杂交自然单元法研究
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摘要
自然单元法(Natural Element Method,简称NEM)是一种近20年才发展起来的数值算法。它采用基于Voronoi图的Sibson或non-Sibonian插值方法构造全域近似函数和试函数。自然单元法具有无网格不依赖网格的优势，同时形函数构造简单，在边界上满足Delta插值性质，是一种发展前景广阔的求解力学和其他工程技术中偏微分方程的数值方法
针对自然单元法无法直接求解节点应力且应力解精度不高的缺陷以及采用高阶连续形函数恢复应力计算效率不高的问题，通过引入应力杂交的思想，采用对应力和位移分别插值，提出了杂交自然单元法。利用杂交自然单元法对弹性力学问题、弹塑性力学问题、粘弹性力学问题以及弹性大变形问题进行了分析。具体工作如下：
将复变量移动最小二乘法引入到自然单元法应力恢复算法中，提出了改进的自然单元法应力恢复算法。与传统应力恢复算法相比，复变量移动最小二乘法待定系数少，可以提高应力恢复算法的计算效率。
采用自然邻点插值对应力和位移分别进行逼近，结合弹性力学的多变量Hellinger-Reissner变分原理，提出了弹性力学的杂交自然单元法。与传统自然单元法相比，杂交自然单元法有着更高的应力计算精度并能够直接求解节点应力。同时，在相同精度条件下，杂交自然单元法比应力恢复算法有着更高计算效率。
在杂交自然单元法的基础上，通过对应力和位移增量分别进行插值，提出了适用于非线性问题的增量形式杂交自然单元法。结合弹塑性问题的Hellinger-Reissner变分原理，提出了弹塑性力学问题的杂交自然单元法，并推导了计算公式。与自然单元法相比，杂交自然单元法有着更高的计算精度，比应力恢复算法更加节省计算时间。
在详细分析了线性粘弹性材料的蠕变特性的基础上，采用各个时间节点上的变形来描述蠕变过程。通过杂交自然单元法与在各个时间节点上建立的Hellinger-Reissner变分原理，推导了适用于线性粘弹性材料蠕变分析的杂交自然单元法。本文方法能够准确的描述材料的蠕变性能；并能够直接求解得到准确的节点应力。
结合大变形问题的特性，推导了增量型杂交自然单元法分析弹性大变形问题的Total Lagrangian格式下表达方式，数值算例中采用了Newton-Raphson增量迭代法，提出了适用于弹性大变形问题分析的杂交自然单元法。算例表明杂交自然单元法有着更高的计算精度和更快的求解效率。
利用增量形式的Hellinger-Reissner变分原理结合岩土工程的特点，推导了适用于岩土工程开挖问题的模拟的杂交自然单元法；同时为了模拟隧道的支护过程提出了适用于本文方法的添加节点的方法。
为了证明本文提出的杂交自然单元法的有效性，采用Matlab编制了数值算例。通过数值算例分析可以看出本文方法的有效性和优越性。
As a new numerical method, Natural Element Method (NEM) has beenproposed at the end of last century. Sibson and non-Sibonian interpolation which arecalled Natural Neighbor Interpolation (NNI) are employed to constitute trial functionin NEM. Because of meshless and easy application, the NEM is one of numericalmethod which has a bright prospect using to solution the partial differentialequations.
Because of the inaccurate stress solution and poor ability to obtain the stress ofnode, the tradition Natural Element Method(NEM) are rarely used to solve complexquestion. In this paper, the hybrid finite element method was introduced into NEMand the hybrid natural element method (HNEM) has been provided. And the HNEMis applied to solve elasticity problems, elastoplasticity problems, larger deflectionproblems and viscoelasticity problems respectively. The main researches of thisthesis are as follows.
Based on the complex variables moving least-square, a new stress recoveryalgorithm for NEM is presented. Compared with the traditional stress recoveryalgorithm, the new method can improve calculation efficiency.
Using NNI to be displacement and stress interpolation function respectively，the HNEM and Hellinger-Reissner variation principle are introduced into analysiselastic problems and the HNEM for elasticity is provided. Comparing with NEM, theHNEM has more precision solution of stress and can obtain the stress of nodesdirectly. In the other hand, the HNEM has higher computational efficiency thanthestress recovery algorithm.
On the basis of HNEM, the incremental form HNEM for nonlinear analysis ispresented. Combined with the incremental Hellinger-Reissner variation principle, theformula of HENM for the elastic-plastic problem is deduced, and is used to analyszethe elastic-plastic mechanics problem. Its numerical precision is better than NEMand the calculation efficiency is higher
The creep properties of linear viscoelastic material are analyzed by usingHNEM. The material property of viscoelastic material has close relations with time,so it is difficult to analyze this problem. Based on the detailed analysis of the linearviscoelastic material characteristics, the creep process of material can be expressed by deformation changes on some time points. The HNEM for linear viscoelasticproblem is presented based on Hellinger-Reissner variation principle. The numericalexamples show that the method in this paper can depict the creep properties ofmaterials accurately, and can obtain the stress of nodes directly and quickly.
The HNEM is applied to elastic large deformation problem in this dissertation.Combining with the incremental Hellinger-Reissner variation principle which accordwith lager deformation analysis the HENM for two-dimensional elastic largedeformation problems is presented based on the total Lagrangian formulation, andthe corresponding formulae are obtained. The Newton-Raphson iteration isemployed in the mumerical implementation. The advantages of HNEM are higherprecision and faster calculation.
Based on the incremental Hellinger-Reissner variation principle and thecharacters of geotechnical engineering，the HNEM has been used to analyze theproblems of geotechnical engineering，such as tunnel excavation. In order tosimulate the process of tunnel support, the method which can be increase nodes hasbeen proposed.
