Proper Generalized Decomposition with time adaptive space separation for transient wave propagation problems in separable domains
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]: where 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 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.
References (42)
- et al.
Generation of controlled delaminations in composites using symmetrical laser shock configuration
Compos. Struct.
(2017) - et al.
A study of composite material damage induced by laser shock waves
Composites A
(2013) - et al.
Development of a laser shock adhesion test for the assessment of weak adhesive bonded cfrp structures
Int. J. Adhes. Adhes.
(2014) - et al.
Development of the symmetrical laser shock test for weak bond inspection
Opt. Laser Technol.
(2019) - et al.
Laser shock adhesion test numerical optimization for composite bonding assessment
Compos. Struct.
(2020) - et al.
Performance of an implicit time integration scheme in the analysis of wave propagations
Comput. Struct.
(2013) - et al.
The bathe time integration method with controllable spectral radius: The -bathe method
Comput. Struct.
(2019) A spectral element method for fluid dynamics: Laminar flow in a channel expansion
J. Comput. Phys.
(1984)- et al.
Dispersion analysis of spectral element methods for elastic wave propagation
Wave Motion
(2008) - et al.
A finite element method enriched for wave propagation problems
Comput. Struct.
(2012)
Space-time finite element methods for second-order hyperbolic equations
Comput. Methods Appl. Mech. Engrg.
The LATIN multiscale computational method and the proper generalized decomposition
Comput. Methods Appl. Mech. Engrg.
Reduced order modeling via pgd for highly transient thermal evolutions in additive manufacturing
Comput. Methods Appl. Mech. Engrg.
A priori model reduction through proper generalized decomposition for solving time-dependent partial differential equations
Comput. Methods Appl. Mech. Engrg.
A proper generalized decomposition for the solution of elliptic problems in abstract form by using a functional Eckart–Young approach
J. Math. Anal. Appl.
Quasistatic analysis of elastoplastic structures by the proper generalized decomposition in a space-time approach
Mech. Res. Commun.
Ideal minimal residual-based proper generalized decomposition for non-symmetric multi-field models – application to transient elastodynamics in space-time domain
Comput. Methods Appl. Mech. Engrg.
Space–time proper generalized decompositions for the resolution of transient elastodynamic models
Comput. Methods Appl. Mech. Engrg.
A new hybrid explicit/implicit in-plane-out-of-plane separated representation for the solution of dynamic problems defined in plate-like domains
Comput. Struct.
An error estimator for separated representations of highly multidimensional models
Comput. Methods Appl. Mech. Engrg.
Lumped mass matrix in explicit finite element method for transient dynamics of elasticity
Comput. Methods Appl. Mech. Engrg.
Cited by (6)
Proper Generalized Decomposition using Taylor expansion for non-linear diffusion equations
2023, Mathematics and Computers in SimulationDevelopment of POD-based Reduced Order Models applied to shallow water equations using augmented Riemann solvers
2023, Computer Methods in Applied Mechanics and EngineeringPGD reduced-order modeling for structural dynamics applications
2022, Computer Methods in Applied Mechanics and EngineeringCitation Excerpt :Different approaches have also been proposed in order to circumvent the difficulty one encounters when using a separated representation in space and time. For instance, an approach that assumes a good space separation, was presented in [13]. It consists in estimating, in an adaptive manner, the number of spatial modes at each time step.
Exploring space separation techniques for 3D elastic waves simulations
2022, Computational Mechanics