Structural topology optimization based on non-local Shepard interpolation of density field

This paper presents a non-local density interpolation strategy for topology optimization based on nodal design variables. In this method, design variable points can be positioned at any locations in the design domain and may not necessarily coincide with elemental nodes. By using the Shepard family of interpolants, the density value of any given computational point is interpolated by design variable values within a certain circular influence domain of the point. The employed interpolation scheme has an explicit form and satisfies non-negative and range-restricted properties required by a physically significant density interpolation. Since the discretizations of the density field and the displacement field are implemented on two independent sets of points, the method is well suited for a topology optimization problem with a design domain containing higher-order elements or non-quadrilateral elements. Moreover, it has the ability to yield mesh-independent solutions if the radius of the influence domain is reasonably specified. Numerical examples demonstrate the validity of the proposed formulation and numerical techniques. It is also confirmed that the method can successfully avoid checkerboard patterns as well as “islanding” phenomenon.