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1、1,Chapter 7 Diffenence Solution to the Question of Plain,2,第七章 平面问题的差分解,3,Chapter 7 Difference Solution to the Questions of Plane,DIFFERENCE SOLUTION TO THE QUESTIONS OF PLAIN,7-1 Derivation of Difference Formula,7-2 Difference Solution to Steady Temperature Field,7-3 Difference Solution to Unsteady
2、 Temperature Field,7-4 Difference Solution to Stress Function,7-5 Example of Difference Solution to Stress Function,7-6 Difference Solution of Stress Function to the Question of Temperature Stress,7-7 Difference Solution to Displacement,7-8 Example of Difference Solution to Displacement,7-9 Displace
3、ment Difference Solution to more Continuous Object,Exercise of Difference Solution to Plane Questions ,4,第七章 平面问题的差分解,平面问题的差分解,7-1 差分公式的推导,7-2 稳定温度场的差分解,7-3 不稳定温度场的差分解,7-4 应力函数的差分解,7-5 应力函数差分解的实例,7-6 温度应力问题的应力函数差分解,7-7 位移的差分解,7-8 位移差分解的实例,7-9 多连体问题的位移差分解,习题课,5,The typical solutions to the theory of
4、elasticity have a certain limits. When the elastic objects boundary conditions and loads are a little complex ，always the rigorous solution to boundary questions of the partial differential equations cant be found。Thus the numerical solutions have an important practical meaning。Difference solution i
5、s one of the numerical solutions。,Difference solution is a method that uses difference equations (algebra equations) instead of basic equations and boundary conditions (sometimes they are differential equations), and translates the solutions to differential equations into algebra equations.,DIFFEREN
6、CE SOLUTION TO THE QUESTIONS OF PLAIN,6,平面问题的差分解,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。,7,7-1 Derivation of Difference Formulation,We make a square grid on the surface of elast
7、ic 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,For example it is in 3-. It only c
8、hanges with the change of coordination of x axes . function f can be opened up into taylor series in the neighbor of point 0：,DIFFERENCE SOLUTION TO THE QUESTIONS OF PLAIN,8,7-1 差分公式的推导,平面问题的差分解,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-上，它只随x坐标的改变而变化。在邻近
9、结点处，函数f可展为泰勒级数如下：,图7-1,9,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：,At point 3，x=x0-h；at point 1, x=x0+h.We can get from (b)：,We can get the d
10、ifference formula from (c) and (d)：,DIFFERENCE SOLUTION TO THE QUESTIONS OF PLAIN,10,平面问题的差分解,我们将只考虑离开结点充分近的那些结点，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：,在结点，x=x0-h；在结点1， x=x0+h。代入(b) 得：,联立（c）、（d），解得差分公式：,11,Similarly,we can get difference formula in the line 4-0-2:,The above（）（）are the basic diffe
11、rence formulas，thus we can get other difference formulas from them as follows ：,Fig.7-2,DIFFERENCE SOLUTION TO THE QUESTIONS OF PLAIN,12,平面问题的差分解,同理，在网线4-0-2上可得到差分公式：,以上（）（）是基本差分公式，从而可导出其它的差分公式如下：,图7-2,13,Difference formulas of () and () can be called as midpoint derivative formulas. Because they us
12、e 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 th
13、at 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 cant use the former.,DI
14、FFERENCE SOLUTION TO THE QUESTIONS OF PLAIN,14,平面问题的差分解,差分公式()及()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。,15,7-2 Difference Solution to Steady Temperature Field,This
15、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：,（c
16、）,（b）,DIFFERENCE SOLUTION TO THE QUESTIONS OF PLAIN,16,7-2 稳定温度场的差分解,平面问题的差分解,本节以无热源的、平面的、稳定的温度场为例，说明差分法的应用。,在无热源的平面稳定场中， ，所以热传导微分方程简化为调和方程 ，即：,（a）,为了用差分法求解，在温度场的域内织成网格，如图7-1所示。在任意一个结点，如在结点0，由差分公式有：,（c）,（b）,17,Substitute it into ，we can get difference equation：,（1）,（1）If all the boundary conditions of a temperature field have first boundary condition，then we
