互补算法在柔性体接触碰撞中的应用
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摘要
本文在国家自然科学基金(编号:10772113)的资助下,开展对互补算法在柔性体的接触碰撞中的应用的研究。
互补问题的最显著特征是含有互补性条件,即要求两组非负变量对应分量的乘积为零。30多年以来,互补问题已经发展成为许多领域非常重要的数学工具,在数学规划、经济、工程及其它学科具有非常广泛的应用。在经济领域中的应用主要有Walrasian平衡、空间价格平衡和对策论模型等;在工程中的应用有接触力学问题、断裂力学问题、弹塑性问题、障碍和自由边界问题、流体弹性动态润滑问题、最优控制问题及交通平衡问题等。也就是说,上述这些问题都可以模型化为互补问题,从而最终归结为互补问题的求解。可以看出,在互补问题众多的应用中,经济和力学是互补问题应用的两个最大的领域。
虽然互补问题已经有许多算法,但一些力学实际问题往往满足不了算法的限定条件,因此需对某些算法做一些适应性改进,在大量阅读文献的基础上,第一章较为全面地综述了互补算法的研究进展。近年来,互补问题的力学应用远远落后于算法发展,把一些新的算法介绍到力学问题求解,既有助于丰富力学问题的算法工具,也有助于扩充互补问题的应用背景。
第二章的前四个算法均为方程组类方法,包括两个光滑牛顿型算法和两个光滑迭代算法。前者借助NCP函数把互补问题转化为等价的非光滑(不可微)方程组,再用带参数的光滑(连续可微)方程组近似这些非光滑方程组,最后用牛顿型方法求解所得到的光滑方程组,希望通过光滑参数趋于零得到原来互补问题的解;后者基于等价不动点格式,构造了一个光滑迭代算法和一个具有有限终止性质的算法,虽然这种迭代算法仅有线性收敛速度,但由于其格式简单、存储量小、保稀疏性、非常易于计算机实现等特点,故较适用于求解大规模稀疏问题。之后给出了两个新的内点算法。本论文从两个不同角度对原一对偶内点方法通常使用的摄动方程组进行了变化,并据此建立了两个不同的内点算法。首先通过对中心化方程实施代数等价变换,得到了新的不同的摄动方程组。我们发现,通过幂变换得到的摄动方程组,可给出彭积明等人提出的大步内点算法的牛顿方程,但代数等价变换的思想要比彭等人算法的思想容易理解得多。在此基础上,我们建立了一个基于幂变换的内点算法。第二个算法利用极大极小(min-max)函数本身所具有的“均化”作用,定义了一个新的邻近度量函数,并以其最优性条件代替中心化方程。这样,在摄动方程本身建立了一种自调节机制,从而使牛顿方向能够根据上次迭代点的信息在各个互补对之间做出自适应的调整。基于改造后的摄动方程组,建立了一个具有自调节功能的内点算法。在每种算法之后用用算例进行了评价。
第三章为柔性体接触碰撞问题的数值求解。利用计算多体动力学中的柔性多体系统的动力学方程,将柔性体的接触碰撞问题转化为线性互补问题,将第二章提出的互补问题算法应用到柔性体接触碰撞问题的数值求解中。之后将数值计算结果与有限元计算的数值结果进行了对照,发现用互补算法解决柔性体的碰撞问题是行之有效的。并且在处理接触点在接触时刻的速度跳跃问题时,用动量守恒计算接触单元的接触初始时刻的速度,然后再进行计算。
互补算法在柔性体的接触碰撞中的应用问题是一个非常具有研究意义的课题,许多方面还需要进一步深入研究和探讨,因此在论文的最后,对本文的研究工作进行了总结,并且对未来的研究问题进行了展望。
This dissertation studies the applications of complementarity algorithms in flexible bodies’contact. This research was funded by the National Natural Science Foundation of China (Grant Nos. 10772113).
The distinguishing feature of a complementarity problem is the set of complementarity conditions, which require that the product of two nonnegative quantities should be zero. Over more than thirty years, this class of problems has become increasingly popular in many fields, such as mathematical programming, economics and engineering. Applications from the field of economics include general Walrasian equilibrium, spatial price equilibrium, and game-theoretic models. In engineering, complementarity problems arise in contact mechanics, fracture mechanics, elastoplastic problems, obstacle and free boundary problems, hydrodynamic lubrication, optimal control problems and traffic equilibrium problems. That is, all the above practical problems can be formulated as complementarity problems and solved by proper algorithms. Economics and mechanics are the two major areas among the numerous applications of complementarity problems.
