连续与非连续变形分析的有限覆盖无单元方法及其应用研究
摘要
岩体的非连续性与非均匀性越来越受到工程界与学术界的重视。其表现在两个方
面，即：（1）岩体强度理论的发展：岩体强度理论从经典的材料力学理论，经历岩体
断裂力学理论，到方兴未艾的岩体损伤力学理论，这一发展过程本身就是对岩体的非
连续性与非均匀性的逐步重视与深入研究的过程。（2）岩体材料特点的认识：岩体既
不是连续体亦非完全离散体，因此可以将岩体抽象为“断续介质”。“断续”与离散及
连续相对应，反映了岩体介于完全连续与离散之间的特点。
岩石在变形、破坏过程中的非连续变形行为计算与数值模拟是岩体力学与工程领
域中一个比较热门的前沿课题。岩体力学的各种新兴数值方法与技术几乎都是围绕这
一中心课题展开研究。
实际岩体结构与材料变形，一般是一个从小变形、损伤演化、宏观裂纹形成与扩
展、直至断裂破碎甚至发生散体刚体运动等大变形、大位移的一系列复杂的渐进变形
与破坏过程。然而，与实际岩体的这种变形破坏特点很不相称的是，岩土数值分析方
法一直呈现两极分化的态势。一方面是以有限元方法（FEM）为主的，基于完全连续
或基体连续（基体是指除岩体中某些宏观非连续界面以外的材料）假说之上的连续性
分析方法；另一方面则是以非连续变形分析（DDA）与离散单元法（DEM）为代表的
基于块体理论、离散介质假说基础之上的完全非连续性分析方法。这一现状使得岩体
的实际变形要么被人为地抽象为连续的，要么被假设为完全非连续的，二者之间似乎
缺乏勾通与联系。笔者认为，造成这一局面的根本原因有以下两个方面：（1）作为勾
通岩体变形与最终破坏的桥梁与纽带、岩体破坏过程分析中最为关键的环节—裂纹
问题，长期以来没有得到成功地解决；（2）缺乏适用于开裂计算的材料连续性与非连
续性的统一数学描述手段，或者说没有能够统一地处理材料的连续与非连续特性，而
且适合于开裂计算的插值构造方法。
近30年来，岩土力学中数值分析方法得到了迅速发展。其中，流形方法（Shi，1992）
和无单元Galerkin法（Belytschko，1994）以其新颖的数值思想、先进的数值技术，得
到了学术界的初步认可和广泛关注。这两种方法有着传统方法所不可替代的突出优
点。流形方法解决了材料连续与非连续性的数学统一表述的问题，使得连续变形分析
与非连续变形分析的统一成为可能。无单元Galerkin法实现了无单元插值，极大地简
单化了前处理工作与裂纹开裂扩展等问题的计算分析。但同时，这两种方法也存在一
些困难与不足。在流形方法中，双重网格可谓一把双刃剑，一方面它构成了流形方法
本身的一大特色，另一方面却不可避免地带来了前处理上的麻烦与裂纹开裂扩展模拟
方面的困难；无单元Galerkin法主耍是以解决前处理的问题而发展起来的一种方法，
该法在岩土力学中的非连续变形问题，如块体运动等的处理上仍有相当的局限性。因
此，对于岩体变形分析而言，流形方法与无单元Gaierkin法相辅相成，互补性非常强。
如果能将二者的优点相结合，或说将两种方法的核心思想或数值技术相融合，则不仅
可以解决前处理与开裂扩展方面的难题，同时也可以将连续与非连续问题的数值分析
相统一。
有限覆蛊无单元方法，采用有限囚租盖技术与多盂权沿动最小二乘法构造连缕与
非连续求解域的插值函戳，是一种可以在统一还学迟近空间形式下处理连续与非连续
问题的无单元方法。
本论文以有限覆盖无单元方法的研究为宗口，以流形方法的研究为出发点和基
础，以流形的“有限租盖思想”的推广应用为主线，分别对高阶流形方法基本原理、
有限覆盖无单元方法的构建、基本原理及具体应用以及广义节点有限元法等相关内客
分别进行了系统而深人的探讨与研究。论文的主要研究内客以及研究成果可概括如
下。
对流形方法的基本思想进行了系统的阐述，对任意高阶流形方法的一般原理进行
了介绍，详细推导出了全一阶多项式召盖函数与全二阶多项式租蛊函数两种形式的高
阶流形方法的具体数学列式（包括单元刚度阵的解析表达式），在统一格式下编制了零
阶、全一阶以及全二阶多项式覆盖函戮的流形方法计算程序，应用程序对具体数值算
例进行了计算。计算结果表明：采用高阶形式的召盖函数可有效地提高流形方法的计
算们度，从而证明了对于复杂问＠采用高阶流形方法的可行性。
流形方法的核心思想是流形的有限租组技术。在深入研究流形方法的基础上，将
流形的“有限覆蛊”思想与无单元Oalerkin法所采用的“无单元插值”技术有机地结
合，发展了一种新的技值分析方法，即有限租盖无单元方法（Finite－Cover－Based
Element－Free Method）。这种方法同时吸取了现有无单元 Galerkin法无需单元离散与连
接和流形方法可针对连续变形与非连续变形问题统一构造插值函数等各自的优点，从
而使得结构与材料的连续变形、裂纹失稳扩展、结构开裂、非连续变形分析、接触分
析甚至多体相互作用等一系列复杂问团的数值模拟成为可能。理论分析与数值算例计
算表明了有限覆?
