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Oi00T HS-005 084OCRITERIA FOR IMPLEMENTATION OFOPTIMUM INTEGRATION ALGORITHM

INTO THE WRECKER PROGRAMVolume II: Technical Final Report

lohn R. TuckexMichael Chi

Chi Associates, Inc.Axlington ,VA

Contract No. OOT H8. 7-01620Contract Amt. $65,275

November 1978 TRANSF0RTATI0N LIBRARY

FINAL REPORT DEC 5 1979

NORTHWESTERN UNIVERSITY.

This document is available t e U.S. Public throu6h the

National Technical n ormation Service,

SPrin67ield, Vir6inia 22161

Fxepaxed Fox

OS. DEPARTMENT OF TRANSPORTATIONNational Highway Traffic Safety Administration

Washington, DC. 20590

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4. Title and Subtitle

Criteria for Implementation of Optimum IntegrationAlgorithm into the WRECKER Program

Technical Final Report Volume II

7. Atrts)

Tucker, John R. and Chi, Michael

5. RePrt Oate

F November 30, l978 6. Pctirming Oigmztin Cd!

>__D.8. Pertming Organizatin RePrt N,

9. Pertetming otgeniztin Name and Address

CHI ASSOCIATES, INC.lOll Arlington Blvd., Suite 3l6Arlington, Virginia 22209

10. Wrb Unit N. (TRAIS)

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OOTHS-7-O1620I). TyPe t RePrt and Perid Cered t2. SPnsring Agency Name and Address

U.S. Oepartment of TransportationNational Highway Traffic Safety Administration2lOO Second Street, SWWashington, O. C. 20590 '

Final Technical Report28 June 77 - 30 Nov. 78

I4. SPnsring Agency Cde

'5. SuPPlementary Ntes

NHTSA Contract Technical Manager: w. Tom Hollowell, N43-l2

t6. Abstrat

Three research versions of the wRECKER program were created by incorporatingnew integration methods in the NRECKER I and II programs.algorithms are the variable time step Newmark, the EPISOOE, and the FU methods.These research versions were tested with seven illustrative problems and the

results show that the new methods are superior to the implicit method used inWRECKER IIand more accurate and more stable than the explicit method used in

The integration

numerical integration method, variabletime step integration methods, Newmarkbeta method, EPISOOE, FU method,NRECKER program, structural dynamics,

vehicle structure modelinq

Oistribution unlimited.available to the public through the

National Technical Information Service,Springfield, Virginia

NRECKER 1- A brief documentation of these research versions is included forready reference. A review of modern methods of numerical integration isappended.

t7. Key Wrds 13. Oistributin Statement

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PREFACE

This final report entitled, "Criteria for Implementation of

Optimum Integration Algorithm into the WRECKER Program," Technical

Final Report Volume II, presents the results of a research project

undertaken from 28 June I977 to 30 November 1978 by Chi Associates,

Inc. (CAI) for the Oepartment of Transportation, National Highway

Traffic Safety Administration (NHTSA) under Contract OOT-HS-7

0l620. The report is presented in two volumes: Volume I, Summary

Final Report, and Volume II, Technical Final Report.

Mr. Tom Hollowell served as the NHTSA Contract Technical Manager.

The CAI Project Manager was Or. Michael Chi and the Project Engineer

was Mr. John R. Tucker.

iii

II.

III.

IV.

TABLE OF CONTENTS

INTROOUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

OEVELOPMENT OF RESEARCH VERSIONS OF NRECKER PROGRAM . . . . . . .. 2

OESCRIPTION OF INTEGRATION METHOOS . . . . . . . . . . . . . . . . 3

ANALYSIS OF RESULTS . . . . . . . . . . . . . . . . . . . . . . . 6

CONCLUSIONS ANO RECOMMENOATIONS . . . . . . . . . . . . . . . . . 15

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

APPENOIX A: TEST CASES . . . . . . . . . . . . . . . . . . . . . 53

APPENOIX B: COMPUTER PROGRAM INPUT MOOIFICATIONS . . . . . . . . 67

APPENOIX C: COMPUTER PROGRAM OUTPUT MOOIFICATIONS . . . . . . . . 7l

APPENOIX O: OESCRIPTION OF PROGRAMS . . . . . . . . . . . . . . . 72

APPENOIX E: STATE-OF-THE-ART REPORT . . . . . . . . . . . . . . .183

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

IO

ll

l2

l3

l4

15

16

17

18

19

20

LIST OF FIGURES

Cantilever Plate, Base Run vs. EPISOOE . . . . . . . . . . 18

Cantilever Plate, Base Run vs. Newmark VTS . . . . . . . . 19

Cantilever Plate, Base Run vs. FU . . . . . . . . . . . . . 20

Cantilever Plate, Implicit vs. Explicit Base Runs . 2l

Cantilever Beam with Slave Nodes, Force in X Oirection

Base Runs vs. EPISOOE . . . . . . . . . . . . . . . 22

Cantilever Beam with Slave Nodes, Force in X OirectionBase Runs vs. Newmark VTS . . . . . . . . . . . . . . . .. 23

Cantilever Beam with Slave Nodes, Force in X OirectionBase Runs vs. FU . . . . . . . . . . . . . . . . . . . . . 24

Cantilever Beam with Slave Nodes, Force in Y OirectionBase Runs vs. EPISOOE . . . . . . . . . . . . . . . . . .. 25

Cantilever Beam with Slave Nodes, Force in Y Oirection

Base Runs vs. Newmark VTS . . . . . . . . . . . . . . . . 26

Cantilever Beam with Slave Nodes, Force in Y OirectionBase Runs vs. FU . . . . . . . . . . . . . . . . . . 27

Cantilever Beam with Slave Nodes, Force in Z Oirection

Base Runs vs. EPISOOE . . . . . . . . . . . . . . . . . .. 28

Cantilever Beam with Slave Nodes, Force in Z Oirection

Base Runs vs. Newmark VTS . . . . . . . . . . . . . . . . 29

Cantilever Beam with Slave Nodes, Force in Z OirectionBase Runs vs. FU . . . . . . . . . . . . . . . . . . . . . 3O

Elastic-Plastic Spring (a), Base Run vs. EPISOOE . . . . . 3l

Elastic-Plastic Spring (a), Base Run vs. Newmark VTS . . . 32

Elastic-Plastic Spring (a), Base Run vs. FU . . . . . 33

Elastic-Plastic Spring (b), Base Runs vs. EPISOOE . . . .. 34

Elastic-Plastic Spring (b), Base Runs vs. Newmark VTS .... 35

Elastic-Plastic Spring (b), Base Runs vs. FU . . . . . . . 36

S-Frame without Strain RateBase Run vs. Newmark VTS . . . . . . . . . . . . . . . . . 37

vii

LIST OF FIGURES (continued)

Figure 2l S-Frame without Strain RateFree Flight vs. Newmark VTS . . . . . . . . . . . . . . . 38

Figure 22 S-Frame without Strain Rate, Base Run vs. FU . . . . . . 39

Figure 23 Frame with Inclined SupportBase Run vs. Newmark VTS . . . . . . . . . . . . . . . . 40

Figure 24 Fixed End Beam with Hinge, Force in Z OirectionBase Run vs. FU . . . . . . . . . . . . . . . . . . . .. 4l

Figure 25 Fixed End Beam with Hinge, Force in O OirectionBase Run vs. FU . . . . . . . . . . .y. . . . . . . . . 42

Figure 26 Fixed End Beam with Hinge, Force in Z OirectionBase Run vs. Newmark VTS . . . . . . . . . . . . . . . . 43

