Diese Arbeit wurde vorgelegt am Lehrstuhl für computergestützte Analyse technischer Systeme

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1 Diese Arbeit wurde vorgelegt am Lehrstuhl für computergestützte Analyse technischer Systeme Dreidimensionale Isogeometrische Elastische Festkörper für Fluid-Struktur Interaktionen Three-Dimensional Isogeometric Elastic Solids for Fluid-Structure Interactions Seminararbeit Seminar Thesis von / presented by Daniel, Hilger Vollständiger Name des/r Prüfer/in Evaluated by Dipl.-Ing Norber Hosters Aachen, January 30, 2017

2 Typ der Arbeit Contents Acronyms List of Figures List of Tables I II III 1. Introduction 1 2. Governing Equations Equations for Fluid Analysis Equations for Structural Analysis Isogeometric Concept Non Uniform Rational B-Splines Methods for Coupling Finite Interpolation Method Validation Structural Stand-Alone Analysis Beam in Channel Flow Conclusion 15 A. Appendix 16 A.1. Weak Form of BVP for Solid References 17

3 Typ der Arbeit I Acronyms Abbreviation ACM CATS CFD FEA FEAFA FSI GLS IGA NURBS Description Aeroelastic Coupling Module Chair for Computational Analysis of Technical Systems Computational Fluid Dynamics Finite Element Analysis Finite Element Analysis for Aeroelasticity Fluid-Structur Interaction Galerkin Least Square Isogeometric Analysis Non Uniform Rational B-Splines

4 Typ der Arbeit II List of Figures 1. Motion of Body in Cartesian Coordinates Isogeometric Concept Transformation of Loads on Boundaries in FSI Scetch of Beam Convergence of Displacement in y-direction Comparison Between Non-Linear and Linear Computations Position of Beam in Channel Position of Beam in Channel

5 Typ der Arbeit III List of Tables 2. Linear Computations First and Second Order Nonlinear Computations First and Second Order

6 Typ der Arbeit 1 1. Introduction In the modern design and construction process it is necessary to develop cheap and light, but still reliable products and machines. In the application of light materials it has to be considered that these material are more easily influenced by their environment. Taking for example the wing of an aircraft, which is deformed due to the flow around it. In order to prevent these materials from failure there can be done experimental but also numerical analysis beforehand to test their reliability. The goal in numerical analysis of materials is to predict an exact and realistic behavior of materials. Therefore the task of this thesis is the implementation of a three dimensional elastic structural element based on the isogeometric concept. The isogeometric concept uses the idea to solve the structural problem on the exact geometry representation of the domain rather than on an approximation. This thesis should provide a basis for further research on three dimensional isogeometric elements and their behavior especially in terms of isogeometric analysis in coupled fluid-structure interactions (FSI) problems. The implemented element is examined on its convergence behavior in stand alone computations and it is tested in a coupled FSI problem. All implementations done in context of this work are extensions to programs that already exist and are provided by the chair of Computational Analysis of Technical Systems (CATS) at the RWTH Aachen. The structural element is therefore added to the existing structural solver Finite Element Analysis for Aeroelasticity (FEAFA) and the coupling module Aeroelastic Coupling Module (ACM) is adapted so that three dimensional isogeometric objects can be used in coupled computations. The flow solver XNS used to compute coupled FSI problems stays unaltered.

