Abstract

This work proposes a novel finite volume paradigm, the face-centred finite volume (FCFV) method. Contrary to the popular vertex (VCFV) and cell (CCFV) centred finite volume methods, the novel FCFV defines the solution on the mesh faces (edges in 2D) to construct locally-conservative numerical schemes. The idea of the FCFV method stems from a hybridisable discontinuous Galerkin (HDG) formulation with constant degree of approximation, thus inheriting the convergence properties of the classical HDG. The resulting FCFV features a global problem in terms of a piecewise constant function defined on the faces of the mesh. The solution and its gradient in each element are then recovered by solving a set of independent element-by-element problems. The mathematical formulation of FCFV for Poisson and Stokes equation is derived and numerical evidence of optimal convergence in 2D and 3D is provided. Numerical examples are presented to illustrate the accuracy, efficiency and robustness of the proposed methodology. The results show that, contrary to other FV methods, the accuracy of the FCFV method is not sensitive to mesh distortion and stretching. In addition, the FCFV method shows its better performance, accuracy and robustness using simplicial elements, facilitating its application to problems involving complex geometries in 3D.

Abstract

The paper presents a new computational framework for the numerical simulation of fast large strain solid dynamics, with particular emphasis on the treatment of near incompressibility. A complete set of first order hyperbolic conservation equations expressed in terms of the linear momentum and the minors of the deformation (namely the deformation gradient, its co-factor and its Jacobian), in conjunction with a polyconvex nearly incompressible constitutive law, is presented. Taking advantage of this elegant formalism, alternative implementations in terms of entropy-conjugate variables are also possible, through suitable symmetrisation of the original system of conservation variables. From the spatial discretisation standpoint, modern Computational Fluid Dynamics code “OpenFOAM” [http://www.openfoam.com/] is here adapted to the field of solid mechanics, with the aim to bridge the gap between computational fluid and solid dynamics. A cell centred finite volume algorithm is employed and suitably adapted. Naturally, discontinuity of the conservation variables across control volume interfaces leads to a Riemann problem, whose resolution requires special attention when attempting to model materials with predominant nearly incompressible behaviour (). For this reason, an acoustic Riemann solver combined with a preconditioning procedure is introduced. In addition, a global a posteriori angular momentum projection procedure proposed in Haider et al. (2017) is also presented and adapted to a Total Lagrangian version of the nodal scheme of Kluth and Després (2010) used in this paper for comparison purposes. Finally, a series of challenging numerical examples is examined in order to assess the robustness and applicability of the proposed methodology with an eye on large scale simulation in future works.

Abstract

A second-order face-centred finite volume method (FCFV) is proposed. Contrary to the more popular cell-centred and vertex-centred finite volume (FV) techniques, the proposed method defines the solution on the faces of the mesh (edges in two dimensions). The method is based on a mixed formulation and therefore considers the solution and its gradient as independent unknowns. They are computed solving an element-by-element problem after the solution at the faces is determined. The proposed approach avoids the need of reconstructing the solution gradient, as required by cell-centred and vertex-centred FV methods. This strategy leads to a method that is insensitive to mesh distortion and stretching. The current method is second-order and requires the solution of a global system of equations of identical size and identical number of non-zero elements when compared to the recently proposed first-order FCFV. The formulation is presented for Poisson and Stokes problems. Numerical examples are used to illustrate the approximation properties of the method as well as to demonstrate its potential in three dimensional problems with complex geometries. The integration of a mesh adaptive procedure in the FCFV solution algorithm is also presented

Abstract

Full Eulerian methods constitute a family of numerical techniques used to simulate fluid-structure interaction problems. In a full Eulerian method, the velocity gradient tensor is used to compute deformation of solid. However, it is difficult to compute solid stress accurately near the interface, where the velocity between fluid and solid changes drastically. In this work, we propose an Eulerian formulation for fluid-structure interaction problems using Lagrangian marker particles with the Reference Map Technique to compute the deformation of solid accurately near material interfaces without using the gradient of the velocity. We illustrate and validate the proposed method through the presentation of various benchmark problems.