【摘要】：彈性力學(xué)問(wèn)題可歸結(jié)為二階耦合橢圓形偏微分方程邊值問(wèn)題。工程中遇到的大部分問(wèn)題都難以得到其解析解。為求解彈性力學(xué)方程,工程實(shí)際中廣泛采用數(shù)值求解技術(shù)。本文提出數(shù)值分析平面彈性問(wèn)題的位移-應(yīng)力混合重心插值配點(diǎn)法。將彈性力學(xué)控制方程表達(dá)為位移和應(yīng)力的耦合偏微分方程組,采用重心插值近似未知量,利用重心插值微分矩陣得到平面問(wèn)題控制方程的矩陣形式離散表達(dá)式。使用重心插值離散位移和應(yīng)力邊界條件,采用附加法施加邊界條件,得到求解平面彈性問(wèn)題的過(guò)約束線性代數(shù)方程組,應(yīng)用最小二乘法求解過(guò)約束方程組,得到平面彈性問(wèn)題位移和應(yīng)力數(shù)值解。對(duì)于不規(guī)則區(qū)域的彈性力學(xué)問(wèn)題,采用重心Lagrange插值正則區(qū)域法,將不規(guī)則區(qū)域嵌入規(guī)則區(qū)域,在規(guī)則區(qū)域上采用重心Lagrange插值近似未知函數(shù)。利用配點(diǎn)法強(qiáng)迫微分方程在離散節(jié)點(diǎn)處精確成立,得到規(guī)則區(qū)域位移-應(yīng)力混合方程組。在不規(guī)則區(qū)域的邊界上取若干節(jié)點(diǎn),由規(guī)則區(qū)域內(nèi)的重心插值插值節(jié)點(diǎn)的未知函數(shù),得到一個(gè)邊界條件的約束代數(shù)方程。將位移-應(yīng)力混合方程的離散方程和邊界條件的約束方程組合成一個(gè)新的過(guò)約束代數(shù)方程組,應(yīng)用最小二乘法求解過(guò)約束方程組,得到平面彈性問(wèn)題位移和應(yīng)力數(shù)值解。本文提供的5個(gè)規(guī)則區(qū)域的數(shù)值算例和4個(gè)不規(guī)則區(qū)域的數(shù)值算例結(jié)果表明:重心Lagrange插值配點(diǎn)法和重心插值正則區(qū)域法的運(yùn)用,可以有效的解決規(guī)則區(qū)域和不規(guī)則區(qū)域的平面彈性問(wèn)題。重心Lagrange插值配點(diǎn)法不僅計(jì)算公式簡(jiǎn)單、節(jié)點(diǎn)適應(yīng)性好、程序通用性強(qiáng)、而且計(jì)算精度非常高。
[Abstract]:The elastic mechanics problem can be reduced to the boundary value problem of the second order coupled elliptic partial differential equation. Most of the problems encountered in the engineering are difficult to obtain its analytical solution. In order to solve elastic equation, numerical solution technology is widely used in engineering practice. In this paper, a displacement-stress mixed center of gravity interpolation method for numerical analysis of plane elastic problems is presented. The governing equations of elasticity are expressed as coupled partial differential equations of displacement and stress. The approximate unknown value of barycentric interpolation is used to obtain the matrix form discrete expression of governing equations of plane problems by using the differential matrix of barycentric interpolation. The boundary conditions of discrete displacement and stress are interpolated by the center of gravity, and the boundary conditions are imposed by the additional method. The overconstrained linear algebraic equations for solving the plane elastic problems are obtained, and the overconstrained equations are solved by the least square method. The displacement and stress numerical solutions of the plane elastic problem are obtained. For the elasticity problem of irregular regions, the barycentric Lagrange interpolation regular region method is used to embed irregular regions into regular regions, and the barycentric Lagrange interpolation is used to approximate unknown functions in regular regions. The collocation method is used to force the differential equation to be accurately established at the discrete nodes, and the displacement-stress mixed equations in the regular region are obtained. By taking some nodes on the boundary of irregular regions and from the unknown functions of barycentric interpolation nodes in regular regions, a constrained algebraic equation with boundary conditions is obtained. The discrete equation of the displacement-stress mixed equation and the constraint equation of the boundary conditions are combined into a new algebraic system of over-constraint. The numerical solution of displacement and stress of the plane elastic problem is obtained by using the least square method to solve the over-constrained equations. The numerical examples of five regular regions and four irregular regions are presented in this paper. The results show that the barycentric Lagrange interpolation method and the barycentric interpolation regular region method are used. It can effectively solve the plane elasticity problem of regular region and irregular region. The barycentric Lagrange interpolation collocation method not only has the advantages of simple calculation formula, good adaptability of nodes, strong generality of program, but also very high calculation precision.
【學(xué)位授予單位】：山東建筑大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2017
【分類(lèi)號(hào)】：O241.82;O343

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