Figure 27 Fixed End Beam with Hinge, Force in O OirectionBase Run vs. Newmark VTS . . . . . . Y . . . . . . . . .. 44

Figure 28 Cylindrical Panel (Tang)Base Runs vs. Newmark VTS . . . . . . . . . . . . . . . 45

Figure A-l 8 Node Cantilever Plate . . . . . . . . . . . . . . . . . 55

Figure A-2 Cantilever Beam with Slave Nodes . . . . . . . . . . . . 57

Figure A-3 S-Frame . . . . . . . . . . . . . . . . . . . . . . . . . 60

Figure A-4 Frame with Inclined Support . . . . . . . . . . . . . . . 62

Figure A-5 Fixed End Beam with Hinge . . . . . . . . . . . . . . . . 64

Figure A-6 Cylindrical Panel . . . . . . . . . . . . . . . . . . . . 66

viii

Table

Table

Table

Table

Table

Table

LIST OF TABLES

CPU Cost Chart . . . . . . . . . . . . . . . . . . . . . . '46

Total Cost Chart . . . . . . . . . . . . . . . . . . . . . 47

Approximate Core Required . . . . . . . . . . . . . . . . 48

Input Specifications for Test Runs, EPISOOE . . . . . . . 49

Input Specifications for Test Runs, Newmark VTS . . . . . 50

Input Specifications for Test Runs, FU . . . . . . . . . . 51

ix

I. INTROOUCTION

The National Highway Traffic Safety Administration (NHTSA) is

mandated to investigate the cause and to mitigate the damage of motor

vehicle collisions. In an effort to analytically model the collision

event, NHTSA sponsored the development of several computer simulation

programs to analyze the response of vehicle structures under crash load

ing. Among the programs, WRECKER is one of the most versatile and most

reliable to date.

/ In order to enhance the usefulness of WRECKER, a research program

was undertaken to validate, test, and improve it. The work contained in

the present report deals with the improvement of the numerical integration

method in WRECKER. In the original form as developed by Nelch gt_al. (l),

a fixed time step Newmark Beta method of explicit form was employed. In

a newer version developed by Yeung and Nelch (2), a fixed time step

implicit Newmark Beta method was incorporated. Base runs using these

methods showed that the implicit method yielded more accurate results but

was considerably more costly to run. The present effort surveyed the

available numerical integration methods and selected three which were

believed to be suitable to the NRECKER program. Three research versions

of WRECKER incorporating the three integration methods were developed,

debugged and tested. The final forms of the program are operational in

the UNIVAC llOB computer at the National Bureau of Standards. Comparison

runs were conducted and analyzed. The relative merits of these new versions

were demonstrated by comparing the results with the base runs by fixed time

step Newmark methods. As an unexpected dividend from this investigation,

several mistakes and shortcomings in the original WRECKER program were

uncovered and noted herein for future benefit.

II. OEVELOPMENT OF RESEARCH VERSIONS OF NRECKER PROGRAM

In the beginning of the contract, a systematic literature search

was conducted to uncover the available numerical integration methods, cul

minating in a state-of-the-art report. Among the fifteen numerical methods

analyzed in the report, three candidates were selected on the basis of

efficiency, accuracy and stability relevant to complex structural dynamics

problems. One method, EPISOOE, was available from the Argonne Code Center

in Chicago as a program package: the other two, the variable-step Newmark

and FU methods, required considerable software development. The numerical

integrators were then thoroughly tested on a stand-alone basis until their

operational status was assured. The next step was to provide necessary

linkages to install the integrators into WRECKER to create three research

versions of the WRECKER program.* A sample problem was then selected and a

benchmark result was obtained, using the fixed time step Newmark methods.

The research versions were run to compare with the benchmark results for

verification. Any discrepancies or abnormalities of the results would be

used for error detection and isolation. Tedious debugging procedures in

this manner were conducted until the research versions were ascertained to

be operational. Production comparison runs were then made with government

specified test problems.

See Appendix E.

+ See Appendix B.

III. OESCRIPTION OF INTEGRATION METHOOS

A. EPISOOE - Program NKEP

The first chosen integrator is the recently developed program package

called EPISOOE (3) which is an acronym for Efficient Program for Integrating

Stiff Ordinary Oifferential Equations. It is a sophisticated, well documented

package of tremendous versatility and capability. In the package, it consists

of two variable-step, variable-order methods, the implicit Adams and the

backward differentiation methods. In all, four different corrector-iteration

procedures are available in this package. The entire package was installed in

the first research version of WRECKER as program NKEP with the full capability

of EPISOOE.

The EPISOOE package was designated to solve a system of first order

ordinary differential equations in standard form, Y(t) = f(Y(t), t) with the

initial conditions Y(to) = Yo. Since the equations of motion in WRECKER

assume the form of a second order system, linkage is needed to transform

the wRECKER's second order equations into an equivalent set of first order

equations. EPISOOE could then be applied to this resulting system.

EPISOOE also requires that the function evaluations, i.e., the values

of f(Y(t), t) for a given time t and known values of Y(t), be isolated and

made available through one subroutine. In the present program, the explicit

formulation of function evaluation as given in NRECKER I (l) was used for its

simplicity. The acceleration values thus determined were used in repeated

iterations through EPISOOE to give highly accurate results and presumably less

cost. The implict formulation of function evaluation as given in WRECKER

II (2) was unsuitable for EPISOOE due to incompatibility of procedures.

It was concluded that the explicit formulation is also cheaper to use

in an otherwise implicit numerical integration method. Its inherent

stability characteristics in general would permit the use of larger

time steps in the solution procedure, which is an additional contributor

to cost-saving.

One complicating factor in the development of this research

version was that within the context of taking a solution step, EPISOOE's

iteration procedure required the storage of the initial conditions

existing at the beginning of that step. This necessitated the creation

of additional programming within EPISOOE.

B. Generalized Newmark Variable Time Step MethodProgram NKVTS

The implicit solution procedure present in MRECKER was modified and

generalized to allow the step size to change, as triggered and controlled by

a user-specified accuracy tolerance. The control was exercised via a

straightforward "half-steps, fullstep" procedure. This necessitated the

saving and restoring of initial conditions existent at the outset of a cycle

in the solution. Since the arrays to be stored were of great length, it

was deemed desirable to save them in a temporary file (labeled "28") whose

size can be adjusted as needed through job control cards at the outset of

a particular run. To enhance the stability of the method, a user-specified

damping can be introduced in the program.

In all test runs of this version, the parameter 8 was always taken

to be %, which is known to make the method unconditionally stable at least

for linear problems.

C. FU's Explicit Method - Program NKFU

This is an explicit, variable step, fourth order procedure for systems

of second order ordinary differential equations, which allows for a user

specified artificial damping term. A detailed elucidation of the formulas

and procedures is available in (4).

The complete process had to be specially programmed to be used

in wRECKER. This is the only explicit numerical integration method studied

in the present effort. The inclusion of an efficient and stable explicit

method was considered to be highly desirable due to its inherent simplicity

and efficiency.

IV. ANALYSIS OF RESULTS

Three research versions of the WRECKER program were tested for seven

different case problems. Oetailed descriptions of these problems are included

in this report as Appendix A. To orient the readers, the following remarks

are given.

First, the proper choice of input specifications can have a

crucial effect on whether the program at hand will calculate the solutions

accurately. As an example, Case 4 was run using the FU version with the

exact same input data as is listed in Table 6, except that the tolerance

used was the much less stringent values of l.-O2. The results deviated

drastically from those of the base run, roughly doubling those magnitudes

shown in Figure 22 obtained using the more appropriate tolerance of l.-O5.

