地下结构断裂破坏分析的无网格流形方法研究
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摘要
近年来无网格方法和数值流形方法以其新颖的计算思想和数值技术得到了力学与工程界的重视。无网格方法是近年来发展起来的一种新兴的数值方法，因其具有不需要网格，只需要节点信息、前处理简单、计算精度高等特点，已成为目前科学和工程计算方法的研究热点之一；而数值流形方法通过引入数学与物理双重网格和有限覆盖技术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得到了统一，非常适合于各种非连续、大变形等问题的分析模拟。然而，在处理复杂非连续问题时（如多裂纹扩展问题），这两种方法均有其各自的缺陷，例如无网格方法一般采用光线法（如可视、衍射准则）来处理域内的不连续问题，这样可能会由于插值点不足而导致试函数难以建立，产生数值解的不稳定性等问题；数值流形方法在处理复杂非连续问题时，由于双重网格的局限性，使得数值流形方法的覆盖系统生成算法十分复杂，严重影响了它的应用。为了克服无网格方法和数值流形方法在处理复杂非连续问题时遇到的困难，本文提出和发展了一种基于单位分解法和有限覆盖技术的、地下结构破坏过程模拟分析的无网格流形MSIM（Meshless ShepardInterpolation Method,简称MSIM）新方法。在该方法中，插值函数除了不受域内不连续面的影响外，还具有高阶完备性、一致性，且可以在需要的节点处具备Kronecker-Delta属性，能够方便准确地施加各种边界条件；克服了传统无网格方法在处理不连续问题时由于采用光线法所遇到的困难及数值流形方法由于网格的存在使得覆盖系统的生成异常复杂的问题。本文主要开展了以下工作：
1、基于单位分解和数值流形方法的有限覆盖理论，建立和发展了基于MSIM插值的、用于地下结构裂纹扩展和破坏过程分析的无网格流形MSIM方法，详细地介绍和推导了相关公式，给出了无网格流形MSIM方法中的不连续问题处理方法和无网格流形MSIM方法数值实现的具体步骤；
2、在无网格流形MSIM方法中引入了J积分法、虚拟裂纹扩展法及虚拟裂纹闭合法等目前常用的几种应力强度因子的计算方法，特别是把目前仅用于有限元计算的虚拟裂纹扩展法、虚拟裂纹闭合法引入到了无网格流形MSIM方法中，并对其相应的裂纹扩展准则选取和裂纹扩展步长的确定等进行了研究；
3、研究和发展了数学覆盖被不连续面切割后的无网格流形MSIM方法物理覆盖系统全自动生成理论与算法；基于Matlab平台，给出了无网格流形覆盖系统自动生成等无网格流形MSIM方法关键模块的实施技术，并编制了可应用于地下结构断裂破坏分析的无网流形MSIM方法的计算程序；
4、利用具有解析解答的或参考解答的含裂纹试件算例，对J积分法、虚拟裂纹扩展法及虚拟裂纹闭合法等各种方法的计算精度和稳定性进行了对比分析研究，并在此基础上对具有试验或数值分析对照结果的各种裂纹扩展算例进行了裂纹扩展模拟分析研究，以验证和测试本文所发展的地下结构断裂破坏分析的无网流形MSIM理论和方法的正确性和可靠性；
5、结合套拱加固的隧道衬砌破坏试验，利用研制的无网格流形MSIM分析程序，对套拱加固后带裂缝衬砌的具体破坏过程进行了分析研究，进一步对本文理论和方法的正确性、有效性进行了验证。
In recent years Numerical Manifold Method (NMM) and Meshless Method (MM)have received worldwide attention in the fields of mechanics and engineering. MM is aclass of new numerical methods developed, only needs the information at nodes, anddoesn’t discretize the domain into a mesh. The advantages of MM are the simplerpre-processing and higher precision. MM is a hot topic of researches on scientific andengineering computation. Based on topological manifold methods and differentiablemanifold methods, NMM has two cover systems, namely the mathematical covers and thephysical covers, so NMM can integrate both continuous analysis and discontinuousanalysis in a united mathematical framework, it is very suitable for simulating the problemsof discontinuity and large deformation. However, several difficulties arise in the realizationand implementation of MM and NMM for dealing with very complex discontinuousdeformation problems such as multi-crack propagation. For example, MM usually uses thediffraction method, the transparency method or the see-through method to deal with thediscontinuity in the domain, which may make it difficult to construct the shape functions asa result of the insufficient interpolation points, thus leading to the instability of thenumerical solution. As for NMM, when dealing with complex discontinuous problems, thelimitation of its dual grids makes the forming algorithm of the cover system verycomplicated, thus seriously hinders the expansion of its application.
