Space time finite element methods in biomedical applications

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1 Space time finite element methods in biomedical applications Olaf Steinbach Institut für Angewandte Mathematik, TU Graz SFB Mathematical Optimization and Applications in Biomedical Sciences LEAD Project Mechanics, Modeling and Simulation in Aortic Disscetion Subproject Parallel Space Time Finite Element Methods 1 / 23

2 FETI Domain Decomposition Methods for the Simulation of Biological Tissues [C. Augustin 214; C. Augustin, G. Holzapfel, OS 214] 2 / 23

3 Cardiac Electromechanics [E. Karabelas 215] Active stress T a 3 / 23

4 Optimal control of Blood Flow in an Artery with Aneurysm [L. John 214] Wall shear stress distribution [L. John, P. Pustejovska, OS 217] 4 / 23

5 Lobe Pumpe [M. Neumüller 213] 5 / 23

6 Space Time Finite Element Methods Formulation in Bochner spaces Formulation in anisotropic Sobolev spaces standard lowest order (piecewise linear) finite elements Heat equation [with M. Zank] Bidomain system [with H. Yang] (Navier ) Stokes [with D. Pacheco] Wave equation [with M. Zank] The aim is to provide an adaptive simulation, simultaneously in space and time, which allows an accurate resolution of wave type solutions, and which allows, i.e. requires, the use of parallel solvers. 6 / 23

7 Dirichlet boundary value problem for the heat equation α t u(x,t) x u(x,t) = f(x,t) for (x,t) Q := Ω (,T), u(x,t) = for (x,t) Σ := Γ (,T), u(x,) = for x Ω. 7 / 23

8 Dirichlet boundary value problem for the heat equation α t u(x,t) x u(x,t) = f(x,t) u(x,t) = u(x,) = for x Ω. Find u L 2 (,T;H 1 (Ω)) H1, (,T;H 1 (Ω)) such that Ω [ ] α t u(x,t)v(x,t)+ x u(x,t) x v(x,t) dxdt = is satisfied for all v L 2 (,T;H 1 (Ω)). for (x,t) Q := Ω (,T), for (x,t) Σ := Γ (,T), Ω f(x,t)v(x,t)dxdt 7 / 23

9 Dirichlet boundary value problem for the heat equation α t u(x,t) x u(x,t) = f(x,t) u(x,t) = u(x,) = for x Ω. Find u L 2 (,T;H 1 (Ω)) H1, (,T;H 1 (Ω)) such that Ω [ ] α t u(x,t)v(x,t)+ x u(x,t) x v(x,t) dxdt = is satisfied for all v L 2 (,T;H 1 (Ω)). for (x,t) Q := Ω (,T), for (x,t) Σ := Γ (,T), Ω f(x,t)v(x,t)dxdt Bilinear form a(u,v) = = Ω Ω [ ] α t u(x,t)v(x,t)+ x u(x,t) x v(x,t) dxdt [ ] α t u(x,t) x u(x,t) v(x,t)dxdt 7 / 23

10 Quasi static Dirichlet problem for u L 2 (,T;H 1 (Ω)) H1, (,T;H 1 (Ω)) x w(x,t) = α t u(x,t) x u(x,t) for (x,t) Q, w(x,t) = for (x,t) Σ. 8 / 23

11 Quasi static Dirichlet problem for u L 2 (,T;H 1 (Ω)) H1, (,T;H 1 (Ω)) x w(x,t) = α t u(x,t) x u(x,t) for (x,t) Q, w(x,t) = for (x,t) Σ. Find w L 2 (,T;H 1 (Ω)) such that x w(x,t) x v(x,t)dxdt = Ω is satisfied for all v L 2 (,T;H 1 (Ω)). Ω [ ] α t u(x,t) x u(x,t) v(x,t)dxdt 8 / 23

