te tangent vector at p along γs), is given by, γp) = dγ d s = d x 1 p d s x + d x 2 1 p d s x + + d x n 2 p d s x. 2.4) n p

16 8 2. EXTERIOR CALCULUS T p M γ 2 p M γ Figure 2.2: Tangent vectors γ of curves γ troug p define te tangent space T p M on differentiable manifold M. Te collection of all possible tangent vectors troug p span a linear vector space, wic we call te tangent space of M at p, denoted by T p M. Te collection of all tangent spaces on M is known as te tangent bundle, denoted by T M wit, T M := p M T p M. 2.5) Now we can construct a coordinate basis at p. Consider n linearly independent curves γ 1,γ 2,,γ n ) troug p given by, γ 1 x 1 ) = x 1,0,,0), γ 2 x 2 ) = 0, x 2,,0),. γ n x n ) = 0,0,, x n ). Te corresponding n tangent vectors define a basis x i at a point p. Definition 2.2 Basis, Sec. 1.3 [18])). x i Given a n-dimensional differentiable manifold M we can define basis vectors at a point p M, x i := 0,,0, d x i,0,,0). 2.6) d s A given basis induces orientation locally on M, i.e te sign of te determinant of te components of te basis is considered positive or negative. We call a manifold M orientable if te orientation is globally consistent. Tis means tat any locally induced orientation at a point p can be mapped continuously along any closed curve in M suc tat te orientation is not reversed at te endpoint in p.

17 2.1. MANIFOLDS AND COORDINATE SPACES 9 A vector v can be expanded in some basis at p, v p =v 1 p) x 1 + v 2 p) x v n p) x n = v i p) i x i. Te action of a vector on a smoot scalar function on te manifold, f C M ), along curve γs), is equivalent to te directional derivative of f in γs), v p f ) = v i p) f i x i = d x i f i d s p x i = d f γs)). d s Note tat tis operation is a derivation. Te dual space of te tangent space is known as cotangent space. Te cotangent space at p, denoted by Tp M, consists of linear functionals acting on te tangent space. Te collection of all cotangent spaces on M is known as te cotangent bundle, and is denoted by T M, wit T M := Tp M. 2.7) p M Te contangent space Tp M is also a linear vector space. We define te basis d x 1,d x 2,,d x n ) for Tp M as te dual of te vector basis in te tangent space T pm. Definition 2.3 Dual basis d x i, Sec. 2.1 [18])). Given a n-dimensional differentiable manifold M we define dual basis vectors at a point p M, suc tat ) d x i x j = δ i j, 2.8) were δ i is te Kronecker delta. Te expansion of a dual vector α, in some dual basis at j p is given by, α p =α p p)d x 1 + α 2 p)d x α n p)d x n = i α i p)d x i. 2 Te action of a dual vector αx) = α 1 x)d x 1 +α 2 x)d x 2 + +α n x)d x n on a vector vx) = v 1 x) + v 2 x) + + v n x) x 1 x 2 x n, is given by, n n αv) = ρ i v j n n n x j d xi = ρ i v j δ i j = ρ i v i. i j Dual vectors, known as a differential 1-forms, will be used as te objects tat old pysical quantities in our formulation of pysics. Te space of differential 1-forms on M is denoted by Λ 1) M ), wit α 1) Λ 1 M ). Scalar functionals, known as differential 0- forms, are denoted by β 0) Λ 0) M ). In our analysis, vectors will be used to describe variables wic are intimately related to te geometric mapping, e.g. deformation and velocity. Differential forms will play te role of pysical variables living on te geometries. In te next section we expand te concept of differential forms, and introduce tools and tecniques required for analysis. i j i

18 10 2. EXTERIOR CALCULUS 2.2. FORMS AND OPERATIONS 2 Te differential 1-form resembles te integrand of a line integral. It is a description of a quantity defined along a line, wic can be measured by integrating it along a curve. Te wedge product, tat will be introduced sortly ereafter, is used to construct iger dimensional forms tat resemble integrands of surface- and volume integrals. In tis section we will introduce te fundamental tools required for differentiation and integration of differential forms. Differentiation of forms is governed by te exterior derivative d. Te exterior derivative is used to differentiate forms, and acts like te grad, curl, or div operator depending on te dimension of te specific form. Te exterior derivative induces important topological connections on te spaces of differential forms, wic is encoded in te DeRam sequence. Proper understanding of its structures is key to te development of structure-preserving discretisations. Te Stokes teorem is te fundamental teorem for integration of forms, and will be introduced at te end of tis section. We will introduce a product operator to build iger dimensional objects like surfaces or volumes. Higer dimensional differential forms can be constructed using te wedge product. Definition 2.4 Wedge product, Sec. 2.5 [18])). Let Λ k), and Λ l) be spaces of k-forms and l-forms on M R n. Te wedge product is te mapping, : Λ k) M ) Λ l) M ) Λ k+l) M ) 2.9) tat satisfies, α k) + β l ) ) γ m) = α k) γ m) + β l) γ m) α k) β l) ) γ m) = α k) β l) γ m) ) f α k) β l) = α k) f β l) = f α k) β l) ) α k) β l) = 1) kl β l) α k), wit α k) Λ k),β l) Λ l ),γ m) Λ m), and f Λ 0). Example 2.1 Action of te wedge product ). Te action of te wedge product is similar to tat of te cross- and dot-product in vector

19 2.2. FORMS AND OPERATIONS 11 calculus. Consider 1-forms α 1),β 1) Λ 1 M ) wit M R 3. example 1: d x 1 d x 1 = d x 1 d x 1 d x 1 d x 1 = 0 example 2: α 1) β 1) = α 1 d x 1 + α 2 d x 2 + α 3 d x 3 ) β 1 d x 1 + β 2 d x 2 + β 3 d x 3 ) = α 1 β 2 d x 1 d x 2 + α 1 β 3 d x 1 d x 3 + α 2 β 1 d x 2 d x 1 + α 2 β 3 d x 2 d x 3 + α 3 β 1 d x 3 d x 1 + α 3 β 2 d x 3 d x 2 = α 1 β 2 α 2 β 1 )d x 1 d x 2 + α 2 β 3 α 3 β 2 )d x 2 d x 3 + α 3 β 1 α 1 β 3 )d x 3 d x 1 example 3: α 1) β 1) γ 1) = α 1 β 2 γ 3 α 2 β 1 γ 3 )d x 1 d x 2 d x 3 + α 2 β 3 γ 1 α 3 β 2 γ 1 )d x 2 d x 3 d x 1 + α 3 β 1 γ 2 α 1 β 3 γ 2 )d x 3 d x 1 d x 2 α 1 α 2 α 3 = det β 1 β 2 β 3 γ 1 γ 2 γ d x1 d x 2 d x Note tat te result α 1) β 1) is a 2-form wit components similar to a b, wic describes a surface in vector calculus. Furtermore, α 1) β 1) γ 1) is a 3-form wit components similar to a b c, wic describes a volume parallelepiped) in vector calculus. Te example reveals a connection between differential forms and geometric objects. Moreover, we observe tat forms resemble integrands of line-, surface-, and volume integrals. In M R 3 ) we associate 3-forms wit volumes, 2-forms wit surfaces, 1-forms wit lines, and 0-forms wit points. Eac differentiable k-form can be interpreted as some quantity tat is to be measured integrated) in some structure, e.g. flux is measured troug surface, and density in volume, and so on. Eac pysical quantity also as some natural orientation wit respect to its underlying structure. We distinguis between tangential inner-oriented) forms, and normal outer-oriented) forms sec. 28 of Burke [19]). A flux is defined normal troug a surface, wile vorticity is defined tangential to te surface. Furter elaboration on te pysical interpretation of differential forms will follow in section 2.5. We present some examples.

20 12 2. EXTERIOR CALCULUS 2 Λ k) geometry example Λ 0) points pressure p 0) = px, y, z) Λ 1) lines velocity v 1) = v x x, y, z)d x + v y x, y, z)d y + v z x, y, z)d z Λ 2) surfaces flux σ 2) = σ z x, y, z)d x d y +σ y x, y, z)d z d x + σ x x, y, z)d y d z Λ 3) volumes density ρ 3) = ρx, y, z)d x d y d z. Definition 2.5 Exterior derivative d, Sec. 2.6 [18])). Differentiation of differential forms is governed by te exterior derivative d. Te exterior derivative d is te mapping, d : Λ k) M ) Λ k+1) M ), 2.10) satisfying, dα k) + β k) ) = dα k) + dβ k) dα k) β m) ) = dα k) β m) + 1) k α k) dβ m) d 2 α k) := d dα k) = 0, Te action of te exterior derivative on a 0-form α 0) Λ 0) is given by, dα 0) := n α x i d xi. 2.11) i=1 Te exterior derivative generalises te grad, curl, and div operators from multivariate calculus. Furtermore, te fact tat d 2 = 0, known as te nilpotency property, is equivalent to te curl grad = 0 and div curl = 0 relations. Te exterior derivative defines a sequence of mappings, known as te DeRam complex. For M R n we ave R Λ 0) d Λ 1) d d Λ n) d Te action of te exterior derivative on various forms is sown below. Example 2.2 Action of te exterior derivative d in R 3 ). Te action of te exterior derivative is very similar to te grad, curl and div operators. Using te properties of te wedge product, and te exterior derivative d, we find for M R 3,

21 2.2. FORMS AND OPERATIONS 13 0-form: dφ 0) = φ x 1 d x 1 + φ x 2 d x 2 + φ x 3 d x 3 1-form: du 1) u1 = d x 1 + u 1 d x 2 + u ) 1 d x 3 d x 1 x 1 x 2 x 3 u2 + d x 1 + u 2 d x 2 + u ) 2 d x 3 d x 2 x 1 x 2 x 3 u3 + d x 1 + u 3 d x 2 + u ) 3 d x 3 d x 3 x 1 x 2 x 3 u3 = u ) 2 d x 2 d x 3 u1 + u ) 3 d x 3 d x 1 u2 + u ) 1 d x 1 d x 2 x 2 x 3 x 3 x 1 x 1 x 2 2-form: dω 2) ω1 = d x 1 + ω 1 d x 2 + ω ) 1 d x 3 d x 2 d x 3 x 1 x 2 x 3 ω2 + d x 1 + ω 2 d x 2 + ω ) 2 d x 3 d x 3 d x 1 x 1 x 2 x 3 ω3 + d x 1 + ω 3 d x 2 + ω ) 3 d x 3 d x 2 d x 3 x 1 x 2 x 3 ω1 = + ω 2 + ω ) 3 d x 1 d x 2 d x 3 x 1 x 2 x 3 2 Hence dφ 0) defines a 1-form wit components of grad operator in Cartesian coordinates, du 1) defines a 2-form wit components of curl operator, and dω 2) defines a 3- form wit component of te div operator. Important topological connections are encoded in te exterior derivative and te DeRam sequence. Connections between exterior derivative s kernel and range, subspaces of te Λ k) s, ave an important role in te existence and uniqueness of PDE solutions. More on te structure of te DeRam sequence can be found in Appendix A. From te nilpotency of te exterior derivative it follows tat if dα k) = β k+1), ten dβ k+1) = 0. Te converse is true on contractible manifolds. Contractible manifolds can be continuously reduced to a point. Examples include any star) domain in Euclidean space. Teorem 2.1 Poincaré Lemma, Sec. 5.4 [18])). Let β k) be a k-form on contractible manifold M, suc tat dβ k) = 0. Tere exists a nonunique α k 1), suc tat, Often α k 1) is referred to as a potential function. dα k 1) = β k). 2.12) Te teorem proofs existence of solutions for balancing relations is pysics. Tese relations, often of te form gradφ) = f, curl ω) = g, and div v) =, are fundamental in

22 14 2. EXTERIOR CALCULUS te description pysics. Furtermore, note tat te teorem proofs existence of potentials for every volume form, as dρ n) = 0 for any volume form ρ n). In vector calculus tis result is known as te surjectiveness of te div operator. 2 Example 2.3 Potential flow). Irrotational inviscid) flow satisfies V = 0, or in terms of differential geometry, d v 1) = 0. Hence by te Poincaré lemma, tere exists a velocity potential function φ 0) Λ 0) s.t. dφ 0) = v 1). Flow solutions satisfying dφ 0) = v 1) are known as potential flow solutions. We will discuss potential flow in more detail in section 2.5. We measure a differential form by pairing it wit an equidimensional submanifold, e.g. a 2-form is measured by pairing it wit some surface S, a 3-form wit some volume V, and so on. We present a few examples. Λ k) geometry example Λ 0) point: x pressure P = x p0) Λ 1) curve: γ circulation Γ = γ v 1) Λ 2) surface: S flux Φ = S σ2) Λ 3) volume: V mass M = V ρ3). Integration is governed troug te generalised Stokes teorem. Teorem 2.2. Generalised Stokes Teorem, Sec. 3.3 [18]) Let α k) Λ k) be a k-form, and M be a k + 1)-dimensional submanifold. Ten, dα k) = α k). 2.13) M Te teorem generalises te classical Stokes, Green s and divergence teorem from multivariate calculus, as well as te fundamental teorem of calculus. Te teorem states tat te boundary operator is te formal adjoint of te exterior derivative d, i.e. M M,dα k) = M,α k). 2.14) Note tat tis result is purely topological. In te next section we will introduce te discrete analogue of te generalised stokes teorem CHAINS AND COCHAINS In discretisation tecniques we find te formation of some a cell complex, i.e. a grid of vertices, edges, faces, and volumes. Tese components, known as k-cells. Using k-cells we can define discrete counter-parts of submanifolds, cains, and differential forms, cocains. Cocains can be integrated by pairing tem wit cains. We can apply te generalised Stokes teorem, Equation 2.13, to tese structures to derive a discrete version of te exterior derivative, te coboundary operator δ. Te coboundary operator satisfies te same topological structure as te exterior derivative. Its action is encoded in incidence matrices E k,k 1). Tese incidence matrices are te foundation of our discretisation approac.

23 2.3. CHAINS AND COCHAINS 15 Definition 2.6 k-cell, Appx. B.a [18])). A k-cell, τ k), is a k-dimensional submanifold of n-dimensional manifold M wic is omeomorpic to k-dimensional unit ball. Essentially k-cells are k-dimensional contractible spatial objects. From ere on we will form a grid of k-cells wic will act as a discrete manifold on wic we can define a discrete counterpart of differential forms. Grids are generally constructed using simplexes, or simplicial structures. In tis work we follow [6, 20] and use k-dimensional cuboids, k-dimensional objects for wic tere exists a bijection to te unit k-cube, {x [ 1,1] k }. Tese cuboids are convenient because of teir ability to old tensorproduct structures, and because Gaussian quadrature rules are defined on its interior. 2 Example 2.4 n-dimensional cuboids). A 3-dimensional cuboid is a convex polytope bounded by 6 quadrilateral faces, eac face is bounded by 4 edges, and eac edge is bounded by 2 nodes. Hence eac 3-dimensional cuboid consists of {8, 12, 6, 1} {0, 1, 2, 3}-dimensional objects. A 4-dimensional cuboid consists of {16, 32, 24, 8, 1} {0, 1, 2, 3, 4}-dimensional objects. Definition 2.7 Cell Complex, Appx. B.a [18])). A cell complex D on a n-dimensional differentiable manifold M is a finite collection of sets of {n,n 1,,0}-cells suc tat, i Te collection of n-cells form a cover M. ii Every face of a k-cell is contained in D. iii Any two k-cells τ k),σ k) D, υ k 1), τ k) σ k) = σ k) = τ k), i.e. τ k),σ k) sare a boundary υ k 1) i.e. tey indicate te same k-cell 2.15) Te cell complex forms te collection of nodes, lines, surfaces, volumes tat is equivalent to grid structures in numerical metods. It will also form a discrete manifold structure on wic we can perform numerical analysis. For a cell complex to act like a manifold, we will need to provide orientation to its k-cells. Oriented k-cells act as a basis. Sets of oriented k-cells are known as k-cains. Tese k-cains can be interpreted as discrete submanifolds. Definition 2.8 k-cains c k), Appx. B.a [18])). Te space of k-cains on a cell complex D, denoted by C k) D), is te space of all sets of k-cells. Hence eac k-cain c k) C k) D) can be written as, c k) = i c i τ k),i, wit τ k) D, 2.16) wit c i te expansion coefficients. We use k-cains to describe k-dimensional pats troug oriented cell complexes. In our analysis we consider coefficients c i {0,±1}, were c i = ±1 provides relative orientation of te i -t k-cell wit respect to a global orientation, and c i = 0 indicates tat i -t k-cell is not witin te considered pat.

24 16 2. EXTERIOR CALCULUS Example cains). Consider te oriented cell complex D sown in Figure 2.3. Closed pats around surfaces s 1, s 2 are described by 1-cains c 1),c 2) C 1) D), 2 c 1 = l 1 + l 6 l 3 l 5, c 2 = l 2 + l 7 l 4 l 6. Te space of C 1) D) is a linear vector space, ence we are able to add c s to construct k) new pats. For example, c 1) + c 2) describes te pat tat encloses s 1 s 2. Next, we introduce te boundary operator δ, wic relates a k-cain to its boundary k 1)-cells. Definition 2.9 Boundary operator, Appx. B.a [18])). Te boundary operator is te mapping, : C k) C k 1), c k) = j c j τ k),j, 2.17) were te boundary operator acting on te j -t k-cell τ k),j is given by, τ k),j = i e i j τ k 1),i, wit, +1, if orientation of τ k 1),i is te same as te boundary of τ k),j e i j = 1, if orientation of τ k 1),i is opposed to tat of te boundary of τ k),j. 0, if τ k 1),i is no face of τ k),j Te boundary operator acting on a k-cell is encoded in an incidence matrix E k 1,k) wit entries E k 1,k)i,j ) given by e j i. Te boundary operator satisfies c k) C k) D), c 0) =0, c k) =0, for wic te latter property translates to matrix notation as E k 2,k 1) E k 1,k) )c k) = 0. Finally, te boundary operator defines a sequence of mappings C 0) C 1) C n) R Example 2.6 Boundary operator ). Consider again te oriented cell complex D in Figure 2.3. We find tat s 1 = l 1 + l 6 l 3 l 5, s 2 = l 2 + l 7 l 4 l 6.

25 2.3. CHAINS AND COCHAINS 17 Now take cains c 2) = [s 1, s 2 ] T, c 1) = [l 1,l 2,,l 7 ] T, and c 0) = [p 1, p 2,, p 6 ] T. could obtain c 1) troug c 1) = E 1,2) c 2), and c 0) troug c 0) = E 0,1) c 1), wit E 0,1) =, E ,2) = 0 1, We 2 were, for example, te 6-t row of E 1,2) corresponds to l 6 for wic orientation is similar to te face of s 1, but opposite to te face of s 2. One could ceck tat E 0,1) E 1,2) c 2) = 0, i.e. te nilpotency c 2) = 0 of te boundary operator is exact in tese discrete structures. Tis is of great importance, as it will preserve te topological relations between exact and closed forms, introduced in te previous section, in te discrete setting. We will now introduce te discrete differential forms, known as cocains, wic assign real values to te k-cains introduced. p 4 l 3 p 5 l 4 p 6 l 5 s 1 l 6 s 2 l 7 p 1 l 1 p 2 l 2 p 3 Figure 2.3: Cell complex D on a 2-dimensional manifold. Positive orientation of k-cells is indicated by direction of te arrows. Definition 2.10 k-cocains c k), Appx. B.b [18])). Te space of k-cocains on a cell complex D, denoted by C k) D), is te space tat is dual to te space of k-cains C k) D). A k-cocain c k) expands in basis τ k), c k) = i c i τ k),i, 2.18) were τ k) is dual to a k-cell suc tat τ k),i τ k),j ) = δ i. Te action of a k-cocain is te j

26 18 2. EXTERIOR CALCULUS 2 mapping c k) : C k) D) R, wit c k),c k) := c k) c k) ) = c i c j τ k),i τ k),j = c i c j. i j i Note tat we could ave cosen any oter dual basis τ k),i tat spans C k). Cocains act as discrete differential forms, as will be sown. We now introduce te coboundary operator δ, wic is te discrete equivalent of te differential operator. Definition 2.11 Coboundary operator δ). Te coboundary operator δ : C k) C k+1) is te formal adjoint of te boundary operator, c k),δc k) = c k),c k). 2.19) It follows tat we can also encode te action of te coboundary operator in an incidence matrix, Hence we define, E k 1,k) c k),c k) = c k),e k 1,k) ) T c k). E k,k 1) := E k 1,k) ) T. 2.20) Te coboundary operator inerits nilpotency property δ δ = 0 from te boundary operator, c k),δδc k) 2.19 = c k),δc k) 2.19 = c k),c k) = 0, and it defines a sequence of mappings, R C 0) δ C 1) δ δ C n) δ. We can now connect te continuous differential forms from te previous section to te discrete cocain structures defined in tis section. Example 2.7 Discretisation of v 1) = dφ 0) ). Consider again te oriented cell complex sown in Figure 2.3. Now consider te equation v 1) = dφ 0), defined on some manifold tat is covered by te cell-complex. We can integrate over any line element l and use Stokes s teorem to find, l v 1) = l dφ 0) 2.13 = l 1 φ 0) Next we discretise te variables v 1),φ 0) troug integration, [ ] T v 1) = v 1), v 1),, v 1), l 1 l 2 l 7 φ 0) = [ φ 0) p 1 ),φ 0) p 2 ),,φ 0) p 6 ) ] T.