The actual deformation and failure process of geo-materials and geo-structures is a
complex progressive evolution involving initially elastic deformation, crack propagation,
large-scale displacement and even movement of a discrete system. However, being only
simulating some stages of the procedure, the current numerical methods in geomachanics are
suitable for either ideal continuum materials, or completely discontinuous media. On the one
hand, Such methods as FEM and BEM are based on the hypothesis of total continuity or
basal materials continuity. On the other hand, there are discontinuous deformation analysis
methods such as DDA and DEM, which are on the basis of the blocky theory and the
assumption that rock mass is discrete media. This existing status makes it necessary that the
actual deformation of rock mass must be represented to be either continuous or completely
discontinuous. A numerical method born with ability in dealing with continuous deformation
problems, discontinuous deformation problems and crack propagation problems seems to be
a blue moon. It seems that there exist two essential factors causing this existing situation. (1)
Being the most important link in the analysis process of rock damage and the ligament
connecting the initial continuous deformation with the final failure of blocky system, the
problem of crack has not been solved thoroughly even recently. (2) There is no an effective
method which can deal with continuous, discontinuous and even crack propagation problems
in a consistent methodology, or there is no an interpolation method which not only Cafl
suitable for characterizing the continuity and discontinuity of materials in a uniform manner
but also does not require mesh generation.
Numerical analysis methods in geomechanics have been developed rapidly in recent
thirty years. As two novel numerical methods for solving boundary-value problems,
Manifold method (Shi, 1990) and element-free Galerkin method (Belytschko, 1994) have
been established elementarily and received worldwide attentions. These two methods own
many strong points and appear to be more superior over the conventional numerical methods
such as finite element methods. By virtue of the finite cover technique of manifold, manifold
method (MM) integrates conventional finite element methods (FEM), discontinuous
deformation analysis (DDA) (Shi 1988) and analytical methods in a united mathematical
framework and can deal with both continuous and discontinuous deformation problems such
as contact and multi-body interaction. Element-free Galerkin method (EFGM) on the basis
of moving least square (MLS) method (Lancaster & Salkaushas, 1981) is very effective in
simulating crack propagation which is a key issue in modeling failure or/and damage
behavior of structures or materials without meshing as required in MM. The kernel of MM is
finite cover technique while mesh-free approalmation (so-called mesh-free technique)
mainly derived from MLS method is a key feature of EFGM. MM and EFOM have own
advantages in handling discontinuous deformation prob1ems.
The main purpose Of the paPer is to exPlore the POssibi1ity tO work out a new numerical
method by combining the finite-cover techeque and mesh-ffee concePt tOgetheL
The researh begins with stUdy of the key idea of MM and goes on with extension and
application of the finite cover technique of MM.
