A level set method for shape optimization in semilinear elliptic problems

We develop a finite-element based level set method for numerically solve shape optimization problems constrained by semilinear elliptic problems. By combining the shape sensitivity analysis and level set method, a gradient descent algorithm is proposed to solve the model problem. Different from solving the nonlinear Hamilton–Jacobi equations with finite differences in traditional level set methods, we solve the linear convection equation and reinitialization equation using the characteristic Galerkin finite element method. The methodology can handle topology as well as shape changes in both regular and irregular design regions. Numerical results are presented to demonstrate the effectiveness of our algorithm as well as to verify symmetry preserving and breaking properties of optimal subdomains.