Practical mechanics problems may not satisfy certain restrictive conditions for some existing algorithms for complementarity problems, though they might be very efficient for some mathematical problems. Therefore, it is necessary to make a few modifications to some existing algorithms so that they are suitable for solving mechanics problems. In chapter 1, the algorithms of complementarity are comprehensively reviewed. The research extent and contents of this dissertation are put forward. Applications of modern algorithms in mechanics dropped far behind the algorithmic developments of complementarity problems. We wish the introduction of some new algorithms to solving mechanics problems should play the role of both enriching the solution tools of mechanics and extending the ranges of complimentarity problems, where by arousing the further interest of researchers from the two fields.
In Chapter 2, two smoothing Newton-type algorithms and two smoothing iterative algorithms are given at first. In the first two algorithms, the complementarily problem is reformulated to non-smooth (non-differentiable) equations using so-called NCP-functions, and then solved by applying Newton-type methods to the smoothened equations. The third algorithm is a smooth iterative method based on an equivalent fixed-point format of the complementarity problem. The forth algorithm is the same as the third one with an addition of finite termination criteria. Although the later two algorithms have only linear rate of convergence, they are especially suitable for large-scale and sparse problems, with features of simple formula, small storage, sparsity preservation and easy implementation. And then two improved interior point algorithms are proposed .They are designed based on the modifications to the standard perturbed system of primal-dual interior-point algorithms from two different angles. The first is based on the algebraically equivalent transformation to the standard centering equation. We discovered that the Newton equations used by Peng Jiming et al. in their long-step interior point algorithms could be derived by a power transformed perturbed system. Our approach of algebraically equivalent transformation is much simpler than the one proposed by Peng et al. in their algorithms. Inspired by this observation, an interior point algorithm based on power transformation is developed. Another interior algorithm is to employ the "homogenizing" effect of min-max function, where the standard centering equation is replaced by the optimality condition of a new proximity measure function. A self-adjusting mechanism is added to the new perturbed system such that the Newton directions of each complementary pairs can be adjusted self-adaptively according to the information of last iterates. We develop a self-adjusting interior point algorithm based on the modified perturbed system.
In chapter 3, we consider finding the numerical solution of flexible bodies’contact problems and the applications of the proposed algorithms. This kind of contact problems can be reduced as a linear complementarily problem using the dynamic equation of computational dynamics of multibody systems. Then we can apply the algorithms proposed in this thesis to the solution of the contact problems. Then compare with the FEM result, we found it is appropriate to consider the contact problems as a linear complementarily problem.
The topic of the applications of complementarity algorithms in flexible bodies’is very significative, where many aspects need further study and more efforts. At conclusion, a summary of work done in this dissertation is given and some problems of interest are also brought forward for future research.
互补问题的最显著特征是含有互补性条件,即要求两组非负变量对应分量的乘积为零。30多年以来,互补问题已经发展成为许多领域非常重要的数学工具,在数学规划、经济、工程及其它学科具有非常广泛的应用。在经济领域中的应用主要有Walrasian平衡、空间价格平衡和对策论模型等;在工程中的应用有接触力学问题、断裂力学问题、弹塑性问题、障碍和自由边界问题、流体弹性动态润滑问题、最优控制问题及交通平衡问题等。也就是说,上述这些问题都可以模型化为互补问题,从而最终归结为互补问题的求解。可以看出,在互补问题众多的应用中,经济和力学是互补问题应用的两个最大的领域。
虽然互补问题已经有许多算法,但一些力学实际问题往往满足不了算法的限定条件,因此需对某些算法做一些适应性改进,在大量阅读文献的基础上,第一章较为全面地综述了互补算法的研究进展。近年来,互补问题的力学应用远远落后于算法发展,把一些新的算法介绍到力学问题求解,既有助于丰富力学问题的算法工具,也有助于扩充互补问题的应用背景。
第二章的前四个算法均为方程组类方法,包括两个光滑牛顿型算法和两个光滑迭代算法。前者借助NCP函数把互补问题转化为等价的非光滑(不可微)方程组,再用带参数的光滑(连续可微)方程组近似这些非光滑方程组,最后用牛顿型方法求解所得到的光滑方程组,希望通过光滑参数趋于零得到原来互补问题的解;后者基于等价不动点格式,构造了一个光滑迭代算法和一个具有有限终止性质的算法,虽然这种迭代算法仅有线性收敛速度,但由于其格式简单、存储量小、保稀疏性、非常易于计算机实现等特点,故较适用于求解大规模稀疏问题。之后给出了两个新的内点算法。本论文从两个不同角度对原一对偶内点方法通常使用的摄动方程组进行了变化,并据此建立了两个不同的内点算法。首先通过对中心化方程实施代数等价变换,得到了新的不同的摄动方程组。我们发现,通过幂变换得到的摄动方程组,可给出彭积明等人提出的大步内点算法的牛顿方程,但代数等价变换的思想要比彭等人算法的思想容易理解得多。在此基础上,我们建立了一个基于幂变换的内点算法。第二个算法利用极大极小(min-max)函数本身所具有的“均化”作用,定义了一个新的邻近度量函数,并以其最优性条件代替中心化方程。这样,在摄动方程本身建立了一种自调节机制,从而使牛顿方向能够根据上次迭代点的信息在各个互补对之间做出自适应的调整。基于改造后的摄动方程组,建立了一个具有自调节功能的内点算法。在每种算法之后用用算例进行了评价。
第三章为柔性体接触碰撞问题的数值求解。利用计算多体动力学中的柔性多体系统的动力学方程,将柔性体的接触碰撞问题转化为线性互补问题,将第二章提出的互补问题算法应用到柔性体接触碰撞问题的数值求解中。之后将数值计算结果与有限元计算的数值结果进行了对照,发现用互补算法解决柔性体的碰撞问题是行之有效的。并且在处理接触点在接触时刻的速度跳跃问题时,用动量守恒计算接触单元的接触初始时刻的速度,然后再进行计算。
互补算法在柔性体的接触碰撞中的应用问题是一个非常具有研究意义的课题,许多方面还需要进一步深入研究和探讨,因此在论文的最后,对本文的研究工作进行了总结,并且对未来的研究问题进行了展望。
This dissertation studies the applications of complementarity algorithms in flexible bodies’contact. This research was funded by the National Natural Science Foundation of China (Grant Nos. 10772113).
