Finite-Volume Methods for Anisotropic Diffusion Problems on Skewed Meshes

Abstract

A direct numerical method for the anisotropic diffusion equation is presented and the balancepoint method and adaptive method are also derived. All these methods are formulated and implemented in the in-house fluid flow solver GTEA. Two test cases, an isotropic problem and an anisotropic problem with exact solution on a skewed mesh, are chosen for comparison and validation. The error and computation time are illustrated. It is concluded that the direct method has the least computation time among all the diffusion problems and that all the methods meet the precision requirement in engineering computation even on skewed meshes.

Meshless fragile points methods based on Petrov‐Galerkin weak‐forms for transient heat conduction problems in complex anisotropic nonhomogeneous media

“…Consequently, the local discretization is consistent for both isotropic and anisotropic materials, whereas the two-point discretization scheme widely used in conventional FVMs is only consistent for isotropic problems. 12,13 The conservation laws are weakly satisfied by the IP numerical flux corrections in the current PG-FPM-2. With appropriate penalty parameters, the PG-FPM-2 shows good consistency and stability in both isotropic and anisotropic media.…”

“…Yet unlike the conventional FVMs, in which the gradient of temperature ∇ u is interpolated on each subdomain boundary independently, in the present PG‐FPM, ∇ u is approximated on the internal points and is assumed constant in the entire subdomain. Consequently, the local discretization is consistent for both isotropic and anisotropic materials, whereas the two‐point discretization scheme widely used in conventional FVMs is only consistent for isotropic problems 12,13 . The conservation laws are weakly satisfied by the IP numerical flux corrections in the current PG‐FPM‐2.…”

“…However, the FVM based a classic two-point flux formula is inconsistent in anisotropic problems. 10,11 Multiple-point flux approximations, [11][12][13] for example, generalized finite difference method, have to be employed. And the same as the FEM, the FVM also suffers from the mesh distortion problem and may have difficulties in analyzing problems with crack developments.…”

Meshless Fragile Points Methods Based on Petrov-Galerkin Weak-Forms for Transient Heat Conduction Problems in Complex Anisotropic Nonhomogeneous Media

“…Yet unlike the conventional FVMs, in which the gradient of temperature ∇ is interpolated on each subdomain boundary independently, in the present PG-FPM, ∇ is approximated on the Internal Points and is assumed constant in the entire subdomain. Consequently, the local discretization is consistent for both isotropic and anisotropic materials, whereas the two-point discretization scheme widely used in conventional FVMs is only consistent for isotropic problems 12,13 . The conservation laws are weakly satisfied by the IP Numerical Flux Corrections in the current PG-FPM-2.…”

“…However, the FVM based a classic two-point flux formula is inconsistent in anisotropic problems 10,11 . Multiple-point flux approximations 12,13,11 , e.g., generalized finite difference method, have to be employed. And the same as the FEM, the FVM also suffers from the mesh distortion problem and may have difficulties in analyzing problems with crack developments.…”