无单元法若干技术研究及其在电磁场计算中的应用
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摘要
无单元法是解决电磁场数值计算问题的一种新的有效方法,在处理小气隙、薄膜等问题上有着独特的优势,可以弥补有限元法处理这些问题的不足。本课题主要针对伽辽金型无单元法的电磁场计算问题展开深入研究。本文主要由以下几部分组成:
第一部分针对无单元法求解效率不高的问题,将多重网格法应用在无单元法上,提出了无单元多重网格法。将多重网格的校正和迭代技术引入到有限单元法的框架中,可提高求解效率,有着广泛的应用,但多重网格法在无单元法上还没有应用。为提高计算效率,本文提出了基于无单元的多重网格法,这种方法在粗、细网格节点解的处理上和粗节点的构造方法及限制算子的确定都与传统的多重网格法不同。实例分析证明,针对同一个用无单元法离散化的分析模型,在达到同样的精度要求下,在不同的自由度下,多重网格法的CPU计算时间要小于ICCG法。这表明无单元多重网格法的计算效率要高于ICCG法。
第二部分针对无单元节点的布置方法进行研究。在使用无单元法求解的过程中,无单元节点的布置适合与否会对解的精度产生很大的影响。本文提出一种新的不规则区域的无单元节点布置方法。该方法首先通过自组织特征映射法来确定背景网格,然后依据无单元节点的布置原则,通过计算点来确定无单元节点。此算法便于生成合理的节点分布,提高求解精度。
第三部分对伽辽金型无单元数值积分问题进行了研究。采用圆形支撑域来构造积分区域,支撑域交叠的部分采用高斯数值积分方法进行积分,依据高斯积分公式,得出了高斯积分点及权重随节点距离变化的关系。该方法不需要背景网格来积分,是一种真正的无单元法。与规则背景网格相比,该方法提高了伽辽金型无单元法数值积分的精度。
第四部分提出了一种建立在重叠型区域分解的基础之上的无单元伽辽金法和有限元耦合模型,来处理定子与转子相对运动问题的方法。该模型将求解区域分成两个子区域:有限元子域、无单元子域。然后依次对无单元子域和有限元子域进行迭代求解,最后得到整个区域的解。该方法的数值解较Belystchko提出的无单元与有限元耦合法更为准确,而且有效地克服了Belystchko提出的耦合法只适用规则交界面的缺点。本文将该方法应用在电机电磁场中,分析得到的结果克服了由于有限元法单元畸变产生的误差。
Element-free Galerkin (EFG) method is a novel effective method for solving electromagnetic computation, which is suitable for dealing with small air gap and membrane problems, which are difficult by finite element method (FEM), so it could make up the defection of FEM. The dissertation studied on the electromagnetic problems by EFG. The dissertation consists of four parts:
Firstly, in order to increase the solving efficiency, the multigrid method was used in EFG, and a multigrid method of element-free Galerkin was presented in the paper. The multigrid iterative technique was extensively introduced to FEM frame, which could increase solving efficiency. In order to resolve low computational efficiency of element-free Galerkin method, the dissertation introduced the accelerated iterative multigrid method to the element-free Galerkin discrete field, proposed a multigrid method of element-free Galerkin (EFG-MG). The disposition on solutions of coarse grid nodes and fine grid nodes, in addition to the construction of coarse nodes and the restriction operator were different from the traditional MG. It was used for numerical computation of electromagnetic field problems, and the high efficiency of element-free Galerkin method was proved by examples. For the same element-free model, the CPU time of MG method is less than ICCG method for different freedom numbers. It indicates that the computational efficiency of MG is higher than ICCG.
Secondly, the dissertation studied on the nodes distribution of EFG. The errors of the solutions are influenced greatly by the distribution of nodes of element-free method when using EFG. In order to reduce the errors of calculation, a new method on distribution of element-free nodes in irregular electromagnetic field domain was developed. In the method, firstly, the self-organizing feature map was used to generate background cells; secondly, the element-free nodes were determined by the evaluation points. The method is convenient for generating reasonable distribution of nodes with high solving accuracy.
Thirdly, the dissertation studied on the numerical integration of EFG. The support domain of circular shape was used to construct integral domain, the Gauss numerical integration methods were used to overlapping area of the supporting domains. Based on the Gaussian integral formula, the curve equations rel ated with the positions of Gaussian integral points and weights to the node distance were derived. The method doesn’t need background cell integration, is a pure element-free method. It is very convenient for numerical integration, by which the solving accuracy is increased.
Finally, the dissertation presented a new coupling model for coupling element-free Galerkin method with finite element method (EFG-FEM). The coupling model was based on overlapping domain decomposition method (DDM), in which the domain was decomposed into FEM subdomain and EFG subdomain; then the FEM subdomain and EFG subdomain were solved in turn by iterative method, until the solutions of the domain were obtained. The solutions are more accurate than Belystchko’s coupling method, and can eliminate influence of irregular interface shape.
