NASA Contractor Report 180893

3-D Inelastic Analysis Methods for Hot Section Components Fourth Annual Status Report for the Period February 14, 1986 to February 14, 1987

Volume I - Special Finite Element Models

S. Nakazawa

May 1988

Prepared for Lewis Research Center Under Contract NAS3-23697

NASA Nafional Aeronautics and %ace Adrninistrafion

(NASA-CR-180893) O N 3-D I N E L A S T I C A H A L Y S15 N 88- 2 15 35 METHODS POR BOT S E C T I G N COHPONENTS. VCLUPld 1: SPECIAL F I N E T E ELEMENT MODELS Annual R e p o r t No. 4, 14 Peb. 1986 - 1 U F e b . 1Sb7 Uaclas (Pratt and Whitney Aircraft) 8 1 p CSCL 2 0 K G3/39 0140242

“/\SA ~dlCrJI -r*CnaulCC ,-,- Report Documentation Page , . 9.T C1”W’ 111“

1 1 Repon No. 2. Government Accession No. 3. Recipmnt’s Catalog No

NASA CR-180893

I 4 Title and Subtitle 5. Report Date

3-D INELASTIC ANALYSIS METHODS FOR HOT SECTION flay 1 988

6. Performing Organization Code t COMPONENTS - FOURTH ANNUAL REPORT ’ Volume I , Special F in i te Element Models

I

’ 7. Authorls)

1 S. Nakazawa

8. Performing organization Report No. 1 PWA-5940-62

9. Performing Organization Name and Address

UNITED TECHNOLOGIES CORPORATION l P r a t t & Whi tney , Commercial Engineering ~ 400 Main S t . , E a s t Har t ford , CT 06108

I 11 Contract or Grant No I

NAS3-23697 I 13. T v ~ e of ReDon and Period Covered

12. Sponsoring Agency Name and Address NATIONAL AERONAUTICS AND SPACE ADMINISTRATION

’ Lewis Research Center 21 000 Brookpark Road C1 evel and, OH 441 35

I Fourth ‘Annual Report 2/14/86 t o 2/14/87 I . . . .

14. Sponsoring Agency Code 1

17 Key Words (Suggested by Author(s1l

1 RTOP 533-04-1A I

18 Distribution Statement I

I 1

15. Supplementary Notes j

P r o j e c t Manager, C. C. Chamis, MS 49-6 NASA-Lewis Research Center; C1 evel and, OH 441 35

This Annual S t a t u s Report presents the r e s u l t s of work performed d u r i n g the f o u r t h ~

y e a r of the 3-D I n e l a s t i c Analysis Methods f o r Hot Sec t ion Components program (NASA ’ Contrac t NAS3-23697). The o b j e c t i v e of t h e program i s t o produce a series o f new computer codes t h a t permit more a c c u r a t e and e f f i c i en t three-dimensional ana lyses of s e l e c t e d h o t s e c t i o n components, i . e . , combustor l i n e r s , t u r b i n e b lades and turbine vanes. The computer codes embody a progress ion of mathematical model s and a r e s t r eaml ined t o take advantage o f geometrical f e a t u r e s , l oad ing c o n d i t i o n s , and ~

forms of mater ia l response t h a t d i s t i n g u i s h each group of s e l e c t e d components. , Volume I o f th is r e p o r t d i scusses the spec ia l f i n i t e e lement models t h a t were developed dur ing the f o u r t h y e a r of the c o n t r a c t e f f o r t .

