In topology optimization, the bisection method is typically used for computing the Lagrange multiplier associated with a constraint. While this method is simple to implement, it leads to oscillations in the objective and could possibly result in constraint failure if proper scaling is not applied. In this paper, we revisit an alternate and direct method to overcome these limitations.

The direct method of Lagrange multiplier computation was popular in the 1970s and 1980s but was later replaced by the simpler bisection method. In this paper, we show that the direct method can be generalized to a variety of linear and nonlinear constraints. Then, through a series of benchmark problems, we demonstrate several advantages of the direct method over the bisection method including (1) fewer and faster update iterations, (2) smoother and robust convergence, and (3) insensitivity to material and force parameters. Finally, to illustrate the implementation of the direct method, drop-in replacements to the bisection method are provided for popular Matlab-based topology optimization codes.

在拓扑优化中，二等分方法通常用于计算与约束关联的拉格朗日乘数。尽管此方法易于实现，但会导致物镜出现震荡，如果不应用适当的缩放比例，可能会导致约束失败。在本文中，我们重新审视了另一种直接方法来克服这些限制。

拉格朗日乘数计算的直接方法在1970年代和1980年代很流行，但后来被更简单的二等分法所取代。在本文中，我们表明直接方法可以推广到各种线性和非线性约束。然后，通过一系列基准问题，我们证明了直接方法相对于二等分方法的多个优点，包括（1）更新迭代次数更少和更快，（2）更平滑和更稳健的收敛性，以及（3）对材料和力参数不敏感。最后，为了说明直接方法的实现，为流行的基于Matlab的拓扑优化代码提供了二等分方法的直接替换。