Time domain boundary elements for dynamic contact problems

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1 Time domain boundary elements for dynamic contact problems Heiko Gimperlein (joint with F. Meyer 3, C. Özdemir 4, D. Stark, E. P. Stephan 4 ) : Heriot Watt University, Edinburgh, UK 2: Universität Paderborn, Germany 3: Universität Stuttgart, Germany 4: Leibniz Universität Hannover, Germany BEM on the Saar 207, Saarbrücken May 29, 207 H. Gimperlein TDBEM for dynamic contact BEM on the Saar 207 / 9

2 Wave equation U : R t Ω c x R 2 t U U = 0 in R Ω c, Ω c = R d \ Ω, U = 0 for t 0. Real problem: Lamé equation for tire dynamics tire does not penetrate road Key engineering problem, analysis challenging H. Gimperlein TDBEM for dynamic contact BEM on the Saar / 9

3 Wave equation U : R t Ω c x R 2 t U U = 0 in R Ω c, Ω c = R d \ Ω, U = 0 for t 0. Real problem: Lamé equation for tire dynamics tire does not penetrate road Scalar model problem: wave equation with contact boundary conditions on G Γ := Ω { U G 0, U ν G h, U G > 0 = U G = h. ν H. Gimperlein TDBEM for dynamic contact BEM on the Saar / 9

4 Wave equation U : R t Ω c x R 2 t U U = 0 in R Ω c, Ω c = R d \ Ω, U = 0 for t 0. Real problem: Lamé equation for tire dynamics tire does not penetrate road Scalar model problem: wave equation with contact boundary conditions on G Γ := Ω Recent works (computational & stabilized FEM): Renard, Hauret, Le Tallec, Wohlmuth,... H. Gimperlein TDBEM for dynamic contact BEM on the Saar / 9

5 Contents U : R t Ω c x R t 2 U U = 0 in R t Ω c x, Ω c = R d \ Ω U = 0 for t 0. Analysis of dynamic contact: variational inequality on Γ := Ω, well-posedness Basics of time domain BEM Numerical analysis of dynamic contact: mixed formulation, first a priori estimates, Uzawa Numerical experiments H. Gimperlein TDBEM for dynamic contact BEM on the Saar / 9

6 Dirichlet-Neumann operator for wave equation U : R t Ω x R t 2 U U = 0 in R t Ω c x, Ω c = R d \ Ω U = 0 for t 0. Dirichlet-Neumann operator on Γ := Ω S(U Γ ) = U ν contact boundary conditions on G Γ (and U = 0 on Γ \ G) { U 0, U ν h, U > 0 = U ν = h. Γ H. Gimperlein TDBEM for dynamic contact BEM on the Saar / 9

7 Dirichlet-Neumann operator for wave equation U : R t Ω x R t 2 U U = 0 in R t Ω c x, Ω c = R d \ Ω U = 0 for t 0. Dirichlet-Neumann operator on Γ := Ω S(U Γ ) = U ν contact boundary conditions on G Γ (and U = 0 on Γ \ G) { { U 0, S(U Γ ) h, U 0, S(U Γ ) h, U > 0 = S(U Γ ) = h. U (S(U Γ ) h) = 0. Γ H. Gimperlein TDBEM for dynamic contact BEM on the Saar / 9

$Dirichlet-Neumann operator for wave equation U : R t Ω x R t 2 U U = 0 in R t Ω c x, Ω c = R d \ Ω U = 0 for t 0.$

8 Dirichlet-Neumann operator for wave equation U : R t Ω x R t 2 U U = 0 in R t Ω c x, Ω c = R d \ Ω U = 0 for t 0. contact boundary conditions on G Γ (and U = 0 on Γ \ G) { U 0, S(U Γ ) h, U > 0 = S(U Γ ) = h. Variational inequality: Find 0 u = U G such that Su, v u h, v u 0 v. { U 0, S(U Γ ) h, U (S(U Γ ) h) = 0. H. Gimperlein TDBEM for dynamic contact BEM on the Saar / 9