27 2.3. CHAINS AND COCHAINS 19 Te v 1), and φ 0) are cocains, eac coefficient assigns a real value to an oriented k- cell, just as a differential form assigns a real value to a manifold. We can now solve te discrete equation using te coboundary operator. Consider any 1-cain c 1), we find v 1) = c 1) c 1) φ 0) v 1),c 1) = φ 0), c 1) = δ φ 0),c 1) v 1) E 1,0) φ 0),c 1) = 0. 2 So we can solve te equation by evaluating v 1) = E 1,0) φ 0). Te result is exact, no approximation as been made in tis derivation, altoug information of te continuous forms ave been lost in te process. In fact, any relation of te form α k) = dβ k 1) can be discretised and solved using tis approac, wic preserves exactness of te exterior derivative, Appendix A. We are able to preserve te exactness of te topological structure of our equations wen we go to te discrete setting. However, te discretisation of te continuous forms to te discrete cocains came at a cost. We were able to conserve te global quantity at te cost of information concerning local beaviour. If we want to go back to a continuous form, we need to reconstruct te solution, performing some kind of interpolation. Bocev and Hyman introduced a framework to encode tis process [6], wic depends on te reduction- and reconstruction-operators. Definition 2.12 Reduction Operator R k) ). Te reduction operator R k) is a mapping R k) : Λ k) M ) C k) D), 2.21) tat satisfies commutativity δr k) = R k+1) d, illustrated below. Λ k) d Λ k+1) C k) R k) δ C k+1) R k+1) Definition 2.13 Reconstruction Operator I k) ). Te reconstruction operator I k) is a mapping I k) : C k) D) Λ k) M ), 2.22) were Λ k) M ) Λk) M ) is an approximate finite-dimensional subspace of Λ k) M ). Te reconstruction operator needs to satisfy te same commutative property, δi k) = I k+1) d, illustrated below.

28 20 2. EXTERIOR CALCULUS C k) δ C k+1) I k) I k+1) 2 Λ k) d Λ k+1) Using te reduction and reconstruction operators it is possible to describe te discretisation process troug te commuting diagram in Figure 2.4. Successful discretisation needs to satisfy te following conditions; i Consistency condition, R k) I k) = I, ii Approximation condition, I k) R k) = I + O p+1 ), were is some measure of grid size, and p te polynomial order of te approximation. We can use operators R, I to define discrete operators acting on cocain c k) C k). Consider any operator acting on Λ k), s.t. : Λ k) Λ l). We can ten define discrete operator troug c k) := R k) I l) c k). Λ k) d Λ k+1) R k) R k+1) C k) δ C k+1) I k) R k) I k+1) R k+1) Λ k) d Λ k+1) Figure 2.4: Commuting diagram of te reconstruction and reduction operators R and I. Classic discretisation tecniques can be described troug te reduction and reconstruction operators. In finite differencing tecniques, one reconstructs using Taylor series expansion, wile in finite volume tecniques, one often reconstructs using polynomial interpolation. In finite element tecniques, forms are reconstructed using basis functions. In finite differencing, and finite volume tecniques, an explicit grid is generated, and one can reduce te continuous forms using integration, as in Example 2.7. Grids structures in finite element tecniques are less obvious. Explicit reduction in finite element tecniques can be performed using dual basis functions or projection operators.

29 2.4. THE CODIFFERENTIAL AND THE INNER PRODUCT THE CODIFFERENTIAL AND THE INNER PRODUCT So far we ave treated relations tat are topological. In te previous section, we ave sown tat te exterior derivative d can be discretised witout loss of its structure. In tis section we will introduce operations tat depend on metric. In te discrete setting local geometry is approximated on some mes, wic induces an approximation error for metric-dependent relations. Fundamental is te Hodge- operator, wic maps an oriented form to its corresponding dual-oriented form. Te Hodge- operator defines an inner-product, wic is a vital tool in variational analysis. 2 Definition 2.14 Hodge- operator, Sec [18])). Te Hodge- operator in M R n is a mapping : Λ k) Λ n k) wit, αx 1,, x n )d x i 1 d x i k = sign αx 1,, x n )d x j 1 d x j n k, 2.23) were {i 1, i k } {1,,n} and {j 1, j n k } its complement, and {, if {j1,, j n k,i 1,,i k } is an even permutation of {1,,n} sign =. +, else Wen applied twice to a differential k-form we find tat α k) = 1) kn k) α k). Te Hodge- defines a complex of sequences of mappings, known as te double DeRam complex, Figure 2.5. d d d d R Λ 0) Λ 1) Λ n) d d d d Λ n) Λ n 1) Λ 0) R Figure 2.5: Double DeRam complex; two dual DeRam sequences are connected troug te Hodge- operator. Example 2.8 Geometrical Interpretation of te Hodge- operator). Te Hodge- transforms differential forms to teir dual forms. Its action doesn t affect te magnitude or direction of te components, but maps te basis to its dual configura-

30 22 2. EXTERIOR CALCULUS tion. Some examples are given, 2 R 2 : 1 = d x 1 d x 2 d x 1 = d x 2 d x 2 = d x 1 R 3 : 1 = d x 1 d x 2 d x 3 d x 1 = d x 2 d x 3 d x 2 d x 1 = d x 3 Now consider te case R 2, for wic te double DeRam complex is sown in Figure 2.6. Any point value becomes a face value and vice versa. A value associated along a line, becomes a value associated troug a line. R p d d d Λ 0) Λ 1) Λ 2) l s s d d d d Λ 2) Λ 1) Λ 0) l p R Figure 2.6: Double DeRam complex in R 2. Te double-deram complex sows tat we can identify two distinct orientations of a pysical quantity on a geometric object. Te top row in Figure 2.6 corresponds to inner-orientated object or tangent orientated). From left to rigt we observe a source in a point, some quantity along a line, rotation in a surface. Te bottom row corresponds to outer-oriented objects or normal oriented). Here we see source troug a face, flux troug a surface, rotation around a point. Topological connections are sown orizontally, wile metric-dependent relations are sown vertically. Definition 2.15 Inner Product, ) M, Sec [18])). Te Hodge- operator defines an L 2 -inner product, ) M : Λ k) Λ k) R by α k),β k) ) M := α k) β k), 2.24) wit α k),β k) Λ k) on M. M

31 2.4. THE CODIFFERENTIAL AND THE INNER PRODUCT 23 Te inner product is essential in variational analysis, and plays a vital role in te construction of a finite-elements framework as we will see in te next capter, capter 3. Wit te Hodge- operator we can define te adjoint of te differential operator, te codifferential d. Definition 2.16 Codifferential d, Sec [18])). Te codifferential d is a mapping d : Λ k) M ) Λ k 1) M ) for wic 2 d α k) = 1) nk 1)+1 d α k) 2.25) Te codifferential defines a sequence of mappings, known as te DeRam complex, wic connects k + 1-forms to k-forms. For M R n we ave Λ 0 Λ 1 Λ n R d d d d It s easy to see tat d is nilpotent, i.e. d d = 0 using te and d 2 properties). Example 2.9 Action of codifferential d in R 3 ). Te action of te exterior derivative is similar to te div, curl and grad operators. Using te properties of te wedge product, and te exterior derivative d, we find for M R 3, 1-form: [ d u 1) u1 = + u 2 + u ] 3 x 1 x 2 x 3 2-form: [ d ω 2) = 1) n+1 ω3 ω ) 2 d x 1 ω1 + ω ) 3 d x 2 ω2 + ω ] 1 )d x 3 x 2 x 3 x 3 x 1 x 1 x 2 2-form: [ ρ d ρ 3) = d x 2 d x 3 + ρ d x 3 d x 1 + ρ ] d x 1 d x 2 x 1 x 2 x 3 Hence d u 1) defines a 0-form wit components of div operator, d ω 2) defines a 1-form wit components of curl operator, and d ρ 3) defines a 2-form wit component of te grad operator. Te codifferential is te adjoint of te exterior derivative. Teorem 2.3 Integration-by-parts, Sec [18])). For forms α k 1) Λ k 1), and β k) Λ k), we ave te relation, α k 1),d β k) ) M = dα k 1),β k) ) M α k 1),β k) ) M, 2.26) Now any differential k-form can be decomposed in ortogonal components tangential and normal to te boundary, i.e. α k) = α n d x n +α t d x t. Hence we can evaluate te boundary integral in terms of tangential and normal components of α k 1),β k), α k 1),β k) ) M = α t,β n dγ. 2.27) M

32 24 2. EXTERIOR CALCULUS Proof can be found in Appendix B. We will often use Teorem 2.3 to reduce continuity constraint on our solution space, or to get rid of te co-differential. 2 We combine te exterior derivative and te codifferential operator to define te Laplace- DeRam operator, wic is a generalised Laplace operator for k-forms. Definition 2.17 Laplace-de-Ram Operator, Sec [18])). For any k-form we define te Laplace-de-Ram operator : Λ k) Λ k) as α k) := dd α k) + d dα k) = d + d ) 2 α k) 2.28) Because d β 0) = 0 for any 0-form β 0), te Laplace-DeRam operator acting on a 0-form reduces to = d d, wic is te equivalence of te scalar Laplacian. Te operators reduces to = dd wen applied to a volume n-form. Te codifferential governs te a topological structure tat mirrors te structure of te exterior derivative. It governs existence of solutions analogue to te Poincaré lemma, Equation In Appendix A we combine properties of bot operators to derive te Hodge decomposition, wic states tere exists an unique decomposition in ortogonal complements for eac k-form. Teorem 2.4 Hodge decomposition, Sec [18])). For any k-form on a closed manifold M tere exists an unique decomposition α k) = dφ k 1) + d ψ k+1) + k), 2.29) were k) is a armonic form, i.e. k) = 0. Note tat tis doesn t imply uniqueness of φ k 1) and ψ k+1). On a contractable manifold te space of armonic forms is empty, i.e. k) = POISSON PROBLEMS Tere exists one-to-one correspondence between PDE s described using vector calculus and differential geometry. In multivariate calculus we distinguis between pysical variables associated wit scalars and tose associated wit vectors. In differential geometry we associate pysical variables wit points, lines, surfaces, and volumes. Any scalar function can be used to describe a point or volume 0,3)-form in R 3, and any vector function can be used to describe a line or surface 1,2)-form. Te duality between a point and a volume, and tat between a surface and a line is encoded in te Hodge - operator. Let s analyse te example of potential flow, wic we introduced in section 2.2. Example 2.10 Potential flow in R 2 ). Te velocity field can be interpreted as a field of lines along wic fluid parcels are transported. Hence we can associate a 1-form wit velocity, i.e. v 1). Now te quantity ω 2) = d v 1) is known as vorticity and is a measure of rotation witin te flow. In te case of irrotational flow, i.e. ω 2) = 0 in our domain, we would get d v 1) = 0. Assuming tat te domain is simply connected, tere exists a potential function φ 0) suc tat dφ 0) = v 1) Teorem 2.1).

33 2.5. POISSON PROBLEMS 25 Consider te quantity given by q 1) = v 1). Te dual form q 1) is defined troug a line like a flux. Mass flux in vector calculus is given by q = ρ v, were ρ is mass density wic we will assume to be constant). It follows tat te 1-form q 1) describes scaled mass flux note tat te minus sign is encoded in te Hodge- operator). Te quantity σ 2) = d q 1) ten describes total flux troug our two-dimensional volumes. If we don t consider any mass source or sink ten we obtain d q 1) = 0. Now again tere exists a potential function ψ 0) suc tat dψ 0) = q 1) Teorem 2.1). Tis potential function is called te stream function. Note tat in R 3 te stream function is a 1-form wit tree components. 2 d d φ 0) v 1) ω 2) d d σ 2) q 1) ψ 0) Figure 2.7: Tonti diagram of te relations governed by potential flow teory. On te top row, inner-oriented relation of te potential function φ 0), velocity v 1), and vorticity ω 2) is sown. On te bottom row, outer-oriented relation of mass production σ 2), mass flux q 1), and te stream function ψ 0) is sown. Te relations are visualised in a Tonti diagram Figure 2.7. Horizontal relations described troug te exterior derivative d are conservation equations. Tey are independent of metric and describe a relation between cange of a quantity witin a domain, and te flux troug te boundary of te domain. Vertical relations describe constitutive equations. Tey depend on metric and material parameters. Examples include Fourier s law q = κ u in eat conduction problems, or Darcy s law q = κ µ p in flow troug porous media. We could combine te relations to obtain te Laplace equation for te potential function φ 0) = 0, or te stream function ψ 0) = 0, wic are relatively easy to solve. Example 2.11 Stokes flow in R 2 ). Consider a Newtonian viscosity µ constant and isotropic), incompressible mass density ρ constant) fluid. Now in te viscous flow regime Re 1, i.e. viscous forces are dominant over advective inertial forces), te flow is governed by equations, { ν v p + f = 0 v = 0, were ν is te kinematic viscosity, v is te velocity, p te pressure, and f any body forces. Te first equation describes conservation of momentum, te second equation describes

34 26 2. EXTERIOR CALCULUS conservation of mass. We can translate to te language of differential geometry. We obtain, { νd d v 1) + d p 0) = f 1) 2 d v 1) = 0. We can obtain an equivalent dual formulation of tese equations by pre-multiplying te equations wit te Hodge-, define dual forms, i.e. q 1) = v 1), p 0) = p 2), and f 1) = g 1), and use te identities of d, d and to simplify te result. For te Stokes flow equations one obtains, { νdd q 1) d p 2) = g 1) d q 1) = 0. Te advantage of tis formulation, is tat te mass conservation constraint d q 1) = 0 depends on te exterior derivative d. Tis means tat a discretisation based on tis formulation can be made suc tat it satisfies conservation of mass exactly pointwise using te incidence matrices introduced in section 2.3). Structure preserving spectral element solution metods for te Stokes flow problem are studied in [21]. Teir discretisation satisfies pointwise divergence-free solution. Te last example sowed tat formulation as consequences in te discrete setting. We can coose te formulation suc tat certain relations are satisfied strongly pointwise, wile oters are weakly satisfied. Te coice of formulation will also ave a consequence on te boundary conditions. Some boundary condition may be enforced strongly essential) in one formulation, wile it may be enforced weakly natural) in a different one. So far we ave not considered te possibility of arbitrary pysical geometries. Often, te quantity of interest will be defined on a geometry troug a mapping. In te finiteelement metod we will need to pull-back differential k-forms to some parent domain on wic te numerical integration rules are defined. Te pull-back operator commutates wit te exterior derivative. In Appendix C we derive appropriate transformations for eac k. Tese relations conserve te topological structure of te exterior derivative on arbitrary domains SUMMARY OF RESULTS Analysis of PDE s troug exterior calculus enables us to distinguis between relations tat depend on metric, and tose tat are topological. Topological relations can be described by algebraic equations, wic can be satisfied exactly in te discrete setting. Proper discretisation of tese relations guarantees numerical stability, and exact beaviour of simulated pysics. Te exterior derivative d is a topological operator, wic describes derivatives of differential forms analogous to te grad-, curl-, and div- operators. In te discrete setting it can be satisfied witout error by construction of incidence matrices. Tis construction preserves te kernel structure of te exterior derivative, wic governs existence of

35 2.6. SUMMARY OF RESULTS 27 PDE solutions. Te kernel structure of te exterior derivative is encoded in te DeRam complex. R Λ 0) d Λ 1) d Λ 2) d Λ 3) d Te Hodge- is a metric-dependent operator, wic maps differential forms to teir dual representation. It is used togeter wit te exterior derivative d to describe elliptic PDE s. Its action needs to be approximated in te discrete setting. In finite volume tecniques its action is approximated troug construction of a dual grid, wereas in finite-element tecniques is is approximated troug inner-product relations. Finally, te Hodge- allows us to map PDE s to a favourable configuration, in wic certain relations or boundary conditions can be satisfied exactly. 2

36

37 3 STRUCTURE PRESERVING ISOGEOMETRIC ANALYSIS "Don t worry, everyting will B-spline" We can use powerful tools from functional analysis to prove well-posedness of elliptic PDE problems. In tis approac one derives a weak formulation of te PDE problem by taking te inner product wit a test function. Te weak formulation of te problem can be converted to a discrete problem by introducing a finite-dimensional basis in wic te unknowns and test-function can be expanded. In isogeometric analysis spline bases from computer aided design are used to discretise te problem. In tis capter we will introduce concepts from functional analysis, and explain te fundamentals of isogeometric analysis. We will construct a discretisation framework using splines tat is compatible wit te DeRam complex. Weak formulations of elliptic PDE problems tat define a symmetric bilinear operator ave been discretised successfully using te finite element metod. Mixed type formulations owever ave posed difficulties. Suc problems are known for spurious oscillations witin te numerical solution. In order to cope wit tese oscillations penalising tecniques were introduced, wic in turn ave a negative effect on te accuracy, Sec. 6.3 [2]). Classical is te work by Brezzi on stability requirements for tese mixed-type formulations [22]. It was sown tat spurious oscillations do not appear wen inf-sub stable finite element spaces are used. Using te concepts introduced in capter 2, we derive appropriate basis functions. Te resulting metod doesn t require penalising tecniques for stability, as sown by [5, 11, 23]. In section 3.1 we introduce fundamentals of functional analysis and introduce a DeRam sequence of Sobolev spaces. We will derive symmetric and mixed formulations of te Poisson problem, and prove well-posedness for bot problem types. Furtermore we will explain ow we can discretise te problem by considering finite-dimensional Sobolev subspaces. Bézier, B-spline, and NURBS- polynomial bases will be introduced in te context of isogeometric analaysis in section 3.2. Tese basis functions will be used 29

38 30 3. STRUCTURE PRESERVING ISOGEOMETRIC ANALYSIS to construct finite-dimensional Sobolev subspaces tat are compatible wit te DeRam complex in section 3.3. We will make a link to concepts introduced in capter 2, and unveil te cocain structure of spline spaces. Finally, in section 3.4, we successfully discretise bot Poisson problems using linear combinations of incidence matrices and mass matrices THE SOBOLEV-DERHAM COMPLEX Finite projection metods, e.g. isogeometric analysis and te finite element metod, can be derived from variational analysis. In tis approac one constructs a weak formulation of te PDE problem by taking te inner product wit some test function. Wellposedness of tese problems can be proven by selecting appropriate trial- and solution spaces. In tis section we construct tese spaces, suc tat tey satisfy te DeRam sequence. We introduce te Poisson problems tat will be studied in tis capter, and derive corresponding weak formulations. We end tis section by explaining ow we weak formulations can be discretised. Consider orientable manifold Ω wit tangent space T Ω and cotangent space T Ω. In section 2.4 we defined an inner product on tese spaces. We use te inner product to define inner product spaces L 2 Λ k). Definition 3.1 Inner product space L 2 Λ k), Sec. 2 [5])). Let Λ k) Ω) denote te space of k-forms on manifold Ω. We define te corresponding space of square integrable forms L 2 Λ k) Ω) as, were, i.e. te norm defined by te inner product. L 2 Λ k) := {α k) Λ k) : α k) L 2 Λ k) < }, 3.1) α k) 2 = α k),α k)) L 2 Λ = α k) α k) 3.2) k) Ω Ω Definition 3.2 Sobolev spaces W d Λ k), Sec. 2 [5])). We define te Sobolev space W d Λ k) Ω) as wit corresponding Sobolev norm, W d Λ k) := {α k) L 2 Λ k) : dα k) L 2 Λ k+1) }, 3.3) α k) 2 W d Λ k) = α k) 2 L 2 Λ k) + dα k) 2 L 2 Λ k+1) 3.4) Note tat W d Λ 0) is te equivalent of te Sobolov space H 1 Ω), W d Λ 1) of Hcurl;Ω), and W d Λ 2) of Hdiv;Ω), i.e. descriptions of Sobolev spaces generally encountered in literature, e.g. [11]. In Ω R 3 tese spaces satisfy te DeRam sequence, i.e. te DeRam Sobolev complex, W d Λ 0) d W d Λ 1) d W d Λ 2) d L 2 Λ 3)

39 3.1. THE SOBOLEV-DERHAM COMPLEX 31 Tis construction can be mirrored to construct Sobolev spaces wit respect to te codifferential d, wit corresponding Sobolev norm, W d Λ k) := {α k) L 2 Λ k) : d α k) L 2 Λ k 1) }, 3.5) α k) Wd Λ k) = αk) L 2 Λ k) + d α k) L 2 Λ k 1) 3.6) Tese spaces incorporates te existence of te DeRam complex given by 3 L 2 Λ 0) W d Λ 1) W d Λ 2) W d Λ 3) d d d Tese spaces act as trial- and solution-spaces in variational analysis of PDE problems. In tis approac one derives a weak formulation of te PDE problem. In tis capter we will study Poisson problems, e.g. tose appearing in potential flow teory, section 2.5. Problem 3.1 Te 0-form Poisson Problem in R 2 ). Consider te domain Ω = [0,1] 2 R 2 wit boundary Ω = Γ d Γ n, were Γ d Γ n =. Given forcing f 0) Λ 0), find solution ψ 0) Λ 0) suc tat, ψ 0) = f 0), ψ 0) = 0, on Γ d 3.7) n ψ 0) = 0, on Γ n A corresponding weak formulation of tis problem can be obtained by taking te inner product wit some test function w 0). Given forcing f 0) L 2 Λ 0), find solution ψ 0) W d Λ 0), suc tat, d w 0),dψ 0)) Ω = w 0), f 0)) Ω, w 0) W d Λ 0) 0, 3.8) were W d Λ 0) 0 is te space of compactly supported functions, Λ 0) 0 := {α0) Λ 0) : wit α 0) = 0 on Γ d }. Te derivation of tis weak formulation, and a proof of well-posedness of te weak formulation is given in Appendix D. In te derivation of te weak formulation we used integration by parts, Teorem 2.3, for two reasons. In te discrete setting we will use a finite span of basis functions of a given polynomial order. Integration by parts reduces continuity requirements on te span of basis functions. Te second reason is tat we now ave an expression tat doesn t contain te codifferential d, but only contains te exterior derivative d, wic we can discretise using te framework introduced in section 2.3.

