# On explicit error estimates for the elastic wave propagation in heterogeneous media

Wen Xu, Régis Cottereau, Bing Tie, Denis Aubry

► To cite this version:

Wen Xu, Régis Cottereau, Bing Tie, Denis Aubry. On explicit error estimates for the elastic wave propagation in heterogeneous media. 12e Colloque national en calcul des structures, CSMA, May 2015, Giens, France. hal-01516629

HAL Id: hal-01516629

https://hal.science/hal-01516629v1

Submitted on 2 May 2017

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.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons CC0 1.0 - Universal - International License
# On explicit error estimates for the elastic wave propagation in heterogeneous media

W. Xu, R. Cottereau, B. Tie, D. Aubry

MSSMat, Ecole Centrale Paris, [wen.xu,regis.cotereau,bing.tie]@ecp.fr

**Abstract** — This work deals with explicit a posteriori error estimates for elastic wave propagation in heterogeneous media. Based on some previous works on acoustic problems, the proposed explicit error estimator is derived in a non-natural $L^\infty$ norm by using several reconstructions of numerically calculated fields. The effectivity of the estimator on uniform meshes and adaptive meshes is studied numerically. The results indicate that the error estimator gives an upper bound to the true error. Improvements for the reconstruction in time and in space are proposed.

**Keywords** — A posteriori error estimate, wave equation, $L^\infty$ norm, effectivity

## 1. A posteriori error estimate for wave equation

### 1.1. Elastic wave propagation model problem

For $t \in (0, T)$ and $x \in \Omega$, we consider the elastic wave propagation problem in an elastic solid $\Omega$: finding $(u(x,t) \times v(x,t)) \in X^2 \times N^2$ with $(u_0, v_0) \in H_0^1(\Omega) \times L^2(\Omega)$ and $f \in N^2$ such that

$$ egin{cases} 
\rho \partial_t v(x,t) - \nabla \cdot \sigma(u) = f & \forall (x,t) \in \Omega \times (0, T) \\ 
\mathrm{Div}(\sigma(\partial_t u(x,t) - v(x,t))) = 0 & \forall (x,t) \in \Omega \times (0, T) \\ 
u = 0 & \forall (x,t) \in \partial\Omega \times (0, T) \\ 
u(x, 0) = u_0(x) & \forall x \in \Omega \\ 
v(x, 0) = v_0(x) & \forall x \in \Omega 
\end{cases} 
\quad (1) $$

We define the space of functions $F(0, T; E) = \{u \in E; \text{ for } \forall t \in (0, T) \text{ and } t \to u(x, \cdot) \text{ is in } F\}$. For example, here we denote: $N^2 = L^2(0, T; L^2(\Omega))$, $X^2 = L^2(0, T; V(\Omega))$. Notice that the second-order elastodynamic equation is transformed here to a first-order hyperbolic system (1) and the equivalence of $\partial_t u$ and $v$ is imposed through the equilibrium of internal forces.

### 1.2. Space and time reconstructions

To be solved, the system (1) is discretized in space by the finite element method and in time by a finite difference scheme. The so-obtained numerical solutions are denoted by $(u_{h\tau}, v_{h\tau})$. With respect to different regularities required by corresponding ingredients of the estimator, we take a time-continuous reconstruction $(\bar{u}_{h\tau}, \bar{v}_{h\tau})$ with Hermitian interpolation:

$$ egin{cases} 
\bar{u}_{h\tau}(x,t) := \frac{(t - t^{n+1})(2t + t^{n+1} - 3t^n)}{\tau^n} u_{h\tau}^n(x) + \frac{(t - t^n)(-2t - t^{n+1} + 3t^n)}{\tau^n} u_{h\tau}^{n+1}(x) \\ 
+ \frac{(t - t^n)(t^{n+1} - t)^2}{\tau^n} v_{h\tau}^n(x) + \frac{(t - t^{n+1})(t^n - t)^2}{\tau^n} v_{h\tau}^{n+1}(x) \\ 
\bar{v}_{h\tau}(x,t) := \frac{t^{n+1} - t}{\tau^n} v_{h\tau}^n(x) + \frac{t - t^n}{\tau^n} v_{h\tau}^{n+1}(x) - \frac{(t - t^n)(t^{n+1} - t)^2}{\tau^n} \frac{v_{h\tau}^{n+2} - 2v_{h\tau}^{n+1} + v_{h\tau}^n}{(\tau^n)^2} (x) 
\end{cases} 
\quad (2) $$
and two field reconstructions in space $(\tilde{\sigma}_{h\tau}, \delta_{h\tau})$:

$$ \tilde{\sigma}_{h\tau}(\cdot, t) \in H(div, \Omega), \quad \forall t \in I_n, \forall 1 \le n \le N \quad (3) \\ (\partial_t \bar{v}_{h\tau} - \nabla \cdot \tilde{\sigma}_{h\tau}^n, 1)_K = (f, 1)_K \quad \text{in } I_n $$

$$ \tilde{\delta}_{h\tau}(\cdot, t) \in H(div, \Omega), \quad \forall t \in I_n, \forall 1 \le n \le N \quad (4) \\ (\partial_{tt} \bar{v}_{h\tau} - \nabla \cdot \tilde{\delta}_{h\tau}, 1)_K = (\partial_t f, 1)_K \quad \text{in } I_n $$

## 1.3. Explicit error bound

The residual method of a posteriori error estimates is exploited with all these reconstructions mentioned. We define the indicator as follows:

- the residual and flux indicator:

$$ \|\eta_R + \eta_{DF}\|_{N^2} = \left\{ \sum_{n=1}^{N} \int_{I_n} \sum_{K \in \mathcal{J}_h} (\eta_{R,K}^n + \eta_{DF,K}^n)^2 dt \right\}^{\frac{1}{2}} \quad (5) $$

with $\eta_{R,K}^n = \frac{c_P h_K c_K}{\sqrt{\lambda_{min}}} \|f - \partial_t \bar{v}_{h\tau} + \nabla \cdot \tilde{\sigma}_{h\tau}^n\|_{0,K}$ in $I_n$ (which represents dynamic volume residues in the interior of each element), $\eta_{DF,K}^n = \frac{1}{\sqrt{\lambda_{min}}} \|\tilde{\sigma}_{h\tau}^n - \sigma(\bar{u}_{h\tau})\|_{0,K}$ in $I_n$ (which represents constraints vectors' jumps of solutions before reconstruction); where $I_n = (t^n, t^{n+1}]$, $\lambda_{min}$ is the smallest eigenvalue of the fourth order tensor $C$, $C_P$ is the Poincaré constant and $C_K$ is the Korn constant;

- the derived residual and flux indicator:

$$ \|\eta_{DE} + \eta_{GV}\|_{N^2} = \left\{ \sum_{n=1}^{N} \int_{I_n} \sum_{K \in \mathcal{J}_h} (\eta_{DE,K}^n + \eta_{GV,K}^n)^2 dt \right\}^{\frac{1}{2}} \quad (6) $$

with $\eta_{DE,K}^n = \frac{c_P h_K c_K}{\sqrt{\lambda_{min}}} \|-\partial_t f\|_{I^n} + \partial_{tt} \bar{v}_{h\tau} - \nabla \cdot \tilde{\sigma}_{h\tau}^n\|_{0,K}$ in $I_n$ (which represents velocity of dynamic volume residues) and $\eta_{GV,K}^n = \frac{1}{\sqrt{\lambda_{min}}} \|-\tilde{\delta}_{h\tau} + \sigma(\partial_t \bar{u}_{h\tau})\|_{0,K}$ (which represents velocity of constraints vectors' jumps);

- the nonconformity indicator:

$$ \|\eta_{VN}\|_{X^2} = \|\partial_t \bar{u}_{h\tau} - \bar{v}_{h\tau}\|_{X^2}; \quad (7) $$

