一種提高平板型殼單元計(jì)算精度的改進(jìn)算法研究
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本文關(guān)鍵詞：一種提高平板型殼單元計(jì)算精度的改進(jìn)算法研究 出處：《大連理工大學(xué)》2016年博士論文 論文類型：學(xué)位論文
更多相關(guān)文章： 有限元 平板型殼單元 單元?jiǎng)偠染仃?/b> 等效節(jié)點(diǎn)載荷 同時(shí)消元回代法
【摘要】：殼體結(jié)構(gòu)是工程應(yīng)用中常見(jiàn)的結(jié)構(gòu)形式之一,廣泛應(yīng)用于土木、水利、汽車、航空航天等各個(gè)工程領(lǐng)域中。由于殼體是具有薄膜和彎曲兩重受力特性的三維特殊結(jié)構(gòu),其控制方程是一組形式復(fù)雜的偏微分方程,直接解析求解是比較困難的。在電子計(jì)算機(jī)出現(xiàn)以前只能計(jì)算一些結(jié)構(gòu)簡(jiǎn)單規(guī)則的問(wèn)題,隨著以計(jì)算機(jī)為工具的有限元方法出現(xiàn)以后,經(jīng)過(guò)眾多學(xué)者的努力,不同類型的殼單元也相繼出現(xiàn)。其中,平板型殼單元是最早發(fā)展起來(lái)的殼單元,它是將殼體離散為一系列折板組成的體系,通過(guò)平面膜單元和平板彎曲單元疊加組合來(lái)模擬殼體的薄膜和彎曲受力狀態(tài)。平板型殼單元以其表達(dá)格式簡(jiǎn)單、應(yīng)用廣泛、計(jì)算效率高、計(jì)算結(jié)果可靠等優(yōu)點(diǎn)在不同類型的殼單元中占有重要的地位,在工程實(shí)際中得到了廣泛的應(yīng)用。但是,平板型殼單元是由平面應(yīng)力單元和平板彎曲單元疊加組合而成的殼單元,采用幾何近似性,用折板面代替殼體曲面。因此,在計(jì)算不規(guī)則殼體結(jié)構(gòu)特別是曲殼結(jié)構(gòu)時(shí)不能很好地模擬曲殼形狀,從而引起較大的誤差,需要將網(wǎng)格合理加密才能達(dá)到一定的精度,這會(huì)引起較大的計(jì)算量。為了進(jìn)一步提高平板型殼單元的計(jì)算性能,本文提出了一種新的單元局部坐標(biāo)系建立方法。該方法在單元的高斯積分點(diǎn)建立多個(gè)局部坐標(biāo)系,并保證每個(gè)局部坐標(biāo)系都位于單元在高斯點(diǎn)處的切平面上,對(duì)每個(gè)單元?jiǎng)偠绒D(zhuǎn)換矩陣進(jìn)行及時(shí)修正,從而可以有效適應(yīng)曲面殼體形狀。為了能夠在新的局部坐標(biāo)系下求得和單元?jiǎng)偠扔嘘P(guān)的參數(shù),同時(shí)又避免頻繁的坐標(biāo)轉(zhuǎn)換造成的計(jì)算誤差,本文還引入了計(jì)算形函數(shù)對(duì)局部坐標(biāo)的導(dǎo)數(shù)和積分轉(zhuǎn)換的雅可比行列式的高精度算法。這種高精度算法直接基于三維曲面整體坐標(biāo)系推導(dǎo)得出,具有較高的計(jì)算精度。利用本文提出的改進(jìn)算法計(jì)算平板型殼單元的剛度矩陣以及等效節(jié)點(diǎn)載荷列陣,都取得了良好的效果。此外,本文還將一種新的線性方程組求解器應(yīng)用到平板型殼單元當(dāng)中,進(jìn)一步提高其計(jì)算性能。本文的主要研究?jī)?nèi)容可以歸納如下：(1)提出了一種計(jì)算平板型殼單元在整體坐標(biāo)系下的單元?jiǎng)偠染仃嚨母倪M(jìn)算法,并將其應(yīng)用到幾種不同類型的平板型殼單元當(dāng)中。通過(guò)將幾種不同類型的平面膜單元和平板彎曲單元疊加組合形成平板型殼單元,對(duì)其計(jì)算性能進(jìn)行討論,并詳細(xì)給出了利用本文的改進(jìn)算法計(jì)算單元?jiǎng)偠染仃嚨睦碚摴?從原理上對(duì)本文方法在提高平板型殼單元計(jì)算性能方面進(jìn)行分析討論。(2)對(duì)本文所提出的改進(jìn)算法在平板型殼單元中的應(yīng)用和實(shí)現(xiàn)進(jìn)行程序開(kāi)發(fā),給出了詳細(xì)的編程流程,并通過(guò)算例對(duì)其計(jì)算精度進(jìn)行比較分析。為了對(duì)本文提出的改進(jìn)算法的計(jì)算性能進(jìn)行驗(yàn)證,將改進(jìn)算法應(yīng)用到幾種典型的平板型殼單元,并對(duì)其進(jìn)行程序?qū)崿F(xiàn),通過(guò)具體的算例對(duì)其正確性和精確性進(jìn)行驗(yàn)證分析。(3)將本文提出的改進(jìn)算法應(yīng)用擴(kuò)展到平板型殼單元等效節(jié)點(diǎn)載荷的計(jì)算上。對(duì)平板型殼單元在不同載荷下等效節(jié)點(diǎn)載荷的計(jì)算進(jìn)行歸納總結(jié),給出了利用本文方法計(jì)算其等效節(jié)點(diǎn)載荷的詳細(xì)的理論公式,并利用程序編制對(duì)其應(yīng)用進(jìn)行數(shù)值實(shí)現(xiàn),通過(guò)具體的算例對(duì)本文方法在計(jì)算等效節(jié)點(diǎn)載荷方面的性能進(jìn)行分析驗(yàn)證。(4)在有限元分析中,引入了一種新的線性方程組直接法求解器——同時(shí)消元回代法,提出了新的剛度矩陣組集方法,并將其應(yīng)用到平板型殼單元問(wèn)題的求解中。給出了新的剛度矩陣的組集方法和詳細(xì)計(jì)算步驟,并結(jié)合同時(shí)消元回代法求解器對(duì)其求解性能進(jìn)行分析討論,給出了程序的編制過(guò)程,通過(guò)具體算例對(duì)其性能進(jìn)行驗(yàn)證。綜上所述,本文提出了一種能夠用于提高平板型殼單元計(jì)算精度的改進(jìn)算法,為平板型殼單元計(jì)算性能的提高提供了一種新的手段。本文所述方法也可以應(yīng)用到其他類型的四邊形平板型殼單元中,具有廣闊的發(fā)展前景。
[Abstract]:The shell structure is one of the common structure in engineering application, widely used in civil engineering, water conservancy, automotive, aerospace and other engineering fields. Because the shell is a special three-dimensional structure stress and bending properties of thin film double, the control equation is a partial differential equation of a group of complex forms, direct analytical solution is more difficult. Some simple rules can only be calculated before problems occur in the electronic computer, with finite element method with the computer as a tool, through the efforts of numerous scholars, shell elements of different types have emerged. Among them, flat shell element is the shell element developed early, it is the case for discrete a series of folded plate system, by plane membrane element and plate bending element combination to simulate the shell membrane and bending stress. Flat shell element lattice with its expression Simple, wide application, high computational efficiency, reliable calculation plays an important role in the shell elements of different types, has been widely used in engineering practice. However, flat shell element is shell element from the plane stress element and plate bending element combination and using geometric approximation of the folding surface instead of curved shell. Therefore, in the calculation of irregular