Firstly, the fundamental of high-order MM is StUdied thoroughIy Generai mathematical
fOrmulations of MM with arbitrny order cover functions are presented. Then MM with the
complete first-order polynomiaI cover functions and with the complete second-order
polynomial cover functions are developed respectiveIy Programs of the
面，即：（1）岩体强度理论的发展：岩体强度理论从经典的材料力学理论，经历岩体
断裂力学理论，到方兴未艾的岩体损伤力学理论，这一发展过程本身就是对岩体的非
连续性与非均匀性的逐步重视与深入研究的过程。（2）岩体材料特点的认识：岩体既
不是连续体亦非完全离散体，因此可以将岩体抽象为“断续介质”。“断续”与离散及
连续相对应，反映了岩体介于完全连续与离散之间的特点。
岩石在变形、破坏过程中的非连续变形行为计算与数值模拟是岩体力学与工程领
域中一个比较热门的前沿课题。岩体力学的各种新兴数值方法与技术几乎都是围绕这
一中心课题展开研究。
实际岩体结构与材料变形，一般是一个从小变形、损伤演化、宏观裂纹形成与扩
展、直至断裂破碎甚至发生散体刚体运动等大变形、大位移的一系列复杂的渐进变形
与破坏过程。然而，与实际岩体的这种变形破坏特点很不相称的是，岩土数值分析方
法一直呈现两极分化的态势。一方面是以有限元方法（FEM）为主的，基于完全连续
或基体连续（基体是指除岩体中某些宏观非连续界面以外的材料）假说之上的连续性
分析方法；另一方面则是以非连续变形分析（DDA）与离散单元法（DEM）为代表的
基于块体理论、离散介质假说基础之上的完全非连续性分析方法。这一现状使得岩体
的实际变形要么被人为地抽象为连续的，要么被假设为完全非连续的，二者之间似乎
缺乏勾通与联系。笔者认为，造成这一局面的根本原因有以下两个方面：（1）作为勾
通岩体变形与最终破坏的桥梁与纽带、岩体破坏过程分析中最为关键的环节—裂纹
问题，长期以来没有得到成功地解决；（2）缺乏适用于开裂计算的材料连续性与非连
续性的统一数学描述手段，或者说没有能够统一地处理材料的连续与非连续特性，而
且适合于开裂计算的插值构造方法。
近30年来，岩土力学中数值分析方法得到了迅速发展。其中，流形方法（Shi，1992）
和无单元Galerkin法（Belytschko，1994）以其新颖的数值思想、先进的数值技术，得
到了学术界的初步认可和广泛关注。这两种方法有着传统方法所不可替代的突出优
点。流形方法解决了材料连续与非连续性的数学统一表述的问题，使得连续变形分析
与非连续变形分析的统一成为可能。无单元Galerkin法实现了无单元插值，极大地简
单化了前处理工作与裂纹开裂扩展等问题的计算分析。但同时，这两种方法也存在一
些困难与不足。在流形方法中，双重网格可谓一把双刃剑，一方面它构成了流形方法
本身的一大特色，另一方面却不可避免地带来了前处理上的麻烦与裂纹开裂扩展模拟
方面的困难；无单元Galerkin法主耍是以解决前处理的问题而发展起来的一种方法，
该法在岩土力学中的非连续变形问题，如块体运动等的处理上仍有相当的局限性。因
此，对于岩体变形分析而言，流形方法与无单元Gaierkin法相辅相成，互补性非常强。
如果能将二者的优点相结合，或说将两种方法的核心思想或数值技术相融合，则不仅
可以解决前处理与开裂扩展方面的难题，同时也可以将连续与非连续问题的数值分析
相统一。
有限覆蛊无单元方法，采用有限囚租盖技术与多盂权沿动最小二乘法构造连缕与
非连续求解域的插值函戳，是一种可以在统一还学迟近空间形式下处理连续与非连续
问题的无单元方法。
本论文以有限覆盖无单元方法的研究为宗口，以流形方法的研究为出发点和基
础，以流形的“有限租盖思想”的推广应用为主线，分别对高阶流形方法基本原理、
有限覆盖无单元方法的构建、基本原理及具体应用以及广义节点有限元法等相关内客
分别进行了系统而深人的探讨与研究。论文的主要研究内客以及研究成果可概括如
下。
对流形方法的基本思想进行了系统的阐述，对任意高阶流形方法的一般原理进行
了介绍，详细推导出了全一阶多项式召盖函数与全二阶多项式租蛊函数两种形式的高
阶流形方法的具体数学列式（包括单元刚度阵的解析表达式），在统一格式下编制了零
阶、全一阶以及全二阶多项式覆盖函戮的流形方法计算程序，应用程序对具体数值算
例进行了计算。计算结果表明：采用高阶形式的召盖函数可有效地提高流形方法的计
算们度，从而证明了对于复杂问＠采用高阶流形方法的可行性。
流形方法的核心思想是流形的有限租组技术。在深入研究流形方法的基础上，将
流形的“有限覆蛊”思想与无单元Oalerkin法所采用的“无单元插值”技术有机地结
合，发展了一种新的技值分析方法，即有限租盖无单元方法（Finite－Cover－Based
Element－Free Method）。这种方法同时吸取了现有无单元 Galerkin法无需单元离散与连
接和流形方法可针对连续变形与非连续变形问题统一构造插值函数等各自的优点，从
而使得结构与材料的连续变形、裂纹失稳扩展、结构开裂、非连续变形分析、接触分
析甚至多体相互作用等一系列复杂问团的数值模拟成为可能。理论分析与数值算例计
算表明了有限覆?
