饱和多孔介质大变形动力及接触分析的耦合对流粒子域插值物质点方法
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摘要
多孔介质力学在许多领域中都有着广泛的应用,常见的岩土材料包括岩石、土壤,以及一些生物材料包括皮肤、肌肉组织和关节软骨等都可以抽象为多孔介质。因此针对多孔介质的研究具有十分重要的意义。本文的研究对象为饱和多孔介质,其作为一种典型的多孔介质,由固体骨架和孔隙流体组成。使用有限元方法可对饱和多孔介质材料进行动力学响应分析,但有限元方法在划分复杂结构网格和处理大变形问题时会出现困难。为了克服这些困难,许多学者提出了一系列的无网格法,作为无网格方法的典型代表的物质点方法可成功模拟饱和多孔介质小变形动力学响应,但由于其插值函数不光滑,在进行大变形情况计算时可能会发生物质点穿越背景网格的误差。广义插值物质点法(GIMP)可以增加插值函数的光滑性,但其计算复杂繁琐。因此有学者提出对流粒子域插值技术(CPDI),针对平行四边形的粒子区域,既可以减少物质点穿越网格的误差又能使计算得到简化。
本文基于u-p形式的控制方程和对流粒子域插值技术,提出了饱和多孔介质大变形动力学响应分析的耦合对流粒子域插值物质点方法(Coupling Convected Particle Domain Interpolation, CCPDI),并发展了适用于耦合对流粒子域插值物质点方法的接触算法,可以进行饱和多孔介质大变形接触分析。本文所发展的方法可有效地避免饱和多孔介质大变形及接触分析过程中物质点跨越网格时所带来的数值误差、准确地捕捉接触网格节点及预测接触时间,典型算例验证了本文方法的正确性和有效性。本文主要内容如下：
第一部分说明了多孔介质的研究背景和研究意义,并对几类无网格方法做简要概括,然后概述了本文主要研究的内容。第二部分针对多孔介质的力学行为模拟介绍了研究多孔介质需要用到的各种物理量和简化计算需要用到的一些假设以及多孔介质的宏观控制方程。
第三部分简要介绍了标准物质点方法(MPM)及其改进算法包括广义插值物质点方法(GIMP)和对流粒子域插值方法(CPDI).广义插值物质点方法是针对解决物质点方法计算大变形问题时物质点穿越网格的误差而提出的,是对标准物质点方法的广义化,但对平行四边形的粒子区域其计算十分复杂。对流粒子域插值方法提出了一种针对平行四边形粒子区域的插值函数简化了计算,提高了计算效率。
第四部分详细介绍了本文提出的耦合对流粒子域插值物质点方法,建立了耦合对流粒子域插值物质点方法u-p形式的控制方程的弱形式并推导其离散求解方程,给出了该方法的计算流程。提出随动载荷施加方法,使载荷施加位置随物体边界一起运动。通过算例与有限元方法和耦合物质点方法的结果对比,验证所提出方法的正确性。
第五部分发展了适用于耦合对流粒子域插值物质点方法的接触算法,使其能够模拟多孔介质的大变形接触问题。将边界粒子域引入判断接触节点算法中,可以方便的描述接触面,并更加准确的预测接触时间。通过一些接触算例验证了所提出方法的正确性。
第六部分是本文的总结和工作展望。附录中包括可进行饱和多孔介质大变形动力学分析的Poro-CCPDI程序整体的流程图以及接触部分的计算流程图。
本文工作得到国家自然科学基金(11072051,90715037,10902021)、教育部111引智计划项目(B08014)、长江学者和创新团队发展计划以及国家重点基础研究发展计划(973计划)项目(2010CB832704)的资助。
The mechanics of porous media has widely applications in many engineering fields, such as geotechnical engineering and biomechanics engineering. Therefore, studies on the dynamic behaviors of saturated porous media have attracted much attention in the past few decades. Saturated porous media which consisted of a skeleton and the fluid in the pore is a typical kind of porous media. The finite element methods (FEMs) have been applied to predict the dynamic response about the solid-fluid problems. Although there are many advantages of the FEMs, the limitations of themselves should not be neglected. One limitation is that large deformation may cause local mesh distortion, which leads to great numerical artifact errors. Another one is that the generation of mesh is quite tricky in three-dimensional objects with complex structure. To overcome these difficulties, many kinds of meshfree/meshless methods have been proposed and the Material Point Method (MPM) is a typical method among them. MPM has been used successfully in simulating the dynamic behaviors of saturated porous media involving small deformation. However, due to the lack of smoothness of the interpolation functions of the original MPM, some numerical noises may occur while material points crossing computational grid boundaries. The generalized interpolation material point method (GIMP) was proposed to improve the smoothness of the interpolation functions, but the procedure is quit complex. To simplify the computational procedure, a modified interpolation technique named Convected Particle Domain Interpolation (CPDI) method was proposed by some scholars.
