Spectral Level Set Methodology in MDO

Abstract

A desirable feature in multidisciplinary design optimization (MDO) is to achieve an acceptable preliminary or initial design in an expeditiously fashion. One way to pursue this goal is to describe the structural boundary using a small number of design variables. We propose an approach to structural optimization that enhances the performance of multidisciplinary design tools because it greatly decreases the number of design variables assigned to structural definition. This approach is based on the Level Set Methods and uses the coefficients of the Fourier series expansion of the level set function as design variables. The global character of these coefficients and the fact that the level set function is realvalued provide the means to reduce the design space dimension. Within the level set framework, the structure is confined to the region in which the level set function is negative. In this way, the zero level set of this function implicitly describes the structural boundary, which can undergo topological changes during the optimization process as the function evolves. The proposed methodology also averts successive mesh generation. We show that an interface can be described to any desirable accuracy using finitely many Fourier modes of the signed distance to the interface. The total error is inversely proportional to the number of Fourier modes. We present and discuss two application examples in topology optimization, namely the design of short and long cantilevers subject to a pontual load. In these cases, we analyze the results obtained with the Spectral Level Set formulation for different parameter values.