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An optimized finite element extrapolating method for 2D viscoelastic wave equation
- Hong Xia^{1} and
- Zhendong Luo^{2}Email authorView ORCID ID profile
https://doi.org/10.1186/s13660-017-1496-7
© The Author(s) 2017
Received: 10 August 2017
Accepted: 4 September 2017
Published: 12 September 2017
Abstract
In this study, we first present a classical finite element (FE) method for a two-dimensional (2D) viscoelastic wave equation and analyze the existence, stability, and convergence of the FE solutions. Then we establish an optimized FE extrapolating (OFEE) method based on a proper orthogonal decomposition (POD) method for the 2D viscoelastic wave equation and analyze the existence, stability, and convergence of the OFEE solutions and furnish the implement procedure of the OFEE method. Finally, we furnish a numerical example to verify that the numerical computing results correspond with the theoretical ones. This signifies that the OFEE method is feasible and efficient for solving the 2D viscoelastic wave equation.
Keywords
MSC
1 Introduction
Let \(\Theta\subset\boldsymbol {R}^{2}\) be a bounded convex polygonal domain with a smooth boundary ∂Θ. We consider the following initial-boundary value problem:
Problem 1
Problem 1 is referred to as a system of viscoelastic wave equations. It has some special and significant physical backgrounds. For instance, it can be used to describe the wave propagation phenomena of actual vibration through a viscoelastic medium (see, e.g., [1, 2]). Although the existence and uniqueness of its analytic solution have been proved (see, e.g., [3–6]), because the viscoelastic wave equation in the real-world engineering applications usually has complex known data or computed domains, the analytical solution cannot be generally solved, so one has to find its solutions numerically. For more than 30 years, it has been attentively studied and many numerical methods for the viscoelastic wave equation have been developed (see, e.g., [5–8]). Among all numerical methods, the finite element (FE) method is considered to be one of the calculating numerical methods with the best theory for the two-dimensional (2D) viscoelastic wave equation (see [8, 9]). Nevertheless, the classical FE methods for the 2D viscoelastic wave equation are some macroscale systems of equations including lots of unknowns, i.e., degrees of freedom, so entail very large computational load in real-world engineering applications. As a consequence, an important issue is how to greatly lessen the number of unknowns of the classical FE methods to reduce the computational load, ease the truncated error amassing, and save CPU time in the numerical computation, while preserving the desired FE solution accuracy.
It has been proved by lots of numerical studies (see, e.g., [10–23]) that the proper orthogonal decomposition (POD) method is a very useful tool to reduce the number of unknowns for numerical models and ease the truncated error amassing in numerical calculations. But most existing reduced-order models, as mentioned, were established via the POD basis formed from the classical numerical solutions at all time nodes, before repetitively computing the reduced-order numerical solutions at the same time nodes, which were some valueless repetitive calculations. Since 2014, some reduced-order FE extrapolating methods based on the POD method for partial differential equations have been established successively by Luo’s team (see, e.g., [24–26]) in order to avert the valueless repeated computations.
However, as far as we know, there has not been any report that the POD method is used to reduce the number of unknowns in the classical FE method for the 2D viscoelastic wave equation. Therefore, in this article, we devote ourselves to building an optimized FE extrapolating (OFEE) method that includes very few unknowns but maintains desired accuracy via the POD method, analyzing the existence, stability, and convergence of the OFEE solutions and verifying the efficiency and feasibility of the OFEE method by some numerical simulations.
