Topological Shape Optimum Design of Structures via X-FEM and Level Set Method

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In this study, a structural topological shape optimization design via the X-FEM and zero level sets is presented. Displacement fields of two-phase topology optimization problems are defined by a weak discontinuity with bisected supports. In order for X-FEM to appropriately be associated with a classical topology optimization algorithm with density design parameters, the design parameters are transformed into control parameters in X-FEM. Then, all elements including enriched elements near material interfaces, which are searched by the control parameters, contain signed distance functions or level set functions at each node. The nodal signed distance functions are bilinearly interpolated, and then, Gauss point signed distance functions determine material properties at Gauss points, i.e. almost complete voids (0.001) or solids (1). Up-wind scheme is used to update the level set functions including zero level set functions, which describe moving material interfaces, followed by shape sensitivity of optimization problems via level set functions. Numerical applications verify the present method produces superior design solutions like smooth material interfaces through considering both mechanically X-FEM and geometrically level set method in comparisons with jagged optimal density distribution results of classical topology optimization.