12 Quasi static Dirichlet problem for u L 2 (,T;H 1 (Ω)) H1, (,T;H 1 (Ω)) x w(x,t) = α t u(x,t) x u(x,t) for (x,t) Q, w(x,t) = for (x,t) Σ. Find w L 2 (,T;H 1 (Ω)) such that x w(x,t) x v(x,t)dxdt = Ω is satisfied for all v L 2 (,T;H 1 (Ω)). Lemma Corollary Ω [ ] α t u(x,t) x u(x,t) v(x,t)dxdt α t u x u 2 L 2 (,T;H 1 (Ω)) = w 2 L 2 (,T;H 1 (Ω)) = a(u,w) α t u x u L 2 (,T;H 1 (Ω)) sup v L 2 (,T;H 1 (Ω)) a(u,v) v L 2 (,T;H 1 (Ω)) 8 / 23

13 Norm equivalence 1 ] [ α t u 2 L 2 2 (,T;H 1 (Ω)) + u 2L 2(,T;H 1(Ω)) α t u x u 2 L 2 (,T;H 1 (Ω)) ] 2 [ α t u 2 L 2 (,T;H 1 (Ω)) + u 2L 2(,T;H 1(Ω)) Lemma 1 α t u 2 2 L 2 (,T;H 1 (Ω)) + u 2 sup L 2 (,T;H 1 (Ω)) v L 2 (,T;H 1(Ω)) a(u,v) v L 2 (,T;H 1 (Ω)) [Schwab, Stevenson 29; Urban, Patera 214; Andreev 213; Mollet 214; OS 215;...] 9 / 23

14 Adjoint variational formulation to find u L 2 (,T;H 1 (Ω)) such that [ ] u(x,t)α t v(x,t)+ x u(x,t) x v(x,t) dxdt = f(x,t)v(x,t)dxdt Ω is satisfied for all v L 2 (,T;H 1 (Ω)) H1, (,T;H 1 (Ω)). Ω 1 / 23

15 Adjoint variational formulation to find u L 2 (,T;H 1 (Ω)) such that [ ] u(x,t)α t v(x,t)+ x u(x,t) x v(x,t) dxdt = f(x,t)v(x,t)dxdt Ω is satisfied for all v L 2 (,T;H 1 (Ω)) H1, (,T;H 1 (Ω)). Operator adjoint formulation L : L 2 (,T;H 1 (Ω)) [L 2 (,T;H 1 (Ω)) H 1,(,T;H 1 (Ω))] Operator primal formulation L : L 2 (,T;H 1 (Ω)) H 1,(,T;H 1 (Ω)) [L 2 (,T;H 1 (Ω))] Ω 1 / 23

16 Adjoint variational formulation to find u L 2 (,T;H 1 (Ω)) such that [ ] u(x,t)α t v(x,t)+ x u(x,t) x v(x,t) dxdt = f(x,t)v(x,t)dxdt Ω is satisfied for all v L 2 (,T;H 1 (Ω)) H1, (,T;H 1 (Ω)). Operator adjoint formulation L : L 2 (,T;H 1 (Ω)) [L 2 (,T;H 1 (Ω)) H 1,(,T;H 1 (Ω))] Operator primal formulation Can we define L : L 2 (,T;H 1 (Ω)) H 1,(,T;H 1 (Ω)) [L 2 (,T;H 1 (Ω))] L : L 2 (,T;H 1 (Ω)) H 1/2, (,T;L 2 (Ω)) [L 2 (,T;H 1 (Ω)) H 1/2, (,T;L2 (Ω))] Ω 1 / 23

17 Model problem u (t) = f(t) for t (,T), u() = Primal variational formulation: Find u H, 1 (,T) such that a(u,v) = u (t)v(t)dt = Solution operator B 1 : H,(,T) 1 L 2 (,T) f(t)v(t)dt for all v L 2 (,T) 11 / 23

18 Model problem u (t) = f(t) for t (,T), u() = Primal variational formulation: Find u H, 1 (,T) such that a(u,v) = u (t)v(t)dt = Solution operator B 1 : H,(,T) 1 L 2 (,T) f(t)v(t)dt for all v L 2 (,T) Dual variational formulation: Find u L 2 (,T) such that u(t)v (t)dt = f(t)v(t)dt for all v H 1,(,T) Solution operator B : L 2 (,T) [H,(,T)] 1 11 / 23

