Numerical Analysis Using M#math

Transcription

1 "msharpmath" revised on Numerical Analysis Using M#math Chapter 9 Eigenvalue Problems 9-1 Existence of Solution and Eigenvalues 9-2 Sturm-Liouville Problems 9-3 Eigenvalues on Stability and Convergence Rate 9-4 Exercises ============================== 9-5 Higher-order ODE and Eigenvalue Problems (optional) Eigenvalue problems occurring in electromagnetics, vibration, structure analysis and so on are related to the boundary value problems of ordinary differential equations. The eigenvalue-finding methods discussed in Chapter 4 provide the necessary platform to solve the eigenvalue problems of ordinary differential equations. Nevertheless, eigenvalues for relevant to an matrix consist of at most eigenvalues. This is in contrast to the fact that the eigenvalue problems are associated with an infinite number of eigenvalues. Therefore, there occurs an intrinsic limitation in obtaining eigenvalues for eigenvalue problems such as the well-known Sturm-Liouville problems. However, fortunately the fundamental eigenvalue that has the smallest absolute value is the most important in applications. And engineering importance diminishes as the absolute value of eigenvalues increases. In this regard, the methods treated in this chapter are of practical importance even though only a finite number of eigenvalues can be determined. First of all, discussed is the situation where an infinite number of solutions are existing. Next, the eigenvalue problems are properly defined. Then, numerical approaches to obtain eigenvalues are discussed. In doing this, the generalized nonlinear differential operator adopted earlier 1

2 is exploited again and the eigenvalue problems are expressed in the following compact form In the last section, the role of eigenvalues is explored in the initial value problems and in the iteration methods. //====================================================================== // syntax in M#math // Umbrella bvp is designed to solve BVPs the general forms of which are cz*{f''''} +cw*{f'''} +cfw*{ff'''} +cuw*{f'f'''} + cvw*{f''f'''} + cww*{f'''f'''} +cv*{f''} +cfv*{ff''} +cuv*{f'f''} + cvv*{f''f''} +cu*{f'} +cfu*{ff'} +cuu*{f'f'} +cf*{f} +cff*{ff} = cs bw*{f'''} + bv*{f''} + bu*{f'} + bf*{f} = bs Syntax bvp.x[n=101,g=1](a,b) ( DE, [ BCs ] ).Spoke.Spoke... x = (a,b).span(n,g=1); bvp [x] ( DE, [BCs] ).Spoke.Spoke... Spokes.peep.relax(relax=0.5).tol/abstol(abstol=1.e-5).reltol(reltol=1.e-5).maxiter(maxiter=500).shift(scalar).plot.plot(f,f',f'',...).return (f,f',f'',...) // peep the iteration procedure // under-relaxation factor // absolute tolerance // relative tolerance // maximum iteration count // eigenvalue shift // plot default data // plot user-defined data // return user-defined data 2

3 .togo (Y=y,Y1=y',...) // takeout each field Section 9-1 Existence of Solution and Eigenvalues In this section, the existence of an infinite number of solutions are discussed focusing on second-order ODEs, and the eigenvalue problems are defined. Existence of an infinite number of solutions. As was discussed in Chapter 8, solutions to boundary value problems are either 1 unique, 2 nonexisting, or 3 of an infinite number. Nevertheless, despite the nonexistence of mathematical solution, a meaningless numerical solution is always obtained due to the inherent limitation of computing with finite number of digits, as was confirmed in Example 8-5. As a beginning step, let us investigate the case where an infinite number of solutions are existing. For a particular second-order ODE, there are two types of boundary conditions with which an infinite number of solutions are obtained. infinite number of solutions with multiple integration constants infinite number of solutions with a single integration constant For the cases of equations (1) and (2), only a trivial solution is obtained numerically. This is because both the ODE and boundary conditions are homogeneous. Homogeneous conditions and eigenvalue problems. As was discussed above, the ODE with non-homogeneous boundary conditions cannot have a trivial solution. In this case, only one non-trivial solution is (1) (2) (3) 3

