自适应比例边界元法及其在弹性力学中的应用
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摘要
本文基于虚功原理建立比例边界元法(SBFEM)的静力和动力平衡方程及其求解方法,将基于应力重构的能量误差指标推广至弹性动力学,发展了高效的网格重分策略和简单、准确的网格映射算法,并进一步考虑了时间离散误差对结构动力响应计算的影响,最终建立了一套比较完整的自适应比例边界元法,并用于一般弹性动力学问题的分析。此外,在Delaunay三角化的基础上发展了多边形比例边界元法,实现了比例边界元网格的全自动划分。
基于虚功原理推导了SBFEM静力平衡方程及其求解方法；比较SPR法和节点平均法重构的节点应力；建立了SBFEM中半解析形式的总能量、能量误差和能量误差指标；确定了h型网格重分策略及自适应算法的流程；比较了在不同能量误差目标值下的自适应网格、计算精度和计算时间；比较了基于自适应SBFEM和FEM求得的应力场。
基于虚功原理推导了SBFEM的动力平衡方程及其求解方法；提出采用Newmark时间积分法求解SBFEM整体平衡方程组；推导了SBFEM的具有半解析形式的动能、应变能、总能量及能量误差,建立了动态能量误差指标；发展了具有半解析特性、简单而准确的SBFEM网格映射方法；确定了动态自适应SBFEM的计算流程；采用自适应SBFEM求解不同结构在爆破荷载、冲击荷载作用下的动力响应,并与常规SBFEM、FEM和自适应FEM等方法作比较,验证了该方法处理一般弹性动力学问题的有效性。
引入超单元”概念,将SBFEM具有半解析形式的动能、应变能、总能量和能量误差推广至子域水平,并提出了相应的子域能量误差判别方法；提出了一种简单、高效的子域重分策略,并与准确的网格映射技术相结合,发展了基于子域重分技术的自适应SBFEM;通过计算冲击荷载作用下的简支梁和悬臂深梁的动力响应,验证了该方法的有效性。
基于Newmark时间积分法的基本假定重构得时域内的线性分布加速度场,推导了能量范数形式的SBFEM时间离散误差及指标,以及相应的时间步长缩放准则和计算流程；发展了一种可自动控制时间离散误差的SBFEM方法。通过计算冲击荷载、爆破荷载作用下三种不同结构的动力响应验证该方法的有效性。
在Delaunay三角化的基础上,以各三角形共用节点或多边形重心为各SBFEM子域的相似中心,直接建立多边形SBFEM的离散网格,从而实现SBFEM处理一般弹性动力学问题的自动建模。通过算例考察了该方法的计算精度对单元尺寸的敏感性,并在结构的全局能量误差中得到反映。
本文旨在深化自适应比例边界元法的理论根基,为其在裂缝扩展模拟等领域的应用奠定重要基础。
The equilibrium equations and their solutions of scaled boundary finite element method (SBFEM) are derived based on the virtual work principle in this thesis, not only for elastostatics but also for elastodynamics. Energy error estimator based on stress recovery technique is extended from elastostatics to elastodyanmics, and an efficient remeshing strategy is given out with a simple but accurate mesh mapping procedure. Furthermore, the error caused by time discretisation is also taken into account, leading to a datively complete adaptive SBFEM for general elstodynamic problems. Additionally, a polygon scaled boundary finite element method is developed based on Delauney Triangulation, making the discritisation in SBFEM fully-automatic.
The equilibrium equations of SBFEM are derived based on virtual work for elastostatics. Two different stress recovery teniques, including superconvergent patch recovery (SPR) and node average technique, are compared. Total energy, energy error and error estimator are given out in semi-analytical formula. An h-type remeshing procedure and a follow chart of the adaptive approach are described in detail. The changes of adaptive meshes, accuracy and computational time are investigated through a numerical experiment. The stress filed calculated by adaptive SBFEM is also compared with that of FEM.
The equilibrium equations of SBFEM are extended to elastodynamics also based on virtual work principle. Newmark intergration scheme is employed to solve the equations. Kenetic energy, strain energy, total energy and energy error are calculated semi-analytically, followed by an energy error estimator developed for elastodynamic cases. A simple but accurate meshing mapping method is derived semi-analytically as well. The follow chart of adaptive SBFEM is described for elasodynamic problems. This method is used to calculate the structures'responses under blast/impact loading, and the results are compared with traditional SBFEM, FEM and adaptive FEM in order to verify the effectiveness of the present method.
The conception of "super element" is introduced and the semi-analytical calculations of kinetic energy, strain energy, total energy and energy error are extended to subdomain level, and a distinguish approach of the subdomain energy error is proposed. A simple subdomain subdivision procedure, combined with the developed mesh mapping method, leads to a new adaptive SBFEM. A simply-supported beam and a cantilever under an impact loading are calculated by this method respectively.
The acceleration field is rebuilt based on the assumption of linear distributed acceleration for Newmark integration scheme. The calculation of the time discretisation error, followed by a local error estimator, is given out. Based on the adjustment of time incresement, an adaptive time-stepping procedure of SBFEM is developed with its follow chart. Several numerical examples are examined based on this method.
A polygon SBFEM is developed based on the Delauney Triangulation. Nodes shared by different triangles or the gravity centres of polygons are taken as the scaling centre of subdomains, a polygon mesh of SBFEM is constructed directly and automaticly. The influence of element size on calculation accuracy is investigated by several numerical examples; meanwhile the total energy is calculated.
