Model Reduction via Discretization : Elasticity and Boltzmann Joachim Schöberl Computational Mathematics in Engineering Institute for Analysis and Scientific Computing Vienna University of Technology IMA Workshop on Structure Preserving Discretizations, Minneapolis, 2014 Joachim Schöberl Page 1
Motivation Solid Mechanics: (with Astrid Pechstein, born Sinwel) Discretize 3D elasticity by a (non-standard) primal method Should behave like a plate/shell method when applied to thin structures Fluid Dynamics: (with Gerhard Kitzler) Discretize Boltzmann equation (in time, space, velocity) Should resemble Euler or Navier Stokes equation, when this is appropriate Joachim Schöberl Page 2
A beam in a beam Reenforcement with E = 50 in medium with E = 1. New mixed FEM, p = 2 Primal FEM, p = 3 Joachim Schöberl Motivation Page 3
Degrees of freedom for TD-NNS elements Mixed elements for approximating displacements and stresses. tangential components of displacement vector normal-normal component of stress tensor Triangular Finite Element: Tetrahedral Finite Element: σ nn u τ σ nn u τ Joachim Schöberl Motivation Page 4
The quadrilateral element Dofs for general quadrilateral element: σ nn Thin beam dofs (σ nn = 0 on bottom and top): uτ Beam stretching components: Beam bending components: u τ mean value σ nn mean value rotation bending moment vertical deflection uτ Joachim Schöberl Motivation Page 5
Hellinger Reissner mixed methods for elasticity Primal mixed method: Find σ L sym 2 and u [H 1 ] 2 such that Aσ : τ τ : ε(u) = 0 τ σ : ε(v) = f v v Dual mixed method: Find σ H(div) sym and u [L 2 ] 2 such that Aσ : τ + div τ u = 0 τ div σ v = f v v [Arnold+Falk+Winther] Joachim Schöberl Helling Reissner mixed methods Page 6
Reduced Symmetry mixed methods Decompose ε(u) = u + 1 2 ( 0 1 1 0 ) curl u = u + ω Impose symmetry of the strain tensor by an additional Lagrange parameter: Find σ [H(div)] 2, u [L 2 ] 2, and ω L skew 2 such that Aσ : τ + u div τ + τ : ω = 0 τ v div σ = fv v σ : γ = 0 γ The solution satisfies u L 2 and ω = curl u L 2, i.e., u H(curl) Arnold+Brezzi+Douglas, Stenberg,... 80s Joachim Schöberl Helling Reissner mixed methods Page 7
Choices of spaces div σ u understood as div σ, u H 1 H 1 = (ε(u), σ) L2 (div σ, u) L2 Displacement u [H 1 ] 2 u [L 2 ] 2 continuous f.e. non-continuous f.e. Stress σ L sym 2 σ H(div) sym non-continuous f.e. normal continuous (σ n ) f.e. The mixed system is well posed for all of these pairs. Joachim Schöberl Helling Reissner mixed methods Page 8
Choices of spaces div σ u understood as div σ, u H 1 H 1 = (ε(u), σ) L2 div σ, u H(curl) H(curl) (div σ, u) L2 Displacement u [H 1 ] 2 u H(curl) u [L 2 ] 2 continuous f.e. tangential-continuous f.e. non-continuous f.e. Stress σ L sym 2 σ L sym 2, div div σ H 1 σ H(div) sym non-continuous f.e. normal-normal continuous (σ nn ) f.e. normal continuous (σ n ) f.e. The mixed system is well posed for all of these pairs. Joachim Schöberl Helling Reissner mixed methods Page 8
The space H(div div) The dual space of H(curl) is H 1 (div): f H(curl) = sup v H(curl) H 1 (div) f, v v H(curl) f, ϕ + z sup ϕ H 1,z [H 1 ] 2 ϕ H 1 + z H 1 div f H 1 + f H 1 We search for σ L sym 2 and div σ H 1 (div). This is equivalent to σ H(div div) := {σ L sym 2 : div div σ H 1 }, where div div ( ) σxx σ xy σ yx σ yy = div ( σxx x σ yx x + σ xy y + σ yy y ) = ij 2 σ ij x i x j H 1 This implies continuity and LBB for div σ, u. Joachim Schöberl Helling Reissner mixed methods Page 9
