弹塑性系统的互补原理及泛惩罚有限元素法
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摘要
本文基于凸分析和非光滑分析的数学理论，对弹塑性介质的位势理论、变分边值问题、极限平衡理论、非线性有限元分析及数值计算方法诸环节进行了较为系统、深入的研究，主要完成如下几项工作：
1．建立了具有弹塑性耦合效应一般介质的共轭塑性公设及相应的广义本构定律，讨论了非凸屈服条件下的加一卸载准则，纠正了经典塑性位势理论中的某些不严格提法。
2、从共轭塑性公设出发，系统地导出了具有强物理非线性本构行为和边界效应边值问题的变分不等原理、对偶一互补变分原理，讨论了系统能量泛函的极值性质和解的存在，唯一性条件，证明了三类变量泛函的驻值性质。
3、提出了力学系统中一条具有认识论上广泛意义的互补对偶原理。
4、建立了理想弹塑性介质变分问题的最小余能原理，证明了一个理论上严格、且具有较大实用价值的罚－对偶变分原理，讨论了强间断条件下刚塑性介质的对偶变分原理。
5、建立了极限平衡理论中严格互补的对偶界限定理，证明了一个满足kuhn—Tucker约束(？)性条件的对偶型变分原理，由此导出了一个中值定理和一个新的下限定理，建立了关于安全因子的有效的罚－对偶变分原理，并予以实际应用。
6、提出了约束变分问题有限元分析的统一方法——泛惩罚有限元素法，此为建立各种类型的有限元格式提供了正确的途径。证明了混合/杂交元和罚函数单元又是其中的泛惩罚函数的二种简单构造。导出了一个有效的罚一对偶有限元格式和其修正格式，这种格式不仅保证了数值解的收敛性，同时克服了单纯罚单元的弊病，具有较高的数值精度和稳定性。并具体构造了一个泛惩罚余能有限元格式。
7、编制了弹塑性分析的泛惩罚有限元程序，提出了一种降低非线性有限元规划问题自由度的计算方法，从而大大提高了计算效率。对工程问题进行了实际应用，大量数值实验结果证明了本文理论分析和数值方法的正确性。
Based on the mathematical theory of nonsmooth analysis, a systematical research has been devoted in this dissertation to problems on the plastic potential theory, variational boundary value problems, limit analysis, nonlinear finite element analysis and numerical procedures etc.. The work carried out includes the following:
1. A couple of generalized conjugate plastic postulates are proposed, which can be used in analysis of elastoplastic coupling for both hard materials(such as usual structural metal) and soft materials (such as excelsior, foam rubber, grain material and some polymer etc.). From these postulates, the corresponding generalized constitutive relations are established, which permit the study of large classes of materials. In addition, the loading-unloading criterion for nonconvex plastic potential is given by use of the concept of generalized gradient in Clarke's sence, and the classical plastic potential is replaced by the generalized superpotentials, which yield the correct and symmetric mathematical expressions of constitutive relations.
2. According to the generalized plastic postulates, two conjugate energy bounding theorems for deformation theories are developed in the form of variational inequalities, which yield the dual-complementary variational principles for general materials including strong physical nonlinearities and discontinuities ( like jumps, locking effects, etc.). The extremum properties of variational functional ( such as Hamiltonian, Lagrangian and psudo-Hamiltonian etc. ) are studied and the existence and uniqueness of solutions for variational boundary value problems are proved by using convex analysis theory.
3. In chapter 5 of this thesis, a true complementary energy variational principle for elasto-perfect plasticity is established with the aid of Fenchel transformation. Based on the property of plastic super-potantial in this principle, two kinds of variational principles are constructed. The first one is a penalty-type principle, which possesses obviously physical significance, and the other one is a duality-type principle, in which the dual variable satisfying the so-called constraint qualification in nonlinear programming is suggested. By joining these two principles together, an interesting penalty-duality variational principle is established, which is not only rigorous in theory, but also of practical value in engineering applications.
