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In conventional level set methods for topology optimization, the evolution of level set function is solved by upwind scheme and needs to be reinitialized frequently throughout the optimization. The upwind scheme is complicated in numerical implementation and the reinitialization is time-consuming and slows down the optimization efficiency. This paper presents a level set method updated with finite difference scheme for structural topology optimization. The velocity field used to update the level set function is interpolated by piecewise basis function in order to incorporate the universal mathematical programming algorithm into the optimization. The diffusion is introduced into the Hamilton-Jacobi equation and the three-step splitting method is adopted to solve the equation. By applying the proposed method, the evolution of the level set can be updated by simple central difference scheme and the procedure of reinitialization is avoided without sacrificing the numerical stability. The classical mean compliance minimization problem with several numerical examples is used to test the validity of the proposed method. [ABSTRACT FROM AUTHOR]

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]

A level-set based multi-material topology optimization method using a reaction diffusion equation is proposed in this paper. Each phase is represented by a combined formulation of different level set functions. This description model is modified from Multi-Material Level Set (MM-LS) topology description model. With a total number of M level set functions, this approach provides a representation of M materials and one void phase (totally M + 1 phases). By this approach, the mathematic model of the multi-material topology optimization problem using a reaction diffusion equation is established. With this model, the geometrical complexity of optimal solutions can be easily controlled by appropriately setting a regularization parameter. Some implementation details for solving this model are also presented in this paper. Finally, several typical numerical examples are shown to confirm the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

This paper presents an accurate implicit re-initialization approach in the framework of a level set method. The improved method includes two schemes for keeping the zero level set unperturbed. The first scheme is to derive and use a new formula for the smoothing parameter in the conventional re-initialization equation based on the principle of the interface not moving. The second scheme is to reduce the local time step to avoid the interface moving across grid points when the sign of the level set function near the interface is changed. The example presented suggests that the new algorithm has a better approximation to the signed distance function and obtains a more accurate interface than the algorithm presented by Sussman et al. (1994). For shape and topology optimization of fluid, the effectiveness of the developed method in this paper was demonstrated by the internal and external flow examples. [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]