In this paper, we propose a numerical technique based on the method of fundamental solutions (MFS) for solving a classical optimal shape design problem. The problem contains a free boundary condition which should be approximated to find the optimal domain for the solution of Laplace equation. For solving the considered optimization problem, we introduce a meshless regularization technique based on the combination of the MFS and application of the Tikhonov’s regularization method and reduce the problem to solve a system of nonlinear equations. A brief sensitivity analysis on model parameters including the position and the size of the subregion D as well the error with boundary conditions is discussed. Numerical simulations while solving several test examples are presented to show the applicability of the proposed method in obtaining satisfactory results.

在本文中，我们提出了一种基于基本解法（MFS）的数值技术，用于解决经典的最佳形状设计问题。该问题包含一个自由边界条件，应该对其进行近似以找到拉普拉斯方程解的最佳域。为了解决所考虑的优化问题，我们引入了基于MFS的无网格正则化技术以及Tikhonov正则化方法的应用，并简化了求解非线性方程组的问题。简要分析模型参数（包括分区D的位置和大小）的敏感性讨论了边界条件的误差。在解决几个测试示例的同时，进行了数值模拟，以显示该方法在获得满意结果中的适用性。