A velocity/stress mixed stabilized nodal finite element for elastodynamics: Analysis and computations with strongly and weakly enforced boundary conditions

Scovazzi, Guglielmo; Song, Tianming; Zeng, Xianyi

doi:10.1016/j.cma.2017.07.018

Cited by 40 publications

(32 citation statements)

References 110 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Hence, it is possible to reformulate (39a)-(39b) using a stabilized formulation, analogous to the one pursued in [46,50,47] in the context of hyperbolic wave systems:…”

Section: Linearized Shallow Waters Equations: Non-dissipative Acoustimentioning

confidence: 99%

“…In this case, boundary conditions can be enforced weakly with the following stabilized variational form, inspired again by the work in [46,50,47], and with close similarities to the variational forms developed in [15,16]: (47) is a discontinuity capturing operator, function of the artificial viscosity ν, which is defined as…”

Section: Nonlinear Shallow Water Equationsmentioning

confidence: 99%

“…These can be treated in a second stage. Substituting (55) in (47) as the solution leads to specific conditions on the discretization to maintain the C-property. Specifically, no inflow or outflow are possible, the integrals involving the term F v i = v i U vanish identically, as well as the volume integrals of the friction source term S f i , as they involve variations of the flow speed.…”

Section: C-property: Preservation Of a Lake At Restmentioning

confidence: 99%

“…We apply our method to the shallow water equations, as a prototypical hyperbolic system. We also consider the linearized limit of the shallow water equations, constituted by the equations of acoustics [46,50,47,20].…”

Section: Introductionmentioning

confidence: 99%

“…We implement the proposed approach for boundary conditions in the context of a stabilized finite element formulation, which takes inspiration specifically from [46,50,47] and more broadly from [15,16,51,26,22,27,28,25,24,23,29,49,40]. The proposed embedded method is not limited in applicability, however, to the specific stabilized method considered here.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

The shifted boundary method for hyperbolic systems: Embedded domain computations of linear waves and shallow water flows

Song

Main

Scovazzi

et al. 2018

Journal of Computational Physics

Self Cite

View full text Add to dashboard Cite

We propose a new computational approach for embedded boundary simulations of hyperbolic systems. Applications are shown for the linear wave equations and for the nonlinear shallow water system. The proposed approach belongs to the class of surrogate/approximate boundary algorithms and is based on the idea of shifting the location where boundary conditions are applied from the true to a surrogate boundary. Accordingly, boundary conditions, enforced weakly, are appropriately modified to preserve optimal error convergence rates. This framework is applied here in the setting of a stabilized finite element method, even though other spatial discretization techniques could have been employed. Accuracy, stability and robustness of the proposed method are tested by means of an extensive set of computational experiments for the acoustic wave propagation equations and shallow water equations. Comparisons with standard weak boundary conditions imposed on grids that conform to the geometry of the computational domain boundaries are also presented. § Team CARDAMOM, INRIA Bordeaux Sud-Ouest, 200 av. de la vieille tour33405 Talence Cedex, France Shifted boundary method pour systèmes hyperboliques:ondes linéaires et équations shallow waterRésumé : On propose une nouvelle approche pour des simulations avec bords immergés pour des systèmes hyperboliques et en particulier les équations shallow water. L'approche proposée consiste en modifier les conditions au bords avec un développement limité permettant d'assurer l'ordre deux avec des embedded boundaries. L'approche est implementé est validée ici dans le cadre d'une méthode de type stabilized finite element sur un très grand nombre de cas tests représentatifs d'applications de propagation de vagues et inondation.

show abstract

“…Hence, it is possible to reformulate (39a)-(39b) using a stabilized formulation, analogous to the one pursued in [46,50,47] in the context of hyperbolic wave systems:…”

Section: Linearized Shallow Waters Equations: Non-dissipative Acoustimentioning

confidence: 99%

Section: Nonlinear Shallow Water Equationsmentioning

confidence: 99%

Section: C-property: Preservation Of a Lake At Restmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

The shifted boundary method for hyperbolic systems: Embedded domain computations of linear waves and shallow water flows

Song

Main

Scovazzi

et al. 2018

Journal of Computational Physics

Self Cite

View full text Add to dashboard Cite

show abstract

Elastoplasticity with linear tetrahedral elements: A variational multiscale method

