二维超圆形颗粒离散元法的算法研究
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摘要
本文采用超圆方程建立复杂形状颗粒的离散元法分析模型,采用几何势方法建立颗粒间及颗粒与边界间的接触判断和接触叠合量求解方程,采用遗传算法求解该方程,由此建立了超圆颗粒的离散元法计算方法。
在课题组研制的二维离散元仿真软件的基础上,编写了二维超圆形颗粒离散元法仿真软件,并将其整合进课题组已研制的二维软件中。二维超圆形颗粒离散元仿真软件分为离散元计算和仿真与性能分析两个模块,离散元计算模块负责计算每个时步颗粒的受力情况、运动速度以及当前的位置。仿真与性能分析模块以图形的方式将离散元计算的结果展示给用户,并提供颗粒所受力场、运动速度场显示、单个颗粒追踪和极值统计等功能。
为验证二维超圆形颗粒离散元法软件的可行性和有效性,以简单图元边界为例,采用白盒测试法对软件的离散元计算模块进行测试,应用黑盒测试法对二维离散元法仿真软件进行整体测试。结果表明,软件的功能已实现。
本文的工作对于完善课题组研制的二维离散元仿真软件,扩大该软件的使用范围均有一定意义。
In the real world, there are granular materials everywhere, such as food particles in agricultural production and agricultural processing, gravel materials in the course of construction gravel materials, medicine and fertilizer granular material in the field of medicine and chemical industry. The nature of granular materials is between solid and fluid, it has a complex physical properties. Research on granular materials with traditional continuous media mechanics method, only put granular materials as a whole and can’t analyze motion of granular materials.
In 1971, Cundall proposed discrete element method to analyze the mechanical properties of granular materials. The basic idea of discrete element method is to abstract granular materials as a group of particles with a certain kind of shape and mass, endue the particle-particle and particle-boundary various physical parameters, analyze the movement of particles and the various physical effects of particles-particle and particles- boundary. The calculation process of discrete element method is divided into a number of time steps, using the dynamic relaxation method, Newton's second law to calculate force conditions, velocity and displacement of each particle at each time step. In addition to analyze motion characteristics of granular materials, the discrete element method can analyze to the flow of chemical reactions and heat transfer process. At present, the discrete element method has become the common methods to research mechanical properties of granular materials and is widely used in geotechnical engineering and equipment, mining engineering and equipment, chemical process and equipment, pharmaceutical engineering and equipment, food and agricultural engineering and equipment research fields.
Currently, the task group has developed a set of design and performance analysis in one, universal, digital design and analysis software. The software put the model of the boundary with CAD, set up a meta based two-dimensional boundary element method for discrete analysis model, process contact inspections of particle-particle and particle-boundary with neighbor search and geometric and numerical methods. Now the two-dimensional discrete element analysis software can simulation several kind of particle with high accuracy.
It is a key issue of how to put the model of particle in discrete element method. It is necessary to model highly accurate in order to accurately simulated the movement of granular materials. Home and abroad, mostly discrete element simulation software use circular particle modeling, it is very simple in contact detection and the calculation of composite volume of round particle, but the circular particle is much different from the actual object, make low accuracy of simulation, the simulation result is quite different from actual result. Currently, the model of particle with complex shape use combination particle model, which the particle consist of with several round with different radius. It calculates the force of combination particle by calculating the force of the round which the combination particle is consists. The combination particle model has high accurate and high quantity of computation. The research on new particle model is important to the development of discrete element analysis method.
Super-quadratics equation is the expansion of elliptic equations in two-dimensional, according to statistics, 80% of the natural world plane image circle equation can be expressed by super-quadratics equation. This feature of super-quadratics equation makes put model of complex shape particle using super-quadratics equation possible. In this paper put model of complex shape particle with super-quadratics equation, use the geometric potential method to process contact inspections of particle- particles and particle-boundary.
The calculation of the value of the composition oval-shaped particles and particles with boundaries using the geometric potential can be summarized as solving a constrained optimization problem. As the complexity of the super-quadratics equation, there is various problems when solve the constrained optimization problem with commonly used optimization method, such as the Newton method, quasi-Newton method, conjugate gradient method and the multiplier method. these algorithms are high demanding of the initial point, and demand the first-order partial differential equations of the target function, the computational complexity is high, and the local convergence of these algorithms are the algorithm is not guaranteed to get the global optimal solution. Discrete element analysis method with high precision requires the use of high-precision global convergence algorithm to solve this problem.
