Optimal grading of TPMS-based lattice structures with transversely isotropic elastic bulk properties

In this work, a topology optimization (TO) based framework for functional grading of triply periodic minimal surfaces (TPMS) based lattice structures is developed, implemented and demonstrated. Material interpolation laws of the gyroid, G-prime and Schwarz-D surfaces are derived by numerical homogenization for transversely isotropic elasticity and are represented as convex combinations of solid isotropic material with penalization (SIMP) and rational approximation of material properties (RAMP) models. These convex combinations are implemented in the TO-based compliance problem with new upper and lower bounds on the density variables representing the volume fraction limits of the lattices. The lower bound on the density variables is treated by introducing a sigmoid filter in the optimization loop forcing densities below the lower boundary towards zero. The optimal density solution is represented by Shepard interpolations or radial basis function networks, which, in turn, are utilized for the thickness grading of the TPMS-based lattices. In addition, the global boundary of the lattice structure is identified by support vector machines. Finally, a standard triangle language (STL) file is generated from the implicit surfaces by using marching cubes, which is utilized for further studies by nonlinear finite element analysis and to set up 3D printing of the optimal component quickly. The framework is demonstrated for the established L-shaped benchmark and the well-known General Electric engine bracket.本研究开发、实现并验证了一种基于拓扑优化（Topology Optimization, TO）的三重周期性极小曲面（Triply Periodic Minimal Surfaces, TPMS）点阵结构功能梯度化框架。针对Gyroid、G'-曲面与施瓦茨-D（Schwarz-D）曲面，通过适配横观各向同性弹性的数值均匀化方法推导其材料插值法则，并将其表示为固体各向同性材料惩罚（Solid Isotropic Material with Penalization, SIMP）模型与材料属性有理近似（Rational Approximation of Material Properties, RAMP）模型的凸组合形式。将此类凸组合应用于基于拓扑优化的柔顺性优化问题中，同时为表征点阵体积分数限值的密度变量设置全新的上下界约束；针对密度变量的下界约束，通过在优化循环内引入Sigmoid滤波器，将低于下界的密度值强制逼近于零。最优密度解通过谢泼德插值（Shepard Interpolations）或径向基函数网络（Radial Basis Function Networks）进行表征，进而用于实现基于TPMS的点阵结构厚度梯度化设计。此外，通过支持向量机（Support Vector Machines, SVM）识别点阵结构的整体几何边界。最终通过移动立方体（Marching Cubes）算法从隐式曲面生成标准三角语言（Standard Triangle Language, STL）文件，可用于后续非线性有限元分析研究，以及快速搭建最优构件的3D打印流程。本框架通过成熟的L形基准测试案例与知名的通用电气（General Electric, GE）发动机支架场景完成了验证演示。