4. The dual bounding theorems and a generalized variational principle for limit analysis are established in chapter 6 of this thesis. Based on the duality-type principle in this problem, a new lower bound theorem is stated, in which, the yield condition is relaxed, and a mean-value theorem is proved which gives the safety factor lying between the upper and lower bounds obtained by classical bounding theorems with the same kinematically and statically admissiol (?) Moreover, a penalty-duality algorethm for the safety factor of limit analysis is developed, and the application of these theorems are illustrated by several examples.
5. As a philosophy by-product, a universal complementary-dual principle of continuum mechanics is advanced in this Ph. D. thesis, which may be described as follows :
Let U be a general displacement space and U~* be a complementary general force space (here, the terminology " complementary " is inphilosophical sence ), then, if there exists a theorem about U, we musthave a dual theorem about U~* if we have a theorem defined on the multiplicative space U×U~*, exchange the place of dual variables in this theorem, then a new dual theorem may be obtained.
All the theorems proposed in this thesis may be considered as applications of this principle.
6. Based on the theory of nonlinear programming, a unified formula----Panpenalty Finite Element Method ---- is established for variational boundary value problems with general constraint conditions. Which provides a correct way to construct various finite element models, it is proved that the hybrid/mixed type model and pure penalty type model are no more than simple construction of the panpenalty function in this formula. An efficient penalty-duality finite element model and its modified model are constructed, the positive definite Hessian matrix of these two models may be obtained for some given penalty factors determined by certain conditions, meanwhile, the disadvantages of pure penalty type model are overcome.
Application of this method is illustrated by examples of comple- mentary energy principle of elasticity, the analysis for incompressible medium ( examples of linear problem with equality constraint ) and the plastic limit analysis (examples of nonliear problem with inequality constraint).
7. A nonlinear programming algorethm is described in chapter 10, which gives an efficient solution of nonlinear finite element equations and a variable metric method based on DFP-BFGS updating is suggested in this algorethm for solving the unconstrained minimization problem. In order to reduce the number of degrees-of-freedom in nonlinear finite element equation, a generalized matrix inverse technique is adopted. And this results in a decrease to a great extant in computer time.
Based on the theories established in this thesis, a computer program PFP ( Panpenalty Finite element Programming ) consisting of about 4000 Fortran statements is developted. Using this program, several engineering problems are calculated, and great number of numerical tests show that the present theories and methods give results with better convergency and higher numerical precision.
1．建立了具有弹塑性耦合效应一般介质的共轭塑性公设及相应的广义本构定律，讨论了非凸屈服条件下的加一卸载准则，纠正了经典塑性位势理论中的某些不严格提法。
2、从共轭塑性公设出发，系统地导出了具有强物理非线性本构行为和边界效应边值问题的变分不等原理、对偶一互补变分原理，讨论了系统能量泛函的极值性质和解的存在，唯一性条件，证明了三类变量泛函的驻值性质。
3、提出了力学系统中一条具有认识论上广泛意义的互补对偶原理。
4、建立了理想弹塑性介质变分问题的最小余能原理，证明了一个理论上严格、且具有较大实用价值的罚－对偶变分原理，讨论了强间断条件下刚塑性介质的对偶变分原理。
5、建立了极限平衡理论中严格互补的对偶界限定理，证明了一个满足kuhn—Tucker约束(？)性条件的对偶型变分原理，由此导出了一个中值定理和一个新的下限定理，建立了关于安全因子的有效的罚－对偶变分原理，并予以实际应用。
6、提出了约束变分问题有限元分析的统一方法——泛惩罚有限元素法，此为建立各种类型的有限元格式提供了正确的途径。证明了混合/杂交元和罚函数单元又是其中的泛惩罚函数的二种简单构造。导出了一个有效的罚一对偶有限元格式和其修正格式，这种格式不仅保证了数值解的收敛性，同时克服了单纯罚单元的弊病，具有较高的数值精度和稳定性。并具体构造了一个泛惩罚余能有限元格式。
7、编制了弹塑性分析的泛惩罚有限元程序，提出了一种降低非线性有限元规划问题自由度的计算方法，从而大大提高了计算效率。对工程问题进行了实际应用，大量数值实验结果证明了本文理论分析和数值方法的正确性。
Based on the mathematical theory of nonsmooth analysis, a systematical research has been devoted in this dissertation to problems on the plastic potential theory, variational boundary value problems, limit analysis, nonlinear finite element analysis and numerical procedures etc.. The work carried out includes the following:
1. A couple of generalized conjugate plastic postulates are proposed, which can be used in analysis of elastoplastic coupling for both hard materials(such as usual structural metal) and soft materials (such as excelsior, foam rubber, grain material and some polymer etc.). From these postulates, the corresponding generalized constitutive relations are established, which permit the study of large classes of materials. In addition, the loading-unloading criterion for nonconvex plastic potential is given by use of the concept of generalized gradient in Clarke's sence, and the classical plastic potential is replaced by the generalized superpotentials, which yield the correct and symmetric mathematical expressions of constitutive relations.