In order to overcome the difficulties of MM and NMM when dealing with complexdiscontinuous problems, a new Meshless Shepard Interpolation Method (MSIM) based onthe partition of unity (PU) and the finite cover theory is employed to solve the problems ofcrack propagation and failure behavior of the underground structures. In the MSIM, Theshape functions are constructed by the partition of unity and the finite cover technology,and thus the shape functions are not affected by discontinuous domains and crack problemscan be more properly treated. The MSIM shape function possesses three distinguishedfeatures: the interpolation property, the arbitrarily high order consistency, and satisfyingKronecker delta property at any desired node. The essential boundary conditions in MSIMcan be treated as easily as they are in FEM. Compared with the conventional MM, theMSIM shape functions are not influenced by the discontinuities in the solution domainbecause of finite cover technology is used. Compared with the mesh-based NMM, the finitecovers and the partition of unity functions of the MSIM are constructed by using a series of nodes. The main works of this thesis are listed as follows:
(1) Base on the partition of unity and the finite cover technology, a new MSIMinterpolation has been developed for the simulation of crack propagation and failurebehavior of the underground structures. Fundamental equations of MISM analysis arededuced. The approach of how to deal with discontinuity in MSIM method are alsointroduced, as well as the specific steps to achieve numerical method applied for crackexpansion analysis.
(2) The J-integral method, the virtual crack extension method and the virtual crackclosure method are introduced into the MSIM method for the computation of the stressintensity factor. Furthermore, the selection of the corresponding crack propagation criterionand the corresponding crack step is investigated.
(3) A simple but efficient algorithm for finite cover automatic generating is introducedinto the MSIM for dealing with complex discontinuous problems. Based on Matlabsoftware, the key implementation technology of finite cover system is proposed, and theMSIM program for concrete crack analysis is developed to implement the meshlessautomatic trace simulating of concrete structure crack expansion.
(4) The accuracy and stability of the J-integral method, the virtual crack extensionmethod and the virtual crack closure method for several crack examples are studied byusing the MSIM method, then comparison of multi-crack propagation simulation betweenMSIM and experimental is carried out. Numerical examples indicate the advantages andthe accuracy of the MSIM for the analysis of crack propagation.
(5) Combined with the experimental study on cracked tunnel lining reinforced withumbrella arch, the failure process analysis of cracked tunnel lining reinforced withumbrella arch is carried out by using MSIM, the property and feasibility of the MSIM isalso tested and verified.