In this article, we use genetic algorithm for solving this optimization problem. The main work in this article is the following:
1. The calculation of the value of the composition oval-shaped particles and particles with boundaries using the geometric potential can be summarized as solving a constrained optimization problem. As the complexity of the super-quadratics equation, it can’t solve this problem effectively with commonly used optimization method. So we solve this problem with genetic algorithm in this article. This article also introduces the basic principles of genetic algorithms and the basic structure of genetic algorithms, and the method improving the efficiency of genetic algorithm.
2. This article describe the principle and algorithm of complex shape particle of discrete element method analysis method, use super-quadratics equations put the DEM analyses model of complex shape particles, process contact inspections and compute the value of the composition using geometric potential and genetic algorithms, the contact inspections of particle-boundary is same to the contact inspections of particle-particle, compute the number of point of intersection between the line and the particle to process contact inspections between the particle and line.
3. This article accomplish the two-dimensional super-quadratics particle discrete element simulation software using VC + + and integrate it into the project team two-dimensional discrete element analysis software developed by task group.
4. Introduced the basic principles of software testing methods, and test the two-dimensional super-quadratics particle discrete element method simulation software. We test the overlap calculation process of particle- particle and particle–boundary by white-box testing, test the accuracy and reliability of software using black-box testing. The results show that the two-dimensional super-quadratics particle discrete element simulation software has realized the expected function.
The work in this article has a certain sense to perfect the two-dimensional digital design and analysis software developed by our team, and to expand the usage of the software.
在课题组研制的二维离散元仿真软件的基础上,编写了二维超圆形颗粒离散元法仿真软件,并将其整合进课题组已研制的二维软件中。二维超圆形颗粒离散元仿真软件分为离散元计算和仿真与性能分析两个模块,离散元计算模块负责计算每个时步颗粒的受力情况、运动速度以及当前的位置。仿真与性能分析模块以图形的方式将离散元计算的结果展示给用户,并提供颗粒所受力场、运动速度场显示、单个颗粒追踪和极值统计等功能。
为验证二维超圆形颗粒离散元法软件的可行性和有效性,以简单图元边界为例,采用白盒测试法对软件的离散元计算模块进行测试,应用黑盒测试法对二维离散元法仿真软件进行整体测试。结果表明,软件的功能已实现。
本文的工作对于完善课题组研制的二维离散元仿真软件,扩大该软件的使用范围均有一定意义。
In the real world, there are granular materials everywhere, such as food particles in agricultural production and agricultural processing, gravel materials in the course of construction gravel materials, medicine and fertilizer granular material in the field of medicine and chemical industry. The nature of granular materials is between solid and fluid, it has a complex physical properties. Research on granular materials with traditional continuous media mechanics method, only put granular materials as a whole and can’t analyze motion of granular materials.
In 1971, Cundall proposed discrete element method to analyze the mechanical properties of granular materials. The basic idea of discrete element method is to abstract granular materials as a group of particles with a certain kind of shape and mass, endue the particle-particle and particle-boundary various physical parameters, analyze the movement of particles and the various physical effects of particles-particle and particles- boundary. The calculation process of discrete element method is divided into a number of time steps, using the dynamic relaxation method, Newton's second law to calculate force conditions, velocity and displacement of each particle at each time step. In addition to analyze motion characteristics of granular materials, the discrete element method can analyze to the flow of chemical reactions and heat transfer process. At present, the discrete element method has become the common methods to research mechanical properties of granular materials and is widely used in geotechnical engineering and equipment, mining engineering and equipment, chemical process and equipment, pharmaceutical engineering and equipment, food and agricultural engineering and equipment research fields.
Currently, the task group has developed a set of design and performance analysis in one, universal, digital design and analysis software. The software put the model of the boundary with CAD, set up a meta based two-dimensional boundary element method for discrete analysis model, process contact inspections of particle-particle and particle-boundary with neighbor search and geometric and numerical methods. Now the two-dimensional discrete element analysis software can simulation several kind of particle with high accuracy.
It is a key issue of how to put the model of particle in discrete element method. It is necessary to model highly accurate in order to accurately simulated the movement of granular materials. Home and abroad, mostly discrete element simulation software use circular particle modeling, it is very simple in contact detection and the calculation of composite volume of round particle, but the circular particle is much different from the actual object, make low accuracy of simulation, the simulation result is quite different from actual result. Currently, the model of particle with complex shape use combination particle model, which the particle consist of with several round with different radius. It calculates the force of combination particle by calculating the force of the round which the combination particle is consists. The combination particle model has high accurate and high quantity of computation. The research on new particle model is important to the development of discrete element analysis method.
