A Multiscale Hybrid-Mixed method for the elastodynamic model in time domain

Transcription

1 1/21 A Multiscale Hybrid-Mixed method for the elastodynamic model in time domain Weslley da Silva Pereira 1 Nachos project-team, Inria Sophia Antipolis - Mediterranée (FR) Inria Sophia Antipolis - Mediterranée October 12, Joint work with MHM research group at LNCC/MCTIC (Brazil).

2 2/21 Outline Motivation MHM for Elastodynamics A Global-Local formulation The MHM method Numerical results An analytical problem Multiscale problem Final conclusions

3 2/21 Motivation

4 3/21 HPC technology Figure : SDumont supercomputer. [ New numerical methods: Precision and robustness; High degree of parallelism; Support to fault tolerance.

5 Elastodynamic problem Equation of motion Constitutive equation ρ tt u σ = f, in Ω (0, T ). σ = C ε, ε := u + ( u)t 2. estação de dados fonte receptores Isotropic media C ε = 2 c µ ε + c λ tr(ε) I Figure : Seismic imaging technique. 4/21

6 MHM - Multiscale Hybrid-Mixed method 2 h Figure : Global mesh T H with elements K. Figure : Local mesh T K h an element K T H. over 2 Harder, C., Paredes, D., and Valentin, F. (2013). A family of Multiscale Hybrid-Mixed finite element methods for the Darcy equation with rough coefficients. J. Comput. Phys., 245: /21

7 6/21 Main contributions Propose of a new MHM method for elastodynamic problems in time domain. Show the numerical validation for method. Show the results of the method in a multiscale scenario.

8 6/21 MHM for Elastodynamics

9 7/21 Partitioning the domain in space and time Ω y t Ω t 0 t - 1 t + 1 t Ω - 2 t + n-1 t - n t + n t - n+1 t t + n T n -1 T Ω x t n + t n-1 I K - = n n-1 (t, t ) n I n Figure : Time partition T n. Figure : Space-time partition.

10 Variational spaces 3 Broken space of displacements: } V = H 1 (T H ) := {w L 2 (Ω) : w K H 1 (K) K T H. Space of tractions: Λ := { } τ n K K, K T H : τ H 0,ΓN (div; Ω; S). Broken products: (, ) TH := K T H (, ) K and (, ) TH := K T H (, ) K. 3 For simplicity, here we consider only homogeneous Dirichlet and Neumann boundary conditions. 8/21

11 9/21 A hybrid variational problem For n = 1,, n T, find u n C(I n ; V) C 1 (I n ; L 2 (Ω)) and λ n C(I n ; Λ) such that d tt (ρu n, w) TH + (σ(u n ),ε(w)) TH + (λ n, w) TH = = (f, w) TH, w V, (µ, u n ) TH = 0, µ Λ, u n (t n 1 ) = u n 1 (t n 1 ), d t u n (t n 1 ) = d t u n 1 (t n 1 ), where u 0 (0) = u 0 and d t u 0 (0) = v 0.

12 9/21 A hybrid variational problem For n = 1,, n T, find u n C(I n ; V) C 1 (I n ; L 2 (Ω)) and λ n C(I n ; Λ) such that d tt (ρu n, w) TH + (σ(u n ),ε(w)) TH + (λ n, w) TH = = (f, w) TH, w V, (µ, u n ) TH = 0, µ Λ, u n (t n 1 ) = u n 1 (t n 1 ), d t u n (t n 1 ) = d t u n 1 (t n 1 ), where u 0 (0) = u 0 and d t u 0 (0) = v 0. Split the solution: u In = u n = u n,λ + u n,f.

13 10/21 Global-Local formulation For n = 1,, n T, find λ n C(I n ; Λ) such that (µ, u n,λ ) Th = (µ, u n,f ) Th, µ Λ. where, for each K I n, u n,λ satisfy { dtt (ρ u n,λ, w) K + (σ(u n,λ ), ε(w)) K = (λ n, w) K, w V(K), and u n,f satisfy u n,λ (t n 1 ) = d t u n,λ (t n 1 ) = 0, d tt (ρ u n,f, w) K + (σ(u n,f ), ε(w)) K = (f, w) K, u n,f (t n 1 ) = u n 1 (t n 1 ), d t u n,f (t n 1 ) = d t u n 1 (t n 1 ). w V(K),

14 11/21 One-level MHM method For n = 1,, n T, find λ n H Λ H such that (µ H, u n,λ H H where, for each K I n, u n,λ H and u n,f H d tt (ρ u n,λ H H satisfy (t n )) Th = (µ H, u n,f H (t n)) Th, µ H Λ H. H satisfy, w) K + (σ(u n,λ H H ), ε(w)) K = = (λ n H, w) K, w V(K), u n,λ H H (t n 1 ) = d t u n,λ H H (t n 1 ) = 0, d tt (ρ u n,f H, w) K + (σ(u n,f H ), ε(w)) K = = (f, w) K, w V(K), u n,f H (t n 1) = u n 1 H d t u n,f H (t n 1) = d t u n 1 H (t n 1), (t n 1). (1) (2)

