Proper Generalized Decomposition with time adaptive space separation for transient wave propagation problems in separable domains

https://doi.org/10.1016/j.cma.2021.113755Get rights and content

Highlights

  • Wave propagation is described with a time adaptive space separation.

  • A Proper Generalized Decomposition algorithm is ran at each time step.

  • The number of modes stabilizes when the waves fill the domain.

  • The computation time per time step linearly increases with the mesh refinement.

Abstract

Transient wave propagation problems may involve rich discretizations, both in space and in time, leading to computationally expensive simulations, even for simple spatial domains. The Proper Generalized Decomposition (PGD) is an attractive model order reduction technique to address this issue, especially when the spatial domain is separable. In this work, we propose a space separation with a time adaptive number of modes to efficiently capture transient wave propagation in separable domains. We combine standard time integration schemes with this original space separated representation for empowering standard procedures. The numerical behavior of the proposed method is explored through several 2D wave propagation problems involving radial waves, propagation on long time analyses, and wave conversions. We show that the PGD solution approximates its standard finite element solution counterpart with acceptable accuracy, while reducing the storage needs and the computation time (CPU time). Numerical results show that the CPU time per time step linearly increases when refining the mesh, even with implicit time integration schemes, which is not the case with standard procedures.

Introduction

Transient wave propagation analyses are often necessary to fully understand the dynamic behavior of a mechanical system or the details of a mechanical process. This is notably the case for laser shock processes that consist in applying a sudden pressure, up to several GPa, locally on the surface of a material part [1], [2]. The outcomes of understanding these processes at the wave propagation level are important to optimize the laser parameters [3], [4] or to determine the mechanical properties of the medium [5], [6]. These laser shock processes are numerically explored through parametric studies on transient wave propagation problems in plate domains, by varying the loading parameters, the material properties or the mechanical behavior laws [7], [8]. It would be beneficial to reduce the computation time and the storage requirements associated to these numerous simulations.

Many numerical tools have been developed over the past decades to solve transient wave propagation problems. The classical approach consists in using a time integration scheme with the Finite Element Method (FEM). It is however difficult to obtain accurate solutions with this method because of numerical dispersion and dissipation, resulting in non-physical wave velocities, period elongations and amplitude decays [9]. In addition, the error between the exact and the numerical solutions may severely increase as the waves travel in the medium, because of spurious numerical high frequency oscillations related to the Gibb’s phenomenon. The numerical solution can straightforwardly be improved by refining the mesh and by carefully selecting the time step according to both the element size and the selected time integration scheme [10]. Instead of refining the mesh, which increases the computation time, another approach is to select an appropriate time integration scheme with good dissipation and dispersion properties. For instance the Bathe method is an effective implicit time integration scheme for computing accurate transient wave propagation solutions [11], [12]. The Spectral Element Method (SEM) has been developed to reduce the numerical dispersion with respect to the standard FEM [13], [14]. Instead of using standard finite elements, high-order Lagrangian-based finite elements with specific integration points are used to build a diagonal mass matrix. Consequently, the SEM can be effective for computing transient wave propagation solutions in explicit time integration. Similarly, Lagrangian-based finite elements enriched with harmonic functions have been proposed to significantly improve the accuracy with respect to standard finite elements [15]. Finally, approaches based on discontinuous finite element formulations have been developed to solve transient wave propagation problems involving sharp gradients. For instance the Time Discontinuous Galerkin method (TDG) using space–time finite elements is effective for reducing high frequency spurious oscillations [16].

