Topological Microstructural Optimization. ANSYS dataset for rectangular cavities

Topological optimization software does not build a model of an object from scratch. It loads a geometric model of a product that was previously manufactured using another method. The process of building a new model is quite time consuming and requires large computational resources. This modeling method is called the finite element method. For each point of the product, the program makes and solves integral equations, taking into account the relationships between all points. The calculation results in a new geometry. The designer can then change something in the model if necessary. The end result is a CAD model. Next, for verification, the model is loaded into other software, where it undergoes a final check for maximum deformations, stresses, and so on. The elastic medium is referred to as “matrix material” in order to distinguish it from the inhomogeneities. The role of the inhomogeneities play cavities, or voids. Several different variants of “density-based” topology optimization procedures are possible for the numerical simulation purposes. In general case, the shapes of cavities could have the arbitrary form and must not be circular or spherical. The isotropic elastic material can turn into the transversely isotropic or orthotropic material for the rectangular cavities. The rectangular voids are oriented in the directions of the coordinate axes x_1 ,x_2 . The sizes a,b of the voids in the directions of the coordinate axes x_1 ,x_2 . The inclusions are located double periodically in the chess-board order. The distances between the centers in both directions are equal respectively A,B. The representative area of material cell, which contains one rectangular cavity, and the area of the cavity are equal correspondingly to A*B. According to definition, porosity is the ratio of area of a cavity to the area of the representative area: p≝ab/AB. The solution determines the optimal shape of the rectangular hole for the extremal elastic energy density. The results be used for the programming of the algorithm of the topology optimization. For the illustration of the technique, the optimization was performed with the commercial finite-element program. For the simulation was chosen the proprietary finite-element system ANSYS 2023.

拓扑优化软件并非从零开始构建物体模型，而是加载通过其他方法预先制造的产品几何模型。构建新模型的过程耗时极久且需大量计算资源，该建模方法即为有限元法（finite element method）。针对产品的每一节点，程序会构建并求解积分方程组，同时兼顾所有节点间的相互作用关系。计算结果将输出新的几何构型，后续设计者可根据实际需求对模型进行修改，最终得到CAD（Computer Aided Design，计算机辅助设计）模型。随后为完成验证工作，该模型将被导入其他软件，以对最大变形、应力等参数进行最终校验。弹性介质被称为“基体材料”，以便与非均匀相区分；非均匀相则由孔洞（voids，即空隙）充当。针对数值模拟场景，可实现多种基于密度的拓扑优化流程变体。通常情况下，孔洞的形状可为任意形式，无需局限于圆形或球形。当存在矩形孔洞时，各向同性弹性材料可转化为横观各向同性或正交各向异性材料。矩形空隙沿坐标轴x₁、x₂方向定向排布，其在x₁、x₂方向的尺寸分别为a与b。夹杂物以棋盘格布局实现双重周期排布，两个方向上的中心间距分别为A和B。包含单个矩形孔洞的代表性材料单元的面积与孔洞面积分别为A·B与ab。根据定义，孔隙率为孔洞面积与代表性单元面积的比值，即p ≜ ab/(AB)。该求解过程可确定对应极值弹性应变能密度的矩形孔洞最优构型，所得结果可用于拓扑优化算法的程序开发。为演示该技术方法，本次优化采用商用有限元软件开展；数值模拟则选用了专有有限元系统ANSYS 2023。