Abstract

Inhomogeneous essential boundary conditions must be carefully treated in the formulation of Reduced Order Models (ROMs) for non-linear problems. In order to investigate this issue, two methods are analysed: one in which the boundary conditions are imposed in an strong way, and a second one in which a weak imposition of boundary conditions is made. The ideas presented in this work apply to the big realm of a posteriori ROMs. Nevertheless, an a posteriori hyper-reduction method is specifically considered in order to deal with the cost associated to the non-linearity of the problems. Applications to nonlinear transient heat conduction problems with temperature dependent thermophysical properties and time dependent essential boundary conditions are studied. However, the strategies introduced in this work are of general application.

Abstract

Imposing essential boundary is a key issue in mesh-free methods. The mesh-free interpolation does not verify the Kronocker delta property and, therefore, the imposition of prescribed values is not as straightforward as for the finite element method. The aim of this paper is to present a general overview on the existing techniques to enforce essential boundary conditions in Galerkin based mesh-free methods. Special attention is paid to the mesh-free coupling with finite elements for the imposition of prescribed values and to methods based on a modification of the Garlekin weak form. Particular examples are used to analyze and compare their performance in different situations.

Abstract

In this work, we propose a method to prescribe essential boundary conditions in the finite element approximation of elliptic problems when the boundary of the computational domain does not match with the element boundaries. The problems considered are the Poisson problem, the Stokes problem, and the Darcy problem, the latter both in the primal and in the dual formulation. The formulation proposed is of variational type. The key idea is to start with the variational form that defines the problem and treat the boundary condition as a constraint. The particular feature is that the Lagrange multiplier is not defined on the boundary where the essential condition needs to be prescribed but is taken as a certain trace of a field defined in the computational domain, either in all of it or just in a region surrounding the boundary. When approximated numerically, this may allow one to condense the DOFs of the new field and end up with a problem posed only in terms of the original unknowns. The nature of the field used to weakly impose boundary conditions depends on the problem being treated. For the Poisson problem, it is a flux; for the Stokes problem, a stress; for the Darcy problem in primal form, a velocity field; and for the Darcy problem in dual form, it is a potential. 

Abstract

Imposing essential boundary conditions is a key issue in mesh-free methods. The mesh-free interpolation does not verify the Kronecker delta property and, therefore, the imposition of prescribed values is not as straightforward as for the finite element method. The aim of this paper is to present a general overview on the existing techniques to enforce essential boundary conditions in Galerkin based mesh-free methods. Special attention is paid to the mesh-free coupling with finite elements for the imposition of prescribed values and to methods based on a modification of the Galerkin weak form. Particular examples are used to analyze and compare their performance in different situations.