In this thesis, the Coupling Convected Particle Domain Interpolation method (CCPDI) is proposed for the dynamic analysis of saturated porous media involving large deformation based on the u-p form governing equations and the interpolation technique of CPDI. Furthermore, on the basis of the proposed CCPDI method, the dynamic contact algorithm is developed to cope with the dynamic contact problem of saturated porous media involving large deformation. The algorithm developed in this thesis could efficiently eliminate the error in simulating large deformation behavior and predicting the contact time. Computational results of several representative examples demonstrate the accuracy and efficiency of the proposed method.
The organization of this thesis is as follows. In the first chapter, the background of this research and a number of meshless methods are briefly summarized. Then, the contents of the thesis are outlined at the end of the chapter. The second chapter introduces several physical quantities used in describing the porous media, some necessary assumption and the governing equation based on the Biot theory. A brief review about several particle methods include the standard MPM, the GIMP and the CPDI method is presented in chapter3.
In the fourth chapter, the procedure of the coupling convected particle domain interpolation method (CCPDI) proposed in the thesis is introduced in details. The weak form is established derived from u-p form governing equations. Furthermore, a tracking method is proposed to deal with the external loads on moving boundaries during the computational process. Several representative examples demonstrate the accuracy and efficiency of the proposed method.
The contact algorithm is developed in chapter5to simulate the dynamic large deformation of saturated porous media. To predict the contact area and the contact time more accurately, the particle domain is utilized in judging the contact background nodes. The validity of the developed algorithm is illustrated through the numerical calculations regarding dynamic contact problem of saturated porous media.
The last chapter includes the conclusions of this thesis and the outlook of further work. In addition, the flow chart of the Poro-CCPDI code, which has been developed in this work and could be used to simulate the dynamic and contact behavior of saturated porous media involving large deformation, can be found in the Appendix.
The thesis is supported by the National Natural Science Foundation of China (11072051,90715037,10902021), the lllproject (B08014), the program for Changjiang Scholar and Innovative Research Team in University (PCSIRT) and the National Key Basic Research Special Foundation of China (2010CB832704).
本文基于u-p形式的控制方程和对流粒子域插值技术,提出了饱和多孔介质大变形动力学响应分析的耦合对流粒子域插值物质点方法(Coupling Convected Particle Domain Interpolation, CCPDI),并发展了适用于耦合对流粒子域插值物质点方法的接触算法,可以进行饱和多孔介质大变形接触分析。本文所发展的方法可有效地避免饱和多孔介质大变形及接触分析过程中物质点跨越网格时所带来的数值误差、准确地捕捉接触网格节点及预测接触时间,典型算例验证了本文方法的正确性和有效性。本文主要内容如下：
第一部分说明了多孔介质的研究背景和研究意义,并对几类无网格方法做简要概括,然后概述了本文主要研究的内容。第二部分针对多孔介质的力学行为模拟介绍了研究多孔介质需要用到的各种物理量和简化计算需要用到的一些假设以及多孔介质的宏观控制方程。
第三部分简要介绍了标准物质点方法(MPM)及其改进算法包括广义插值物质点方法(GIMP)和对流粒子域插值方法(CPDI).广义插值物质点方法是针对解决物质点方法计算大变形问题时物质点穿越网格的误差而提出的,是对标准物质点方法的广义化,但对平行四边形的粒子区域其计算十分复杂。对流粒子域插值方法提出了一种针对平行四边形粒子区域的插值函数简化了计算,提高了计算效率。
第四部分详细介绍了本文提出的耦合对流粒子域插值物质点方法,建立了耦合对流粒子域插值物质点方法u-p形式的控制方程的弱形式并推导其离散求解方程,给出了该方法的计算流程。提出随动载荷施加方法,使载荷施加位置随物体边界一起运动。通过算例与有限元方法和耦合物质点方法的结果对比,验证所提出方法的正确性。
第五部分发展了适用于耦合对流粒子域插值物质点方法的接触算法,使其能够模拟多孔介质的大变形接触问题。将边界粒子域引入判断接触节点算法中,可以方便的描述接触面,并更加准确的预测接触时间。通过一些接触算例验证了所提出方法的正确性。
第六部分是本文的总结和工作展望。附录中包括可进行饱和多孔介质大变形动力学分析的Poro-CCPDI程序整体的流程图以及接触部分的计算流程图。
本文工作得到国家自然科学基金(11072051,90715037,10902021)、教育部111引智计划项目(B08014)、长江学者和创新团队发展计划以及国家重点基础研究发展计划(973计划)项目(2010CB832704)的资助。
The mechanics of porous media has widely applications in many engineering fields, such as geotechnical engineering and biomechanics engineering. Therefore, studies on the dynamic behaviors of saturated porous media have attracted much attention in the past few decades. Saturated porous media which consisted of a skeleton and the fluid in the pore is a typical kind of porous media. The finite element methods (FEMs) have been applied to predict the dynamic response about the solid-fluid problems. Although there are many advantages of the FEMs, the limitations of themselves should not be neglected. One limitation is that large deformation may cause local mesh distortion, which leads to great numerical artifact errors. Another one is that the generation of mesh is quite tricky in three-dimensional objects with complex structure. To overcome these difficulties, many kinds of meshfree/meshless methods have been proposed and the Material Point Method (MPM) is a typical method among them. MPM has been used successfully in simulating the dynamic behaviors of saturated porous media involving small deformation. However, due to the lack of smoothness of the interpolation functions of the original MPM, some numerical noises may occur while material points crossing computational grid boundaries. The generalized interpolation material point method (GIMP) was proposed to improve the smoothness of the interpolation functions, but the procedure is quit complex. To simplify the computational procedure, a modified interpolation technique named Convected Particle Domain Interpolation (CPDI) method was proposed by some scholars.
