The MHM method for the second-order elastodynamic model

(1)

HAL Id: hal-01840081

https://hal.inria.fr/hal-01840081

Preprint submitted on 31 Jul 2018

HAL is a multi-disciplinary open access

archive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

The MHM method for the second-order elastodynamic

model

Weslley da Silva Pereira, Claire Scheid, Frédéric Valentin

To cite this version:

Weslley da Silva Pereira, Claire Scheid, Frédéric Valentin. The MHM method for the second-order elastodynamic model. 2018. �hal-01840081�

(2)

The MHM method for the second-order

elastodynamic model

Weslley da Silva Pereira1

Joint work with

Claire Scheid2 and Frédéric Valentin1 1_{Laboratório Nacional de Computação Científica (BR)}

2_{Inria Sophia Antipolis - Mediterranée (FR)}

July 13, 2018

(3)

Outline

Introduction Hybridization procedure The MHM method Numerical results An analytical problem Multiscale problem

Final conclusions and remarks

(4)

Introduction

(5)

Elastodynamic problem

Equation of motion and the constitutive equation

ρ∂ttu − ∇· σ =f, σ =Cε , ε :=

∇u + (∇u)T

2 with u(x, 0) = u0(x), ∂tu(x, 0) = v0(x) and u|∂Ω= 0.

Figure : Seismic imaging technique.

Figure : Layered media.

(6)

MHM - Multiscale Hybrid-Mixed method

1

Figure : TH with elements K .

h

Figure : Local mesh ThK.

1

C. Harder, D. Paredes, and F. Valentin.A family of Multiscale Hybrid-Mixed finite element methods for the Darcy equation with rough coefficients.J. Comput. Phys., 245:107–130, 2013

(7)

Hybridization procedure

(8)

1. Partition the domain Ω × [0, T ]

V := w ∈ L2(Ω) : w |K ∈ H1(K ) ∀K ∈ TH , Λ := τ nK|∂K, ∀K ∈ TH : τ ∈ H(div; Ω; S) . t t t t t t t t Ω t t 0 1 1 2 n-1 n n n+1 n -1 n T T + + + + -I_n K I =_n (t , t )_n-1 _n T_H H Notation: • (·, ·)K - L2(K ) product;

• h·, ·i_∂K - dual product

H−12(∂K ) × H 1 2(∂K ); • (·, ·)∂T_H:=P K ∈THh·, ·i∂K; • u(n):= u|In. 5/20

(9)

2. A hybrid variational problem

Find (u, λ) that, for n = 1, · · · , nT, satisfies

           dtt(ρu(n), w )K+ (σ(u(n)), ε(w ))K−hλ(n), w i∂K =(f , w )K, ∀w ∈ H1(K ), (µ, u(n))∂T_H= 0, ∀µ ∈ Λ, u(n)(tn−1) =u(n−1)(tn−1), dtu(n)(tn−1) =dtu(n−1)(tn−1),

where un:= u(n)(tn) = u(tn) and vn:= dtu(n)(tn) = dtu(tn) .

Theorem

The problem is well-defined for

f ∈ H1(0, T ; L2(Ω)), u0, v0 ∈ H10(Ω) and σ(u0) ∈ H(div; Ω; S).

The solution is unique in

C (0, T ; V) ∩ C1(0, T ; L2(Ω)) × L2(0, T ; Λ). Moreover, u solves the original problem and

λ = σ(u)nK on ∂K , ∀K ∈ TH.

(10)

3. Split the solution

u(n)=Tn(λ(n))+Tˆn(zn), zn:= (un−1, vn−1, f ).

where Tn and ˆTn are linear operators

Tn: L2(In; Λ) → L2(In; L2(Ω)), ˆ

Tn: V × L2(Ω) × L2(In; L2(Ω)) → C (In; V) ∩ C1(In; L2(Ω)).

For each K × In, u(n)_λ := Tn(λ(n)) is the solution of

( dtt(ρ u (n) λ , w )K+ (σ(u (n) λ ), ε(w ))K= hλ(n), w i∂K, ∀w ∈ H1(K ), u(n)_λ (tn−1) = dtu(n)_λ (tn−1) = 0,

and u(n)z := ˆTn(zn) is the solution of      dtt(ρ u(n)z , w )K+ (σ(u (n) z ), ε(w ))K = (f , w )K, ∀w ∈ H1(K ), u(n)z (tn−1) = un−1, dtu(n)z (tn−1) = vn−1. 7/20

(11)

Global-Local formulation

2

Global-Local formulation

Find λ that, for n = 1, · · · , nT, satisfies

(µ, Tn(λ(n)))∂TH = −(µ, ˆTn(zn))∂TH, ∀µ ∈ Λ,

where

zn:= (un−1, vn−1, f ),

un:= Tn(λ(n))|tn+ ˆTn(zn)|tn,

vn:= dtTn(λ(n))|tn+ dtTˆn(zn)|tn,

This is equivalent to the hybrid problem using

u(n):= Tn(λ(n)) + ˆTn(zn).

2

A. T. Gomes, D. Paredes, W. Pereira, R. Souto, and F. Valentin.A Multiscale Hybrid-Mixed Method for the Elastodynamic Model with Rough Coefficients.In XXXVIII CILAMCE, 2017

(12)

The MHM method

(13)

One-level MHM method

Global discretization • ΛH ⊂ Λ and basis {ψi} mΛ i =1. • _P₀(In; ΛH) ⊂ L2(In; Λ). u(n)_H := mΛ X j=1 αnjTn(ψj) + ˆTn(zH,n)

For n = 1, · · · , nT , find αn∈ RmΛ such that mΛ X j=1 αnj(ψi, Tn(ψj)|tn)∂TH=−(ψi, ˆTn(zH,n)|tn)∂TH, ∀i = 1, · · · , mΛ, where zH,n:= (uH,n−1, vH,n−1, f ), uH,n:= u (n) H (tn), vH,n:= dtu (n) H (tn)

and initial conditions

uH,0:= u0 and vH,0:= v0.