4 numerically obtained even though there exist an infinite number of solutions [the case of equation (3)]. The eigenvalue problems for the ODE over an interval treat only homogeneous boundary conditions and only nontrivial solutions. In the above, are not zero simultaneously. The homogeneous boundary condition guarantees that a constant multiple of an eigenfunction is also an eigenfunction. Of course, the ODE relevant to the eigenvalue problem is type of homogeneous differential equation. Eigenvalue problems, definition and property. Especially for the secondorder ODE, the eigenvalue problems are generalized as the Sturm-Liouville problem the detail of which will be discussed in the next section. Here, the characteristics of eigenvalue problems are addressed briefly. Let us find the mathematical solution to the following second-order ODE with homogeneous differential equation and boundary conditions (4) (5) First for the case of, the general solution is given as so that from the boundary condition. And from the boundary condition, we must have. However, since, it follows that. In other words, a trivial solution is the only possible solution. Next for the case of, from the general solution and from the boundary conditions, the only possible solution is found to be the trivial solution. Last for the case of, the general solution (6) (7) 4

5 is utilized to find from the boundary condition. This means that. And the remaining boundary condition enforces that (8) which can be satisfied by. At this point, the corresponding value is called the eigenvalue, and the corresponding solution is called the eigenfunction. Meanwhile, it is conventional that the integration constant is set to be unity, unless otherwise specified. From the above simple example, the eigenvalue problems are understood to exhibit the following characteristics. homogeneous ODE homogeneous boundary condition infinite number of eigenvalues eigenfunction corresponding to an eigenvalue, a constant multiple of, is also an eigenfunction the fundamental eigenvalue of smallest absolute value Next, let us investigate the trend of eigenfunctions in response to the increase of their absolute values. The graphical solution to the above-studied eigenvalue problem (9) is shown in Figure

6 Figure 9-1 Eigenfunctions corresponding to The eigenfunction corresponding to the smallest absolute eigenvalue is especially termed to be the fundamental eigenfunction. As shown in Figure 9-1, the fundamental eigenfunction does not have any root that satisfies inside the domain. Similarly, the second, third and fourth eigenfunctions have one, two and three roots of, respectively, inside the domain. This property is valid for other types of boundary conditions. For example, the profiles of eigenfunctions corresponding to the boundary conditions are displayed in Figure 9-2. It can be easily seen that the number of roots of is the same as that shown in Figure 9-1. Figure 9-2 Eigenfunctions corresponding to Section 9-2 Sturm-Liouville Problems The Sturm-Liouville problem, the most famous eigenvalue problem of second-order ODE, is studied in this section due to its practical importance in engineering and science. 6

7 Self-adjoint form. A standard second-order linear differential equation written as (10) can be recast into the so-called self-adjoint form by multiplying to both sides of the equation (11) The above self-adjoint form serves as the generic form producing an infinite number of orthogonal functions, and is used to define the Sturm-Liouville problem. Table 9-1 Sturm-Liouville Problem and eigenfunction expansion differential equation: boundary conditions: at each side or at both sides (periodic) for, is a real continuous function for an interval,, eigenvalue: all the eigenvalues are real and of infinite number eigenfunction: two distinct eigenfunctions are orthogonal, eigenfunction expansion: Sturm-Liouville problem. For the interval equation is defined by the following self-adjoint form, the Sturm-Liouville (12) 7

8 where a parameter is called the eigenvalue. In the above, are all real-valued and continuous over an interval. Especially over an interval. The self-adjoint form in equation (12) and the boundary conditions as given in Table 9-1 consist of the boundary value problem called the Sturm- Liouville problem. Frequently, the conditions 1, 2, 3 in Table 9-1 are combined together as (13) where and are not zeros simultaneously. The condition 4 in Table 9-1 is related to the singular Sturm-Liouville problem, and the condition 5 to the periodic Sturm-Liouville problem. In contrast, the regular Sturm-Liouville problem is defined by the conditions 1, 2 and 3. Discretization of Sturm-Liouville equation. The following Sturm- Liouville equation (14) can be discretized for the internal grids such that (15) where the central difference approximation over the interval shown in Figure 9-3 is employed. By rearranging, we have (16) which is further standardized to represent a typical eigenvalue problem (17) 8

9 Then, by utilizing the tridiagonal form of matrix found by the inverse power method., eigenvalues can be easily Figure 9-3 Grid configuration for discretization of Sturm-Liouville equation (Example 1) For a Sturm-Liouville problem,,, find the fundamental eigenvalue and the fundamental eigenfunction by numerical treatment. The given ODE belongs to the Sturm-Liouville problem and is characterized by. Re-writing equation (16) and inserting gives (18) where 5 uniform subintervals are used, i.e.. And the boundary condition at can be discretized from equation (8-47) such that or (19) 9