19 Model problem u (t) = f(t) for t (,T), u() = Primal variational formulation: Find u H, 1 (,T) such that a(u,v) = u (t)v(t)dt = Solution operator B 1 : H,(,T) 1 L 2 (,T) f(t)v(t)dt for all v L 2 (,T) Dual variational formulation: Find u L 2 (,T) such that u(t)v (t)dt = f(t)v(t)dt for all v H 1,(,T) Solution operator B : L 2 (,T) [H,(,T)] 1 Question B s : [L 2 (,T),H,(,T)] 1 s [[H,(,T)] 1,L 2 (,T)] s for s (,1),s = 1 2 Related work: [Fontes 1996; Langer, Wolfmayr 213; Larsson, Schwab 215] 11 / 23

20 Solution operator Adjoint operator B 1 : H 1,(,T) L 2 (,T) B 1 : L 2 (,T) [H 1,(,T)] u,b 1v L2 (,T) = B 1 u,v L2 (,T) for all u H 1,(,T),v L 2 (,T) 12 / 23

21 Solution operator Adjoint operator B 1 : H 1,(,T) L 2 (,T) B 1 : L 2 (,T) [H 1,(,T)] u,b 1v L2 (,T) = B 1 u,v L2 (,T) for all u H 1,(,T),v L 2 (,T) Define Eigenvalue problem A := B 1B 1 : H 1,(,T) [H 1,(,T)] Au = λu 12 / 23

22 Solution operator Adjoint operator B 1 : H 1,(,T) L 2 (,T) B 1 : L 2 (,T) [H 1,(,T)] Define u,b 1v L2 (,T) = B 1 u,v L2 (,T) for all u H 1,(,T),v L 2 (,T) Eigenvalue problem Au,v L2 (,T) = B 1 u,b 1 v L2 (,T) = A := B 1B 1 : H 1,(,T) [H 1,(,T)] Au = λu t u(t) t v(t)dt = λ u(t)v(t)dt u (t) = λu(t) for t (,T), u() =, u (T) = ( π ) t v k (t) = sin 2 +kπ T, λ k = 1 ( π ) 2, T 2 2 +kπ k =,1,2,3, / 23

23 u H, 1(,T) ( π ) t u(t) = u k sin 2 +kπ T, u k = 2 π ) t u(t)sin( T 2 +kπ T dt k= 13 / 23

24 u H, 1(,T) ( π ) t u(t) = u k sin 2 +kπ T, u k = 2 π ) t u(t)sin( T 2 +kπ T dt k= Time derivative (B 1 u)(t) = u (t) = 1 T k= ( π ( π ) t u k )cos 2 +kπ 2 +kπ T 13 / 23

25 u H, 1(,T) ( π ) t u(t) = u k sin 2 +kπ T, u k = 2 π ) t u(t)sin( T 2 +kπ T dt k= Time derivative (B 1 u)(t) = u (t) = 1 T B 1 u,v L 2 (,T) = 1 T = 1 T = 1 2 if we choose k= l= k= k= ( π ( π ) t u k )cos 2 +kπ 2 +kπ T ( π )cos( π ) t u k 2 +kπ 2 +kπ T v(t)dt k= ( π ) π ) t ( π ) t u k u l cos( 2 +kπ 2 +kπ T cos 2 +lπ T dt u 2 k ( π 2 +kπ ) = u 2 H 1/2, (,T) π ) t v(t) = u l cos( 2 +lπ T, l= 13 / 23

26 Transformation operator ( π ) t (H T u)(t) = u k cos 2 +kπ T, u(t) = π ) t u k sin( 2 +kπ T, k= k= 14 / 23

27 Transformation operator ( π ) t (H T u)(t) = u k cos 2 +kπ T, u(t) = π ) t u k sin( 2 +kπ T, k= k= Lemma H T : H s,(,t) H ș (,T), H T u H ș (,T) = u H s, (,T) 14 / 23