Second, the overall costs of runs are influenced to some extent

by the amount of core space needed for storing and executing the system.

A depiction of the comparative approximate storage requirements of the

various programs is given in Table 3, where IBANK refers to space used to

store instructions and OBANK is space allocated for the storage of data.

It is thus to be expected that the overhead costs for the EPISOOE version,

for example, will regularly surpass those of the FU version owing to the

greater amount of coding used for monitoring and iterating-to-convergence

to within the requested accuracy bound.

Third, the choice of initial step size for any variable step

solution procedure will have a direct bearing on the total cost of the run.

For example, if an unnecessarily small first step size is used, excess

execution time is spent iterating while the sizes of successive steps are

gradually increased to a point where a more appropriate and efficient step

we write l.-O2 for I. x 10-2.

size is obtained.

Finally, any version of the NRECKER program has the major part of

its costs directly proportional to the number of function evaluations,

i.e., acceleration calculations, which must be performed in the solution

process, no matter how simple the problem. This is another factor which

contributes to increased costs when too small a step size is employed

initially, for the number of calculations which must be performed to

compute an acceleration for a step size H is the same no matter what

value is given to H.

with these generalities in mind, we now proceed to the specifics.

Case 1 - Eight Node Cantilever Plate

A. Accuracy

Figures l, 2 and 3 show the comparisons of the accelerations at

nodes 4 and 8 in Z direction. The acceleration values were chosen as a

basis of comparison since acceleration is a primary variable. If the

acceleration agrees, the velocity and displacement would follow suit

without problems. The choice of nodes 4 and 8 was made because there the

maximum accelerations take place. In the aforementioned figures, it is

clear that the agreement between the base run of the implicit method and

all three research verions is acceptable. The agreement is best for

EPISOOE results, although there is seemingly "oscillation" behavior at

node 4. This is, in fact, only the codes reaction to a cumulative

error which exceeds the tolerance and the effect of the code in automatically

correcting it. This exemplifies how the prooram uses a built-in algorithm

to reduce the step size and obviate the error accumulation in the

solution.

The agreement for the FU version is good. This is remarkable for

an explicit method, keeping in mind its impressive economy in running costs.

The Newmark method apparently yields good results at the initial

time period but deteriorates rapidly as time progresses, especially for

node 4. The results can undoubtedly be improved a good deal by imposing a

more stringent error tolerance. However, because of the cost factor, it

was not attempted since the outcome would not have been cost-competitive.

The comparison between the two base runs was quite poor_

In reference to Figure 4, it is reasonable to conclude that the results in

the explicit version are faulty, as evidenced by rather severe

oscillatory behavior that cannot be accounted for by physical reasons.

B. Stability

No evidence of instability in any of the research versions was

found. Oscillation, for which stability is suspect, was rampant in the

explicit version base run at node 4.

C. Cost

All three methods are much cheaper to use than the implicit version

of the fixed time step Newmark method used in base runs. we must remark in

passing, however, that the result by the Newmark version must be improved by

reducing time step size which would somewhat increase the cost. As

previously mentioned, the explicit version base run gave faulty results

owing to oscillation. Nevertheless, the comparison in cost is interesting.

Tables l and 2 show that the FU method cost only slightly more than half

of the corresponding cost for the explicit version base run but gave

vastly superior results. EPISOOE, due to its sophistication (iteration

and information storage), was slightly higher in cost than the explicit

version base run.

Case 2 - Cantilever Beam with Slave Nodes

A. Accuracy

Figures 5 through 13 show the comparison of forces at node I in

the X, Y and Z directions. It is clear that the agreement with the base

runs was excellent in the X and Z directions. In the Y direction,

however, only the results in the FU version agree with those in the base

runs. we remark in passing that the amount of forces in the Y direction

should theoretically be identically zero and appear in minimal quantities

as a reflection of round-off errors. Consequently, we believe that

the Newmark and EPISOOE versions actually gave better results than the

base runs and the FU method for these components. we should also note

that the acceleration in the 2 direction for the Newmark version changes

sign in each successive time step. The exact cause of this is unknown.

Investigation showed that this was not caused by the variable time step

algorithm and was present in the original version of WRECKER. In a future

effort, this program error should be uncovered and corrected.

8. Stability

There is no indication of any stability problem.

C. Cost

The cost comparison as shown in Tables I and 2 is similar to Case I

in that the FU method shows a dominance in economy for both CPU and total

costs. The EPISOOE is slightly more expensive than the explicit version

and much less than the implicit version of the base run. The Newmark

versions is the most expensive and, as shown in Table 2, the total cost

exceeded that for the implicit version of the base run by a wide margin.

we attribute this high cost to the excessive data file storage necessary for

time step change. It should also be noted that the explicit version base

run gives excellent results and is much more cost effective than the

implicit version.

Case 3 - Elastic Plastic Spring

A. Accuracy

Figures l4 through l9 show that in both situations, (a) and (b),

accuracy is totally acceptable.

B. Stability

Again, there is no indication of any stability problems.

C. Cost

The higher cost of the Newmark VTS'in situation (a) compared to

the implicit base run is due to this test being started with an

inordinately small step size, 2.-O4, so as to observe whether the code

"stabilized" the choice of step size with the stringent error tolerance of

l.-O5. It did at H = 1.6-O3.

A run made of the Newmark VTS version applied to situation (a)

starting with a larger step size and with a more lenient tolerance, say

l.-O4, would be considerably cheaper.

The comparison of the costs for the FU and EPISOOE runs appears

at first to be incongruous with their performance heretofore. But the

explanation of why the FU runs are slightly more expensive than those for

EPISOOE is simple: in each of the two situations a much smaller starting

step size was used for the PU version (see Table 6). Hence, more

iterations were required to increase the step size to its allowed maximum.

Case 4 - S-Frame without Strain Rate

A. Accuracy

For this problem, EPISOOE failed to start properly and no results

are available. The Newmark version yields satisfactory results in the

VTS refers to the new variable time step version.

10

beginning of the simulation interval, as shown in Figure 20. Near the

end, the test run deviated significantly from that of the base run. we

attribute this discrepancy to the error bound which, being a constant

for the entire period, was too large for the beginning period. An overly

stringent control would not be advisable, however, because that would

cause excessive computer running time when the displacement became

large.

Figure 2l shows the comparison for the cases involving velocity

decaying effect at the initial period of time and no decay effect

(free flight). The Newmark version apparently runs consistent with

the results of the free flight base run. This insensitivity to the

decaying function must be attributed to insufficient error tolerance,

especially at the early period of the run.

The FU version again gives very accurate results (see Figure 22).

Finally, the failure of the EPISOOE version indicates that

there is an unresolved problem with the NKEP code in initializing

conditions with simultaneous input from a velocity time function and

the subroutine FORCE while taking the very first step in the solution

process. It seems to be endemic only to this means of providing

simultaneous input, since no difficulty arises in any of the other test

cases run.

8. Stability

No instability was indicated in either research version.

C. Cost

Since cost comparisons for this problem required extrapolation

for each method, the costs presented in the tables can only be

considered approximate. Nonetheless, there are considerable cost

ll

savings for bgth_the research versions over the explicit base run. we

suggest, however, that the values for the Newmark VTS version are

conservative since they are based on the known exact cost for the last

thirdof the simulation interval where the step size is settled and

changes infrequently. The more frequent changes in step size that are

known to have taken place in the first third of the simulation entail a

higher price, wherein each step change necessitates the re-evaluation

and inversion of the effective stiffness matrix.