40 32 3. STRUCTURE PRESERVING ISOGEOMETRIC ANALYSIS 3 Problem 3.2 Te 2-form Poisson Problem in R 2 ). Consider te domain Ω = [0,1] 2 R 2 wit boundary Ω = Γ d Γ n, were Γ d Γ n =. Given forcing f 2) Λ 0), find solution ψ 2) Λ 2) suc tat, ψ 2) = f 2), ψ 2) = 0, on Γ d 3.9) t ψ 2) = 0, on Γ n As before, we could multiply wit some test-function w 2), and apply integration by parts, to obtain. d w 2),d ψ 2)) Ω = w 2), f 2) ) Ω. 3.10) Tis formulation won t do us any good in te discrete setting, because it contains te codifferential d. Alternatively we write Equation 3.9 as an equivalent system of first order differential equations, d ψ 2) = v 1), d v 1) = f 2), ψ 2) t ψ 2) = 0, on Γ d = 0, on Γ n 3.11) Te corresponding weak formulation of tis system can be obtained taking inner products wit test functions q 1) and w 2), and apply integration by parts. Given forcing f 2) L 2 Λ 2), find solutions v 1) W d Λ 1) and ψ 2) W d Λ 2), suc tat, { d q 1),ψ 2)) Ω = q 1), v 1)) Ω, q1) W d Λ 1) 0, w 2),d v 1)) Ω = w 2), f 2)) Ω, w 2) L 2 Λ 2). 3.12) Te derivation of tis weak formulation, and a proof of well-posedness of te weak formulation is given in Appendix D. Well-posedness of tis formulation depends explicitly on te kernel structure of te Sobolev DeRam complex, W d Λ 0) d W d Λ 1) d L 2 Λ 2) To ensure well-posedness of te discretised problem, discrete function spaces need to be constructed tat satisfy te same structure. In section 3.3 ow we can do so. In te example of te 0-form Poisson equation, Example D.1, we obtained weak formulation of te form, Find u V, suc tat av,u) = F v), for all v V, were S and V are te solution- and trial function spaces, a, ) a bilinear form, and F ) a functional. In te Galerkin discretisation approac we take a finite-dimensional

41 3.1. THE SOBOLEV-DERHAM COMPLEX 33 subspace V V, and take te solution space equation to te trial S := V. In our case V will consist of te space spanned by te isoparametric basis, wic we will introduce in te next section, section 3.2. We define te finite-dimensional solution u suc tat it satisfies, Find u S, suc tat av,u ) = F v ), for all v V. Suc discretisation approac is known as a conforming metod. A discrete solution is sougt tat satisfies te continuous PDE problem. Well-posedness of te problem was proven for any element v V, and ence for any v V. Tis implies well-posedness of te discrete problem in te case of symmetric operator problems as te Poisson problem. Note tat well-posedness of te 2-form problem depends on te kernel structure of te Sobolev DeRam complex, wic needs to be mirrored by discrete spaces to prove well-posedness in te discrete setting. 3 R W d Λ 0) d d d W d Λ 1) L 2 Λ n). Now, because, au, v ) = F v ), and au, v ) = F v ), for all v V, We ave tat au, v ) au, v ) = au u, v ) = F v ) F v ) = 0. Tis is known as te Galerkin ortogonality. Te global error ɛ = u u ) is bounded. Teorem 3.1 Cea s Lemma, Sec. 1.5 [2])). Consider te finite-dimensional form of variational problem tat satisfies te conditions of Teorem D.3 or Teorem D.4. Te approximate finite element solution u to te analytical solution u V of te weak problem under consideration, is bounded by u u C min v V u v. If te finite-dimensional solution space is te span of a piecewise polynomial basis of polynomial order p, ten min u v C u) s, v V were is a measure of mes size, and C u) a positive constant tat depends on smootness of te solution u. Constant s depends on smootness of solution and polynomial order of te basis. Te teorem states tat te approximate solution converges to te exact solution in te Sobolev norm. We call te order of convergence optimal wen s = p. In te next section we will introduce isoparametric bases. Te span of tese bases will be used as our finite-dimensional solution- and trial spaces.

42 34 3. STRUCTURE PRESERVING ISOGEOMETRIC ANALYSIS BASIS FUNCTIONS AND ISOGEOMETRIC ANALYSIS In te restriction to finite-dimensional function-spaces one approximates pysical quantities as a linear combination of basis functions. Te callenge is ten to find te appropriate coefficients, suc tat te combination approximates te pysical solution. Te basis functions are generally defined on a simple regular mes, wic we call te parameter space. However, most often te pysical problem tat we aim to solve, is defined on some kind of geometry, wic we call te pysical space. To solve te problem in te pysical space, it is required to map te basis from te parent domain to te pysical domain. Te mapping can also be approximated as a linear combination of basis function wen it is not explicitly known. In mecanics we often find an intimate relation between geometry and te pysics. Accelerations, velocities and displacements affect te geometry on wic te equations need to be solved. Take for example a problem were we would solve for te displacements of some kind of structure or material at a given period of time. We would ten like to update our geometry using te found displacements. If te displacements and te mapping are defined in a different basis, we would need to interpolate or project our solution from one basis to te oter. In many cases solutions sensitive to errors in te geometry, e.g. tin-sell dynamics, boundary layers in fluid dynamics. Te interpolation errors can ave a significant impact on te approximate solution. To circumvent tis problem, one would need to use te same basis for te mapping as te one used for analysis. Tis is known as te isoparametric approac. In te modern day design process owever, geometries are dictated troug CAD Computer Aided Design). Many CAD programs use Bézier curves, B-splines and NURBS Non-uniform rational B-splines) to represent geometries. Te concept of using te same basis for analysis as te one used for te mappings is known as te isogeometric approac. Te isogeometric paradigm was pitced by te Huges group in 2005 to bridge te gap between CAD design and finite elements analysis [8]. In tis section we will introduce te framework of isogeometric analysis. We will first introduce various bases, Béziers, B-splines, and NURBS, tat are used to describe geometries in CAD. Tese basis functions will be te fundamental tool in our analysis. Definitions and formulas tat will be introduced sortly can be found in te book by Piegl and Tiller, [24]. Bézier curves are widely used in computer grapics, because tey can be manipulated intuitively, and are able to represent smoot geometries. A Bézier curve is a linear combination of Bernstein polynomials, wic can be efficiently evaluated troug te recursive decasteljau Algoritm. Definition 3.3 Bézier Curves, Sec. 1.3 [24])). A Bézier curve of polynomial order p is te mapping C t) : R R d given by, C ξ) = p λ B i,p ξ)p i, 0 ξ 1, 3.13) i=0 were λ i,p is te i -t Bernstein polynomial basis function, and coefficients {P i } wit

43 3.2. BASIS FUNCTIONS AND ISOGEOMETRIC ANALYSIS 35 P i R d are te control points. Te i -t Bernstein polynomial is defined troug, λ i,p ξ) = ) p ξ) i 1 ξ) p i. 3.14) i Te polygon tat interpolates te control points linearly is known as te control polygon or control net. Te curve lies in te convex ull of its control polygon, and te curves ends are tangent to te control polygon. Finally, te Bézier curves are variation diminising, i.e. tey preserve te local monotonicity. Note tat tere exists a cange of basis to express a Bézier curves in Lagrange polynomials instead of Bernstein polynomials. Lagrange interpolation polynomials are classically used in FEA. 3 Success of te Bézier basis is due to tese properties, wic enable for intuitive design. Classical polynomial interpolation does not satisfy tese properties. Te disadvantage of Bézier curves is tat tey cannot be controlled locally, i.e. canging one coefficients will affect te wole curve Bernstein polynomials ave full support over parameter domain ξ [0,1]). One solution would be to use curves tat consist of multiple Bézier segments, so called composite Bézier curves. Te drawbacks of tese curves are tat continuity is difficult to acieve over segments, and tat te number of control points increase wit te polynomial degree p. Tese difficulties are overcome by te use of B- spline curves. B-splines carry te same advantageous properties as Bézier curves, and ave local support. Definition 3.4 B-spline Curves, Sec [24])). A B-spline curve of polynomial order p is te mapping C ξ) : R R d suc tat, C ξ) = n λ i,p ξ)p i, ξ R, 3.15) i=1 were λ i,p is te i -t B-spline basis function, and P i R d te i -t control point. A B- spline is defined by its knot vector troug te Cox-de-boor recursive formula. A knot vector is a non-decreasing sequence of real numbers Ξ = {ξ 1,,ξ n+p+1 }, suc tat te entries ξ i, known as knots, satisfy ξ i ξ i+1, for i = 1,,n + p + 1. Furtermore, we require te first and final entry to be non-equal, and eac ave multiplicity of p + 1, i.e. open knot vector. Te i -t B-spline basis function λ i,p is ten defined troug te recursion, p = 0, λ i,0 ξ) = { 1, if ξi ξ < ξ i+1 0, else p > 0, λ i,p ξ) = ξ ξ i λ i,p 1 ξ) + ξ i+p+1 ξ λ i+1,p 1 ξ), 3.16) ξ i+p ξ i ξ i+p+1 ξ i+1 were we define ξ ξ i ξ i+p ξ = 0 if ξ i i+p ξ i = 0, and ξ i+p+1 ξ ξ i+p+1 ξ = 0 if ξ i+1 i+p+1 ξ i+1 = 0. By construction te B-spline basis functions are a partition of unity i.e. tey sum up to 1), and te i -t B-spline basis function is supported on te interval [ξ i,ξ i+p+1 ). Te continuity of a basis function can be reduced by using repeated knot values. Te Bernstein

44 36 3. STRUCTURE PRESERVING ISOGEOMETRIC ANALYSIS polynomial basis of order p can be obtained troug te recursion by taking knot vector Ξ = {0,,0,1,,1}. Hence te space of Bézier curves is contained witin te space of B-spline curves, Figure 3.1. Te derivative of a B-spline basis function is given by, d dξ λ i,pξ) = p p λ i,p 1 ξ) λ i+1,p 1 ξ). 3.17) ξ i+p ξ i ξ i+p+1 ξ i+1 3 NURBS B-splines Béziers Figure 3.1: Set topology of functions used in computational geometry. Example 3.1 Example basis). Consider knot vector {0,0,0, 1 2,1,1,1}. Using Equation 3.16 we find te B-spline basis for p = {0,1,2}, Figure 3.2. p = 0 λ 1,0 ξ) = λ 2,0 ξ) = λ 5,0 ξ) = λ 6,0 ξ) = 0 { 1, if 0 ξ < 1 λ 3,0 ξ) = 2 0, else { 1, if 1 λ 4,0 ξ) = 2 ξ < 1 0, else p = 1 λ 1,1 ξ) = λ 5,1 ξ) = 0 { 1 2ξ, if 0 ξ < 1 λ 2,1 ξ) = 2 0, else 2ξ, if 0 ξ < 1 2 λ 3,1 ξ) = 2 2ξ, if 0 ξ < 1 0, else { 2ξ 1, if 1 λ 4,1 ξ) = 2 ξ < 1 0, else p = 2 { 1 2ξ) 2, if 0 ξ < 1 λ 1,2 ξ) = 2 0, else 2ξ2 3ξ), if 0 ξ < 1 2 λ 2,2 ξ) = 1 ξ)2 2ξ), if 0 ξ < 1 0, else 2ξ 2, if 0 ξ < 1 2 λ 3,2 ξ) = 3ξ 1)2 2ξ), if 0 ξ < 1 0, else { 2ξ 1) 2, if 1 2 ξ < 1 λ 4,2 ξ) = 0, else

45 3.2. BASIS FUNCTIONS AND ISOGEOMETRIC ANALYSIS λ0ξ) 0.5 λ1ξ) 0.5 λ2ξ) ξ ξ ξ 3 Figure 3.2: Non-zero B-spline basis functions for knot vector Ξ = {0,0,0, 1 2,1,1,1} for polynomial order p = 0 left), p = 1 center), and p = 2 rigt). A B-spline basis, wic consists of piecewise polynomials, cannot be used to describe conic curves, i.e. yperbola, parabola, and ellipses. An additional dimension of weiging coefficients can be used to construct tese conic sections. Tese weiged B- splines are NURBS, Non-uniform rational B-splines). Definition 3.5 NURBS Curves, Sec. 4.2 [24])). A NURBS curve of polynomial order p is te mapping C : R R d suc tat, C ξ) = n ρ i,p ξ)p i, ξ R, 3.18) i=1 were λ i,p is te i -t rational basis function, and P i R d te i -t control point. te rational basis functions are defined troug teir knot vector, and a vector of weigts w R n+1. Te i -t rational basis function ρ i,p is given by, ρ i,p ξ) = λ i,p ξ)w i n i=1 λ i,pξ)w i, 3.19) were λ i,p is te i -t B-spline basis function of polynomial order p. Te coefficients P w = [P i i, w i ] T R d+1 are elements of te d + 1 projective space. Te resulting NURBS curve is ten defined in te d model space. For unitary weigts, w i = 1, te NURBS curve becomes a B-spline curve. Hence te space of B-spline curves is contained witin te space of NURBS curves, Figure 3.1. Example 3.2 NURBS Curve - Construction of a Circle). In tis example we sow ow we can construct a circular curve using a rational spline basis. Let knot vector Ξ, weigts w, and coefficients P be Ξ = {0,0,0, 1 4, 1 4, 2 4, 2 4, 3 4, 3 } 4,1,1,1, { } w = 1, 2,1, 2,1, 2,1, 2,1, [ ] P =,

46 38 3. STRUCTURE PRESERVING ISOGEOMETRIC ANALYSIS Take B-spline basis polynomial p = 2. Now te set of rational basis functions ρ i,p ξ), Figure 3.3, can be constructed using Equation 3.19 and te Cox-de-Boor algoritm, Equation Note tat repeated knots in Ξ affect te continuity of te basis functions. Te multiplicity of an interior knot m produces a B-spline or rational basis function wit continuity p m). Furtermore, note tat any B-spline or NURBS curve interpolates te control net were te basis functions ave C 0 continuity, Figure ρ2ξ) ξ } Figure 3.3: Set of rational basis functions for p = 2, knot vector Ξ = {0,0,0, 4 1, 1 4, 2 4, 2 4, 3 4, 3 4,1,1,1 weigts w = { 1, 2 2,1, 2 2,1, 2 2,1, } 2 2,1.,and set of Now Equation 3.18 can be used to compute te NURBS curve red), Equation In te figure it is sown ow te weigting curve W ξ) = n i=1 λ i,pξ)w i projects te B- spline curve defined by p i=0 λ i,pξ)w i P i onto te circle z x p λ i, p ξ)w i i = 0 y Cξ) Control net Figure 3.4: NURBS circular curve red) wit corresponding 9-point control net black). Te weiging curve blue) projects a B-spline curve onto te circle.

47 3.2. BASIS FUNCTIONS AND ISOGEOMETRIC ANALYSIS z x y Figure 3.5: Cone surface wic is constructed by taking tensor product of circular curve from Example 3.2 and a linear curve. Higer order geometries, i.e. surfaces and volumes, can be constructed by taking tensor products of tese fundamental curves, e.g. Figure 3.5. An alternative way is te use of T-splines. Te control net of a T-spline surface can ave T -saped joints, wile te control net of a NURBS surface must be te topological counterpart of a Cartesian mes can only ave crossroad intersections). Te development of unstructured spline geometries is an active field of researc. Patces of NURBS geometries can be sewn togeter to construct complexer objects. A B-spline basis can be refined witout modifying te geometry. In one approac one inserts new knots to te knot vector, wic is known as knot insertion. Anoter approac is to increase te multiplicity of te knots, and increasing te degree of basis functions, wic is known as degree elevation. Tese refinement strategies are analogous to - and p- refinement in classical finite elements metod. In a tird refinement strategy, known as k-refinement, one combines knot and insertion and degree elevation. Tere does not exist an analogy for tis approac in FEA. Definition 3.6 Knot Insertion, Sec [25])). Given knot vector Ξ = {ξ 1,ξ 2,,ξ n+p+1 }, and extended knot vector wit m additional knots, Ξ = {ξ 1 = ξ 1, ξ 2,, ξ n+m+p+1 = ξ n+p+1 }, suc tat Ξ Ξ. Te control points of te extended knot vector P can be obtained from te original control points P troug, P = T p P, 3.20)

48 40 3. STRUCTURE PRESERVING ISOGEOMETRIC ANALYSIS were te transformation matrix T p is defined troug te recursive formula, T 0 i,j = { 1, ξi [ξ j,ξ j +1 ) 0, oterwise ξ T q+1 i+q ξ j = T q i,j ξ j +q ξ i,j + ξ j +q+1 ξ i+q T q, for q = 0,1,, p ) j ξ j +q+1 ξ i,j +1 j +1 3 Definition 3.7 Order Elevation, Sec. 2.1 [25])). Given a knot vector Ξ = {ξ 1,ξ 2,,ξ n+p+1 }. Te multiplicity of te knots are increased by one witout adding new knot values. Te new control points can be determined using te transformation matrix T defined in Definition 3.6. Tis leads to te following framework of isogeometric analysis. Given a geometric mapping defined in NURBS, B-spline, or Bézier expansions, one can refine te geometry using te strategies described above to obtain a mes suitable for analysis. Note tat te geometry is exactly contained in te coarsest mes and preserved as te mes is refined. Tis is different from classical finite-element approaces in wic refinement requires interpolation of te CAD geometry wit interpolation errors as te consequence). Furtermore, note tat NURBS must be refined in te projective space. Finally, one can link a NURBS geometry to a FEA code troug Bézier extraction. Bézier extraction is te process of transforming te basis to obtain a basis of composite Bézier bases. Tese can be transformed element-by-element to a Lagrangian basis, wic is exploited in many classical FEA codes DERHAM CONFORMING SPLINE SPACES In line wit te isoparametric paradigm we express quantities of interest in terms of te geometric bases introduced in te previous section. Following various works, [8, 11, 13, 26], we start out wit B-splines to set up a framework for R 1. B-splines are cosen because of teir use in representing geometries and because of teir controllable smootness. Te control net of a linear combination of splines form a cocain structure on wic we can differentiate using incidence matrices. We first construct spline spaces in R 1 tat satisfy te DeRam sequence. Tis framework can be extended to arbitrary dimensions by taking tensor products of te spline spaces for te R 1 case. Definition 3.8 Spline Space S p n ). Let Ξ be an arbitrary knot vector of dimension n, and p some polynomial order tat is supported by Ξ. Te span of te set of basis functions defined by te set {p,ξ} troug Equation 3.16 is denoted by, S p Ξ := span{λ i,p} n i= ) In our analysis we will limit ourselves to uniform knot vectors. A knot vector is uniform if te knot values are uniformly distributed. Its basis is defined troug te combination of its dimension, n, and polynomial order p. We define te span of suc basis functions as, S p n := {S p : Ξ open and uniform, wit dimξ) = n}. 3.23) Ξ

49 3.3. DERHAM CONFORMING SPLINE SPACES 41 Tese spline spaces will be used as building blocks to define our finite dimensional spaces. First we will define te spaces in R 1 satisfying te sequence, W d Λ 0) d L 2 Λ 1) Definition 3.9 Te Space of 0-forms Λ 0) in R1 ). Consider some scalar function, i.e. some 0-form f 0) W d Λ 0) in R 1. An approximation of tis function, f 0) W d Λ 0) 0), is a B-spline curve of polynomial order p, i.e. f ξ) : R R wit, f 0) n ξ) := λ i,p ξ)f i, ξ R, 3.24) i=1 or W d Λ 0) := S p n, were f i R is te i -t expansion coefficient or control point notice te similarity wit Equation 3.15). Note tat a polynomial order of at least p = 1 is required to satisfy te Sobolev continuity constraints. We will make use te following notation, 3 f 0) = f T λ p, wit f T = [f 0, f 1,, f n ] te vector of expansion coefficients, and λ T = [λ p 0,λp 1,,λp n] te vector of basis functions. Next consider te 1-form v 1) L2 Λ 1) 1) tat satisfies v = d f 0). Te question is, ow to define its space L 2 Λ 1)? Do we let L2 Λ 1) := S p n as well? We can apply te exterior derivative on te 0-form f 0) to obtain te appropriate basis for v 1), ) n d f 0) ξ) = p λ i,p 1 ξ)dξ f i f i 1 ). i=2 ξ i+p ξ i A proof of tis result is provided in Appendix E. If we let v i = f i f i 1 ), and use B- p spline basis functions of a polynomial degree lower wit scaling ξ i+p ξ, ten te relation i v 1) = d f 0) is exact. Definition 3.10 Te Space of 1-forms Λ 1) in R1 ). Consider te 1-form v 1) L 2 Λ 1) in R 1. An approximation of tis function, v 1) L2 Λ 1), is a B-spline curve of polynomial order p 1, i.e. v 1) ξ) : R R wit, n v 1) ξ) := L 2 Λ 1) i=2 := S p 1 p λ i,p 1 ξ)v i dξ, 3.25) ξ i+p ξ i n, We distinguis between te basis functions tat are used to expand 0-forms and 1- forms, and call tem nodal- and edge type basis functions in line wit te work by Bocev and Gerritsma, [7].