7 Typ der Arbeit 2 2. Governing Equations As already mentioned in the introduction computational analysis of FSI problems covers the analysis of flow fields, structure dynamics and additionally the coupling of their solutions. The fluid in FSI problems discussed in this thesis are incompressible Newtonian fluids and the structure is represented by solids with an elastic material behavior Equations for Fluid Analysis The flows considered in this thesis are assumed to be time dependent, incompressible, Newtonian fluids. Under these assumptions the balance of mass and momentum can be written in the form commonly known as the Navier-Stokes equation for incompressible fluids depending on the velocity field v and the pressure field p. ( ) v ρ + (v ) v = p + ν v + ρf in Ω ]0, T [, (1a) t v = 0 in Ω ]0, T [, (1b) v (x, t) = v D (x, t) Γ D ]0, T [, (1c) n σ (x, t) = t (x, t) Γ N, t ]0, T [, (1d) v (x, 0) = v 0 (x) in Ω, (1e) where ρ is the fluid density, ν the viscosity, f a gravitational force field and σ denotes the Cauchy stresses. The boundary conditions are split into Dirichlet and Neumann boundary conditions given by equations (1c) and (1c). Equations (1) are solved by flow solver XNS provided by CATS. As a solution approach a space-time discretization with a Galerkin Least Square (GLS) stabilization, which is desribed in detail by Behr [2] and Donea [5] Equations for Structural Analysis In context of this thesis the structural analysis is considered for three-dimensional incompressible elastic solids that experience small strains but large deformations. The deformation and movement of a body is described by equations (2) which are derived from the momentum and mass balance under the assumption of incompressibility [6]. ρ 2 u t σ = b 2 x in Ω (t) ]0, T [, (2a) u (x, t) = u D (x, t) on Ω D (t) ]0, T [, (2b) σ n = t on Ω N (t) ]0, T [, (2c) u (x, 0) = u 0 (x) in Ω (0). (2d)

8 Typ der Arbeit 3 The physical quantities contained in these equation are the displacement u, the corresponding acceleration 2 u, the Cauchy stresses in the material σ and the body forces t 2 in b. The weak formulation to this boundary value problem (BVP) is obtained by multiplying with a test function δu V := {δu H 1 (Ω (t)), δu = 0 on Ω D (t)} which can be interpreted as the virtual displacement resulting in the principle of virtual work. Now find a solution u S := {u H 1 (Ω (t)), u = u D on Ω D (t)} so that it holds: ( ) ρ 2 u δu dω t 2 t (div (σ)) δu dω t = b δu dω t. (3) Ω t Ω t Ω t Applying divergence theorem and the chain rule for differentiation on equation (3) it can be reformulated yielding equation (4), compare (A.1). ( ) ρ 2 u δu dω Ω t t 2 t + σ : ε (δu) dω t Ω t = b δu dω t (σ n) δu d Ω t. (4) Ω t Ω t In equation (4) ε denotes the Almansi strains, which form a stress strain pair together with the Cauchy stresses. Considering large deformations of the body throughout the loading process, the strains cannot assumed to depend linearly on the displacement. Due to this nonlinearity the deformed state of the body is unknown and has to be computed as part of an incremental solution process. Therefore we have to distinguish between two states of the body, first the reference configuration, where the body is in its known undeformed state and second the unknown deformed state, called the current configuration, cmp fig (1), [8]. The mapping between two configurations is described by the function Φ (X, t). The mapping is analogously described by the deformation gradient, which is determined as part of the solution process. Thereby the deformation gradient is a measure for the deformation of the current configuration with respect to the reference solution and is defined in the following way [8]: F = x X = x i X j = (X i + u i ) X j (5)

9 Typ der Arbeit 4 Y, y Ω X Φ (X, t) ω x Current Configuration Z, z Reference Configuration Figure 1: Motion of Body in Cartesian Coordinates X, x In this thesis the total Lagrangian solution approach is used where we solve the boundary value problem (2) on the known reference configuration of the body. Therefore physical quantities that are present in the current configuration are transfered to the reference configuration. Relevant quantities that need to be transferred are the stresses and strains. The deformation gradient from equation (5) is used for mapping these quantities. This results in a new strain and stress measure. The Cauchy stress on the reference configuration is called the second Piola-Kirchhoff stress, equation (6) and the corresponding strain is the Green-Lagrange strain, equation (7) [6]. S := det (F) F 1 σf T 1, (6) E := ( F T F I ). (7) As a stress strain pair they are conjugate to the Cauchy stress and Almansi strain tensor used in equation (3), so that equation (3) can be expressed on the domain of the reference configuration [6]: ( ) ρ 2 u δu dω Ω 0 t S : E (δu) dω 0 Ω 0 = b δu dω0 (S n) δu d Ω 0. (8) Ω 0 Ω 0 In order to close all unknown terms in equation (8) the Green-Lagrange strains have to be defined and a model for the second Piola-Kirchhoff stresses is introduced. The strains are defined in dependency of the deformation gradient resulting in a nonlinear dependency on the displacement: E := 1 2 ( F T F I ) = 1 2 ( ( (X + u) X ) T ( ) ) (X + u) I. (9) X