Continuity properties of the space H(div div) Lemma: Let σ be a piece-wise smooth tensor field on the mesh T = {T } such that σ nt H 1/2 ( T ). Assume that σ nn = n T σn is continuous across element interfaces. Then there holds div σ H(curl). Proof: Let v be a smooth test function. div σ, v := σ : v = { T = { div σ v T T T T T div σ v σ nτ v τ } + E T } σ n v E [σ nn ] v }{{} n =0 div σ L2 (T ) v L2 (T ) + σ nτ H 1/2 ( T ) v τ H 1/2 ( T ) C(σ) v H(curl) By density, the continuous functional can be extended to the whole H(curl): div σ, v = { } div σ v σ nτ v τ T T T Joachim Schöberl Helling Reissner mixed methods Page 10
The TD-NNS-continuous mixed method Assuming piece-wise smooth solutions, the elasticity problem is equivalent to the following mixed problem: Find σ H(div div) and u H(curl) such that { Aσ : τ + T div τ u } T τ nτu τ T { T div σ v T σ nτv τ } T = 0 τ = f v v Proof: The second line is equilibrium, plus tangential continuity of the normal stress vector: (div σ + f)v + [σ nτ ]v τ = 0 v T T E E Since the space requires continuity of σ nn, the normal stress vector is continuous. Element-wise integration by parts in the first line gives (Aσ ε(u)) : τ + τ nn [u n ] = 0 τ T T E E This is the constitutive relation, plus normal-continuity of the displacement. Tangential continuity of the displacement is implied by the space H(curl). Joachim Schöberl Helling Reissner mixed methods Page 11
The de Rham complex curl div H(curl) H(div) L 2 H 1 W h curl div Vh Q h S h satisfies the complete sequence property range( ) = ker(curl) range(curl) = ker(div) on the continuous and the discrete level. Important for stability, error estimates, preconditioning,... Joachim Schöberl Exact sequences Page 12
The 3-step exact sequence H 1 H 2 (T ) H(curl) [H 1 (T )] 2 σ T ( ) H(div div) div H 1 (div) div H 1 with the stress operator σ(v) = ( vy { y 1 vy 2 sym x + v y x v x x } ). The composite operators are airy(w) = σ( w) = ( 2 w y 2 sym w x y w x 2 ) div σ(v) = 1 2 Curl curl v There holds range(σ( )) = ker(div) range(div σ( ) = ker(div) Joachim Schöberl Exact sequences Page 13
Discrete sequence H 1 H 2 (T ) H(curl) [H 1 (T )] 2 σ T H(div div) W h Vh σ T Σ h Construction of conforming finite elements Start with C 0 -continuous finite elements for H 1 H 2 (T ) Build finite elements for H(curl) and H(div div) using the discrete sequence Joachim Schöberl Exact sequences Page 14
Finite Elements for H 1 H 1 H 2 (T ) H(curl) [H 1 (T )] 2 σ T H(div div) W h Vh σ T Σ h H 1 -basis functions on the triangle Vertex basis functions Edge basis functions Element basis functions φ V = λ V φ E i φ T ij = u i(x, y)v j (y) Joachim Schöberl Exact sequences Page 15
Finite Elements for H(curl) H 1 H 2 (T ) H(curl) [H 1 (T )] 2 σ T H(div div) W h Vh σ T Σ h H(curl)-basis functions on the triangle [JS + Zaglmayr, 05] Edge-element basis functions N αβ 0 = λ α λ β λ α λ β High-order edge basis functions ψ E i = φ E i+1 Element basis functions φ T ij = (u iv j ), u i v j u i v j, N αβ 0 v j Joachim Schöberl Exact sequences Page 16
Finite Elements for H(div div) H 1 H 2 (T ) H(curl) [H 1 (T )] 2 σ T H(div div) W h Vh σ T Σ h H(div div)-basis functions on the triangle Edge basis functions ϕ E i = σ( φ E i ) (divergence free) Element basis functions σ( (u i v j )) (divergence free) u i σ( v j ), v j σ( u i ) Joachim Schöberl Exact sequences Page 17
