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&**',?  :&4 ? 53"!8835?3"? (

n

increases, calculating

formidable as calculating

Fortunately, there are simple ways of estimating these values. For example, an

upper bound on the largest eigenvalue can be obtained from the infinity norm max j A;I :: lsisn

llAll"'

(11.55)

An estimate of the smallest eigenvalue can be calculated using the relatively simple inverse iteration procedure presented in Section

11.6.3

12.8.4.

Error Estimates and Preconditioning An approximate condition number is often all that is needed in estimating potential

[A] is represented in the computation by p 10 p, and similarly, s is the resulting number of solution, llcSxll!llxll 10 s. From Equation 11.48, the

error. Suppose that the coefficient matrix decimal places, that is

llcSAll!llAll

correct decimal places in the

=

=

11.6 11.6.4

Errors

333

Detecting, Controlling, and Correcting Error We close this section on error analysis by shifting our attention from predicting possible

error to detecting and, if possible, controlling and even correcting for numerical errors actually committed.

. . Perhaps the most obvious and commonly used procedure for error detection 1s sub­

stituting the solution vector obtained, say,

{xA}, back in the original system of equations {bA}· If the elements of {bA} differ substantially from those of the original right-hand side vector {b}, it is clear that and calculating the corresponding right-hand side vector

significant errors have occurred in the solution process and we have not found the true solution

{xrl· Unfortunately, the converse is not necessarily true; small differences {bA} and {b} do not prove that {xA} is accurate. By the mere definition of ill­ conditioning given earlier, several solutions which may vary widely can provide a {bA} that is acceptably close to {b}. In this same regard, according to between

(11.61) the elements of

[At1

may be so large-another sign of possible ill-conditioning-that

a small discrepancy in the right-hand sides

a wide range in solution vectors.

{c5b}

may be severely amplified, producing

In a structural analysis, back substituting the solution vector is the equivalent of

performing an equilibrium check. Although this check may be inconclusive, extending

it to a check for displacement compatibility will confirm that a solution is the correct one for a given idealization. Where a complete equilibrium and compatibility check

can be a time consuming process, a distributed sampling of checks can often confirm

the merit of a solution.

When a poorly conditioned problem is encountered, several measures can be used

to correct or at least control the amount of numerical error. One scheme is to use

scaled6 equations as a criterion for employing partial pivoting (see Section

11.2.1).

Solution accuracy may then be improved by interchanging rows of original coefficient

values so that the divisor is always the most significant coefficient in the submatrix

column of the unknown being eliminated. This concept may be extended to complete

pivoting, where columns are also interchanged so that the dividing coefficient is the

largest in the entire submatrix of coefficients to be operated on. In complete pivoting,

symmetry is preserved.

Another method for increasing solution accuracy is to employ a technique known

as iterative improvement (Ref.

11.12). After factoring [A) according to algorithms such 11.15, the solution {x1} to Equation 11.1 is first computed. The residual of this solution {r1} is then computed according to Equation 11.31. Using {r1} and the original decomposition of [A], a solution {z1) is obtained for as Equations

11.13

or

(11.62) A more accurate solution to the original system of equations can then be furnished by

(11.63) The process would then be repeated until the change in the solution

{z')

becomes

negligible. In a structural analysis, this approach is equivalent to computing residual 6Scaling refers to normalizing the equations so that the maximum coefficient in a row equals unity. Because scaling may introduce rounding error, it is recommended that the scaled coefficients be used only as a criterion for pivoting and the original coefficient values be retained within the elimination and substitution computations.

11. 7

With

{xi}= Ll.50

1.19

{ } { } m { �} 0.Ql 0.01 O.Ol 0.01

{ri} = {b} - [A]{xi} =

{zi} = [A] {ri} =

(x,1 = {xd

("} = {bl

11.7

335

0.18Y

0.89

-I

Problems

0.50 0.19 O.ll 0.82

'" ' =

+

[A}{x,J =

indi"ting that

{x,]

'' tho trno 'olution.

PROBLEMS 11.1

Solve the following systems of equations by (1) Gauss elimination; (2) the Cho­

lesky method; and (3) the root-free Cholesky method. 64x1 + 32x2

(a)

- 16x3

32x1 + 416x2 + 72x3 -l6x1 + 72x2 (b)

170

=

+ 120x3

5

=

25x1

+

5x2

-

2.5x3

5x1

+

l7x2

+

ll.5x3

-2.5x1 + ll.5x2 + 13.25x3

11.2

140

=

=

=

=

60

-8 11

Attempt to solve the following system of equations by the Cholesky method.

What can be concluded? 20x1 + l0x2 - 40x3 l0x1 + l5x2 + 30x3 -40x1 + 30x2 - 20x3

=

=

=

100 -110 -300

11.3

Solve the equations of Problem 11.2 by the root-free Cholesky method.

11.4

Solve the equations of Problem 11.1 by (1) Gauss-Seidel iteration; and (2) Jacobi

iteration. Obtain a solution which is accurate to within three significant figures.