10 Typ der Arbeit 5 In this thesis the Saint Venant-Kirchhoff model is chosen to model the second Piola- Kirchhoff stress, which simply assumes a linear relation between the stresses and the strains based on Hook s law of elasticity: S = λ trace (E) + 2 µ E = 4 C E vect. (10) In equation (10) λ and µ are the Lamé constants, 4 C is constant material tensor of fourth order and E vect is the Green-Lagrange strain tensor written in vector notation exploiting the symmetry properties of Green-Lagrange strains [1]. With the choice of an appropriate basis for the solution space S and a finite discretization equation (8) and further lineraization, can be written in a matrix form [1]. As a basis for the discretization we use the isogeometric concept which is introduced in section (2.2.1). M U (i) + t 0K U (i) = t+ t R t 0 F (i 1). (11) Here M denotes the mass matrix, U (i) the acceleration at incremental step i, K the stiffness matrix at time t, U (i) the increment of the deformation, t+ t R the external loads of timestep t + t and t 0F (i 1) the incremental internal forces based on the displacement at timestep t [1] Isogeometric Concept In order to solve the equation (8) on the solution space S it is necessary to construct a basis to span the solution space. In classical finite element analysis (FEA) the solutions space is approximated by a set of basis functions which are then used to approximate the geometry. In the isogeometric concept however a basis is chosen that exactly represents the geometry of a body which is then used to approximate the solution space, cmp. figure (2). Classical FE: Geometry Solution Field imposed on Isogeometric Analysis: Geometry Solution Field Figure 2: Isogeometric and Isoparametric Concept [4] In this thesis the basis to fulfill the isogeometric concept is constructed from Non Uniform Rational B-Splines (NURBS), which fulfill the sufficient condition for basis convergence [4] namely

11 Typ der Arbeit 6 C 0 continuity in the element boundaries, C 1 continuity in the element interior, completness Non Uniform Rational B-Splines As the name implies NURBS are constructed from Non Uniform B-Splines. B-Splines describe curves, surfaces or volumes whose shape is defined by a parametric representation. For curves the parametric space is one dimensional and contains a set of n + p + 1 coordinates which are stored in the so called knot vector Ξ = {ξ 1, ξ 2,...ξ n+p+1 }. Thereby the parametric space is divided into n + p knots, that can be compared to elements in standard finite elements. Here n is the number of basis functions used to construct the spline corresponding to the number of non-zero elements and p is the polynomial degree of the basis functions. The spline is now constructed from recursively defined basis functions N i,p (ξ) starting from piecewise constant functions (p = 0) { 1 if ξ i ξ ξ i+1, N i,0 (ξ) =, (12) 0 else. and for higher degrees of the polynomial basis functions N i,p (ξ) = ξ ξ i ξ i+p ξ i N i,p 1 (ξ) + ξ i+p+1 ξ ξ i+p+1 ξ i+1 N i+1,p 1 (ξ). (13) The basis functions possess the following properties homogenity of functions, pointwise nonnegative over entire domain, partition of unity, s.t. ξ holds n i=1 N i,p (ξ) = 1. Further it can be remarked, that each basis function has a support on p + 1 knots and therefore p 1 continues derivatives across knots boundaries. In classical FE the continuity across element boundaries is only C 0 continues. The desire to construct arbitrary geometries including circles, spheres and conics sections based on B-Spline basis function requires further modification of the basis, so that piecewise rational basis functions are obtained. Therefore, the B-Spline basis is weighted by a weighting function W (ξ) yielding: R p i (ξ) = N i,p w i W (ξ) (14)