Super-quadratics equation is the expansion of elliptic equations in two-dimensional, according to statistics, 80% of the natural world plane image circle equation can be expressed by super-quadratics equation. This feature of super-quadratics equation makes put model of complex shape particle using super-quadratics equation possible. In this paper put model of complex shape particle with super-quadratics equation, use the geometric potential method to process contact inspections of particle- particles and particle-boundary.
The calculation of the value of the composition oval-shaped particles and particles with boundaries using the geometric potential can be summarized as solving a constrained optimization problem. As the complexity of the super-quadratics equation, there is various problems when solve the constrained optimization problem with commonly used optimization method, such as the Newton method, quasi-Newton method, conjugate gradient method and the multiplier method. these algorithms are high demanding of the initial point, and demand the first-order partial differential equations of the target function, the computational complexity is high, and the local convergence of these algorithms are the algorithm is not guaranteed to get the global optimal solution. Discrete element analysis method with high precision requires the use of high-precision global convergence algorithm to solve this problem.
In this article, we use genetic algorithm for solving this optimization problem. The main work in this article is the following:
1. The calculation of the value of the composition oval-shaped particles and particles with boundaries using the geometric potential can be summarized as solving a constrained optimization problem. As the complexity of the super-quadratics equation, it can’t solve this problem effectively with commonly used optimization method. So we solve this problem with genetic algorithm in this article. This article also introduces the basic principles of genetic algorithms and the basic structure of genetic algorithms, and the method improving the efficiency of genetic algorithm.
2. This article describe the principle and algorithm of complex shape particle of discrete element method analysis method, use super-quadratics equations put the DEM analyses model of complex shape particles, process contact inspections and compute the value of the composition using geometric potential and genetic algorithms, the contact inspections of particle-boundary is same to the contact inspections of particle-particle, compute the number of point of intersection between the line and the particle to process contact inspections between the particle and line.
3. This article accomplish the two-dimensional super-quadratics particle discrete element simulation software using VC + + and integrate it into the project team two-dimensional discrete element analysis software developed by task group.
4. Introduced the basic principles of software testing methods, and test the two-dimensional super-quadratics particle discrete element method simulation software. We test the overlap calculation process of particle- particle and particle–boundary by white-box testing, test the accuracy and reliability of software using black-box testing. The results show that the two-dimensional super-quadratics particle discrete element simulation software has realized the expected function.
The work in this article has a certain sense to perfect the two-dimensional digital design and analysis software developed by our team, and to expand the usage of the software.
引文
[1] Williams J R, Pentland A P. Super-quadratics and modal for discrete elements in interactive design. Engineering Computation, 1992, 9: 115~127
[2] John H Holland. Adaptation in Natural and Artificial Systems: An Introductory Analysis with Applications to Biology, Control, and Artificial Intelligence.
[3]胡毓达.实用多目标最优化.上海:上海科学技术出版社. 1990.4.
[4]陈国良.遗传算法及其应用.北京:人民邮电出版社. 1996.
[5]席裕庚,柴天佑,恽为天.遗传算法综述.控制理论与应用. 1996, 1(6).
[6]周明,孙树栋,彭炎午.基于遗传模拟退火算法的机器人路径规划.航空学报, 1998, 2(1).
[7]孙树栋,曲彦宾.遗传算法在机器人路径规划中的应用研究.西北工业大学学报. 1998, 16(2).
[8]田盛丰.人工智能原理与应用.北京理工大学出版社, 1993.
[9] De Jong K A. An Analysis of the Behavior of a Class of Genetic Adaptive Systems. Ph D Dissertation, University of Michigan, No.76-9381, 1975.
[10] Davis L. Adapting Operater Probability in Genetic. In: Proc. of 3rd Int. conf. on Genetic Algorithms, Morgan Kaufmann, 1989, 61-69
[11] Whitley D. al. Genitor II: A Distributed Genetic Algorithm. J Expt Ther Intell 1990.2:189~214
[12] Goldberg D E. Genetic Algorithm in Search. Optimization and Machine Learning. Addison-Welsey, 1989.
[13]周明,孙树.遗传算法原理及应用.北京:国防工业出版社. 1999.
[14] Holsteins R B Artificial Genetic Adaptation in Computer Control System. Ph D Dissertation, University of Michigan, 1971.
[15] Bagley J D. The Behavior of Adaptive System which Employ Genetic and Correlation Algorithm. Dissertation Abstracts International, 1967, 28(12).
[16] Brindle A. Genetic Algorithm for Function Optimization. Ph D, University of Alberta, 1981.
