Imposing Dirichlet boundary conditions in hierarchical Cartesian meshes by means of stabilized Lagrange multipliers

Abstract

This is the pre-peer reviewed version of the following article: Tur, M., Albelda, J., Nadal, E. and Ródenas, J. J. (2014) AbstractThe use of Cartesian meshes independent of the geometry has some advantages over the traditional meshes used in the finite element method. The main advantage is that their use together with an appropriate hierarchical data structure reduces the computational cost of the Finite Element analysis. This improvement is based on the substitution of the traditional mesh generation process by an optimized procedure for intersecting the Cartesian mesh with the boundary of the domain and the use efficient solvers based on the hierarchical data structure. One major difficulty associated to the use of Cartesian grids is the fact that the mesh nodes do not, in general, lie over the boundary of the domain, increasing the difficulty to impose Dirichlet boundary conditions. In this paper, Dirichlet boundary conditions are imposed by means of the Lagrange multipliers technique. A new functional has been added to the initial formulation of the problem that has the effect of stabilizing the problem. The technique here presented allows for a simple definition of the Lagrange multipliers field, that even allow us to directly condense the degrees of freedom of the Lagrange multipliers at element level.

CutFEM: Discretizing geometry and partial differential equations

“…The use of Nitsche's method for unfitted interface problems and fictitious domain methods has been developed in [5-13, 28, 29, 48]. Other related approaches based on Lagrange multipliers or discontinuous Galerkin methods have been suggested in the following works [14][15][16][17][18][19][20].…”

A modified perturbed Lagrangian formulation for contact problems

“…In [51] an implicit definition of the multiplier field was introduced for 2D elements, based on the value of the multiplier at the quadrature points of the surface used to numerically evaluate the boundary integrals. The Dirichlet boundary conditions in immersed boundary elements problem was analyzed in this work.…”

“…However, as the problem is a minimization with respect to the displacements and n · σ(u)n linearly depends on this field, the negative sign may cause a non desired behavior of the stabilization term. A possible solution with standard finite elements is to bound this term by taking a sufficiently large value for κ so that this integral can be bounded by the strain energy [51]. In the case of immersed boundary elements there are some difficulties involved in bounding this term, particularly in meshes with cut elements that have a very small volume/surface ratio, as pointed out in [25].…”

“…In [19] the idea of condensing the Lagrange multipliers to obtain a simplified method for immersed boundaries was introduced. It was also used in [51] using the quadrature points. The same ideas were applied in [5] using a stabilizing field defined in the volume of the element instead of its surface.…”

Cartesian grid FEM (cgFEM): High performance h-adaptive FE analysis with efficient error control. Application to structural shape optimization

“…In this case we do not use any stabilization procedure because for the particular case of the examples used in this thesis it is not necessary. An improvement of the proposed method including a stabilization term is recently introduced in [56]. Consider that the solution of the problem (2.8) is equivalent to the following one, where we introduce the Lagrange multipliers field λ to impose the Dirichlet boundary conditions:…”