Topology optimization of planar and plate structures using conformal mapping

This paper proposes a new method to determine the optimum topology of planar and plate structures using conformal mapping. First it is proved that two invariants of stresses, which are the sum and difference of principal stresses, satisfy the Laplace equation. Additionally, it is proved that two invariants of bending moments are the sum and difference of principal bending moments, and is proved that they satisfy the Laplace equation under certain condition. Finally, we show that corresponding relationships between fluid mechanics and electromagnetism are valid to the theory of planar elasticity and plate bending. From these considerations, a convenient design method to optimize topologies of planar and plate structures for complicate design domains is proposed using conformal mapping. Several numerical examples of optimum topologies are treated by the proposed method. Through numerical results the effectiveness and validity of proposed method are discussed quantitatively and qualitatively.