A high-order discontinuous Galerkin approximation to ordinary differential equations with applications to elastodynamics

Antonietti, Paola F.; Mazzieri, Ilario; Santo, Niccolò Dal; Quarteroni, Alfio

doi:10.1093/imanum/drx062

Cited by 25 publications

(25 citation statements)

References 55 publications

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“…For this purpose, several high order numerical schemes on unstructured meshes were introduced in the past. A series of explicit high order discontinuous Galerkin (DG) schemes for elastic wave propagation on unstructured meshes was proposed in [34,35,36,37,38,39], while the concept of space-time discontinuous Galerkin schemes, originally introduced and analyzed in [40,41,42,43,44,45,46] for computational fluid dynamics (CFD), was later also extended to linear elasticity in [47,48,49]. The space-time DG method used in [49] is based on the novel concept of staggered discontinuous Galerkin finite element schemes, which was introduced for CFD problems in [50,51,52,53,53,54,55,56].…”

Section: Introductionmentioning

confidence: 99%

A simple diffuse interface approach on adaptive Cartesian grids for the linear elastic wave equations with complex topography

Tavelli

Dumbser

Charrier

et al. 2019

Journal of Computational Physics

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In most classical approaches of computational geophysics for seismic wave propagation problems, complex surface topography is either accounted for by boundary-fitted unstructured meshes, or, where possible, by mapping the complex computational domain from physical space to a topologically simple domain in a reference coordinate system. However, all these conventional approaches face problems if the geometry of the problem becomes sufficiently complex. They either need a mesh generator to create unstructured boundary-fitted grids, which can become quite difficult and may require a lot of manual user interactions in order to obtain a high quality mesh, or they need the explicit computation of an appropriate mapping function from physical to reference coordinates. For sufficiently complex geometries such mappings may either not exist or their Jacobian could become close to singular. Furthermore, in both conventional approaches low quality grids will always lead to very small time steps due to the Courant-Friedrichs-Lewy (CFL) condition for explicit schemes. In this paper, we propose a completely different strategy that follows the ideas of the successful family of high resolution shock-capturing schemes, where discontinuities can actually be resolved anywhere on the grid, without having to fit them exactly. We address the problem of geometrically complex free surface boundary conditions for seismic wave propagation problems with a novel diffuse interface method (DIM) on adaptive Cartesian meshes (AMR) that consists in the introduction of a characteristic function 0 ≤ α ≤ 1 which identifies the location of the solid medium and the surrounding air (or vacuum) and thus implicitly defines the location of the free surface boundary. Physically, α represents the volume fraction of the solid medium present in a control volume. Our new approach completely avoids the problem of mesh generation, since all that is needed for the definition of the complex surface topography is to set a scalar color function to unity inside the regions covered by the solid and to zero outside. The governing equations are derived from ideas typically used in the mathematical description of compressible multiphase flows. An analysis of the eigenvalues of the PDE system shows that the complexity of the geometry has no influence on the admissible time step size due to the CFL condition. The model reduces to the classical linear elasticity equations inside the solid medium where the gradients of α are zero, while in the diffuse interface zone at the free surface boundary the governing PDE system becomes nonlinear. We can prove that the solution of the Riemann problem with arbitrary data and a jump in α from unity to zero yields a Godunov-state at the interface that satisfies the free-surface boundary condition exactly, i.e. the normal stress components vanish. In the general case of an interface that is not aligned with the grid and which is not infinitely thin, a systematic study on the distribution of the volume fraction function inside the interfac...

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Section: Introductionmentioning

confidence: 99%

A simple diffuse interface approach on adaptive Cartesian grids for the linear elastic wave equations with complex topography

Tavelli

Dumbser

Charrier

et al. 2019

Journal of Computational Physics

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“…For the time integration of the system of second order ordinary differential equations (19), we employ the leap-frog method [91], that it widely employed time integration scheme for the numerical simulation of elastic waves propagation, see for example [21,31,67,81]. Other time integration techniques based on high-order time marching scheme can be employ to discretize in time the (visco)elasdodynamics equation, such as Runge-Kutta methods [64], the ADER-DG method [65] or the space-time DG discretization [12]. With this aim, we subdivide the time interval (0, T ] into N T subintervals of amplitude ∆t = T /N T and we denote by U i ≈ U(t i ) the approximation of U at time t i = i∆t, i = 1, 2, .…”

Section: Fully Discrete Formulationmentioning

confidence: 99%

Numerical modeling of seismic waves by discontinuous spectral element methods

et al. 2018

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We present a comprehensive review of Discontinuous Galerkin Spectral Element (DGSE) methods on hybrid hexahedral/tetrahedral grids for the numerical modeling of the ground motion induced by large earthquakes. DGSE methods combine the exibility of discontinuous Galerkin meth-ods to patch together, through a domain decomposition paradigm, Spectral Element blocks where high-order polynomials are used for the space discretization. This approach allows local adaptivity on discretization parameters, thus improving the quality of the solution without affecting the compu-tational costs. The theoretical properties of the semidiscrete formulation are also revised, including well-posedness, stability and error estimates. A discussion on the dissipation, dispersion and stability properties of the fully-discrete (in space and time) formulation is also presented. Here space dis-cretization is obtained based on employing the leap-frog time marching scheme. The capabilities of the present approach are demonstrated through a set of computations of realistic earthquake scenar-ios obtained using the code SPEED (http://speed.mox.polimi.it), an open-source code specifically designed for the numerical modeling of large-scale seismic events jointly developed at Politecnico di Milano by The Laboratory for Modeling and Scientific Computing MOX and by the Department of Civil and Environmental Engineering.

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“…In order to explore this idea we have employed the algorithm introduced by Antonietti and Mazzieri in [37]. This algorithm is based on a space semi-discretization of second order hyperbolic problems that results in a high-order DG scheme.…”

Section: Numerical Algorithmmentioning

confidence: 99%

Space–time Trefftz-DG approximation for elasto-acoustics

Barucq¹,

Calandra

Diaz³

et al. 2018

Applicable Analysis

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Discontinuous finite element methods have proven their numerical accuracy and flexibility, but they are still criticized for requiring high number of degrees of freedom for computation. There have been some advances with Hybridizable Discontinuous Galerkin (HDG) approximations but essentially in the time-harmonic regime, because the time integration requires implicit schemes. Another possibility to explore seems to be Trefftz methods, which distinguish themselves by the choice of basis functions: they are local solutions of initial equation. Thus, in case of time-dependent problems, space-time meshes are required. Classical Trefftz-type approximation uses the exact solutions of Acoustic System (AS) and Elastodynamic System (ES) taken with different frequencies, in order to obtain better numerical errors. We have computed a polynomial basis using the Taylor expansions of generating exponential functions which are the exact solutions of the initial acoustic and elastodynamic systems. This basis, compared to the classical one based on trigonometric functions, requires less degrees of freedom for the same level of accuracy. In the present work, we develop the theory for coupled Elasto-Acoustic System (EAS). We confirm well-posedness of the problem, based on the a priory error estimates in mesh dependent norms. We consider a space-time polynomial basis to implement numerically the method and to perform some numerical experiments showing promising results.

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A high-order discontinuous Galerkin approximation to ordinary differential equations with applications to elastodynamics

Cited by 25 publications

References 55 publications

A simple diffuse interface approach on adaptive Cartesian grids for the linear elastic wave equations with complex topography

A simple diffuse interface approach on adaptive Cartesian grids for the linear elastic wave equations with complex topography

Numerical modeling of seismic waves by discontinuous spectral element methods

Space–time Trefftz-DG approximation for elasto-acoustics

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