Arbitrary high order accurate space–time discontinuous Galerkin finite element schemes on staggered unstructured meshes for linear elasticity

Tavelli, Maurizio; Dumbser, Michael

doi:10.1016/j.jcp.2018.03.038

Cited by 24 publications

(16 citation statements)

References 102 publications

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“…The AMR grid is not at all aligned with the free surface boundary and remains always locally Cartesian (with h-adaptivity). Furthermore, the time step size in our approach is not affected by the so-called small cell problem or sliver element problem, as it would have been the case for Cartesian cut-cell methods or low quality unstructured meshes and which usually requires a special treatment [36,49]. In our diffuse interface approach, the eigenvalues of the PDE system are independent of α and also our mesh can be chosen independently of α and almost independently of the geometry of the problem to be solved (apart from local h adaptivity used in regions of strong gradients of α).…”

Section: Scattering Of a Plane Wave On A Circular Cavitymentioning

confidence: 99%

“…In this test case we to study the two dimensional tilted Lamb problem. We take the same setup as used in [25,106,49]. The physical domain Ω = {(x, y) ∈ R 2 | 0 ≤ x ≤ 4000 , 0 ≤ y ≤ 2000 + tan (θ)x} contains a free surface with a tilt angle of θ = 10 • , so that the boundary is not grid aligned when using a Cartesian mesh along the coordinate axes.…”

Section: D Tilted Lamb Problemmentioning

confidence: 99%

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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: Scattering Of a Plane Wave On A Circular Cavitymentioning

confidence: 99%

Section: D Tilted Lamb Problemmentioning

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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“…An alternative consists in the family of fully implicit and semi-implicit space-time DG methods, see e.g. [19,82,98,99,[109][110][111]120,121]. Another different option that leads to high order explicit fully-discrete one-step schemes, and which is followed in this paper, combines ideas of the ADER approach of Toro and Titarev, originally developed within the finite volume framework [20,114,117,118], with space-time DG methods.…”

Section: Introductionmentioning

confidence: 99%

On Thermodynamically Compatible Finite Volume Methods and Path-Conservative ADER Discontinuous Galerkin Schemes for Turbulent Shallow Water Flows

et al. 2021

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In this paper we propose a new reformulation of the first order hyperbolic model for unsteady turbulent shallow water flows recently proposed in Gavrilyuk et al. (J Comput Phys 366:252–280, 2018). The novelty of the formulation forwarded here is the use of a new evolution variable that guarantees the trace of the discrete Reynolds stress tensor to be always non-negative. The mathematical model is particularly challenging because one important subset of evolution equations is nonconservative and the nonconservative products also act across genuinely nonlinear fields. Therefore, in this paper we first consider a thermodynamically compatible viscous extension of the model that is necessary to define a proper vanishing viscosity limit of the inviscid model and that is absolutely fundamental for the subsequent construction of a thermodynamically compatible numerical scheme. We then introduce two different, but related, families of numerical methods for its solution. The first scheme is a provably thermodynamically compatible semi-discrete finite volume scheme that makes direct use of the Godunov form of the equations and can therefore be called a discrete Godunov formalism. The new method mimics the underlying continuous viscous system exactly at the semi-discrete level and is thus consistent with the conservation of total energy, with the entropy inequality and with the vanishing viscosity limit of the model. The second scheme is a general purpose high order path-conservative ADER discontinuous Galerkin finite element method with a posteriori subcell finite volume limiter that can be applied to the inviscid as well as to the viscous form of the model. Both schemes have in common that they make use of path integrals to define the jump terms at the element interfaces. The different numerical methods are applied to the inviscid system and are compared with each other and with the scheme proposed in Gavrilyuk et al. (2018) on the example of three Riemann problems. Moreover, we make the comparison with a fully resolved solution of the underlying viscous system with small viscosity parameter (vanishing viscosity limit). In all cases an excellent agreement between the different schemes is achieved. We furthermore show numerical convergence rates of ADER-DG schemes up to sixth order in space and time and also present two challenging test problems for the model where we also compare with available experimental data.

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“…4, is of the kind of the space-time DG schemes, see e.g., [87,107,125,126]. Following [117,119,120], we have to define the space-time basis functions as an extension of the spatial basis functions introduced in the previous paragraph.…”

Section: Extension To the Space-time Basis Functionsmentioning

confidence: 99%

A Novel Staggered Semi-implicit Space-Time Discontinuous Galerkin Method for the Incompressible Navier-Stokes Equations

Romeo

Dumbser

Tavelli

2020

Commun. Appl. Math. Comput.

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A new high-order accurate staggered semi-implicit space-time discontinuous Galerkin (DG) method is presented for the simulation of viscous incompressible flows on unstructured triangular grids in two space dimensions. The staggered DG scheme defines the discrete pressure on the primal triangular mesh, while the discrete velocity is defined on a staggered edge-based dual quadrilateral mesh. In this paper, a new pair of equal-order-interpolation velocity-pressure finite elements is proposed. On the primary triangular mesh (the pressure elements), the basis functions are piecewise polynomials of degree N and are allowed to jump on the boundaries of each triangle. On the dual mesh instead (the velocity elements), the basis functions consist in the union of piecewise polynomials of degree N on the two subtriangles that compose each quadrilateral and are allowed to jump only on the dual element boundaries, while they are continuous inside. In other words, the basis functions on the dual mesh are built by continuous finite elements on the subtriangles. This choice allows the construction of an efficient, quadraturefree and memory saving algorithm. In our coupled space-time pressure correction formulation for the incompressible Navier-Stokes equations, the arbitrary high order of accuracy in time is achieved through the use of time-dependent test and basis functions, in combination with simple and efficient Picard iterations. Several numerical tests on classical benchmarks confirm that the proposed method outperforms existing staggered semi-implicit space-time DG schemes, not only from a computer memory point of view, but also concerning the computational time. Keywords Incompressible Navier-Stokes equations • Semi-implicit space-time discontinuous Galerkin schemes • Staggered unstructured meshes • Space-time pressure correction method • High-order accuracy in space and time Mathematics Subject Classification 65M60 * M.

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Arbitrary high order accurate space–time discontinuous Galerkin finite element schemes on staggered unstructured meshes for linear elasticity

Cited by 24 publications

References 102 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

On Thermodynamically Compatible Finite Volume Methods and Path-Conservative ADER Discontinuous Galerkin Schemes for Turbulent Shallow Water Flows

A Novel Staggered Semi-implicit Space-Time Discontinuous Galerkin Method for the Incompressible Navier-Stokes Equations

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