Case 5 - Frame with Inclined Roller Support

A. Accuracy

Attempts made to employ the FU and EPISOOE versions to this

case (and the next two) were futile as neither one was able to track any

of the correct solution paths for more than a very short initial period.

Methods using the explicit (i.e., WRECKER 1) form of evaluating the

accelerations, even in conjunction with implicit numerical integration

formulas, as those in EPISOOE, cannot perform static or quasi-static

loading simulations.

The Newmark VTS version handled the problem with no difficulty

(see Figure 23) and was unconstrained by the accuracy request in that it

doubled the step size at every possible time until it was limited by

the value of HMAX.

B. Stability

There was no stability problem in the Newmark VTS run.

C. Cost

The higher cost of the research version is entirely due to

starting the calculations with the overly conservative initial step size

After two restarts.

of l.-O3 (see Table 5). The step was doubled at each possible

instance in the run until it would have exceeded the maximum of l.-Ol,

and thereafter that value was used. A larger initial step was not used,

nor were steps larger than l.-Ol allowed because the runs made for the

next case (chronologically made prior to those of this case), which is

a similar sort of situation, indicated that step sizes larger than 2.-O3

there yield instability in the accelerations. By running the Newmark VTS

for this case with HMIN=H=l.Ol and setting HMAX=5.-Ol, say, the research

version will certainly compare more favorably with the implicit base run

done with H=l.-Ol.

when the base run is made with H=l.-O3, the advantage of the

Newmark VTS procedure is self-evident.

Case 6 - Fixed End Beam with Oiscontinuity Oevice (Hinge)

A. Accuracy

Again, for this quasi-static situation, neither the EPISOOE

version nor the PU version is applicable. What little FU could handle,

prior to losing the solution entirely, is depicted in Figures 24 and 25.

when the Newmark VTS was applied to this model, the results

seemed acceptable in comparison to those of the H=l.-Ol implicit base

run, except that the accelerations which were being printed differed

markedly between the two runs. The implicit base run was re-done with

a step of H=l.-O3, and the accelerations produced there were generally

in good agreement with those for the research version.

Figures 26 and 27 exhibit that the forces at node l for the

first implicit base run and the run of the research version are in

perfect agreement.

13

Case 7 - Clamped Cylindircal Panel (Tang)

A. Accuracy

Again, neither the FU nor the EPISOOE version is applicable to

this quasi-static situation.

The Newmark VTS version successfully tracks the solution for

about two-thirds of the entire simulation interval and then develops an

oscillation in the computed force in the Z direction at node 1, the point

of loading, which quickly causes the results to go awry, ending in the

code asserting that the effective stiffness matrix has become singular

and thus uninvertible. Figure 28 shows only those results which are

present prior to the oscillating load overwhelming the calculations. For

these, the agreement is good.

B. Stability

The oscillation which develops in the computed loading force

appears to be an instability of some sort, but it is from all evidence

available an inherited characteristic, and not generic to the improvements

made in this contract effort.

C. Cost

The extrapolated cost of this research run for the entire

simulation interval is given in Tables 1 and 2. The corresponding amounts

for the implicit base run(s) were not available.

14

V. CONCLUSIONS ANO RECOMMENOATIONS

A. Conclusions

Three different integration methods were implemented into the

NRECKER program in substitution for the existing Newmark fixed time step

method. Although a detailed assessment of these integration methods must

await additional, more extensive testing, the limited experience gained on

a variety of problems does indicate that all the research versions using

new integration methods yield satisfactory, and sometimes excellent results.

The new integration methods generally were vastly superior to the

fixed time step implicit Newmark method employed in the original WRECKER

program. They are comparable to the original explicit version of the

Newmark fixed time step method in terms of memory requirement, time and

cost considerations. However, they proved to be of greater accuracy and

stability than the original explicit method.

Limited experience shows that the FO method (variable time step,

explicit) is best for general dynamic problems. It is not applicable for

static and quasi-static problems for which a variable time step Newmarkmethod is recommended. I

The EPISOOE version of the research NRECKER program is comparable

in accuracy and stability to the implicit fixed time step Newmark method

for complicated nonlinear dynamic problems, but is vastly superior in terms

of running efficiency. Unfortunately, due to its built-in sophistication,

it is severely penalized by the high cost of function evaluations in

NRECKER and so it does not compete well with simpler explicit methods.

15

Also, the EPISOOE version was found to be unsuitable for use in the static

and quasi-static problems. This is because the NRECKER I procedure used

for evaluating accelerations was not accurate enough to satisfy

EPISOOE for these problems. In spite of the fact that EPISOOE employs

a sophisticated implicit Adams predictor-corrector integration technique, it

could not overcome this difficulty. Some difficulties have been experienced

during the program development in choosing the input, such as initial time

step size and the bounds of permissible size of subsequent time steps,

the amount of artificial damping, etc., due to the lack of general guidelines.

The proper choice of the various parameters can undoubtedly improve the

program efficiency. Most of the runs for this project were made with

guesstimates for the input, which may have been quite poor and thus the

final results here should be considered as first indications of the merit

and potential of the research versions of the WRECKER program.

Some general guidelines of choosing these input parameters, especially

the initial time step size and error tolerance at the beginning of a run

should be made available. However, the deduction of the guidelines may

have to rely on a painstaking accumulation of experience in running the

programs with a wider variety of problems.

B. Recommendations

I. The FU method apparently has great potential, at least for the

problems included herein. Therefore, a more systematic investigation of the

effect of damping to its stability should be conducted.

2. Analyze iteration-to-convergence methods and incorporate the

most suitable one in the Newmark VTS version of WRECKER.

16

3. Improve the starting procedure in the EPISOOE version of NRECKER

program such that it can admit inputs of the velocity time function and the velo

city decay function simultaneously in the S-Frame problem.

4. Make a systematic investigation of the algorithm for error control

procedure and criteria with special emphasis on (a) the selection of the key

components to be monitored and (b) the weighting of error estimates.

5. Construct a coupling procedure to link two or more versions of

the WRECKER program so that the best usage can be made to solve any given

problem by a combination of different methods. A monitoring algorithm should

be created to switch from one version to another automatically for maximum

running efficiency.

6. Certain errors in the WRECKER I and II program such as the inconsist

ent sign appearing in successive time step in Cantilever Beam with Slave Nodes

problem and oscillatory behavior in loading in the Clamped Cylindrical Panel

probem should be debugged and corrected.

7. Improve the modeling technique so that structures of high complexity

such as a beam-plate assemblage can be readily simulated by NRECKER programs.

-5

6x105

Node4: Node8:

ACCELERATION

ol.l

YImplicitC)EPISOOE

0Implicit

4}EPISOOE

0'123A567O91):!)

TIMEISECI

FigureI.CantileverPlate,AccelerationatNodes4and8inZOirection

.I

6x10

Node4:xImplicitONewmarkVTS 5--Node8:oImplicitANewmarkVTS

Node

1.3-0|

0-+A=e--

o oTa

013,w

doo

0lNode@

2Eo

=MM-..-,-e--i

A

.

00115k5g7Q90:

TIME(SEC)~

Figure2-CantileverPlate,AccelerationinZOirectionatNodes4and8

.L

6x10

Node4:

5.

Node8:

.Node

2:.2.

0o

u: L)2

(Node@

011.11012a1.s1'.ME(SEC)

Figure3:CantileverPlate,AccelerationinZOirectionatNodes4and8

xImplicit 0ImplicitOM80

0EU AFU

2IMAMAvrtH,Ii

Afaa]AAA

02

Node4:x----xImplicitxxxxxxExplicit Node8:o----oImpliciteeeeeeExplicit

ACIELJERATHJN

IIIlll

01234SI1I(

TINISECI

Figure4-CantileverPlate,AccelerationatNodes4and8inZOirection,BaseRuns

1?

