A coupled topology and domain shape optimization framework is presented that is based on incorporating the shape design variables of the design domain in the Solid Isotropic Material with Penalization topology optimization method. The shape and topology design variables are incrementally updated in a sequential fashion, using a staggered numerical update scheme. Non-Uniform Rational B-Splines are employed to parameterize the shape of the design domain. This not only guarantees a highly accurate description of the shape boundaries by means of smooth basis functions with compact support, but also enables an efficient control of the design domain with only a few control points. Furthermore, the optimization process is performed in a computationally efficient way by applying a gradient-based optimization algorithm, for which the sensitivities can be computed in closed form. The usefulness of the coupled optimization approach is demonstrated by analyzing several benchmark problems that are subjected to different types of initial conditions and domain bounds. The variation in simulation results denotes that a careful construction of the initial design domain is necessary and meaningful.

提出了一种基于拓扑结构优化方法的固体各向同性材料中设计域形状设计变量合并的拓扑和域形状优化耦合框架。使用交错的数值更新方案，以连续的方式增量更新形状和拓扑设计变量。非均匀有理B样条用于参数化设计域的形状。这不仅可以通过具有紧凑支持的平滑基础功能来保证形状边界的高精度描述，而且还可以仅使用几个控制点就可以有效控制设计领域。此外，通过应用基于梯度的优化算法，可以以计算有效的方式执行优化过程，可以以封闭形式计算灵敏度。通过分析受到不同类型的初始条件和域边界的几个基准问题，可以证明耦合优化方法的有用性。仿真结果的变化表明，仔细设计初始设计域是必要且有意义的。