I

I

NASA FORM 1626 OCT 86

PREFACE

This Annual Status Report, Volume I, describes the r e s u l t s o f work performed on Task VB (Special F i n i t e Element Models) dur ing the f o u r t h year o f the NASA Hot Section Technology (HOST) program, "3-D I n e l a s t i c Analysis Methods f o r Hot Section Components" (Contract NAS3-23697). The f i n i t e element code, MHOST (MARC HOST), i s developed f u r t h e r based on mixed i t e r a t i v e s o l u t i o n procedures w5ose concepts and basic technology were establ ished i n the f i r s t and second year e f f o r t s . The technology i s extended i n the c u r r e n t phase t o incorporate the e f f e c t o f m u l t i p l e embedded s i n g u l a r i t i e s i n generic modeling regions. S p e c i f i c a l l y , the l o c a l mesh ref inement technology and the special f unc t i on representat ion f o r s ingu la r behavior o f s t r a i n are developed based on mixed element concepts. The l o c a l mesh ref inement algor i thm, r e f e r r e d t o as the subelement i t e r a t i o n , u t i l i z e s conventional l i n e a r and h igher order polynomial representat ions f o r s p a t i a l d i s c r e t i z a t i o n .

The program i s being conducted under the d i r e c t i o n o f D r . C. C. Chamis o f t he NASA-Lewis Research Center. Prime con t rac to r a c t i v i t i e s a t Uni ted Technologies Corporation are managed by Dr . E. S. Todd. Subcontractor e f f o r t s on f i n i t e element tasks a t MARC Analysis Research are l e d by D r . J. C. Nagtegaal.

PRECEDLNG PAGE BLANK NOT FILMED

iii

TABLE OF CONTENTS

Sect ion

1 .0 SUMMARY

Page

1

2.0 LITERATURE SURVEY

2.1 In t roduc t ion 2.2 Var ia t iona l Formulation and Element Technology 2.3 A P o s t e r i o r i Error Estimates and Post-Processing 2.4 Nonlinear Analysis of S o l i d s and S t r u c t u r e s 2.5 Concl udi ng Remarks

3.0 GLOBAL SOLUTION STRATEGY AND ELEMENT FORMULATION

3.1 In t roduc t ion 3.2

3.3 G1 obal Sol u t i on A1 gor i thms 3.4 Large Displacement Analysis

Semi d i screte Finite E l ement Equations and Temporal D i s c r e t i z a t i o n

4.0 COMPUTER PROGRAM DEVELOPMENT AND VALIDATION

4.1 In t roduc t ion 4.2 Computer Code Architecture 4.3 User I n t e r f a c e 4.4 Val i d a t i o n

4.4.1 Performance Improvement by the Profile So lve r 4.4.2 Vibrat ion Analysis o f a Rota t ing Beam 4.4.3 Large Displacement Analysis 4.4.4 Applicat ion o f the Subelement Method f o r I n e l a s t i c

Analysis

5.0 REFERENCES

DISTRIBUTION LIST

LISTING 1 INPUT DATA FOR A LARGE DISPLACEMENT ANALYSIS BY PLANE STRESS ELEMENTS

LISTING 2 GLOBAL FINITE ELEMENT MODEL

LISTING 3 SUBELEMENT MODEL

PRECEDING PAGE BLANK NOT FILMED

2

8

8 a

1 4 1 5

1 9

1 9 1 9 27 37 37 40 42 62

75

79

42

64

67

V

LIST OF ILLUSTRATIONS

Figure Number

1

2

3

4

8

9

1 0

11

1 2

1 3

1 4

1 5

Ti t le

Architecture of the MHOST Code Version 4.2

F in i te Element tlesh for Problem 1 (Turbine Blade Model w i t h P1 atform)

Finite Element Mesh f o r Problem 2 (Buckling Analysis of a Cyl inder )

Finite Element Mesh fo r Problem 3 (Composite Laminate Fan Blade)

Val i d a t i o n Model f o r Cent r i fuga l S t i f f e n i n g

E l a s t i c-P1 a s t i c Membrane ; Deformed Shape

Elast ic-P1 a s t i c Membrane: Contours o f Equivalent P l a s t i c S t r a i n

Elas t ic -Pl a s t i c Membrane: Contours of Mi ses S t r e s s

Compression of S o l i d Axisymmetric Cyl inder; Deformed Shape ( Hourgl assi ng 1

Axisymmetric Upset t ing: Equiva len t Mises Tensile S t r e s s MHOST Vs. MARC Resul t s

