This article presents an educational code written in FreeFEM, based on the concept of topological derivative together with a level-set domain representation method and adaptive mesh refinement processes, to perform compliance minimization in structural optimization. The code is implemented in the framework of linearized elasticity, for both plane strain and plane stress assumptions. As a first-order topology optimization algorithm, the topological derivative is in fact used within the numerical procedure as a steepest descent direction, similar to methods based on the gradient of cost functionals. In addition, adaptive mesh refinement processes are used as part of the optimization scheme for enhancing the resolution of the final topology. Since the paper is intended for educational purposes, we start by explaining how to compute topological derivatives, followed by a step-by-step description of the code, which makes the binding of the theoretical aspects of the algorithm to its implementation. Numerical results associated with three classic examples in topology optimization are presented and discussed, showing the effectiveness and robustness of the proposed approach.

本文介绍了用 FreeFEM 编写的教育代码，基于拓扑导数的概念以及水平集域表示方法和自适应网格细化过程，以在结构优化中执行合规性最小化。对于平面应变和平面应力假设，该代码是在线性化弹性框架下实施的。作为一阶拓扑优化算法，拓扑导数实际上在数值过程中用作最速下降方向，类似于基于成本泛函梯度的方法。此外，自适应网格细化过程被用作优化方案的一部分，以提高最终拓扑的分辨率。由于本文旨在用于教育目的，我们首先解释如何计算拓扑导数，然后是对代码的逐步描述，它将算法的理论方面与其实现结合起来。介绍并讨论了与拓扑优化中三个经典示例相关的数值结果，显示了所提出方法的有效性和鲁棒性。