《弹性力学》（双语版） 第七章 平面问题的差分解

f+-2 h2 Similarly we can get difference 845 formula in the line 4-0-2 11301913 (3) 2h 2f/2+f4-2f0 (4) 0 2h Fig 7-2 The above(1-(4) are the basic difference formulas thus we can get other difference formulas from them as follows (6+f)-(f3+f7)(5) Oxy 4h 11

11 (2) 2 2 1 3 0 0 2 2 h f f f x f + − =           Similarly,we can get difference formula in the line 4-0-2: (4) 2 2 (3) 2 2 4 0 0 2 2 2 4 0 h f f f y f h f f y f + − =           − =           The above（１）—（４）are the basic difference formulas， thus we can get other difference formulas from them as follows ： Fig.7-2 [( ) ( )] (5) 4 1 6 8 5 7 0 2 2 f f f f x y h f = + − +           

平二的盖分 01=+-2 h2 845 同理,在网线40-2上可得到差分公式: 11301913 (3) 2h 2f/2+f4-2f0 (4) 0 2h 图72 以上(1)-(4)是基本差分公式,从而可导出其它 的差分公式如下: (6+f)-(f3+f7)(5) axoy。4h 12

12 (2) 2 2 1 3 0 0 2 2 h f f f x f + − =           同理，在网线4-0-2上可得到差分公式： (4) 2 2 (3) 2 2 4 0 0 2 2 2 4 0 h f f f y f h f f y f + − =           − =           以上（１）—（４）是基本差分公式，从而可导出其它 的差分公式如下： 图7-2 [( ) ( )] (5) 4 1 6 8 5 7 0 2 2 f f f f x y h f = + − +           

DIRFERENCESOLUTLONTOTHEQUESTONSOFPLAIN [6J-4(f+f)+(+f1 5=方一20+++D)+(+++6 6f-4(2+f4)+(f0+f12) Difference formulas of(1) and(3) can be called as midpoint derivative formulas. because they use the function value of two crunodes whose interval is 2h to express the first derivative value id The formula which uses the function value of three border upon crunodes to express the first derivative value of a endpoint can be called endpoint derivative formula We must point out that midpoint derivative has a higher precision than endpoint because the former reflects the change of function of both sides of the crunodes. But the later only reflects one side of the crunodes. So we always try our best to use the former and 3 only use the later because we cant use the former

13 [6 4( ) ( )] 1 [4 2( ) ( )] (6) 1 [6 4( ) ( )] 1 4 0 2 4 1 0 1 2 0 4 4 4 0 1 2 3 4 5 6 7 8 0 2 2 4 4 0 1 3 9 1 1 0 4 4 f f f f f y h f f f f f f f f f f x y h f f f f f f x h f = − + + +           = − + + + + + + +            = − + + +           Difference formulas of (１) and (３) can be called as midpoint derivative formulas. Because they use the function value of two crunodes whose interval is 2h to express the first derivative value of the midpoint. The formula which uses the function value of three border upon crunodes to express the first derivative value of a endpoint can be called endpoint derivative formula. We must point out that midpoint derivative has a higher precision than endpoint. Because the former reflects the change of function of both sides of the crunodes. But the later only reflects one side of the crunodes. So we always try our best to use the former , and only use the later because we can’t use the former

平二的盖分 =,46f6-4(f+f)+(+千) Or h af ao2。h 46-2(+f2+f+f)+(+f+f1+/)6) h 16f6-42+f4)+(f0+f2) 差分公式(1)及(3)是以相隔2h的两结点处的函数值来表 示中间结点处的一阶导数值,可称为中点导数公式。 以相邻三结点处的函数值来表示一个端点处的一阶导数 值,可称为端点导数公式。 应当指出:中点导数公式与端点导数公式相比,精度较 高。因为前者反映了结点两边的函数变化,而后者却只反映 了结点一边的函数变化。因此,我们总是尽可能应用前者, 而只有在无法应用前者时才不得不应用后者

14 [6 4( ) ( )] 1 [4 2( ) ( )] (6) 1 [6 4( ) ( )] 1 4 0 2 4 1 0 1 2 0 4 4 4 0 1 2 3 4 5 6 7 8 0 2 2 4 4 0 1 3 9 1 1 0 4 4 f f f f f y h f f f f f f f f f f x y h f f f f f f x h f = − + + +           = − + + + + + + +            = − + + +           差分公式(１)及(３)是以相隔2h的两结点处的函数值来表 示中间结点处的一阶导数值，可称为中点导数公式。 以相邻三结点处的函数值来表示一个端点处的一阶导数 值，可称为端点导数公式。 应当指出：中点导数公式与端点导数公式相比，精度较 高。因为前者反映了结点两边的函数变化，而后者却只反映 了结点一边的函数变化。因此，我们总是尽可能应用前者， 而只有在无法应用前者时才不得不应用后者

DIRFERENCESOLUTLONTOTHEQUESTONSOFPLAIN 87-2 Difference Solution to Steady Temperature Field This section we discuss the no heat source, plane and, steady temperature field and explain the application of difference method In the no heat source plane and, steady temperature field 00 0 a=' ot= 0, so Heat conduction differential equation can be Smp/i fied as harmonic equation- V2T=0 aT a 0 (a) In order to use difference method, we make grids in the temperature field. Just as Fig.7-1o At any node, for example at node 0, we can get the follows from difference formula: 02T)71+73-2T ax2 h2 (b) 27)T2+-27 (c) 15 h

15 §7-2 Difference Solution to Steady Temperature Field This section we discuss the no heat source, plane and , steady temperature field and explain the application of difference method. In order to use difference method，we make grids in the temperature field. Just as Fig.7-1。At any node，for example at node 0，we can get the follows from difference formula： 2 1 3 0 0 2 2 2 h T T T x T + − =           2 2 4 0 0 2 2 2 h T T T y T + − =           （c） （b） In the no heat source, plane and, steady temperature field ,so Heat conduction differential equation can be simplified as harmonic equation, 0 2 2 2 2 =   +   y T x T （a） 0, 0, = 0   =   =   t T z T t  0 2  T =