50 42 3. STRUCTURE PRESERVING ISOGEOMETRIC ANALYSIS Definition 3.11 Nodal- & Edge-type basis functions, λ 0),λ 1) ). Te B-spline basis functions tat are used to expand 0-forms are nodal-type basis functions, and te basis functions tat are used to expand 1-forms are edge-type basis functions, [7]. Consider te relation v 1) = d f 0) tat,, wit f 0) W d Λ 0) 1) and v L2 Λ 1), suc 3 wit, λ 0) f 0) = f T λ 0), 3.26) v 1) = v T λ 1) dξ, 3.27) ξ) := λ i i,p ξ), 3.28) ) λ 1) p ξ) := λ i i,p 1 ξ). 3.29) ξ i+p ξ i Te relation between te expansion coefficients, v i = f i f i 1 ), is encoded by te incidence matrix E 1,0), Definition 2.11, suc tat v = E 1,0) f. Te nodal-type basis functions form a partition of unity. Te edge-type basis functions ave unit mass, proof in Appendix E. Tis property reduces integration of discrete volume forms to summation of te expansion coefficients, e.g. ) n n v 1) = λ 1) ξ)v i i dξ = v i Ω Ω i=2 An example pair of nodal- and edge-type basis functions is sown in Figure 3.6. Note tat edge-type basis functions cannot be derived for te NURBS basis, as te exterior derivative does not map a NURBS basis onto anoter NURBS basis. Tis means we cannot unify te structure-preserving approac completely wit isogeometric analysis. In tis work we will only consider fixed geometries on wic violation of te isoparametric of isogeometric paradigm doesn t induce additional error. Let us reflect for a moment and compare our framework to te cocain structure introduced in section 2.3. Differentiation of te 0-form is encoded in te same incidence matrix as te action of te coboundary operator δ, indicating tat te control net is in fact a cocain structure. Hence, reduction and reconstruction operators, R k) and I k) can be defined tat satisfy commuting properties sown in Figure 3.7. Buffa and Hiemstra, [11, 27], make use of a dual basis to define te action of te reduction operator. i=2 δ C 0) C 1) I 0) R 0) I 1) R 1) W d Λ 0) d L 2 Λ 1) Figure 3.7: Commuting diagram of te reconstruction and reduction operators, cocain spaces C k), and te spline spaces Λ k).

51 3.3. DERHAM CONFORMING SPLINE SPACES λ 0) ξ) ξ 8 3 λ 1) ξ) ξ Figure 3.6: Pair of nodal- top) and edge-type bottom) basis functions wit p = 2 and knot vector } Ξ = {0,0,0, 4 1, 4 2, 3 4,1,1,1. Te nodal-type basis form a partition of unity, and te edge-type basis functions ave unit mass. Definition 3.12 Dual basis µ j k) ). A dual basis function function µ j is defined suc tat, k) µ j k) λk) i = δ j i, 3.30) wic allows te definition of te reduction and reconstruction operator troug application of te basis function or its dual. n R k) α i λ k) i i=1 ) := I k) α) := α T λ k) = n α i δ j i = α j = α, 3.31) i=1 n i=1 α i λ k) i 3.32) Te B-spline dual basis owever is impractical for various reasons. It is not complete, not integrable using standard quadrature rules, and tere is no analytical expression for its basis functions, Sec. 5.1 [25]). Its existence proves te connection wit te cocain structure. We extend tis framework to R 3 using tensor products of nodal- and edge-type basis functions. Definition 3.13 Te Space of k-forms Λ k) in R3 ).

52 44 3. STRUCTURE PRESERVING ISOGEOMETRIC ANALYSIS Let f 0) W d Λ 0), v 1) W d Λ 1), and ω 2) W d Λ 2). We define, 3 suc tat, W d Λ 0) p,p,p :=Sn,n,n, W d Λ 1) :=S p 1,p,p S p,p 1,p S p,p,p 1 W d Λ 2) :=S p 1,p 1,p S p,p 1,p 1 S p 1,p,p 1 L 2 Λ 3) :=S p 1,p 1,p 1, f 0) = λ 0) i ξ)λ 0) j η)λ 0) k ζ) ) f i,j,k v 1) = λ 1) i ξ)λ 0) j η)λ 0) k ζ) ) v ξ i,j,k dξ + λ 0) i ξ)λ 1) j η)λ 0) k ζ) ) v η i,j,k dη + λ 0) i ξ)λ 0) j η)λ 1) k ζ) ) v ζ i,j,k dζ ) ω 2) = λ 1) ξ)λ 1) η)λ 0) i j k ζ) ω ξ,η dξ dη i,j,k + ) λ 0) ξ)λ 1) η)λ 1) i j k ζ) ω η,ζ dη dζ i,j,k + ) λ 1) ξ)λ 0) η)λ 1) i j k ζ) ω ζ,ξ dζ dξ. i,j,k Applying te exterior derivative to tese forms yields, d f 0) = λ 1) i ξ)λ 0) + λ 0) i ξ)λ 1) η)λ 0) j k η)λ 0) j k d v 1) = λ 1) i ξ)λ 1) j η)λ 0) ζ) ) fi,j,k f i 1,j,k ) dξ ζ) ) fi,j,k f i,j 1,k ) dη + ) λ 0) ξ)λ 0) η)λ 1) fi i j k ζ) ),j,k f i,j,k 1 dζ, ) )v k ζ) ξ i,j,k vξ i,j 1,k vη i,j,k + vη dξ dη i 1,j,k + ) λ 0) ξ)λ 1) η)λ )v 1) i j k ζ) η i,j,k vη i,j,k 1 vζ i,j,k + vζ dη dζ i,j 1,k + ) λ 1) ξ)λ 0) η)λ )v 1) i j k ζ) ζ i,j,k vζ i 1,j,k vξ i,j,k + vξ dζ dξ, i,j,k 1 dω 2) = λ 1) i ξ)λ 1) j η)λ 1) k ζ) ) ω ξ,η i,j,k ωξ,η i,j,k 1 + ωη,ζ i,j,k ω η,ζ i 1,j,k + ωζ,ξ i,j,k ωζ,ξ i,j 1,k ) dξ dη dζ. Te coefficients of any derivative dα k) is obtained by applying an incidence matrix to te coefficients of α k). Te defined spline spaces satisfy te exact sequence, W d Λ 0) d d d W d Λ 1) W d Λ 2) L 2 Λ 3)

53 3.4. DISCRETISATION OF THE POISSON PROBLEMS 45 We will use a simplified notation, φ k), to denote basis functions in R n. For example, we let, wit, f 0) = f T φ 0) = f i,j,k φ 0) i,j,k ξ,η,ζ), and, wit, φ 0) i,j,k = λ0) i ξ)λ 0) η)λ 0) j k ζ), v 1) =v ξ ) T φ 1) ξ dξ + v η ) T φ 1) η dη + v ζ ) T φ 1) ζ dζ, = [ v ξ ) T v η ) T v ζ ) T ] φ 1) ξ dξ φ 1) η dη dζ =v T φ 1) φ 1) ξ i,j,k = λ 1) i ξ)λ 0) φ 1) ζ η)λ 0) j k ζ), 3 Differentiation of a form is ten written as, dω 2) = d ω T φ 2)) = E 3,2) ω) T φ 3) DISCRETISATION OF THE POISSON PROBLEMS Te framework of structure-preserving isogeometric analysis enables us to discretise Poisson problems tat ave been introduced in section 3.1. Te resulting linear system consists of a combination of incidence matrices, and mass matrices. In tis section we will introduce mass matrices, and sow tat we can solve bot Poisson problems wit optimal convergence rates. Definition 3.14 Mass Matrices M k,k) ). Te mass matrix, M k,k), is te linear mapping M k,k) : C k) C k). It is a Gramian of basis functions, i.e. te Hermitian matrix of inner products wit entries M k,k) given by, i,j M k,k) = i,j φ k) i ),φ k) j. Ω Te mass matrix is symmetric and positive definite. Te mass matrix inerits a sparsity pattern by troug te local support of te basis functions. Local support can be exploited for element-wise construction of te mass matrix more details on te elementwise construction of B-spline mass matrices can be found in sec of te IGA book, [25]). Gauss-Legendre quadrature is used to evaluate te integrals in te assembly routine.

54 46 3. STRUCTURE PRESERVING ISOGEOMETRIC ANALYSIS Example 3.3 Mass matrix of 1-Forms M 1,1) in R 2 ). 3 M 1,1) = φ 1),φ 1)) Ω = φ 1) φ 1) = = Ω Ω Ω [ φ 1) ξ φ 1) η dη [ φ 1) ξ φ1) dξ ])[ φ 1) dη ]) T ξ φ 1) η dξ ξ )T φ 1) η φ 1) η ) T ]) dξ dη Example 3.4 Discretisation of te 0-form Poisson problem). Consider te problem described in Example D.1 on domain Ω = [0,1] 2 R 2. We let Γ n =, and take f 0) = 2π 2 sinπx)sinπy). Te problem is ten, given f 0) L 2 Λ 0), find ψ 0) W d Λ 0), suc tat d w 0),dψ 0)) Ω = w 0), f 0)) Ω, w 0) W d Λ 0) 0. Te exact solution of tis problem is given by, We let W d Λ 0) p,p := Sn,n, suc tat, ψ 0) = sinπx)sinπy). w 0) = w T φ 0), ψ 0) = ψt φ 0). Te discrete problem consists of finding ψ 0) W d Λ 0) suc tat, d w 0),dψ0) )Ω = w 0), f 0)) 0), w Ω W d Λ 0) 0. Well-posedness for te discrete problem follows from well-posedness of te continuous problem, proven in Example D.1. w T E 1,0) ) T M 1,1) E 1,0) ψ = w T φ 0), f 0) ) Ω. Te Diriclett boundary conditions are essential boundary conditions and need to be enforced strongly. Numerical solution for W d Λ 0) = S 2,2 6,6 is sown in Figure 3.8a. Convergence for various refinement levels are sown in Figure 3.9a. Observed convergence rates are optimal wit respect to cosen spline polynomial order. Example 3.5 Discretisation of te 2-form Poisson problem). Now consider te problem described inexample D.2 on domain Ω = [0,1] 2 R 2. We let

55 3.4. DISCRETISATION OF THE POISSON PROBLEMS ξ η ψ 0) ξ η ψ 2) 3 a) ψ 0) W d Λ 0) b) ψ 2) L 2 Λ 2) Figure 3.8: Reconstructed solution of te 0-form left) and 2-form rigt) Poisson problem, Example 3.4 and Example 3.5. DeRam conforming spline space were taken wit W d Λ 0) := S p,p n,n, wit polynomial order p = 2 and dimension n = ψ 0) ψ 0) L p = 1 p = p = 3 p = ψ 2) ψ 2) L p = p = 2 p = 3 p = a) ψ 0) ψ 0) L 2 b) ψ 2) ψ 2) L 2 Figure 3.9: L 2 -Error convergence results for te 0-form left)- and 2-form rigt) Poisson problem, Example 3.4 and Example 3.5, for various -and p-refinement levels. Measured convergence rates are optimal wit respect to spline polynomial order. Note tat te space of ψ 0) is equipped wit te Sobolev norm, altoug te weaker convergence in te L 2 norm is sown left).

56 48 3. STRUCTURE PRESERVING ISOGEOMETRIC ANALYSIS Γ n =, and take f 2) = 2π 2 sinπx)sinπy)d x d y. Te problem ten consists of finding te set v 1),ψ 2) ) W d Λ 1),L 2 Λ 2) ), suc tat { d q 1),ψ 2)) Ω q 1), v 1)) Ω = 0, q1) W d Λ 1) 0, w 2),d v 1)) Ω = w 2), f 2)) Ω, w 2) L 2 Λ 2) 0, for a given f 2) L 2 Λ 2). Te exact solution of ψ 2) is given by, 3 We let W d Λ 1) ψ 2) = sinπx)sinπy)d x d y. p 1,p := Sn 1,n S p,p 1 n,n 1, and L2 Λ 2) p 1,p 1 := Sn 1,n 1 suc tat, q 1) = q T φ 1), w 2) = w T φ 2) v 1) = v T φ 1), ψ 2) = ψt φ 2). Te discrete problem ten consists of finding v 1),ψ2) ) W d Λ 1),L2 Λ 2) ) )Ω d q 1) w 2),ψ2),d v 1) ) Ω q 1), v 1) Ω = 0, q1) W d Λ 1) 0, = w 2), f 2)) 2), w Ω L2 Λ 2) 0. ), suc tat Well-posedness of te discrete problem depends on te exactness of te DeRam complex wic is satisfied by te discrete cocain) spaces. Te resulting system is given by, [ q T w T ][ M 1,1) E 2,1) ) T M 2,2) ][ ] [ ] v 0 M 2,2) E 2,1) = ψ w T φ 2), f 2) ) Ω Note tat te resulting system is symmetric. Te Diriclett boundary conditions are natural boundary conditions and are weakly enforced. A numerical solution corresponding to te coice of W d Λ 0) = S 2,2 6,6, wic lets L2 Λ 2) = S 5,5 1,1 ), is sown in Figure 3.8b. Observed convergence rates are optimal wit respect to cosen spline polynomial order, Figure 3.9b SUMMARY OF RESULTS Well-posedness of variational formulations of mixed-type elliptic PDE problems depend on te topological structure of te DeRam complex. One guarantees existence and uniqueness of te solution in bot te continuous and te discrete setting by selecting an appropriate set of function spaces, i.e. Sobolev spaces tat satisfy te DeRam complex. Finite-dimensional polynomial subspaces are taken to discretise quantities of interest. In isogeometric analysis spans of finite numbers of B-spline basis functions are taken to define te function spaces in te discrete setting. One distinguises between nodaland edge-type B-spline basis functions to discretise 0-forms and 1-forms in R 1. In R n tensor products of combinations of tese basis functions are taken to construct k-forms.

57 3.5. SUMMARY OF RESULTS 49 Derivatives of discrete forms can be obtained by applying an incidence matrix to its vector of coefficients. Te Hodge- is discretised troug te inner product matrices, known as mass matrices. Te resulting linear system consists of a combination of incidence matrices E and mass matrices M. Te system can be inverted to obtain te coefficients of te discrete solution. 3

58

59 4 SCALAR TRANSPORT Hyperbolic PDE s involve transport of quantities wit finite transport velocities. In tis capter we aim to expand te framework derived for elliptic PDE problems to be able to construct structure-preserving discretisation tecniques for yperbolic PDE s. We build towards discretisation of te non-linear convective term of te Euler equations by studying scalar transport problems. In section 4.1 we expand our toolbox of exterior calculus to be able to formulate and analyse coordinate invariant representations of yperbolic problems. We introduce te Lie derivative L v and interior product i v, wic describe ow differential forms evolve as tey are transported along some flow wit te velocity vector field v. Te scalar transport PDE s tat will be studied in tis capter are presented in section 4.2. In section 4.3 we discuss various strategies for te discretisation of te interior product. We coose to discretise transport problems on a static Eulerian) mes using a space-time discretisation approac THE INTERIOR PRODUCT AND THE LIE DERIVATIVE We introduce additional concepts from exterior calculus to describe transport penomena. For teorems and notation we rely again on Frankel s geometry of pysics, [18]. Consider some initial concentration at a point p, a 0-form ρ 0) 0 p) := ρ0) p, t 0 ), tat is subjected to some flow. Te problems described in tis capter consist of finding te distribution, ρ 0) p, t), as it advected along wit te flow. We can find te velocity of te te concentration at p by differentiating its coordinate mapping wit respect to time. Moreover, given its velocity, we can find its patway, te integral curve, troug integration. Existence of tese relations is stated in Teorem 4.1. Teorem 4.1 Fundamental Teorem on Vector Fields, Sec. 1.4 [18])). Let v : U R n be a differentiable vector field on some manifold U M. For eac point 51

60 52 4. SCALAR TRANSPORT p U tere is te integral curve γ : ɛ, ɛ) U, suc tat, { dγt) d t γ0) = p. = vγt)), 4.1) Furtermore, tere is a set U p, te neigbourood of p, and a differentiable mapping Φ : U p ɛ,ɛ) R n, suc tat te curve φ t q) := Φq, t) satisfies te differential equation, 4 t φ t q) = vφ t q)), 4.2) for all time t ɛ,ɛ) and points q U p. Te family of mappings {φ t } is known as local flow. Te integral and differential flow-vector field relations are illustrated in Figure 4.1. Note tat te velocity-vector v at q is an element of te tangent space T M q. φ t q) vφ 0 q)) γt) q p vγ0)) U p Figure 4.1: Local flow in U p. Te velocity field defines an integral curve γt) in p wit vp) = dγt) d t. t=0 Moreover, te local flow {φ t } in U p defines a velocity field v troug differentiation, e.g. t φ 0q) = vφ 0 q)). We can describe te cange of a differential form along te flow of a vector field troug te action of te differential operator, known as te Lie-derivative. Definition 4.1 Lie-derivative L v, Sec. 4.2 [18])). Let v T M be te velocity vector field wit flow {φ t }. Let α k) Λ k) M ) be a differentiable k-form. Te Lie-derivative is te mapping L v : Λ k) Λ k), for wic, L v α k) := d d t [φ t αk) ] t=0, 4.3) were φ t denotes te pull-back mapping, Definition C.1. Te Lie-derivative satisfies te following relation, L v α k) β l) ) = L v α k) β l ) + α k) L v β l). 4.4) Te Lie-derivative and te exterior derivative measure te derivative of some differential k-form. Tey are related troug Cartan s formula, L v α k) = di v + i v d)α k), 4.5)

61 4.1. THE INTERIOR PRODUCT AND THE LIE DERIVATIVE 53 were i v is te interior product. Te interior product defines a contraction of a differentiable form wit a vector field. Definition 4.2 Interior product i v, Sec. 2.9 [18])). Te interior product is te mapping i v : Λ k) Λ k 1) defined by, i v f 0) = 0, i v α 1) = α 1) v), 4 wit properties, i v α k) β l ) ) = i v α k) β l) + 1) k α k) i v β l), i v i w α k) = i w i v α k), for vector fields v, w. Nilpotency follows from te skew-symmetry, i.e. i 2 v := i v i v = 0. Note tat te Lie-derivative commutes wit bot te interior product i v and te exterior derivative d, wic can easily be derived from Cartan s formula, Equation 4.5, and nilpotency of bot operators. Example 4.1 Action of te interior product i v ). Consider vector field v on M R 3, v = n i v i x i, and differential forms u 1) Λ 1), ω 2) Λ 2), and σ 3) Λ 3). Te action of te interior

