This paper employs a new tangential derivative of boundary integral equation for the optimization problems in the acoustic field with a objective function involving tangential derivatives of the sound pressure on the boundary. The level set method is adopted to generate the topological structure by updating the level set function which defines the boundary of the material domain with its zero contour line. The hyper singular integral is directly derived and singular terms are canceled due to the form of the tangential derivative at the boundary. The topological derivative is derived through the adjoint variable method(AVM) and the most of the unknowns in the variation of objective function can be canceled by evaluating the adjoint field. However, one of the terms which includes the variation of the tangential derivative of the sound pressure is evaluated using integration by parts. The remaining part having unknown variation of sound pressure is neglected by extending the objective function defined boundary by one elements at its start and end points. Numerical implementations demonstrate the effectiveness and correctness of the proposed method for topology optimization problems with the objective function involving tangential derivative quantities.

本文采用边界积分方程的切向导数求解声场优化问题，其目标函数涉及边界上声压的切向导数。采用水平集方法，通过更新定义材料域与零等高线边界的水平集函数来生成拓扑结构。超奇异积分是直接导出的，并且由于边界处切向导数的形式而取消了奇异项。拓扑导数是通过伴随变量法(AVM)导出的，目标函数变化中的大部分未知数可以通过对伴随场的评估来消除。然而，包括声压的切向导数变化的项之一是使用分部积分来评估的。通过将目标函数定义的边界在其起点和终点扩展一个元素来忽略具有未知声压变化的剩余部分。数值实现证明了所提出的拓扑优化问题方法的有效性和正确性，目标函数涉及切向导数。