10m

Figure5.

animplicit 'Explicit >EPISOOE

TIME(SEC)

CantileverBeamwithSlaveNodes,ForceinXOirectionatNodel

0x66

4:102'f

-5_xImplicit 'Explicit

QNewmarkVTS

TIME(SEC)

Figure6.CantileverBeamwithSlaveNodes,ForceinXOirectionatNodel

.

4:152

Figure7.

xhnpUcExplicit

0EU

TIMEISECI

CantileverBeamwithSlaveNodes,ForceinXdirectionatNode1

in?

iJxIJ

-:mo

--_xHnpHcH'Explicit 4EPISOOE

-2,.-_.

(I) LIJ u_IOU

. .-

'5'

'0'

IIlo9']ll

o12aI.ss1a9

TIMEISEC)

Figure8.CantileverBeamwithSlaveNodes,ForceinYOirectionatNodel

'6

.

-30

.xlmpUcHExplicii

-JO

--.x' lmpUcM'Expudi

0EU(Explicit)

-2?

(D u:

U-

a:8_1_

TIMEISECI

Figure10,CantileverBeamwithSlaveNodes,ForceinYOirectionatNodel

LZ

'3

3X10. FORCES

xImblici! Explicit 41>EPISOOE/

TIME(SECl

Figurell.CantileverBeamwithSlaveNodes,ForceinZOirectionatNodel

.

10110

3810

' xImplicit 'ExpUcH

C)NewmarkVTS(Implicit)

2. .

FORCES

TIME(SEC)

Figure12.CantileverBeamwithSlaveNodes,ForceinZOirectionatNodel

.2

3X10

ximPlicit Explicit

OFU(Explicit)

FORCES

0100'6

TIMEISECI

Figurel3.CantileverBeamwithSlaveNodes,ForceinZOirectionatNodel

O.

6x10

5-

0EPISOOE1:Implicit

W

:2Q

2o

n:

2:3ETI'

onUU.

2_e..

(5

_o

l

(II |(I_ _l_2

01Z3I.56789OX

TIME(SEC)

Figure14-Elastic-PlasticSpring(a),AccelerationatNode2inXOirection

_,

26x10 ACCELERATMN

0NewmarkVTS

xImplicit

_O

aO

Q

Ri

O

- O

Q

Q

I.l)J'c)(1l-2

O1234S6781):)

TIME(SEC)

Figure15.Elastic-PlasticSpring(a),AccelerationatNode2inXOirection

.

13:0

2

6x10

'0FU

!Implicit

ACCELERATIONU

ll

'0 0t

o2~o

o 0t

.O

10o

0(;

IIIIItlI

O1Z3L56789

TIMEISECI

Figure16.Elastic-PlasticSpring(a),AccelerationatNode2inXOirection

-2

..

0.8

DISPLACENENT(INCH)

1LBmace-0.45a103nan-on

1INCH',2.54CH

F

aJUUNETLM-->

u-MASS-0.2xxrs-scczmen

-0.30x103!

s

0.10.4tO

EPISOOE

__._EXPLICITCOOEAt-0.001SEC _IMPLICITCOOEAt-0.001SEC

IlII1l1)lI1l1

0.20.30.40.50.6 I0.7

TIME(SEC)

FIGURE17.RESPONSE01-ANELASTIC-PLASTICSPRING,(b)

u-seczrtcu

(7.

DISPLACENEN..(INCH)

1LBroacz-0.45x103Newton

1INCH-2.54on

300F

M_

2

KIPS-_ECINCH.

soit10iisiFron-secicii

H-HASS'O '0WM

0.10.af

DNewmarkVTS

___EXPLICITCODEAt:-0.00].SEC _IMPLICITCODEAtI0.001SEC

1IliI1I1rI#

FIGURE)8.

.30.40.50.6 I0.70.8

TIME(SEC)

nesronseorANELASTIC-PLASTICSPRING,(b)

SE:

0.8

DISPLACI-TNENT(INCH)

1LBroace-4.45x103

1INCHI2.56CH

NEWTON

0.2FIGURE19.

-JUUNRW\-M

F

a

MIMASSI0.2

I0.35x103

0.10.4F0EU

_..__EXPLICITCOOEAtI0.001SEC

1Il1I1l10.30.140.50.6

TIME(SEC)

RESPONSEorANELASTIC-PLASTICSPR?NC,(b)

KIPS-SEC2INCH0u-sec

I0.7

IMPLICItICODEAtI0.00).SEC

1

#cm

.:

ONewmarkVTS

'1-_- UExplicit

-2.__._

aO

(A)xC)

\Aw>\

3,73,o\ e.O\

tn.\

E5C)

_.

-5-_-

lI{I -6.I.aL1L

O2I.6810

TIMEISECI

Figure20-S-FramewithoutStrainRate,OisplacementinXOirectionatNodel7

0ExplicitBaseRun

1}FreeFlight ()NewmarkVTS

MSPLCEMENT

U

l

-

-5-_-

-6-('i1.

0zI.sa\O12u.16a20m3

TIME(SE(3)

Figure21.S-FramewithoutStrainRate,OisplacementinXOirectionatNodel7

.E

{)FU

DExplicit

FUN

'liii

TIMEISEC)

Figure22.S-FramewithoutStrainRate,OisplacementinXdirectionatNodel7

'OISPLA'CEMENT'

4x15

OISPLICEMENT

u: x Implicit D Newmark VTS

w: o Implicit A Newmark VTS

I I o 2 4 6 5 P

STATE LOAD IN LB

Figure 23. Frame with Inclined Support, Horizontalor Vertical Oisplacement, u or w, at the

Roller Support

4()

0:103

3x10

0FU(Explicit)

xImplicit

FORCES

I)l1III

2aI.sa7

TIME(sec)

Figure24.FixedEndBeamwithHinge,ForceinZOirectionatNode1

IV

aP.._..,-_~w_-.-_.---'I-.---_,-_--._III.--I

10TU(Explicit)2

oImplicit,.

Gr

L

2o

in.'

aI

g9 u,1"

/"

1/

i

0235

TIMEISEC)

Figure25.FixedEndBeamwithHinge,ForceinbyOirectionatNodel

Z?

13x15

-3:10lN

FORCES

XNewmarkVTS

()Implicit

I

Figure26.

FixedEndBeamwithHinge,ForceatNodelinZOirection

5

TIMEISECI

.17

LX10

xNewmarkVTSF+

4?Implicit

a(./"

.,/

//////q(//

'0

m1/....