Finite Element Analysis Analysis in discrete norms: v 2 V h = T ε(v) 2 T + E h 1 [v n ] 2 L 2 (E) τ 2 Σ h = τ 2 L 2 + E h τ nn 2 L 2 (E). Continuous and inf-sup stable. By saddle-point theory: for m k. u u h Vh + σ σ h Σh ch m ε(u) H m By adding the stabilization term h 2 T (div σ, div τ) T, to the bilinear-form (and consistent rhs) the method is robust as ν 1/2. T Joachim Schöberl Exact sequences Page 18
Unit square, left side fixed, vertical load, adaptive refinement σ P 1 2 dof σ nn per edge conforming non-conforming u P 1 2 dof u τ 1 dof u τ u P 2 3 dof u τ 2 dof u τ ν = 0.3: ν = 0.4999: 1 0.1 order 1 order 1, nc order 2 order 2, nc 1 0.1 order 1 order 2 error stress 0.01 0.001 error stress 0.01 0.001 1e-04 1e-04 1e-05 10 100 1000 10000 100000 1e+06 1e+07 dofs 1e-05 10 100 1000 10000 100000 1e+06 1e+07 dofs Joachim Schöberl Exact sequences Page 19
Curved elements fixed left top, pull right top Elements of order 5 σ xx Joachim Schöberl Exact sequences Page 20
Reissner Mindlin Plates Energy functional for vertical displacement w and rotations β: ε(β) 2 A 1 + t 2 w β 2 MITC elements with Nédélec reduction operator: ε(β) 2 A 1 + t 2 w R h β 2 Mixed method with σ = A 1 ε(β) H(div div), β H(curl), and w H 1 : L(σ; β, w) = 1 2 σ 2 A + div σ, β t 2 w β 2 Joachim Schöberl Thin structures Page 21
Reissner Mindlin Plates and Thin 3D Elements Mixed method with σ = A 1 ε(β) H(div div), β H(curl), and w H 1 : L(σ; β, w) = σ 2 A + div σ, β t 2 w β 2 Reissner Mindlin element: 3D prism element: σ nn β τ w σ nn u τ Joachim Schöberl Thin structures Page 22
Tensor-product Finite Elements Thin domain: ω R 2, I = ( t/2, t/2), Ω = ω I. FE-space for displacement: L xy k+1 = {v H 1 (ω) : v T P k+1 } N xy k = {v H(curl, ω) : v T P k } L z k+1 = {v H 1 (I) : v T P k+1 } N z k = {v L 2 (I) : v T P k } Tensor-product Nédélec space: V k = N xy k L z }{{ k+1 L xy } k+1 N k z }{{} u xy u z Regularity-free quasi-interpolation operators (Clement) which commute: I xy k+1 : L 2(ω) L xy k+1, Qxy k : L 2(ω) N xy k : I xy k+1 = Qxy k Tensor product interpolation operator: Q k = Q xy k Iz k+1 }{{} u xy I xy k+1 Qz k }{{} u z Joachim Schöberl Thin structures Page 23
Anisotropic Estimates Thm: There holds ε(u u h ) 2 T + F T h 1 op [u n ] 2 F + σ σ h 2 c { h m xy m xyε(u) + h m z m z ε(u) } 2 Proof: Stability constants are robust in aspect ratio (for tensor product elements) Anisotropic interpolation estimates (H 1 : Apel). E.g., the shear strain components 2 ε xy,z (u Q k u) L2 = z (u xy I z Q xy u xy ) + xy (u z I xy Q z u z ) L2 = (I Q xy Q z ) ( ) z u xy + xy u z L2 h m xy m x ε xy,z (u) 0 + h m z m z ε xy,z (u) L2 Thesis Astrid Pechstein, [A. Pechstein + JS 11], [A. Pechstein + JS 12] Joachim Schöberl Thin structures Page 24
Hybridization: Implementation aspects The following two methods are (essentially) equivalent: hybridization of mixed method (Arnold+Brezzi) Introduce Lagrange parameter λ n to enforce continuity of σ nn. Its meaning is the displacement in normal direction. Continuous / hybrid discontinuous Galerkin method (Cockburn+Gopalakrishnan+Lazarov) Displacement u is strictly tangential continuous, HDG facet variable (= normal displacement) enforces weak continuity of normal component. Anisotropic error estimates from mixed methods can be applied for HDG method! Joachim Schöberl Thin structures Page 25
