The paper proposes a density gradient based approach to topology optimization under design-dependent boundary loading. Since there is no explicit boundary representation in density-based topology optimization, the design-dependent boundary loads are implicitly imposed through a domain integration of spatial gradient of the density field. The density gradient based loading is formally derived and justified based on the smoothed boundary method for numerical solution of PDE. The Heaviside projection, density filtering and local refinement are combined to efficiently control the thickness of the loading interface and the number of elements within it. For problems with both loading and non-loading boundary, an auxiliary density field is introduced to keep track of the loading boundary. Both 2D and 3D heat conduction problems, linear elasticity problems and coupled thermoelastic problems under design-dependent boundary loading are presented to illustrate the effectiveness and efficiency of the method.

本文提出了一种基于密度梯度的拓扑优化方法，该方法在设计依赖的边界载荷下进行。由于在基于密度的拓扑优化中没有明确的边界表示形式，因此通过密度场空间梯度的域积分隐式地施加了依赖于设计的边界载荷。基于平滑边界法对PDE进行数值求解，正式推导并证明了基于密度梯度的载荷。Heaviside投影，密度过滤和局部优化相结合，可以有效地控制加载界面的厚度和其中的元素数量。对于既有加载边界又有非加载边界的问题，引入了一个辅助密度场来跟踪加载边界。2D和3D导热问题，