,

u/

*//

1_/

4/

O/

o/1ll.Ii

0123L567

TIMEISEC)

Figure27.FixedEndBeamwithHinge,ForceinByOirectionatNode1

Vb

CENTERLOADIIbl)

15m 3

LEGEND

WeeFuu.PANELWRECKER ,..x,,-i/4PANELWRECKER

TABLE ICPU COST CHART

(Oollars)

PROBLEM CASES BASE RUNS EPISOOE NEWMARK VTS FU

I. 8 Node Cant. Plate (I) $23.17 $ 7.79(E) $ I.53 $T 71 $ .85

2. Cant. Beam with (I) S 2.34t S 2.14+Slave Nodes (E) $ .78 $ .88 $ .56

3. Elastic-Plastic (I) S .83 $ l.47Spring a) .((E) UN $ .43 $ .57

b) UN $ .87 S 4.89 $ 1.O4

4. S-Frame w/o $926.02Strain Rate (E) $948.95 00 $68T.56

5. Frame with (I) S .98 NA S 4.26 NAInclined Support ( = 01)

(I) $ 45.76( = OOl)

6. Fixed End Beam (I) S .98 NA $ .99 NAwith Hinge ( =.Ol)

(I) $ 40.20( = 001)

7. (Tang) Cylindrical UN NA $l39.28 NAPanel

Implicit base run, E = Explicit base runCost information unavailableC 2

IIIIIIII

Quasi-static problem for which an explicit-type method is not applicableMethod is not starting properly for this input data

Value has been extrapolated from that for a partial run to represent costat the end of the entire simulation interval for comparison purposes.t Alternating sign in acceleration in Z direction at Node 4

46

TABLE 2TOTAL COST CHART

(Oollars)

PROBLEM CASES BASE RUNS EPISOOE NEWMARK VTS FU

l. 8 Node Cant. Plate (I) 5 34.92 $ 28.41(E) S 3.56 $3.91 $ 2.31

2. Cant. Beam with (I) $ 5.00+ $ 9.69Slave Nodes (E) $ 2.26 $2.90 $ 1.83

3. Elastic-Plastic (I) $ 2.83 $ 6.62Spring {a) (5) UN $1.83 $ 1.95

0) UN $2.71 $ 13.61 $ 2.53

4. S-Frame w/o $1060.36Strain Rate (E) $1321.29 00 $1233.05

5. Frame with (1) $ 2.82 NA $ 17.54 NAInclined Support (H= 01)

) $ 76.60(H= 001)

6. Fixed End Beam (1) $ 5.94 NA $ 2.69 NAwith Hinge (H=.Ol)

I $ 69.38(H=.001)

7. (Tang) Cylindrical UN NA $ 219.53* NAPanel

I = Implicit base run, E = Explicit base run.UN = Cost information unavailable.NA = Quasi-static problem for which an explicit-type method is not applicable.60 = Method is not starting properly for this input data.

Value has been extrapolated from that for a partial run to represent costat the end of the entire simulation interval for comparison purposes.+ Alternating sign in acceleration in Z direction at Node 4.

(7

TABLE 3

NEWMARK VTS

FU

EPISOOE

IMPLICIT OREXPLICIT BASE RUN

APPROXIMATE CORE REQUIREO

IBANK(WOROS)

(7224

16830

?9684

16425

OBANK(WOROS)

80247

6093]

81O49

6l907

TOTAL(NOROS)

97471

7776)

l00733

78332

TABLE 4

INPUT SPECIFICATIONS FOR TEST RUNS

EPISOOE

CASE HSTART TOLERANCE TIMENO

l. Cantilever Plate l.-O6 l.-O4 l.-O4

2. Cant. Beam with 2.5-O7 l.-O4 l.-O5

3. Spring (a) 5.-O3 l.02 2.-O2

(b) 1.-O3 1.-O4 1.

4. S-Frame 09

5. Frame withInclined Support (NA)

6. Fixed End Beamwith Hinge (NA)

7. Tang Cylindrical

Panel (NA)

NA = Method is not applicable.

60 = Method does not properly set up initial conditions from input data.

49

TABLE5

INPUTSPECIFICATIONSFORTESTRUNS

NEWMARKVTS

CASEHMINHMAXHSTARTTOLERANCEARTOAMP(Y)TIMENO

I.CantileverPlateI.-O6I.03T.-O6T.-O40.I.O4

2.Cant.Beamwith

SlaveNodesl.25-07l.-O3l.2507l.-O5O.l.-O5 3.Spring(a)2.-O4l.-Ol2.-O4l.-O50.2.-O2(b)2.-O4l.-Ol2.-O4l.-O4O.l.

4.S-Framel.-O55.-O35.-O4l.-O2O.2.-O2

5.Framewith

InclinedSupportl.-O6l.-Oll.-O3l.-O5O.l.

6.FixedEndBeam

withHingel.-O25.Ol5.O2l.-O3O.l.

7.TangCylindrical

Panell.-O23.2-Oll.-O2l.O2O.l.

OS

INPUTSPECIFICATIONFORTESTRUNS

FU

CASEHMINHMAXHSTARTTOLERANCEARTOAMP(Y)TIMENOl.CantileverPlatel.-O6l.-O3l.-O6l.-O4lOO.l.-O4

2.Cant.Beamwith

SlaveNodes2.5-O7l.-O32.5-O7l.-O4lOO.l.-O5 3.Spring(a)2.-O4l.-Ol1.-O5l.-O410.2.-O2(b)2.-O4l.-Ol5.-O4l.-O20.l.

4.S-Framel.-O61.-O3l.-O6l.-O50.2.-O2

5.Framewith

InclinedSupport(NA)

6.FixedEndBeam

withHinge(NA)2.5-O616-O4l.-O5l.-O3O.l.

7.longCylindrical

Panel(NA)

X

Attemptmadetorunthisexplicitmethodforquasi-staticsituation.

_,

REFERENCES

welch, R.E., R.W. Bruce, and T. Belytschko, "Finite Element Analysisof Automotive Structures Under Crash Loadings, Vol. II." Final

Technical Report for Oepartment of Transportation Contract No. OOTHS-lO5-3-697, March 1976.

Yeung, K.S. and R.E. welch, "Automobile Structures Under Crash Loading."Summary Final Report of Oepartment of Transportation Contract No.

OOT-HS-6-Ol364, October l977.

Hindmarsh, A.C. and G.O. Byrne, "EPISOOE: An Experimental Packagefor the Integration of Systems of Ordinary Oifferential Equations."Lawrence Livermore Laboratory Report UCIO-30ll2, May l975.

Fu, C.C., "A Method for the Numerical Integration of the Equationsof Motion Arising from a Finite-Element Analysis." Journal ofApplied Mechanics, September l970.

52

APPENOIX A

TEST CASES

1. 8 Node Cantilever Plate

Cantilever Beam with Slave Nodes

Elastic-Plastic Spring (a) and (b)

S-Frame without Strain Rate under Impact Load

Frame with Inclined Roller Support

GUT-DOOM

Fixed End Beam with Oiscontinuity Oevice (Hinge)

7. Clamped Cylindrical Panel (Tang)

53

Case 1: 8 Node Cantilever Plate u

A steel plate, 3" x l x 0.1", is modeled with 6 triangular plate

elements and 8 nodes as shown in Figure A-l.The plate is cantilevered from

one end (the 1-5 edge) and a load of 5 1bf is suddenly applied at time t = 0

to each of the nodes (nodes 4 and 8) at the free end.

The material properties were

modulus of elasticity, E = 30 x 106 psi

yield stress, 0y = 210 psi

first plastic modulus, Ep = 20 x 106 psi

Poisson ratio, 0 = 0.3

Base runs were made using the implicit procedure and the explicit

procedure, each with 100 steps of size H = 10's.

54

IQEIQL

m .oom

93332.

33m

mm

Case 2: Cantilever Beam with Slave Nodes

A three-beam steel box section frame is cantilevered at one end,

and the displacement of the free end in the Z direction is specified by

an input time function table. The center beam element is offset .1" from

the axis of beam elements. The endpoints of the offset element are connected

by undeformable rigid links to the other two elements. In the nomenclature

of the NRECKER program, the endpoints of the offset element are "slave"

nodes to "master" nodes on the other two. Material properties are those

for mild steel, i.e.,

30 x 106 psi"'1 ,modulus of elasticity,

Poisson ratio, v 0.3

The implicit and explicit base runs were each made for 50 steps of

length H = 2.5 x 10'7.