11.5

Solve the equations of Problem 11.lb by (1) Gauss-Seidel iteration; and (2)

Gauss-Seidel iteration with overrelaxation

11.6

f3

=

1.4. Use

?

=

5%.

For the equations of Problem 11.2, perform five iterations of the Gauss-Seidel

method. Based on this solution and the conclusion made in Problem 11.2, comment on the apparent effectiveness of using this scheme.

References 11.15

337

Solve the following system of equations and discuss the condition of the system:

11.16

1 .00 x1 + 0.50x2 + 0.33x3

=

6

0.50x1 + 0.33x2 + 0 . 25 x3

=

4

0.33x 1 + 0.25x2 + 0. 20 x3

=

2

The following system of equations is similar to that in Problem 11.15. Solve it

exactly by retaining the fractional notation. Compare your solution with that obtained in Problem 11.15. Approximately how many digits would have to be used in a Gauss elimination solution using decimal notation in order to obtain four-digit accuracy? x,

+

!xi !xi

+ +

!x2 !x2 ! x2

+ + +

! x3 ! x3 !x3

=

6

=

4

=

2

Repeat Example 11.9 using Equations 11.41 and 11.52 to calculate the condi­

11.17

tion numbers.

11.18

The structure shown consists of three axial force members in series. The forces

are Pb

=

Pd

=

1 and P.

=

0. The stiffnesses are kab

=

kc,1

=

1 and kbc

=

1 X 104•

(1) Calculate the displacement vector using Gauss elimination and four-digit arithmetic (truncate to four digits after every operation-be sure that truncation is done and not merely on the calculator's display-this may require reentry after every operation). (2) Repeat using eight-digit arithmetic. (3) Based on these solutions, discuss the con­ dition of the system.

Problem 11.18 11.19

c

Use Equations 11.52 or 11.53 and Equation 11.56 to estimate the truncation

error for Problem 11.18. Compare with results obtained in Problem 11.18.

11.20 Precondition the coefficient matrix of Problem 11.18, and repeat Problems 11.18 and 11.19. 11.21

Repeat Problems 11.18 to 11.20 with kab

=

kct1

=

1 X 104 and kbc

=

1.

REFERENCES 11.1

L. Fox, An Introduction to Numerical Linear Algebra, Oxford University Press,

New York , 1965.

11.2

G. H. Golub and C. F. Van Loan, Matrix Computations, 3rd edition, The Johns Hopkins University Press, Baltimore, Md., 1996.

11.3

K. J. Bathe, Finite Element Procedures, Prentice-Hall, Englewood Cliffs, N.J.,

11.4

1996. J. B. Scarborough, Numerical Mathematical Analysis, 6th edition, The Johns

11.S

J. L. Buchanan and P. R. Turner, Numerical Methods and Analysis, Mc Graw­

11.6

Hill, New York, 1992. G e orge and J. W. H. Liu, Computer Solution of Large Sparse Positive Definite Systems, Prentice-Hall, Englewood Cliffs, N.J., 1981.

Hopkins University Press, Baltimore, Md., 1966.

A.

338

Chapter 11

Solution of Linear Algebraic Equations

11.7

K. J. Bathe and E. L. Wilson, Numerical Methods in Finite Element Analysis,

Prentice-Hall, Englewood Cliffs, N.J., 1976.

11.8

G. H. Paulino, I. F. M. Menezes, M. Gattass, and S. Mukherjee, "Node and Element Resequencing Using the Laplacian of a Finite Element Graph," Intl.

JI. Num. Meth. in Engr., Vol. 37, No. 9, 1994, pp. 1511-1555. 11.9

B. M. Irons, "A Frontal Solution Program for Finite Element Analysis," Intl JI

for Num. Methods in Engineering, 2, 5-32, 1970.

11.10 J. H. Wilkinson, Rounding Errors in Algebraic Processes, Prentice-Hall, Engle­ wood Cliffs, N.J., 1963.

11.11 R. A. Rosanoff, J. F. Gloudeman, and S. Levy, "Numerical Conditioning of Stiffness Matrix Formulations for Framed Structures,"

Technical Report

AFFDL-TR-68-150, USAF Flight Dynamics Laboratory, Wright-Patterson Air Force Base, Ohio, 1968.

11.U G. Forsythe and C. B. Moler, Computer Solution of Linear Algebraic Systems, Prentice-Hall, Englewood Cliffs, N.J., 1967.

U.4

Automatic Load Incrementation

351

In these equations, N is the total number of unknown displacement compo� ents,

is the kth element of the incremental displacemeni vector units of

d!J.k>

the value of

6.ref

d!J.k {d.:i{}, and dependmg on the

is taken as either the largest component of translation

or rotation within the total displacement vector

{.:i;}.

Using any of these norms, a convergence criterion of

llell

(12.25)

s t

can be used where the acceptable tolerance tis usually on the order of

10 2

to

10 6,

depending on the desired accuracy. As an alternative to comparing displacements, reasonable equilibrium criteria may also be based on assessing the unbalanced load vector (Ref. 12.6) or by studying the increment of internal work (Ref. 12.4).