The main distinctions between the OFEE method and the other existing reduced-order FE extrapolating methods built on the POD method (see, e.g., [24–26]) consist in the fact that the viscoelastic wave equation not only contains three second-order derivative terms of time and of spatial variables but also includes two mixed derivative terms of time (first-order) and spatial variables (second-order) so that either the modeling of the OFEE method or the demonstration of the existence, stability, and convergence of the OFEE solutions faces more difficulties and requires more techniques than the existing other aforementioned reduced-order FE extrapolating methods. However, the OFEE method has some specific applications. Though an optimized splitting positive definite mixed FE extrapolation (OSPDMFEE) model based on the POD technique for the 2D viscoelastic wave equation is developed in [27], it has three unknown functions and the OSPDMFEE model has more degrees of freedom than the current OFEE format, so that its theoretical analysis and numerical simulations have more difficulties than the current OFEE method. It is worth mentioning that we can discuss the existence, stability, and convergence of the reduced-order FE solutions by means of the classical FE theory. Especially, the OFEE method only employs the classical FE solutions at the initial very few time nodes to formulate the POD basis and build the OFEE format so that it does not have repetitive calculations, such as done in references [24–27]. Consequently, it is a development and an improvement of the existing aforementioned ones (see, e.g., [10–23]).
The remaining content of the article is organized as follows. In Section 2, we first present the classical FE method for the 2D viscoelastic wave equation and analyze the existence, stability, and convergence of the classical FE solutions. In Section 3, we develop the OFEE method via the POD method for the 2D viscoelastic wave equation, analyze the stability and convergence of the OFEE solutions, and furnish the implement procedure of the OFEE method. Next, in Section 4, we use some numerical simulations to verify the efficiency and feasibility of the OFEE method. Finally, in Section 5, we summarize our main conclusions.
2 The classical FE method for the 2D viscoelastic wave equation
2.1 Generalized solution for the 2D viscoelastic wave equation
The following arisen Sobolev spaces as well as their norms are well known (see [28]).
For convenience, we write \(U=H^{1}_{0}(\Theta)\). Thus, by using Green’s formula for the 2D viscoelastic wave equation, we obtain the following variational formulation:
Problem 2
For Problem 2, we have the following result.
Theorem 1
Proof
Because Problem 2 is a system of linear equations as regards the unknown function u, in order to prove the existence and uniqueness of solutions for Problem 2, it is necessary to prove that Problem 2 has only the zero solution when \(f(x,y,t)=\varphi _{0}(x, y)=\varphi_{1}(x,y)=0\).
2.2 Semi-discrete format as regards time for the 2D viscoelastic wave equation
Let N be a positive integer, \(\Delta t=T/N\) the time step size, and \(t_{i}=i\Delta t\). If we use \((u^{n+1}-u^{n})/ (2\Delta t)\) to approximate \(u_{t}\) and \((u^{n+1}-2u^{n}+u^{n-1})/\Delta t^{2}\) to approximate \(u_{tt}\) for the 2D viscoelastic wave equation, we obtain the following semi-discrete formulation of time:
Problem 3
For Problem 3, we have the following.
Theorem 2
Proof
Because Problem 3 is a system of linear equations as regards the unknown function \(u^{n}\), in order to prove the existence and uniqueness of solutions for Problem 3, it is necessary to prove that Problem 3 has only the set of zero solutions when \(f(x,y,t)=\varphi_{0}(x, y)=\varphi_{1}(x,y)=0\).
2.3 Classical fully discrete FE method for the 2D viscoelastic wave equation
Thus, the fully discrete FE formulation for the 2D viscoelastic wave equation (1) is as follows:
Problem 4
For Problem 4, we have the following.
Theorem 3
Proof
(i) The existence and uniqueness of the solution sequence for Problem 4 .
Problem 5
(ii) The stability of the solution sequence \(\{u_{h}^{n}\}_{n=1}^{N}\) for Problem 4 , i.e., inequality ( 23 ).
(iii) Convergence of the solution sequence for Problem 4 .