12 Typ der Arbeit 7 with W (ξ) = n N i,p (ξ) w i. (15) i=1 Curves, surfaces and volumes of arbitrary shape can now be constructed as a linear combination of rational basis functions equation (14) and a set of coordinates of the physical space, called control points B i. For a one dimensional parametric space this yields a curve embedded in a three dimensional physical space. C (ξ) = n R p i (ξ) B i, (16) i=1 or for volumes V (ξ) = n i=1 m o j=1 k=1 R p,q,r i,j,k (ξ) B i,j,k. (17) The NURBS basis from equation (14) fulfills the isogeometric concept, namely can exactly represent any arbitrary geometry and additionally can be used to span the solution space, so that equation (11) can be derived from equation (8) Methods for Coupling In order to be able to describe a fluid-structure interaction problem the previously presented boundary value problems (1) and (3) have to be coupled. The interaction between fluid and the structure happens along a common boundary, which is denoted by Γ F S. Along this boundary kinematic (19) and dynamic (20) boundary conditions for both sides have to be related [3]. t, x Γ F S : u F (t, x) = u S (t, x), (18) u F (t, x) t, x inγ F S : = u S (t, x), (19) t t t, x Γ F S : n σ F (t, x) = n σ S (t, x). (20) In equations (19) u denotes the displacement for the fluid (subscript F ) and the structure (subscript S). Consequential the boundary condition for velocity and acceleration can be obtained. In equation (20) σ represents surface stresses on the boundaries of each domain. In general the temporal and spatial discretization of the structure and the fluid are not

13 Typ der Arbeit 8 equal, so that it is required to introduce a mapping between the different discretizations. In case of the temporal coupling it is only to mention here that a strong coupling approach is used, that iteratively solves the structure and flow problems with updated boundaries until the boundary conditions in equation (19) - (20) are satisfied. In case of the spatial discretization a mapping between the boundary points of the different meshes has to be introduced. Therefore the loads and displacements of each mesh point on the boundary of the fluid have to be distributed to the points of the structure and vice versa. One method to do this is the finite interpolation method [3]. Ω S Γ F S ξ j Ω F F j,f Load ξ Mesh Point CP of NURBS Figure 3: Transformation of Loads on Boundaries in FSI Finite Interpolation Method The finite interpolation method generates a weighted mapping between the control points from the NURBS, describing the boundary of the structure and the boundary points of the fluid mesh. Thereby the weighting functions of the NURBS object R i ( ξj ) are exploited to distribute the influence of each fluid node to p+1 control points on the structure. Therefor it is necessary to match the position of each mesh point of the fluid to its parametric coordinate ξ on the boundary spline or in case that the mesh point is not located exactly on the spline to the position closest to it. Equation (21) shows the

14 Typ der Arbeit 9 finite interpolation method applied to the loads on the boundaries, which analogously can be done for the displacement, velocity or acceleration [3]. F i,s = N ( ) F j,f R i ξj j=1 (21)

