基于点密度插值的拓扑优化密度过滤方法研究

RESEARCH ON TOPOLOGICAL OPTIMIZATION DENSITY FILTERING METHOD BASED ON NODAL DENSITY INTERPOLATION

在基于密度描述的拓扑优化方法中,棋盘格式和灰度单元等数值不稳定问题严重影响着优化构型的可制造性和实用性.为了解决这些问题,提出了一种改进的密度过滤方法.该方法首先将正方形单元中心点处密度作为设计变量,并利用点插值方法获取设计域内光滑连续的材料密度分布.然后,将单元平均密度作为过滤后的单元物理密度用于结构刚度计算.因此,改进的密度过滤仍是一种线性过滤方法,仅需在经典密度过滤基础上进行简单修改.本文方法的核心在于利用了Voronoi图的最邻近原则,该原则使得目标单元中心处的点密度对该单元内任意点的密度插值权重最大.进一步地,通过在点插值权重函数中引入一个优化参数,达到控制灰度单元比例和确保收敛稳定性的目的.与Heaviside映射过滤不同,该优化参数并不会增加优化问题的非凸性,导致优化求解的收敛性变差.同时,改进的密度过滤天然具有保体积特性,有效避免了类似Heaviside映射过滤中迭代振荡现象的发生.最小柔顺化问题的算例结果表明,改进的密度过滤通过优化参数的改变,既可避免棋盘格式,又能获得接近理想的0~1分布的优化结果.而且,相较于Heaviside映射过滤,其在求解过程中具有更好的鲁棒性及优化效率.

In density-based topology optimization methods,the numerical instability problems,such as chessboard patter and gray scale elements affect the manufacturability and practicality of optimized structures seriously.In order to eliminate these problems,the paper proposes a modified density filtering method.The proposed method first takes the densities at the center points of square elements as the design variables,and uses the nodal interpolation method to obtain a smooth and continuous material density distribution within the design domain.Then,the average densities of elements are used as the filtered physical densities for structural stiffness calculation.Therefore,the modified density filter is still a linear filtering method,which only requires simple modifications on the basis of classical density filter.The key of the modified density filter lies in utilizing the nearest neighbor principle of Voronoi diagram,which maximizes the density interpolation weight of the nodal density at the center of the target element for any point within the element.Furthermore,by introducing an optimization parameter into the nodal interpolation weight function,the goals of controlling the proportion of grayscale elements and ensuring convergence stability are achieved.Unlike the Heaviside mapping filters,the added optimization parameter does not increase the non-convexity of the optimization problem,resulting in a decrease in the convergence of the optimization solution.And the modified density filter naturally has the characteristic of volume preserving,effectively avoiding the occurrence of iterative oscillation phenomena similar to the Heavisidetype filters.The numerical results of the compliance minimization problem show that the modified density filter can avoid chessboard patter and obtain optimization result close to the ideal 0~1 distribution by only changing the optimization parameter.Moreover,compared to the Heaviside mapping filters,the proposed method has better robustness and optimization efficiency in the solving process.