Remark 1
The full FE formulation Problem 4 is directly built from the semi-discrete formulation Problem 3 with respect to time such that one can bypass the semi-discrete formulation with respect to spatial variables and its theoretical analysis becomes simpler. Thus, as long as \(f(x,y,t)\), \(\varphi_{0}(x, y)\), \(\varphi_{1}(x, y)\), ε, γ, time step k, the spatial mesh size h, and the FE subspace \(U_{h}\) are assigned, we attain the solution sequence \(\{u_{h}^{n} \}_{n=1}^{N}\subset U_{h}\) by solving Problem 4. We take the subsequence \(\{u_{h}^{n}\} _{n=1}^{L}\) from the initial L solutions of \(\{u_{h}^{n} \}_{n=1}^{N}\) as snapshots (in general, \(L\ll N\) and \(\sqrt{L}<5\), for example, \(L=20\), \(N=200\)).
3 The OFEE format for the 2D viscoelastic wave equation
3.1 Formulations of the POD basis and establishment the OFEE format
Proposition 4
Lemma 5
Thus, by means of \(U^{d}\), the OFEE format for the 2D viscoelastic wave equation is described as follows:
Problem 6
Remark 2
It is easily seen that Problem 4 at each time node includes \(N_{h}\) unknowns (where \(N_{h}\) is the number of vertices of triangles in \(\Im_{h}\)), whereas Problem 6 at the same time node contains only d unknowns (\(d\ll l\leq L\ll N\ll N_{h}\)). For real-world engineering problems, the number \(N_{h}\) of vertices of triangles in \(\Im _{h}\) can easily reach a few millions, while d is only the number of the major eigenvalues and is very small (for example, in Section 4, \(d=6\), but \(N_{h}\geq4\times10^{4}\)). Problem 6 here is the OFEE format for the 2D viscoelastic wave equation. In particular, Problem 6 employs only the initial few known L solutions of Problem 4 used to extrapolate other \(N-L\) solutions, and has no repetitive computations. The first L OFEE solutions are obtained by projecting the first L classical FE solutions into the POD basis, while the other remaining (\(N-L\)) OFEE solutions are obtained by extrapolating and iterating equation (44). Therefore, it is completely different from the existing POD-based reduced-order formulations.
3.2 The error estimations of the OFEE solutions
In the following, we employ the classical FE method to deduce the error estimations of OFEE solutions for the 2D viscoelastic wave equation. We have the following main result.
Theorem 6
Proof
(a) The existence and uniqueness of solutions \(u_{d}^{n}\) for Problem 6 .
When \(n=1,2, \ldots, L\), it is obvious that Problem 6 has a unique solution subset \(\{u_{d}^{n}\}_{n=1}^{L}\) obtained by (43).
(b) The stability of the sequence of solutions \(u_{d}^{n}\) for Problem 6 .
(c) The convergence of the sequence of solutions \(u_{d}^{n}\) for Problem 4 .
Remark 3
- (1)
It is known from Theorem 6 that, in order to not adversely affect accuracy, it is necessary to take L as \(L\ll N\), for example, we usually take L such that \(\sqrt{L}<5\). Thus, it is unnecessary to extract total transient solutions at all time nodal points \(t_{n}\) as snapshots such as done in [19, 20].
- (2)The error \((L\sum_{j=d+1}^{l}\lambda_{j} )^{1/2}\) in Theorem 6 gives some indication as to how to choose the number d of the POD basis, namely, it is only necessary to meet$$\Biggl(L\sum_{j=d+1}^{l} \lambda_{j} \Biggr)^{1/2}\leq\max\bigl\{ \Delta t^{2}, h^{k}\bigr\} . $$
3.3 The implement procedure of the OFEE format
Solving the OFEE format, i.e., Problem 6, requires the following seven steps:
Step 2. Formulate the snapshot matrix \(\boldsymbol {A}=({A}_{ij})_{L\times L}\), where \({A}_{ij}=(\nabla u_{h}^{i}, \nabla u_{h}^{j})\) and \((\cdot,\cdot)\) is the \(L^{2}\)-inner product.
Step 3. Find the eigenvalues \(\lambda_{1}\geq\lambda_{2}\geq \cdots\geq\lambda_{l}>0\) (\(l=\operatorname{dim}\{u_{h}^{n}: 1\leq n\leq L\}\)) of A and the corresponding eigenvectors \(\boldsymbol {v}^{j}=(a_{1}^{j},a_{2}^{j},\ldots, a_{L}^{j})\) (\(j=1,2,\ldots,l\)).