The actual deformation and failure process of geo-materials and geo-structures is a
complex progressive evolution involving initially elastic deformation, crack propagation,
large-scale displacement and even movement of a discrete system. However, being only
simulating some stages of the procedure, the current numerical methods in geomachanics are
suitable for either ideal continuum materials, or completely discontinuous media. On the one
hand, Such methods as FEM and BEM are based on the hypothesis of total continuity or
basal materials continuity. On the other hand, there are discontinuous deformation analysis
methods such as DDA and DEM, which are on the basis of the blocky theory and the
assumption that rock mass is discrete media. This existing status makes it necessary that the
actual deformation of rock mass must be represented to be either continuous or completely
discontinuous. A numerical method born with ability in dealing with continuous deformation
problems, discontinuous deformation problems and crack propagation problems seems to be
a blue moon. It seems that there exist two essential factors causing this existing situation. (1)
Being the most important link in the analysis process of rock damage and the ligament
connecting the initial continuous deformation with the final failure of blocky system, the
problem of crack has not been solved thoroughly even recently. (2) There is no an effective
method which can deal with continuous, discontinuous and even crack propagation problems
in a consistent methodology, or there is no an interpolation method which not only Cafl
suitable for characterizing the continuity and discontinuity of materials in a uniform manner
but also does not require mesh generation.
Numerical analysis methods in geomechanics have been developed rapidly in recent
thirty years. As two novel numerical methods for solving boundary-value problems,
Manifold method (Shi, 1990) and element-free Galerkin method (Belytschko, 1994) have
been established elementarily and received worldwide attentions. These two methods own
many strong points and appear to be more superior over the conventional numerical methods
such as finite element methods. By virtue of the finite cover technique of manifold, manifold
method (MM) integrates conventional finite element methods (FEM), discontinuous
deformation analysis (DDA) (Shi 1988) and analytical methods in a united mathematical
framework and can deal with both continuous and discontinuous deformation problems such
as contact and multi-body interaction. Element-free Galerkin method (EFGM) on the basis
of moving least square (MLS) method (Lancaster & Salkaushas, 1981) is very effective in
simulating crack propagation which is a key issue in modeling failure or/and damage
behavior of structures or materials without meshing as required in MM. The kernel of MM is
finite cover technique while mesh-free approalmation (so-called mesh-free technique)
mainly derived from MLS method is a key feature of EFGM. MM and EFOM have own
advantages in handling discontinuous deformation prob1ems.
The main purpose Of the paPer is to exPlore the POssibi1ity tO work out a new numerical
method by combining the finite-cover techeque and mesh-ffee concePt tOgetheL
The researh begins with stUdy of the key idea of MM and goes on with extension and
application of the finite cover technique of MM.
Firstly, the fundamental of high-order MM is StUdied thoroughIy Generai mathematical
fOrmulations of MM with arbitrny order cover functions are presented. Then MM with the
complete first-order polynomiaI cover functions and with the complete second-order
polynomial cover functions are developed respectiveIy Programs of the
引文
[1] 孙广忠著．岩体结构力学．北京：科学出版社，1988．
[2] 陶振宇．试论岩石力学的最新进展．力学进展，1992，22(2)：161～172．
[3] 常士骠．我国岩土工程回顾与展望．水文地质工程地质，1993，20(1)：23～26．
[4] 哈秋聆．三峡工程中的若干力学问题．力学进展，1994，24(4)：433～439．
[5] 杨志法．关于岩石力学当前发展战略的一些看法．岩土工程学报，1994，16(1)：112～113．
[6] 孙钧．岩石力学的若干进展．面向21世纪的岩石力学与工程—中国岩石力学与工程学会第四次学术大会论文集，北京：中国科学技术出版社，1996．
[7] 谢和平，刘夕才，王金安．关于21世纪岩石力学发展战略的思考．岩土工程学报，1996，18(4)：98～102．
[8] 傅冰骏．国际岩石力学发展动向．岩石力学与工程学报，1997，16(2)：195～196．
[9] 顾宝和．岩土工程勘察技术现状及发展问题述评．工程勘察，1998，(4)：3～8．
[10] 郭书泰．工程地质勘察与岩土工程技术发展现状与展望．探矿工程，1999，(1)：8～11．
[11] 王芝银．岩土工程发展动向．西安矿业学院学报，1999，19(增刊)：1～4，45．
[12] 杨更社．我国岩石力学的研究现状及其进展，西安矿业学院学报，1999，19(增刊)：5～11．
[13] 李荣强，陈善雄．岩土工程勘察的现状和趋势．岩土工程界，1999，(9)：13～18．
[14] 唐辉明，腾伟福．岩土工程数值模拟新方法．地质科技情报，1998，17(增刊2)：41～48．
[15] 唐春安．岩石破坏裂过程数值模拟方法发展的若干问题．中国CSRM软岩工程专业委员会第二届学术大会论文集．10～16．
[16] 刘怀恒．岩土力学数值方法的应用及发展．岩土力学，1989，10(2)：14～21．
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