2. According to the generalized plastic postulates, two conjugate energy bounding theorems for deformation theories are developed in the form of variational inequalities, which yield the dual-complementary variational principles for general materials including strong physical nonlinearities and discontinuities ( like jumps, locking effects, etc.). The extremum properties of variational functional ( such as Hamiltonian, Lagrangian and psudo-Hamiltonian etc. ) are studied and the existence and uniqueness of solutions for variational boundary value problems are proved by using convex analysis theory.
3. In chapter 5 of this thesis, a true complementary energy variational principle for elasto-perfect plasticity is established with the aid of Fenchel transformation. Based on the property of plastic super-potantial in this principle, two kinds of variational principles are constructed. The first one is a penalty-type principle, which possesses obviously physical significance, and the other one is a duality-type principle, in which the dual variable satisfying the so-called constraint qualification in nonlinear programming is suggested. By joining these two principles together, an interesting penalty-duality variational principle is established, which is not only rigorous in theory, but also of practical value in engineering applications.
4. The dual bounding theorems and a generalized variational principle for limit analysis are established in chapter 6 of this thesis. Based on the duality-type principle in this problem, a new lower bound theorem is stated, in which, the yield condition is relaxed, and a mean-value theorem is proved which gives the safety factor lying between the upper and lower bounds obtained by classical bounding theorems with the same kinematically and statically admissiol (?) Moreover, a penalty-duality algorethm for the safety factor of limit analysis is developed, and the application of these theorems are illustrated by several examples.
5. As a philosophy by-product, a universal complementary-dual principle of continuum mechanics is advanced in this Ph. D. thesis, which may be described as follows :
Let U be a general displacement space and U~* be a complementary general force space (here, the terminology " complementary " is inphilosophical sence ), then, if there exists a theorem about U, we musthave a dual theorem about U~* if we have a theorem defined on the multiplicative space U×U~*, exchange the place of dual variables in this theorem, then a new dual theorem may be obtained.
All the theorems proposed in this thesis may be considered as applications of this principle.
6. Based on the theory of nonlinear programming, a unified formula----Panpenalty Finite Element Method ---- is established for variational boundary value problems with general constraint conditions. Which provides a correct way to construct various finite element models, it is proved that the hybrid/mixed type model and pure penalty type model are no more than simple construction of the panpenalty function in this formula. An efficient penalty-duality finite element model and its modified model are constructed, the positive definite Hessian matrix of these two models may be obtained for some given penalty factors determined by certain conditions, meanwhile, the disadvantages of pure penalty type model are overcome.
Application of this method is illustrated by examples of comple- mentary energy principle of elasticity, the analysis for incompressible medium ( examples of linear problem with equality constraint ) and the plastic limit analysis (examples of nonliear problem with inequality constraint).
7. A nonlinear programming algorethm is described in chapter 10, which gives an efficient solution of nonlinear finite element equations and a variable metric method based on DFP-BFGS updating is suggested in this algorethm for solving the unconstrained minimization problem. In order to reduce the number of degrees-of-freedom in nonlinear finite element equation, a generalized matrix inverse technique is adopted. And this results in a decrease to a great extant in computer time.
Based on the theories established in this thesis, a computer program PFP ( Panpenalty Finite element Programming ) consisting of about 4000 Fortran statements is developted. Using this program, several engineering problems are calculated, and great number of numerical tests show that the present theories and methods give results with better convergency and higher numerical precision.
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