基于径向积分法的多边形及多面体有限元方法
发布时间：2018-07-29 09:12
【摘要】：随着数值模拟中所面临的问题的多样化,要求有限元方法使用的单元不再局限于三角形、四边形、四面体以及六面体单元等规则单元。任意多边形单元和多面体单元作为新的单元形式,由于其边数和面数的任意性,在划分复杂模型网格时比常规单元更加灵活方便,能够更好地模拟材料与结构的热、力学性能,并可以能够更好地处理一些如断裂,裂纹扩展等复杂问题。然而,任意多边形和多面体单元因其几何形状的任意性很难构造出多项式形式插值函数,导致积分计算单元刚度阵和载荷向量时比较困难。本文针对任意多边形及多面体单元积分困难的问题,做了以下研究：首先,以Wachpress插值函数作为多边形单元的形函数,发展了二维多边形单元的径向积分法用于计算单元刚度阵和载荷向量。对于任意形状及尺寸的多边形单元,使用径向积分法将多边形单元计算刚度阵和载荷向量过程中的面积分,转化成沿着单元边界的线积分进行计算,避免因多边形单元积分域不规则以及积分函数的非多项式形式带来的难题。实际工程应用当中,多边形单元可区分为常规单元(三角形单元和四边形单元)和任意多边形单元。考虑到本文发展方法与已有有限元程序的通用性,本工作中对于常规单元仍然直接使用高斯积分进行计算。其次,以Floater插值函数作为多面体单元的形函数,提出了三维任意形状多面体单元的径向积分法用于计算单元刚度阵和载荷向量。对于复杂的任意多面体单元,计算过程中,我们使用两次径向积分法进行对单元积分域的转换。第一次,使用径向积分法将任意多面体单元的体积分转换成沿着单元表面的面积分。类似于多边形单元的区分方式,将转换后的积分面区分成常规(三角形及四边形)积分面和其他复杂类型积分面。对于后者,我们再次将径向积分法进一步集成到单元积分中,将面积分转换成沿着单元边上的线积分。经过上述两次转化之后,对于带有多边形面的三维多面体单元,其单元刚度阵以及载荷向量的计算,最终转化为多面体棱边上的线积分之和。最后,使用两个多边形单元算例和三个多面体单元算例验证本文所提方法的计算精度和有效性。二维分片试验用于验证本文方法对任意多边形单元的计算精度；对带孔平板的分析用于验证径向积分法对四边形单元的计算精度；三维悬臂梁用于验证积分点数对计算精度的影响；三维分片试验用于验证本文方法对任意多面体单元的计算精度；削角立方八面体结构几何形状复杂,用于验证本文方法对于复杂多面体单元的计算精度。数值算例结果表明,在积分点数相同的情况下本文所提方法计算精度高于常用的三角化方法。需要指出的是,相比文献中给出的其他二维多边形单元及三维多面体单元的积分方法,本文发展的积分法在积分过程中不需要将多边形和多面体单元切割成小的三角形和四面体子单元,只需在常规单元和单元边上进行积分,程序实施简单、通用性强、计算精度高。
[Abstract]:With the diversification of the problems faced in the numerical simulation, the element used by the finite element method is no longer limited to the triangles, quadrilateral, tetrahedron and hexahedral elements. The arbitrary polygon and polyhedral elements are used as new element forms, and the complex model grid is divided because of the arbitrariness of the number of sides and the number of surfaces. It is more flexible and convenient than conventional units. It can better simulate the thermal and mechanical properties of materials and structures, and can better deal with complex problems such as fracture and crack propagation. However, arbitrary polygons and polyhedron elements are difficult to construct polynomial interpolation functions because of the arbitrary geometry of their geometry, resulting in integral calculation. In this paper, the following research is made on the problems of the difficulty of integrating the arbitrary polygon and the polyhedron element. First, the Wachpress interpolation function is used as the shape function of the polygon element, and the radial integral method of the two-dimensional polygon element is developed to calculate the stiffness matrix and the load vector of the unit. The polygon element with the shape and size of the polygon is used to calculate the area of the polygon element in the stiffness matrix and the load vector process by the radial integral method. It can be converted into a line integral along the boundary of the element to avoid the difficult problems caused by the irregular integral domain of the polygon element and the non multi term form of the integral function. The polygon element can be divided into regular elements (triangular element and quadrilateral element) and arbitrary polygon element. Considering the generality of this method and the existing finite element program, the Gauss integral is still used directly for the conventional unit in this work. Secondly, the Floater interpolation function is used as the form function of the polyhedral element. The radial integral method of the three-dimensional arbitrary shape polyhedron element is used to calculate the stiffness matrix and the load vector of the unit. For the complex arbitrary polyhedral element, we use the two radial integral method to convert the element integral domain. The first time, the volume of the arbitrary polyhedral element is divided by the path integral method. Change the area along the surface of the unit. Similar to the division of polygon units, the converted integral surface area is divided into conventional (triangular and quadrilateral) integral surfaces and other complex types of integration surfaces. For the latter, we further integrate the radial integral method into the unit integral, converting the area into the edge of the unit. After the above two transformation, the calculation of the element stiffness matrix and the load vector for the three-dimensional polyhedral element with polygon surface is finally converted into the sum of the line integral on the polyhedral edge. Finally, the calculation accuracy of the proposed method is verified by using two polygon elements and three polyhedral elements. And effectiveness. The two-dimensional slice test is used to verify the calculation accuracy of this method for arbitrary polygon units. The analysis of the plate with holes is used to verify the accuracy of the radial integral method for the quadrilateral element; the three-dimensional cantilever beam is used to verify the effect of the number of points on the calculation precision; the three dimensional slice test is used to verify the method of this paper. The computational accuracy of the polyhedron element and the complex geometric shape of the cuboid eight surface body are used to verify the calculation accuracy of this method for complex polyhedron elements. The numerical example shows that the calculation accuracy of this method is higher than the common triangulation method in the case of the same integral point number. It should be pointed out that the comparison of the literature is compared with the literature. The integral method of other two-dimensional polygon elements and three dimensional polyhedron elements is given in this paper. The integral method developed in this paper does not need to cut polygons and polyhedron elements into small triangles and tetrahedral subunits in the integration process. It only needs to be divided on the conventional unit and the edge of the unit. The program is simple, versatile and accurate. High.