Axi symmetri c Upse t t ing : E i g h t Node Sol i d Model

Load Def l ec t ion Curves of a Clamped Square Plate Under Uniform Pressure Loading

Global and Subelement Meshes f o r an Embedded Hole

Global F i n i t e Element Mesh

Elastic S t r e s s D i s t r i b u t i o n f o r a Plate w i t h a Hole

Page

20

39

-

39

40

41

49

51

53

56

57

59

61

62

63

71

v i

Tab1 e Number

LIST OF TABLES

Ti t le

Updated Lagrangian A1 gor i t h m

MHOST Analysis Capabi 11 t y

Element Parameters

Performance Tests f o r the P r o f i l e Solver So lu t ion Package i n the MHOST Code

V i b r a t i o n Frequency-Canti 1 e v e r Beam Under Centri fuga1 Loading (Column A: Without S t i f f e n i n g ; Column B: W i t h S t i f f e n i n g and Centrifugal Mass)

Nodal S t r e s s Concentrat ion Fac to r a t the Edge o f a C i r c u l a r Hole After One I t e r a t i o n

E l a s t i c - P l a s t i c Analysis of a Square P l a t e w i t h a Circular Hole; Nodal S t r e s s a t the Edge of the C i r c u l a r Hole a t Maximum SCF Location ( P o i n t A )

Page

1 6

28

36

38

-

41

73

74

v i i

NOMENCLATURE

A 1 p h a be t i c a 1 Symbo 1

a

aj

E B

C

C

cs d

D

L)i jkl e

E

f

fj F

9

G

GI h

h

I

J

Description

Acceleration vector

Acceleration vector component

Discrete gradient operator

Strain-displacement matrix

Geometric constant

Damping matrix

Strain projection operator

Differential operator

Material modulus matrix

Fourth order tensor component of material modulus

Nodal strain vector

Young's modulus

Body force vector

Body force vector component

Nodal force vector

Gravity acceleration vector

Diagonalized Gramm matrix for finite element basis

Ith diagonal entry of diagonalized Gramm matrix

Thickness for plates and shells

Surface loading vector

Functional indicator

Jacobian matrix for the isoparametric transformation

K Displacement stiffness matrix

viii

NOMENCLATURE (continued)

Alphabetical Symbol

L

L

M

n

N

9 P

9

r

R

R

ROT

S

S

S

t

t

tj

U

U

V

V

Description

Lagrangian functional

Centrifugal effects matrix

Mass matrix

Unit normal vector

Finite element basis function

Cartesian component of the unit normal

Load at a given point

Pressure loading

Radial coordinate for axisymmetric geometry

Radius

Nodal Residual vector

Rotation tensor

Line search parameter

Nodal stress vector

Surf ace

Time

Traction vector

Traction vector component

Displacement vector

Right stretch tensor

Displacement vector component

Velocity vector

Space for admissible displacement variation

ix

NOMENCLATURE (continued)

Alphabetical Symbol

W

W

X

I xj

Greek Symbol

[a1

P

6 , A

V

r €1

ij r

5 9 1 1 9 s

Y

A

V

Description

Vector for BFGS update

Velocity vector component

Lateral deflection

Vector for BFGS update

Position vector

Cartesian component of the position vector

Description

Yield surface center location

Newmark constant

Incremental indicators

Gradient operator

Strain tensor

Strain tensor component

Isoparametric coordinates

Load factor for the arc-length method

Eigenvalue parameter

Poisson's ratio

Stress tensor

Stress tensor component

Material density

Angular (rotational) speed

Problem domain

Boundary o f the domain

X

NOMENCLATURE (continued)

Sub- and Superscripts

c (subscript)

e

E, EL

G (subscript)

h

i, j, k, ...