15 Typ der Arbeit Validation The validation of the implemented three dimensional nonlinear isogeometric element is split into two steps. First it is validated with respect to the convergence of the solution of stand alone computations for structural analysis and in a second validation step this element is then used in a coupled FSI computation. Proleptic it is to mention here that the second validation will not be observed with respect to physical correctness of the computed solution, but rather be a test that shows that coupled computations are possible with this newly introduced element Structural Stand-Alone Analysis The physical setting that is used to validate the convergence of the implemented element is shown in figure (4). It is based on a paper published by Turek et al. [7]. g z x y 0.02 [m] 0.02 [m] 0.35 [m] Figure 4: Scetch of Beam A three dimensional cantilever beam is defleceted under a gravitational bodyforce of g T = (0, 2, 0) [ ] m s acting on it. Its size corresponds to the values given in the scetch 2 (4). The beam has a density of ρ = 1000 [ ] kg m, a Poisson ration of ν = 0.4 and a 3 Young s modulus of E = [ ] 6 kg ms. In all computations the deflection for the tip 2 of the beam is examined. In the reference configuration the observed point is located at x = (0.0, 0.0, 0.35) T [m] In this thesis the results of the computations are presented for different number of elements used to discretize the beam with first and second order basis functions. Further the results for a nonlinear formulation that considers large deformations are compared to the results for a linear formulation. The results for the different computations are presented in tables (2) -(3). The beam is evaluated for a discretization with 140, 560, 1260, 2240, 5040 and 8960 elements. In figure (5) the residual for the displacement in y-direction with respect to a referential

16 Typ der Arbeit 11 Number of Elements Degree Basis Function Resolution Displacement x-direction Displacement y-direction Displacement z-direction 140 O2 35x2x O2 70x4x O2 105x6x O2 140x8x O2 210x12x O2 280x16x Number of Elements Table 2: Linear Computations First and Second Order Degree Basis Function Resolution Displacement x-direction Displacement y-direction Displacement z-direction 140 O1 35x2x O1 70x4x O1 105x6x O1 140x8x O1 210x12x O1 280x16x O2 35x2x O2 70x4x O2 105x6x O2 140x8x O2 210x12x O2 280x16x Table 3: Nonlinear Computations First and Second Order solution is displayed for first and second order basis functions with different discretizations levels of the domain. The green lines indicate the solution with first degree basis functions whereas the red lines indicate the second order basis functions. It can be observed that the solutions tend to converge against one solution. Further it can be observed that an increase in the order of the basis function leads to better convergence of the solution at the same level of discretization, especially for a rough discretization of the structure.

17 Typ der Arbeit Nonlinear 1st Order Basis Nonlinear 2nd Order Basis Residual of y-displacement Number of Elements Figure 5: Convergence of Displacement in y-direction Additionally to the convergence observations the solutions of linear and non-linear computations are displayed in figure (6). The graph in combination with tables (2)-(3) highlights that large deformations cannot be neglected in the modelation of the stressstrain relations. Therefore the linear approach converges towards a different solution than the nonlinear approach. It can be observed from table (2) that the linear approach results in no deformation of the beam in x-direction which does not match experimental observation and contradicts with conservation of mass and the assumption of incompressibility.

18 Typ der Arbeit Linear 2nd Order Basis Nonlinear 2nd Order Basis Residual of y-displacement Number of Elements Figure 6: Comparison Between Non-Linear and Linear Computations 3.2. Beam in Channel Flow The coupled problem that is examined in this thesis is a steady state problem of a beam that is located in the center of a channel flow as shown in figure (7). As boundary conditions the beam surface and channel floor consider no slip boundaries, whereas the channel walls consider slip boundaries. As an inflow condition a parabolic inflow is chosen with a Reynolds number of Re = 10, density of ρ = 1 [ ] kg m and a viscosity of 3 η = 1 [ kg s m] m 1.2 m 0.03 m 0.03 m 2.6 m 0.3 m 1.3 m z x 1.3 m 2.6 m y x 1.3 m 2.6 m Figure 7: Position of Beam in Channel The result of that steady state problem is displayed in figure (8). Both solutions to the structural problem and the flow problem converged to a steady state solution meaning they reached a state of equilibrium for the displacement of the beam and the flow around the displaced body. The beam is slightly bent into direction of the flow. The

19 Typ der Arbeit 14 streamlines indicate the influenced flow field of the fluid flowing around the beam. Obtaining a converged steady state for the flow around the beam shows that FSI computations with the implemented isogeometric element are possible. Figure 8: Position of Beam in Channel