The computation time and the storage requirements of these methods significantly increase when rich discretizations are used to study the details of the wave propagation. Reduced order modeling is an active field of research offering promising approaches to deal with this issue. In the present study we do not intend to propose a new model order reduction technique. We focus on the Proper Generalized Decomposition (PGD) which is an attractive technique when the geometry of the domain is separable, as for the laser shock simulations described above. The PGD has been developed to overcome the so-called curse of dimensionality when it comes to solve a multi-dimensional problem [17], [18]. This technique consists in computing on the fly a separated form representation of the solution, directly from the weak formulation of the problem. The main idea is to solve several problems of lower dimensions than the initial problem to alleviate the computational costs. When it comes to solve time-dependent partial differential equations, the PGD is usually implemented by considering the space–time separation presented in [19]: u(x,y,t)i=1MSi(x,y)Ti(t)where u represents a time-dependent scalar field defined over a 2D space domain. This space–time separation avoids using a time incremental solver, by analogy with the Proper Orthogonal Decomposition (POD). If the length of the sum M is small (between 10–100 for instance) then the time dependent solution can be stored with significantly less storage capacities. Unlike the POD that requires snapshots of the solution [20], no prior knowledge on the solution needs to be known with the PGD. Over the last decade, the space–time separation has led to a plurality of demonstrations that the PGD computes accurate solutions faster than standard incremental solvers. This technique has been applied to a wide range of problems in different branches of physics: in rheology [21], thermal analysis [22], kinetic theory of complex fluids [23], and even in quantum chemistry [24]. From a mathematical standpoint, several works [25], [26] proved convergence results of the PGD algorithm with the space–time separation for problems involving elliptic operators. In elastodynamics, the space–time separation has been used to compute vibration responses up to the medium frequency domain [27], [28]. However, difficulties have been experienced when the problem equations involve hyperbolic operators [23], which is typically the case with a wave equation. Therefore very few works have addressed the topic of wave propagation with the PGD. In [29], [30] the authors presented a space–time separation within a TDG framework to remove spurious oscillations associated to transient wave propagation simulations. They applied their method to a 2D transient elastodynamic problem with a rectangular geometry, and they reported poor convergence results when using a standard PGD algorithm. To improve the numerical performances in terms of storage gain, the authors proposed a recompactification strategy to compute a lower rank approximation of the solution. However, this technique requires an experimented analyst to select the so-called auxiliary rank parameter of the method. In addition, difficulties are reported to compute the solution faster than with a standard TDG technique. As mentioned in their work, the computational cost associated to their method could be improved by considering a complete variables separation strategy, both in space and in time. However, to our knowledge, this approach has not been investigated.

In this context, the aim of this paper is not proposing an algorithm making radically faster or better, the existing algorithms already perform well, and from that point of view the one here proposed reduces storage needs and speed-up calculations. Here the main and crucial ingredient is the space separation, because nowadays engineered materials may involve thin coatings, geometries sometimes composed of extremely fine plies, and then when solutions are rich throughout the also rich domain thickness, 3D discretizations are sometimes unattainable. This is particularly the case for laser shock processes on composite laminate targets [3], [5], [7]. In this study, we combine standard integrators with an original space separated representation for empowering standard procedures. The proposed methodology is here applied to 2D domains, with stringent loading configurations for the space separation, to analyze its numerical behavior. Therefore, the results do not reflect the entire potential of the method for reducing the CPU time and the storage needs for computationally intensive problems. Future research will be devoted to 3D transient elastodynamic problems in composite laminates.

The paper is organized as follows. In Section 2 we present a time adaptive space separation strategy. We discuss the main features of this approach and we highlight its limitations. We present an implementation of this formulation to the 2D scalar wave equation, but the same methodology can be employed for the 2D elastodynamics equations. Three commonly employed time integration schemes for wave propagation analyses are implemented: the central difference method (explicit, conditionally stable), the Newmark trapezoidal rule and the Bathe method (implicit, unconditionally stable). In Section 3 we explore the numerical behavior of the proposed PGD algorithm on several 2D problems involving radial waves, multiple reflections, propagation on long time analyses, and wave conversions. We compare the obtained results with the exact solution when available, and to numerical solutions computed with a standard finite element method using the same time integration scheme. Finally we discuss the numerical performances of the proposed approach in terms of storage gain and computation time.

Section snippets

Formulation of the method

In this section we introduce a new PGD formulation adapted to transient wave propagation problems in separable domains. The main idea is to perform a space separation at each time increment with an adaptive number of modes. Then we compare the advantages and the limitations of this approach with respect to PGD methods based on a conventional space–time separation. Finally we present the implementation of this formulation to the 2D scalar wave equation.

Numerical results

In this section, we present the performances of the time adaptive space separation for several transient wave propagation problems. First, we consider a pre-stressed membrane with a concentrated impact loading in its center. We study the convergence properties of the PGD solution with comparisons to the exact solution and to a standard 2D finite element solution. Three different time integration schemes are tested. Then, we solve a water ripple propagation problem and we focus on long time

Conclusion

In this work, we proposed a time adaptive space separation strategy to solve transient wave propagation problems in separable domains. The separated form representation of the solution is computed on the fly with a PGD algorithm ran at each time increment. The numerical results show that the PGD solution is enriched when needed, as the waves are generated by the loading and travel in the domain. The number of modes in the PGD solution stabilizes when long time analyses are considered. The time

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

This work was supported by a government funding under the project FUI, France MONARQUE.

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