Two variational formulations for elastic domain decomposition problems solved by SGBEM enforcing coupling conditions in a weak form
The solution of Boundary Value Problems of linear elasticity using a domain decomposition approach (DDBVPs) is considered. Some theoretical aspects of two new energy functionals, adequate for a formulation of symmetric Galerkin boundary element method (SGBEM) applied to DDBVPs with non-conforming meshes along interfaces, are studied. Considering two subdomains ?A ? A and ?B ? B , the first functional, E(uA,uB) E ( u A , u B ) , is expressed in terms of subdomain displacement fields, and the second one, ?(uA,uB,tA,tB) ? ( u A , u B , t A , t B ) , in terms of unknown displacements and tractions defined on subdomain boundaries. These functionals generalize the energy functionals studied in the framework of the single domain SGBEM, respectively, by Bonnet [Eng Anal Boundary Elem 1995;15:93102] and Polizzotto [Eng Anal Boundary Elem 1991;8:8993]. First, it is shown that the solution of a DDBVP represents the saddle point of the functional E . Second, it is shown that the solution of an SGBEM system of boundary integral equations for a DDBVP corresponds to the saddle point of the functional ? ? . Then, the functional ? ? is considered for the finite-dimensional spaces of discretized boundary displacements and tractions showing that the solution of the SGBEM linear system of equations represents the saddle point of ? ? , generalizing in this way the boundary minmax principle, introduced by Polizzotto, to SGBEM solutions of DDBVPs. Finally, a relation between both energy functionals is deduced.
domain decomposition symmetric galerkin boundary element method saddle point boundary integral equations minmax principle variational principles potential energy non-matching meshes