In the present contribution a mixed least-squares finite element method (LSFEM) based on the Theory of Porous Media (TPM) is presented. In detail, we investigate an incompressible binary model consisting of the phases solid and liquid. The main idea is based on the modeling of saturated porous structures. The resulting finite element is a four-field formulation in terms of solid displacements, liquid pressure, mixture stresses and a new variable related to the pressure gradient. The conforming discretization of the unknowns in the spaces H(div) and H 1 is realized by vector-valued Raviart-Thomas and standard Lagrange functions. Finally, a numerical example for liquid saturated porous structures considering an incompressible, linear elastic material behavior at small deformations, demonstrates the applicability of the LSFEM approach to the TPM.
The Modeling of non-Newtonian fluids is an essential issue for many industrial applications, for example in the field of chemistry, bioengineering and medical science. In this contribution we present a least-squares (LS) finite element approach to model steady flow of incompressible non-Newtonian fluids based on the Navier-Stokes equations. The application of the least-squares finite element method (LSFEM) especially in the case of fluid mechanics is motivated by some theoretical advantages compared to the well-known (mixed) Galerkin method. The LSFEM is not restricted to the LBB-condition and results in a minimization problem with symmetric positive definite equation systems also for differential equations with non self-adjoint operators. Furthermore, the LSFEM provides an inherent a posteriori error estimator without additional costs, which enables an efficiency enhancement through adaptive mesh refinements. In this contribution the first-order LS formulation in terms of stresses and velocities, as introduced in [4], is extended to consider the nonlinear dependence of the viscosity on the shear rate of the fluid. Therefore, the well-known Carreau model for generalized Newtonian fluids, see for instance [1], is investigated. For the approximation of the independent variables, velocities and stresses, we apply conforming discretizations in H 1 and H(div) using standard Lagrange interpolation polynomials and vector-valued Raviart-Thomas interpolations functions. A flow through a square domain with exact solution is considered to validate the proposed schemes with respect to accuracy and efficiency.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.