Shell structure R = 0.5, t = 0.005 σ P 2, u P 3 stress component σ yy Joachim Schöberl Thin structures Page 26
Real Life Application Geometry Displacement uy Deformed geometry, stress σxx Interior stress Joachim Scho berl Applications Page 27
Contact problem an application Undeformed bear Stress, component σ 33 Joachim Schöberl Applications Page 28
Coriolis mass flow meter (S. Stingelin, Endress-Hauser) Monolithic fluid structure interaction velocity x-component in fluid for different viscosities: ν = 0.1 ν = 10 ν = 100 (A + iωνb ω 2 M)u = f, +div-free constraint A.. elasticity op, B.. viscosity op, M.. mass op, and small viscosity ν Simulation with hybrid DG in fluid and elasticity. p = 2...4, N = 200k...500k, Time = 20 sec... 2 min Joachim Schöberl Applications Page 29
Reduced Basis Method (RBM) (A + iωνb ω 2 M)u = f Compute snapshots u j = u(ω j ) for j = 1... m, and project onto V m = span{u j } 1e-11 1e-12 1e-13 3 modes 4 modes 7 modes Eigenvalues from RBM: response 1e-14 1e-15 1e-16 1e-17 1e-18 modes EV 1 EV 2 3 693 - i 0.136 2992 - i 6.62 4 693 - i 0.170 3003 - i 1.85 7 693 - i 0.166 3002 - i 1.73 with Anna Zechner 1e-19 500 1000 1500 2000 2500 3000 3500 4000 frequency Joachim Schöberl Boltzmann Equation Page 30
Flow around NACA airfoil Joachim Schöberl Boltzmann Equation Page 31
Particle distribution function : f = f(t, x, v) 0 Boltzmann Equation with time t [0, T ], position x Ω R d, and velocity v R d. f(t, x, ṽ)dṽ d x B(x, x) B(v, v) is expected number of particles in B(x, x) having velocity in B(v, v). Macroscopic quantities: density ρ(t, x) = velocity V (t, x) = 1 ρ temperature T (t, x) = 1 ρ f(t, x, v) dv vf(t, x, v) dv v V 2 f(t, x, v) dv Joachim Schöberl Boltzmann Equation Page 32
Boltzmann Equation Evolution of f: f t (t, x, v) + div x(vf) = 1 Kn Q(f) Kn... Knudson number, mean free path between collisions t, x, v Quadratic collision operator: Q ( f(t, x, ) ) (v) = R d B(v, w, e ) [ f(v )f(w ) }{{} S d 1 gain ] f(v)f(w) de }{{} dw loss with pre-collision velocities v and w such that v + w = v + w conservation of momentum of 2 particles v 2 + w 2 = v 2 + w 2 conservation of kinetic energy of 2 particles v v + w 2 e [C. Cercignani. Mathematical methods in kinetic theory, 1990] Joachim Schöberl Boltzmann Equation Page 33
Conservation Properties There holds R 3 Q(f)ϕ(v) dv = 0 for ϕ {1, v 1, v 2, v 3, v 2 }. This corresponds to conservation of mass, momentum, and energy. There holds for f 0 2 v V Q(f) = 0 f(v) = ρe T The kernel of Q(.) consists of Maxwell distributions (=Gaußians), determined by total mass, velocity and temperature. Joachim Schöberl Boltzmann Equation Page 34
Variational formulation of spatially homogeneous BE Find: f C 1 ([0, T ], L 2,e v 2 /T ): forall t (0, T ] and ϕ L 2,e v 2 /T. R d f t ϕ dv = R d Q(f)ϕ dv Has a unique solution converging to the equilibrium Maxwell distribution. Numerical approximation by a Petrov-Galerkin method: f(t) V N := e v V 2 /T Π N ϕ W N := Π N Choose L 2,e v 2-orthogonal polynomials: Hermite polynomials Joachim Schöberl Boltzmann Equation Page 35
Efficient implementation of the collision operator Rewrite multiple integral of collision operator: Q(f)ϕ dv = = B(v, w, e )[f(v )f(w ) f(v)f(w)]de dwdv B(ṽ, w, e )[f (v) (ṽ )f (v) ( ṽ ) f (v) (ṽ)f (v) ( ṽ)] de dṽdv with mean velocity v = v+w 2 and relative velocities ṽ = v v and ṽ = v v, and f (v) := f( v). Transform between three sets of basis-functions nodal in Hermite points: cheap multiplication and component-wise shifts tensor product Hermite: cheap transformation to Laguerre polar coordinate Laguerre: diagonalization of S d 1..de Total costs of collision operator application for d = 2 is O(N 5 ) multiplications (within matrix-matrix multiplications). Joachim Schöberl Boltzmann Equation Page 36
