Two kinds of design variables, i.e., pseudo-density variables associated with the framework structure and location design variables associated with connected components are involved in the layout design of multi-component systems. Although sensitivities with respect to the first ones can easily be carried out as in topology optimization, the semi-analytical method (SAM) is often used for sensitivity analysis with respect to the location design variables. Due to the geometric perturbation of the finite element mesh, the latter can then be regarded as a geometric perturbation model (GPM). In this paper, we propose a material perturbation model (MPM) using fixed finite element (FE) mesh for sensitivity analysis with respect to location design variables. The material discontinuity across the boundary between each component and the framework structure is smoothed approximately by means of a modified Heaviside function. When a location design variable of a certain component is perturbed, attached finite elements to the component boundary are assumed to undertake only a shift of material properties while the finite element mesh itself remains geometrically unchanged. As a result, analytical sensitivities with respect to location design variables are achieved as easily as for pseudo-density variables. The computing efficiency is thus improved because the velocity field for the mesh perturbation in the semi-analytical scheme is no longer needed. The MPM is illustrated by means of numerical tests, especially the design optimization of 3D multi-component systems.
Topology optimization is one of the most motivated and challenging topics in the past decades and has been successfully applied to a variety of problems including designs for structural compliance, natural frequencies and buckling loads etc. [1]. Different schemes such as the homogenization-based model, solid isotropic material with penalization (SIMP), rational approximation of material properties (RAMP) and level-set method etc. were established [2], [3], [4], [5], [6] and [7].
The layout design of complex multi-component systems studied here can be regarded as an extension of existing topology optimization and is attracting much attention [8], [9], [10], [11], [12] and [13]. As shown in Fig. 1(a) and (b), the involved components might act as some functional or electronic devices with specific mechanical properties. When they are considered as non-designable elastic solids or voids with fixed positions, the problem can be dealt with directly by means of the classic topology optimization methodology. However, the challenging design is to perform the location optimization of the involved components and topology optimization of the framework structure simultaneously, as illustrated in Fig. 1(c). On the one hand, the given components should be properly embedded in a limited design space without overlap to satisfy the compactness or other geometric and physical conditions. On the other hand, an optimal configuration of the framework structure has to be figured out inside the system.
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Fig. 1.
Illustration of multi-component problem: (a) problem description, (b) topology design optimization and (c) topology optimization with movable components.
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If the layout design is concerned with the system compliance minimization, it is necessary to calculate its sensitivities with respect to pseudo-density design variables of the framework structure and location design variables of the components. In previous implementations [10], [11], [12] and [13], the sensitivities related to pseudo-density variables on the so-called density points are calculated in an analytical way. However, the sensitivities related to location design variables resorted to the semi-analytical approach, which is relatively time-consuming especially for large-scale 3D problems because the mesh shape perturbation has to be determined at each perturbation of each location design variable. In some cases, GPM may also result in mesh distortions and lead to analysis failures. Recently, the superelement technique is adopted to model the components and a sensitivity analysis approach based on it is proposed in favor of computing efficiency [14]. Even though it has been shown that the superelement technique based sensitivity approach greatly improves computing efficiency and is capable of dealing with a large number of components in 2D situation, it is still incapable of solving large-scale 3D problems for both reasons of low computing efficiency and parametric programming complexity.
In the structural optimization community, the Heaviside function and its modified versions were widely used in the level-set method [15] and [16] and density filters of SIMP based methods [17], [18] and [19]. In level-set method based topology optimizations, the Heaviside function was used to avoid regenerating the element mesh when the boundary changes during the iterative design process. In order to obtain a black and white solution, i.e., pseudo-density design variables converge to 0 or 1, Guest et al. [17] proposed a Heaviside function based density filter strategy. Later, Sigmund reformulated the Heaviside operator and proved that the Heaviside filters provide the best and most discrete designs among all morphology based schemes [18]. Based on the previous work, a volume preserving density filter based on Heaviside functions was proposed for a better efficiency and stability in optimization by Xu et al. [19]. Recently, the Heaviside function was also employed to describe the nonlinear relationship between stress and material modulus to smooth the constitutive discontinuity [20].
In this paper, we present an explicit and analytical sensitivity analysis approach with respect to location design variables based on the modified Heaviside function. The latter is used to smooth approximately the discontinuity of material properties over adjacent elements attaching the boundary of each movable component. Meanwhile, the perturbation effect of each location design variable is modeled as a material shift over a fixed FE mesh, i.e., the so-called MPM. Numerical tests prove that the MPM is easy to implement, reliable and able to improve the optimization efficiency greatly.
This work proposed a new sensitivity analysis approach with respect to the location design variables for the integrated layout design of multi-component systems. Two different sensitivity analysis models, i.e., MPM and GPM are illustrated and compared. It proves that the MPM using Heaviside function is applicable for the stiffness layout design. Compared with the GPM, the great advantage is the modeling simplicity with a fixed finite element mesh in sensitivity analysis. Further investigations have to be made for stress design problems.