62 54 4. SCALAR TRANSPORT product on tese forms is given by, i v u 1) = 1-form: n i v i ) ) n x i u j d x j j = v i u j i j = i,j i v i u i 4 i v ω 2) = i v σ 3) = 2-form: n i j,k j <k v i x i ) n ω j k d x j d x k j,k j <k n n = v j ω j k d x k v k ω j k d x j j,k j <k d x 1 d x 2 d x 3 = det ω 23 ω 13 ω 12 v 1 v 2 v 3 3-form: n i v i x i ) σd x 1 d x 2 d x 3 = σv 1 d x 2 d x 3 σv 2 d x 1 d x 3 + σv 3 d x 1 d x 2 We observe tat te action of te interior product wit vector field v is similar to te wedge product wit some differential form related to v we recognize components of te vector dot-, cross-, and scalar product). In fact we will be able to relate vector fields to differential forms of te same dimension, and re-express te action of te interior product as a wedge product on forms. Te dimension of a vector field v defined on M R n equals n. We can formulate isomorpisms to transform te vector field to differential forms. Differential 1-forms, and n 1)-forms ave te same dimension as v wic makes tem suitable candidates for te proposed relations. To map to te vector field to a 1-form we will make use of te metric tensor, and to go to a n 1)-form, we will make use te interior product acting on te n-form basis. Definition 4.3 Metric tensors g i j, and g i j, Sec. 2.1 [18])). Consider te basis x j and its dual basis d x i on M R n. Te metric tensors g i j, and g i j are te symmetric differentiable matrices, g i j = x i, x j, 4.6) g i j = d x i,d x j, 4.7)

63 4.1. THE INTERIOR PRODUCT AND THE LIE DERIVATIVE 55 were te bilinear operation, denotes te pairing. It s easy to see tat g i j g i j = δ j i, from wic it follows tat g i j = g 1. Te metric tensors relate te basis to its dual basis, i j x i = x j δ j i = x j xi,d x j = x i, x j d x j = g i j d x j 4.8) d x i = d x j δ j i = d x j d x i, x j = d x i,d x j x j = g i j x j 4.9) Note tat te metric tensor equals a diagonal matrix on ortogonal grids. Definition 4.4 Vector - Form isomorpisms). Given vector field v on M R n, we can relate v to an inner-oriented 1-form v 1) troug te metric tensor, v = v j x j = ) v j g j i d x i = g i j v )d j x j = v 1), 4.10) j j i i j were te components of v 1) are given by j g i j v j. We can relate v to an outer oriented n 1)-form by applying te interior product to te n-form basis, 4 v n 1) = i v vol n), 4.11) were vol n) := d x 1 d x n is te volume-form basis. Components of te n 1)- form are as sown in Example 4.1. In literature te vector-form isomorpisms are often referred to as musical isomorpisms, defining te operator to map te vector to a form, and te operator to map te form to a vector. Finally, we can rewrite te action of te interior product wit vector v on a k-form α k) as, i v α k) = 1) kn k) α k) v 1) ), 4.12) were v 1) is te 1-form related to v troug Equation 4.10, proof of tis relation can be found in Hirani s work on discrete exterior calculus, [28]). We can combine tis relation wit Equation 4.11 to find te relation between te 1-form v 1) and te n 1)-form v n 1), from wic it follows tat, v n 1) = i v vol n) 4.12 = vol n) v 1) ) = 1 v 1) ) = v 1), 4.13) i v α k) = 1) kn k) α k) v n 1) ), 4.14) We introduce te isomorpisms because tey can be used to express te action of te interior product in terms of differential forms, for wic we ave developed te discretisation approac in capter 3. We are left wit a coice, will we use te 1-form v 1) or te n 1)-form v n 1)? In te problems tat we will study we will assume incompressible flow, wic implies tat conservation of mass is governed by divv = 0. We can relate divv to v n 1) by taking te exterior derivative of Equation 4.11, d v n 1) = di v vol n) = divv)vol n). 4.15)

64 56 4. SCALAR TRANSPORT It follows tat te incompressibility constraint divv = 0 equals d v n 1) = 0 and d v 1) = 0. In te discrete setting we are able to preserve te exactness of te exterior derivative d, wic makes te isomorpism to v n 1) te favourable option. However, as we will see, te 1-form isomorpism can be used to derive a sligtly simplified representation of te interior product s adjoint. Next we introduce adjoints of te interior product and te Lie derivative, wic will be used to manipulate weak formulations later on. Definition 4.5 Cointerior product, i v ). Te cointerior product is te mapping i v : Λ k) Λ k+1) defined by, i v := 1)nk+1) 1 i v. 4.16) 4 Te action of te cointerior product i v on differential form αk) can be written as te action of a wedge product, i v αk) = 1) k α k) v 1) ), 4.17) were v 1) is te 1-form related to te vector field v. Te Cointerior product is te adjoint of te interior product, β k 1),i v α k)) = iv M βk 1),α k)) 4.18) M Proofs of tese relations are provided in Appendix B. Definition 4.6 Co-Lie derivative L v ). Te Co-Lie derivative is te mapping L v : Λk) Λ k) defined by, L v := 1)kn+1)+1 L v. 4.19) Te co-lie derivative is related to te codifferential and te cointerior product troug a relation mirroring Cartan s formula, Equation 4.5, Te co-lie derivative is te adjoint of te Lie-derivative, L v αk) = d i v + i v d )α k). 4.20) β k),l v αk)) M = L v β k),α k)) M β k),i v αk)) M i v β k),α k)) M. 4.21) In te upcoming section we will use te interior product and Lie-derivative to formulate te equations governing advection problems. We will present te model problems on periodic domains tat will be studied in tis capter TRANSPORT PROBLEMS In tis section we derive te governing equations for scalar transport problems. We present some test problems, wic will be analysed in te sections to come. Scalar transport is governed by te conservation of mass. Let ρ n) denote te density of a given quantity. Te total mass M is given by, M = ρ n), 4.22) V t)

65 4.2. TRANSPORT PROBLEMS 57 were V t) M is some volume. Application of te transport teorem leads to te governing equations describing scalar transport. A formal derivation of te governing equations can be found in Appendix G. Teorem 4.2 Transport Teorem). Te time-rate of cange of a k-form α k) Λ k) paired to a k-dimensional sub-manifold V M at some time-instant t R is given by, d α k) α k) = + L v α ). k) 4.23) d t V t) V t) t Teorem 4.3 Scalar Transport Equations). Te equation for conservation of mass of some quantity wit density distribution ρ n) tat is advected along wit some incompressible fluid wit velocity vector field v is given by, V t) ρ n) ) + L v ρ n) = 0, 4.24) t 4 for some volume V t) M. Te velocity vector field satisfies te incompressibility constraint, divv = 0 or d v n 1) = 0, were v n 1) is te n 1)-form associated wit v. We can derive alternative formulations of te governing transport equation, Equation 4.24, by considering dual formulation or incorporating te incompressibility constraint, i.e. te non-conservative form. All four formulations are derived in Appendix G. Consider te dual formulation of te non-conservative formulation, ρ 0) t + L v ρ 0) = ) We coose to define te problems on periodic domains. A solution leaving one part of te domain, enters te domain on te oter side. We denote periodic domains wit reverse brackets, ], [. Note tat a periodic domain in R 1 as te topology of a circle, wereas a periodic domain in R 2 as te topology of a torus, Figure 4.2. Te bencmark problems tat we consider, were introduced in te book by Donea and Huerta Sec 3.11 [2]). Figure 4.2: Periodic domain in R 2 as te topology of a torus.

66 58 4. SCALAR TRANSPORT Problem 4.1 Scalar advection-diffusion in R 1 ). Consider te space-time manifold Q = Ω R +, were Ω :=]0,1[ is te periodic unitinterval. Te incompressibility condition in R 1 is satisfied if te velocity field is constant, i.e v = U x, wit U R. Given some diffusion coefficient ɛ R +, and as initial solution ρ 0) x,0) := ρ 0 x) at t = 0 te sinusoid, 4 { 1 ρ 0 x) = cos π σ x x 0) )), if x x 0 σ, 0, elsewere 4.26) Te initial solution is compactly supported on [x 0 σ, x 0 +σ], as maximum equal to 1 at x 0, and is continuously differentiable. It as mass ρ 0 d x = σ, and moment ρ 2 0 d x = 3σ 4. We take x 0 = 1 2 and σ = 0.2, Figure 4.3a. Te problem ten consists of finding ρ0) x, t) Λ 0) Q) suc tat, { t ρ 0) + i v dρ 0) ɛd dρ 0) = 0 ρ 0). 4.27) x,0) = ρ 0 x) We will mainly focus on te case of pure advection, ɛ = 0, for wic te exact solution is given by ρ 0) x, t) = ρ 0 x U t), i.e. te solution traverses te domain wit velocity U. We will sow tat diffusion as a stabilizing effect on te numerical solution.

67 4.2. TRANSPORT PROBLEMS ρ0x) x a) Equation 4.26 wit x 0 = 0.5 and σ = ρ 0 x, y) x y b) Equation 4.28 wit x 0, y 0 ) = 0.3,0.3, and σ = 0.2. Figure 4.3: Initial condition for te advection problems in R 1 top) and R 2 bottom). Te initial condition is continuously differentiable, is compactly supported, and as unit maximum at its center location. Problem 4.2 Scalar advection in R 2 ). Consider te space-time manifold Q = Ω R +, were Ω =]0,1[ 2 is te periodic unitsquare. As an initial condition we take te tensor product of te sinusoids, Equation 4.26, ρ 0) x, y,0) = ρ 0 x)ρ 0 y). 4.28) Te initial solution is compactly supported on [x 0, y 0 ) ± σ,σ)], as its maximum equal to 1 located at x 0, y 0 ), and is continuously differentiable. It as mass ρ 0 d x d y = σ 2, and moment ρ 2 0 d x d y = 9 16 σ2. We take initial position x 0, y 0 ) = 0.3,0.3), and σ = 0.2. Te problem ten consists of finding ρ 0) x, t) Λ 0) Q) suc tat, { t ρ 0) + i v dρ 0) = 0 ρ 0) x, y,0) = ρ 0 x, y),

68 60 4. SCALAR TRANSPORT for a given velocity field v. We will consider two different incompressible flow fields, te circular velocity-field, and te Rudman vortex. We define te circular velocity field, v = 2 y 1 ) 2 x + 2 x 1 ) 2 y. 4.29) 4 One can verify tat div v = 0. By using te polar coordinate transformation x 1 2, y 1 2 ) = r cosθ,r sinθ), one can sow te radial component of te flow equals zero. Furtermore we find tat v = 2r. Hence, te fluid parcel wit initial position on r traverses te orbit γ = r, wit period T = 2πr 2r = π[s]. Iso-contours of te stream function are sown in Figure 4.4a, indicating te circular pats, and te increase in velocity wit r. Note tat te circular velocity field is discontinuous on a periodic domain. Te second test case tat we introduce is te Rudman vortex velocity field, [29]. Te Rudman vortex is defined as, v = A sinπx)cosπy) x A cosπx)sinπy) y, 4.30) for some constant A R. Te Rudman vortex is a continuously differentiable incompressible flow-field tat deforms volumes, making it a more difficult test case in te discrete setting. Iso-contours of te Rudman vortex are sown in Figure 4.4b. Te distance between 2 iso-contours varies, ence any parcel tat is advected in te streamtube bound by tese iso-contours will alternatively accelerate and decelerate along te way y 0.5 y x a) ψ 0) = [x 2 x) + y 2 y)] x b) ψ 0) = A π sinπx)sinπy) Figure 4.4: Iso-contours of te stream function ψ 0) corresponding to left) te circular velocity field, Equation 4.29, and rigt) te Rudman vortex, Equation In te next sections we will discretise tese problems using a space-time metod. Most classical metods first discretise te spatial dependencies to obtain a semi-discrete system of differential algebraic equations. Often tese equations are solved using an explicit, implicit, or combined multi-stepping algoritm. In a space-time finite element metod one expands variables in bot space- and time-dependent basis functions. One

69 4.3. DISCRETISATION OF TRANSPORT PROBLEMS 61 advantage of tis approac is tat implementation of time dependencies of te considered problem become trivial. Togeter wit te advantages of isogeometric analysis, space-time discretisations would be igly advantageous for moving-boundary problems, or fluid structure interactions. Anoter advantage is te possibility to locally) refine in bot space and time, e.g. as sown in te work by Abedi et al. [30]. Te advantages of space-time metods come at a cost owever. Te additional temporal dimension ave a uge impact on bot assembly cost and solve cost. Our motivation to coose for space-time approac is tat it is consistent wit our framework, and because it is convenient for rigorous stability analysis. It motivates us to introduce te following notation. Definition 4.7 Space-time Lie-derivative L X, Sec 4.3 [18])). Consider te space-time domain Q = Ω T, wit spatial domain Ω R n), and T = [0, t] te considered time-interval. Te space-time Lie-derivative is te mapping L X : Λ k) Ω) Ω) tat satisfies, 4 L X α k) = d i X + i X d )α k) 4.31) = c αk) t + L v α k), 4.32) were constant c is te velocity of te time-frame usually we take c = 1), X te spacetime velocity-vector X := c t + v, and d te space-time exterior derivative, dα k) = d t αk) t + dα k). 4.33) 4.3. DISCRETISATION OF TRANSPORT PROBLEMS Solutions on periodic domains will need to be reconstructed using a periodic spline basis. A set of periodic node- and edge-type basis is sown in Figure 4.5. Suc basis can be obtain from on ordinary B-spline basis troug knot insertion, Appendix H. In tis approac, we obtain transformation matrices tat maps a standard set of B-spline basis functions to a set of periodic B-spline basis functions. Note tat te dimension of te node-type basis corresponding to a periodic basis is equal to te dimension of its corresponding edge-type basis, i.e. a circular grid as an equal amount of nodes and edges. In Appendix H we sow ow a periodic basis can be constructed.

70 62 4. SCALAR TRANSPORT 1.0 λ 0) ξ) ξ 4 4 λ 1) ξ) ξ Figure 4.5: Te set of top) nodal- and bottom) edge-type periodic B-splines, λ 0) ξ) and λ 1) ξ), i i corresponding to W d Λ 0) := P 4 2. Proposed strategies to discretise te Lie-derivative can rougly be divided into 3 categories, static, dynamic, and mixed approaces. In te static approac Eulerian), te mes is fixed, and fluxes are approximated using interpolation- or projection metods. In te dynamic approac Lagrangian), te flow is approximated, and te mes is advected wit te flow. Dynamic approaces require periodical re-mesing to cope wit accumulated deformations. In te mixed approac Semi-Lagrangian), te mes is advected to approximate fluxes. Most dynamic- and mixed approaces rely on te idea of extrusion, e.g. Bossavit, [31], Heumann & Hiptmair, [32], and Mullen et al. [33]. In tese approaces one approximates te extruded volume E v V, t), E v V, t) = V τ), 4.34) τ [0,t] and use an upwind/downwind approac to approximate integral fluxes, Figure 4.6. Anoter approac was explored in te tesis works by Pala and Rebelo, [34, 35]. Tey approximated fluxes by pulling back quadrature points over some time-step. Te dynamicand mixed approaces rely on te quality of te mapping. Te development of volumepreserving mappings, corresponding to an incompressible flow-field, would be of great benefit for tese approaces. Altoug tese approaces are equally valid strategies to discretise transport penomena, we will pursuit a static strategy.

71 4.3. DISCRETISATION OF TRANSPORT PROBLEMS 63 V t) v V 4 Figure 4.6: Approximation of te extruded volume E v V, t). To discretise te action of te interior product in a static approac, we need to find a suitable vector-basis tat mimics te pairing of a vector and a differential form. Gerritsma et al. try to define te Lie-derivative on cocain structures using pointwise contractions, [36]. Tis requires point-wise storage of bot function-values and derivatives, e.g. using ermite polynomials. However, we wis to expand te framework introduced in capter 3 using nodal- and edge-type B-splines to discretise differential forms. Instead of trying to find a proper basis for vectors to discretise te interior product, we look at its adjoint, te wedge product between differential forms. Example 4.2 Discrete Wedge Product). Consider te discretisation of i v ρ 2) on Ω R 2. We can derive a weak formulation by taking te inner product wit test function w 1), w 1),i v ρ 2)) 4.18 Ω = w 1) v 1),ρ 2)) Ω = σ 2) ρ 2)) Ω, were v 1) is te 1-form associated wit v, and σ 2) = w 1) v 1). In te discrete setting we take, w 1) = w x i,j λ1) i j v 1) = kl v x k,l λ1) k x)λ 0) i j y)d x + i j x)λ0) y)d x + l kl w y i,j λ0) x)λ 1) i j v y k,l λ0) k y)d y x)λ1) y)d y l

72 64 4. SCALAR TRANSPORT 4 Te 2-form σ 2) 1) := w v 1) is ten given by, σ 2) = w 1) v 1) = w x i,j v y k,l λ0) k x)λ0) y)λ 1) j i j kl w y i,j v x k,l λ0) i j kl = i j kl = i j kl = i j = i j x)λ 0) i l y)λ 1) k x)λ 1) i l y)d x d y x)λ1) y)d x d y j w x i,j v y k,l φ0) k,j φ2) i,l d x d y w y i,j v x k,l φ0) i,l φ2) k,j d x d y ) w x i,l v y k,j w y k,j v x i,l φ 0) k,l φ2) i,j d x d y [ kl ] ) w x i,l v y k,j w y k,j v x i,l φ 0) φ 2) k,l i,j d x d y σ i,j x, y)φ 2) i,j d x d y. Te result is different from te 2-forms tat we defined in capter 3. From te example we see tat te wedge product does not send te discrete forms to te rigt space, and does not satisfy te desired commutativity sown in te diagram, Figure 4.7. To map te wedge product to te rigt space we can use projection or interpolation tecniques. Λ k) Λ l) Π k) Λ k+l) Π l) Λ k) Π k+l) Λ l) Λ k+l) Figure 4.7: desired commutativity of te wedge product Definition 4.8 Contraction Matrices C k 1,k) v ). Te contraction matrix, C k 1,k) v C k 1,k) v are given by, )i,j ) C k 1,k) v = i,j, is te linear mapping C k 1,k) : C k) C k 1). Its entries φ k 1) i v,i v φ k) j )Ω = φ k 1) i v 1) ),φk) j, Ω 4.35) were v 1) = Π1) v 1) wit v 1) is te 1-form associated wit te vector v. Te continuous velocity-field v can also be used if it is known. Te contraction matrix approximates te interior product. Contraction matrices only approximate te interior product s nilpotency, wereas te incidence matrices satisfy te nilpotency of te exterior derivative exactly.