(14)

Two-level MHM method

For n = 1, · · · , nT , find αn∈ RmΛ such that mΛ X j=1 αnj(ψi, η (n) j )∂TH =−(ψi, η (n) 0 )∂TH, ∀i = 1, · · · , mΛ, where zh,n:= (uh,n−1, vh,n−1, f ) , uh,0≈ u0 and vh,0≈ v0 . 10/20

(15)

Two-level MHM method

• uh,n ∈ H1(TH), ∀n = 1, · · · , nT (H1(Ω)-non-conforming)

• Strong symmetric σh

• High-order local approximations in space Vh(K )

• Use of different local parameters (h, p) and

(∆τ = ∆t_nτ, γ, β) to different K × In

• Energy-conservative local method ⇒ energy-conservative

MHM

(16)

Numerical results

(17)

An analytical problem

Isotropic homogeneous media with exact solution u = [u1, u2, u3]T:

( _u

1(t, x , y , z) = 1₂(1 − cos(ωt)) sin(2πx ) sin(2πy ) sin(2πz) ,

u2(t, x , y , z) = −1₂(1 − cos(ωt)) sin(2πx ) sin(2πy ) sin(2πz) ,

u3(t, x , y , z) = 1₂(1 − cos(ωt)) sin(πx ) sin(πy ) sin(πz) .

ΛH :=µ_H∈ Λ : µH|F ∈ [P`(F )] 2 for all F ∈ ∂TH Vh(K ) := vh∈ C (K ) : vh|τ ∈ [Pk(τ )]3 for all τ ∈ ThK 12/20

(18)

Convergence results

Figure : Experiments with ` = 3 and k = 5.

10-6 10-5 10-4 10-3 10-2 10-1 100 101 10-1 ₁₀0 H ||u-uh||0 ||u-uh||1 ||v-vh||0 ||v-vh||1 ||σ-σh||div O(H) O(H2) O(H3) 10-5 10-4 10-3 10-2 10-1 100 101 10-1 ₁₀0 Δt ||u-uh||0 ||u-uh||1 ||v-vh||0 ||v-vh||1 ||σ-σh||div O(Δt2)

Error convergence observed numerically:

• O(∆t2) in all norms.

• O(H`+2) in the L2(Ω)-norm, O(H`+1) in the H1(TH)-norm

and O(H`) in the div -norm.

(19)

Multiscale scenario

3

Figure : The physical domain divided in layers. Problem: • Flat case. • Single shot. • Two-dimensional problem at a slice (inline = 0 m).

• Free surface boundary

condition.

3

J. de la Puente.HPC4E Seismic Test Suite, 2016.Copyright Josep de la Puente (Barcelona Supercomputing Center) 2016. Licenced under the Creative-Commons Attribution-ShareAlike 4.0 International License

(20)

MHM configuration

• Edges divided in 4 equally spaced parts.

• _P1 _{approximation in each part of edge.}

• _{Local meshes with 4096 P}3 elements (h = H/64).

(21)

MHM solution

Figure : Isolines of |uh| solution at t = 0.15.

(22)

Global energy

Eh:= 1 2 X K ∈TH (ρvh, vh)K + (σh, ε(uh))K 0 1x10-7 2x10-7 3x10-7 4x10-7 5x10-7 0 0.5 1 1.5 2 t E_h 17/20

(23)

Comparison with a Galerkin solution

Figure : Isovalues of the von Mises stress for MHM (top) and a reference solution (bottom), at t = 0.15.

(24)

Final conclusions and remarks

• The MHM method provides a natural level of parallelism.

• It allows local mesh refinement and high-order approximations.

• Optimal convergence in time and super convergence in space

for displacement, velocity and stress.

• It conserves energy depending on the local level.

• It is robust on non-aligned mesh with interfaces.

Ongoing work:

• Local time step validation.

• Error and stability analysis.

(25)

Thank you!

Funding and resources:

(26)

[1] J. de la Puente.HPC4E Seismic Test Suite, 2016. Copyright Josep de la Puente (Barcelona Supercomputing Center) 2016. Licenced under the Creative-Commons Attribution-ShareAlike 4.0 International License.

[2] A. T. Gomes, D. Paredes, W. Pereira, R. Souto, and

F. Valentin. A Multiscale Hybrid-Mixed Method for the

Elastodynamic Model with Rough Coefficients. In XXXVIII

CILAMCE, 2017.

[3] C. Harder, A. L. Madureira, and F. Valentin.A Hybrid-Mixed

Method for Elasticity. ESAIM: M2AN, 50(2):311–336, 2016.

[4] C. Harder, D. Paredes, and F. Valentin.A family of Multiscale

Hybrid-Mixed finite element methods for the Darcy equation

with rough coefficients. J. Comput. Phys., 245:107–130, 2013.

(27)

[5] C. Harder and F. Valentin.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 401–433.Springer, 2016.

[6] W. S. Pereira and F. Valentin.A Locking-Free MHM Method

for Elasticity. In Proceeding Series of the Brazilian Society of

Computational and Applied Mathematics, volume 5, 2017.

[7] P. A. Raviart, J. M. Thomas, P. G. Ciarlet, and J.-L. Lions.

Introduction à l’analyse numérique des équations aux dérivées partielles. Masson, Paris, 1983.