9 Analytic set-up: Space-time Sobolev spaces space time anisotropic Sobolev spaces H r σ(r +, H s (Γ)), σ > 0: H r σ(r +, H s (R d )) defined using Fourier Laplace transform { ψ : supp ψ R + R d, R+iσ dω } R dξ ω 2r ( ω 2 + ξ 2 ) s Fψ(ω, ξ) 2 < d H r σ(r +, H s (R d )) supported distributions H. Gimperlein TDBEM for dynamic contact BEM on the Saar / 9

10 Analytic set-up: Space-time Sobolev spaces space time anisotropic Sobolev spaces H r σ(r +, H s (Γ)), σ > 0: H r σ(r +, H s (R d )) defined using Fourier Laplace transform { ψ : supp ψ R + R d, R+iσ dω } R dξ ω 2r ( ω 2 + ξ 2 ) s Fψ(ω, ξ) 2 < d H r σ(r +, H s (R d )) supported distributions Variational inequality: Find 0 u H 2 σ (R +, H 2 (G)) such that Su, v u h, v u 0 v H 2 σ (R +, H 2 (G)). H. Gimperlein TDBEM for dynamic contact BEM on the Saar / 9

11 Existence of solutions Ω = R d, G Γ bounded polygon. Existence of solutions: Lebeau Schatzmann 84, students of Eskin, including Cooper 99. Theorem (Cooper, 99): For h H 3 2 σ (R +, H 2 (Γ c )) the contact problem admits a unique solution u H 2 σ (R +, H 2 (Γ c )) +. H. Gimperlein TDBEM for dynamic contact BEM on the Saar / 9

12 Existence of solutions Ω = R d, G Γ bounded polygon. Existence of solutions: Lebeau Schatzmann 84, students of Eskin, including Cooper 99. Theorem (Cooper, 99): For h H 3 2 σ (R +, H 2 (Γ c )) the contact problem admits a unique solution u H 2 σ (R +, H 2 (Γ c )) +. Basic strategy: elliptic regularization: S S ε( 2 t + R d ) solution exists for ε > 0 by standard theory control ε 0 + using improved coercivity estimate for flat Γ (from Fourier analysis): ϕ 2 2, 2,σ, σ Sϕ, ϕ C ϕ 2 0, 2,σ, H. Gimperlein TDBEM for dynamic contact BEM on the Saar / 9

13 Discretization Γ = M i= Γ i (quasi-uniform) triangulation V p h piecewise polynomial functions of degree p on Γ = M i= Γ i (continuous if p ) [0, T) = L n= [t n, t n ), t n = n( t) V q t piecewise polynomial functions of degree q in time (continuous and vanishing at t = 0 if q ) tensor products in space-time: V p,q h, t = Vp h Vq t Ṽ p,q h, t subspace vanishing at boundary H. Gimperlein TDBEM for dynamic contact BEM on the Saar / 9

14 A priori error analysis Variational inequality: Su, v u h, v u 0 v H 2 σ (R +, H 2 (Γc )). Discretized variational inequality in V p,q h, t : S h, t u h, t, v h, t u h, t h, v h, t u h, t 0 v h, t Ṽ p,q h, t. Crucial: not S t. H. Gimperlein TDBEM for dynamic contact BEM on the Saar / 9

15 A priori error analysis Variational inequality: Su, v u h, v u 0 v H 2 σ (R +, H 2 (Γc )). Discretized variational inequality in V p,q h, t : S h, t u h, t, v h, t u h, t h, v h, t u h, t 0 v h, t Ṽ p,q h, t. Theorem Let h H 3 2 σ (R +, H 2 (G)). Then u u t,h 2 2, 2,σ, σ inf ( h S σ u 0 ϕ t,h Ṽ p,q 2,,σ u ϕ t,h 2 2, 2,σ, h, t + u ϕ t,h 2 2, 2,σ, + O(S S h, t)). Assume u H 2 +ϵ σ (R +, H 2 +ϵ (G)) for some ϵ > 0, S = S h, t. Then u u t,h 2 2, 2,σ, (hϵ + ( t) 2 +ϵ ) u 2 +ϵ, 2 +ϵ,σ + (h2ϵ + ( t) 2ϵ ) u 2 2 +ϵ, 2 +ϵ, H. Gimperlein TDBEM for dynamic contact BEM on the Saar / 9