[17] Rosenberg R S. Simulation of Genetic Populations with Biochemical Propreties. Ph D Dissertation, University of Michigan, 1967.
[18] Goldberg D E Messy Genetic Algorithm: Motivation Analysis And First Result. Complex System, 1989, 3:493-530
[19]刘勇.非数值并行算法(二)--遗传算法.北京:北京科学出版社. 1995.
[20] Bathurst R J, Rothenburg L. Micromechanical aspects of isotropic granular assemblies with linear contact interactions. Appl. Mech. Am. Soc. Mech. Engrs., 1988, 55: 17~23
[21] Williams J R, Pentland A P. Super quadratics and modal dynamics for discrete elements ininteractive design. Engineering Computations, 1992, 9: 115~127
[22] Buchholtz V, Poschel T. A vectorized algorithm for molecular dynamics of short-range interacting particles. International Journal of Modern Physics C, 1993, 4(5): 1049～1057
[23] Ting J M. An ellipse based micromechanical model for angular granular materials. Proc ASCE Eng. Mech. Specialty Conf, Columbus, OH.2.1991, 1214~1218
[24] Xiao Shan Lin. Short Communication Contact Detection Algorithm for three- Dimensional ellipsoids in discrete element modeling. International journal. For numerical and analytical method in geomemechanics. 1995, 9:653~659
[25] Ting J M. An ellipse-based discrete element modeling of granular materials. Computers and Geotechnics(1993).
[26] Ng T T. numerical simulations of granular soil using elliptical particles. Microstructure Characterization in Constitutive Modeling of Metals and Granular Media. The ASME Summer Mech. and Materials Conf. Tempe, AZ, 1992, 95~118
[27] Holsteins R B Artificial Genetic Adaptation in Computer Control System. Ph D Dissertation, University of Michigan, 1971.
[28] Bagley J D. The Behavior of Adaptive System which Employ Genetic and Correlation Algorithm. Dissertation Abstracts International, 1967, 28(12)
[29] Brindle A. Genetic Algorithm for Function Optimization. Ph D, University of Alberta, 1981.
[30] Rosenberg R S. Simulation of Genetic Populations with Biochemical Propreties. Ph D Dissertation, University of Michigan, 1967.
[31] Goldberg D E Messy Genetic Algorithm: Motivation Analysis And First Result. Complex System, 1989, 3:493-530
[32] Holland J H, Reitman T S Cognitive System Based on Adaptive Algorithms. In: Waterman D A & Hayes-Roth F Eds.Pattern New York: Academic Press. 1978.
[33] Davis L D. Handbook of Generic Algorithms. Van Nostrand Reinhold, 1991
[34] Koza J R Genetic Programming, on the Programming of Computers by Mean of Natural Selection Mit Press, 1992.
[35] Michalewicz Z. Genetic Algorithms and Opitimal Contral Problem. Proc.of 29th IEEE Conf. on Decision and Contral, 1990.
[36] Back T. The Interaction of Mutation Rate, Selection and Self Adaptation within a Genetic Algorithm. Parallel Problem Solving from Nature 2, North Holland, 1992.
[37] Cavicchio D J. Reproductive Adaptive Plans. In: Proc. of the ACM 1972 Annual Conf., 1972.
[38] Booker L B, Goldberg D E, Holland J H. Classifier Systems and Genetic Algorithms. Artificial Intelligence, 1989, 40
[39] Syswrda G. Uniform Crossover in Genetic Algorithm. proc. of 3rd Int. Conf. on Genetic algorithms, Morgan Kaufmann,1989.
[40]唐焕文.实用最优化方法.大连:大连理工大学出版社. 2004.01.
[2] John H Holland. Adaptation in Natural and Artificial Systems: An Introductory Analysis with Applications to Biology, Control, and Artificial Intelligence.
[3]胡毓达.实用多目标最优化.上海:上海科学技术出版社. 1990.4.
[4]陈国良.遗传算法及其应用.北京:人民邮电出版社. 1996.
[5]席裕庚,柴天佑,恽为天.遗传算法综述.控制理论与应用. 1996, 1(6).
[6]周明,孙树栋,彭炎午.基于遗传模拟退火算法的机器人路径规划.航空学报, 1998, 2(1).
[7]孙树栋,曲彦宾.遗传算法在机器人路径规划中的应用研究.西北工业大学学报. 1998, 16(2).
[8]田盛丰.人工智能原理与应用.北京理工大学出版社, 1993.
[9] De Jong K A. An Analysis of the Behavior of a Class of Genetic Adaptive Systems. Ph D Dissertation, University of Michigan, No.76-9381, 1975.