In order to show the efficiency of the HNME in the dissertation, the MATLABcodes of the methods above have been written. The numerical examples are provided,and the validity and efficiency of these methods are demonstrated.
针对自然单元法无法直接求解节点应力且应力解精度不高的缺陷以及采用高阶连续形函数恢复应力计算效率不高的问题，通过引入应力杂交的思想，采用对应力和位移分别插值，提出了杂交自然单元法。利用杂交自然单元法对弹性力学问题、弹塑性力学问题、粘弹性力学问题以及弹性大变形问题进行了分析。具体工作如下：
将复变量移动最小二乘法引入到自然单元法应力恢复算法中，提出了改进的自然单元法应力恢复算法。与传统应力恢复算法相比，复变量移动最小二乘法待定系数少，可以提高应力恢复算法的计算效率。
采用自然邻点插值对应力和位移分别进行逼近，结合弹性力学的多变量Hellinger-Reissner变分原理，提出了弹性力学的杂交自然单元法。与传统自然单元法相比，杂交自然单元法有着更高的应力计算精度并能够直接求解节点应力。同时，在相同精度条件下，杂交自然单元法比应力恢复算法有着更高计算效率。
在杂交自然单元法的基础上，通过对应力和位移增量分别进行插值，提出了适用于非线性问题的增量形式杂交自然单元法。结合弹塑性问题的Hellinger-Reissner变分原理，提出了弹塑性力学问题的杂交自然单元法，并推导了计算公式。与自然单元法相比，杂交自然单元法有着更高的计算精度，比应力恢复算法更加节省计算时间。
在详细分析了线性粘弹性材料的蠕变特性的基础上，采用各个时间节点上的变形来描述蠕变过程。通过杂交自然单元法与在各个时间节点上建立的Hellinger-Reissner变分原理，推导了适用于线性粘弹性材料蠕变分析的杂交自然单元法。本文方法能够准确的描述材料的蠕变性能；并能够直接求解得到准确的节点应力。
结合大变形问题的特性，推导了增量型杂交自然单元法分析弹性大变形问题的Total Lagrangian格式下表达方式，数值算例中采用了Newton-Raphson增量迭代法，提出了适用于弹性大变形问题分析的杂交自然单元法。算例表明杂交自然单元法有着更高的计算精度和更快的求解效率。
利用增量形式的Hellinger-Reissner变分原理结合岩土工程的特点，推导了适用于岩土工程开挖问题的模拟的杂交自然单元法；同时为了模拟隧道的支护过程提出了适用于本文方法的添加节点的方法。
为了证明本文提出的杂交自然单元法的有效性，采用Matlab编制了数值算例。通过数值算例分析可以看出本文方法的有效性和优越性。
As a new numerical method, Natural Element Method (NEM) has beenproposed at the end of last century. Sibson and non-Sibonian interpolation which arecalled Natural Neighbor Interpolation (NNI) are employed to constitute trial functionin NEM. Because of meshless and easy application, the NEM is one of numericalmethod which has a bright prospect using to solution the partial differentialequations.
Because of the inaccurate stress solution and poor ability to obtain the stress ofnode, the tradition Natural Element Method(NEM) are rarely used to solve complexquestion. In this paper, the hybrid finite element method was introduced into NEMand the hybrid natural element method (HNEM) has been provided. And the HNEMis applied to solve elasticity problems, elastoplasticity problems, larger deflectionproblems and viscoelasticity problems respectively. The main researches of thisthesis are as follows.
Based on the complex variables moving least-square, a new stress recoveryalgorithm for NEM is presented. Compared with the traditional stress recoveryalgorithm, the new method can improve calculation efficiency.
Using NNI to be displacement and stress interpolation function respectively，the HNEM and Hellinger-Reissner variation principle are introduced into analysiselastic problems and the HNEM for elasticity is provided. Comparing with NEM, theHNEM has more precision solution of stress and can obtain the stress of nodesdirectly. In the other hand, the HNEM has higher computational efficiency thanthestress recovery algorithm.
On the basis of HNEM, the incremental form HNEM for nonlinear analysis ispresented. Combined with the incremental Hellinger-Reissner variation principle, theformula of HENM for the elastic-plastic problem is deduced, and is used to analyszethe elastic-plastic mechanics problem. Its numerical precision is better than NEMand the calculation efficiency is higher
The creep properties of linear viscoelastic material are analyzed by usingHNEM. The material property of viscoelastic material has close relations with time,so it is difficult to analyze this problem. Based on the detailed analysis of the linearviscoelastic material characteristics, the creep process of material can be expressed by deformation changes on some time points. The HNEM for linear viscoelasticproblem is presented based on Hellinger-Reissner variation principle. The numericalexamples show that the method in this paper can depict the creep properties ofmaterials accurately, and can obtain the stress of nodes directly and quickly.
The HNEM is applied to elastic large deformation problem in this dissertation.Combining with the incremental Hellinger-Reissner variation principle which accordwith lager deformation analysis the HENM for two-dimensional elastic largedeformation problems is presented based on the total Lagrangian formulation, andthe corresponding formulae are obtained. The Newton-Raphson iteration isemployed in the mumerical implementation. The advantages of HNEM are higherprecision and faster calculation.
Based on the incremental Hellinger-Reissner variation principle and thecharacters of geotechnical engineering，the HNEM has been used to analyze theproblems of geotechnical engineering，such as tunnel excavation. In order tosimulate the process of tunnel support, the method which can be increase nodes hasbeen proposed.
In order to show the efficiency of the HNME in the dissertation, the MATLABcodes of the methods above have been written. The numerical examples are provided,and the validity and efficiency of these methods are demonstrated.
引文
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