结构有限元分析神经网络计算研究
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摘要
由于传统计算机本身固有的计算与存储之间的“瓶颈”障碍，许多工程力学问题因计算规模大而得不到求解，许多新问题和悬而未决的老问题，由于计算问题等原因还没有突破性进展。因此，需要发展新型的计算力学理论与方法。
随着确定性结构的数值计算理论日益完善，人们不再满足于对确定性结构的分析，开始在结构分析中考虑结构的不确定因素。结构的不确定性分为两类，一类为随机性，另一类为模糊性。与确定性结构分析相同，对不确定结构的分析主要采用有限元计算方法。到目前为止，随机有限元法的研究已基本成熟。而由于模糊方程组和区间方程组尚没有一个令人满意的求解方法，制约了有限元法在具有模糊参数结构的分析计算中的应用。
鉴于上述原因，开展结构分析的实时计算和模糊有限元的求解方法的研究，不仅具有重要的理论意义，而且具有十分重要的工程应用价值。
针对上述问题，在国家自然科学基金(高技术新概念新构思探索)、教育部优秀青年教师资助计划、高等学校全国100篇优秀博士学位论文作者专项基金等的资助下，本文对弹塑性力学问题的动静态的实时计算、模糊有限元的求解方法等问题进行了系统和深入的研究，取得了以下成果：
(1)根据有限元总刚矩阵经修正后具有正定性的特点以及弹性体势能函数的具体形式，将饱和模式的线性系统(简称为LSSM系统)引入到有限元的神经网络计算中，在理论上实现了有限元神经网络计算的无误差求解。给出了有限元的神经网络计算的电路实现。
(2)特征值问题是结构动态分析计算的关键之一。本文应用Reyleigh极小值原理，将神经网络的能量函数的极小点对应于广义特征值问题的极小特征值所对应的特征向量，在神经网络向着能量函数极小点运动的同时得到了极小特征向量的精确解答。从特征值的变分特性出发，给出了基于罚函数法的其他特征值的神经网络求解方案。从而在理论上给出了所有特征值的神经网络求解方法。
(3)在实际工程中，塑性变形是经常发生的。与弹性结构分析相比，对发生塑性变形结构的分析需要更多的计算时间，因此塑性结构的实时计算更为重要。针对已有方法的局限性，以塑性理论中的二阶段最小势能变分原理为基础，对弹塑性力学问题的神经网络有限元计算方法进行了研究，提出了基于二阶段最小势能变分原理的弹塑性力学问题的神经网络求解方法，建立了塑性力学问题的有限元神经网络计算模型。
(4)针对模糊方程组的求解技术尚不完善，模糊有限元方程组的计算存在着计算量大且将得到的解代回原方程时方程组的两边不平衡等问题，提出了基于模糊系数规划的模糊有限元求解方法。此方法将模糊系数规划与线弹性力学的行为本质——线弹性物体的受力平衡过程为一个二次方程的能量极小化过程相结合，不但物理意义明确，而且经计算机仿真表明，按该方法设计计算，从工程角度看，有利于提高系统的安全储备。
(5)在基于模糊系数规划的模糊有限元求解方法的基础上，利用神经网络的优化计算能力给出了一种模糊有限元的神经网络计算方法，从而形成了一套较合理的具有模糊参数结构的实时分析计算方法。
(6)研究了基于模糊变分原理的对模糊宗量进行二阶摄动展开的的摄动模糊有限
元计算方法。对现有文献中没有由单刚矩阵集成总刚矩阵这一过程进行了补充完善。通
过实例计算和与已有模糊有限元计算方法的比较，说明了该方法的有效性。并指出基于
模糊变分原理的摄动模糊有限元法是在小扰动情况下进行的一种近似求解方法。当总刚
矩阵又、总体载荷向量f等不能展开为各模糊源的有限次(一般取为两次)的表达式时，
随着模糊摄动量的增大，其截断误差将不能接受，此时若采用摄动模糊有限元法进行计
算，其分析将会失败。
Many problems of engineering mechanics that have large computation scale haven't been solved due to the bottleneck between the computation and memory of the Von Neumann computer. And there still haven't breakthrough in some new or pendent problems because of such reason as computation, etc. Therefore, new theory and method of computational mechanics need to be developed to solve above problems.