Colliding double-peak: Polynomial order in v-space is 50, thus ndof = 2601 5000 time-steps RK2, simulation time: 20 minutes Joachim Schöberl Boltzmann Equation Page 37
Discretization of full Boltzmann operator f ( v ) t + div x,v f = Q(f) 0 Transport term is an advection term in R 2d with divergence-free wind (v, 0). Use Discontinuous Galerkin with upwind discretization in R 2d. element-trial space: V T = M v,t Π (x) Π (x) element-test space: V T = Π (x) Π (x) The Maxwellian M v,t on each element T is defined by means of the macroscopic quantities. Joachim Schöberl Boltzmann Equation Page 38
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References for Boltzmann Stochastic numerics for Boltzmann: Rjasanow + Wagner, 94 fast collision by FFT: Bobylev+Rjasanow 94, Pareschi+Perthame 96, Pareschi+Russo 2000 polynomial - spectral method for collision: Kitzler+JS 13 (preprint), Fonn+Grohs+Hiptmair 14 DG + Boltzmann: B.A.De Dios + J.A. Carillo + C.-W. Shu 12, Kitzler+JS 14 DG + Vlasov-Poisson : Cheng+Gamba+Majorana+Shu, 11++ Joachim Schöberl Boltzmann Equation Page 40
Ongoing work and visions By means of an adaptive choice of local v-order, the method should resemble Euler equations or Navier Stokes equations, when they are (locally) accurate enough. Time integration by operator splitting. The current explicit time-stepping method shall be replaced by an implicit/explicit time-stepping method, where the stiffness due to the collision operator should be treated implicitly. Spatial DG for the linear Boltzmann transport operator should give insight for the design of numerical fluxes for Euler and Navier-Stokes equations. Joachim Schöberl Boltzmann Equation Page 41
References Doug N. Arnold. Discretization by finite elements of a model parameter dependent problem, Numer. Math., 81 Doug N. Arnold. An interior penalty finite element method with discontinuous elements, SINUM, 82 Doug N. Arnold, F. Brezzi, and J. Douglas, Jr.PEERS: a new mixed finite element for plane elasticity. Japan J. Appl. Math., 84. Doug N. Arnold and F. Brezzi. Mixed and nonconforming finite element methods: implementation, postprocessing and error estimates., 85 Doug N. Arnold and R. S. Falk. The boundary layer for the Reissner-Mindlin plate model. SIAM J. Math. Anal.,90 S. M. Alessandrini, Doug N. Arnold, R. S. Falk, and A. L. Madureira. Dimensional reduction for plates based on mixed variational principles. In Shells..., 97 Doug N. Arnold and A. Mukherjee. Tetrahedral bisection and adaptive finite elements. In M. Bern, J. Flahery, and M. Luskin, editors, Grid generation and adaptive algorithms, volume 113 of IMA Vol. Math. Appl, 99 Doug N. Arnold, R. S. Falk, and R. Winther. Multigrid in H(div) and H(curl). Numer. Math., 85, 2000. Doug N. Arnold, F. Brezzi, B. Cockburn, and L. D. Marini. Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal., 2002 Doug N. Arnold, R. S. Falk, and R. Winther. Finite element exterior calculus, homological techniques, and applications. Acta Numer., 2006 Joachim Schöberl Boltzmann Equation Page 42
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