The adjoint method allows efficient calculation of the gradient with respect to the design variables of a topology optimization problem. This method is almost exclusively used in combination with traditional Finite-Element-Analysis, whereas Fourier-based solvers have recently shown large efficiency gains for homogenization problems. In this paper, we derive the discrete adjoint method for Fourier-based solvers that employ compatibility projection. We demonstrate the method on the optimization of composite materials and auxetic metamaterials, where void regions are modelled with zero stiffness.

翻译：联合方法使得能够有效计算与地形优化问题设计变量有关的梯度,这种方法几乎完全与传统的有限元素分析结合使用,而基于Fourier的解决方案最近显示,在同质化问题上效率提高很大。在本文中,我们为使用兼容性预测的基于Fourier的解决方案得出了离散连接方法。我们展示了优化复合材料和辅助元材料的方法,在这些区域中,空虚区域以零硬度为模型。