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In the present paper, the task of designing a compliant mechanism is accomplished by solving a multi-objective topology optimization problem. The objective function includes a target displacement at the output port and compliances of two load cases. The design variables include two types of boundaries, i.e., the traction free boundary and the Dirichlet boundary. The topology optimization of compliant mechanism and its support is solved by using a level set method. Two level set functions are used to represent a compliant mechanism and the two types of boundaries. Each of the two boundaries is represented by a part of the zero level set of one level set function. Evolution of the two boundaries is modeled by two independent Hamilton–Jacobi equations, thus the shape and topology of the two boundaries change independently. The shape derivatives are derived by using a Lagrangian function and the adjoint method. The finite element analysis is done by modifying a fixed background mesh, and the artificial weak material is not used. Numerical examples in two dimensions are investigated. [ABSTRACT FROM AUTHOR]

Abstract: This paper proposes a new topology optimization method, which can adjust the geometrical complexity of optimal configurations, using the level set method and incorporating a fictitious interface energy derived from the phase field method. First, a topology optimization problem is formulated based on the level set method, and the method of regularizing the optimization problem by introducing fictitious interface energy is explained. Next, the reaction–diffusion equation that updates the level set function is derived and an optimization algorithm is then constructed, which uses the finite element method to solve the equilibrium equations and the reaction–diffusion equation when updating the level set function. Finally, several optimum design examples are shown to confirm the validity and utility of the proposed topology optimization method. [ABSTRACT FROM AUTHOR]

We present a level-set based topology optimization algorithm for design optimization problems involving an arbitrary number of different materials, where the evolution of a design is solely guided by topological derivatives. Our method can be seen as an extension of the algorithm that was introduced in Amstutz and Andrä (2006) for two materials to the case of an arbitrary number M of materials. We represent a design that consists of multiple materials by means of a vector-valued level set function which maps into R M − 1. We divide the space R M − 1 into M sectors, each corresponding to one material, and establish conditions for local optimality of a design based on certain generalized topological derivatives. The optimization algorithm consists in a fixed point iteration striving to reach this optimality condition. Like the two-material version of the algorithm, also our method possesses a nucleation mechanism such that it is not necessary to start with a perforated initial design. We show numerical results obtained by applying the algorithm to an academic example as well as to the compliance minimization in linearized elasticity. • Multi-material topology optimization based on topological derivatives. • Arbitrary number of materials. • Local topological optimality condition for multi-material design. • Application to structural optimization. [ABSTRACT FROM AUTHOR]