Development and validation of a numerical topology optimization scheme for two and three dimensional structures

A novel finite element topology optimization procedure is presented based on the application of probability density and cumulative distribution functions. The procedure utilizes a family of Beta functions with constant probability mean which provide a smooth transition from a uniform to a bi-modal density distribution while conserving constant mean density and therefore constant mass. Validation of the method is demonstrated for several well-known two-dimensional minimum-weight structures. A general minimum-weight cylindrical structural layout for the support of any combination of axial and torsional loading has been developed to provide a test case for three dimensional numerical topological optimization. It is observed that this solution presents a challenge, especially for cases where the axial load is significantly larger than the torsional loading. For these cases, slender members are an essential part of the optimal topology.