This paper proposes a new methodology for computing boundary sensitivities in level set topology optimization using the discrete adjoint method. The adjoint equations are constructed using the discretized governing field equations. The objective function is differentiated with respect to the boundary point movement for computing boundary sensitivities using the discrete adjoint equations. For this purpose, we present a novel approach where we perturb the boundary implicitly by locally modifying the level set function around a given boundary point. These local perturbations are combined with the derivatives of the objective function with respect to the volume fractions of individual elements to compute boundary sensitivities. This enables the circumvention of smoothing or interpolation methods typically used in level set topology optimization to compute sensitivities; and improves the accuracy of the sensitivities and the convergence characteristics. We demonstrate the effectiveness of our method in the context of stress minimization and stress constrained topology optimization problems for orthogonal bracket design under multiple load cases.

本文提出了一种使用离散伴随方法在水平集拓扑优化中计算边界敏感度的新方法。使用离散的控制场方程构造伴随方程。使用离散的伴随方程，针对边界点移动对目标函数进行微分，以计算边界灵敏度。为此，我们提出了一种新颖的方法，其中我们通过在给定边界点周围局部修改级别集函数来隐式地扰动边界。这些局部扰动与目标函数相对于单个元素的体积分数的导数组合在一起，以计算边界敏感度。这样可以规避通常在水平集拓扑优化中使用的平滑或插值方法来计算灵敏度。并提高了灵敏度和收敛特性的准确性。我们在应力最小化和应力约束的拓扑优化问题中证明了我们的方法的有效性，该问题对于多载荷情况下的正交支架设计而言是正确的。