Design Of Topology Optimization For Cross Section Using Moving Morphable Components Approach

Thin-walled beams are extensively applied in the various engineering structures,in which the conceptual design of cross-sectional shape and topology is the most important issue.Traditional topology optimization methods cannot obtain the optimized results with the thin-walled features to satisfy the stamping process.Therefore,this paper proposes a thin-walled cross-sectional design method using the moving morphable components(MMC)approach.In this paper,the cross-sectional area is selected as the objective function of topology optimization,and the bending and torsional moments of inertia are taken as the constraint functions to establish the optimization formulations.The arbitrary shape cross-sectional analytical equations of the bending and torsional moments of inertia are given for the Euler-Bernoulli beams.The cross-sectioanal bending moments of inertia equations are obtained according to the Euler-Bernoulli beam theory;The cross-sectioanal stress function is solved by using the semi-inverse method,and the cross-sectioanal torsional moment of inertia equation is obtained by referring to the solution of the circular cross section bar in the elastic mechanics.The arbitrary shape cross-sectional numerical solution equation of the bending and torsional moments of inertia are deduced in this paper,in which the cross-sectional stress function related to the numerical solution equation of torsional moment of inertia is solved by using the finite element method,which is established by the differential equation and boundary condition.The element stiffness matrix and element node matrix of bilinear element and arbitrary shape convex quadrilateral element are derived in this paper.Moreover,this paper introduces the topology optimization method of MMC approach,and calculates the numerical equations and sensitivity analysis equations of the cross-sectional area,bending and torsional moments of inertia under the MMC framework.The finite element algorithm for the cross-sectional bending and torsional moments of inertia,and the topology optimization method of thin-walled cross section are also programmed in this paper.The accurancy of finite element solution for cross-sectional bending and torsional moments of inertia and the feasibility of topology optimization for the thin-walled cross section using the MMC approach are verified through the numerical examples.The optimized results show that the method proposed in this paper can realize the theoretical process of thin-walled cross section from scratch,and the optimized thin-walled cross section can satisfy the stamping process after post-processing.