A unified simultaneous shape and topology optimization method for multi-material laminated shell structures

In this paper, a simultaneous shape and topology optimization method is presented for designing multi-material structures. The whole shape and the layer's material distributions of a laminated shell structure composed of multi-materials are optimized. The free-form optimization method for shells and the generalized solid isotropic material with penalization (GSIMP) method are respectively employed and combined effectively for shape and topology optimization. Shape along with fictitious homogenized-density variations are used as design variables and simultaneously determined. In other words, the optimal topology is determined in the variable design surface optimized by shape optimization. Compliance is used as the objective functional and minimized under the volume and the area constraints for each material. The optimal design problem is formulated as a distributed-parameter optimization problem, and the sensitivity functions with respect to shape and density variations are theoretically derived. Both the optimal shape and density variations are determined with the unified H-1 gradient method, where the sensitivity functions are respectively applied as the Robin condition, to the design surface and the domain in order to determine the optimal shape and topology simultaneously. Several numerical results including a comparison with the non-simultaneous methods are presented to show the effectiveness of the proposed method. With the proposed method, the optimal lighter and stiffer multi-material laminated shell structure can be obtained without any design parameterization, free of numerical instabilities such as checkerboard pattern and zigzag shape problems.