融合连续体结构拓扑优化ICM法提升与扩展变密度法

Improve the variable density method (abbreviated as VD) by the ICM method. Comparison of two methods: (1) design variables: VD has artificial densities. ICM has independent topology variables. Their commonality is that their design variables all are continuous on the interval [0, 1]. (2) performance functions: VD has a penalty function of the material elastic modulus for the artificial densities. ICM has a filter function of the stiffness matrix for the topology variables. Their commonality is that their performance functions are all power functions. The analogy analysis can be carried on due to their commonalities exists. The variable density method will be improved and expanded by merging the conceptions and approaches of the ICM method, such as independent topology variables, mapping process, etc. Following aspects will be researched: (1) A more rational explanation to penalty function will be presented; and the import of the artificial density into optimization model will be more legal. (2) Expand a penalty function of the elastic modulus to multiple penalty functions, such as penalty function of weight, allowable stress, mass matrix, geometry matrix, and so on. Not only solve the strength or stiffness topology optimization problem, but also solve dynamic or stable topology optimization problem. (3) Search for the relationships of penalty functions with fast or slow penalty effect. (4) Explore more forms of penalty functions, such as the exponential function, the modified Sigmoid function and the sine function. Function with fast convergence will be adopted. (5) Search for the model and solution of more reasonable penalty functions to speed up the convergence. (6) Determine the correspondence of the economic index and the performance index for the objective function and constraints. (7) A full range variable density method for not only SIMP model, but also the RAMP model, SINH model, etc. This study has theoretical and application value: more reasonable model, more stable iteration process, clearer configuration.

以ICM法推进变密度法（VD）。比较：（1）设计变量：VD人工密度，ICM拓扑变量，共性-皆[0 1]区间连续。（2）性能函数：VD弹性模量为人工密度的惩罚函数，ICM刚度阵为拓扑变量的过滤函数，共性-皆幂函数。因有共性故可类比，融合ICM拓扑变量、映射等概念和方法提升与拓展VD。研究：（1）理性解释惩罚函数，使人工密度引入合法。（2）将一个弹性模量惩罚函数拓展为包括重量、许用应力、质量阵、几何阵的多个惩罚函数，不仅解决强度或刚度问题，还解决动力、稳定问题。（3）诸多惩罚函数的快与慢关系。（4）将惩罚函数拓展为更广泛函数，如指数、修正Sigmoid和正弦函数，采用收敛快的形式。（5）寻求合理惩罚函数的建模与求解，加快收敛。（6）经济指标和性能指标同目标函数和约束条件的定位。（7）全方位的变密度法，SIMP法、RAMP法、SINH法等。本研究有理论和应用价值：模型合理，求解稳健，构型清晰。

本项目在连续体结构拓扑优化中，将ICM方法在概念、理论及算法等方面优势，融合进该领域研究非常广泛的变密度方法，主动提升与扩展了该方法。研究进展如下：① 还原变密度法在结构拓扑优化上应有的独立层次，提升变密度法人工密度及惩罚概念到独立拓扑变量及映射反演概念的层次上，使变密度法建立在更合理、更坚实的理论基础之上；② 拓展变密度法中单一的材料弹性模量惩罚函数，引入重量、许用应力、质量阵、几何阵等多个惩罚函数，克服单一弹性模量惩罚函数的局限性，使变密度法更灵活处理应力、频率及稳定等拓扑优化问题；③ 融合ICM法中幂函数形式外的过滤函数如指数、修正Sigmoid等，研究诸多惩罚函数的惩罚的“快”与“慢”关系，寻求到更为高效的惩罚模型；融合ICM法的建模与求解方法，提出了更为高效的求解一类可分离凸规划的对偶显式模型DP-EM方法；使变密度法求解过程更稳健、更快速，最优拓扑构型更加清晰；④ 解决了经济指标和性能指标是否存在同目标函数和约束条件的定位问题；提出了全新的互逆规划理论，应用于结构拓扑优化建立合理化模型；使变密度法建立更合理的优化模型。通过融合ICM法的优势，提升了变密度法，使之在“高效”、“稳健”、“边界光滑”、“拓扑清晰”四个问题上全面得到改善，模型更为合理，方法更为实用。项目研究计划外，① 将传统安全概念扩展到破损-安全概念，研究了考虑破损-安全的结构拓扑优化，克服了传统结构拓扑优化所得拓扑构型冗余度不足的缺陷，使得结构在局部破损的情况下有足够的安全冗余。② 基于结构拓扑优化，提出了一种树状结构拓扑创构新方法，克服了以往树状结构创构设计依赖于一些设计参数设定的不足；应用结构拓扑优化于高层建筑大型支撑体系的拓扑优化设计，验证了建立合理化模型的必要性。研究成果发表论文14篇，其中SCI、EI 源刊8 篇；作学术报告19人次；获国际奖1人次；在国际知名出版公司Elsevier出版学术专著1本。