Step 4. For the error \(\delta=O(\Delta t^{2},h^{k})\) needed, decide the number d of the POD basis satisfying \((L\sum _{j=d+1}^{l}\lambda_{j})^{1/2}\leq\delta\).
Step 5. Produce the POD basis \(\psi_{j}=\sum _{i=1}^{L}a_{i}^{j}u_{h}^{i}/{\sqrt{L\lambda_{j}}}\) (\(j=1,2,\ldots,d\)).
Step 7. If \(\|u_{d}^{n-1}- u_{d}^{n}\|_{1}\geq\| u_{d}^{n}-u_{d}^{n+1}\|_{1}\) (\(n=L, L+1,\ldots, N-1\)), then \(u_{d}^{n}\) (\(n=1,2, \ldots, N\)) are the OFEE solutions for Problem 6 satisfying the desired accuracy. Else, i.e., if \(\|u_{d}^{n-1}- u_{d}^{n}\|_{1} <\|u_{d}^{n}- u_{d}^{n+1}\|_{1}\) (\(n=L, L+1,\ldots, N-1\)), let \(W_{i}=u_{d}^{i}\) (\(i=n-L, n-L+1, \ldots, n-1\)) and return to Step 2.
Remark 4
Though the OFEE solutions of Problem 6 are theoretically ensured with an accuracy of order \(O(\Delta t^{2},h^{k})\) (if \(\Delta t=O(h)\)), due to error accumulation in the computational process, the actual numerical solutions may contain a larger error than theoretically predicted. Therefore, in order to obtain numerical solutions with the desired computing accuracy, it is best to add Step 7; if the computing accuracy is unsatisfactory, improvements of numerical solutions can be made by renewing the snapshots and the POD basis. This explains why the OFEE format is superior to the classical SPDMFE method.
4 Numerical simulations
In this section, we furnish a numerical example to illustrate that the results of numerical computation are concordant with our theoretical analysis and also demonstrate the feasibility and efficiency of the OFEE format for the 2D viscoelastic wave equation.
We first divide the domain Θ̄ into \(200\times200\) small squares with side length \(\triangle x=\triangle y=10^{-2}\). Then we link the diagonal of the square to divide each square into two triangles and each in the same direction. Further, we adopt local refining meshes such that the scale of meshes on \([0.65, 1.3]\times[2, 2.03]\) and nearby \((x,2)\) (\(0\leq x \leq2\)) are one-third of the meshes nearby \((x,0)\) (\(0\leq x \leq2\)), forming the triangularization \(\Im_{h}\). Thus \(h=\sqrt{2}\times 10^{-2}\). In order to satisfy \(k=O(h)\), we take the time step size \(k=10^{-2}\). The MFE space \(U_{h}\) is taken as piecewise linear polynomials.
5 Conclusions
In this article, we use the POD technique to build the OFEE format for the 2D viscoelastic wave equation. We first extract snapshots from the initial few L (\(L\ll N\)) classical FE solutions for the 2D viscoelastic wave equation. Next, we constitute the POD basis of snapshots by means of the POD method. Then the FE subspaces of the classical FE format are replaced with the subspaces spanning the most main POD bases to build the OFEE formulation for the 2D time-dependent conduction-convection problem. Finally, we deduce the existence, uniqueness, stability, and convergence of the OFEE solutions of the 2D viscoelastic wave equation and furnish the implement procedure for the OFEE format. Comparing the numerical simulation errors with the theoretical errors we have verified that the theoretical errors are concordant with the computing errors, thus validating both the feasibility and efficiency of the OFEE format.
Declarations
Acknowledgements
This research was supported by the National Science Foundation of China (grant 11671106) and the Fundamental Research Funds for the Central Universities (grant 2016MS33).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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