The distinguishing feature of a complementarity problem is the set of complementarity conditions, which require that the product of two nonnegative quantities should be zero. Over more than thirty years, this class of problems has become increasingly popular in many fields, such as mathematical programming, economics and engineering. Applications from the field of economics include general Walrasian equilibrium, spatial price equilibrium, and game-theoretic models. In engineering, complementarity problems arise in contact mechanics, fracture mechanics, elastoplastic problems, obstacle and free boundary problems, hydrodynamic lubrication, optimal control problems and traffic equilibrium problems. That is, all the above practical problems can be formulated as complementarity problems and solved by proper algorithms. Economics and mechanics are the two major areas among the numerous applications of complementarity problems.
Practical mechanics problems may not satisfy certain restrictive conditions for some existing algorithms for complementarity problems, though they might be very efficient for some mathematical problems. Therefore, it is necessary to make a few modifications to some existing algorithms so that they are suitable for solving mechanics problems. In chapter 1, the algorithms of complementarity are comprehensively reviewed. The research extent and contents of this dissertation are put forward. Applications of modern algorithms in mechanics dropped far behind the algorithmic developments of complementarity problems. We wish the introduction of some new algorithms to solving mechanics problems should play the role of both enriching the solution tools of mechanics and extending the ranges of complimentarity problems, where by arousing the further interest of researchers from the two fields.
In Chapter 2, two smoothing Newton-type algorithms and two smoothing iterative algorithms are given at first. In the first two algorithms, the complementarily problem is reformulated to non-smooth (non-differentiable) equations using so-called NCP-functions, and then solved by applying Newton-type methods to the smoothened equations. The third algorithm is a smooth iterative method based on an equivalent fixed-point format of the complementarity problem. The forth algorithm is the same as the third one with an addition of finite termination criteria. Although the later two algorithms have only linear rate of convergence, they are especially suitable for large-scale and sparse problems, with features of simple formula, small storage, sparsity preservation and easy implementation. And then two improved interior point algorithms are proposed .They are designed based on the modifications to the standard perturbed system of primal-dual interior-point algorithms from two different angles. The first is based on the algebraically equivalent transformation to the standard centering equation. We discovered that the Newton equations used by Peng Jiming et al. in their long-step interior point algorithms could be derived by a power transformed perturbed system. Our approach of algebraically equivalent transformation is much simpler than the one proposed by Peng et al. in their algorithms. Inspired by this observation, an interior point algorithm based on power transformation is developed. Another interior algorithm is to employ the "homogenizing" effect of min-max function, where the standard centering equation is replaced by the optimality condition of a new proximity measure function. A self-adjusting mechanism is added to the new perturbed system such that the Newton directions of each complementary pairs can be adjusted self-adaptively according to the information of last iterates. We develop a self-adjusting interior point algorithm based on the modified perturbed system.
In chapter 3, we consider finding the numerical solution of flexible bodies’contact problems and the applications of the proposed algorithms. This kind of contact problems can be reduced as a linear complementarily problem using the dynamic equation of computational dynamics of multibody systems. Then we can apply the algorithms proposed in this thesis to the solution of the contact problems. Then compare with the FEM result, we found it is appropriate to consider the contact problems as a linear complementarily problem.
The topic of the applications of complementarity algorithms in flexible bodies’is very significative, where many aspects need further study and more efforts. At conclusion, a summary of work done in this dissertation is given and some problems of interest are also brought forward for future research.
