A 3D cell-centered ADER MOOD Finite Volume method for solving updated Lagrangian hyperelasticity on unstructured grids

TLDR: In this article, a cell-centered Lagrangian finite volume (FV) discretization is combined with the a posteriori multidimensional Optimal Order Detection (MOOD) limiting strategy to ensure robustness and stability at shock waves with piecewise linear spatial reconstruction.

About: This article is published in Journal of Computational Physics.The article was published on 2022-01-15 and is currently open access. It has received 4 citations till now. The article focuses on the topics: Hyperelastic material & Discretization.

Figures

Fig. 1. Sketch of the Lagrangian-Eulerian mapping relating a material Lagrangian point X at t = 0 and a spatial Eulerian one x at t > 0 through Φ, and its Jacobian F(X, t) = ∂Φ

Fig. 14. Twisting column — Time evolution of non-dimensionalised height of the column measured at initial point xT = (0, 0, 6) (left) — Numerical dissipation (88) of second order scheme (center) — Percentage of bad cells detected at each time step (right).

Fig. 9. Cantilever thick beam — Comparison between the MOOD cascades: P1 → P0 (LAM P0-lim) and P1 → Plim1 → P0 (LAM P1-lim) for the computed numerical dissipation (88) as a function of time (left) and percentage of bad cells detected at each time step (right).

Fig. 19. L-shaped block test problem — From top-left to bottom-right: time t = 2.5, 5.0, 7.5, 10.0, 12.5 and 15 s.

Table 3 Maximal value of B for the first 75000 time-steps of the 2D and 3D tests cases in this paper.

Fig. 20. L-shaped block test problem — Angular momentum Ac in x (green), y (red) and z (blue) directions as a function of time.

Frequently asked questions

Q1. What have the authors contributed in "A 3d cell-centered ader mood finite volume method for solving updated lagrangian hyperelasticity on unstructured grids" ?

In this paper, the authors present a conservative cell-centered Lagrangian Finite Volume scheme for solving the hyperelasticity equations on unstructured multidimensional grids. The starting point of the present approach is the cell-centered FV discretization named EUCCLHYD and introduced in the context of Lagrangian hydrodynamics. This strategy has been successfully tested in an hydrodynamics context and the present work aims at extending it to the case of hyperelasticity. A relatively large set of numerical test cases is presented to assess the ability of the method to achieve effective second order of accuracy on smooth flows, maintaining an essentially non-oscillatory behavior and general robustness across discontinuities and ensuring at least physical admissibility of the solution where appropriate. These test cases feature material bending, impact, compression, non-linear deformation and further bouncing/detaching motions. 

Q2. What future works have the authors mentioned in the paper "A 3d cell-centered ader mood finite volume method for solving updated lagrangian hyperelasticity on unstructured grids" ?

A plan for future work involves the introduction of plasticity into this hyperelasticity model. The authors also plan to investigate the high-order extension over curvilinear simplicial grids of the present FV discretization. Another direction of evolution would be to add some ArbitraryLagrangian-Eulerian capability and the possibility to let two elastic materials interacting with each other, for instance impacting, deforming and further detaching from each others.