《弹性力学》（双语版） 第七章 平面问题的差分解

平二的盖分 弹性力学的经典解法存在一定的局限性,当弹性体的边 界条件和受载情况复杂一点,往往无法求得偏微分方程的边 值问题的解析解。因此,各种数值解法便具有重要的实际意 义。差分法就是数值解法的一种 所谓差分法,是把基本方程和边界条件(一般均为微分 方程)近似地改用差分方程(代数方程)来表示,把求解 微分方程的问题改换成为求解代数方程的问题

6 弹性力学的经典解法存在一定的局限性，当弹性体的边 界条件和受载情况复杂一点，往往无法求得偏微分方程的边 值问题的解析解。因此，各种数值解法便具有重要的实际意 义。差分法就是数值解法的一种。 所谓差分法，是把基本方程和边界条件（一般均为微分 方程）近似地改用差分方程（代数方程）来表示，把求解 微分方程的问题改换成为求解代数方程的问题

DIRFERENCESOLUTLONTOTHEQUESTONSOFPLAIN 87-1 Derivation of Difference Formulation We make a square grid on the surface of elastic object, by using two group lines which are parallel to the coordinate axes and the distance of two parallel lines Is h. Shown in Fig. 7-1 Suppose ff(x,y) is a continue function in elastic object. This function is in a line which is parallel to x axes Fig-1 For example it is in 3-0-1. It only changes with the change of coordination of x axes function f can be opened up into taylor series in the neighbor of point 0: (x-x)+ f=fo+ax o 2!(、ax2 (x-x0)+ (x-x0) 0 7

7 §7-1 Derivation of Difference Formulation We make a square grid on the surface of elastic object，by using two group lines which are parallel to the coordinate axes and the distance of two parallel lines is h . Shown in Fig. 7-1. Suppose f=f(x,y) is a continue function in elastic object . This function is in a line which is parallel to x axes. Fig.7-1 ( ) ... 3! 1 ( ) 2! 1 ( ) 3 0 0 3 3 2 0 0 2 2 0 0 0 − +           − +            − +        = + x x x f x x x f x x x f f f For example it is in 3-０-１. It only changes with the change of coordination of x axes . function f can be opened up into taylor series in the neighbor of point 0：

平间二的盏分拿 §7-1差分公式的推导 我们在弹性体上,用相隔等间距 h而平行于坐标轴的两组平行线织成正 方形网格,如图7-1。 设f=fxy)为弹性体内的某一个连 续函数。该函数在平行于x轴的一根网 线上,例如在3-0-1上,它只随x坐 标的改变而变化。在邻近结点0处, 函数何展为泰勒级数如下: 图7-1 =+(9)(x-x)+ (x-x0)-+ (x-x0)3+ Ox o 2!(a 3 8

8 §7-1 差分公式的推导 我们在弹性体上，用相隔等间距 h而平行于坐标轴的两组平行线织成正 方形网格，如图7-1。 设f=f(x,y)为弹性体内的某一个连 续函数。该函数在平行于x轴的一根网 线上，例如在3-０-１上，它只随x坐 标的改变而变化。在邻近结点０处， 函数f可展为泰勒级数如下： 图7-1 ( ) ... 3! 1 ( ) 2! 1 ( ) 3 0 0 3 3 2 0 0 2 2 0 0 0 − +           − +            − +        = + x x x f x x x f x x x f f f

DIRFERENCESOLUTLONTOTHEQUESTONSOFPLAIN We will only think of those points which are very near to point 0. It means that x-xo is sufficient small. So three or more power of (X-xo) can be eliminated. The above formulation can be simplified as f=fo (x-xo+ (x-x0)2 (b) 2!(ax At point 3, xxo-h; at point 1, x-Xo +h We can get from (b) =n/0)h2(03f 十 ax)。2(ax (c) sf=fo+ af hlaf oo We can get the difference formula from(c)and(d) af fi-f 2h 9

9 We will only think of those points which are very near to point 0. It means that x-x0 is sufficient small. So three or more power of（x-x0）can be eliminated .The above formulation can be simplified as： ( ) ( ) 2! 1 ( ) 2 0 0 2 2 0 0 0 x x b x f x x x f f f −            − +        = + At point 3，x=x0 -h；at point 1, x=x0+h.We can get from (b)： ( ) 2 0 2 2 2 0 3 0 c x h f x f f f h            +        = − ( ) 2 0 2 2 2 0 1 0 d x h f x f f f h            +        = + We can get the difference formula from (c) and (d)： (1) 2 1 3 0 h f f x f −  =       

平二的盖分 我们将只考虑离开结点0充分近的那些结点,即(xx0) 充分小。于是可不计(xx0)的三次及更高次幂的各项,则上 式简写为: f=+(x-x)+ Ox (ar2/(x-x)2 在结点3,xx0-h;在结点1,xx0+h。代入(b)得: f3=f6 0)h2(02f 十 ax)。2(ax (c) =+19)+h2(a2 (d) 联立(c)、(d),解得差分公式: af fi-f 2h 10

10 我们将只考虑离开结点０充分近的那些结点，即（x-x0） 充分小。于是可不计（x-x0）的三次及更高次幂的各项，则上 式简写为： ( ) ( ) 2! 1 ( ) 2 0 0 2 2 0 0 0 x x b x f x x x f f f −            − +        = + 在结点３，x=x0 -h；在结点1， x=x0+h。代入(b) 得： ( ) 2 0 2 2 2 0 3 0 c x h f x f f f h            +        = − ( ) 2 0 2 2 2 0 1 0 d x h f x f f f h            +        = + 联立（c）、（d），解得差分公式： (1) 2 1 3 0 h f f x f −  =       