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This study aims to optimize topology of structures at macro and micro scales, simultaneously, by using a level set method in an isogeometric analysis (IGA) framework. To achieve this, equilibrium and homogenization equations in the model are solved by IGA method. The level set functions are defined over a grid in parameter space of associating b-splines of the IGA model. Therefore, control net of the model and level set grid are separated and there is no need to refine the control net for having smooth boundaries. Sensitivity analyses for both scales are performed to calculate the velocity of boundary points and the level set functions are updated by solving reaction-diffusion equations. Finally, several 2D and 3D examples with different geometry and boundary conditions are provided to show performance and efficiency of the method. Obtained results show good agreement with examples in literature in terms of both topology and final value of objective function. Also, by using IGA level set method, smooth boundaries are achieved in the final topology of micro and macro structures. • Isogeometrical topology optimization is used to design structures in two micro and macro scales, simultaneously. • Homogenization equations are solved by isogeometric analysis. • Sensitivity analyses for two scales are performed and level set functions are updated through reaction-diffusion equations. • Numerical examples show time effectiveness and performance of the proposed method. [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 work uses the zero-level contour of a parameterized level set function, a linear combination of cubic B-spline basis functions, to express the structural profile in structural topology optimization. Together with mean compliance, diffusion energy is minimized under a volume constraint to control the structural complexity. The design variables, namely the coefficients of cubic B-spline basis functions, are updated by solving the reaction–diffusion equation within a finite element analysis framework. The bisectional algorithm accurately calculates the Lagrangian multiplier of the volume constraint in each iteration. In addition to expressing the optimized structure smoothly, the proposed method is highly efficient. For instance, it only takes 20 iterations to solve the cantilever and MBB beams in 2D. For 3D optimization, we obtain several elegant bridge designs using nearly one million elements, demonstrating the great potential of the proposed method for practical applications. [Display omitted] • Smoother structures expression and fewer optimization iteration steps. • Flexible structural complexity control. • Avoid re-initialization of the level set function. • A simple convolution process in retrieving the level set function. • Promising engineering applications. [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]