Integrated shape and topology optimization with variable design domain for stiffening structures

This paper presents an approach to create stiffeners for thick-walled 3D linear elastic structures by using variable design domain topology optimization. The layout and shape of stiffeners are simultaneously designed within the variable design domain which is optimized by shape optimization at each iteration. The SIMP-based topology optimization determines the generative regions of stiffeners within the variable design domain, and the shape optimization determines the optimal detailed shape while sequentially growing the stiffeners. Under the volume and equilibrium equation constraints, compliance is minimized to stiffen a structure. After formulating this design optimization problem as a distributed-parameter optimization problem, the sensitivity functions for shape and topology are derived using the Lagrange multiplier method, the material derivative method and the adjoint method. The derived sensitivity functions are applied to the vector and scalar types of H¹ gradient method to determine the optimal shape and topology of stiffeners. The both types of H¹ gradient method serve, enabling the achievement of a smooth optimal external shape while concurrently addressing potential issues related to grayscale and checkerboard patterns, as well as reducing the objective function. The effectiveness of this method is demonstrated through several design examples.