Dirichlet boundary conditions in sequences of Cartesian grids using a stabilized Lagrange multipliers technique based on recovered tractions

Autores UPV
Año
CONGRESO Dirichlet boundary conditions in sequences of Cartesian grids using a stabilized Lagrange multipliers technique based on recovered tractions

Abstract

The procedures used in the standard FEM to apply the boundary conditions cannot be used within the framework of immersed boundary methods, as in these methods the FE nodes do not generally lay on the boundary. The case of the Neumann boundary conditions is solved taking into account that the elements can be cut by the integration surface not necessarily on element faces, as they do in the standard FEM. However, the case of the Dirichlet boundary conditions is much more complex. To solve the problem, a common alternative is to use the Lagrange multiplier technique. Nevertheless, it can be difficult to find a compatible discretization of displacements and multipliers that satisfies the InfSup condition. In practice, the main problem appears when the number of Lagrange multipliers is too high, which can cause undesired oscillations in the Lagrange multiplier field. This is an open problem, and several approaches based on stabilized methods have been presented recently. The alternative implemented in this work consists of a stabilized Lagrange multiplier method suitable for cases where a sequence of successively h-refined hierarchical Cartesian grids meshes [1] is used in the analysis. In this contribution the stabilization term are the tractions evaluated from the recovered field [2] on the Dirichlet boundary. The solution for each mesh i is obtained via an efficient iterative process (the system matrix remains constant through the process) where the recovered tractions are updated in each step [3]. The hierarchical data structure of the Cartesians grids allows to easily and efficiently project the recovered tractions evaluated in mesh i-1 to the current mesh i in order to initialize the iterative process. REFERENCES [1] E. Nadal, J. J. Ródenas, J. Albelda, M. Tur, J. E. Tarancón, F. J. Fuenmayor. Efficient Finite Element Methodology Based on Cartesian Grids: Application to Structural Shape Optimization. Abstract and Applied Analysis. Article ID 953786, 19 pages. 2013 [2] J.J. Ródenas, M. Tur, F.J. Fuenmayor, A. Vercher, Improvement of the Superconvergent Patch Recovery technique by the use of constraint equations: the SPR-C technique. IJNME. 70(6): 705-727. 2007 [3] M. Tur, J. Albelda, O. Marco and J. J. Ródenas, Stabilized method of imposing Dirichlet boundary conditions using a smooth stress field, CMAME, under review.