73 4.3. DISCRETISATION OF TRANSPORT PROBLEMS 65 Example 4.3 Computation of C 0,1) v and C 1,2) v ). Let v = u x x, y) x +u y x, y) y be te velocity vector field on Ω R 2. Te 1-form contraction matrix C 0,1) v is given by, C 0,1) v = φ 0),i v φ 1)) Ω = φ 0) i v φ 1) = = Ω Ω Ω φ 0) )[ u x φ 1) x Te 2-form contraction matrix C 1,2) v C 1,2) v u y φ 1) y ]) T d x d y [ φ 0) u x φ 1) x ) T φ 0) u y φ 1) y ) T ])d x d y is given by, = φ 1),i v φ 2)) Ω = φ 1) i v φ 2) Ω [ ]) φ 1) [ ux x d x φ 2) ]) T d x = Ω φ 1) y d y u y φ 2) d y [ ]) φ 1) x u y φ 2) ) T = u x φ 2) ) T d x d y Ω φ 1) y 4 We can use a combination of incidence matrices E k,k 1) and contraction matrices, to differentiate te Lie-derivative as sown in te following example. C k 1,k) v Example 4.4 Discretisation of te scalar advection-diffusion equation). Consider te scalar advection-diffusion problem on te space-time domain Q = Ω T, were Ω =]0,1[ is te unit-interval periodic domain, Problem 4.1, and T = [0, t] te considered time-interval. Note tat te space-time domain Q as te topology of a cylinder. We obtain a weak formulation of Equation 4.27 by taking te inner product wit testfunction w 0) W d Λ 0) 0 Q), wit w 0) = 0 at t = 0, and integrate-by-parts te diffusive term, w 0),L X ρ 0)) Q = ɛ d w 0),dρ 0)) Q, wit L X te space-time Lie-derivative, and X = c t + u x. To discretise te system we take W d Λ 0) p,p := S n,n. Te 0-form basis is te product of periodic splines in te spatial dimension and ordinary splines in te temporal dimension, φ 0) i,j x, t) = λ i x)λ j t). We let, and find, w 0) = w T φ 0), ρ 0) = ρt φ 0), ) w 0),i X dρ 0) = w T C 0,1) Q X ) d w 0),dρ0) = w T E 1,0) Q Ω E 10) ρ ) T M 1,1) E 1,0) Ω ρ,

74 66 4. SCALAR TRANSPORT were E 1,0) is te incidence matrix acting on spatial edges only. Note tat [ Ω ] E 1,0)) = E 1,0) T Ω,E1,0) T, wit E 1,0) acting on temporal edges. Te resulting system can be T written as, C 0,1) E 1,0)) ρ = ɛ X E 1,0) Ω ) T M 1,1) E 1,0) ρ Note tat te initial condition is an essential boundary condition wic needs to be enforced. For a few example runs we take spline spaces wit dimension n = 24, and polynomial order p = 2. We consider te following 3 cases, Ω 4 1. Pure advection wit U = 1, c = 1, and ɛ = 0 2. Advection-diffusion wit U = 1, c = 1, and ɛ = Slowed time-frame wit U = 1, c = 1 2, and ɛ = Te results are sown in Figure 4.8. In every case mass was conserved wit macine precision. In te first case solution traversed te domain once. Spurious oscillations are observed at various locations troug space and time. Tese oscillations or wiggles are caracteristic in iger-order approximations of advective problems. Tese wiggles accumulate over time destabilizing te numerical solution. Te next section is devoted to te analysis and control of tese oscillations. In te second problem diffusion was added, causing te distribution to smear out as it travels troug space and time. Te action of diffusion is, unlike convection, governed by symmetric operations causing te smearing to take place in bot spatial directions. Spurious oscillations will smear out for tis case. Some classical solution metods add artificial diffusion, or exploit te dissipative beaviour of a sceme to control oscillations. Te disadvantage of tese approaces is tat tey also smear out te desired solution. In te final case we alved te velocity of te time frame. Tis as te consequence tat it takes twice as long to reac te end-state. We observed tat te distribution traverses te domain two times, and tat te smearing reduces te distribution to te near constant value of σ = 0.2. To reduce te cost of assembly and solving we take a partition of te domain in temporal direction. Consider space-time domain Q = [Ω, T ] wit T = [0, t]. We take a uniformly distributed sequence of time-instants, {0 = t 0, t 1, t 2,..., t N = t}, suc tat T = k=0[t k, t k+1 ], and t = [t k+1 t k ]. Te domain is now divided into N 1 space-time slabs. Te system can be assembled- and solved in a sequence, one slab at a time. Te initial condition of a new slab equals te outflow solution of te previous slab, wic gives C 0 continuity in-between slabs. Alternatively, a discontinuous-galerkin metod can be used to enforce continuity over slabs. Example 4.5 Circular Transport using Space-Time Slabs). Consider te circular transport problem on a periodic domain introduced in Problem 4.2. We let T = [0,π], i.e. 1 period, and divide Q in N = 8 space-time slabs Q = Ω T wit

75 4.3. DISCRETISATION OF TRANSPORT PROBLEMS 67 4 Figure 4.8: Solutions for scalar advection diffusion equation on space-time manifold Q = Ω [0,1] for n = 24, p = 2. Tree cases were considered. Upper figures correspond to {u,c,ɛ} = {1,1,0}, center figures to {u,c,ɛ} = {1,1,0.001}, and lower figures to {u,c,ɛ} = {1, 1 2,0.001}. Left column sows iscontours of te distribution as it travels troug space-time. Te rigt column sows surface plots. Colors/eigt indicate te value of te density ρ 0) ). temporal lengt T = π/8. We obtain te weak formulation of te space-time problem on a slab by multiplying wit some test-function w 0) Λ 0) 0 Q), w 0),L X ρ 0)) Q = 0, w 0) Λ 0) 0 Q), were te boundary condition at t = t k is enforced strongly at eac slab. We define te discrete problem by taking Λ 0) = S 2,2,2 24,24,8 S 8 2 0), suc tat w = w T φ 0) and ρ 0) =

76 68 4. SCALAR TRANSPORT ρ T φ 0). We find system of equations, C 0,1) X E 1,0) ρ = 0. 4 Te system needs to be solved slab-by-slab using te final solution at Q k as initial solution for Q k+1. Results are sown Figure 4.9a. Te sape and te eigt of te sinusoid are reasonably well preserved wit some error at its outer contour, and some deformation at its lower- and left side. Again, erroneous wiggles are observed over te domain. Tese wiggles form a circular pattern along te sinusoids patway. Tese oscillations generally ave a large amplitude at te outer region were te transport velocity is te igest. In te next section we aim to get a better understanding of tese oscillations, and try to control tem.

77 4.3. DISCRETISATION OF TRANSPORT PROBLEMS 69 y x a) Reconstructed solution, ρ 0) y x b) Error, e 0) ρ Figure 4.9: Contour plots of approximate solution ρ 0) and corresponding error e0) ρ using a standard Galerkin approac to solve te circular advection problem after one period, T = π. Spatial temporal spline spaces Λ 0) 2,2 := P 24,24 S 8 2, number of slabs k = 8.

78 70 4. SCALAR TRANSPORT SUMMARY OF RESULTS Te Lie-derivative L v describes variations of a k-form along te flow of some vector field v. It is related to te exterior derivative d and te interior product i v troug Cartan s formula. Te interior product i v describes te contraction of some k-form wit a vector field v, and it is a topological operator. Tese operators are used to construct a coordinate invariant representation of yperbolic PDE s. Te vector field v can be related to a 1-form v 1), wic can be discretised in te existing framework using nodal- and edge-type B-spline basis functions. Tis construction introduces Hodge- dependencies, inducing error in te discretisation, i.e. te exact structure of te interior product is not conserved. Te contraction matrix C v is te resulting inner product matrix of te interior product. Te resulting system consists of a linear combination of incidence, mass, and contraction matrices, E, M, and C v. Te discrete solution suffers from spurious oscillations, wic grow over time.

79 5 EULER EQUATIONS Te incompressible Euler equations in fluid dynamics describe balance of mass, momentum, and energy in absence of viscosity and termal conductivity wen variations of density are small, i.e. incompressible fluids. Solutions of tese equations describe te pressure, velocity, termal fields in a region of interest. Te incompressible Euler equations ave a ric geometric structure, wic studied in te classical work by Arnold, [15]. Te Euler equations contain various symmetries and invariants. In our discretisation we aim to preserve tis structure, and conserve invariants in our simulations. In section 5.1 we present an coordinate invariant representation of te Euler equations, and discuss various formulations. Te discretisation approac is discussed in section 5.2. Te spatial discretisation can be constructed, suc tat te invariants are conserved at te semi-discrete level. Te time integrator must be carefully selected to satisfy tis conservation at eac discrete time level. In section 5.3 we verify our results by running various simulations THE EULER EQUATIONS Te Euler equations govern conservation of mass, momentum, and energy under te assumptions of incompressible and inviscid flow. Solutions of tese equations describe te motion of idealised fluids. Under te assumption of incompressible flow, variations in mass density ρ n) are assumed to be small. For tis case te mass and momentum equations can be solved independently from te energy equation. Te assumption is generally used to model liquids and subsonic gas flows i.e. low Mac number flows). Under te assumption of inviscid flow, viscous effects and sear stresses are assumed to be zero. For tis case, boundary layers and turbulent effects are neglected. Tis assumption is used to model convection-dominates flow i.e. ig Reynolds number flows) outside boundary layers. Te conservation of momentum of an incompressible and inviscid fluid flow in R n is 71

80 72 5. EULER EQUATIONS given by, ) v j v t + v j i x i = 1 p ρ x j + b j, 5.1) wit v i te components of te velocity field wit j = {1,,n}, p te pressure, ρ te mass density, and b i components of body force. In Appendix I we derive a coordinate invariant form of te Euler equations expressed. Teorem 5.1 Euler Equations for Incompressible Flow, Sec. 4.3c [18])). Te equations for conservation of mass and momentum for an ideal incompressible fluid on manifold M R n are given by, 5 v 1) t d v 1) = 0 5.2) + L v v 1) = d f 0), 5.3) were v is te velocity-vector field, v 1) te velocity 1-form associated wit v, and f 0) given by, d p f 0) = ρ 1 2 v 1) 2 + φ 0), wit pressure p, mass density ρ, and potential φ 0). We can solve for f 0) using te pressure Poisson equation, wic can be obtained by taking te codifferetial d of Equation I.3, and substitute te incompressibility constraint, Equation I.2, d L v v 1) = f 0). 5.4) An equivalent system of equations can be derived by taking te exterior derivative d of tese equations. Teorem 5.2 Vorticity Formulation of te Euler Equations). Te equations for conservation of mass and momentum for an ideal incompressible fluid on manifold M R n are given by, ω 2) t ψ 2) = ω 2) 5.5) + L v ω 2) = 0, 5.6) were ψ 2) is te stream function wit d ψ 2) = v 1), and vorticity ω 2) = d v 1). Te Euler equations ave a Hamiltonian structure, wic was studied in te classical work by Vladimir Arnold [15]. Te equations govern various symmetries and invariants. In te discrete setting we aim to preserve tese structures and conserve te mentioned quantities over time. Te Euler equations satisfy conservation of integral circulation Γ,

81 5.1. THE EULER EQUATIONS 73 vorticity Ω, and elicity H, d d t Γ = d d t d d t Ω = d d t d d t H = d d t γt) v 1) = 0, 5.7) ω 2) = 0, 5.8) At) V t) 3) = 0, 5.9) were 3) = v 1) ω 2) is te elicity 3-form. Formal derivation of tese results is given in Appendix I. Note tat conservation of circulation and vorticity express an equal result, one can derive one from anoter by applying te generalised Stokes teorem, Equation Helicity is not te only quadratic invariant conserved by te Euler equations. Te equations also govern conservation of kinetic-energy K, d 1 d t K = d d t 2 and in R 2 te conservation of enstropy S, v 1), v 1)) V t) = 0, 5.10) 5 d 1 d t S = d d t 2 ω 2),ω 2)) V t) = ) Proof of tese relations is given in Appendix I. Te exact part of te Lie derivative, i.e. di v v 1), can be included in te rigt-andside term to minimize te amount of terms tat need to be discretised. We define te energy ead, g 0) := f 0) + i v v 1), 5.12) suc tat te momentum equation, Equation I.3, can be written as, v 1) t + i v d v 1) = d g 0), 5.13) wic is known as Gromeka-Lamb s formulation sec [37]). We can obtain dual formulations of te Euler equations by applying te Hodge - operator, and substituting te adjoint co-differential d and co-lie-derivative L v, wic ave been introduced in section 2.4 and section 4.1. In order to avoid confusion wit v 1) in R 2 we denote q 1) = i v vol n) as te n 1)-form associated wit te velocity vector field v. Teorem 5.3 Dual Formulations of te Euler equations). Te dual equations for te conservation of mass and momentum for an ideal incompressible fluid on manifold M R n are given by, q n 1) t d q n 1) = ) i v ωn 2) = d g n), 5.15)

82 74 5. EULER EQUATIONS were v is te velocity-vector field, q n 1) = v 1) te forms associated wit v, ω n 2) = 1) n 1 d q n 1) is te dual vorticity form, and g n) = g 0) te energy ead, Equation Te dual pressure Poisson equation is given by, di v ωn 2) = g n). 5.16) Te stream function is given by ψ n 2), suc tat dψ n 2) = q n 1). Te dual vorticity formulation is given by, Note tat d i v ωn 2) = i ω q1). ω n 2) t 1) n 1 ψ n 2) = ω n 2) 5.17) d i v ωn 2) = ) 5 In te continuous setting primal and dual formulation are equivalent. In te discrete setting, owever, as we ave learned in te previous capters, te coice of formulation as an impact on te conservative properties. Te exactness of te exterior derivative d can be conserved. Te primal formulation enables exactness of te vorticity constraint dd v 1) = dω 2) = 0 in te discrete setting, wereas te dual formulation enables exactness of te incompressibility constraint ddψ n 2) = d q n 1) = 0. Note tat te equations introduced need to be solved in pairs in order to solve for all unknowns. Te pressure Poisson equation, Equation 5.16, needs to be solved togeter wit te momentum equations, Equation 5.15, to solve for unknowns q 1) and g n). Similarly te stream function Poisson equation, Equation 5.17, needs to be solved wit te vorticity equation, Equation 5.18, to solve for unknowns ω n 2) and ψ n 2). We will refer to tese as te momentum-pressure equations, and te vorticity-stream equations DISCRETISATION OF THE EULER EQUATIONS Exactness of conservation of mass, i.e. te incompressibility constraint, is required in te discrete setting in order to satisfy conservation of some oter quantities as we will sow later on. Tis motivates us to discretise te dual formulations, were te incompressibility is governed by te exterior derivative d, rater tan te codifferential d. In tis section we will derive discretisations of te momentum and vorticity formulation. Te semi-discrete system conserves integral mass, vorticity, kinetic energy and enstropy proof in Appendix J). Te time discretisation tecnique must carefully be selected to conserve tese quantities over time. Definition 5.1 discretisation of te Momentum-Pressure Formulation). Consider te periodic unit square domain, Ω =]0,1[ 2. We obtain a weak-formulation of te pressure Poisson equation and te velocity equation, Equation 5.16 and Equation 5.15, by taking te inner product wit test functions r 2) Λ 2) 0 Ω) and s1) Λ 1) 0 Ω), { r 2),diω q1)) Ω = r 2),dd g 2)) Ω, s 1), t q 1)) Ω s 1),iω q1)) Ω = d s 1), g 2)). Ω

83 5.2. DISCRETISATION OF THE EULER EQUATIONS 75 Note tat te pressure-poisson equations needs to be solved by taking a system of first order equations as was done for te 2-form Poisson equation, Example 3.5. We define te discrete solution by taking, r 2) = r T φ 2), s 1) = st φ 1), g 2) = g T φ 2), q 1) = q T φ 1), wic leads to te non-linear time-dependent system of equations, [ M 1,1) E 2,1) ) T M 2,2) E 2,1) ] [ g ] [ ] 0 = E 2,1) M 1,1)) 1 C 1,1) ω q 5.19) 1,1) q M t C1,1) ω q = E 2,1)) T M 2,2) g. 5.20) Te semi-discrete system of non-linear equations satisfies conservation of mass, vorticity, kinetic energy, and enstropy. Proof is presented in Appendix J. However, not every time discretisation preserves te invariance of tese quantities over time. Numerical scemes tat preserve structures of Hamiltonian systems are symplectic integrators and energy-conserving integrators. Pala and Gerritsma studied and derived tese integrators for Hamiltonian ODE systems from a structure-preserving discretisation framework, [38]. Te simplest of tese metods is te implicit midpoint metod. 5 Definition 5.2 Implicit Midpoint Metod). Given time-dependent system of non-linear equations, { d yt) d t = Ay)yt) wit, yt k ) = y k, 5.21) were y k denotes te solution at time-instant t k. Te implicit midpoint metod is given by, y k+1 y k t = Aỹ)ỹ, 5.22) wit ỹ = 1 ) 2 y k+1 + y k, and time-step t = tk+1 t k. Te implicit midpoint metod is second order accurate in time. Application of te implicit midpoint metod to te discrete momentum equation, Equation J.2, yields te system of non-linear equations, M 1,1) q k+1 q k t We define te following algoritm, 1 ) 1 2 C1,1) ω q k+1 + q k = E 2,1) ) T M 2,2) ) g 2 k+1 + g k. 5.23) 1. Initialise: Given solution q k, ω k, g k at time level t k. Let initially ω := ω k.

84 76 5. EULER EQUATIONS 2. Iterate: Assemble contraction matrix C 1,1) ω. Simultaneously solve te discrete momentum equation and te pressure-poisson equation, Equation 5.23 and Equation J.1, to obtain solutions q k+1 and g k+1. Evaluate ω k+1, let ω := 1 2 ω k+1 + ω k ). Iterate tis step until set convergence criterion is met. Conservation of mass depends on pairing of C 1,1) ω q and g, wic can be acieved by simultaneously solving for q k+1 and g k+1. Conservation of integral vorticity is independent of te coice on time-integrator. It depends on te exactness of te exterior derivative, wic, by construction, is satisfied by E 1,0). Conservation of kinetic energy and enstropy are preserved using te implicit midpoint metod can be cecked using te relations in Appendix J). However, enstropy conservation depends on correct matcing of ω and q, wic requires convergence of te linearisation tecnique. We let our convergence criterion depend on te enstropy error, i.e. we iterate until 5 S t k+1 ) S t 0 ) τ, 5.24) S t 0 ) were τ is te desired tolerance, e.g. τ = 10 6, and S te discrete enstropy, S = ω T E 1,0)) T M 1,1) q. 5.25) Definition 5.3 discretisation of te Vorticity-Stream Formulation). Consider te periodic unit-square domain, Ω =]0,1[ 2. We obtain weak-formulations of te stream-function Poisson equation and te vorticity equation, Equation 5.17 and Equation 5.18, by taking te inner product wit test functions r 0), s 0) Λ 0) 0 Ω), { dr 0),dψ 0)) Ω = r 0),ω 0)) Ω s 0), t ω 0)) Ω i v d s 0),ω 0)) Ω = ) We define te discrete solution by taking, r 0) = r T φ 0), s 0) = st φ 0), ψ 0) = ψt φ 0), ω 0) = ωt φ 0), wic leads to te non-linear time-dependent system of equations, M 0,0) ω + E 1,0)) T M 1,1) E 1,0) ψ = ) 0,0) ω M t C 0,1) v E 1,0)) T ω = ) We will prove tat te discrete equations satisfy te relevant conservation laws tat we derived in te previous section, section 5.1.

85 5.3. RESULTS, AND ANALYSIS 77 Application of te implicit midpoint metod to te discrete vorticity equation, Equation J.5, yields te system of non-linear equations, M 0,0) ω k+1 ω k t We define te following algoritm. 1 2 C 0,1) ṽ E 1,0)) T ωk+1 + ω k ) = ) 1. Initialise: Given solution q k, ω k, g k at time level t k. Let initially q := q k. 2. Iterate: Assemble contraction matrix C 0,1) ṽ. Solve te discrete vorticity equation, Equation 5.29, to obtain ω k+1. Ten solve te discrete stream-function Poisson equa- tion, Equation 5.27, to obtain solution stream-function ψ k+1. Evaluate q k+1, and let q := 1 ) 2 q k+1 + q k. Iterate tis step until set convergence criterion is met. Te advantage of tis formulation is tat conservation of mass is satisfied by construction of te stream-function. Terefore, it is not required to solve te vorticity- and Poisson equations simultaneously. Again, conservation of integral vorticity is independent on te coice of time integrator. Conservation of kinetic-energy and enstropy are preserved using te implicit midpoint metod. However, conservation of kinetic-energy depends on te non-linear solution, and requires convergence of te linearisations tecnique. We let te convergence criterion depend on te kinetic-energy error, i.e. we set, 5 K t k+1 ) K t 0 ) τ, 5.30) K t 0 ) were τ is te desired tolerance, and K te discrete kinetic-energy, K = ψ T M 0,0) ω. 5.31) It was sown by Pala and Gerritsma tat te non-linear systems,equation 5.23 and Equation 5.29, can be combined into two systems of quasi-linear equations wic can be solved staggered at time [17]. Tis approac combines te mass, vorticity, and enstropy conserving properties of te vorticity formulation wit te vorticity, and kinetic-energy conserving properties of te momentum formulation, and conserves all quantities witout costly iterative tecniques. In te next section we validate our results by introducing some test problems RESULTS, AND ANALYSIS Te problem of two co-rotating Taylor vortices was introduced by Mullen et al. to analyze te performance of structure-preserving discretisations. Two equipotent vortices will split apart above some critical distance. Non-conserving metods fail to reproduce tis beaviour after wic te vortices merge togeter into one big vortex, [33]. We will look at te problem of

86 78 5. EULER EQUATIONS Example 5.1 Taylor Vortices). Consider te Euler equations on Ω =]0,1[ 2. As initial condition we consider a distribution of N = 2 Taylor vortices of equal strengt, wic induce vorticity, ωx) = 2 i=1 Uσ i a 2 r r i ) 2 a 2 ) exp r x) r i ) 2 a 2 ), 5.32) 5 were U = 1 is te maximal tangential velocity, a = te vortex strengt, σ i = ±1, te direction of rotation positive for counter-clockwise), and r x) r i ) 2 te squared distance to te i -t core. We place initial cores near te center of te domain, x i, y i ) = {0.4, 0.5),0.6, 0.5)}, Equation Te cores will start to circle around te center. Te vortices deform in a stretcing motion, wic causes a tail to be formed at eac vortex. Te cores briefly touc, but te cores ave gained too muc momentum to merge. Instead tey drift apart. Note tat stabilized numerical tecniques would induce momentum loss, wic could fail in capturing tis beaviour. We considered two configurations to sow tat te conservative properties are independent of discretisation parameters. In te first configuration we used a spline basis of polynomial order p = 1 wit dimension n = 48, and used a time-step t = 1/16. In te second configuration we let p = 2 and n = 64, and let t = 1/32. Te solutions at t = 1.5 are sown in Figure 5.1. Bot configurations were able to capture te non-merging motion. Te resulting distributions are rougly te same. However, te cores corresponding to te first configuration are less compact. We also observe more local extrema in te centre region and in te tail regions. Finally, tere is a difference in te core locations. Te vortex-stream formulation governs static conservation of mass, and dynamic conservation of vorticity, kinetic energy, and enstropy. We compare te dynamically conserved quantities to teir initial value, wereas te static quantity is evaluated at eac instant. Results of bot configurations are sown in Figure 5.1. Bot configurations conserve enstropy, vorticity, and mass wit great precision. Kinetic energy is bounded wit relative error less tan 1 percent. Te quadratic quantities, i.e. kinetic energy and enstropy, are better conserved by te second configuration.

