發(fā)布時(shí)間：2018-12-12 23:13

【摘要】：弱形式求積元法（簡(jiǎn)稱求積元）把數(shù)值積分、數(shù)值微分與單元形式緊密地結(jié)合在一起，形成了一套獨(dú)具特色的數(shù)值計(jì)算方法。它針對(duì)問(wèn)題的弱形式描述，，在劃分可積域后直接引入數(shù)值積分和數(shù)值微分來(lái)離散問(wèn)題，得到代數(shù)方程組以求解問(wèn)題。由于采用了數(shù)學(xué)離散的思想，求積元法不用直接構(gòu)造形函數(shù)，在構(gòu)造高階單元等方面展現(xiàn)出其強(qiáng)大的能力。求積元法所構(gòu)造的高階板單元的計(jì)算效率很高，使用靈活方便，計(jì)算結(jié)果可靠，后處理精度高，是工程數(shù)值計(jì)算問(wèn)題的一個(gè)理想選擇。本文在課題組多年來(lái)對(duì)弱形式求積元法的研究基礎(chǔ)上研究了薄板等需要滿足邊界C1類連續(xù)的板問(wèn)題，主要內(nèi)容包括： 1、薄板的線性分析。利用Gauss-Lobatto積分、微分求積和廣義微分求積構(gòu)造了可處理任意四邊形問(wèn)題的薄板位移型單元，該單元能夠?qū)崿F(xiàn)單元組集、各種形式邊界條件的施加，并能夠滿足解的位移協(xié)調(diào)性要求。在與有限元計(jì)算結(jié)果的對(duì)比中，該單元顯現(xiàn)出了明顯的高計(jì)算效率。本文嘗試了通過(guò)單元重構(gòu)的方法進(jìn)一步增強(qiáng)求積單元的靈活性，成功地解決了不同單元組集的問(wèn)題。 2、高階板的線性分析。在薄板線性單元的基礎(chǔ)上構(gòu)造了針對(duì)Reddy三階板模型以及Kant三階板模型的求積單元，展示了弱形式求積元法對(duì)于高階板這類位移場(chǎng)較復(fù)雜的問(wèn)題同樣能夠得到準(zhǔn)確可靠的計(jì)算結(jié)果。通過(guò)對(duì)厚板的計(jì)算進(jìn)一步明確了高階板模型的應(yīng)用范圍。 3、薄板的幾何非線性分析。對(duì)于幾何非線性分析這種位移場(chǎng)連續(xù)性較好，且計(jì)算量大的問(wèn)題非常適合采用弱形式求積元進(jìn)行計(jì)算。本文利用弱形式求積元法計(jì)算了von Kárman薄板大撓度問(wèn)題，得到可靠的計(jì)算結(jié)果，計(jì)算效率明顯強(qiáng)于有限元。求積單元對(duì)于非線性屈曲問(wèn)題也能夠得到準(zhǔn)確的結(jié)果。 4、薄板的彈塑性分析�；谒苄栽隽坷碚摌�(gòu)造了用于薄板彈塑性分析的高階求積單元。該單元在計(jì)算理想彈塑性材料的薄板問(wèn)題時(shí)表現(xiàn)出強(qiáng)大的能力，甚至可以進(jìn)行極限分析。 數(shù)值算例表明求積元法在處理C1類連續(xù)板問(wèn)題時(shí)計(jì)算結(jié)果可靠、效率與傳統(tǒng)有限元相比優(yōu)勢(shì)明顯，且有著足夠的靈活性以滿足工程結(jié)構(gòu)的數(shù)值計(jì)算需要。
[Abstract]:The weak form quadrature element method (abbreviated as quadrature element) combines the numerical integral, numerical differential and the element form closely, and forms a set of unique numerical calculation methods. Aiming at the weak formal description of the problem, the numerical integral and numerical differential are directly introduced to discretize the problem after dividing the integrable domain, and the algebraic equations are obtained to solve the problem. Because of the idea of mathematical discretization, the quadrature element method does not need to construct the shape function directly, and it shows its powerful ability in constructing higher order element and so on. The high order plate element constructed by the quadrature element method is an ideal choice for engineering numerical calculation because of its high efficiency, flexibility and convenience, reliable calculation results and high post-processing accuracy. In this paper, based on the research of weak form quadrature element method in our research group, we study the problems of thin plates and other continuous plates which need to satisfy boundary C1 class. The main contents are as follows: 1. Linear analysis of thin plates. By using Gauss-Lobatto integral, differential quadrature and generalized differential quadrature, a thin plate displacement-type element which can deal with any quadrilateral problem is constructed. This element can realize the application of the set of elements and various forms of boundary conditions. And it can meet the displacement coordination requirement of the solution. Compared with the results of finite element calculation, the element shows obvious high computational efficiency. This paper attempts to further enhance the flexibility of quadrature units by means of cell reconstruction, and successfully solves the problem of different sets of units. 2. Linear analysis of higher order plates. The quadrature element for Reddy third order plate model and Kant third order plate model is constructed on the basis of thin plate linear element. It is shown that the weak form quadrature element method can also obtain accurate and reliable results for the more complicated problems of displacement field such as high order plates. Through the calculation of thick plate, the application scope of high order plate model is further clarified. 3. Geometric nonlinear analysis of thin plates. For the geometric nonlinear analysis, the continuity of the displacement field is good, and the problem of large amount of calculation is very suitable to use the weak form quadrature element to calculate the displacement field. In this paper, the weak form quadrature element method is used to calculate the large deflection problem of von K 謾 rman thin plate. The reliable results are obtained, and the computational efficiency is obviously higher than that of the finite element method. The quadrature element can also obtain accurate results for nonlinear buckling problems. 4. Elastoplastic analysis of thin plates. Based on the theory of plastic increment, a high order quadrature element for elastoplastic analysis of thin plates is constructed. The element can be used to calculate the thin plate problem of ideal elastic-plastic material, and it can even be used for limit analysis. Numerical examples show that the quadrature element method is reliable in dealing with C _ 1 continuous plate problems, has obvious advantages over the traditional finite element method, and has sufficient flexibility to meet the needs of numerical calculation of engineering structures.
【學(xué)位授予單位】：清華大學(xué)
【學(xué)位級(jí)別】：博士
【學(xué)位授予年份】：2013
【分類號(hào)】：TU311.4

【參考文獻(xiàn)】

相關(guān)期刊論文 前3條

1 李明瑞;梁板殼的幾何大變形——從近似的非線性理論到有限變形理論[J];力學(xué)與實(shí)踐;2003年03期

2 王永亮，王鑫偉;多外載聯(lián)合作用下圓板的非線性彎曲[J];南京航空航天大學(xué)學(xué)報(bào);1996年01期

3 夏飛;伍小平;李鴻晶;;基于弱形式求積元的Timoshenko梁彈塑性分析[J];信陽(yáng)師范學(xué)院學(xué)報(bào)(自然科學(xué)版);2010年04期

本文編號(hào)：2375410