The implicit base run exhibits an as yet unexplained alternating

sign for the acceleration in the Z direction at node 4 (the free end).

The cantilever beam with slave nodes is depicted in Figure A-2.

56

>_~

Y

1.01 3 EFLs1 t1E5 6

Global Coordinate System

2

T I, l/64

'2

1///

LBeam Cross-Section

Figure A-2Cantilever Beam with Slave Nodes

.)

57

Case 3: Elastic-Plastic Spring (a) and (b)

This example is intended to illustrate the elastic-plastic

spring element. There are only two nodes and one element: the spring

element's undeformed length is l0 inches, and it lies along the x-axis.

Node l is fixed and massless, and node 2 is constrained to move along

the x-axis without rotations, and has a lumped mass of .2 lbf-secz/in

(a weight of 77.28 lbf). The spring's material properties are

(a) (b)

elastic stiffness, E = lOO lbf/in same as (a)

first yield force, 0y1= 200 lbf same as (a)

first plastic stiffness, Ep1= 33.3 lbf/in same as (a)

second yield force, oy2= O. lbf 300 lbf

second plastic stiffness, Ep2= 250. lbf/in O lbf/in

ultimate strength, ou = 300 lbf 3lO lbf

strain rate coefficient, 0 = 33.3 0

A piecewise linear force function, f, is supplied in user-coded

subroutine FORCE. Applied to node 2, the function is defined by

3000 t, O _t _0.1

f(t) = lOOO(.4-t), 0.1 |A t _O.4

O, 0.4 t|/

58

Case 4: S-Frame without Strain Rate

Half of a two member S-shaped frame of mild steel is modeled using

l6 beam elements of rectangular cross-section. The open walled beam cross

section is represented with l0 segments. The global coordinate system and

typical dimensions are presented in FigureA-3. Nodes I through T7 are

constrained by symmetry boundary conditions and are given an initial

velocity of 322.5 in/sec toward the barrier. The velocity at Node l decays

to zero in .1 msec according to the function

v = 322.5 (.OOOl-TIME)2/lO-8.

The half-masses of the front and rear transverse connecting links are

lumped at the end nodes.

modulus of elasticity, E = 30 x lO6 psi

yield stress, oyl= 36 x 103 psi

first plastic modulus, Ep1= 30 x lO3 psi

ultimate stress, 0U = 100 x 103 psi

Poisson's ratio, 0 = 0.33

second yield stress, oy2= 50 x 103 psi

second plastic modulus, Ep2= 30 x l03 psi.

The explicit base run was made with a step size of H = 5 x l07

and terminated on a I80 min (SUP) limit soon after printing out at time

T = .Ol3623 for cycle 27250. The total simulation interval goes to T = .02.

59

17

FrameCenterline

RigidFront

ConnectingLink

Totalwgt.1.51bf

Z1k

727/2

SectionA-A

TypicalBeamCross

Section

ifoJsi

A

1015

10.19 FigureA-3S-Frame

RigidRear

Connecting

Link

Totalwgt.8.51bf

Case 5: Frame with Inclined Roller Support

A steel two-member frame with one leg rigidly mounted and the

other lying on an inclined support plane is modeled by two box section

beam elements. The inclined support is modeled by a single spring

element. The entire assembly (FigureA-4) is initially in the x-z

plane, and is subjected to an in plane load at the frame apex (node 2)

of IO lbf/sec. Base run computations are in implicit mode with l0 time

steps of size At = O.1 sec.

The box section beam elements are of mild steel with (elastic)

material properties:

30 x 106 psimodulus of elasticity, E

0.3 uPoisson ratio,

The spring element has a stiffness of 30 x lO4lbf/in.

6)

Spring Element

2 u Global Coordinate System

T/64

___.__>~.

'

Ei. uw __,_was-aFrame Cross-Section

5,.

FigureA-4 Frame with Inclined Support

62

Case 6: Fixed End Beam with Oiscontinuity Oevice (Hinge)

A mild steel box section beam is modeled with three beam elements

and four nodes. The x-axis of the global coordinate system lies along

the longitudinal centerline of the beam, as shown in Figure A-5.

A moment release about the y-axis (transverse) has been incor

porated at 50% of the length of the center beam element. A z-force of

100 in/sec is imposed at node 3 through an input time function table.

The base run computations were done in implicit mode using ten

time steps of size At = 0.1 sec, and using elastic material properties:

namely

elastic modules, E 30 x lO6psi

Poisson's ratio, 0 = 0.3

63

MomentAReleaseabout y axis at

this section

:)

x I 1/64(.1.44

Figure A-5Fixed End Beam with Oiscontinuity Oevice

64

Case 7: Clamped Cylindrical Panel (Tang)

This model aims to compute the quasistatic response of a

26" x 28" x 0.125 cylindrical steel panel segment (50" radius of

curvature) subjected to a point load of I500 lbf at its center (Figure A

6a). A mesh consisting of 36 nodes and 50 triangular plate elements is

used to represent one quadrant of the l panel (Figure6b). Symmetrical

boundary conditions are applied along the two "material" edges, and the

panel is loaded with a consistent load of 375 lbf applied in l00 load

steps of 3.75 lbf each. Since in this one-quarter panel model the load

is applied at the corner boundary node number one, the shape function

associated with that node has essentially a of the support it would have

in the full panel model, and so its coefficient (the value of the

generalized force at that node) should be a of the desired full panel

load.

65

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n.d=Q1anm_

wwzmd

0,a . m am

M~

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wmzmd

3mm:

mm

APPENOIX B

COMPUTER PROGRAM INPUT MOOIFICATIONS

A description is presented of those changes and additions to the

program input for WRECKER which are needed to run each of the three

versions developed in this project.

B.1. Input Format Changes for WKEP

l. On card 1, the entry in column 1 must always be E.

2. On card 2, the variable MXSTEP is not used, and the variable

OELT now provides the initial time step (which is thereafter

automatically adjusted by the EPISOOE logic).

3. On card 3, the variable KONTRL(5) is not used.

4. Two additional cards should be placed at the end of the data:

Card No. Format Variable Name Oescription

l3 6El0.6 HMIN (Unused)HMAX (Unused)EXTRA (Unused)EVTS3 Local error tolerance

TIMENO End of solution interval

l3.1 315 IERROR Error control indicatorMF Method flagINOEX Status flag

EVTSB is used only on the first call to subroutine ORIVE. Estimates

of the relative local error are kept less than EVTS3 in root-mean-square

(RMS) norm. Errors in Yi are divided by YMAX(i) to get relative errors,

where the vector YMAX is computed in subroutine ORIVE as described below

under IERROR. and the vector Y = (Yi) is made up of the displacements

and velocities Y = (X, X).

Oescribed in detail below and in comment cards at the start of thelisting of subroutine ORIVE.

67

IERROR has the following values and meanings:

I For absolute error control. YMAX(i) is always I.

2 For relative error control. YMAX(i) is always IYiI, using

the latest computed values of Yi.

3 For a semi-relative control, defined as follows.

YMAX(i) is initialized as IYiI, or I if Yi(to) = 0. Then

YMAX(i) is updated after each step to be the larger of its

current value and IYiI. (Thus if Yi(to) f 0, YMAX(i) is the

largest value of |Y1| seen so far.) Errors in Yi are then

controlled relative to YMAX(i).