12.4

AUTOMATIC LOAD INCREMENTATION The size of the load ratio employed in each increment of the analysis can have a dramatic effect on the solution. In the single-step methods, proper selection of

dA is

the only means for controlling drift-off error. In the iterative schemes, a poor definition of

dJ\.l

could result in the solution not converging within a practical number of itera­

tions. In both methods, an excessively small load ratio may result in significant com­ putational effort with a negligible increase in accuracy. To provide assistance in de­

termining dAJ, there are several types of automated procedures that may be employed. Two of these schemes &re presented. The first may be applied to both the single-step and iterative methods. The second is intended only for iterative procedures. In both schemes, it is assumed that the size of the load ratio for the first increment

dA�

first iteration of the first increment earlier, a value of

10%-20%

dA1

or the

has been prescribed by the analyst. As stated

of the anticipated maximum applied load is generally

satisfactory. Section 12.6 also provides load ratio size constraints that are required specifically for a material nonlinear analysis based on the plastic hinge method.

12.4.1

Change in Stiffness The load ratio

dA at any point in the analysis should reflect the corresponding state of

nonlinearity. A simple scalar measure of the degree of nonlinearity can be obtained from a

current stiffness parameter given by s 1

=

(Ref.

12.7)

{d�ir!Prerl {d.:il}T{Pref}

(12.26)

Since the parameter S; will always have an initial value of unity, stiffening or softening of. the structural system will be indicated by values greater than or less than one, respectively. With the exception of bifurcation points, S; will become zero at a limit

point. For a more sophisticated and perhaps more rigorous measure of nonlinearity, Reference 12.3 provides details on a

generalized stiffness parameter.

Using either Equation 12.26 or the stiffness parameter of Reference 12.3, load ratios for a single-step method or the first cycle of an iterative method can be obtained from (12.27) where

d>.1

is the load ratio prescribed at the start of the analysis, and the exponent y 0.5 to 1 (Ref. 12.1). Strategies for selecting the appropriate sign in Equation 12.27 will be discussed in Section 12.7.

typically equals

12.5

Element Result Calculations

353

vector of axial forces, end shears, and moments that is usually needed for design. In the second case,

[k]

must be in local coordinates and it follows that {F] will be also.

We may illustrate the extension of this conventional procedure to nonlinear analysis by considering the element of Figure 12.7. The first sketch, Figure 12.7a, illustrates the element's orientation and the forces acting on it at the start of a linear step of a single­ step or iterative procedure. Figure 12.7b shows its orientation at the end of the step with the corresponding forces referred to the initial local axes. Figure 12.7c is of the

(c)

(b)

(a) Figure 12.7 Element end forces.

U.6

Increment of element forces:

{dF;}

L2

=

5

100

2

Plastic Hinge Constraints

355

400 Y

5

Forces at end of step referred to final configuration:

2Fa

{1F} {TF) with

with

=

=

L8

20

{1F)

+

1

2

=

=

{dF}

15°,

20°,

=

[''Y]

(2-y]

L10

=

=

[

15ooy

20

8

500

'°'

[

�

15

si

15

sin 15

0

0

1

0.940

0.342 0.940

From Equations 12.32 and 12.33

{2F}

=

19oo y

O] [

cos 15

0 42

;

25

10

600

25

=

0.9M

0.259

0.259

0.966

0

0

�]

0

(Tr){IF}

=

(2r]('r)T

10

12.14

25

24.03

600 10 25 1900

600 12.14 24.03 1900

This method of force recovery is straightforward in application and conventional in the sense that it is a direct extension of the standard linear elastic procedure. It will be recognized, however, that there is nothing in it to distinguish between displacements resulting from rigid body motion and those due to deformation: It is an approximate approach to force recovery. Methods that do distinguish between the two sources of displacement are discussed in Appendix B and the limitations of this conventional one are explored. It is shown that, for small strain, moderate displacement analysis, the rigid body motion effects are normally small. For this reason, and because it has been found adequate in many examples of both elastic and inelastic nonlinear analysis, conventional force recovery in conjunction with a single step method is considered suitable for general framework analysis. In cases of highly nonlinear structures, an iterative analysis method and the force recovery procedure defined in Appendix B are recommended.

12.6

PLASTIC HINGE CONSTRAINTS For the material analysis model developed in Section 10.2, the load ratio is typically reduced to prevent plastic hinge formation from occurring within a load increment,

and thereby avoid any accompanying abrupt changes in stiffness. To calculate a load

12.8

Critical Load Analysis-An Eigenproblem

369

A first estimate of the minimum eigenvalue is

Results of the second iteration:

{y2J

=

{lJ] 1a7i A2

= =

Ll.598 x 10-6

0.2425

L uo9 x 10-9

3.689

6.o3o x 10-4f

2.852

L8.578 x 10-10

=

596.161f

4.662 x 10-4r

0.750

and according to Equation 12.51

1

e = a

1.371

0.750

0.150

1

100%

=

82.8%