In this thesis, the Coupling Convected Particle Domain Interpolation method (CCPDI) is proposed for the dynamic analysis of saturated porous media involving large deformation based on the u-p form governing equations and the interpolation technique of CPDI. Furthermore, on the basis of the proposed CCPDI method, the dynamic contact algorithm is developed to cope with the dynamic contact problem of saturated porous media involving large deformation. The algorithm developed in this thesis could efficiently eliminate the error in simulating large deformation behavior and predicting the contact time. Computational results of several representative examples demonstrate the accuracy and efficiency of the proposed method.
The organization of this thesis is as follows. In the first chapter, the background of this research and a number of meshless methods are briefly summarized. Then, the contents of the thesis are outlined at the end of the chapter. The second chapter introduces several physical quantities used in describing the porous media, some necessary assumption and the governing equation based on the Biot theory. A brief review about several particle methods include the standard MPM, the GIMP and the CPDI method is presented in chapter3.
In the fourth chapter, the procedure of the coupling convected particle domain interpolation method (CCPDI) proposed in the thesis is introduced in details. The weak form is established derived from u-p form governing equations. Furthermore, a tracking method is proposed to deal with the external loads on moving boundaries during the computational process. Several representative examples demonstrate the accuracy and efficiency of the proposed method.
The contact algorithm is developed in chapter5to simulate the dynamic large deformation of saturated porous media. To predict the contact area and the contact time more accurately, the particle domain is utilized in judging the contact background nodes. The validity of the developed algorithm is illustrated through the numerical calculations regarding dynamic contact problem of saturated porous media.
The last chapter includes the conclusions of this thesis and the outlook of further work. In addition, the flow chart of the Poro-CCPDI code, which has been developed in this work and could be used to simulate the dynamic and contact behavior of saturated porous media involving large deformation, can be found in the Appendix.
The thesis is supported by the National Natural Science Foundation of China (11072051,90715037,10902021), the lllproject (B08014), the program for Changjiang Scholar and Innovative Research Team in University (PCSIRT) and the National Key Basic Research Special Foundation of China (2010CB832704).
引文
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[2]HUYGHE J M, VON CAMPEN D H, Arts T, et al. A two-phase finite element model of the diastolic left ventricle [J]. Journal of Biomechanics,1991,24(7):527-538.
[3]OOMENS C W J, VAN CAMPEN D H. A mixture approach to the mechanics of skin [J]. Journal of Biomechanics,1987,20:877-885.
[4]MAK A F T, HUANG L, WANG Q. A biphasic poroelastic analysis of the flow dependent subcutaneous tissue pressure and compaction due to epidermal loadings:issues in pressure core [J]. Journal of Biomechanical Engineering,1994,116:421-429.
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[8]TERZAGHI K. The shearing resistance of saturated soils and the angle between the planes of shear [C]. International conference on soil mechanics and foundation engineering, 1, Cambridge,1936, MA Proc I,54-56.
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[23]WIECKOWSKI Z, YOUN S K, YEON J H. A particle-in-cell solution to the silo discharging problem [J]. International Journal for Numerical Methods in Engineering,1999, 45:1203-1225.
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[31]BARDENHAGEN S G, BRACKBILL J U, SULSKY D. The material point method for granular materials[J]. Computer Methods in Applied Mechanics and Engineering,2000, 187:529-541.
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[2]HUYGHE J M, VON CAMPEN D H, Arts T, et al. A two-phase finite element model of the diastolic left ventricle [J]. Journal of Biomechanics,1991,24(7):527-538.