With the increasingly perfecting of numerical calculation theory of definite structure, people are not satisfied with the analysis of definite structure, and begin to take the uncertain factors of structure into account when analyzing the structure. Uncertainty of structure can be classed into two different categories. One is stochastic uncertainty. The other is fuzzy uncertainty. Finite element method is adopted to analyze the uncertain structure the same as to analyze the definite structure. Up to now, the research of stochastic finite element is mature basically. However, the application of finite element in analysis and computation of uncertain structure is to be limited because there hasn't a satisfying solving method to fuzzy equations and interval equations. So, studying on real-time computation of structural analysis and on the solving method of fuzzy finite element not only has important theory significance, but also has important application value in engineering.
Point to above problems, under the financial support of the National Natural Science Foundation (exploration of high tech and new concept and new conceive), the Excellent Young Teachers Program of Ministry of Education and National Excellent Doctoral Dissertation Special Foundation, the static and dynamic real-time computation of elasticity-plastic mechanics, solving method of fuzzy finite element and other problems were studied in this paper. And some achievement was gained as following:
(1) Based on the positive definiteness of system stiffness matrix of finite element that was modified and the form of potential energy function of elastic body, the linear system of saturation mode (LSSM) was introduced into the neural computation of finite element, by which the no-error solving of finite element neural net computation was realized in theory. The circuit realization of neural computation of finite element was presented.
(2) Eigenvalue is very important to dynamic analysis and computation of structure. Based on the theory of Reyleigh minimum, the minimum of energy function of neural network was mapped to the eigenvector that was mapped to the minimal eigenvalue of the generalized eigenvalue problem, by which the precise solution of minimal eigenvalue was gained while the neural network moving to the minimum of energy function. Based on the variational characteristic of eigenvalue, the neural network solving method of other eigenvalues based on penalty function method was presented. Thus, the neural network solving method of all eigenvalues was presented in theory.
(3) In practical engineering, plastic deformation occurs frequently. Compared with elastic structural analysis, more time is used to analyze the plastic deformation structure. So, the real-time computation of plastic structure is more important than that of elastic structure.
Point to the limitation of existing method, the neural computation method of finite element of elasticity-plastic mechanics was studied on the base of variational principle of the second order minimal potential energy in plastic theory. The neural network solving method of elasticity-plastic mechanics based on variational principle of the second order minimal potential energy was presented, and the neural network computation model of finite element of plastic mechanics was given.
(4) Due to the imperfectness of solving the fuzzy equations, complexity in computation of fuzzy finite element equations and imbalance in the two sides of fuzzy finite element equations while the solution was lead back to the original equations, the fuzzy finite element method based on fuzzy coefficient programming was
随着确定性结构的数值计算理论日益完善，人们不再满足于对确定性结构的分析，开始在结构分析中考虑结构的不确定因素。结构的不确定性分为两类，一类为随机性，另一类为模糊性。与确定性结构分析相同，对不确定结构的分析主要采用有限元计算方法。到目前为止，随机有限元法的研究已基本成熟。而由于模糊方程组和区间方程组尚没有一个令人满意的求解方法，制约了有限元法在具有模糊参数结构的分析计算中的应用。
鉴于上述原因，开展结构分析的实时计算和模糊有限元的求解方法的研究，不仅具有重要的理论意义，而且具有十分重要的工程应用价值。
针对上述问题，在国家自然科学基金(高技术新概念新构思探索)、教育部优秀青年教师资助计划、高等学校全国100篇优秀博士学位论文作者专项基金等的资助下，本文对弹塑性力学问题的动静态的实时计算、模糊有限元的求解方法等问题进行了系统和深入的研究，取得了以下成果：
(1)根据有限元总刚矩阵经修正后具有正定性的特点以及弹性体势能函数的具体形式，将饱和模式的线性系统(简称为LSSM系统)引入到有限元的神经网络计算中，在理论上实现了有限元神经网络计算的无误差求解。给出了有限元的神经网络计算的电路实现。
(2)特征值问题是结构动态分析计算的关键之一。本文应用Reyleigh极小值原理，将神经网络的能量函数的极小点对应于广义特征值问题的极小特征值所对应的特征向量，在神经网络向着能量函数极小点运动的同时得到了极小特征向量的精确解答。从特征值的变分特性出发，给出了基于罚函数法的其他特征值的神经网络求解方案。从而在理论上给出了所有特征值的神经网络求解方法。
(3)在实际工程中，塑性变形是经常发生的。与弹性结构分析相比，对发生塑性变形结构的分析需要更多的计算时间，因此塑性结构的实时计算更为重要。针对已有方法的局限性，以塑性理论中的二阶段最小势能变分原理为基础，对弹塑性力学问题的神经网络有限元计算方法进行了研究，提出了基于二阶段最小势能变分原理的弹塑性力学问题的神经网络求解方法，建立了塑性力学问题的有限元神经网络计算模型。
