假设位移拟协调有限元及其在精确几何分析中的应用
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摘要
有限元是一种重要的数值仿真分析方法,在工业领域中的设计、校核和生命周期检测等多个方面发挥巨大作用,深刻地改变了工业领域的方法和思想。拟协调有限元是有限元中十分重要的一种方法,其特点是同时弱化平衡方程和几何方程,与传统有限元相比更加灵活、有效。拟协调单元广泛应用于多个工业领域,在结构分析,尤其是板壳结构分析中发挥着巨大的作用。因此,对拟协调有限元的研究具有重要的理论研究和工程应用价值。
本文以拟协调有限元为研究对象,从单元构造和算法理论等方面进行了研究,主要工作可分为两部分。第一部分结合弹性力学平面问题和板壳问题对拟协调元进行了研究,完善了拟协调有限元的列式框架,建立了系统的单元构造理论和单元性能分析方法,构造了一系列有效的单元,应用到工业领域分析中。通过对拟协调有限元的研究,提出了几何方程中微分算子的弱导数和“泰勒展开校核”收敛性检验方法,强调了有限元中基函数的作用,深化了有限元中“协调性”要求的理解。第二部分,将拟协调有限元推广到精确几何分析领域,提出精确几何拟协调分析方法。该方法不再需要传统的有限元网格,可以由几何模型数据直接进行分析,为下一代的几何设计-有限元分析一体化的仿真分析系统提供有效算法。自主开发了基于几何数据的分析框架,并构造了一系列有效的单元。精确几何拟协调分析从变分原理和逼近空间两个角度,区别于以等几何分析为代表的精确几何分析方法。
本文对拟协调元的单元构造方法进行了系统的研究。完善了拟协调有限元中位移场和应变场试探函数的选取规则,强化了位移场和应变场的联系,解决了拟协调有限元中位移形函数的计算问题,便于单元一致质量阵和一致载荷阵的计算,使单元稳定性增强,具有更好的收敛性能。本文对算法理论进行了研究,提出几何方程中微分算子的弱导数,针对有限元中重要的收敛性问题,提出了单元应变的泰勒展开校核方法,可以有效地检查单元的收敛速度。打破了传统有限元中“协调性”等诸多列式禁区,提供了一个统一的、有效的列式准则。将其总结为“假设位移拟协调有限元”方法。
按照假设位移拟协调有限元方法,本文构造了一系列结构分析单元,为工业领域应用提供了分析工具。本文构造的平面四边形单元在直角坐标系下直接列式,解决了有限元中长期存在的三角形单元和四边形单元列式理论不统一的问题。该单元不需要借助于等参坐标和数值积分,具有显式的刚度矩阵,是一个简单、高效的单元。本文将其应用到板材件的一步逆成形分析中,得到比传统四节点等参单元精度更好、效率更高的结果。本文构造的四边形板壳单元具有很好的收敛性,在大量标准算例中与其它著名单元结果进行了对比,证实了其具有较好的实用价值。
“精确几何分析”是指利用计算机辅助设计中的几何模型(CAD模型)直接进行仿真分析。精确几何分析中不需要将几何模型转化为有限元网格模型的步骤,相对传统有限元仿真分析,其明显优势在于避免网格划分,融合现有的计算机辅助设计(CAD)和仿真分析(CAE),极大地简化工业设计/分析流程。同时,精确几何分析可以保证分析模型中的几何是精确的,对壳体屈曲分析、飞行器周围流体分析等几何敏感的问题,具有先天的相对传统有限元分析的优势。
利用假设位移拟协调有限元,研究精确几何分析问题,提出“精确几何拟协调分析”方法。与等几何分析等其它的精确几何分析方法相比,本方法打破了“等参”的分析框架,采用多项式基函数逼近物理场,充分利用多项式简洁、便于计算的特性。同时仍然采用非均匀有理B样条函数精确地表示几何场,适应精确几何分析要求。利用假设位移拟协调有限元框架,采用应变弱化技术,对位移场和应变场同时进行逼近,并选用完备的逼近函数,提高了单元的精度。
利用ACIS几何造型引擎,自主开发了精确几何分析程序框架,可以输入、修改并输出标准的几何模型数据。基于精确几何拟协调分析,实现了一维柱、梁单元,二维平面单元、平板单元等一系列分析模块。精确几何拟协调分析发展了拟协调元的算法理论,为精确几何分析引入了新的技术手段。
本文从单元构造框架和单元算法理论等方面发展了拟协调有限元,提出了“假设位移拟协调有限元”和“精确几何拟协调分析”方法,构造了一系列有效的单元并将其应用到工业实践中。本文在单元算法理论、单元构造框架等基础理论问题的研究是对有限元理论的发展,本文在“精确几何分析”方面的工作适应几何设计-仿真分析一体化的要求,具有重要的学术和工业应用价值。
Finite element method is the most important numerical analysis method, which plays great role in design, analysis and the life circle monitoring of industry products. The design method and concept of industry are deeply influenced by the finite element method. Quasi-conforming finite element method is an important and characteristic paradigm in finite element method. In quasi-conforming finite element method, the strain-displacement relationships are weakened along with the equilibrium equations; as a result the method is flexible and practicable, which can unify the conforming and non-conforming frame. The elements formulated with quasi-conforming are widely used in multiple industry areas, especially in the analysis of structure. Therefore, the research in quasi-conforming finite element method is import for its great science as well as the industry application value.