15 The MHM method For n = 1,, n T, find λ n R m Λ such that A λ n = f n, where A ij := (ψ i, u ψ j h ) T H and fi n := (ψ i, u n,f h ) T H. Discretization details Uniform global time partition time independent A. λ n λ n H := m Λ i=1 λn i ψ i, using Galerkin method, and each basis function has support in single face. H (t n) u ψ j h and u n,f H (t n) u n,f h, using Galerkin method in space and Newmark method in time. u n,ψ j Global approximations u n Hh := v n Hh := m Λ j=1 λ n j v ψ j h m Λ j=1 λ n j u ψ j h + un,f h, + v n,f h, σ n Hh K := C ε(u n Hh K ). 12/21

16 12/21 Numerical results

17 13/21 An analytical problem Ω = [0, 1] [0, 1] [0, 1], c µ = 0.4, c λ = 0.4 and ρ = 1. Exact solution u = [u 1, u 2, u 3 ] T is given by u 1 (t, x, y, z) = 1 2 (1 cos(ωt)) sin(2πx) sin(2πy) sin(2πz), u 2 (t, x, y, z) = 1 2 (1 cos(ωt)) sin(2πx) sin(2πy) sin(2πz), u 3 (t, x, y, z) = 1 2 (1 cos(ωt)) sin(πx) sin(πy) sin(πz), where ω = 2µρ 1 π. Homogeneous Dirichlet boundary condition.

18 14/21 Convergence results u-u Hh 0 u-u Hh 1 v-v Hh 0 v-v Hh H σ-σ Hh div O(H) O(H 2 ) O(H 3 ) u-u Hh 0 u-u Hh 1 v-v Hh Δt v-v Hh 1 σ-σ Hh div O(Δt 2 ) Figure : Convergence curves with respect to H (left), using P 1 approximation over the faces, and to Δt (right), using P 3 approximation over the faces.

19 Multiscale scenario 4 Figure : The physical domain divided in layers. Adapted from [de la Puente, 2016]. Problem: Flat case. Single shot. Two-dimensional problem at a slice (inline = 0 m). Free surface boundary condition. 4 de la Puente, J. (2016). HPC4E Seismic Test Suite. Copyright Josep de la Puente (Barcelona Supercomputing Center) Licenced under the Creative-Commons Attribution-ShareAlike 4.0 International License 15/21

20 MHM configuration Edges divided in 4 equally spaced parts. P 1 approximation in each part of edge. Local meshes with 4096 P 3 elements (h = H/64). Figure : Non-aligned global mesh with 341 elements (H = L/8) and the boundary of the layers in white. 16/21

21 17/21 MHM solution Figure : Isolines of u Hh solution at t = 0.15.

22 Global energy E n Hh := 1 2 K T h [ K ρv n Hh v n Hh + K ] C ε(u n Hh) : ε(u n Hh) t E Hh Figure : Total energy history in the multiscale case. 18/21

23 Comparison with a Galerkin solution Figure : Isovalues of the von Mises stress for MHM (top) and a reference solution (bottom), at t = /21

24 20/21 Final remarks A new multiscale method was proposed for 2D and 3D elastodynamic problems, following the MHM methodology. The method allows, in a simple way, local mesh refinement. The results show linear convergence in time and super convergence in space for displacement, velocity and stress. Ongoing work: Local time step implementation. Validation of absorbing boundary conditions. Mathematical theory for the method.

25 21/21 Acknowledgements Thank you!

26 22/21 Referências I [de la Puente, 2016] de la Puente, J. (2016). HPC4E Seismic Test Suite. Copyright Josep de la Puente (Barcelona Supercomputing Center) Licenced under the Creative-Commons Attribution-ShareAlike 4.0 International License. [Harder et al., 2016] Harder, C., Madureira, A. L., and Valentin, F. (2016). A Hybrid-Mixed Method for Elasticity. ESAIM: M2AN, 50(2): [Harder et al., 2013] Harder, C., Paredes, D., and Valentin, F. (2013). A family of Multiscale Hybrid-Mixed finite element methods for the Darcy equation with rough coefficients. J. Comput. Phys., 245:

27 23/21 Referências II [Harder and Valentin, 2016] Harder, C. and Valentin, F. (2016). Foundations of the MHM method, volume 114 of Lecture Notes in Computational Science and Engineering, chapter Building Bridges: Connections and Challenges in Modern Approaches to Numerical Partial Differential Equations, pages Springer. [Pereira and Valentin, 2017] Pereira, W. S. and Valentin, F. (2017). A Locking-Free MHM Method for Elasticity. In Proceeding Series of the Brazilian Society of Computational and Applied Mathematics, volume 5. [Raviart et al., 1983] Raviart, P. A., Thomas, J. M., Ciarlet, P. G., and Lions, J.-L. (1983). Introduction à l analyse numérique des équations aux dérivées partielles. Masson, Paris.