Curvilinear finite elements for Lagrangian hydrodynamics

Dobrev, Veselin; Ellis, Truman Everett; Kolev, Tzanio V.; Rieben, Robert N.

doi:10.1002/fld.2366

Cited by 72 publications

(54 citation statements)

References 6 publications

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“…The reader is reminded that on triangular/tetrahedral grids the finite element methods described in [1,40,39,12,13], and the compatible finite difference scheme described in [7,24] are exactly the same algorithm. This is because the gradients of the velocity computed with a staggered discretization are constant on each cell of a triangular grid, and, consequently, a staggered finite element, finite volume, or compatible finite difference formulation collapse to identical discrete operators.…”

Section: Introductionmentioning

confidence: 99%

Lagrangian shock hydrodynamics on tetrahedral meshes: A stable and accurate variational multiscale approach

Scovazzi

2012

Journal of Computational Physics

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Section: Introductionmentioning

confidence: 99%

Lagrangian shock hydrodynamics on tetrahedral meshes: A stable and accurate variational multiscale approach

Scovazzi

2012

Journal of Computational Physics

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“…The ODE for the geometry (27) and the physical system of the PDE (23) are solved together by an iterative procedure, which stops when the residuals of the two systems are less than a prescribed tolerance tol (typically tol ≈ 10 −12 ). For the linear homogeneous case, at most (M + 1) iterations are needed to reach convergence, [34].…”

Section: δT = Cfl Minmentioning

confidence: 99%

“…The equations of Lagrangian hydrodynamics can be solved with high order of accuracy using continuous finite elements, as proposed in [27][28][29]70,74], while a high order discontinuous Galerkin framework has been applied to Lagrangian schemes in [56,[84][85][86]. In [18,58] Cheng and Shu used a third order accurate essentially non-oscillatory (ENO) reconstruction operator to develop the first high order Godunov-type Lagrangian finite volume schemes ever.…”

Section: Introductionmentioning

confidence: 99%

An Efficient Quadrature-Free Formulation for High Order Arbitrary-Lagrangian–Eulerian ADER-WENO Finite Volume Schemes on Unstructured Meshes

Boscheri

Dumbser

2015

J Sci Comput

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In this paper we present a new and efficient quadrature-free formulation for the family of cell-centered high order accurate direct arbitrary-Lagrangian-Eulerian one-step ADER-WENO finite volume schemes on unstructured triangular and tetrahedral meshes that has been developed by the authors in a recent series of papers (Boscheri et al. in J Comput ). High order of accuracy in time is obtained by using a local space-time Galerkin predictor on moving curved meshes, while a high order accurate nonlinear WENO method is adopted to produce high order essentially non-oscillatory reconstruction polynomials in space. The mesh is moved at each time step according to the solution of a node solver algorithm that assigns a unique velocity vector to each node of the mesh. A rezoning procedure can also be applied when mesh distortions and deformations become too severe. The space-time mesh is then constructed by straight edges connecting the vertex positions at the old time level t n with the new ones at the next time level t n+1 , yielding closed space-time control volumes, on the boundary of which the numerical flux must be integrated. This is done here with a new and efficient quadrature-free approach: the space-time boundaries are split into simplex sub-elements, i.e. either triangles in 2D or tetrahedra in 3D. This leads to space-time normal vectors as well as Jacobian matrices that are constant within each sub-element. Within the space-time Galerkin predictor stage that solves the Cauchy problem inside each element in the small, the discrete solution and the flux tensor are approximated using a nodal space-time basis. Since these space-time basis functions are defined on a reference element and do not change, their integrals over the simplex sub-surfaces of the space-time reference control volume can be integrated once and for all analytically during a preprocessing step. The resulting integrals are then used together B M. Dumbser 123 J Sci Comput with the space-time degrees of freedom of the predictor in order to compute the numerical flux that is needed in the finite volume scheme. We apply the high order algorithm presented in this paper to the equations of hydrodynamics obtaining convergence rates up to fourth order of accuracy in space and time. A set of classical Lagrangian test problems has been solved and the results have been compared with the ones given by the original formulation of the algorithm (Boscheri and Dumbser 2013, 2014). The efficiency has been monitored and measured for each test case and the new quadrature-free schemes were up to 3.7 times faster than the ones based on Gaussian quadrature.Keywords Arbitrary-Lagrangian-Eulerian (ALE) · Finite volume schemes · Quadraturefree flux integration · WENO reconstruction on moving unstructured meshes · High order of accuracy in space and time · Local rezoning · Hydrodynamics

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“…(U n+1 +U n ) is used to preserve total energy from time t n to time t n+1 [7,8]. The pressure unknowns is computed directly by equation of state:…”

Section: Time Discretizationmentioning

confidence: 99%

“…Different approaches can be used for the solution of hydrodynamics equations: Eulerian [3,2], Lagrangian [14,7] or ALE [1,16]. In this work, a Lagrangian finite element approach is exploited to describe strong evolving material interfaces and shocks propagations.…”

Section: Introductionmentioning

confidence: 99%

A Lagrangian Finite Element Method for the Simulation of 3d Compressible Flows

Cremonesi¹,

Frangi²

2016

Proceedings of the VII European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS Congress 2016)

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Abstract. The numerical solution of compressible fluid flows is of paramount importance in many industrial and engineering applications. Compared to the classical fluid dynamics, the introduction of the fluid compressibility changes the formulation of the problem and consequently its computational treatment.Among the possible numerical solutions of compressible flow problems, the finite element method has always been privileged. However, the standard Eulerian approaches with fixed domain are not particularly suited to represent the strong shock waves and the significant movement of the external boundaries. On the contrary, in problems characterized by evolving surfaces, Lagrangian approaches can be very effective.The governing equations of compressible flow problems are mass, momentum and energy conservation. These equations are discretized in the spirit of the Lagrangian Particle Finite Element Method (PFEM). The strong distortions of the mesh, typical of the Lagrangian approaches, are managed with a continuous remeshing of the computational domain. The nodal unknowns are velocities, density and internal energy. To fully exploit the potential of continuous remeshing, only nodal variables are stored and consequently only linear interpolation are used. In addition, an artificial viscosity has been introduced to stabilize the formation and propagation of shock waves. Finally, explicit time integration of the governing equations enables a highly efficient solution of the discretized problem.The proposed approach has been validated against typical benchmarks of gas dynamics in the presence of strong shock waves. A very good agreement has been shown in all the tests proving the excellent accuracy and versatility of the proposed method.

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Curvilinear finite elements for Lagrangian hydrodynamics

Cited by 72 publications

References 6 publications

Lagrangian shock hydrodynamics on tetrahedral meshes: A stable and accurate variational multiscale approach

Lagrangian shock hydrodynamics on tetrahedral meshes: A stable and accurate variational multiscale approach

An Efficient Quadrature-Free Formulation for High Order Arbitrary-Lagrangian–Eulerian ADER-WENO Finite Volume Schemes on Unstructured Meshes

A Lagrangian Finite Element Method for the Simulation of 3d Compressible Flows

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