【学位授予单位】：大连理工大学
【学位级别】：硕士
【学位授予年份】：2015
【分类号】：O241.82
本文编号：2152204
[Abstract]:With the diversification of the problems faced in the numerical simulation, the element used by the finite element method is no longer limited to the triangles, quadrilateral, tetrahedron and hexahedral elements. The arbitrary polygon and polyhedral elements are used as new element forms, and the complex model grid is divided because of the arbitrariness of the number of sides and the number of surfaces. It is more flexible and convenient than conventional units. It can better simulate the thermal and mechanical properties of materials and structures, and can better deal with complex problems such as fracture and crack propagation. However, arbitrary polygons and polyhedron elements are difficult to construct polynomial interpolation functions because of the arbitrary geometry of their geometry, resulting in integral calculation. In this paper, the following research is made on the problems of the difficulty of integrating the arbitrary polygon and the polyhedron element. First, the Wachpress interpolation function is used as the shape function of the polygon element, and the radial integral method of the two-dimensional polygon element is developed to calculate the stiffness matrix and the load vector of the unit. The polygon element with the shape and size of the polygon is used to calculate the area of the polygon element in the stiffness matrix and the load vector process by the radial integral method. It can be converted into a line integral along the boundary of the element to avoid the difficult problems caused by the irregular integral domain of the polygon element and the non multi term form of the integral function. The polygon element can be divided into regular elements (triangular element and quadrilateral element) and arbitrary polygon element. Considering the generality of this method and the existing finite element program, the Gauss integral is still used directly for the conventional unit in this work. Secondly, the Floater interpolation function is used as the form function of the polyhedral element. The radial integral method of the three-dimensional arbitrary shape polyhedron element is used to calculate the stiffness matrix and the load vector of the unit. For the complex arbitrary polyhedral element, we use the two radial integral method to convert the element integral domain. The first time, the volume of the arbitrary polyhedral element is divided by the path integral method. Change the area along the surface of the unit. Similar to the division of polygon units, the converted integral surface area is divided into conventional (triangular and quadrilateral) integral surfaces and other complex types of integration surfaces. For the latter, we further integrate the radial integral method into the unit integral, converting the area into the edge of the unit. After the above two transformation, the calculation of the element stiffness matrix and the load vector for the three-dimensional polyhedral element with polygon surface is finally converted into the sum of the line integral on the polyhedral edge. Finally, the calculation accuracy of the proposed method is verified by using two polygon elements and three polyhedral elements. And effectiveness. The two-dimensional slice test is used to verify the calculation accuracy of this method for arbitrary polygon units. The analysis of the plate with holes is used to verify the accuracy of the radial integral method for the quadrilateral element; the three-dimensional cantilever beam is used to verify the effect of the number of points on the calculation precision; the three dimensional slice test is used to verify the method of this paper. The computational accuracy of the polyhedron element and the complex geometric shape of the cuboid eight surface body are used to verify the calculation accuracy of this method for complex polyhedron elements. The numerical example shows that the calculation accuracy of this method is higher than the common triangulation method in the case of the same integral point number. It should be pointed out that the comparison of the literature is compared with the literature. The integral method of other two-dimensional polygon elements and three dimensional polyhedron elements is given in this paper. The integral method developed in this paper does not need to cut polygons and polyhedron elements into small triangles and tetrahedral subunits in the integration process. It only needs to be divided on the conventional unit and the edge of the unit. The program is simple, versatile and accurate. High.
【学位授予单位】：大连理工大学
【学位级别】：硕士
【学位授予年份】：2015
【分类号】：O241.82
【参考文献】
相关期刊论文 前1条
1 郭瑜超;黄河;何俊;吴存利;段世慧;;特殊多边形蜂窝结构有效传热系数研究[J];导弹与航天运载技术;2014年04期
，本文编号：2152204
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