16 Mixed formulation for improved energy conservation Variational inequality: Su, v u h, v u 0 v H 2 σ (R +, H 2 (Γc )). Lagrange multiplier λ = Su h indicates contact / forces Mixed formulation: Find (u, λ) H 2 σ (R +, H 2 (G)) H 2 σ (R +, H 2 (G)) + such that { (a) Su, v λ, v = h, v (b) u, µ λ 0, for all (v, µ) H 2 σ (R +, H 2 (G)) H 2 σ (R +, H 2 (G)) +. H. Gimperlein TDBEM for dynamic contact BEM on the Saar / 9

17 Mixed formulation for improved energy conservation Variational inequality: Su, v u h, v u 0 v H 2 σ (R +, H 2 (Γc )). Lagrange multiplier λ = Su h indicates contact / forces Discretized mixed formulation: Find (u h, t, λ h2, t 2 ) Ṽ, h, t V 0,0,+ h 2, t 2 such that { (a) Su h, t, v h, t λ h2, t 2, v h, t = h, v h, t (b) u h, t, µ h2, t 2 λ h2, t 2 0, for all (v h, t, µ h2, t 2 ) Ṽ, h, t V 0,0,+ h 2, t 2. Both the continuous and discretized variational inequalities admit unique solutions. H. Gimperlein TDBEM for dynamic contact BEM on the Saar / 9

18 Mixed formulation for improved energy conservation Discretized mixed formulation: Find (u h, t, λ h2, t 2 ) Ṽ, h, t V 0,0,+ h 2, t 2 such that { (a) Su h, t, v h, t λ h2, t 2, v h, t = h, v h, t (b) u h, t, µ h2, t 2 λ h2, t 2 0, for all (v h, t, µ h2, t 2 ) Ṽ, h, t V 0,0,+ h 2, t 2. Error analysis based on: Theorem (space time inf sup) Let C > 0 sufficiently small, and max{h, t } min{h 2, t 2 } α > 0 such that for all λ t2,h 2 : < C. Then there exists v t,h sup, λ t2,h 2 α λ t2,h v t,h v t,h 2 0, 0, 2,σ, 2,σ. H. Gimperlein TDBEM for dynamic contact BEM on the Saar / 9

19 Mixed formulation for improved energy conservation Discretized mixed formulation: Find (u h, t, λ h2, t 2 ) Ṽ, h, t V 0,0,+ h 2, t 2 such that { (a) Su h, t, v h, t λ h2, t 2, v h, t = h, v h, t (b) u h, t, µ h2, t 2 λ h2, t 2 0, for all (v h, t, µ h2, t 2 ) Ṽ, h, t V 0,0,+ h 2, t 2. The inf-sup condition implies a priori estimates similar to Brezzi Hager Raviart 78: Theorem (a priori error estimate) λ λ t2,h 2 0,,σ inf λ λ t2,h 2 2 0, λ 2,σ + ( t ) 2 u u t,h 2, 2,σ, t2,h 2 u u t,h 2, 2,σ, σ inf v t,h u v t,h 2, 2,σ, { } + inf λ t2,h 2 λ λ 2, 2,σ + λ t2,h 2 λ t2,h 2 2, 2,σ t2,h 2 H. Gimperlein TDBEM for dynamic contact BEM on the Saar / 9

20 Proof of a priori estimate: λ λ t2,h 2 Inf-sup condition: For all λ t2,h 2 Algebra α λ t2 h 2 λ λ t2,h t2,h 2 0,,σ sup 2 λ t2,h 2, v t,h 2 v t,h v t,h 0, 2,σ, S(u t,h = sup u), v t,h + λ λ t2,h 2, v t,h v t,h v t,h 0, First term: duality and inverse inequality in time 2,σ,. S(u t,h u), v t,h S σ (u t,h u) 2, 2,σ v t,h 2, 2,σ, Second term: duality u t,h u 2, 2,σ, ( t ) 2 v t,h 0, 2,σ,. λ λ t2,h 2, v t,h λ λ t2,h 2 0, 2,σ v t,h 0, 2,σ,. Combine into estimate for λ λ t2,h 2 0, 2,σ. H. Gimperlein TDBEM for dynamic contact BEM on the Saar / 9

