力学仿真中的应变软化问题数模分析与神经计算力学研究
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摘要
本文由题为“应变软化问题的数模分析”和“神经计算力学”上下两篇组成，论文的上
篇主要针对应变软化问题在理论、模型和算法上存在的问题进行了研究与探讨；下篇则主
要将神经网络理论引进到力学仿真的全过程中，力争建立计算力学中的一个新的研究分支
——神经计算力学。在论文的上篇主要做了如下一些工作：
一、在理论上，图解说明了应变软化材料满足应变空间下的Ilyushin假设；证明了
在小变形条件下考虑材料的应变软化特性时，结构也有可能出现解的非唯一性，同时给
出了一种用于结构分叉分析的耦合判断准则；最后提出了一种求解结构极限荷载的数值
方法。
二、在计算模型上，针对为了客观模拟应变软化现象而提出的多种模型，证明了基于
虚拟裂纹与基于裂纹带的两种内嵌软化带模型在本质上是一致的；指出有些非局部模型
并不能解决数值模拟应变软化问题时存在的网格依赖性和零能量消耗问题。
三、在算法上，指出已有的各种初应力法加速方案对于应变软化问题都不能保证迭代
过程的收敛性；提出了一种新的初应力加速方法，它不仅适用于弹塑性应变硬化问题，
而且适用于应变软化问题。
四、利用内嵌软化带模型进行了溪落渡拱坝的开裂分析，得到了一些有益的结论。
在论文的下篇则主要做了如下一些工作：
一、首先，简单阐述了一些基本的神经网络模型，讨论了这些网络模型具有的特点：
而后提出了“神经计算力学”——这一计算力学的新研究分支，给出了它的定义和具体
的研究内容。
二、分别提出了有限元网格自动生成和施工过程仿真的神经算法，利用它们可以实
现网格的自适应调节与在一套计算网格下对不同结构工况进行连续分析。
三、提出了一种直接利用结构荷载与位移的信息，提取材料本构关系的神经网络模
型与相应算法，完善了利用神经网络进行材料本构关系提取的理论体系。
四、从理论上建立了用于结构优化的神经网络的设计准则，给出了一个一般性设计
方法；针对Hopfield网络提出了一种新的能量函数，将Hopfield网络结构拓展到跨导不
对称的情况下，建立了用于刚度矩阵不对称的结构求解的改进Hopfield网络，建全了反
馈神经网络在结构分析中的应用体系。
This paper comprises two parts: one is named "Numerical Analysis of Strain-Softening" and the other is named "Neural Computational Mechanics". The first part of the paper studies the problems on theory x computational methods and computational models of strain-softening. The second part puts neural network into mechanics simulation, and a new branch of computational mechanics called "The neural computational mechanics" is built.
In the first part of this paper some works are performed as follows:
1. In theory, It is graphically illustrated that strain-softening materials satisfy the IPyushin hypothesis at the space of strain. The non-Uniqueness of solution to structure considering material strain-softening wills occur evens if under conditions of small deformation. At the same time, a coupled-principle that is used to judge the bifurcation of structure is put forward. At last a numerical method to obtain the limit load of structure is given.
2. In terms of computational models for simulating strain-softening, It is showed that the finite element with inner fictitious crack and the finite element with cracking band are the same in essence, and that some non-local models can not overcome the problems of mesh's dependence and zero consumes of energy.
3. As to computational methods, it is showed that every accelerated initial stiffness scheme can not make sure of the converge of iteration while considering material's strain softening. A new method that fits strain-harden as well as strain-softening is given.
4. With the finite element with inner softening band, the cracking of Xi Luodu arch-dam is analyzed. Some useful conclusions are arrived at.
In the second part of this paper some works are performed as follows:
1. At first, some basic neural networks are introduced, and their characteristics are discussed. The neural computational mechanics-a new branch of computational mechanics- is put forward, and the definition, the system and the research direction are given.
2. The neural algorithms for automatic generation of FE mesh and simulation of construction stages are put forward. With them the FE mesh can be adjusted automatically and the structure of different stages may be analyzed with a FE mesh.
3. A new neural network and algorithm used to draw the constitutive relation of material from information of the force and displacement is put forward. The system of drawing the constitutive relation of material by neural network is improved.
4. The design principle of neural network for structure's optimum is researched in theory and a general design method is given. A new enejgy function of Hopfield network is put forward. On the basis of this function, the unnecessary of Kuadao symmetry is proved. The modified Hopfield network that can solve the equations of unsymmetrical stiffness matrix is designed.
