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In the present paper, an integrated paradigm for topology optimization on complex surfaces with arbitrary genus is proposed. The approach is constructed based on the two-dimensional (2D) Moving Morphable Component (MMC) framework, where a set of structural components are used as the basic units of optimization, and computational conformal mapping (CCM) technique, with which a complex surface represented by an unstructured triangular mesh can be mapped into a set of regular 2D parameter domains numerically. A multi-patch stitching scheme is also developed to achieve an MMC-friendly global parameterization through a number of local parameterizations. Numerical examples including a saddle-shaped shell, a torus-shape shell and a tee-branch pipe are solved to demonstrate the validity and efficiency of the proposed approach. It is found that compared with traditional approaches for topology optimization on 2D surfaces, optimized designs with clear load transmission paths can be obtained with much fewer numbers of design variables and degrees of freedom for finite element analysis (FEA) via the proposed approach.

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