33 research outputs found
Non-negative mixed finite element formulations for a tensorial diffusion equation
We consider the tensorial diffusion equation, and address the discrete
maximum-minimum principle of mixed finite element formulations. In particular,
we address non-negative solutions (which is a special case of the
maximum-minimum principle) of mixed finite element formulations. The discrete
maximum-minimum principle is the discrete version of the maximum-minimum
principle.
In this paper we present two non-negative mixed finite element formulations
for tensorial diffusion equations based on constrained optimization techniques
(in particular, quadratic programming). These proposed mixed formulations
produce non-negative numerical solutions on arbitrary meshes for low-order
(i.e., linear, bilinear and trilinear) finite elements. The first formulation
is based on the Raviart-Thomas spaces, and is obtained by adding a non-negative
constraint to the variational statement of the Raviart-Thomas formulation. The
second non-negative formulation based on the variational multiscale
formulation.
For the former formulation we comment on the affect of adding the
non-negative constraint on the local mass balance property of the
Raviart-Thomas formulation. We also study the performance of the active set
strategy for solving the resulting constrained optimization problems. The
overall performance of the proposed formulation is illustrated on three
canonical test problems.Comment: 40 pages using amsart style file, and 15 figure
The cutoff method for the numerical computation of nonnegative solutions of parabolic PDEs with application to anisotropic diffusion and lubrication-type equations
The cutoff method, which cuts off the values of a function less than a given
number, is studied for the numerical computation of nonnegative solutions of
parabolic partial differential equations. A convergence analysis is given for a
broad class of finite difference methods combined with cutoff for linear
parabolic equations. Two applications are investigated, linear anisotropic
diffusion problems satisfying the setting of the convergence analysis and
nonlinear lubrication-type equations for which it is unclear if the convergence
analysis applies. The numerical results are shown to be consistent with the
theory and in good agreement with existing results in the literature. The
convergence analysis and applications demonstrate that the cutoff method is an
effective tool for use in the computation of nonnegative solutions. Cutoff can
also be used with other discretization methods such as collocation, finite
volume, finite element, and spectral methods and for the computation of
positive solutions.Comment: 19 pages, 41 figure
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
An anisotropic mesh adaptation method for the finite element solution of heterogeneous anisotropic diffusion problems
Heterogeneous anisotropic diffusion problems arise in the various areas of
science and engineering including plasma physics, petroleum engineering, and
image processing. Standard numerical methods can produce spurious oscillations
when they are used to solve those problems. A common approach to avoid this
difficulty is to design a proper numerical scheme and/or a proper mesh so that
the numerical solution validates the discrete counterpart (DMP) of the maximum
principle satisfied by the continuous solution. A well known mesh condition for
the DMP satisfaction by the linear finite element solution of isotropic
diffusion problems is the non-obtuse angle condition that requires the dihedral
angles of mesh elements to be non-obtuse. In this paper, a generalization of
the condition, the so-called anisotropic non-obtuse angle condition, is
developed for the finite element solution of heterogeneous anisotropic
diffusion problems. The new condition is essentially the same as the existing
one except that the dihedral angles are now measured in a metric depending on
the diffusion matrix of the underlying problem. Several variants of the new
condition are obtained. Based on one of them, two metric tensors for use in
anisotropic mesh generation are developed to account for DMP satisfaction and
the combination of DMP satisfaction and mesh adaptivity. Numerical examples are
given to demonstrate the features of the linear finite element method for
anisotropic meshes generated with the metric tensors.Comment: 34 page
The effect of weight loss on lameness in obese dogs with osteoarthritis
This paper describes the effect of weight loss on lameness in obese dogs with osteoarthritis (OA). Fourteen obese client-owned dogs with clinical and radiographic signs of OA participated in an open prospective clinical trial. After a screening visit and a visit for collection of baseline data, the dogs were fed a restricted-calorie diet over a study period of 16 weeks that incorporated six follow-up visits. At each visit, body weight and pelvic circumference were measured and severity of lameness was assessed using a numeric rating scale (NRS), a visual analogue scale (VAS) and kinetic gait analysis. This is the first study to assess both subjectively and objectively, the effect of weight loss alone on lameness in obese dogs with OA. The results indicate that body weight reduction causes a significant decrease in lameness from a weight loss of 6.10% onwards. Kinetic gait analysis supported the results from a body weight reduction of 8.85% onwards. These results confirm that weight loss should be presented as an important treatment modality to owners of obese dogs with OA and that noticeable improvement may be seen after modest weight loss in the region of 6.10 – 8.85% body weight