【摘要】：隨著數(shù)值模擬中所面臨的問題的多樣化,要求有限元方法使用的單元不再局限于三角形、四邊形、四面體以及六面體單元等規(guī)則單元。任意多邊形單元和多面體單元作為新的單元形式,由于其邊數(shù)和面數(shù)的任意性,在劃分復雜模型網(wǎng)格時比常規(guī)單元更加靈活方便,能夠更好地模擬材料與結(jié)構(gòu)的熱、力學性能,并可以能夠更好地處理一些如斷裂,裂紋擴展等復雜問題。然而,任意多邊形和多面體單元因其幾何形狀的任意性很難構(gòu)造出多項式形式插值函數(shù),導致積分計算單元剛度陣和載荷向量時比較困難。本文針對任意多邊形及多面體單元積分困難的問題,做了以下研究：首先,以Wachpress插值函數(shù)作為多邊形單元的形函數(shù),發(fā)展了二維多邊形單元的徑向積分法用于計算單元剛度陣和載荷向量。對于任意形狀及尺寸的多邊形單元,使用徑向積分法將多邊形單元計算剛度陣和載荷向量過程中的面積分,轉(zhuǎn)化成沿著單元邊界的線積分進行計算,避免因多邊形單元積分域不規(guī)則以及積分函數(shù)的非多項式形式帶來的難題。實際工程應用當中,多邊形單元可區(qū)分為常規(guī)單元(三角形單元和四邊形單元)和任意多邊形單元�？紤]到本文發(fā)展方法與已有有限元程序的通用性,本工作中對于常規(guī)單元仍然直接使用高斯積分進行計算。其次,以Floater插值函數(shù)作為多面體單元的形函數(shù),提出了三維任意形狀多面體單元的徑向積分法用于計算單元剛度陣和載荷向量。對于復雜的任意多面體單元,計算過程中,我們使用兩次徑向積分法進行對單元積分域的轉(zhuǎn)換。第一次,使用徑向積分法將任意多面體單元的體積分轉(zhuǎn)換成沿著單元表面的面積分。類似于多邊形單元的區(qū)分方式,將轉(zhuǎn)換后的積分面區(qū)分成常規(guī)(三角形及四邊形)積分面和其他復雜類型積分面。對于后者,我們再次將徑向積分法進一步集成到單元積分中,將面積分轉(zhuǎn)換成沿著單元邊上的線積分。經(jīng)過上述兩次轉(zhuǎn)化之后,對于帶有多邊形面的三維多面體單元,其單元剛度陣以及載荷向量的計算,最終轉(zhuǎn)化為多面體棱邊上的線積分之和。最后,使用兩個多邊形單元算例和三個多面體單元算例驗證本文所提方法的計算精度和有效性。二維分片試驗用于驗證本文方法對任意多邊形單元的計算精度；對帶孔平板的分析用于驗證徑向積分法對四邊形單元的計算精度；三維懸臂梁用于驗證積分點數(shù)對計算精度的影響；三維分片試驗用于驗證本文方法對任意多面體單元的計算精度；削角立方八面體結(jié)構(gòu)幾何形狀復雜,用于驗證本文方法對于復雜多面體單元的計算精度。數(shù)值算例結(jié)果表明,在積分點數(shù)相同的情況下本文所提方法計算精度高于常用的三角化方法。需要指出的是,相比文獻中給出的其他二維多邊形單元及三維多面體單元的積分方法,本文發(fā)展的積分法在積分過程中不需要將多邊形和多面體單元切割成小的三角形和四面體子單元,只需在常規(guī)單元和單元邊上進行積分,程序?qū)嵤┖唵巍⑼ㄓ眯詮�、計算精度高�?br/>[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.
【學位授予單位】：大連理工大學
【學位級別】：碩士
【學位授予年份】：2015
【分類號】：O241.82

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相關(guān)期刊論文 前1條

1 郭瑜超;黃河;何俊;吳存利;段世慧;;特殊多邊形蜂窩結(jié)構(gòu)有效傳熱系數(shù)研究[J];導彈與航天運載技術(shù);2014年04期

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本文編號：2152204