- the two initial condition indicators:

$$ \eta_{CI,1} = \frac{1}{\sqrt{2}} \|v_0 - \bar{v}_{h\tau}^0\|_{0,\Omega}; \quad (8) $$

$$ \eta_{CI,2} = \frac{1}{\sqrt{2}} \|u_0 - \bar{u}_{h\tau}^0\|_{a,\Omega}. $$

We can derive the error estimator with these indicators:
$$ \sqrt{\|v - \bar{v}_{h\tau}\|_{N^\infty}^2 + \|u - \bar{u}_{h\tau}\|_{X^\infty}^2} \qquad (9) $$

$$ \le 2 \left\{ \begin{aligned} & \sup_n \{\|\eta_R^n + \eta_{DF}^n\|_{0,\Omega}\} + \|\eta_R^0 + \eta_{DF}^0\|_{0,\Omega} + \sqrt{T}(\|\eta_{VN}\|_{X^2}^2 + \|\eta_{DE} + \eta_{GV}\|_{N^2}^2) \\ & + \sqrt{2}\{\|\eta_R + \eta_{DF}\|_{N^2}\|\eta_{VN}\|_{X^2} + \eta_{CI,1}^2 + \eta_{CI,2}^2\}^{\frac{1}{2}} \end{aligned} \right\} $$

We notice that all the indicators are fully computable from $(u_{h\tau}, v_{h\tau})$ and we can distinguish and estimate separately the different error components.

## 2. Numerical results

A one-dimensional elastic rod is considered as a numerical example. The left end of the rod is submitted to a pressure loading that evolves in time as a truncated Ricker-type signal while its right end is left free. See Figure 1.

<figure><img src="image_2.png" /><figcaption>Figure 1 – One-dimensional rod example</figcaption></figure>

We consider two cases: homogeneous material with uniform mesh and heterogeneous material with uniform and adaptive meshes; then we compare the estimated and actual errors in heterogeneous case and we analyze the different terms of estimator: term1 = $2\sup_n \{\|\eta_R^n + \eta_{DF}^n\|_{0,\Omega}\}$; term2 = $2\|\eta_R^0 + \eta_{DF}^0\|_{0,\Omega}$; term3 = $2\sqrt{T}\|\eta_{VN}\|_{X^2}$; term4 = $2\sqrt{T}\|\eta_{DE} + \eta_{GV}\|_{N^2}$; term5 = $\sqrt{2}\|\eta_R + \eta_{DF}\|_{N^2}\|\eta_{VN}\|_{X^2}$.

<figure><img src="image_3.png" /><figcaption>Figure 2.1 – Heterogeneous case with uniform mesh</figcaption></figure>
<figure><img src="image_4.png" /><figcaption>Figure 2.2 – Heterogeneous case with adaptive mesh</figcaption></figure>

In Figure 3(a), we notice that the estimated error gives a computable upper bound on the unknown error to actual error in all cases; but obviously, the effectivity index, as the ratio of the estimated and actual error, is too large and it cannot converge to 1 as the computational effort grows. In Figure 3(b), we find that the first term $||η_{DE} + η_{GV}||_N^2$ and the third term $sup\{||η_R^n + η_{DF}^n||_{0,Ω}\}$ contribute the most to the estimated error and their performance is determined by the reconstructions in time and in space.

### 3. Conclusion

We develop an explicit a posteriori error estimator for elastic wave propagation in heterogeneous media. Further research works are needed by taking account higher order reconstructions in time $(\bar{u}_{hτ}, \bar{v}_{hτ})$ and in space $(\tilde{\sigma}_{hτ}, \tilde{\delta}_{hτ})$, which are two key points to improve the numerical results.

### References

[1] A. Ern and M. Vohralík. A posteriori error estimation based on potential and flux reconstruction for the heat equation, SIAM J. Numer. Anal., 198-223, 2010.

[2] C. Johnson. Discontinuous Galerkin finite element methods for second order hyperbolic problems, Comput. Methods Appl. Mech. Engrg., 117-129, 1993.

[3] D. Aubry, D. Lucas, B. Tie. Adaptive strategy for transient/coupled problems applications to thermoelasticity and elastodynamics, Comput. Methods Appl. Mech. Engrg., 41-50, 1999.

[4] E. H. Georgoulis, O. Lakkis and C. Makridakis. A posteriori $L^∞(L^2)$-error bounds for finite element approximations to the wave equation, IMA J. Numer. Anal., 1245-1264, 2013.

[5] F. Ibrahima. Estimation d'erreur pour des problèmes de propagation d'ondes en milieux élastiques linéaires hétérogènes [Error estimate for wave propagation problems in linear elastic heterogeneous media], Internship Report, Ecole Centrale Paris, 2011.