shell structure especially the shell structure can well simulate the curved shell shape, which causes the great error, will need to reach a reasonable grid encryption precision, it will cause a large amount of calculation. In order to further improve the computational performance of the flat plate the shell element, the paper puts forward a new method of element local coordinate. The method established a local coordinate system in units of the Gauss integral points, and to ensure that each local coordinate system are located in the unit The tangent plane at Gauss point, timely correction of each element stiffness matrix conversion, which can effectively adapt to the surface of the shell shape. In order to obtain new parameters in local coordinates and unit stiffness, and avoid frequent calculation error caused by coordinate transformation, this paper also introduces high precision algorithm for calculating the shape functions of the local coordinate of the derivative and integral transformation of Jacobi determinant. Derived 3D coordinates the overall accuracy of the algorithm directly based on high precision. Using this improved algorithm to calculate flat shell element stiffness matrix and the equivalent nodal load matrix, have achieved good results. In addition, this paper also will be a new linear equation solver is applied to the flat shell element, further improve its performance. The main research work of this paper Content can be summarized as follows: (1) put forward an improved algorithm for calculating flat shell element in the global coordinate system of the element stiffness matrix, and applied to several different types of flat shell element. The plane membrane element and plate bending element combination of several different types of flat form shell element, the calculation of performance are discussed, and gives the detailed theoretical calculation formula of element stiffness matrix using the improved algorithm, from the principle of this method in improving the performance of flat shell element are discussed. (2) and the improved algorithm of the proposed application in flat shell the unit of program development, detailed programming process is given, and an example of the calculation accuracy were analyzed. In order to calculate the performance of the proposed algorithm is verified, will be changed In the algorithm is applied to several typical flat shell element, and carries on the program, through a numerical example to validate its correctness and accuracy. (3) to expand the application of improved algorithm is proposed in this paper to calculate the flat shell element equivalent node loads. The calculation of flat shell element under different load equivalent node loads are summarized, with the theoretical formula for calculating the equivalent node load is given by using this method, and the application program of the numerical implementation, through specific examples to verify the feasibility of this method in the calculation of equivalent node load performance aspects. (4) in the Co. element analysis, introduced a new direct method of linear equations solver and elimination back generation method, puts forward the stiffness matrix assembly method, and its application to the flat shell element problem In solving the stiffness matrix is given. The new assembly method and the detailed calculation steps, and at the same time combined with the elimination back solver are discussed on the generation method of solving the performance, preparation process are given, through specific examples to verify its performance. In summary, this paper presents a can be used for the improved algorithm can improve the calculation precision flat shell element, which provides a new method for the flat shell element to improve the performance. The method can also be applied to the quadrilateral flat shell element of other types, and has broad prospects for development.
【學(xué)位授予單位】：大連理工大學(xué)
【學(xué)位級(jí)別】：博士
【學(xué)位授予年份】：2016
【分類號(hào)】：TP301.6
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