1、基于单位分解和数值流形方法的有限覆盖理论，建立和发展了基于MSIM插值的、用于地下结构裂纹扩展和破坏过程分析的无网格流形MSIM方法，详细地介绍和推导了相关公式，给出了无网格流形MSIM方法中的不连续问题处理方法和无网格流形MSIM方法数值实现的具体步骤；
2、在无网格流形MSIM方法中引入了J积分法、虚拟裂纹扩展法及虚拟裂纹闭合法等目前常用的几种应力强度因子的计算方法，特别是把目前仅用于有限元计算的虚拟裂纹扩展法、虚拟裂纹闭合法引入到了无网格流形MSIM方法中，并对其相应的裂纹扩展准则选取和裂纹扩展步长的确定等进行了研究；
3、研究和发展了数学覆盖被不连续面切割后的无网格流形MSIM方法物理覆盖系统全自动生成理论与算法；基于Matlab平台，给出了无网格流形覆盖系统自动生成等无网格流形MSIM方法关键模块的实施技术，并编制了可应用于地下结构断裂破坏分析的无网流形MSIM方法的计算程序；
4、利用具有解析解答的或参考解答的含裂纹试件算例，对J积分法、虚拟裂纹扩展法及虚拟裂纹闭合法等各种方法的计算精度和稳定性进行了对比分析研究，并在此基础上对具有试验或数值分析对照结果的各种裂纹扩展算例进行了裂纹扩展模拟分析研究，以验证和测试本文所发展的地下结构断裂破坏分析的无网流形MSIM理论和方法的正确性和可靠性；
5、结合套拱加固的隧道衬砌破坏试验，利用研制的无网格流形MSIM分析程序，对套拱加固后带裂缝衬砌的具体破坏过程进行了分析研究，进一步对本文理论和方法的正确性、有效性进行了验证。
In recent years Numerical Manifold Method (NMM) and Meshless Method (MM)have received worldwide attention in the fields of mechanics and engineering. MM is aclass of new numerical methods developed, only needs the information at nodes, anddoesn’t discretize the domain into a mesh. The advantages of MM are the simplerpre-processing and higher precision. MM is a hot topic of researches on scientific andengineering computation. Based on topological manifold methods and differentiablemanifold methods, NMM has two cover systems, namely the mathematical covers and thephysical covers, so NMM can integrate both continuous analysis and discontinuousanalysis in a united mathematical framework, it is very suitable for simulating the problemsof discontinuity and large deformation. However, several difficulties arise in the realizationand implementation of MM and NMM for dealing with very complex discontinuousdeformation problems such as multi-crack propagation. For example, MM usually uses thediffraction method, the transparency method or the see-through method to deal with thediscontinuity in the domain, which may make it difficult to construct the shape functions asa result of the insufficient interpolation points, thus leading to the instability of thenumerical solution. As for NMM, when dealing with complex discontinuous problems, thelimitation of its dual grids makes the forming algorithm of the cover system verycomplicated, thus seriously hinders the expansion of its application.
In order to overcome the difficulties of MM and NMM when dealing with complexdiscontinuous problems, a new Meshless Shepard Interpolation Method (MSIM) based onthe partition of unity (PU) and the finite cover theory is employed to solve the problems ofcrack propagation and failure behavior of the underground structures. In the MSIM, Theshape functions are constructed by the partition of unity and the finite cover technology,and thus the shape functions are not affected by discontinuous domains and crack problemscan be more properly treated. The MSIM shape function possesses three distinguishedfeatures: the interpolation property, the arbitrarily high order consistency, and satisfyingKronecker delta property at any desired node. The essential boundary conditions in MSIMcan be treated as easily as they are in FEM. Compared with the conventional MM, theMSIM shape functions are not influenced by the discontinuities in the solution domainbecause of finite cover technology is used. Compared with the mesh-based NMM, the finitecovers and the partition of unity functions of the MSIM are constructed by using a series of nodes. The main works of this thesis are listed as follows:
(1) Base on the partition of unity and the finite cover technology, a new MSIMinterpolation has been developed for the simulation of crack propagation and failurebehavior of the underground structures. Fundamental equations of MISM analysis arededuced. The approach of how to deal with discontinuity in MSIM method are alsointroduced, as well as the specific steps to achieve numerical method applied for crackexpansion analysis.
(2) The J-integral method, the virtual crack extension method and the virtual crackclosure method are introduced into the MSIM method for the computation of the stressintensity factor. Furthermore, the selection of the corresponding crack propagation criterionand the corresponding crack step is investigated.
(3) A simple but efficient algorithm for finite cover automatic generating is introducedinto the MSIM for dealing with complex discontinuous problems. Based on Matlabsoftware, the key implementation technology of finite cover system is proposed, and theMSIM program for concrete crack analysis is developed to implement the meshlessautomatic trace simulating of concrete structure crack expansion.
(4) The accuracy and stability of the J-integral method, the virtual crack extensionmethod and the virtual crack closure method for several crack examples are studied byusing the MSIM method, then comparison of multi-crack propagation simulation betweenMSIM and experimental is carried out. Numerical examples indicate the advantages andthe accuracy of the MSIM for the analysis of crack propagation.
(5) Combined with the experimental study on cracked tunnel lining reinforced withumbrella arch, the failure process analysis of cracked tunnel lining reinforced withumbrella arch is carried out by using MSIM, the property and feasibility of the MSIM isalso tested and verified.
引文
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