[10] Davis L. Adapting Operater Probability in Genetic. In: Proc. of 3rd Int. conf. on Genetic Algorithms, Morgan Kaufmann, 1989, 61-69
[11] Whitley D. al. Genitor II: A Distributed Genetic Algorithm. J Expt Ther Intell 1990.2:189~214
[12] Goldberg D E. Genetic Algorithm in Search. Optimization and Machine Learning. Addison-Welsey, 1989.
[13]周明,孙树.遗传算法原理及应用.北京:国防工业出版社. 1999.
[14] Holsteins R B Artificial Genetic Adaptation in Computer Control System. Ph D Dissertation, University of Michigan, 1971.
[15] Bagley J D. The Behavior of Adaptive System which Employ Genetic and Correlation Algorithm. Dissertation Abstracts International, 1967, 28(12).
[16] Brindle A. Genetic Algorithm for Function Optimization. Ph D, University of Alberta, 1981.
[17] Rosenberg R S. Simulation of Genetic Populations with Biochemical Propreties. Ph D Dissertation, University of Michigan, 1967.
[18] Goldberg D E Messy Genetic Algorithm: Motivation Analysis And First Result. Complex System, 1989, 3:493-530
[19]刘勇.非数值并行算法(二)--遗传算法.北京:北京科学出版社. 1995.
[20] Bathurst R J, Rothenburg L. Micromechanical aspects of isotropic granular assemblies with linear contact interactions. Appl. Mech. Am. Soc. Mech. Engrs., 1988, 55: 17~23
[21] Williams J R, Pentland A P. Super quadratics and modal dynamics for discrete elements ininteractive design. Engineering Computations, 1992, 9: 115~127
[22] Buchholtz V, Poschel T. A vectorized algorithm for molecular dynamics of short-range interacting particles. International Journal of Modern Physics C, 1993, 4(5): 1049～1057
[23] Ting J M. An ellipse based micromechanical model for angular granular materials. Proc ASCE Eng. Mech. Specialty Conf, Columbus, OH.2.1991, 1214~1218
[24] Xiao Shan Lin. Short Communication Contact Detection Algorithm for three- Dimensional ellipsoids in discrete element modeling. International journal. For numerical and analytical method in geomemechanics. 1995, 9:653~659
[25] Ting J M. An ellipse-based discrete element modeling of granular materials. Computers and Geotechnics(1993).
[26] Ng T T. numerical simulations of granular soil using elliptical particles. Microstructure Characterization in Constitutive Modeling of Metals and Granular Media. The ASME Summer Mech. and Materials Conf. Tempe, AZ, 1992, 95~118
[27] Holsteins R B Artificial Genetic Adaptation in Computer Control System. Ph D Dissertation, University of Michigan, 1971.
[28] Bagley J D. The Behavior of Adaptive System which Employ Genetic and Correlation Algorithm. Dissertation Abstracts International, 1967, 28(12)
[29] Brindle A. Genetic Algorithm for Function Optimization. Ph D, University of Alberta, 1981.
[30] Rosenberg R S. Simulation of Genetic Populations with Biochemical Propreties. Ph D Dissertation, University of Michigan, 1967.
[31] Goldberg D E Messy Genetic Algorithm: Motivation Analysis And First Result. Complex System, 1989, 3:493-530
[32] Holland J H, Reitman T S Cognitive System Based on Adaptive Algorithms. In: Waterman D A & Hayes-Roth F Eds.Pattern New York: Academic Press. 1978.
[33] Davis L D. Handbook of Generic Algorithms. Van Nostrand Reinhold, 1991
[34] Koza J R Genetic Programming, on the Programming of Computers by Mean of Natural Selection Mit Press, 1992.
[35] Michalewicz Z. Genetic Algorithms and Opitimal Contral Problem. Proc.of 29th IEEE Conf. on Decision and Contral, 1990.
[36] Back T. The Interaction of Mutation Rate, Selection and Self Adaptation within a Genetic Algorithm. Parallel Problem Solving from Nature 2, North Holland, 1992.
[37] Cavicchio D J. Reproductive Adaptive Plans. In: Proc. of the ACM 1972 Annual Conf., 1972.
[38] Booker L B, Goldberg D E, Holland J H. Classifier Systems and Genetic Algorithms. Artificial Intelligence, 1989, 40
[39] Syswrda G. Uniform Crossover in Genetic Algorithm. proc. of 3rd Int. Conf. on Genetic algorithms, Morgan Kaufmann,1989.
[40]唐焕文.实用最优化方法.大连:大连理工大学出版社. 2004.01.
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