I, J, K, ... IYAX

PL

13 (subscript)

S (subscript)

t (superscript

T (subscript)

T (superscript)

- 1 ( superscript 1

030

DescriPt ion

Centr if uga 1 effects indicator

Quantities defined on each element

Elastic response

Geometric effects indicator

Finite element approximation

Indices for vector and tensor component (when used as subscripts); iteration and incrementation counter (when used as superscripts, also sometimes when used as subscripts with bold face indicators)

Nodal point counter

Maximum number

Plastic response

Recovery process indicator

Shear, transverse shear component

Time indicator

Tangent

Transpose o f matrix

Inverse of matrix

Initial state indicator

Derivative with respect to spatial (location) coordinate

Derivative with respect to time

xi

1.0 SUMMARY

The Special Fini te Element Models portion of the HOST (Hot Section Technology) 3-D Inel a s t i c Analysis Methods program i s d i v i d e d in to two 24-month segments: a base program, and an option program exercised a t the discretion of the Government. Versions 1 and 2 of MHOST (MARC-HOST) were developed d u r i n g the base program (Tasks IB and IIB). The MHgST codeemploys both shell and so l id (hexahedral ) elements i n a mixed i t e r a t ive solution framework. This code provides comprehensive capabi l i t ies for investigating 1 ocal ( s t r e s s / s t r a in ) and global (vibration and buckling modes) behavior of turbine engine hot section components. In the development of MHOST, advantage has been taken of the technical expertise of engineers a t MARC Analysis Research Corporation. T h i s has led t o the construction of sophisticated algorithms and codes which may reduce computer time and memory requirements for three-dimensional analyses.

Version 4 of the MHOST code has been developed under Task VB of the HOST 3-D Inelastic Analysis Methods option program. These l a t e s t development e f fo r t s have concentrated on the consol idation of the numerical technology created and implemented for the basic mixed i t e r a t ive solution strategy. In addition, the subelement solution strategy designed to capture 1 ocal behavior re1 ated t o embedded d i sconti n u i t i e s has been i nvesti gated further.

Efforts have been devoted to maintain the computer program on d i f fe ren t computer systems and to improve qual i ty. An in-core prof i le solution subsystem has been added, rep1 acing the or i g i nal band matri x sol u t i on. The computational procedures for eigenvalue extraction and t ransient time integration have been improved and been made compatible w i t h this new solution package.

Enhancement of the solution capabi l i t ies of the MHOST code include the large displacement option to update nodal coordinates, the s t r e s s s t i f fening option fo r quasi-static and eigenvalue analysis and the centrifugal mass option for a structure rotating a t a h i g h angular velocity.

Algorithms developed for and implemented into the MHOST code u t i l i z ing the mixed global so lu t ion strategy and subel ement techniques are discussed i n detail i n this report i n conjunction w i t h remarks on programming considerations.

2.0 LITERATURE SURVEY

This sec t i on reviews 1 i t e r a t u r e on the a p p l i c a t i o n o f nonl inear f i n i t e element methods t o the i n e l a s t i c analys is o f t u r b i n e engine h o t sect ion components. It i s no t intended t o discuss here the s ta te -o f - the -a r t f i n i t e element technol ogy b u t t o provide a snapshot o f the c u r r e n t t rends i n research and development f o r t he pe r iod s ince the l a s t annual r e p o r t [NAKAZAWA (198611. An attempt i s made i n t h i s survey t o p o s i t i o n the technology developed under the HOST con t rac t i n the computational mechanics perspective.

2.1 INTRODUCTION

The major areas o f research i n the improvement o f computational performance and a p p l i c a b i l i t y o f f i n i t e elements have continued t o be: i

1. Mixed and h y b r i d formulat ions designed t o improve the accuracy o f f i n i t e elements

2. G1 obal sol u t i o n procedures f o r 1 i near and nonl i nea r f i n i t e e l ement equations t o reduce both memory requirements and the number o f arithmetic operations with particular emphasis on iterative solution s t r a t e g i e s

3. Post-processing methods designed n o t on ly t o est imate e r r o r s accurate ly a p o s t e r i o r i , l ead ing t o adaptive mesh ref inement s t ra teg ies, b u t a lso t o e x t r a c t more in format ion o u t o f a f i n i t e e l emen t sol u t i on .