Abboud

Scovazzi

2018

Numerical Meth Engineering

Self Cite

View full text Add to dashboard Cite

Summary We present a computational framework for the simulation of J2‐elastic/plastic materials in complex geometries based on simple piecewise linear finite elements on tetrahedral grids. We avoid spurious numerical instabilities by means of a specific stabilization method of the variational multiscale kind. Specifically, we introduce the concept of subgrid‐scale displacements, velocities, and pressures, approximated as functions of the governing equation residuals. The subgrid‐scale displacements/velocities are scaled using an effective (tangent) elastoplastic shear modulus, and we demonstrate the beneficial effects of introducing a subgrid‐scale pressure in the plastic regime. We provide proofs of stability and convergence of the proposed algorithms. These methods are initially presented in the context of static computations and then extended to the case of dynamics, where we demonstrate that, in general, naïve extensions of stabilized methods developed initially for static computations seem not effective. We conclude by proposing a dynamic version of the stabilizing mechanisms, which obviates this problematic issue. In its final form, the proposed approach is simple and efficient, as it requires only minimal additional computational and storage cost with respect to a standard finite element relying on a piecewise linear approximation of the displacement field.

show abstract

Novel quadratic Bézier triangular and tetrahedral elements using existing mesh generators: Applications to linear nearly incompressible elastostatics and implicit and explicit elastodynamics

Kadapa

2018

Numerical Meth Engineering

View full text Add to dashboard Cite

In this paper, we present novel techniques of using quadratic Bézier triangular and tetrahedral elements for elastostatic and implicit/explicit elastodynamic simulations involving nearly incompressible linear elastic materials. A simple linear mapping is proposed for developing finite element meshes with quadratic Bézier triangular/tetrahedral elements from the corresponding quadratic Lagrange elements that can be easily generated using the existing mesh generators. Numerical issues arising in the case of nearly incompressible materials are addressed using the consistent B-bar formulation, thus reducing the finite element formulation to one consisting only of displacements. The higher-order spatial discretization and the nonnegative nature of Bernstein polynomials are shown to yield significant computational benefits. The optimal spatial convergence of the B-bar formulation for the quadratic triangular and tetrahedral elements is demonstrated by computing error norms in displacement and stresses.The applicability and computational efficiency of the proposed elements for elastodynamic simulations are demonstrated by studying several numerical examples involving real-world geometries with complex features. Numerical results obtained with the standard linear triangular and tetrahedral elements are also presented for comparison. 544KADAPA of finite elements, triangular/tetrahedral elements are often the favorable choice because of the ease of mesh generation, even for complicated industrial geometries, and the availability of numerous commercial and open-source mesh generation software suites. In the past few decades, a plethora of numerical techniques has been developed for performing simulations using triangular and tetrahedral elements, as the task of mesh generation with triangular and tetrahedral elements is well established.Nevertheless, in spite of the tremendous amount of research work that has gone into the development of numerical techniques for obtaining accurate numerical solutions using triangular/tetrahedral elements, robust and efficient techniques for performing simulations using these elements are still lacking, especially for explicit elastodynamic simulations involving nearly incompressible material models that are commonly encountered in soils, a majority of polymeric materials, and biological tissues. This lack of development is due to the fact that triangular/tetrahedral Lagrange elements suffer from quite a few disadvantages that limit their applicability to explicit elastodynamic simulations consisting of nearly incompressible elastic and elastoplastic material models. The important disadvantages of triangular/tetrahedral Lagrange elements that motivated the present research work are as follows.• Due to the approximation of constant strain across the element, linear triangular/tetrahedral Lagrange elements with pure displacement formulation exhibit a very stiff behavior when compared with the linear quadrilateral/hexahedral Lagrange elements. 1 Therefore, to obtain numerical results of suffici...

show abstract

A velocity/stress mixed stabilized nodal finite element for elastodynamics: Analysis and computations with strongly and weakly enforced boundary conditions

Cited by 40 publications

References 110 publications

The shifted boundary method for hyperbolic systems: Embedded domain computations of linear waves and shallow water flows

The shifted boundary method for hyperbolic systems: Embedded domain computations of linear waves and shallow water flows

Elastoplasticity with linear tetrahedral elements: A variational multiscale method

Novel quadratic Bézier triangular and tetrahedral elements using existing mesh generators: Applications to linear nearly incompressible elastostatics and implicit and explicit elastodynamics

Contact Info

Product

Resources

About