非协调数值流形方法研究
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摘要
本文以复合材料细观力学数值模拟为背景,以数值流形方法为基础,对其进行了改进和拓展,将数值流形方法推广至三维,并提出了非协调数值流形方法,通过数值算例验证了本文方法的正确性和有效性。主要取得了以下成果:
将覆盖位移函数用自然坐标表示,对四节点四边形数值流形方法提出了改进措施,使得在一般非规则有限数学覆盖网格下,求解单元刚度矩阵的数值积分变得较简单,提高了计算效率。
通过引入公共面法,重新定义了数值流形方法的接触判断,使接触判断的计算量大大减少,建立了三维数值流形方法。此项工作充实了数值流形方法的研究范围,拓宽了数值流形方法的应用领域。
在流形单元总体位移场上附加非协调位移基本项,使单元位移函数趋于完全,提出了非协调数值流形方法。该方法改进了流形单元的计算精度,提高了流形单元的计算效率。
对Wilson非协调位移基本项进行改进,得到了非协调函数生成的一般公式,给出了对应于零阶覆盖和一阶覆盖的非协调流形单元的显式表达,方便于工程应用。
对非协调数值流形方法进行了稳定性和收敛性分析,完善了非协调数值流形方法的理论基础。并对其稳定性和收敛性进行了数值验证,数值试验表明,新单元构造过程简单,对单元畸变不敏感,计算精度可以得到一定提高,从而证明了本文所提方法的正确性和有效性。
将非协调数值流形方法应用于稳态温度场的数值计算,推导了势问题的非协调数值流形方法。数值算例表明,非协调数值流形方法得到的温度场近似分布,能够很好地反映真实温度场分布的特性。
数值流形方法使用有限覆盖技术,将连续与非连续问题统一在一起,成功地模拟了断续节理岩体的裂纹扩展。本文提出的非协调数值流形方法是在传统的数值流形方法基础上,引入非协调项而得到,因此,非协调数值流形方法不仅继承了数值流形方法的优点,而且提高了数值流形方法的计算精度和效率。作为对裂纹扩展问题的进一步讨论,将非协调数值流形方法应用于求解应力强度因子的有效性,并对张拉型裂纹问题进行了数值模拟。
The background of the researches is numerical simulation of particle-reinforced composites. Based on numerical manifold method, this paper improves and extends the method. A new numerical method named incompatible numerical manifold method is presented. To illustrate the feasibility and stability of the present approach, numerical examples are provided. The main contributions can be listed as follows:The covered displacement function expressed by the natural coordinate, the four-nodal quadrangular numerical manifold method is improved. It makes much easy the integration of the four-nodal quadrangular numerical manifold element in irregular covered mathematical mesh form. The results have been found more accurate than that of the finite element method.Using Common Plane techniques to detect and categorize contacts between three-dimensional blocks of any arbitrary shape, the contact detection method is reconstructed to reduce computation. The three-dimensional numerical manifold method is presented. The work enriches and extends the research domain of numerical manifold method.The incompatible displacement term is added to total displacement function, it makes displacement function tend to entire, establishes the improved manifold method named incompatible numerical manifold method. This method can greatly increase calculating accuracy and computing efficiency without adding generalized degrees of freedom.Wilson incompatible displacement function term is improved, generalized formula of incompatible displacement function is deduced. In order to apply this method to engineering, we give an explicit treatment of zero-order coved function and one-order coved function.In order to perfect theoretic system of incompatible numerical manifold method, its stability and convergence are analyzed. Some numerical examples are provided to illustrate this method. The conclusion shows that this method produces highly accurate and stable results.
将覆盖位移函数用自然坐标表示,对四节点四边形数值流形方法提出了改进措施,使得在一般非规则有限数学覆盖网格下,求解单元刚度矩阵的数值积分变得较简单,提高了计算效率。
通过引入公共面法,重新定义了数值流形方法的接触判断,使接触判断的计算量大大减少,建立了三维数值流形方法。此项工作充实了数值流形方法的研究范围,拓宽了数值流形方法的应用领域。
在流形单元总体位移场上附加非协调位移基本项,使单元位移函数趋于完全,提出了非协调数值流形方法。该方法改进了流形单元的计算精度,提高了流形单元的计算效率。
对Wilson非协调位移基本项进行改进,得到了非协调函数生成的一般公式,给出了对应于零阶覆盖和一阶覆盖的非协调流形单元的显式表达,方便于工程应用。
对非协调数值流形方法进行了稳定性和收敛性分析,完善了非协调数值流形方法的理论基础。并对其稳定性和收敛性进行了数值验证,数值试验表明,新单元构造过程简单,对单元畸变不敏感,计算精度可以得到一定提高,从而证明了本文所提方法的正确性和有效性。
将非协调数值流形方法应用于稳态温度场的数值计算,推导了势问题的非协调数值流形方法。数值算例表明,非协调数值流形方法得到的温度场近似分布,能够很好地反映真实温度场分布的特性。
数值流形方法使用有限覆盖技术,将连续与非连续问题统一在一起,成功地模拟了断续节理岩体的裂纹扩展。本文提出的非协调数值流形方法是在传统的数值流形方法基础上,引入非协调项而得到,因此,非协调数值流形方法不仅继承了数值流形方法的优点,而且提高了数值流形方法的计算精度和效率。作为对裂纹扩展问题的进一步讨论,将非协调数值流形方法应用于求解应力强度因子的有效性,并对张拉型裂纹问题进行了数值模拟。
The background of the researches is numerical simulation of particle-reinforced composites. Based on numerical manifold method, this paper improves and extends the method. A new numerical method named incompatible numerical manifold method is presented. To illustrate the feasibility and stability of the present approach, numerical examples are provided. The main contributions can be listed as follows:The covered displacement function expressed by the natural coordinate, the four-nodal quadrangular numerical manifold method is improved. It makes much easy the integration of the four-nodal quadrangular numerical manifold element in irregular covered mathematical mesh form. The results have been found more accurate than that of the finite element method.Using Common Plane techniques to detect and categorize contacts between three-dimensional blocks of any arbitrary shape, the contact detection method is reconstructed to reduce computation. The three-dimensional numerical manifold method is presented. The work enriches and extends the research domain of numerical manifold method.The incompatible displacement term is added to total displacement function, it makes displacement function tend to entire, establishes the improved manifold method named incompatible numerical manifold method. This method can greatly increase calculating accuracy and computing efficiency without adding generalized degrees of freedom.Wilson incompatible displacement function term is improved, generalized formula of incompatible displacement function is deduced. In order to apply this method to engineering, we give an explicit treatment of zero-order coved function and one-order coved function.In order to perfect theoretic system of incompatible numerical manifold method, its stability and convergence are analyzed. Some numerical examples are provided to illustrate this method. The conclusion shows that this method produces highly accurate and stable results.
引文
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