Both modern numerical technol ogy and advanced computing machinery now a1 1 ow the a p p l i c a t i o n o f f i n i t e elements t o a wider range o f problem areas. The use o f l a r g e meshes f o r represent ing f i n e d e t a i l i n engineer ing problems has become very valuable. Sophist icated numerical methods have been developed t o deal w i t h combined nonl i nea r k inemat ics and ma te r ia l responses i n a h i g h l y e f f i c i e n t manner.

This sec t i on concentrates on the repo r t s and a r t i c l e s which have appeared since the previous 1 i t e r a t u r e survey repo r t s [NAKAZAWA (1 986) Section 3.1, and

. FYHRE, HUGHES (198311.

2.2 VARIATIONAL FORMULATION AND ELEMENT TECHNOLOGY

I n t e r e s t continues t o remain h igh i n using the mixed v a r i a t i o n a l formulat ion as a basis f o r developing f i n i t e element methods. The Hu-Washizu p r i n c i p l e has become the most popular v a r i a t i o n a l formulat ion from which numerical methods are derived. It seems the most natura l approach, f o r instance, from which t o der ive a1 g o r i thms f o r probabi l i s t i c s t r u c t u r a l analysis. Eva1 ua t i on methods f o r f i n i t e elements based upon mixed v a r i a t i o n a l p r i n c i p l e s have a l so been developed recen t l y . I n modern l i t e r a t u r e , the words 'mixed' and ' hyb r id ' , as

2

appl ied t o f i n i t e elements, are often used interchangeably and ambiguously. Seemingly, the mixed methods derived from piecewise discontinuous stress representations are usually referred to as ' hybrid' formulations, while those derived from independent s t ra in interpolations as well as other mixed formulations are referred t o as 'mixed' forms. I t i s interest ing t o note moreover tha t the word 'hybrid' i s even used by a certain sector of computational mechanics a s referring to a c lass of methods combining the f i n i t e element and boundary integral formulations.

Interest i n the Hu-Washizu principle has been revived i n recent years by the untimely death of one of the original investigators and the subsequent publication of his memorial volume [PIAN (1984)l. Util ization of this principle for the analysis of existing methodology, such as the numerically integrated displacement formulation [ZIENKIEWICZ, NAKAZAWA (1 98411, appears frequently i n the 1 i terature. BELYTSCHKO (1 986) presents an excellent review of the technology available for thick plate and shell element formulations based on the Rei ssner-Mi ndl i n theory.

Efforts t o use the mixed Hu-Washizu f i n i t e element equations as a d r i v i n g mechanism for the i t e r a t ive solution of l inear and nonlinear problems continue as a par t of the HOST project, as reported i n progress reports by WILSON, BAK, NAKAZAWA, BANERJEE (1984, 19851, and NAKAZAWA (1986). Also, a number of papers discuss various aspects of this methodol ogy [ NAKAZAWA, NAGTEGAAL (1 9861, and NAKAZAWA , SP I EGEL ( 1 986 11.

W i t h par t icular application t o t ransient dynamics, a version of the mixed i t e r a t ive solution algorithm combined w i t h a second order, single step time integration operator has been proposed and applied t o a number of l i nea r e l a s t i c problems by ZIENKIEWICZ, LI, NAKAZAWA (1986).

An analysis of a cer ta in c lass of mixed f i n i t e elements i s discussed by LE TALLAC (1 986 1. An i t e r a t ive a1 gori t h m based on the Hu-Washizu pri nci pl e i s derived for rate-independent deviatoric p las t ic i ty . The e l a s t i c s t r a in i s used as a variable instead of the total strain. T h i s formal a l terat ion of the formulation, i n conjunction w i t h the radial return algorithm being hardwired to t h e analysis, enabled Le Tallac t o discuss the solution existence ques t ion ; this then a1 1 ows h im t o consider the possible formal i za t ion of convergence character is t ics .

