Stress sensitivity analysis constitutes an essential problem in gradient-based structural shape optimization. Unlike the traditional grid perturbation method (GPM), a general material perturbation method (MPM) using a fixed mesh is originally developed to simplify the sensitivity analysis scheme in this work. A domain function is introduced to characterize the boundary perturbation, whose effect is considered by correcting simultaneously stiffness matrices and stresses of elements attaching the perturbed boundary. Implementations of the MPM on shape optimization of plane stress, axisymmetric, 3D and thin-walled curved shell problems show that the proposed method has the advantage of efficient and explicit computing of stress sensitivities.
Shape optimization is one of the most challenging problems in structure designs. Many researchers have devoted themselves in the community [1], [2], [3], [4], [5] and [6]. As the most efficient algorithms, the difficult and essential part of gradient-based algorithms is sensitivity analysis. Usually, it is carried out by using the grid perturbation method (GPM) to obtain the so-called velocity field [7], [8] and [9] for the determination of modified nodal locations in response to the boundary perturbation. Different approaches such as mesh mapping, Laplacian morphing, boundary nodal perturbation and physical approach using fictive displacement [10], [11], [12], [13] and [14] were largely utilized to this aim. The underlying assumption is that any boundary perturbation only changes the mesh shape of a domain, i.e. nodal positions, while the material property of each element is firmly attributed to the element mesh and remains unchanged.
To simplify the design and sensitivity analysis procedure as easily as in topology optimization, attempts were made to extend the concept of Fixed Grid (FG) representation [15] into shape optimization problems [16]. As illustrated in Fig. 1, a rectangular base domain is defined to envelop the considered structure. One such base domain is fully discretized into a structured finite element mesh that can be classified into three subsets: elements inside the structure domain, outside the structure domain and crossed by the domain boundary. The mesh is fixed not only at the step of sensitivity analysis but also at all iteration steps. The main concern is about how to deal with the elements crossed by the moving boundaries. Usually, averaged material properties weighted by the area fraction are assigned to the concerned boundary elements and then used to calculate the stiffness matrix. Dunning et al. [17] proposed a weighted least squares method to improve the accuracy of responses with a weighting function based on both area-fraction and the distance of the sampling point to the boundary. As indicated in their work, the Area-fraction weighted Fixed Grid (AFG) approach may cause poor sensitivity computation of the elemental stress and further lead to some unstable problem of optimization convergence. Local stresses thus obtained are poorly approximated even for a much refined mesh [15] so that the accuracy of corresponding stress sensitivities is worse than expected. This is why the FG representation was mainly limited to the minimization of the structural compliance.
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Fig. 1.
Distinction between the GPM with a moving mesh and the FG representation with a fixed mesh: (a) the GPM with a moving mesh; (b) the FG reprentation with a fixed mesh.
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In recent years, stress-based optimization in the framework of a fixed mesh receives much attention. García and Steven improved the stress accuracy greatly [18] by employing a FG global/local analysis, i.e., refining local mesh around the boundary. Kim and Chang [19] studied the stress sensitivities using a fixed mesh as Eulerian representation for shape optimization. The developments of XFEM [20], [21], [22] and [23] and IGA [4] and [24] techniques are expected to provide more facilities for shape optimization.
In this paper, the so-called MPM is developed to perform stress sensitivity analysis. The main difference between the proposed method and existing ones is twofold. First, only the structure domain is meshed. The fixed grid approach is only used for sensitivity analysis, while the mesh is updated to track the revised boundary in the usual way during shape optimization iterations. The mesh updating relies upon the preprocessor of the commercial finite element software to make sure that the mesh quality and the FE analysis are qualified for the downstream optimization. This mesh strategy was used in the work of Xia et al. [25] to minimize the structural compliance by changing the layout of components. Second, accuracies of stress sensitivities are improved through stress corrections of specific boundary elements attaching the moving boundary. It is shown that proper corrections of both element stiffness matrices and stress components are essential to achieve a good accuracy of stresses and their sensitivities. A variety of examples are employed to illustrate the MPM.
In this paper, a material perturbation method (MPM) using a fixed mesh is proposed for stress sensitivity analysis and shape optimization. A domain function is formulated to describe the boundary perturbation effect upon the elements. Stress corrections based on the domain function are carried out to ensure the accuracies of the effective stresses and their sensitivities caused by the material discontinuity. It is shown that only the stress components parallel to the moving boundary need to be corrected. A comparative study is systematically carried out among the GPM, BGPM and MPM. The MPM demonstrates its advantage in avoiding the sophisticated velocity field calculation for the shape perturbations of nodal locations. Representative numerical examples are tested for 2D, axisymmetric, 3D and thin-walled curved shell problems and stable convergences for stress-based shape optimization are efficiently achieved. Stress sensitivities related to the MPM are found to be similar to those of the BGPM. Meanwhile, results show that stress corrections have a remarkable effect on the accuracy of stress sensitivities in order that their values are bounded and kept with correct signs.
