Symmetric Variational Formulation of BIE for Domain Decomposition Problems in Elasticity – An SGBEM Approach for Nonconforming Discretizations of Curved Interfaces

Abstract

An original approach to solve domain decomposition problems by the symmetric Galerkin boundary element method is developed. The approach, based on a new variational principle for such problems, yields a fully symmetric system of equations. A natural property of the proposed approach is its capability to deal with nonconforming discretizations along straight and curved interfaces, allowing in this way an independent meshing of non-overlapping subdomains to be performed. Weak coupling conditions of equilibrium and compatibility at an interface are obtained from the critical point conditions of the energy functional. Equilibrium is imposed through local traction (Neumann) boundary conditions prescribed on a subdomain situated at one side of the interface, and compatibility is imposed through local displacement (Dirichlet) boundary conditions prescribed on the other subdomain situated at the opposite side of the interface. No additional unknowns such as Lagrange multipliers are introduced. An SGBEM code for 2D elastic domain decomposition problems has been implemented. The effectiveness of the approach developed is documented by numerical examples involving non-matching linear boundary element meshes at the interfaces, where the accuracy is analyzed by comparing the numerical results obtained versus the analytical solutions and by evaluating the convergence rate of the error in the (discretized) integral L2-norm and maximum-normfor h-refinements of boundary element meshes. Finally, a theoretical analysis of a problem with an interior and an exterior subdomain is introduced to explain the observed behaviour of numerical results.