摘要
设计是企业产品创新的源头，是制造业核心竞争力的关键所在。产品创新设计不仅要满足产品结构形状的需求，而且还要满足结构性能要求。当前，大量企业产品设计应用三维CAD软件进行几何建模，采用基于限元法的CAE软件进行结构性能分析，但CAD模型与CAE模型之间的模型转换、模型简化、网格剖分等前处理过程需耗费大量的时间，影响产品设计效率。为此，本文重点对边界元法和快速多极边界元法进行深入研究，提出一种GPU并行的快速多极边界元方法，并应用于产品结构性能分析。该方法能有效简化传统有限元分析前处理过程，提高产品设计效率，同时也为新一代产品设计CAD/CAE软件提供一种可行的集成化方法。论文主要研究工作如下：
(1)对三维弹性力学问题的边界元法进行研究，包括边界积分方程建立、单元积分方法、角点问题处理、边界面应力计算及GMRES迭代求解算法等。针对角点处面力不连续问题，提出了一种边界条件相关的混合单元法，并利用三维模型BREP表达中角点拓扑关系实现了混合单元的自动生成。与现有混合单元角点处理算法相比，该方法仅在位移约束角点处采用非连续单元，有效减少非连续单元引起的附加自由度，降低结构分析问题求解规模。
(2)对快速多极边界元法的算法原理进行研究，提出了一种节点单元双重信息自适应结构树的构建方法，实现了高阶边界单元积分的快速计算。基于该方法建立的双重信息快速多极边界元法可将边界元法的时间和空间复杂度由O(N2)降到O(N)，且单元积分计算量仅为采用全局节点法和节点分片法的快速多极边界元法的三分之一。此外，将快速多极边界元法与给定边界条件结合，提出了一种适用于快速多极边界元法的刚体位移特解法，解决了1/r2奇异积分和自由项系数的求解问题。
(3)针对快速多极边界元法中多极展开向局部展开系数传递(M2L)计算过程存在效率低的问题，本文对基于指数展开的新型快速多极边界元法进行探索，研究表明该方法在展开阶次较大时才能达到高的计算精度及显著的加速效果，且需额外增加存储。为此，本文进一步研究子层结点向父层结点越层传递的M2L优化改进方法，实验结果显示，越层传递M2L方法不需增加额外内存，且加速效果与展开阶次无关，有利于结构性能分析的快速计算。
(4)充分利用边界单元及自适应结构树结点的固有并行特征，提出了一种基于CUDA架构的自适应快速多极边界元GPU并行算法，对快速多极边界元法中多极展开、多极展开系数传递、多极展开向局部展开系数传递、局部展开系数传递以及近场节点单元积分计算进行加速。实验结果表明，该算法不仅具有显著的加速效果，而且对不同形状的三维模型均具有良好的适应性，有效提高产品结构性能分析效率。
最后，以上述理论研究为基础，对集成化CAD/CAE产品设计软件技术及系统架构进行研究，结合现有自主知识产权的三维参数化特征建模软件InterSolid，采用Visual C++集成开发环境，研制开发了集成化CAD/CAE设计分析原型系统软件。并以此为基础，针对不同形状、不同复杂程度的典型三维产品实例进行结构分析工程计算验证，实验结果表明，本文所提出的理论及算法具有计算效率高、求解规模大、适应性强等优势，具有良好的工程应用前景。
Design is the source of product innovation in enterprises and plays a key role in the corecompetitiveness of manufacturing industry. The innovative design of products not onlysatisfies the demands of structural shape, but also meets the requirements of structuralperformance. Nowadays, CAD and finite element method based CAE softwares are used ingeometrical modeling and structural performance analysis respectively in most productdesigns of enterprises, but it takes much time in the pre-processings between CAD modelsand CAE models such as model conversion, model simplification and mesh generation. Thuspre-processing simplification and CAD/CAE integration are of great importance to theimprovement of product design efficiency. In this dissertation, the boundary element method(BEM) and the fast multipole boundary element method (FMBEM) are intensively studied,and a GPU parallel adaptive FMBEM is presented and used in structural performance analysisof products. Such method can simplify the pre-processings and improve the efficiency ofproduct design, which provides an available method for CAD/CAE software integration. Themajor research works and contributions of this dissertation can be concluded as follows:
(1) The BEM for3D elasticity problems has been studied in detail, including theestablishment of boundary integral equations, the element integral methods, the treatment ofcorner problem, the stress computing on boundary surfaces and generalized minimal residualmethod (GMRES) etc. In order to address the problem that tractions are discontinuous atcorners, a BEM with mixed boundary elements related to boundary conditions is proposedand the mixed boundary elements can be automatically generated according to the topologicalrelation of model’s BREP at corners. Comparing with other BEMs with mixed boundaryelements, our method distributes non-conforming elements only at the displacement-givencorners, which reduces the extra degrees of freedom introduced by the non-conformingelements and the solving scale of structure analysis.
(2) The algorithm theory of the FMBEM is studied. Then a construction method of thedual-information adaptive tree containing both node and element information is presented,which can computes the integrals of high-order elements efficiently. Based on this method, the dual-information FMBEM can reduce the BEM’s complexities of both time and spacefrom O(N2) to O(N), and the computational complexity of element integrals is only one thirdof the FMBEM with the global node method or node patch method. Furthermore, a rigid bodymovement method for FMBEM is proposed by combining the FMBEM with given boundaryconditions, which is used to compute the1/r2singular integrals and the free-term coefficients.
(3) For the purpose of improving the efficiency of the moment-to-local (M2L) translation,the new FMBEM based on the exponential-expansion is researched. The results show thatsuch method obtains high accuracy and remarkable acceleration only if the number ofexpansion terms is large, and increases the memory cost. Therefore, the child-to-parent M2Ltranslation optimization method is studied as an alternative. The examples show that thechild-to-parent M2L can accelerate all the expansion terms almost without memory increment,which is more suitable for structural performance analysis.
(4) A CUDA based GPU parallel algorithm of the adaptive FMBEM is presentedaccording to the intrinsic parallelism of boundary elements and knots of the adaptive tree,which accelerates moment expansion, moment-to-moment translation, M2L translation,local-to-local translation and near field direct computing in the FMBEM. The experimentalexamples show that the parallel algorithm not only obviously accelerates the FMBEM butalso obtains load balancing for models with different shapes, which effectively improves theefficiency of structural performance analysis of products.
Finally, on the basis of above proposed methods, the software technique and systemarchitecture of CAD/CAE integrated product design have been studied. The CAD/CAEintegrated prototype system software has been developed using Visual C++by combiningwith the3D modeling software InterSolid which owns proprietary intellectual property rights.Whereafter,3D engineering examples with different shapes and complexities are analyzed bythe software to verify their structures. The results show that the theories and algorithms in thisdissertation have advantages of high computing efficiency, large solving scale and strongadaptability, which have promising future in engineering.
引文
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