第一部分针对无单元法求解效率不高的问题,将多重网格法应用在无单元法上,提出了无单元多重网格法。将多重网格的校正和迭代技术引入到有限单元法的框架中,可提高求解效率,有着广泛的应用,但多重网格法在无单元法上还没有应用。为提高计算效率,本文提出了基于无单元的多重网格法,这种方法在粗、细网格节点解的处理上和粗节点的构造方法及限制算子的确定都与传统的多重网格法不同。实例分析证明,针对同一个用无单元法离散化的分析模型,在达到同样的精度要求下,在不同的自由度下,多重网格法的CPU计算时间要小于ICCG法。这表明无单元多重网格法的计算效率要高于ICCG法。
第二部分针对无单元节点的布置方法进行研究。在使用无单元法求解的过程中,无单元节点的布置适合与否会对解的精度产生很大的影响。本文提出一种新的不规则区域的无单元节点布置方法。该方法首先通过自组织特征映射法来确定背景网格,然后依据无单元节点的布置原则,通过计算点来确定无单元节点。此算法便于生成合理的节点分布,提高求解精度。
第三部分对伽辽金型无单元数值积分问题进行了研究。采用圆形支撑域来构造积分区域,支撑域交叠的部分采用高斯数值积分方法进行积分,依据高斯积分公式,得出了高斯积分点及权重随节点距离变化的关系。该方法不需要背景网格来积分,是一种真正的无单元法。与规则背景网格相比,该方法提高了伽辽金型无单元法数值积分的精度。
第四部分提出了一种建立在重叠型区域分解的基础之上的无单元伽辽金法和有限元耦合模型,来处理定子与转子相对运动问题的方法。该模型将求解区域分成两个子区域:有限元子域、无单元子域。然后依次对无单元子域和有限元子域进行迭代求解,最后得到整个区域的解。该方法的数值解较Belystchko提出的无单元与有限元耦合法更为准确,而且有效地克服了Belystchko提出的耦合法只适用规则交界面的缺点。本文将该方法应用在电机电磁场中,分析得到的结果克服了由于有限元法单元畸变产生的误差。
Element-free Galerkin (EFG) method is a novel effective method for solving electromagnetic computation, which is suitable for dealing with small air gap and membrane problems, which are difficult by finite element method (FEM), so it could make up the defection of FEM. The dissertation studied on the electromagnetic problems by EFG. The dissertation consists of four parts:
Firstly, in order to increase the solving efficiency, the multigrid method was used in EFG, and a multigrid method of element-free Galerkin was presented in the paper. The multigrid iterative technique was extensively introduced to FEM frame, which could increase solving efficiency. In order to resolve low computational efficiency of element-free Galerkin method, the dissertation introduced the accelerated iterative multigrid method to the element-free Galerkin discrete field, proposed a multigrid method of element-free Galerkin (EFG-MG). The disposition on solutions of coarse grid nodes and fine grid nodes, in addition to the construction of coarse nodes and the restriction operator were different from the traditional MG. It was used for numerical computation of electromagnetic field problems, and the high efficiency of element-free Galerkin method was proved by examples. For the same element-free model, the CPU time of MG method is less than ICCG method for different freedom numbers. It indicates that the computational efficiency of MG is higher than ICCG.
Secondly, the dissertation studied on the nodes distribution of EFG. The errors of the solutions are influenced greatly by the distribution of nodes of element-free method when using EFG. In order to reduce the errors of calculation, a new method on distribution of element-free nodes in irregular electromagnetic field domain was developed. In the method, firstly, the self-organizing feature map was used to generate background cells; secondly, the element-free nodes were determined by the evaluation points. The method is convenient for generating reasonable distribution of nodes with high solving accuracy.
Thirdly, the dissertation studied on the numerical integration of EFG. The support domain of circular shape was used to construct integral domain, the Gauss numerical integration methods were used to overlapping area of the supporting domains. Based on the Gaussian integral formula, the curve equations rel ated with the positions of Gaussian integral points and weights to the node distance were derived. The method doesn’t need background cell integration, is a pure element-free method. It is very convenient for numerical integration, by which the solving accuracy is increased.
Finally, the dissertation presented a new coupling model for coupling element-free Galerkin method with finite element method (EFG-FEM). The coupling model was based on overlapping domain decomposition method (DDM), in which the domain was decomposed into FEM subdomain and EFG subdomain; then the FEM subdomain and EFG subdomain were solved in turn by iterative method, until the solutions of the domain were obtained. The solutions are more accurate than Belystchko’s coupling method, and can eliminate influence of irregular interface shape.
引文
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