20 Typ der Arbeit Conclusion In context of this thesis a three dimensional elastic isogeometric element was successfully implemented in the existing structural solver FEAFA. Beside the implementation a short convergence study was done to examine the quality of the implementation. The study showed satisfying results and underlined the necessity of considering large deformations in the stress models that were used. Further the results highlighted that especially for rough discretizations of the structure higher order basis functions lead to a strong improvement of the computed solution. Though it was not part of this thesis future work can examine wether the isogeometric approach is advantageous to the standard finite element method in terms of higher order basis functions on rough discretizations. Apart from the structural stand alone computation a coupled FSI problem was successfully computed for a steady state problem. The necessary extensions for load transformations between three dimensional isogeometric objects and grid points of the fluid discretization were included in the coupling module ACM. The corectness of this mapping was tested on basis of the steady state computation, which showed satisfying results. The results obtained from the stand alone computation and the steady state problem form a good basis for prospective computations of FSI problems based on three dimensional IGA structures. Although that this thesis showed some promising results further testing in extended convergence test or test with special geometries,e.g. cones or spheres have to be examined for this element typ. Also the use in FSI problem was only tested for a simple steady test case and has not yet tackled the problems that might occur for unsteady computations or flows at high Reynolds numbers.

21 Typ der Arbeit 16 A. Appendix A.1. Weak Form of BVP for Solid Ω t ( ) ρ 2 u t 2 δu dω t (div (σ)) δu dω t = b δu dω t. (22) Ω } t Ω {{} t ( ) Using the chain rule for differentiation on the term in (*): div (σ) δu = div ( σ T δu ) σ grad (δu), (23) and apply the divergence theorem ( div (σ) δu dω t = σ T δu ) n d Ω t = (σ n) u d Ω t, (24) Ω t Ω t Ω t results in ( ) ρ 2 u t 2 Ω t δu dω t σ : grad (δu) dω t + Ω t (σ n) δu d Ω t = Ω t b δu dω t. Ω t (25) With the condition of symmetric Cauchy stresses the following relation holds and can be replaced in the above equation: σ : grad (δu) = ( σ i,j ( δu) i,j = 1 σ i,j ( δu) 2 i,j + ) σ j,i ( δu) j,i i,j i,j i,j = σ : sym (grad (δu)) = σ : ε (δu) (26) Inserting this relation yields: ( ) ρ 2 u δu dω Ω t t 2 t + σ : ε (δu) dω t Ω t = b δu dω t (σ n) δu d Ω t. (27) Ω t Ω t

22 Typ der Arbeit 17 References [1] K. J. Bathe. Finite-Elemente-Methoden: Matrizen und lineare Algebra, die Method der finite Element, Lösungen von Gleichgewichtsbedingungen und Bewegungsgleichungen. Springer-Verlag Berlin Heidelber New York Tokyo, [2] M. Behr. Stabilized Finite Element Method for Incompressible Flows with Emphasis on Moving Boundaries and Interfaces. PhD thesis, University of Minnesota, [3] M. B. Blomberg. Last- und verformungsprojektion zwischen diskreten und splinebasierten oberflächendarstellungen im bereich fluid-struktur-interaktionen. Master s thesis, RWTH Aachen, Lehrstuhl für computergestütze Analyse technischer Systeme, [4] J. A. Cottrell, T. R. J. Hughes, and Y. Bazilevs. Isogeometric Analysis towards Integration of CAD and FEA. John Wiley & Sons, [5] J. Donea and A. Huerta. Finite Element Method for Flow Problems. Wiley, [6] B. Markert. Porous Media Viscoelasticity with Application to Polymeric Foams. PhD thesis, Institut für Mechanik Universität Stuttgart, [7] S. Turek, J. Hron, and et al. Numerical benchmarking of fluid-structure interaction: A comparison of different discretization and solution approaches. Lecture Notes in Computational Science and Engineering, [8] O.C. Zienkiewicz and R.L. Taylor. The Finite Element Method for Solid and Structural Mechanics. Elsevier, 2005.