87 5.3. RESULTS, AND ANALYSIS 79 y t = x y t = x Figure 5.1: Vorticity ω 0) at t = 1.5 of Taylor vortices on a periodic domain. Spline basis n = 48, p = 1, and time-step t = 1/32 left), and n = 64, p = 2, t = 1/64 rigt). 5

88 80 5. EULER EQUATIONS 1.0 t = t = 0.25 y 0.5 y x 1.0 t = x 1.0 t = y 0.5 y x 1.0 t = x 1.0 t = y 0.5 y x x Figure 5.2: Vorticity ω 0) at various time-instants of simulated motion of an initial distribution of co-rotating Taylor vortices on a periodic domain. Results were obtained using spline basis of polynomial order p = 2 and dimension n = 64, and time-step size t = 1/64.

89 5.3. RESULTS, AND ANALYSIS St) S0) S0) Ωt) Ω0) Ω0) t t Kt) K0) K0) p=1 p= t Φt) t 5 Figure 5.3: Enstropy error upper left), vorticity error upper rigt), kinetic energy error lower left), and total mass flux lower rigt) over time for te co-rotating Taylor vortices on a periodic domain. Spline basis n = 48, p = 1, and time-step t = 1/32 blue), and n = 64, p = 2, t = 1/16 green) St) S0) S0) Ωt) t t Kt) K0) K0) Φt) t t Figure 5.4: Enstropy error error upper left), vorticity upper rigt), kinetic energy error lower left), and total mass flux lower rigt) over time for te counter-rotating Taylor vortices on a semi-periodic domain. Spline basis n = 64, p = 2, and time-step t = 1/16 were used.

90 82 5. EULER EQUATIONS Example 5.2 Solid Wall Boundary). So far we ave only considered periodic domains. In tis example we introduce solid wall boundaries. Consider cylindrical domain Ω = ]0,1[ [0,1], wic is periodic in x and non-periodic in y. Boundaries Ω = {x, y) Ω, y {0, 1}} are impenetrable walls, wic requires normal velocity components to be equal to zero, i.e. q x 1) = vd x = 0 at Ω. Te weak formulation of te vorticity-stream formulation including boundary terms is given by, { dr 0),dψ 0)) Ω + r 0),dψ 0)) Ω = r 0),ω 0)) Ω s 0), t ω 0)) Ω i v d s 0),ω 0)) Ω + s 0),i v ω0)) Ω = 0 5 Te boundary terms tat appear in te stream function Poisson equation can be used to satisfy tangential flow conditions. Te bounder terms in te vorticity equation can be used to describe te in- and outflow of vorticity. In te case of te mentioned solid-wall boundary conditions, evaluation of tese natural boundary conditions yields, r 0),dψ 0)) Ω = s 0),i v ω0)) Ω = Ω Ω r 0) dψ 0) = r ψ Ω x d y r ψ Ω y d x s 0) iv ω0) = sωud y + sωvd x Ω Ω In- and outflow boundary conditions are essential boundary conditions. Te impenetrable wall condition corresponds to zero inflow. It follows tat te stream function ψ needs to be constant at te boundaries, ψ x = q x 1) Ω) = 0 Ω ψ Ω) = C, were we take C = 0. As an initial condition we consider a distribution of two counterrotating Taylor vortices, i.e. σ i = {+1, 1} wit cores locations, x i, y i ) = {0.4,0.75),0.6,0.75)}. Oter parameters are taken as introduced in Equation To discretise te system we take a spline basis wit p = 2 and n = 64, and let time-step t = 1/32. Results are sown in Figure 5.5. Te counter-rotating cores induce an upward velocity, wic transports te cores to te upper wall. Note tat te outer regions of te vortices are rotating in opposite direction, wic induces a downward velocity transporting te outer regions to te lower wall. Near te wall te cores start travelling parallel to te wall in opposite direction until tey meet again, were tey induce a downward velocity wic transports te cores downward. Note tat te solid wall acts as a counter-rotating core. Te integral quantities kinetic energy, vorticity, enstropy, and mass flux were measured over time, Figure 5.4. Again we observe tat enstropy, vorticity and mass flux are conserved wit macine precision.

91 5.4. SUMMARY OF RESULTS t = t = y 0.5 y x t = x t = y 0.5 y x x Figure 5.5: Vorticity ω 0) at various time-instants of simulated motion of an initial distribution of counter-rotating Taylor vortices on a semi-periodic domain wit upper- and lower solid wall boundaries SUMMARY OF RESULTS Te incompressible Euler govern conservation of mass, and momentum. Te equations ave a ric geometric structure, governing various symmetries and invariants. In R 2 te integral quantities mass, vorticity, kinetic-energy, and enstropy are conserved over time. Te equivalent vorticity formulation can be derived by taking te exterior derivative of te momentum equations. Te dual formulation must be taken, suc tat te equation for te conservation of mass depends on te exterior derivative, wic can be satisfied witout error in te discrete setting. Conservation of te integral quantities inges on te exactness of te exterior derivative d and te interior product i v. Te introduced framework ensures tat tis structure is preserved in te semi-discrete system of equations. A symplectic or energy conserving time integrator must be cosen to conserve te integral quantities over time.

92

93 6 CONCLUSION & FUTURE DIRECTIONS Tis tesis is te result of a researc project on discretisation tecniques for CFD. Te project was divided in tree objectives. Te first objective was to study te existing existing structure-preserving IGA framework for elliptic PDE s. Te second objective was to expand tis teoretical framework to discretise yperbolic PDE s, and te tird objective was to construct a structure-preserving IGA discretisation for te incompressible Euler equations, capter 5. Recently, some attention as been focused on structure-preserving discretisation tecniques witin te scientific community. In tese tecniques special care is taken in te preservation of fundamental matematical structures, symmetries, and invariants in te discretisation process. Exterior calculus proves to be insigtful in identifying relevant topological dependencies. In exterior calculus we describe elliptic PDE s wit te topological derivative operator d and te metric dual Hodge- operator. Conservation of te important topological structure of te derivative operator d lies at te eart of te structure-preserving framework. Te framework is of teoretical relevance. It explains te correctness of using dual grids in finite-volume metods, and allows derivation of inf-sub stable basis functions in finite-element metods. Te extension to yperbolic PDE s is not trivial. A superior teory to discretise te convective terms and te vector fields is unavailable at tis moment. However, by carefully selecting te geometric formulation of te PDE, it is possible to conserve te nilpotency property of te interior product. Tis proved to be te key ingredient in te construction of an energy-conserving discretisation for te incompressible Euler equations in R 2. Tese discrete Euler equations beave as te continuous Euler equations wen te mes as sufficient resolution to represent relevant vortical structures, and wen spurious oscillations are witin bounds. It needs to be verified if tis construction is valid for arbitrary geometries. Te next step is to develop a structure-preserving discretisation for te incompressible Euler equations in R 3. It will be callenging to conserve integral elicity. Anoter 85

94 86 6. CONCLUSION & FUTURE DIRECTIONS step migt be te discretisation of te compressible Euler equations, for wic tere exists a wide range of formulations and discretisaions. I wonder if it is possible to identify an optimal formulation, or some fundamental construction, tat needs to be satisfied at te discrete level. Tese directions will improve our understanding of discretisations and improve accuracy of simulations.

95 A DERHAM COHOMOLOGY Te DeRam sequence as important topological relations tat governs existence of differential forms on manifolds. In coomology we reduce tese relations to topological structures, i.e. cains and cocains. In section 2.3 we ave sown tat te codifferential δ is te topological analogue of te exterior derivative d. In tis capter we introduce te structures corresponding to te exterior derivative d, te coboundary operator δ, and te codifferential d. We will present te Poincaré lemma and te Hodge decomposition. Definition A.1 Closed Forms, Sec. 5.2 [18])). A k-form α k) is closed if dα k) = 0. Te space of closed k-forms Z d Λ k) is te kernel of te exterior derivative, i.e. Z d Λ k) := {α k) Λ k) : dα k) = 0} Λ k). Definition A.2 Exact Forms, Sec. 5.2 [18])). A k form α k) is exact if tere exists a β k 1) s.t. α k) = dβ k 1). Te space of exact k-forms B d Λ k 1) is te range of te exterior derivative, i.e. B d Λ k 1) := dλ k 1) Λ k). It follows tat every exact form is closed, i.e. if, β k 1) s.t. α k) = dβ k 1) ten dα k) = ddβ k 1) = 0. Te topological structure is depicted in Figure A.1. Te converse is true on contractible manifolds. Contractible spaces can be continuously reduced to a point. Examples include any star) domain in Euclidean space. Teorem A.1 Poincaré Lemma, Sec. 5.4 [18])). On a contractible manifold M every closed k-form is exact, i.e. α k) Z d Λ k) M ), tere exists β k 1) Λ k 1) M ) s.t. α k) = dβ k 1). 87

96 88 A. DERHAM COHOMOLOGY A Λ k 1) Λ k) Λ k+1) Z d Λ k) d d B d Λ k 1) HΛ k) Figure A.1: Poincaré Lemma, Te range of te exterior derivative B d Λ k 1) is contained in te kernel Z d Λ k). On contractible domains te converse is also true, for wic ten B d Λ k 1) = Z d Λ k). Te teorem proofs existence of solutions for balancing relations is pysics. Tese relations, often of te form gradφ) = f, curl ω) = g, and div v) =, are fundamental in pysics. From te teorem it also follows tat for any volume form α n), tere exist a potential function. Since dλ n) = 0, we ave tat Λ n) = Z d Λ n), ence α n) Λ n) = Z d Λ n), β n 1) B d Λ k 1) s.t. dβ n 1) = α n). A.1) Tus te exterior derivative acting on Λ n 1) is surjective on contractible domains surjectivity of te operator). Note tat tis is only true if and only if te boundary conditions satisfy te compatibility condition. Finally, te teorem implies tat te kernel of Λ k) is equal to dλ k 1) on contractible manifolds, i.e. B d Λ k 1) = Z d Λ k). Hence we can decompose te space of k-forms Λ k) into Λ k) M ) = Z d Λ k) Z C d Λk), A.2) were Z C d denotes te complement of Z d Λ k) in Λ k). On non-contractible domains we can define te space of k-forms wic are closed but not exact. We call suc forms Harmonic forms. Definition A.3 Harmonic forms, Sec [18])). Te space of armonic k-forms H Λ k) is defines as H Λ k) := {α k) Z d Λ k) : α k) B d Λ k 1) }. On contractible domains H Λ k) =. Te space of closed k-forms can be written as te sum of exact k-forms united wit te space of armonic k-forms, Z d Λ k) = B d Λ k 1) H Λ k). If we combine tis expression wit Equation A.2 we obtain, Λ k) M ) = B d Λ k 1) H Λ k) Z C d Λk). Tis result is known as te Hodge decomposition. Te Hodge decomposition enables us to decompose differential forms in algebraic complements.

97 89 Definition A.4 Co-closed Forms, Sec [18])). A k-form α k) is co-closed if d α k) = 0. Te space of co-closed k-forms Z d Λ k) is te kernel of te codifferential, i.e. A Z d Λ k) := {α k) Λ k) : d α k) = 0} Λ k). Definition A.5 Co-exact Forms, Sec [18])). A k form α k) is co-exact if β k+1) s.t. α k) = d β k+1). Te space of co-closed k-forms B d Λ k+1) is te range of te codifferential, i.e. B d Λ k+1) := d Λ k+1) Λ k) Definition A.6 Harmonic Form, Sec [18])). In line wit our previous definition, section 2.2, we can define te space of armonic forms as follows, H Λ k) := {α k) Z d Λ k) : α k) B d Λ k 1) } H Λ k) := {α k) Z d Λ k) : α k) B d Λ k+1) }. Tese definitions can be united as follows. A k-form α k) is armonic if α k) = 0. Te space of armonic k-forms is defined as, H Λ k) := {α k) Λ k) : dα k) = d α k) = 0}. As in section 2.2 we can decompose te space of k-forms in ortogonal components, Λ k) M ) = B d Λ k 1) H Λ k) Z C d Λk) Λ k) M ) = B d Λ k+1) H Λ k) Z C d Λ k). Te decompositions can be unified by taking Z C d Λk) = B d Λ k+1) and Z C d Λ k) = B d Λ k 1). Tis leads to a more useful definition of te Hodge decomposition. Teorem A.2 Hodge decomposition, Sec [18])). For any k-form on a closed manifold M tere exists an unique decomposition α k) = dφ k 1) + d ψ k+1) + k), were k) is a armonic form. Note tat tis doesn t imply uniqueness of φ k 1) and ψ k+1). On a contractable manifold te space of armonic forms is empty, i.e. k) = 0. Te coboundary operator δ induces te same topological structure on cocain complexes. We can define cocain spaces analogue to te spaces of closed-, exact-, and armonic form. B δ C k 1) := δc k 1), Z δ C k) := {c k) C k) : δc k) = }, HC k) := {c k) Z δ C k) : c k) B δ C k 1) }, Te topological structure of tese spaces is sown in equivalent to te one sown in Figure A.1. Te topology of te exterior derivative is preserved in discrete cocain structures. Tis structure governs discrete counterparts of te Poincaré lemma, tat depends on te exactness of te coboundary operator.

98 B DERIVATION OF ADJOINT RELATIONS In section 2.4 we introduced te L 2 -inner product troug te action of te Hodge- operator. In tis capter we derive te various adjoint relations tat ave been used trougout te work. Let L : Λ k) M ) Λ l) M ) be a linear operator. Te adjoint operator is te liner operator L : Λ l ) M ) Λ k) M ) tat satisfies, α k),l β l)) = L α k),β l)), B.1) M M for some α k) Λ k) M ) and β l) Λ l) M ). Te exterior derivative d, te interior product i v, and te Lie derivative L v do not depend on metric. Teir respective adjoints, te codifferential d, te cointerior product iv, and te co-lie derivative L v do explicitly depend on te metric-dependent Hodge- operator. Te adjoint relations can be used to obtain a more favourable weak formulation. Note tat in many derivations we use te following property of te Hodge- operator, α k) = 1) kn k). 1. Te codifferential: d α k) = 1) nk 1)+1 d α k) B.2) Let α k 1) Λ k 1), and β k) Λ k), now dα k 1) β k) ) =dα k 1) β k) + 1) k 1 α k 1) d β k) ) =dα k 1) β k) + 1) k 1 α k 1) 1) n k+1)k 1) d β k) =dα k 1) β k) + 1) n k)k 1) α k 1) d β k) =dα k 1) β k) + 1) n k)k 1) α k 1) 1) nk 1) 1 d ) β k) =dα k 1) β k) + 1) kk 1) 1 α k 1) d β k) =dα k 1) β k) α k 1) d β k). 90

99 91 In te last line we used tat kk 1) is even for all k. By rearranging te terms, integrating over manifold M, and using Teorem 2.2 and Definition 2.15, we find, α k 1) d β k) = dα k 1) β k) M α k 1),d β k)) dα = k 1),β k)) M M M M M α k 1) β k) α k 1) β k). B.3) B Te adjoint relation is obtained wen te boundary term reduces to zero. 2. Te cointerior product: i v := 1)nk+1) 1 i v B.4) Let α k) Λ k), β k 1) Λ k 1), and v be a vector field, and v 1) te 1-form associated wit v. Substitution te 1-form isomorpism, Equation 4.10, in te definition of i v yields, i v βk 1) = 1) nk 1+k 1)n k+1) β k 1) v 1) ) = 1) nk 1+k 1)n k+1)+k 1)n k+1)+kn k) β k 1) v 1) ) = 1) k2 1 β k 1) v 1) ) = 1) k 1 β k 1) v 1) ), Taking te inner product of β k 1) and i v α k) yields, β k 1),i v α k)) M = M β k 1) i v α k) 4.10 = 1) kn k) M β k 1) α k) v 1)) = 1) kn k)+n k+1)k 1) β k 1) α k) v 1) M = 1) kn k)+n k+1)k 1)+n k) β k 1) v 1) α k) = 1) k 1 β k 1) v 1),α k)) M = iv βk 1),α k)) M M Te cointerior product i v is te adjoint of te interior product i v. β k 1),i v α k)) M = i v βk 1),α k)) M B.5) 3. Te co-lie derivative: L v := 1)kn+1)+1 L v. B.6)

100 92 B. DERIVATION OF ADJOINT RELATIONS Te co-lie derivative is related to te codifferential and cointerior product tat is similar Cartan s formula, Equation 4.5. Let α k) Λ k) and v some vector field. We find, B L v αk) 4.5 = 1) kn+1)+1 di v + i v d) α k) ) = 1) kn+1)+1 1) n k 1)k+1) d i v + 1) n k+1)k 1) i v d α k) = 1) kn+1)+1 1) n k 1)k+1)+n d i v + 1)n k+1)k 1)+n i v d ) α k) =d i v + i v d )α k). Using adjoint relations Equation B.3 and Equation B.5 we find, β k),l v αk)) M B.7 = β k),d iv αk)) + β k),iv d α k)) M M dβ k),iv β αk)) k),iv αk)) + M M i v β k),d α k)) M = = i v dβ k),α k)) β k),iv αk)) M M + di v β k),α k)) i v β k),α k)) M M 4.5 = L v β k),α k)) β k),iv αk)) i v β k),α k)). M M M B.7) B.8) B.9) Te adjoint relation is obtained wen te boundary terms reduce to zero.

101 C GENERALISED PIOLA TRANSFORMATIONS In tis capter we derive te pull-back operators for k-forms in R 3. Tese transformations commutate wit te exterior derivative, suc tat te DeRam structure is preserved on arbitrary geometries. Note tat tese transformations are also useful wen determining integrals on some parent domain. Definition C.1 Pull-back operator φ, Sec. 2.7 [18])). Consider smoot bijective coordinate mapping φ : N M. Consider local coordinates x i for manifold M and ξ j for manifold N, suc tat x = φξ) = xξ). Let α 0) Λ 0) M ) be some 0-form. We define pull-back operator φ to be equal to φ α 0) )ξ) := αxξ)). Te pull-back operator satisfies te following properties, φ dα 0) ) = dφ α 0) ) φ α k) β l) ) = φ α k) ) φ β l) ) We can use tese properties to derive pull-back operators for arbitrary k-forms in R 3. Let β 1) be some 1-form. φ β 1) = φ β i d x i ) = β i xξ))dφ x i ) = β i xξ)) xi ξ j dξ j = βdξ j = β 1). or β = βf, were F = F i j = xi is te deformation gradient. Now for some 2-form ω 2) we ξ j 93

102 94 C. GENERALISED PIOLA TRANSFORMATIONS find, φ ω 2) = φ ω i j d x i d x j = ω i j xξ))φ d x i ) φ d x j ) = ω i j xξ)) xi ξ k x j ξ l dξ k dξ l = ω 2), C wit i j, and k l. We find tat xi dξ k dξ l defines te adjugate of te deformation gradient matrix, i.e. x j ξ k ξ l ω = ωadjf ) = ωdetf )F 1 ). Note tat tis relations is famous as te Piola stress) transformation. Finally, for some 3-form ρ 3) we find, φ ρ 3) = ρxξ)) xi ξ l x j ξ m x k ξ n dξ l dξ m dξ n, were we obtain ρ = ρdetf ). Tese transformations for te components of k-forms commute wit te exterior derivative, suc tat any DeRam complex in some pysical space is also satisfied in some pull-backed parameter space. Te transformations are known as te generalized Piola transformations. We can pull-back integrals using te general pull-back formula. Teorem C.1 General pull-back formula, Sec. 3.1 [18])). Consider again te smoot bijective coordinate mapping φ : N M. Let U N, suc tat φu ) M. Let α k) Λ k) M ) be a k-form. Te integral of αk) on φu ) can be pulled-back to U, α k) = φ α k). C.1) φu ) U

103 D WELL-POSEDNESS OF THE POISSON PROBLEM Well-posedness of elliptic PDE problems can be analysed in te powerful Hilbert space setting. In tis setting, weak formulations of te PDE problems are analysed. We will introduce some fundamental properties and teorems tat enable us to prove existence and uniqueness to te PDE problems. Furtermore we prove well-posedness for te Poisson problems tat were introduced in capter 3. Lemma D.1 Caucy-Scwarz inequality). Any two k-forms α k),β k) L 2 Λ k) satisfy te inequality α k),β k) ) α k) L 2 Λ k) βk) L 2 Λ k) D.1) Lemma D.2 Generalised Poincaré inequality, proof in [39])). Consider a k-form α k) W d Λ k) Ω) on contractible manifold Ω, i.e. te space of armonic k-forms is empty. Ten α k) satisfies, α k) L 2 Λ k) C dαk) L 2 Λ k+1), D.2) for some constant C. Te Poincaré inequality allows us to define an equivalent Sobolev norm α k) Wd Λ k := dαk L 2 Λ k+1). Definition D.1 Weakly coercive). Let H 1, H 2 denote Hilbert spaces. A bilinear form B, ) : H 1 H 2 R is weakly coercive if were α R. Bu, v) sup α u H1, u H 1, v H 2 v H2 and sup u H 1 Bu, v) u H1 α v H2, v H 2, 95

104 96 D. WELL-POSEDNESS OF THE POISSON PROBLEM Definition D.2 Coercive). A bilinear form B, ) : H H R is coercive if Bu,u) α u 2 H, u H. Definition D.3 Continuous). A bilinear form B, ) : H 1 H 2 R is continuous if were c R. Bu, v) c u H1 v H2, D Te Lax-Milgram teorem is a powerful teorem to prove existence and uniqueness of solution to coercive PDE problems. We will also consider indefinite variational problems for wic well-posedness can be proved using Brezzi s teorem, Teorem D.4. Teorem D.3 Lax-Milgram). Let u, v H, and let F : H R define a bounded linear functional, and let a, ) : H H R be a continuous, coercive, bilinear form. Ten!u H s.t. au, v) = F v), v H. Problem D.1 Mixed Variational Problem). Given a pair of Hilbert spaces H 1,H 2, bilinear forms a, ) : H 1 H 1 R, b, ) : H 1 H 2 R, and bounded linear functionals F ) : H 1 R, and G ) : H 2 R. We ten seek u H 1, p H 2 s.t. { au, v) bv, p) = F v), bu, q) = Gq), D.3) v H 1, q H 2. Teorem D.4 Inf-Sup Condition - Brezzi s Teorem Sec. 2.1 [22])). Let Z = {v H 1 : bv, q) = 0, q H 2 }, let bilinear form a, ) be continuous on H 1 H 1 and coercive on Z Z. Tere exists an unique solution u, p) H 1 H 2 for Problem D.1 if and only if bilinear form b, ) is continuous on H 1 H 2 and if bv, p) sup C p H2. v H 1 v H1 We will apply tese teorems to two example problems in te framework of exterior calculus using te DeRam Sobolev Complex. Te first problem, te 0-form Poisson problem, is associated wit unconstrained optimization of a quadratic functional. Tis type of problem as classically been solved successfully using Rayleig-Ritz approac. In tis approac te problem is converted to a set of discrete minimization problems wit resulting linear problem Au = f, were A is a symmetric positive definite SPD) matrix.