This thesis aims at extending the theory of adaptive SBFEM and making a good prepare for its applications in engineering, especially for crack propagation modeling.
基于虚功原理推导了SBFEM静力平衡方程及其求解方法；比较SPR法和节点平均法重构的节点应力；建立了SBFEM中半解析形式的总能量、能量误差和能量误差指标；确定了h型网格重分策略及自适应算法的流程；比较了在不同能量误差目标值下的自适应网格、计算精度和计算时间；比较了基于自适应SBFEM和FEM求得的应力场。
基于虚功原理推导了SBFEM的动力平衡方程及其求解方法；提出采用Newmark时间积分法求解SBFEM整体平衡方程组；推导了SBFEM的具有半解析形式的动能、应变能、总能量及能量误差,建立了动态能量误差指标；发展了具有半解析特性、简单而准确的SBFEM网格映射方法；确定了动态自适应SBFEM的计算流程；采用自适应SBFEM求解不同结构在爆破荷载、冲击荷载作用下的动力响应,并与常规SBFEM、FEM和自适应FEM等方法作比较,验证了该方法处理一般弹性动力学问题的有效性。
引入超单元”概念,将SBFEM具有半解析形式的动能、应变能、总能量和能量误差推广至子域水平,并提出了相应的子域能量误差判别方法；提出了一种简单、高效的子域重分策略,并与准确的网格映射技术相结合,发展了基于子域重分技术的自适应SBFEM;通过计算冲击荷载作用下的简支梁和悬臂深梁的动力响应,验证了该方法的有效性。
基于Newmark时间积分法的基本假定重构得时域内的线性分布加速度场,推导了能量范数形式的SBFEM时间离散误差及指标,以及相应的时间步长缩放准则和计算流程；发展了一种可自动控制时间离散误差的SBFEM方法。通过计算冲击荷载、爆破荷载作用下三种不同结构的动力响应验证该方法的有效性。
在Delaunay三角化的基础上,以各三角形共用节点或多边形重心为各SBFEM子域的相似中心,直接建立多边形SBFEM的离散网格,从而实现SBFEM处理一般弹性动力学问题的自动建模。通过算例考察了该方法的计算精度对单元尺寸的敏感性,并在结构的全局能量误差中得到反映。
本文旨在深化自适应比例边界元法的理论根基,为其在裂缝扩展模拟等领域的应用奠定重要基础。
The equilibrium equations and their solutions of scaled boundary finite element method (SBFEM) are derived based on the virtual work principle in this thesis, not only for elastostatics but also for elastodynamics. Energy error estimator based on stress recovery technique is extended from elastostatics to elastodyanmics, and an efficient remeshing strategy is given out with a simple but accurate mesh mapping procedure. Furthermore, the error caused by time discretisation is also taken into account, leading to a datively complete adaptive SBFEM for general elstodynamic problems. Additionally, a polygon scaled boundary finite element method is developed based on Delauney Triangulation, making the discritisation in SBFEM fully-automatic.
The equilibrium equations of SBFEM are derived based on virtual work for elastostatics. Two different stress recovery teniques, including superconvergent patch recovery (SPR) and node average technique, are compared. Total energy, energy error and error estimator are given out in semi-analytical formula. An h-type remeshing procedure and a follow chart of the adaptive approach are described in detail. The changes of adaptive meshes, accuracy and computational time are investigated through a numerical experiment. The stress filed calculated by adaptive SBFEM is also compared with that of FEM.
The equilibrium equations of SBFEM are extended to elastodynamics also based on virtual work principle. Newmark intergration scheme is employed to solve the equations. Kenetic energy, strain energy, total energy and energy error are calculated semi-analytically, followed by an energy error estimator developed for elastodynamic cases. A simple but accurate meshing mapping method is derived semi-analytically as well. The follow chart of adaptive SBFEM is described for elasodynamic problems. This method is used to calculate the structures'responses under blast/impact loading, and the results are compared with traditional SBFEM, FEM and adaptive FEM in order to verify the effectiveness of the present method.
The conception of "super element" is introduced and the semi-analytical calculations of kinetic energy, strain energy, total energy and energy error are extended to subdomain level, and a distinguish approach of the subdomain energy error is proposed. A simple subdomain subdivision procedure, combined with the developed mesh mapping method, leads to a new adaptive SBFEM. A simply-supported beam and a cantilever under an impact loading are calculated by this method respectively.
The acceleration field is rebuilt based on the assumption of linear distributed acceleration for Newmark integration scheme. The calculation of the time discretisation error, followed by a local error estimator, is given out. Based on the adjustment of time incresement, an adaptive time-stepping procedure of SBFEM is developed with its follow chart. Several numerical examples are examined based on this method.
A polygon SBFEM is developed based on the Delauney Triangulation. Nodes shared by different triangles or the gravity centres of polygons are taken as the scaling centre of subdomains, a polygon mesh of SBFEM is constructed directly and automaticly. The influence of element size on calculation accuracy is investigated by several numerical examples; meanwhile the total energy is calculated.
This thesis aims at extending the theory of adaptive SBFEM and making a good prepare for its applications in engineering, especially for crack propagation modeling.
引文
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