28 Transformation operator (H T u)(t) = Lemma ( π ) t u k cos 2 +kπ T, u(t) = π ) t u k sin( 2 +kπ T, k= k= H T : H s,(,t) H ș (,T), H T u H ș (,T) = u H s, (,T) Inverse transformation operator T v)(t) = ( π ) t v k sin 2 +kπ T, v(t) = ( π ) t v k cos 2 +kπ T (H 1 k= k= 14 / 23

29 Transformation operator (H T u)(t) = Lemma ( π ) t u k cos 2 +kπ T, u(t) = π ) t u k sin( 2 +kπ T, k= k= H T : H s,(,t) H ș (,T), H T u H ș (,T) = u H s, (,T) Inverse transformation operator (H 1 Lemma T v)(t) = ( π ) t v k sin 2 +kπ T, v(t) = ( π ) t v k cos 2 +kπ T k= t H T u = H 1 T tu k= 14 / 23

30 Transformation operator (H T u)(t) = Lemma ( π ) t u k cos 2 +kπ T, u(t) = π ) t u k sin( 2 +kπ T, k= k= H T : H s,(,t) H ș (,T), H T u H ș (,T) = u H s, (,T) Inverse transformation operator T v)(t) = ( π ) t v k sin 2 +kπ T, v(t) = ( π ) t v k cos 2 +kπ T (H 1 Lemma k= t H T u = H 1 T tu k= Lemma: For u H 1/2, (,T) and w H 1/2, (,T) we have H T u,w L2 (,T) = u,h 1 T w L 2 (,T) 14 / 23

31 Corollary t u,h T v (,T) = H T u, t v (,T) 15 / 23

32 Corollary t u,h T v (,T) = H T u, t v (,T) Lemma v,h T v L2 (,T) > for all v H 1/2, (,T) 15 / 23

33 Corollary t u,h T v (,T) = H T u, t v (,T) Lemma Lemma v,h T v L2 (,T) > for all v H 1/2, (,T) (H T u)(t) = 1 [ T 2T 1 sin ( π 2 ) s t + T 1 sin ( π 2 ) s+t T ] u(s) ds 15 / 23

34 Corollary t u,h T v (,T) = H T u, t v (,T) Lemma Lemma v,h T v L2 (,T) > for all v H 1/2, (,T) (H T u)(t) = 1 [ T 2T 1 sin ( π 2 ) s t + T 1 sin ( π 2 ) s+t T ] u(s) ds Corollary: For T (H u)(t) = 1 π u(s) s t 2s s +t ds Hilbert transformation Construction of optimal test functions [Demkowicz, Gopalakrishnan 211] 15 / 23

35 Model problem u (t) = f(t) for t (,T), u() = Variational formulation: Find u H 1/2, (,T) such that t u,v (,T) = f,v (,T) for all v H 1/2, (,T) 16 / 23

36 Model problem u (t) = f(t) for t (,T), u() = Variational formulation: Find u H 1/2, (,T) such that t u,v (,T) = f,v (,T) for all v H 1/2, (,T) Variational formulation: Find u H 1/2, (,T) such that t u,h T v (,T) = f,h T v (,T) for all v H 1/2, (,T) 16 / 23

37 Model problem u (t) = f(t) for t (,T), u() = Variational formulation: Find u H 1/2, (,T) such that t u,v (,T) = f,v (,T) for all v H 1/2, (,T) Variational formulation: Find u H 1/2, (,T) such that Theorem t u,h T v (,T) = f,h T v (,T) for all v H 1/2, (,T) t u,h T u (,T) = u 2 H 1/2, (,T) for all u H 1/2, (,T) Corollary u 1/2 H, (,T) sup t u,v (,T) for all u H 1/2, (,T) v v H 1/2, (,T) 1/2 H, (,T) 16 / 23

38 Galerkin FEM: Find u h V h := S 1 h (,T) H1/2, (,T) such that t u h,h T v h (,T) = f,h T v h (,T) for all v h V h Linear system K h u = f, K h = K h > Numerical example: u(t) = 8t 3 8t 2 +2t, t (,1) L N u u h L 2 eoc u u h L 2 eoc λ min λ max κ / 23