(4)针对模糊方程组的求解技术尚不完善，模糊有限元方程组的计算存在着计算量大且将得到的解代回原方程时方程组的两边不平衡等问题，提出了基于模糊系数规划的模糊有限元求解方法。此方法将模糊系数规划与线弹性力学的行为本质——线弹性物体的受力平衡过程为一个二次方程的能量极小化过程相结合，不但物理意义明确，而且经计算机仿真表明，按该方法设计计算，从工程角度看，有利于提高系统的安全储备。
(5)在基于模糊系数规划的模糊有限元求解方法的基础上，利用神经网络的优化计算能力给出了一种模糊有限元的神经网络计算方法，从而形成了一套较合理的具有模糊参数结构的实时分析计算方法。
(6)研究了基于模糊变分原理的对模糊宗量进行二阶摄动展开的的摄动模糊有限
元计算方法。对现有文献中没有由单刚矩阵集成总刚矩阵这一过程进行了补充完善。通
过实例计算和与已有模糊有限元计算方法的比较，说明了该方法的有效性。并指出基于
模糊变分原理的摄动模糊有限元法是在小扰动情况下进行的一种近似求解方法。当总刚
矩阵又、总体载荷向量f等不能展开为各模糊源的有限次(一般取为两次)的表达式时，
随着模糊摄动量的增大，其截断误差将不能接受，此时若采用摄动模糊有限元法进行计
算，其分析将会失败。
Many problems of engineering mechanics that have large computation scale haven't been solved due to the bottleneck between the computation and memory of the Von Neumann computer. And there still haven't breakthrough in some new or pendent problems because of such reason as computation, etc. Therefore, new theory and method of computational mechanics need to be developed to solve above problems.
With the increasingly perfecting of numerical calculation theory of definite structure, people are not satisfied with the analysis of definite structure, and begin to take the uncertain factors of structure into account when analyzing the structure. Uncertainty of structure can be classed into two different categories. One is stochastic uncertainty. The other is fuzzy uncertainty. Finite element method is adopted to analyze the uncertain structure the same as to analyze the definite structure. Up to now, the research of stochastic finite element is mature basically. However, the application of finite element in analysis and computation of uncertain structure is to be limited because there hasn't a satisfying solving method to fuzzy equations and interval equations. So, studying on real-time computation of structural analysis and on the solving method of fuzzy finite element not only has important theory significance, but also has important application value in engineering.
Point to above problems, under the financial support of the National Natural Science Foundation (exploration of high tech and new concept and new conceive), the Excellent Young Teachers Program of Ministry of Education and National Excellent Doctoral Dissertation Special Foundation, the static and dynamic real-time computation of elasticity-plastic mechanics, solving method of fuzzy finite element and other problems were studied in this paper. And some achievement was gained as following:
(1) Based on the positive definiteness of system stiffness matrix of finite element that was modified and the form of potential energy function of elastic body, the linear system of saturation mode (LSSM) was introduced into the neural computation of finite element, by which the no-error solving of finite element neural net computation was realized in theory. The circuit realization of neural computation of finite element was presented.
(2) Eigenvalue is very important to dynamic analysis and computation of structure. Based on the theory of Reyleigh minimum, the minimum of energy function of neural network was mapped to the eigenvector that was mapped to the minimal eigenvalue of the generalized eigenvalue problem, by which the precise solution of minimal eigenvalue was gained while the neural network moving to the minimum of energy function. Based on the variational characteristic of eigenvalue, the neural network solving method of other eigenvalues based on penalty function method was presented. Thus, the neural network solving method of all eigenvalues was presented in theory.
(3) In practical engineering, plastic deformation occurs frequently. Compared with elastic structural analysis, more time is used to analyze the plastic deformation structure. So, the real-time computation of plastic structure is more important than that of elastic structure.
Point to the limitation of existing method, the neural computation method of finite element of elasticity-plastic mechanics was studied on the base of variational principle of the second order minimal potential energy in plastic theory. The neural network solving method of elasticity-plastic mechanics based on variational principle of the second order minimal potential energy was presented, and the neural network computation model of finite element of plastic mechanics was given.
(4) Due to the imperfectness of solving the fuzzy equations, complexity in computation of fuzzy finite element equations and imbalance in the two sides of fuzzy finite element equations while the solution was lead back to the original equations, the fuzzy finite element method based on fuzzy coefficient programming was
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