105 97 Example D.1 0-form Poisson problem). Consider te domain Ω = [0,1] 2 R 2 wit boundary Ω = Γ d Γ n, were Γ d Γ n =. Given f 0) Λ 0), find ψ 0) Λ 0) suc tat, ψ 0) = f 0), ψ 0) = 0, on Γ d n ψ 0) = 0, on Γ n We obtain a weak formulation of tis problem by taking te inner product wit some test function w 0) Λ 0) 0. Were Λ0) 0 is te space of compactly supported 0-forms according, Λ 0) 0 := {α0) Λ 0) : tr α 0) = 0 on Γ d }. We use integration by parts for two reasons. First, we would like to reduced te continuity constraints on te considered trial space. Secondly, we would like to get rid of te codifferential d. We find, D w 0),d dψ 0)) Ω = w 0), f 0)) Ω d w 0),dψ 0)) Ω = w 0), f 0)) Ω + w 0),dψ 0) ) Ω. For tis case te boundary integral reduces to zero. We ence obtain te following weak formulation. Given f 0) L 2 Λ 0), find ψ 0) W d Λ 0) s.t. d w 0),dψ 0)) Ω = w 0), f 0)) Ω, w 0) W d Λ 0) 0. We can rewrite tis problem in variational formulation, defining bilinear form a, ) and functional F ), a, ) : W d Λ 0) W d Λ 0) R : a, ) := d,d ) Ω F ) : W d Λ 0) R : F ) :=, f 0) ) Ω. Given ψ 0) L 2 Λ 0), find ψ 0) W d Λ 0) suc tat, aw 0),ψ 0) ) = F w 0) ), w 0) Λ 0) 0. Lax-Milgram proves existence of unique solution if a, ) is coercive, and continuous, and F ) is bounded. Using Caucy-Scwarz CS), and Poincaré PC), we find aw 0),ψ 0) ) = d w 0),dψ 0) ) Ω C S d w 0) dψ 0) <, aw 0), w 0) ) = d w 0),d w 0) ) Ω = d w 0) 2 PC C w 0) 2, F = sup w 0) 1 F w 0) ) C S sup w 0) 1 w 0) f 0) = f 0) <. Note tat we used te boundedness of norms tat exists by definitions of function spaces L 2 Λ 0) and W d Λ 0). Hence by Lax-Milgram tere exists an unique solution ψ 0) W d Λ 0) s.t. aw 0),ψ 0) ) = F w 0) ), w 0) W 0 d Λ0).

106 98 D. WELL-POSEDNESS OF THE POISSON PROBLEM In te second problem we will consider te volume, n-form, Poisson problem. Tis problem is equivalent to constrained optimization of a quadratic functional. Tese problems are solved numerically using a mixed Galerkin formulation. Example D.2 n-form Poisson problem, Sec. 2.3 [23])). Consider te domain Ω = [0,1] 2 R 2 wit boundary Ω = Γ d Γ n, were Γ d Γ n =. Given f 2) Λ 2), find ψ 2) Λ 2) suc tat, ψ 2) = f 2), in Ω, ψ 2) = 0, on Γ d, t ψ 2) = 0, on Γ n D Again we multiply wit some test-function w 2) and perform integration by parts for reasons mentioned before to obtain, d w 2),d ψ 2)) Ω = w 2), f 2)) Ω. We could use a similar roadmap as for te 0-form Poisson case, using Lax-Milgram to prove well-posedness, and taking te dual Sobolev space W d Λ 2). However, in te finite projection metod, we will only be able to satisfy one of te DeRam Sobolov complexes simultaneously. Hence we want to limit ourselves to te use of te following sequence of function spaces in proving well-posedness of te problem. W d Λ 0) d W d Λ 1) d L 2 Λ 2) Consider te equivalent system of first-order-differential equations, { d ψ 2) = v 1), d v 1) = f 2) A weak formulation can be obtained taking inner product wit q 1) W d Λ 1) 0 and w 2) L 2 Λ 2) respectively, and performing integration by parts, were Λ 1) 0 := {α1) Λ 1) : tr α 1) = 0 on Γ n }. We obtain, { d q 1),ψ 2)) Ω + q 1),ψ 2)) Ω = q 1), v 1)) Ω, q1) W d Λ 1) 0, w 2),d v 1)) Ω = w 2), f 2)) Ω, w 2) L 2 Λ 2). Note tat te boundary integral reduces to zero due to te induced boundary conditions. We can define bilinear forms a, ), and b, ) and functional F ), a, ) : Λ 2) Λ 2) R : a, ) :=, ) Ω b, ) : W d Λ 1) L 2 Λ 1) R : b, ) := d, ) Ω F ) : W d Λ 0) R : F ) :=, f 0) ) Ω.

107 99 Now we can rewrite te problem in te variational formulation. Given f 2) L 2 Λ 2), find ψ 2) L 2 Λ 2) and v 1) W d Λ 1) suc tat, { aq 1), v 1) ) bq 1),ψ 2) ) = 0, q 1) Λ 1) 0 bv 1), w 2) ) = F w 2) ), w 2) Λ 2). Let Z d Λ 1) = {q 1) W d Λ 1) : d q 1) = 0}. Note tat te exterior derivative on n 1)-forms is surjective, Equation A.1, wic implies tat for any ψ 2) L 2 Λ 2), tere exists some q ψ 1) W d Λ 1), suc tat ψ 2) = d q ψ 1), and ψ2) = d q ψ 1). Brezzi s teorem, Teorem D.4, can now be used to prove well-posedness of tis problem. We need to prove tat a, ) is coercive on Z d Λ 1) Z d Λ 1), tat b, ) is continuous, and tat te inf-sup condition is satisfied for b, ). aq 1), q 1) ) = q 1) 2 W d Λ 1) = q 1) L 2 Λ 1) + d q 1) L 2 Λ 2), D bq 1),ψ 2) ) = d q 1),ψ 2) ) C S d q 1) ψ 2) < sup bq1),ψ2) ) q 1) W d Λ 1) q 1) bq1) ψ,ψ2) ) ψ2) 2 q ψ 1) = q ψ 1) 1 C ψ2), were we used tat q 1) ψ PC C d q 1) ψ = C ψ2). Hence tere exists an unique solution set v 1),ψ 2) ) W d Λ 1),L 2 Λ 2) ). Note tat well-posedness depended on te exact sequence tat is satisfied by te Sobolev DeRam complex. In our discretization approac we must ensure tat our finite-dimensional function spaces satisfy te exact sequence to prove well-posedness in te discrete setting. In te 0-form Poisson example tis was not te case wic poses less restrictions on te coice of finite-dimensional function spaces in te discretization process.

108 E DERIVATION OF EDGE FUNCTIONS Te finite-dimensional function spaces are constructed by taking spans of combinations of nodal- and edge-type basis functions as is sown in section 3.3. In tis capter we derive te edge-type basis function and prove its unit mass property. Consider te 1-form v 1) L2 Λ 1) 1) tat satisfies v = d f 0). Te question is, ow do we define L 2 Λ 1)? Do we let L2 Λ 1) on te 0-form f 0) yields te following, := S p n as well? Application of te exterior derivative d f 0) ξ) = d n λ i,p ξ)f i )dξ dξ i= n ) p p = λ i,p 1 ξ) λ i+1,p 1 ξ) f i )dξ i=1 ξ i+p ξ i ξ i+p+1 ξ i+1 p = p λ 1,p 1 ξ)dξ f 1 λ 2,p 1 ξ)dξ f 1 ξ p ξ 0 ξ p+1 ξ 1 p + λ 2,p 1 ξ)dξ f 2 ξ p+1 ξ 1 p λ n,p 1 ξ)dξ f n 1 ξ n+p ξ n p p + λ n,p 1 ξ)dξ f n λ n+1,p 1 ξ)dξ f n ξ n+p ξ n ξ n+p+1 ξ n+1 ) n p = λ i,p 1 ξ)dξ f i f i 1 ) ξ i+p ξ i i=2 Were we used tat ξ p ξ 0 = ξ n+p+1 ξ n+1 = 0. If we let v i = f i f i 1 ), and use B- p spline basis functions of a polynomial degree lower wit scaling ξ i+p ξ, ten te relation i v 1) = d f 0) is exact. 100

109 101 Unit mass. Take v 1) = d f 0) wit, f i = { C, if i < j C + 1, if i j, for some constant C R, suc tat v i = δ j i. Now, ξj +p ξ j λ 1) j ξ)dξ = = ξj +p ξ j ξj +p n i=2 λ 1) ξ)v i i ξ j v 1) = ξj +p ) dξ d f 0) ξ j dξ = f 0) ξ j +p ξ j = C + 1) C = 1 Tis property reduces integration of 1-forms to mere summation of te expansion coefficients, i.e. for some v 1) L2, Ω v 1) = Ω n i=2 λ 1) ξ)v i i ) dξ = n v i i=2 E

110 F CONSTRUCTION OF BOUNDED COCHAIN PROJECTIONS We want to construct projection operators, Π k), tat commute wit te exterior derivative conform, Λ k) d Λ k+1) C k) Π k) δ C k+1) Π k+1) Suc commuting projection operators are known as cocain projection operators. Bounded cocain operators are fundamental in analysis. Tese projection operators allow us to provide estimates of te discrete solution wit respect to te continuous solution. More on bounded cocain projectors can be found in te work by Falk and Winter, [5, 40]. Construction of suc operators is not trivial, standard Galerkin L 2 projections of a k- form and k +1-form do not yield te desired commutativity. However, one can construct suc projection by introducing a Poisson problem. An example is given in Appendix F. Example F.1 Construction of bounded cocain projections for ω 2) = d v 1) in R 2 ). Consider te pair v 1) W d Λ 1) and ω 2) L 2 Λ 2) on Ω = [0,1] 2 given by, v 1) = 1 4π cos2πx)sin2πy)d x 1 4π sin2πx)cos2πy)d y ω 2) = d v 1) = sin2πx)sin2πy)d x d y. We want to define bounded projection operators satisfying commutativity δπ 1) v 1) = Π 2) d v 1). We let W d Λ 1) := S 2,3 7,8 S 3,2 8,7 and L2 Λ 2) := S 2,2 7,7. To define Π2) we consider 102

111 103 standard L 2 projection, i.e. s 2),ω2) )Ω = s 2),ω2)) Ω, s2) L2 Λ 2), Using te existence of q 1), suc tat, s 2) = d q 1), Equation A.1, and using ω 2) = d v 1), we find an equivalent formulation, d q 1) 1),d v )Ω = d q 1),d v 1)), Ω q1) W d Λ 1). Integration by parts yields, q 1), v 1) )Ω d q 1), v 1) ) Ω = q 1),d v 1)) d q 1) Ω, v 1)), Ω q1) W d Λ 1). Hence we let, Π 2) : L 2 Λ 2) C 2), ω 2) Π 2) ω 2) wit, s 2),ω2) )Ω = s 2),ω2)), Ω s2) L2 Λ 2) F Π 1) : W d Λ 1) C 1), v 1) Π 1) v 1) wit, q 1), v 1) )Ω = q 1), v 1)), Ω q1) W d Λ 1) To reduce continuity constraints, conform te cosen space W d Λ 1), we will work we te integrated-by-parts version of Π 1). We obtain systems of equations, M 2,2) ω = φ 2),ω 2)) Ω E 2,1) ) T M 2,2) E 2,1) v E 2,1)) T ) φ 2), v 1) = E 2,1)) T φ 2),d v 1)) Ω Ω E 2,1)) T φ 2), v 1)) Ω. A reconstruction of δπ 1) v 1) is sown in Figure F.1. We find, δπ 1) v 1) Π 2) d v 1) L 2 = e 15, wit, v 2 dimv) L 2 := v i 2. i

112 104 F. CONSTRUCTION OF BOUNDED COCHAIN PROJECTIONS ω 2) x y Figure F.1: Reconstruction of bounded cocain projection δπ 1) v 1). F

113 G DERIVATION OF TRANSPORT PROBLEMS We can derive te integral transport equations by analysing ow some k-form α k) paired to some k-dimensional sub-manifold V M evolves as it is advected wit some velocity vector field v. Let {φ t } be te flow associated wit v, suc tat V t) = φ t V. Teorem 4.1. From Teorem C.1 it follows tat, V t) α k) = φ t αk). V G.1) We can derive te transport teorem by differentiating tis relation wit respect to t yields, d d t V t) α k) 1 t) = lim α k) t + t) t 0 t V t+ t) 1 = lim t 0 t G.1 = lim = 4.3 = t t 0 1 φ t V = φ t V V t) V t) ) α k) t) α k) t + t) α k) t) V t+ t) V t+ t) + α k) t) V t+ t) φ t+ t [αk) t + t) α k) t)] + V α k) t + t) α k) t) lim t 0 t α k) + φ t t L v α k) V α k) t + L v α k) ). + φ t V V t) V ) α k) t) ) [φ t+ t φ t ]αk) t) φ t lim 1 α k) t) t 0 t 105

114 106 G. DERIVATION OF TRANSPORT PROBLEMS Te time-rate of cange of a k-form α k) Λ k) paired to a k-dimensional sub-manifold V M at some time-instant t R is tus given by, d α k) α k) = + L v α ). k) G.2) d t V t) V t) t Example G.1 Conservation of mass, incompressible flow, and te advection-diffusion equation). Consider a volume V t) M R n filled wit some quantity wit mass density ρ n). Te quantity s mass is given by, M = ρ n). V t) Conservation of mass implies d d t M = 0. Applying Teorem 4.2 yields, 0 = d d t M = d ρ ρ n) n) = + L v ρ ), n) G.3) d t V t) V t) t Using Cartan s formula and te fact tat dρ n) = 0, we find tat L v ρ n) = di v ρ n). We can re-express te volume-form, ρ n) = ρ vol n). Now, di v ρ n) = di v ρ vol n) = dρi v vol n) 4.14 = dρv n 1) = d q n 1), G were q n 1) := ρv n 1) is te mass flux. Hence, V t) ρ n) t = d q n 1) = q n 1), V t) V t) G.4) i.e. te time-rate of cange of mass in te volume is balanced by te flux of te material troug te volume s boundary. Additionally, we may want to take into account diffusive fluxes. Diffusive fluxes satisfy some constitutive law of te form, q n 1) diffusive = ɛd ρ 3), were te material parameter ɛ > 0 is te rate of diffusion. Examples of tese constitutive laws are discussed in section 2.5. Te total flux is ten equal to te sum of te advective flux and te diffusive flux, q n 1) = i v ρ 3) ɛd ρ 3). Substituting tis relation into Equation G.4 yields, V t) V t) ρ n) = di v ɛd )ρ 3) t V t) ρ n) ) + L v ρ n) ɛ ρ n) = 0, t were we assumed ɛ to be constant. Te result is known as te scalar advection-diffusion equation. Note tat in te case of convection, wen ρ 3) is te mass density of te flow itself, tere cannot be an additional diffusive term. In tat case all information governing

115 107 te flow of te material is contained in its velocity v. In te case of a convection problem, conservation of mass is governed by, V t) ρ n) ) + L v ρ n) = 0. G.5) t Under te assumption of incompressible flow, variations of mass density ρ n) are negligible witin te volume V t), i.e. derivatives of ρ are assumed to be zero. For tis case we find tat, 0 = = = V t) ) ρn) t + L v ρ n) dρv n 1) V t) V t) ρd v n 1). Hence an incompressible smoot flow in V t) satisfies d v n 1) = 0. We will refer to tis condition as te incompressibility constraint. Note tat by te Poincaré lemma, Teorem 2.1, tere exists some potential function ψ n 2), suc tat dψ n 2) = v n 1). Tis potential function is known as te stream function. Example G.2 Stream function ψ n 2) ). We can retrieve te pat of a fluid parcel by determining iso-surfaces of te stream function. Consider some fluid parcel wit mass density ρ n) tat is advected along some flow wit velocity field v on M. Along its pat we find tat, G L v φ t ρ n) 4.3 = d d t [φ t φ t ρ n) ] = d [ ρ n) ] d t t=0 = 0. Hence it follows tat di v ρ n) = d q n 1) = 0. By te Poincaré lemma, Teorem 2.1, tere exists a potential function ψ n 2), suc tat dψ n 2) = q n 1). Note tat te stream function satisfies te equation i v dψ n 2) = i v q n 1) = i v i v ρ n) = 0. Now consider te case of incompressible flow, for wic te tere exists a stream function ψ n 2), suc tat dψ n 2) = v n 1). Hence, L v ρ n) = dρ 0) v n 1) ) = dρ 0) v n 1) + ρ 0) d v n 1) = dρ 0) dψ n 2) Let γ be a n 1)-dimensional iso-contour of te stream function ψ n 2), i.e. a submanifold on wic ψ n 2) equals some constant. Satisfying te conservation of mass, Equation G.5, along tis iso-contour yields, 0 = ρn) t + L v ρ n) = ρn) t + dρ 0) dψ n 2), x γ.

116 108 G. DERIVATION OF TRANSPORT PROBLEMS Hence te fluid parcel remains constant along γ, and γ is its pat. If te flow field is time dependent, ten iso-contours are streaklines. Now let A be te n 1)-dimensional surface tat is bounded by various iso-contours of ψ n 2), e.g. as sown in Figure G.1. Ten te integral flux troug A is constant, q n 1) = dψ n 2) = ψ n 2) = C. A A A C 2 C 3 A C 1 C 4 Figure G.1: Surface A red) bounded by iso-surfaces blue) of te stream function ψ 1) = {C 1,C 2,C 3,C 4 }. Te integral flux is constant for any area bounded by te iso-surfaces, A q2) = A ψ1) = C 1 C 2 C 3 +C 4. G From te principle of conservation of mass of some quantity wit mass density ρ n) tat is transported along some incompressible flow wit velocity field v we can formulate 4 equivalent formulations, ρ n) t ρ n) t ρ 0) t ρ 0) t + L v ρ n) = 0, G.6) L v ρn) = 0, G.7) + L v ρ 0) = 0, G.8) L v ρ0) = 0, G.9) were ρ n) = ρ vol n) = ρ 0) vol n) or ρ 0) = ρ n). Equation G.6 and Equation G.9 are known as conservative formulations of te conservation of mass, wereas Equation G.7 and Equation G.8, are known as non-conservative formulations. Proof. Equation G.6 follows from te integral conservation of mass, Equation G.5, under te assumption of smootness witin V t). We can derive Equation G.9 by taking te Hodge- of Equation G.6, L v ρ n) = L v ρ 0) 4.19 = L v ρ0).