10 Also, the boundary condition at is simply written as (20) These two boundary conditions are inserted into equation (18), and the following linear equations are obtained. (21) By defining the matrix (22) and by using the Fadeev-Leverrier method, the characteristic equation is derived as (refer to the note ) the solution of which gives (23) (24) Indeed, for the given matrix, the same result can be obtained by the QR iteration method also. However, the above method is frequently used only for small matrices, and does not utilize the tridiagonal form. Therefore, it is desirable to use the inverse power method and the eigenvalue shifting. In order to exploit the inverse power method, by putting and solving, we have 10

11 (25) The above-obtained and is solved to give is again substituted into the right-hand side of (26) This procedure is repeated to obtain the following result iter 1/lambda f[1] f[2] f[3] f[4] from which the converged solution is found to be (27) The fundamental eigenvalue shows an error of 3.6% compared with the exact value (see a note in ). And the fundamental eigenfunction is shown as below [the boundary value is determined by using equation (19)]. 11

12 The exact eigenvalue can be determined from the eigenfunction is determined to be where. M#math to find the exact eigenvalue is as follows. and #> double ftn(x) = tan(x*pi)+x; #> w =.bisect(0.51,0.99, ftn); lam = w^2; // eigenvalue w = lam = #> bvp.x(0,pi) ( {y''}+{%y}=0, [ {y}-{y'}=0, {y}=0 ] ).togo(u = y); ans = #> x = (0,pi).span(6).tr; #> [ x, u.tr.rowskip(-20), sin(w*(pi-x)), sin(w*(pi-x))/ ]; ans = [ ] [ ] [ ] [ ] [ ] [ ] Use of generalized differential operator. Here, the following generalized differential operator (28) or the linearized differential operator as in equation (8-65) 12

13 (29) is employed to express the eigenvalue problem as (30) Meanwhile, the differential operator is thought to be discretized to be (31) as was discussed in Chapter 8. Therefore, the eigenvalue problem (32) can be discretized as above. At this point, it is observed that the term in the right-hand side of equation (32) contributes nothing to the discretization of the differential operator. The eigenvalues are obtained in sequence of smaller magnitudes by using the inverse power method and eigenvalue shifting. For example, the Sturm-Liouville equation can be expanded and modified as (33) where the coefficients of the differential operator are defined to be (34) (Example 2) The Bessel equation transformed into the Sturm-Liouville problem as below is For the case of and, find the three smallest eigenvalues. The boundary condition is given as. Use the form for numerical solution. 13

14 Re-writing the equation in the form with, we have (35) By the way, it is actually unnecessary to specify the boundary condition at due to the nature of singular Sturm-Liouville problem. Indeed, in order to exploit the compact form, it is required to specify the boundary condition at. Therefore, an additional condition is imposed considering the symmetry (in reality, corresponds to the center of the cylinder). M#math is written as below #> double eiglam(s) { Lam = bvp.x[201](0,1) ( {y''} +1/x*{y'}-1/x^2*{y}+{%y} = 0, [ {y'}=0, {y}=0 ] ).shift(s);; return sqrt(lam); } #> eiglam(0); eiglam.lam = ans = #> eiglam(50); eiglam.lam = ans = #> eiglam(100); eiglam.lam = ans = then the result is It can be seen that use of 200 subintervals gives very accurate eigenvalues. The exact eigenvalues are obtained from the following M#math. #> solve.x (.J_1(x) = 0 ).span(0.01,11); ans = [ e-008 ] [ e-009 ] [ e-008 ] 14

15 Section 9-3 Eigenvalues on Stability and Convergence Rate In contrast to the situations where the main objective is to find eigenvalues, it also happens that eigenvalues are not required to be sought even though they play an underlying serious effect. This can be illustrated with stiff differential equations and iteration methods on matrix equations. Eigenvalues associated with initial value problems. It was already discussed in section 7-8 that the stiffness of the differential equations is determined from the eigenvalues of the Jacobian matrix given in equation (7-91). In solving initial value problems, eigenvalues of the Jacobian matrix have the following nature. greatest eigenvalue determines step sizes due to stability smallest eigenvalue determines the number of steps for integration Therefore, there occurs a severe difficulty if a ratio of the greatest to smallest eigenvalues differ by several orders of magnitudes. An example is presented below to discuss this behavior. (Example 3) When a numerical solution to the following coupled linear equations is obtained by the Runge-Kutta method with, discuss the numerical stability. First, the coupled linear equations are written in the following matrix form (36) and M#math is written as #> double y1e(x) = 2*exp(-2*x)-exp(-1000*x); #> double y2e(x) = -exp(-2*x)+exp(-1000*x); #> matrix odesol(double h) { 15