This thesis is on the research of quasi-conforming finite element method. The work is composed of two parts. First, the assumed displacement quasi-conforming finite element framework is proposed based on the research of elastic plane and plate/shell problems. The element construction and performance analysis of quasi-conforming finite element method is improved. In the second part, the exact-geometry based analysis is introduced to quasi-conforming framework. A new method, the exact geometry based quasi-conforming analysis is proposed. With this method, the mesh is not needed for the analysis process. The method provides an important algorithm for the next generation analysis system which can unify the geometry design and analysis.
In the first part of this thesis, the element construction method and algorithm theory are researched. The assumed displacement method is proposed to improve the selection rule for displacement and strain trial functions. With this method, the connection between displacement and strain fields are strengthened, which qualify the elements with better robustness and convergence performance. The calculation for shape function of displacement field is implemented in the quasi-conforming method, which is important for calculation for consistent mass matrix and load vector. The general derivative of the strain-displacement relation is proposed. For the convergence problem, the Taylor expansion test is introduced which can check the convergence rate of element formulation. This part of work is summarized as the "assumed displacement quasi-conforming finite element method"
With assumed displacement quasi-conforming finite element method, a series of structural analysis elements are formulated which are applicable for industrial analysis. A four-node element for plane problem is directly formulated in rectangular coordinate system, and the isoparametric transformation and numerical analysis which is essential for traditional finite element method is avoidable. The element is simple and efficient with explicit stiffness matrix. The formulation is used in the inverse analysis of panel stamping and the performance is better than4-node isoparametric element with great efficiency. The plant and shell elements are formulated with assumed displacement quasi-conforming method. The elements have great performance in convergence rate and extensive tests are carried out for the complex structural analysis. The results by present plate and shell elements are compared with results from commercial analysis software package and the analytical solution, which shows that present elements are precise and applicable.
Exact geometry based analysis is a new method which apply the geometry model directly for analysis and the process for generating finite element mesh from geometry model is avoided. The advantagement of exact geometry based analysis compared with traditional finite element method is that the computer aided design and analysis can be unified and the industry design and analysis processes are greatly improved. For problems which are sensitive of the geometry appearance, such as the buckling analysis of shell and the fluid analysis around the aircraft, exact geometry based analysis is especially suitable and can improve the precision.
In the second part of this thesis, the exact geometry concept is applied in quasi-conforming finite element method and the exact geometry based quasi-conforming analyisis is proposed. In this method the isoparametric framework is broken and the polynomial basis functions are used to simulate the physical fields and non-uniform rational B spline for precise geometry model. The strain-displacement equations are softened and the complete set of basis functions are used to simulate the physical fields. With exact geometry based quasi-conforming method the precision of analysis is improved, which provides an alterate technique choise for exact geometry based analysis.
With the geometry modeling core "ACIS", an exact geometry based analysis software framework is constructed, which is able to input, modify and output the standard geometry models. With exact geometry based quasi-conforming analysis, the one dimensional column element, beam element, two dimensional plane element and plate element are developed.
