555 research outputs found
Modeling anisotropic diffusion using a departure from isotropy approach
There are a large number of finite volume solvers available for solution of isotropic diffusion equation. This article presents an approach of adapting these solvers to solve anisotropic diffusion equations. The formulation works by decomposing the diffusive flux into a component associated with isotropic diffusion and another component associated with departure from isotropic diffusion. This results in an isotropic diffusion equation with additional terms to account for the anisotropic effect. These additional terms are treated using a deferred correction approach and coupled via an iterative procedure. The presented approach is validated against various diffusion problems in anisotropic media with known analytical or numerical solutions. Although demonstrated for two-dimensional problems, extension of the present approach to three-dimensional problems is straight forward. Other than the finite volume method, this approach can be applied to any discretization method
Enforcing the non-negativity constraint and maximum principles for diffusion with decay on general computational grids
In this paper, we consider anisotropic diffusion with decay, and the
diffusivity coefficient to be a second-order symmetric and positive definite
tensor. It is well-known that this particular equation is a second-order
elliptic equation, and satisfies a maximum principle under certain regularity
assumptions. However, the finite element implementation of the classical
Galerkin formulation for both anisotropic and isotropic diffusion with decay
does not respect the maximum principle.
We first show that the numerical accuracy of the classical Galerkin
formulation deteriorates dramatically with increase in the decay coefficient
for isotropic medium and violates the discrete maximum principle. However, in
the case of isotropic medium, the extent of violation decreases with mesh
refinement. We then show that, in the case of anisotropic medium, the classical
Galerkin formulation for anisotropic diffusion with decay violates the discrete
maximum principle even at lower values of decay coefficient and does not vanish
with mesh refinement. We then present a methodology for enforcing maximum
principles under the classical Galerkin formulation for anisotropic diffusion
with decay on general computational grids using optimization techniques.
Representative numerical results (which take into account anisotropy and
heterogeneity) are presented to illustrate the performance of the proposed
formulation
Finite-difference schemes for anisotropic diffusion
In fusion plasmas diffusion tensors are extremely anisotropic due to the high temperature and large magnetic field strength. This causes diffusion, heat conduction, and viscous momentum loss, to effectively be aligned with the magnetic field lines. This alignment leads to different values for the respective diffusive coefficients in the magnetic field direction and in the perpendicular direction, to the extent that heat diffusion coefficients can be up to 10 to the 12 th times larger in the parallel direction than in the perpendicular direction. This anisotropy puts stringent requirements on the numerical methods used to approximate the MHD-equations since any misalignment of the grid may cause the perpendicular diffusion to be polluted by the numerical error in approximating the parallel diffusion. Currently the common approach is to apply magnetic field-aligned coordinates, an approach that automatically takes care of the directionality of the diffusive coefficients. This approach runs into problems at x-points and at points where there is magnetic re-connection, since this causes local non-alignment. It is therefore useful to consider numerical schemes that are tolerant to the misalignment of the grid with the magnetic field lines, both to improve existing methods and to help open the possibility of applying regular non-aligned grids. To investigate this, in this paper several discretisation schemes are developed and applied to the anisotropic heat diffusion equation on a non-aligned grid.</p
Addressing Integration Error for Polygonal Finite Elements Through Polynomial Projections: A Patch Test Connection
Polygonal finite elements generally do not pass the patch test as a result of
quadrature error in the evaluation of weak form integrals. In this work, we
examine the consequences of lack of polynomial consistency and show that it can
lead to a deterioration of convergence of the finite element solutions. We
propose a general remedy, inspired by techniques in the recent literature of
mimetic finite differences, for restoring consistency and thereby ensuring the
satisfaction of the patch test and recovering optimal rates of convergence. The
proposed approach, based on polynomial projections of the basis functions,
allows for the use of moderate number of integration points and brings the
computational cost of polygonal finite elements closer to that of the commonly
used linear triangles and bilinear quadrilaterals. Numerical studies of a
two-dimensional scalar diffusion problem accompany the theoretical
considerations
Numerical results for mimetic discretization of Reissner-Mindlin plate problems
A low-order mimetic finite difference (MFD) method for Reissner-Mindlin plate
problems is considered. Together with the source problem, the free vibration
and the buckling problems are investigated. Full details about the scheme
implementation are provided, and the numerical results on several different
types of meshes are reported
- …