21 Proof of a priori estimate: u u t,h Estimate for u without inf-sup: Use coercivity u t,h v t,h 2 2, 2,σ, σ S(u t,h v t,h ), u t,h v t,h Then Galerkin orthogonality, mapping properties, etc. standard! H. Gimperlein TDBEM for dynamic contact BEM on the Saar 207 / 9

22 Computing the Dirichlet-Neumann operator S(u Γ ) = du dν Computation through layer potentials: S = W + (K 2 )V (K 2 ) Γ Kφ(t, x) = K φ(t, x) = R + Γ R + Γ Wφ(t, x) = Here G(t, x) = δ(t x ) 4π x R + Γ G (t τ, x y) φ(τ, y) dτ ds y, n y G (t τ, x y) φ(τ, y) dτ ds y, n x 2 G n x n y (t τ, x y) φ(τ, y) dτ ds y. a fundamental solution of wave equation. H. Gimperlein TDBEM for dynamic contact BEM on the Saar / 9

23 Computing the Dirichlet-Neumann operator S(u Γ ) = du dν Computation through layer potentials: S = W + (K 2 )V (K 2 ) Γ Weak formulation of Su = h H 3 2 σ (R +, H 2 (Γ)) on flat screen: Find (u, v) H 2 σ (R +, H 2 (Γ)) H 2 σ (R +, H 2 (Γ)) such that Wu + (K )v, ϕ = h, ϕ, 2 (K )u + Vv, Ψ = 0, 2 for all (ϕ, Ψ) H 2 σ (R +, H 2 (G)) H 2 σ (R +, H 2 (G)). H. Gimperlein TDBEM for dynamic contact BEM on the Saar / 9

24 Computing the Dirichlet-Neumann operator S(u Γ ) = du dν Computation through layer potentials: Γ S = W + (K 2 )V (K 2 ) Discretized weak formulation of Su = h on flat screen: Find (u h, t, v h, t ) V, V, such that Wu h, t (K 2 )v h, t, ϕ h, t = h, ϕ h, t, (K 2 )u h, t + Vv h, t, t Ψ h, t = 0, for all (ϕ h, t, Ψ h, t ) V,0 V,0. Note: No t in first equation S coercive on flat screen, important later. H. Gimperlein TDBEM for dynamic contact BEM on the Saar / 9

25 Uzawa algorithm Discretized mixed formulation: { (a) Su h, t, v h, t λ h2, t 2, v h, t = h, v h, t (b) u h, t, µ h2, t 2 λ h2, t 2 0, Two algorithms: Uzawa in space-time / Uzawa as time stepping scheme H. Gimperlein TDBEM for dynamic contact BEM on the Saar / 9

26 Uzawa algorithm Discretized mixed formulation: { (a) Su h, t, v h, t λ h2, t 2, v h, t = h, v h, t (b) u h, t, µ h2, t 2 λ h2, t 2 0, Two algorithms: Uzawa in space-time / Uzawa as time stepping scheme Space-time Uzawa algorithm Choose ρ > 0. Set k = 0, y (0) = 0. WHILE stopping criterion not satisfied solve: Sx (k) = h + y (k) compute: y (k+) = Pr K (y (k) ρx (k) ), where (Pr K y) i = max{y i, 0} k k + ENDWHILE H. Gimperlein TDBEM for dynamic contact BEM on the Saar / 9

27 Uzawa algorithm Two algorithms: Uzawa in space-time / Uzawa as time stepping scheme Space-time Uzawa algorithm Choose ρ > 0. Set k = 0, y (0) = 0. WHILE stopping criterion not satisfied solve: Sx (k) = h + y (k) compute: y (k+) = Pr K (y (k) ρx (k) ), where (Pr K y) i = max{y i, 0} k k + ENDWHILE Recall: Sx (k) = h + y (k) is a linear system with matrix ( W ((K) 2 I) ) (K 2 I) V H. Gimperlein TDBEM for dynamic contact BEM on the Saar / 9