39 Dirichlet boundary value problem for the heat equation α t u(x,t) x u(x,t) = f(x,t) u(x,t) = u(x,) = for x Ω. for (x,t) Q := Ω (,T), for (x,t) Σ := Γ (,T), Find u H 1,1/2 ;, := L 2 (,T;H 1 (Ω)) H1/2, (,T;L 2 (Ω)) such that Ω [ ] α t u(x,t)v(x,t)+ x u(x,t) x v(x,t) dxdt = f(x,t)v(x,t)dxdt Ω is satisfied for all v H 1,1/2 ;, := L 2 (,T;H 1 (Ω)) H1/2, (,T;L2 (Ω)). 18 / 23

40 Dirichlet boundary value problem for the heat equation α t u(x,t) x u(x,t) = f(x,t) u(x,t) = u(x,) = for x Ω. for (x,t) Q := Ω (,T), for (x,t) Σ := Γ (,T), Find u H 1,1/2 ;, := L 2 (,T;H 1 (Ω)) H1/2, (,T;L 2 (Ω)) such that Ω [ ] α t u(x,t)v(x,t)+ x u(x,t) x v(x,t) dxdt = f(x,t)v(x,t)dxdt Ω is satisfied for all v H 1,1/2 ;, := L 2 (,T;H 1 (Ω)) H1/2, (,T;L2 (Ω)). Theorem 1 2 u H 1,1/2 ;, (Q) sup v H 1,1/2 ;, (Q) a(u,v) v H 1,1/2 ;, (Q) 18 / 23

41 Galerkin FEM: Find u h V h := Q 1 h (Q) H1,1/2 ;, (Q) such that t u h,h T v h Q + x u h, x H T v h L 2 (Q) = f,h T v h Q for all v h V h Numerical example u(x,t) = sin 5π 4 t sinπx for (x,t) (,1) (,2), α = 1 N x N t dof h x h t u u h L 2 eoc u u h H 1 eoc / 23

42 Heat equation with nonlinear reaction term e.g. Schlögl model, Nagumo equation, Monodomain equation t u x u + 1 ε (u3 u) =, Q = (17,19) (17,19) (,5), u(x 1,x 2,t) = 1 2 ( 1 tanh x ) 1 st 2, ε =.38, s = 3/( 2ε) 2ε #Dofs u u h L2(,T;H 1(Ω)) eoc / 23

43 Convergence history Figure: uniform ( + ), octasection ( ), bisection ( ), ideal ( ) 21 / 23

44 Adaptive refinement in space time Figure: Adaptive meshes: octasection (left) and bisection (right). 22 / 23

45 References 1. M. Neumüller, O. Steinbach: Refinement of flexible space time finite element meshes and discontinuous Galerkin methods. Comput. Visual. Sci. 14 (211) O. Steinbach: Space time finite element methods for parabolic problems. Comput. Meth. Appl. Math. 15 (215) O. Steinbach, H. Yang: An algebraic multigrid method for an adaptive space time finite element discretization. Lecture Notes in Comput. Sci., 1665, Springer, Cham, pp , O. Steinbach, H. Yang: Comparison of algebraic multigrid methods for an adaptive space time finite element discretization of the heat equation in 3D and 4D. Numer. Linear Algebra Appl. 25 (218), no. 3, e2143, 17 pp. 5. O. Steinbach, H. Yang: A space time finite element method for the linear bidomain equations, submitted, O. Steinbach, M. Zank: A stabilized space time finite element method for the wave equation, accepted, O. Steinbach, H. Yang: Space time finite element methods for parabolic evolution equations: Discretization, a posteriori error estimation, adaptivity and solution, submitted, O. Steinbach, M. Zank: Coercive space time finite element methods for initial boundary value problems, to be submitted. 23 / 23

Space time finite and boundary element methods

Space time finite and boundary element methods Olaf Steinbach Institut für Numerische Mathematik, TU Graz http://www.numerik.math.tu-graz.ac.at based on joint work with M. Neumüller, H. Yang, M. Fleischhacker,