16 } x = (0,h).march(6); return ode[x] ( y1' = 996*y1+1996*y2, y2' = -998*y1-1998*y2, y1 = 1, y2 = 0 ).return( x, y1/y1e(x), y2/y2e(x) ); #> odesol(0.01); ans = [ #IND ] [ ] [ ] [ e e+008 ] [ e e+011 ] [ e e+014 ] #> odesol(0.05); ans = [ #IND ] [ ] [ ] [ ] [ ] [ e+006 ] #> odesol(0.001); ans = [ #IND ] [ ] [ ] [ ] [ ] [ ] From the above results, it can be seen that solutions with diverge, whereas a considerably accurate solution is obtained with the smallest step size of. The reason why these results are appearing is that the eigenvalues associated with the corresponding matrix (37) differ in two orders of magnitude, i.e.. The greatest eigenvalue limits the step size through the Euler's stability condition 16

17 (38) The convergence of numerical solutions depends on whether the stability condition is satisfied or not. The exact solution to (Example 3) is given below. A proper use of M#math is utilizing Spoke '.stiff(tol=0.01)' #> ode.x[1000](0,1) ( y1' = 996*y1+1996*y2, y2' = -998*y1-1998*y2, y1=1, y2=0 ).stiff.return(x,y1,y2).skip(10).plot; Eigenvalues occurring in iteration methods. When the Gauss-Seidel iteration method is used to solve coupled linear equations, the diagonal dominance should be determined a priori as was discussed in section 3-6. The convergence characteristics of iteration methods is greatly affected by distribution of the eigenvalues of associated matrix. (Example 4) Solve the following coupled linear equation by the Gauss-Seidel iteration method until the error falls below. The diagonal elements of the given matrix are greater than the off-diagonal elements, and therefore the diagonal dominance is satisfied. This implies that solution can be obtained by iteration. To apply the Gauss-Seidel iteration method, the given equations are modified as (39) 17

18 By initializing and performing the first iteration, we have (40) Subsequent iterations can be performed by using the following M#math #> (a,b,c,d) = (3.2,3,5,5.2);; (x,y) = (0,0);; a=3.2; #> for.i(1,1000) { } r1 = 7-b*y;; // a*x + b*y = 7 r2 = -2-c*x;; // c*x + d*y = -2 x = r1/a;; y = r2/d;; err = 7-a*x-b*y + -2-c*x-d*y ;; [ i, x, y, err ]; if( err < 1.e-7) break; // 354 iterations are required ans = [ ] ans = [ ] ans = [ ]... ans = [ e-007 ] ans = [ e-008 ] Then, a final solution can be obtained by a total of 354 iterations. Even though the given matrix is just a simple matrix, it requires more than 300 iterations, the reason of which stems from the underlying effect of the eigenvalue of the given matrix. This will be discussed in detail below. Role of eigenvalues in iteration methods. In order to examine the iteration method from the standpoint of eigenvalues, matrices, are introduced as (41) then, the iteration given in equation (39) can be written as 18 (42)

19 where is given as an initial guess at the start of iteration. Meanwhile, the error at the th iteration (43) is defined, and equation (42) is recast into (44) or By utilizing the above expression, the error at the (45) th iteration becomes (46) or (47) If the matrix is selected to be, then. This means that the solution is obtained by the direct method since iteration is not required. Otherwise the error at the -th itration is calculated recursively from equation (47) such that (48) where the matrix is a key factor determining the convergence rate. In the case of (Example 4), the matrix is (49) 19

20 the eigenvalues of which are and their absolute values are very close to the unity (see the note in ). If the absolute value of eigenvalues of matrix exceeds the unity, the error diverges as iterations continue. In other words, by letting be the absolute eigenvalue close to the unity, the magnitude of error diminishes as (50) Therefore, convergence is very slow when, whereas convergence is considerably accelerated when. For example, for the following case (51) the matrix is calculated to be (52) the eigenvalues of which are. As a result, the error diminishes rapidly as iterations continue. Actual computation shows that iter x y residue where the rate of residue decreasing is about a factor multiplied by, i.e. a decreasing rate of one-tenth per iteration is found. M#math for equation (52) #> I =.I(2); A = [ 30,3; 5,50 ]; B = [ 30; 0,50 ]; #> (I-A*B.inv).eig; 20