[3]OOMENS C W J, VAN CAMPEN D H. A mixture approach to the mechanics of skin [J]. Journal of Biomechanics,1987,20:877-885.
[4]MAK A F T, HUANG L, WANG Q. A biphasic poroelastic analysis of the flow dependent subcutaneous tissue pressure and compaction due to epidermal loadings:issues in pressure core [J]. Journal of Biomechanical Engineering,1994,116:421-429.
[5]SIMON B R, WU J S S, CARTLON M W, ET AL. Structural models for human spinal motion segments based on a poroelastic view of the intervertebral disk [J]. Journal of Biomechanical Engineering,1985,107:327-335.
[6]BOER R D, Reflection on the development of the theory of porous media [J]. Applied Mechanics Reviews,2003,56(6):27-42.
[7]WASHBURN E W. The dynamics of capillary flow [J]. Phys. Rev.,1921,17:273-283.
[8]TERZAGHI K. The shearing resistance of saturated soils and the angle between the planes of shear [C]. International conference on soil mechanics and foundation engineering, 1, Cambridge,1936, MA Proc I,54-56.
[9]BOWEN R M. Incompressible porous media models by use of the theory of mixtures [J]. Journal of Engineering Science,1980,18:1129-1148.
[10]BOER R D. Highlights in the historical development of the porous media theory:toward a consistent macroscopic theory [J]. Applied Mechanics Reviews,1996,49(4):201-262.
[11]LIU W K, BELYTSCHKO T, CHANG H. An arbitrary Lagrangian-Eulerian finite element method for path-dependent materials [J]. Computer Methods in Applied Mechanics and Engineering,1986,58:227-246.
[12]BELYTSCHKO T, KRONGAUZ Y, ORGAN D, et al. Meshless methods:an overview and recent developments [J]. Computer Methods in Applied Mechanics and Engineering,1996, 139:3-47.
[13]BABUSKA I, MELLENK J M. The partition of unity method [J]. International Journal for Numerical Methods in Engineering,1997,40:727-758.
[14]DEMKOWICZ L, ODEN J T. An adaptive characteristic petrov-galerkin finite element method for convection-dominated linear and nonlinear parabolic problems [J]. Journal of Computational Physics,1986,67:188-213.
[15]ATLURI S N, ZHU T. New concepts in meshless methods [J]. International Journal for Numerical Methods in Engineering,2000,47:537-556.
[16]HARLOW F H. The particle-in-cell computing method for fluid dynamics [J]. Methods in Computational Physics,1964,3:319.
[17]BRACKBILL J U. The ringing instability in particle-in-cell calculations of low-speed flow [J]. Journal of Computational Physics,1988,75:469-492.
[18]BURGESS D, SULSKY D, BRACKBILL J U. Mass matrix formulation of the FLIP particle-in-cell method [J]. Journal of Computational Physics,1992,103:1-15.
[19]BRACKBILL J U, RUPPEL H M. FLIP:a method for adaptively zoned, particle-in-cell calculations in two dimensions [J]. Journal of Computational Physics,1986, 65:314-343.
[20]BRACKBILL J U, KOTHE D B, RUPPEL H M. FLIP:a low dissipation, particle-in-cell method for fluid flow [J]. Computer Physics Communications,1988,48:25-38.
[21]SULSKY D, CHEN Z, SCHREYER H L. A particle method for history-dependent materials [J]. Computer Methods in Applied Mechanics and Engineering,1994,118:179-196.
[22]ZHANG H W, WANG K P, CHEN Z. Material point method for dynamic analysis of saturated porous media under external contact/impact of solid bodies [J]. Computer Methods in Applied Mechanics and Engineering,2009,198(17-20):1456-1472.
[23]WIECKOWSKI Z, YOUN S K, YEON J H. A particle-in-cell solution to the silo discharging problem [J]. International Journal for Numerical Methods in Engineering,1999, 45:1203-1225.
[24]YORK A R, SULSKY D, SCHREYER H L. The material point method for simulation of thin membranes [J]. International Journal for Numerical Methods in Engineering,1999, 44:1429-1456.
[25]ZHOU S, STORMONT J, CHEN Z. Simulation of geomembrane response to settlement in landfills by using the material point method [J]. International Journal for Numerical and Analytical Methods in Geomechanics,1999,23:1977-1994.
[26]SULSKY D, SCHREYER H L. Axisymmetric form of the material point method with applications to upsetting and Taylor impact problems [J]. Computer Methods in Applied Mechanics and Engineering,1996,139:409-429.
[27]SULSKY D, ZHOU S J, SCHREYER H L. Application of a particle-in-cell method to solid mechanics [J]. Computer Physics Communications,1995,87:236-252.
[28]BARDENHAGEN S G. Energy Conservation error in the material point method for solid mechanics. Journal of Computational Physics,2002,180:383-403.
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