用序列二次规划可以高效率地解决许多工程结构的优化问题。一类问题将原来的非线性规划用力学处理和Taylor展开化为一系列二次规划问题，在目标函数可分离的情况下，通过Kuhn-Tucker条件进一步化为以Lagrange乘子为变量的准无约束的二次规划，在目标函数不可分离情况下，则直接求解原变量的近似二次规划，经过序列迭代，可以求解工程结构的静、动力优化问题。另一类问题是将原问题化为正多项式的几何规划，再将其对偶问题化为二次规划，序列迭代求解之，可以求解目标函数、约束函数非线性程度高的工程问题，对按设计规范公式进行设计的一类问题尤为有效

基于有限元概念，本文研究了结构响应对于设计变量的严格解析解，以空间桁架为例，得到了位移显式的解析表达式——一个关于截面设计变量的有理函数，并进行了严格的数学论证；进而推出了有益于结构敏度分析、结构优化的若干结论，针对含有位移显式的优化模型，提出了采用广义几何规划解法的建议；最后计算了若干简例。

本文在文[4]的基础上，按“有无复合体”模型对应力约束下多工况平面连续体结构拓扑优化的求解进行研究，提出了“工况影响系数”的概念，用于“病态荷载”问题的识别和解决，获得了令人满意的数值结果。

There are three difficulties in the topology optimization problem of continuum structure: (1) The problem under multiple load case is not easy to be approached than one under single load case, because the former becomes a multiple objective problem based on compliance objective function. (2) The problem with local constraints, such as elemental stress limit, is not easy to be solved than one with global constraints, such as displacement or frequency limits, because sensitivity analysis of the former has too expensive computation. (3) The problem with a phenomenon of ill-conditioned load, which is similar with ill-conditioned stiffness matrix in the structural analysis, is not easy to get reasonable final topological structure, because the former is difficult to consider different influences between the loads with small forces and the loads with big forces, and some of topology paths of transferring small forces may disappear during the process of iteration. To overcome above difficulties, five measures are adopted below: (1) Topology optimization model is established by ICM (Independent Continuous Mapping) method. Difficulties caused by multiple objective function of compliance are overcome owing to taking weight as objective function. (2) Based on the von Mises criteria in theory of elastic failure, all element's stress constraints are transformed into a structural energy constraint, namely, a global constraint substitutes for lots of local constraints. (3) To approach of structural energies under multiple load cases, the process of optimization is divided into three steps: <a> the structural energy under the load case corresponding to maximal structural energy subjected to a weight constraint is minimized to converge a minimum energy; <b> according to the minimum energy, a formula based on numerical experience is obtained to determine the allowable structural energy under each load case; <c> an optimization model with the weight function subjected to all allowable structural energies is established and solved to search the optimum topological structure. (4) The phenomenon of the ill-conditioned load is divided to classify into three cases: <a> ill-condition exists between load cases, but not within each load case; <b> ill-condition exists within some load case; <c> ill-condition exists not only between load cases, but also within some load case. (5) A strategy based on strain energy is proposed to adopt ICM method with stress globalization, and the problems of the above mentioned three cases of ill-conditioned load are solved in term of different complementary approaches one by one. Several numerical examples show that the topology path of transferring forces can be obtained more easily by substituting global strain energy constraints for local stresses constraints, and the problem of ill-conditioned load can be solved well by the weighting method which take structural energy as weighting coefficient.

直径300mm硅片的生产技术是当今硅材料生产研究的重要方向之一，而晶体生长界面的形状、温度分布、晶体中氧的浓度和均匀性等对熔体流动状态十分敏感，采用实验的方法来测量熔体的流动、温度场分布是很困难的，因此很难通过实验的方法获得熔体的流动是如何影响晶体生长的质量的，而数值模拟能提供熔体流动、温度分布等详细内容，为单晶硅的生长提供有利的指导，本文采用低雷诺数的K-ε紊流模型，对直径300mm的大直径单晶硅生长进行了数值模拟，通过熔体在有、无勾形磁场作用时的流场、温度场的分析，阐明了勾形磁场影响熔体流动的机理。