21 ans = [ 0.1 ] [ -0.1 ] Section 9-4 Exercises =========================== 1 For an interval of, the Sturm-Liouville problem is given as. Find the fundamental eigenvalue for the following homeogenous boundary conditions. (a) (b) (c) 2 For an interval of, the Sturm-Liouville equation known as the Bessel function is given as Find the fundamental eigenvalue for the following boundary conditions (a) (b) (c) 3 When an axial load of is applied, the curvature of the slender rod can be determined from where. The final result becomes an eigenvalue problem written as For the two cases shown in the accompanying figure, find the axial road where values are given as. 21

22 4 An axisymmetric vibration of a circular membrane is governed by the following differential equation For the case of, find the fundamental eigenvalue. 5 The most well-known Sturm-Liouville problems are as follows (a) Bessel equation (b) Legendre equation (c) Tchebyshev equation (d) Hermite equation 22

23 (e) Laguerre equation // // end of Lectures // Section 9-5 (optional) Higher-order ODE and Eigenvalue Problems Eigenvalue problems associated with higher-order ODEs can be solved in a similar way. In this section, eigenvalue related to the bending and vibration of a beam is considered which is characterized by the fourth-order ODEs. Bending of a beam. When a distributed load is applied on a beam, the vertical displacement of the beam is governed by the following fourth-order differential equation (53) where is the modulus of elasticity, is the moment of inertia of the cross section. Especially, the product of is termed the flexural rigidity. The boundary conditions for the beam shown in Figure 9-4 are (54) (55) By converting the ODE in equation (53) into the form, we have (56) 23

24 where in general. Figure 9-4 Bending of a beam with a fixed end However, the boundary condition at takes a different form other than equation (55), depending on the state in which the right end of the beam is located. (Example 5) Find the displacement of a beam when a distributed load is applied to the beam. Here, boundary conditions are given as, and, (units are omitted for convenience). Find a numerical solution with 100 subintervals. Since is a constant, solution can be found easily by using equation (53) with the following program #> double ye(x) { L = 8; Lpi = L/pi; kei = 0.001; a2 = 0.5*L/pi^3; return kei*( Lpi^3*(Lpi*sin(x/Lpi)-x) + a2*x^2*(3*l-x) ); } #> kei = 0.001; L = 8; #> bvp.x(0,l) ( {y''''}-kei*sin(pi*x/l) = 0, [ {y}=0, {y'}=0, {y}=0, {y''}=0 ] ).return( x, ye(x), y/ye(x) ).rowskip(-10); A use of 100 subintervals produces a fairly exact solution as below. 24

25 kei = L = 8 ans = [ #INF ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ #INF ] The exact solution is given as Vibration of beam. Now, the dynamic behavior related to the vibration of a beam is expressed by the following partial differential equation (57) where is the distributed load on the beam, is the mass per length. Consider the case where the displacement is given as from which equation (57) becomes ( ) (58) (59) If to be were constants, the exact solution to the above equation can be found 25

26 (60) from the boundary conditions [equations (54) and (55)]. (Example 6) For the case of, find the smallest three eigenvalues for the eigenvalue problem given in equation (59). Numerical solutions should be obtained with from 10 to 100 subintervals by adding 10 subintervals after each run. Find the effect of grid density on the eigenvalue obtained numerically. Exact solutions are found to be, and (see the note below). Since is a constant, the form of differential operator is very simple. M#math is written as #> L = 8; EIm = 1000; s = 0; // change s for other eigenvalues #> row = []; #> for.nx(10,100, 10) { Lam = bvp.x[nx+1](0,l) ( -EIm*{f''''}+{%f} = 0, [ {f} = 0, {f'} = 0, {f''}=0, {f'''}=0 ] ).shift(s); row _= [ nx, sqrt(lam)];; // vertical concatenation } #> row; row = [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] Comparing with the exact solution, the result corresponding to 100 subintervals gives a consistency up to 4 significant digits. Finding other eigenvalues can be carried out by changing the shifintg value in the above M#math code. 26

27 // // end of file //