Parametric structural shape & topology optimization with a variational distance-regularized level set method

Abstract

The signed distance function (SDF) gives the shortest distance from a given point to the boundary, and the sign indicates whether this point is inside or outside the closed boundary or enclosed region. The SDF property is highly preferred in classical level set methods to maintain the numerical stability during the topology optimization process and provide a metric for the distance-based interpolation of different material properties. In conventional level set methods, a common way of achieving a level set function with the signed distance property is to periodically implement the so-called reinitialization scheme by solving an additional Hamilton–Jacobi partial differential equation. However, such reinitialization scheme is implemented outside the optimization loop with the optimization process suspended, which may shift the optimization result and bring convergence issues. In this paper, a double-well potential functional is employed for distance regularization inside the topology optimization loop, which can enforce the signed distance property of the level set function in a narrow band along the design boundaries while keeping the level set function flat in the rest area of the computational domain. The radial basis function (RBF) based parameterization technique is combined with mathematical programming to improve the performance of the proposed distance-regularized topology optimization method in handling problems with non-convex objective functions and multiple constraints. The flatness of the level set function in the material region also enables easy creation of new holes to the design in the topology optimization process. Both 2D and 3D benchmark examples are employed to demonstrate the validity of the proposed method.