(A) study on the effective non-matching interface treatment methods in multi-structure problems and their applications to topology optimization

Abstract

In this thesis, a study on the effective non-matching interface treatment methods based on a mortar method is introduced. The mortar method imposes displacement compatibility conditions along the interface using Lagrange multipliers. However, in this method, due to the introduction of the Lagrange multipliers, the system equation loses the positive-definiteness and becomes numerically unstable. To solve these numerical problems, in this thesis, the condensed mortar method is proposed by explicitly eliminating the Lagrange multipliers as well as the overlapped interface displacements in the finite element formulation. Therefore, the standard form as in the general structural formulation consisting of only essential parts of displacements is obtained. Among the mortar methods, the localized mortar method is a scheme that introduces a fictitious frame to precisely impose interface compatibility condition. In the localized mortar method, previously developed schemes of defining frame nodes are easy to apply when the interface is geometrically simple. However, these schemes are difficult to apply when the geometry of the interface is complicated especially in three-dimensional cases. Therefore, to overcome this limitation, new allocation schemes for defining the frame nodes are proposed. By using the developed methods, a topology optimization for problems where two adjoined subdomains are joined or contacted is studied. First, to adequately impose the interface compatibility condition in the non-matching interface, the condensed mortar method is used. When the condensed mortar method is applied, the Lagrange multipliers and overlapped interface displacements are disappeared so that the overall system can be handled in the same manner as in the usual structural problems. Therefore, the conventional topology optimization process can be used without any modifications or considerations even when non-matching interfaces exist. When two meshes with different degrees of refinement are joined along the interface, the results of the topology optimization show great mesh-dependency. For alleviating the dependency, a modified filtering scheme is developed by introducing the area of each element in the existing weight function. Through the proposed topology optimization approach, several examples with non-matching interface meshes are presented.