引文
[1] S.C.Billups, K.G.Murty. Complementarity problems. Journal of Computational and Applied Mathematics, 2000,124:303-318
[2] P.T.Harker, J.S.Pang. Finite-dimensional variational inequality and nonlinear complementarity problems:a survey of theory,algorithms and applications. Mathematical Programming,1990,48:161-220
[3]洪嘉振。计算多体系统动力学。北京:高等教育出版社,1999(2003重印)
[4] R.W.Cottle, J.S.Pang, and R.E.Stone. The linear Complementarity Problem, Academic Press, Boston,MA,1992
[5] K.G.Murty. Linear Complementarity, Linear and Nonlinear Programming, Heldermann Verlag, Berlin, 1988,Free public access at http://www.personal.engin.umich.edu/~Murty/book/LCPbook/index.html
[6] N.Karmarkar. A new polynomial-time algorithm for linear programming. Combinatorica, 1984,4:373-395
[7] S.J.Wright.Primal-dual interior-point methods.SIAM.Philadelphia,PA,1997
[8] O.L.Mangasarian,Equivalence of the complementarity problem to a system of nonlinear equations,SIAM Journal on Applied Mathematics,1976,31:89-92
[9] B.C.Eaves. On the basic theorem of complementarity. Mathematical Programming, 1971,1:68-75
[10]修乃华,高自友。互补问题算法的新近展。数学进展,1999,28(3):193-210。(Xiu Naihua, Gao Ziyou. The new advances in methods for complementarity problem. Advances in Mathematics.1999,28(3):193-210.(in Chinese))
[11] M.C.Ferris, C.Kanzow. Complementarity and related problems: A survey. Ln: P.M.pardalos and M.G.C.Resende (eds): Handbook on Applied Optimization[C], pages 514-530. Oxford University Press,New York,2002
[12] Qi Liqun, Sun Defeng, Nonsmooth equations and smoothing methods.Progress in Optimization: Contribution from Australian. Eberhard A, Glover B and Ralph D (eds.). Kluwer Academic Publisher.Norwell, MA. USA,1998
[13] B.Chen, R.T.Hanker. A non-interior continuation method for linear complementarity problems.SIAM Journal on Matrix Analysis and Applicatons,1993,14:1168-1190
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[2] P.T.Harker, J.S.Pang. Finite-dimensional variational inequality and nonlinear complementarity problems:a survey of theory,algorithms and applications. Mathematical Programming,1990,48:161-220
[3]洪嘉振。计算多体系统动力学。北京:高等教育出版社,1999(2003重印)
[4] R.W.Cottle, J.S.Pang, and R.E.Stone. The linear Complementarity Problem, Academic Press, Boston,MA,1992
[5] K.G.Murty. Linear Complementarity, Linear and Nonlinear Programming, Heldermann Verlag, Berlin, 1988,Free public access at http://www.personal.engin.umich.edu/~Murty/book/LCPbook/index.html
[6] N.Karmarkar. A new polynomial-time algorithm for linear programming. Combinatorica, 1984,4:373-395
[7] S.J.Wright.Primal-dual interior-point methods.SIAM.Philadelphia,PA,1997
[8] O.L.Mangasarian,Equivalence of the complementarity problem to a system of nonlinear equations,SIAM Journal on Applied Mathematics,1976,31:89-92
[9] B.C.Eaves. On the basic theorem of complementarity. Mathematical Programming, 1971,1:68-75
[10]修乃华,高自友。互补问题算法的新近展。数学进展,1999,28(3):193-210。(Xiu Naihua, Gao Ziyou. The new advances in methods for complementarity problem. Advances in Mathematics.1999,28(3):193-210.(in Chinese))
[11] M.C.Ferris, C.Kanzow. Complementarity and related problems: A survey. Ln: P.M.pardalos and M.G.C.Resende (eds): Handbook on Applied Optimization[C], pages 514-530. Oxford University Press,New York,2002
[12] Qi Liqun, Sun Defeng, Nonsmooth equations and smoothing methods.Progress in Optimization: Contribution from Australian. Eberhard A, Glover B and Ralph D (eds.). Kluwer Academic Publisher.Norwell, MA. USA,1998
[13] B.Chen, R.T.Hanker. A non-interior continuation method for linear complementarity problems.SIAM Journal on Matrix Analysis and Applicatons,1993,14:1168-1190
[14] C.Kanzow.Some noninterior continuation methods for linear complementarity problems,SIAM Journal on Matrix Analysis and Applicatons,1996,17:851-868
[15] J.V.Burke, Song Xu.The global linear convergence of a noninterior path following algorithm for linear complementarity problems. Mathematics of Operations Research,1998,23:719-734
[16] G.Maier. A matrix structural theory piecewise linear elastoplasticity with interacting yield planea.Meccanica.1970.5:54-56
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