This thesis is the inheritance and development for quasi-conforming finite element method. The formulation framework and theory of quasi-conforming are improved, and a set of efficient elements are formulated. The exact geometry concept is implemented in quasi-conforming method and a new method is proposed. The work of this thesis provides both theory basis and also technique storage for numerical analysis.
本文以拟协调有限元为研究对象,从单元构造和算法理论等方面进行了研究,主要工作可分为两部分。第一部分结合弹性力学平面问题和板壳问题对拟协调元进行了研究,完善了拟协调有限元的列式框架,建立了系统的单元构造理论和单元性能分析方法,构造了一系列有效的单元,应用到工业领域分析中。通过对拟协调有限元的研究,提出了几何方程中微分算子的弱导数和“泰勒展开校核”收敛性检验方法,强调了有限元中基函数的作用,深化了有限元中“协调性”要求的理解。第二部分,将拟协调有限元推广到精确几何分析领域,提出精确几何拟协调分析方法。该方法不再需要传统的有限元网格,可以由几何模型数据直接进行分析,为下一代的几何设计-有限元分析一体化的仿真分析系统提供有效算法。自主开发了基于几何数据的分析框架,并构造了一系列有效的单元。精确几何拟协调分析从变分原理和逼近空间两个角度,区别于以等几何分析为代表的精确几何分析方法。
本文对拟协调元的单元构造方法进行了系统的研究。完善了拟协调有限元中位移场和应变场试探函数的选取规则,强化了位移场和应变场的联系,解决了拟协调有限元中位移形函数的计算问题,便于单元一致质量阵和一致载荷阵的计算,使单元稳定性增强,具有更好的收敛性能。本文对算法理论进行了研究,提出几何方程中微分算子的弱导数,针对有限元中重要的收敛性问题,提出了单元应变的泰勒展开校核方法,可以有效地检查单元的收敛速度。打破了传统有限元中“协调性”等诸多列式禁区,提供了一个统一的、有效的列式准则。将其总结为“假设位移拟协调有限元”方法。
按照假设位移拟协调有限元方法,本文构造了一系列结构分析单元,为工业领域应用提供了分析工具。本文构造的平面四边形单元在直角坐标系下直接列式,解决了有限元中长期存在的三角形单元和四边形单元列式理论不统一的问题。该单元不需要借助于等参坐标和数值积分,具有显式的刚度矩阵,是一个简单、高效的单元。本文将其应用到板材件的一步逆成形分析中,得到比传统四节点等参单元精度更好、效率更高的结果。本文构造的四边形板壳单元具有很好的收敛性,在大量标准算例中与其它著名单元结果进行了对比,证实了其具有较好的实用价值。
“精确几何分析”是指利用计算机辅助设计中的几何模型(CAD模型)直接进行仿真分析。精确几何分析中不需要将几何模型转化为有限元网格模型的步骤,相对传统有限元仿真分析,其明显优势在于避免网格划分,融合现有的计算机辅助设计(CAD)和仿真分析(CAE),极大地简化工业设计/分析流程。同时,精确几何分析可以保证分析模型中的几何是精确的,对壳体屈曲分析、飞行器周围流体分析等几何敏感的问题,具有先天的相对传统有限元分析的优势。
利用假设位移拟协调有限元,研究精确几何分析问题,提出“精确几何拟协调分析”方法。与等几何分析等其它的精确几何分析方法相比,本方法打破了“等参”的分析框架,采用多项式基函数逼近物理场,充分利用多项式简洁、便于计算的特性。同时仍然采用非均匀有理B样条函数精确地表示几何场,适应精确几何分析要求。利用假设位移拟协调有限元框架,采用应变弱化技术,对位移场和应变场同时进行逼近,并选用完备的逼近函数,提高了单元的精度。
利用ACIS几何造型引擎,自主开发了精确几何分析程序框架,可以输入、修改并输出标准的几何模型数据。基于精确几何拟协调分析,实现了一维柱、梁单元,二维平面单元、平板单元等一系列分析模块。精确几何拟协调分析发展了拟协调元的算法理论,为精确几何分析引入了新的技术手段。
本文从单元构造框架和单元算法理论等方面发展了拟协调有限元,提出了“假设位移拟协调有限元”和“精确几何拟协调分析”方法,构造了一系列有效的单元并将其应用到工业实践中。本文在单元算法理论、单元构造框架等基础理论问题的研究是对有限元理论的发展,本文在“精确几何分析”方面的工作适应几何设计-仿真分析一体化的要求,具有重要的学术和工业应用价值。
Finite element method is the most important numerical analysis method, which plays great role in design, analysis and the life circle monitoring of industry products. The design method and concept of industry are deeply influenced by the finite element method. Quasi-conforming finite element method is an important and characteristic paradigm in finite element method. In quasi-conforming finite element method, the strain-displacement relationships are weakened along with the equilibrium equations; as a result the method is flexible and practicable, which can unify the conforming and non-conforming frame. The elements formulated with quasi-conforming are widely used in multiple industry areas, especially in the analysis of structure. Therefore, the research in quasi-conforming finite element method is import for its great science as well as the industry application value.