28 Uzawa algorithm Two algorithms: Uzawa in space-time / Uzawa as time stepping scheme Space-time Uzawa algorithm Choose ρ > 0. Set k = 0, y (0) = 0. WHILE stopping criterion not satisfied solve: Sx (k) = h + y (k) compute: y (k+) = Pr K (y (k) ρx (k) ), where (Pr K y) i = max{y i, 0} k k + ENDWHILE Lemma The space-time Uzawa algorithm converges, provided that 0 < ρ < 2C σ. Here C σ is the coercivity constant of S. H. Gimperlein TDBEM for dynamic contact BEM on the Saar / 9

29 Uzawa algorithm Two algorithms: Uzawa in space-time / Uzawa as time stepping scheme Space-time Uzawa algorithm Choose ρ > 0. Set k = 0, y (0) = 0. WHILE stopping criterion not satisfied solve: Sx (k) = h + y (k) compute: y (k+) = Pr K (y (k) ρx (k) ), where (Pr K y) i = max{y i, 0} k k + ENDWHILE We often use a reformulation of the variational inequality as a variational inequality in every time step. efficient Uzawa in every time step. It converges in every time step. H. Gimperlein TDBEM for dynamic contact BEM on the Saar / 9

30 Experiment : Dirichlet-Neumann equation Su = h on Γ = S 2 for times [0, 5], where h = Neumann data of u(t, r) = ( 3 π(4 t) 4 cos( 2 ) + 4 cos(π(4 t)))[h(4 t) H( t)]. H = Heaviside function. Approximation on uniform icosahedral meshes with t h 0.6. Plot L 2 ([0, T] Γ)-error vs. degrees of freedom 0 0 Dirichlet to Neumann equation relative L 2 error in space time DOF H. Gimperlein TDBEM for dynamic contact BEM on the Saar / 9

31 Experiment 2: Contact Su h on flat screen Γ = [ 2, 2] 2 {0} G = [, ] 2 {0}, 0 < t < 5, t x = 0.7 h = e 2t t 4 cos(2πx) cos(2πy)χ [ 0.25,0.25] (x)χ [ 0.25,0.25] (y) relative L 2 ([0, T] Γ)-error against benchmark with 2800 triangles, t = Space Time Uzawa relative L 2 error in space time DOF H. Gimperlein TDBEM for dynamic contact BEM on the Saar / 9

32 Experiment 2: Uzawa for contact Su h Γ = [ 2, 2] 2 {0} G = [, ] 2 {0}, 0 < t < 6, t = 0. h = e 2t t 4 cos(2πx) cos(2πy)χ [ 0.25,0.25] (x)χ [ 0.25,0.25] (y), 3200 triangles Evolution of the relative difference between solutions in L 2 (Γ) between time step Uzawa and space-time Uzawa. Difference increases sharply around the onset of contact. 0 4 Space Time vs. Time Step Uzawa on 3200 triangles 0 6 relative error Time H. Gimperlein TDBEM for dynamic contact BEM on the Saar / 9

33 Experiment 3: Su h with non-flat contact Γ = [ 2, 2] 3 Γ c = 3 faces (top, left, back), 0 < t < 4, t x = 0.7 h as before, on every contact face relative L 2 ([0, T] Γ)-error against benchmark 0 Space Time Uzawa relative error in space time DOF H. Gimperlein TDBEM for dynamic contact BEM on the Saar / 9

34 Experiment 3: Su h with non-flat contact H. Gimperlein (a) t=3 (b) t=5 (c) t=5.5 (d) t=6 TDBEM for dynamic contact BEM on the Saar / 9

35 Conclusions & References first a priori analysis of contact problems, using TDBEM convergence of Galerkin and mixed methods for flat contact (only case where existence of solution is known) based on inf-sup condition in space-time and refined coercivity estimates for Dirichlet-Neumann convergence of space-time Uzawa and time step Uzawa Analysis and numerics widely open: adaptivity, friction, stabilized methods,... Final remark: Theory and numerics extend to punch problems with the Neumann-Dirichlet operator. HG, Meyer, Oezdemir, Stephan, Time domain boundary elements for dynamic contact problems, preprint. H. Gimperlein TDBEM for dynamic contact BEM on the Saar / 9

Time domain boundary elements for dynamic contact problems

Time domain boundary elements for dynamic contact problems Heiko Gimperlein Fabian Meyer Ceyhun Özdemir Ernst P. Stephan (dedicated to Erwin Stein on the occasion of his 85th birthday) Abstract This article