篇主要针对应变软化问题在理论、模型和算法上存在的问题进行了研究与探讨；下篇则主
要将神经网络理论引进到力学仿真的全过程中，力争建立计算力学中的一个新的研究分支
——神经计算力学。在论文的上篇主要做了如下一些工作：
一、在理论上，图解说明了应变软化材料满足应变空间下的Ilyushin假设；证明了
在小变形条件下考虑材料的应变软化特性时，结构也有可能出现解的非唯一性，同时给
出了一种用于结构分叉分析的耦合判断准则；最后提出了一种求解结构极限荷载的数值
方法。
二、在计算模型上，针对为了客观模拟应变软化现象而提出的多种模型，证明了基于
虚拟裂纹与基于裂纹带的两种内嵌软化带模型在本质上是一致的；指出有些非局部模型
并不能解决数值模拟应变软化问题时存在的网格依赖性和零能量消耗问题。
三、在算法上，指出已有的各种初应力法加速方案对于应变软化问题都不能保证迭代
过程的收敛性；提出了一种新的初应力加速方法，它不仅适用于弹塑性应变硬化问题，
而且适用于应变软化问题。
四、利用内嵌软化带模型进行了溪落渡拱坝的开裂分析，得到了一些有益的结论。
在论文的下篇则主要做了如下一些工作：
一、首先，简单阐述了一些基本的神经网络模型，讨论了这些网络模型具有的特点：
而后提出了“神经计算力学”——这一计算力学的新研究分支，给出了它的定义和具体
的研究内容。
二、分别提出了有限元网格自动生成和施工过程仿真的神经算法，利用它们可以实
现网格的自适应调节与在一套计算网格下对不同结构工况进行连续分析。
三、提出了一种直接利用结构荷载与位移的信息，提取材料本构关系的神经网络模
型与相应算法，完善了利用神经网络进行材料本构关系提取的理论体系。
四、从理论上建立了用于结构优化的神经网络的设计准则，给出了一个一般性设计
方法；针对Hopfield网络提出了一种新的能量函数，将Hopfield网络结构拓展到跨导不
对称的情况下，建立了用于刚度矩阵不对称的结构求解的改进Hopfield网络，建全了反
馈神经网络在结构分析中的应用体系。
This paper comprises two parts: one is named "Numerical Analysis of Strain-Softening" and the other is named "Neural Computational Mechanics". The first part of the paper studies the problems on theory x computational methods and computational models of strain-softening. The second part puts neural network into mechanics simulation, and a new branch of computational mechanics called "The neural computational mechanics" is built.
In the first part of this paper some works are performed as follows:
1. In theory, It is graphically illustrated that strain-softening materials satisfy the IPyushin hypothesis at the space of strain. The non-Uniqueness of solution to structure considering material strain-softening wills occur evens if under conditions of small deformation. At the same time, a coupled-principle that is used to judge the bifurcation of structure is put forward. At last a numerical method to obtain the limit load of structure is given.
2. In terms of computational models for simulating strain-softening, It is showed that the finite element with inner fictitious crack and the finite element with cracking band are the same in essence, and that some non-local models can not overcome the problems of mesh's dependence and zero consumes of energy.
3. As to computational methods, it is showed that every accelerated initial stiffness scheme can not make sure of the converge of iteration while considering material's strain softening. A new method that fits strain-harden as well as strain-softening is given.
4. With the finite element with inner softening band, the cracking of Xi Luodu arch-dam is analyzed. Some useful conclusions are arrived at.
In the second part of this paper some works are performed as follows:
1. At first, some basic neural networks are introduced, and their characteristics are discussed. The neural computational mechanics-a new branch of computational mechanics- is put forward, and the definition, the system and the research direction are given.
2. The neural algorithms for automatic generation of FE mesh and simulation of construction stages are put forward. With them the FE mesh can be adjusted automatically and the structure of different stages may be analyzed with a FE mesh.
3. A new neural network and algorithm used to draw the constitutive relation of material from information of the force and displacement is put forward. The system of drawing the constitutive relation of material by neural network is improved.
4. The design principle of neural network for structure's optimum is researched in theory and a general design method is given. A new enejgy function of Hopfield network is put forward. On the basis of this function, the unnecessary of Kuadao symmetry is proved. The modified Hopfield network that can solve the equations of unsymmetrical stiffness matrix is designed.
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