In this paper we present a mixed stabilized finite element formulation that does not lock and also does not exhibit unphysical oscillations near the incompressible limit. The new mixed formulation is based on a multiscale variational principle and is presented in two different forms. In the first form the displacement field is decomposed into two scales, coarse-scale and fine-scale, and the fine-scale variables are eliminated at the element level by the static condensation technique. The second form is obtained by simplifying the first form, and eliminating the fine-scale variables analytically yet retaining their effect that results with additional (stabilization) terms. We also derive, in a consistent manner, an expression for the stabilization parameter. This derivation also proves the equivalence between the classical mixed formulation with bubbles and the Galerkin least-squares type formulations for the equations of linear elasticity. We also compare the performance of this new mixed stabilized formulation with other popular finite element formulations by performing numerical simulations on three well known test problems.
SUMMARYIn this paper, we consider anisotropic diffusion with decay, which takes the form (x)c(x)− div [D(x)grad[c(x)]] = f (x) with decay coefficient (x) 0, and diffusivity coefficient D(x) 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 diffusions with decay does not respect the maximum principle. Put differently, the classical Galerkin formulation violates the discrete maximum principle (DMP) for diffusion with decay even on structured computational meshes.We first show that the numerical accuracy of the classical Galerkin formulation deteriorates dramatically with an increase in for isotropic media and violates the DMP. However, in the case of isotropic media, the extent of violation decreases with the mesh refinement. We then show that, in the case of anisotropic media, the classical Galerkin formulation for anisotropic diffusion with decay violates the DMP 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.
Abstract. We consider the tensorial diffusion equation, and address the discrete maximumminimum 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. It is well-known that the classical finite element formulations (like the single-field Galerkin formulation, and Raviart-Thomas, variational multiscale, and Galerkin/least-squares mixed formulations) do not produce non-negative solutions (that is, they do not satisfy the discrete maximumminimum principle) on arbitrary meshes and for strongly anisotropic diffusivity coefficients.In this paper we present two non-negative mixed finite element formulations for tensorial diffusion equations based on constrained optimization techniques. 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 the second non-negative formulation is based on the variational multiscale formulation. For the former formulation we comment on the effect of adding the non-negative constraint on the local mass balance property of the Raviart-Thomas formulation.We perform numerical convergence analysis of the proposed optimization-based non-negative mixed formulations. 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.
Abstract. Much of the work on flow through porous media, especially with regard to studies on the flow of oil, are based on "Darcy's law" or modifications to it such as Darcy-Forchheimer or Brinkman models. While many theoretical and numerical studies concerning flow through porous media have taken into account the inhomogeneity and anisotropy of the porous solid, they have not taken into account the fact that the viscosity of the fluid and drag coefficient could depend on the pressure in applications such as enhanced oil recovery. Experiments clearly indicate that the viscosity varies exponentially with respect to the pressure and the viscosity can change, in some applications, by several orders of magnitude. The fact that the viscosity depends on pressure immediately implies that the "drag coefficient" would also depend on the pressure.In this paper we consider modifications to Darcy's equation wherein the drag coefficient is a function of pressure, which is a realistic model for technological applications like enhanced oil recovery and geological carbon sequestration. We first outline the approximations behind Darcy's equation and the modifications that we propose to Darcy's equation, and derive the governing equations through a systematic approach using mixture theory. We then propose a stabilized mixed finite element formulation for the modified Darcy's equation. To solve the resulting nonlinear equations we present a solution procedure based on the consistent Newton-Raphson method. We solve representative test problems to illustrate the performance of the proposed stabilized formulation. One of the objectives of this paper is also to show that the dependence of viscosity on the pressure can have a significant effect both on the qualitative and quantitative nature of the solution.
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