This thesis is on the research of quasi-conforming finite element method. The work is composed of two parts. First, the assumed displacement quasi-conforming finite element framework is proposed based on the research of elastic plane and plate/shell problems. The element construction and performance analysis of quasi-conforming finite element method is improved. In the second part, the exact-geometry based analysis is introduced to quasi-conforming framework. A new method, the exact geometry based quasi-conforming analysis is proposed. With this method, the mesh is not needed for the analysis process. The method provides an important algorithm for the next generation analysis system which can unify the geometry design and analysis.
In the first part of this thesis, the element construction method and algorithm theory are researched. The assumed displacement method is proposed to improve the selection rule for displacement and strain trial functions. With this method, the connection between displacement and strain fields are strengthened, which qualify the elements with better robustness and convergence performance. The calculation for shape function of displacement field is implemented in the quasi-conforming method, which is important for calculation for consistent mass matrix and load vector. The general derivative of the strain-displacement relation is proposed. For the convergence problem, the Taylor expansion test is introduced which can check the convergence rate of element formulation. This part of work is summarized as the "assumed displacement quasi-conforming finite element method"
With assumed displacement quasi-conforming finite element method, a series of structural analysis elements are formulated which are applicable for industrial analysis. A four-node element for plane problem is directly formulated in rectangular coordinate system, and the isoparametric transformation and numerical analysis which is essential for traditional finite element method is avoidable. The element is simple and efficient with explicit stiffness matrix. The formulation is used in the inverse analysis of panel stamping and the performance is better than4-node isoparametric element with great efficiency. The plant and shell elements are formulated with assumed displacement quasi-conforming method. The elements have great performance in convergence rate and extensive tests are carried out for the complex structural analysis. The results by present plate and shell elements are compared with results from commercial analysis software package and the analytical solution, which shows that present elements are precise and applicable.
Exact geometry based analysis is a new method which apply the geometry model directly for analysis and the process for generating finite element mesh from geometry model is avoided. The advantagement of exact geometry based analysis compared with traditional finite element method is that the computer aided design and analysis can be unified and the industry design and analysis processes are greatly improved. For problems which are sensitive of the geometry appearance, such as the buckling analysis of shell and the fluid analysis around the aircraft, exact geometry based analysis is especially suitable and can improve the precision.
In the second part of this thesis, the exact geometry concept is applied in quasi-conforming finite element method and the exact geometry based quasi-conforming analyisis is proposed. In this method the isoparametric framework is broken and the polynomial basis functions are used to simulate the physical fields and non-uniform rational B spline for precise geometry model. The strain-displacement equations are softened and the complete set of basis functions are used to simulate the physical fields. With exact geometry based quasi-conforming method the precision of analysis is improved, which provides an alterate technique choise for exact geometry based analysis.
With the geometry modeling core "ACIS", an exact geometry based analysis software framework is constructed, which is able to input, modify and output the standard geometry models. With exact geometry based quasi-conforming analysis, the one dimensional column element, beam element, two dimensional plane element and plate element are developed.
This thesis is the inheritance and development for quasi-conforming finite element method. The formulation framework and theory of quasi-conforming are improved, and a set of efficient elements are formulated. The exact geometry concept is implemented in quasi-conforming method and a new method is proposed. The work of this thesis provides both theory basis and also technique storage for numerical analysis.
引文
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