MF is the method flag. It is used only for input, and used only on

the first call, unless INOEX = -l. The allowed values of MF are l0,

l2, I3, 20, 22 and 23. MF is an integer with two decimal digits,

METH and MITER (MF = l0METH + MITER). (MF can be regarded as the ordered pair

(METH, MITER).) METH is the basic method indicator, with the following values

and meanings:

I For the variable-step, variable-order implicit Adams method,

suitable for non-stiff problems:

2 For the variable-step, variableorder BOF method suitable for

stiff problems.

MITER is the corrector iteration method indicator, with the following values

and meanings:

0 for functional (fixed point) iteration:

_I

cannot be used with NRECKER:

2 for chord method with Jacobian generated internally:

3 for chord method with diagonal approximation to Jacobian.

68

INOEX is an integer flag used by subroutine ORIVE to distinguish what

the status is at the time ORIVE is called. Initially, it should be 1. Other

circumstances are described in comment cards at the beginning of the listing of

ORIVE in Appendix O, which should be read by any user of this method.

The runs made for this report always used IERROR=3, MF=10, and INOEX=1.

B.2. Input Format Changes for WKVTS

1. On card 1, the entry in column 1 must always be I.

2. On card 2, the variable MXSTEP is not used, and the variable OELT

now provides the initial time step (which is thereafter automatically

adjusted by the programming logic).

3. On card 3, the variable KONTRL(4) now represents the artificial

damping term v in the form GAMMA1000. All test runs in this

report used GAMMA = 0.

4. One additional card must be placed at the end of the data:

Card No. Format Variable Name Oescription

13 6E10.6 HMIN Minimum step size permittedHMAX Maximum step size permitted

EXTRA (Unused)EXTRA (Unused)EOISP Local error toleranceTIMENO End of solution interval

B.2.1. External File Requirement for WKVTS

A second temporary scratch file - Logical Unit 28 - is used in the

solution process, and must be made available via the job control language

prior to program execution.

8.3. Input Format Changes for WKFU

1. On card 1, the entry in column 1 must always be E.

2. On card 2, the variable MXSTEP is not used, and the variable OELT

69

now provides the initial time step (which is thereafter

automatically adjusted by the programming logic).

One additional card must be placed at the end of the data:

Card No. Format Variable Name Oescription

13 6El0.6 HMIN Minimum step size permittedHMAX Maximum step size permitted

EVTSl Local error toleranceART OAMP Artificial damping term

EXTRA (Unused)TIMENO End of simulation interval

70

APPENOIX C

COMPUTER PROGRAM OUTPUT MOOIFICATIONS

All three new versions of WRECKER provide appropriate echoes of

the additional input from card group 13. Also, the NKEP version gives

an additional line of information at the end of the printing of an output

cycle. There, the current values of the step size used, the latest order

(of numerical integration formula) used, the number of time steps taken

by EPISOOE to that point (which will, generally, be greater than the number

of cycles which have occurred), and the number of function evaluations

(i.e., acceleration calculations) to that point are given.

71

APPENOIX O

OESCRIPTION OF PROGRAMS

0.1. Oescription of wKEP

0.1.1. Subroutines Removed from WRECKER

The following NRECKER subroutines were superfluous for the merging

of EPISOOE with WRECKER and so were deleted for the research version development:

ASSMBL, BOUNO, CRVTBL, EFFSTF, EPTSTF, ESOLV, ETOG, FORMK, INCOOE, KAOO, LOCSB,

MCHB, MOOIFY, NTOG, PLSTF, SLTOMR, SOLVE, TRIANG, ZERORC.

0.1.2. New Subroutines Oescriptions

The following single precision subroutines in WKEP are new or modified

(as indicated by N or M), and listings of them are given after these brief

descriptions.

(N) AOJUST

Common BlocksEPCOMlEPCMlD

A subroutine from EPIS0OE which is called by TSTEP to adjust the

history array Y when the integration order is reduced.

(N) M

Common BlocksEPCOMlEPCMlD

A subroutine from EPISOOE which is called by TSTEP to set the

coefficients that are used there, both for the basic integration step and

for error control.

on 95Common Blocks

An EPISOOE subroutine called by PSET which performs the LU decomposi

tion of a matrix.

72

(N) OIFFUN

Common BlocksWORK

FUNKEOFNOYNAMJUNK

FBLOK

BEAMSEPCOMlSTF

OIFSTR

A CAI developed subroutine required by the EPISOOE package whichis

called by TSTEP, and also by PSET if MITER=2 (see Appendix B). It

computes the vector V = (X, X) for given values of T = Time and Y

= (X,X), after conditions at the start of a solution cycle have been

restored.

(N) eweCommon Blocks

EPCOMlEPCOM2EPCOM3EPCOM4EPCOMSEPCOM6EPCOM7EPCOM8EPCOM9STRJUNK

FBLOKBEAMSSTFOYNAMOIFBLANK

The EPISOOE managing subroutine, called by SOLEP, which drives the

integration process through one cycle by calling the step-by-step inte

gration routine TSTEP, checks certain input for errors, and writes

error messages. It also saves the prevailing conditions existent

73

(N)

(N)

at the beginning of a cycle for later restoration in OIFFUN during the

iteration procedure performed in TSTEP. It has been amended to prevent

interpolation of calculated results.

FUNKEV

Common BlocksBLANKMASS

FUNFUNKE

A CAI developed subroutine which is called by OIFFUN to perform

a function evaluation (acceleration calculation) at a given time for a

given set of initial conditions.

INTERP

Common BlocksEPCOMl

An EPISOOE subroutine which normally would compute interpolated

solution values by a call from ORIVE. It is not used, but kept so that

the EPISOOE package is intact in wKEP.

MAIN

Common BlocksWORK

BLANKCONTRL

OYNAMSTR

BEAMS

FUNKE

OFN

The main program from NRECKER which has been modified to call

SOLEP after calls to REAOIN and ASSBLE.

PEOERV

Common Blocks

A dummy version of a subroutine required by EPISOOE provided

It is never actually called.merely to satisfy the loader.

74

00 PlCommon Blocks

EPCOMlEPC0M2

EPCOM4EPCOMSEPCOM6EPCOM7EPCOM8

An EPISOOE subroutine called by TSTEP if MITER=2 (see Appendix B)

which sets up and processes a Jacobian dependent coefficient matrix for solution

of the linear algebraic system generated by the chord corrector iteration method.

(M) REAOIN

Common BlocksENGYSTR

OYNAMOUTPACONTRL

BEAMSFUN

VTS

The WRECKER subroutine was modified to permit input and echo of

the parameters necessary to utilize EPISOOE.

(M) RESTRT

Common BlocksJUNK

OYNAMSTRBLANKENGYFBLOKBEAMS

EPCOM9

The WRECKER subroutine, modified to enable checkpointing and

restarting with the EPISOOE version.

75

(N) --

Common Blocks

An EPISOOE subroutine called by TSTEP if MITER is 2, which solves

linear algebraic systems for which the matrix was processed by OEC.

(N) SOLEP

Common BlocksBGOVTSJUNK

CONTRLSTRSTF

OYNAMBEAMS

FUNKE

OIFEPCOM9FUNOUTPA

A CAI developed subroutine which is the linkage, along with

OIFFUN and FUNKEV, between WRECKER and EPISOOE. In this subroutine

the solution via EPISOOE is mandated by looping around a call to

ORIVE. Within the loop the results are printed by a call to OUTPUT.

(N) TSTEP

Common BlocksEPCOMlEPCOM2EPCOM3

EPCOM4EPCOMSEPCOM6EPCOM7EPCOM8EPCOM9EPCMl0

An EPISOOE subroutine called by ORIVE which takes a single step in

the integration and performs the control of local error (including the

choice of step size and order) for that step.

O.1.3. New Subroutines Listings

76

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