The focus of this paper is on multi-material topology optimization (MMTO), where the objective is to not only compute the optimal topology, but also the distribution of two or more materials within the topology. In the popular density-based MMTO, the underlying pseudo-density fields are typically represented using an underlying mesh. While mesh-based MMTO ties in well with mesh-based finite element analysis, there are inherent challenges, namely the extraction of thin features, and the computation of the gradients of the density fields.

The objective of this paper is to present a neural network (NN) based MMTO method where the density fields are represented in a mesh-independent manner, using the NN’s activation functions, with the weights and biases associated with the NN serving as the design variables. Then, by relying on the NN’s built-in optimization routines, and a conventional finite element solver, the MMTO problem is solved.

The salient features of the proposed method include: (1) thin features can be extracted through a simple post-processing step, (2) gradients and sensitivities can be computed accurately through back-propagation, (3) the NN construction implicitly guarantees the partition of unity between constituent materials, (4) the NN designs often exhibit better performance than mesh-based designs, and (5) the number of design variables is relatively small. Finally, the proposed framework is simple to implement, and is illustrated through several examples.

本文的重点是多材料拓扑优化（MMTO），其目的不仅在于计算最佳拓扑，而且还在于计算拓扑中两种或两种以上材料的分布。在流行的基于密度的MMTO中，通常使用基础网格来表示基础伪密度字段。尽管基于网格的MMTO与基于网格的有限元分析很好地结合在一起，但存在固有的挑战，即，提取细小的特征以及计算密度场的梯度。

本文的目的是提出一种基于神经网络（MMTO）的MMTO方法，该方法使用NN的激活函数以与网格无关的方式表示密度场，并将与NN相关的权重和偏差用作设计变量。 。然后，依靠NN内置的优化例程和常规的有限元求解器，可以解决MMTO问题。

所提出方法的显着特征包括：（1）可以通过简单的后处理步骤提取薄特征；（2）通过反向传播可以准确计算梯度和灵敏度；（3）NN构造隐含地保证了划分考虑到构成材料之间的统一性，（4）NN设计通常表现出比基于网格的设计更好的性能，并且（5）设计变量的数量相对较少。最后，所提出的框架易于实现，并通过几个示例进行了说明。