For the evaluation of element formulations, as well a s being the f i r s t step o f f ini te element code validation, the patch t e s t invented by Irons and documented i n BAZELEY, CHEUNG, IRONS, ZIENKIEWICZ (1966) has been used by many researchers and commercial code developers. For instance, NAGTEGAAL, NAKAZAWA, TATEISHI (1986) discuss the development of an eight-node shell element based on the assumed s t r a in approach and demonstrate i t s val idi ty by performing an extensive s e t of patch t e s t calculations. The concept of patch test i s re-examined i n a recent paper .by TAYLOR, SIMO, ZIENKIEWICZ, CHAN (19861, where the equivalence between the sat isfact ion of the patch t e s t and the consistency and s t a b i l i t y of f i n i t e element approximations i s discussed. Note tha t the argument by STUMMEL (1980) seriously questioning the val idi ty of the patch t e s t i s countered i n this paper.

3

An important extension o f the patch t e s t i s proposed i n a more recent paper by ZIENKIEWICZ, QU, TAYLOR, NAKAZAWA (1986). Assuming t h a t t he consistency o f the approximations i s s a t i s f i e d , the t e s t focuses upon the s t a b i l i t y o f mixed f i n i t e e l ements imp1 i e d by the Babuska-Brezzi condi ti on [BABUSKA (1 973 ) , and BREZZI (1 974) 1. The procedure i nvol ves count ing the m i nimum possi b l e number o f a c t i v e degrees-of-freedom i n a patch and comparing t h i s number w i t h the maximum poss ib le number o f c o n s t r a i n t degrees-of-freedom. If no cons t ra in t s are imposed on nodal stresses and s t r a i n s , t he mixed i t e r a t i v e s o l u t i o n procedure w i t h continuous i nterpol a t i o n passes the t e s t and i s assumed stab1 e. An extension o f t h i s s t a b i l i t y patch t e s t t o the three f i e l d formulat ions o f Hu-Washizu type and t o p la tes and s h e l l s der ived from the Reissner-Mindl in theory i s being i nves t i ga ted by ZIENKIEWICZ (1986) and h i s co-workers.

The r e l a t i o n between the s t a b i l i t y patch t e s t and the mathematical ly der ived Babuska-Brezzi cond i t i on i s a1 so being act1 ve ly i nves t i gated.

Use o f the most general mixed v a r i a t i o n a l formulat ion o f t he Hu-Washizu type i n element development continues t o be i n the mainstream o f research a c t i v i t i e s . Despite the s l i g h t confusion i n terminology, a fam i l y o f simple h igh performance elements has been developed u t i l i z i n g , consciously o r unconsciously, the mixed method concepts.

I n t e r e s t i n the equal order i n t e r p o l a t i o n f o r mixed f i n i t e element processes is d e f i n i t e l y growing. A number o f papers and repo r t s have been publ ished n o t on ly v i a t h i s p r o j e c t b u t a l so by the research group a t Swansea. Also, a number o f associated academic research p r o j e c t s have been i n i t i a t e d i n the l a s t couple o f years. It i s a n t i c i p a t e d t h a t the number o f pub l i ca t i ons on the u t i l i z a t i o n o f the equal order i n t e r p o l a t i o n mixed method and i t s i t e r a t i v e s o l u t i o n s t ra tegy w i l l increase considerably i n the next few years.

2.3 A POSTERIORI ERROR ESTIMATES AND POST-PROCESSING

The methodology used f o r computing the approximation e r r o r i n a f i n i t e element s o l u t i o n has been improved s i g n i f i c a n t l y i n the l a s t few years. Also, the adaptive mesh refinement algor i thms based on the s p a t i a l d i s t r i b u t i o n o f e r r o r i n d i c a t o r s have been developed and app l i ed t o a wide range o f l i n e a r and nonl inear problems. A book on t h i s subject , compiled by Gago, represents present s ta te -o f - the -a r t techno1 ogy [BABUSKA, ZIENKIEWICZ, GAGO, OLIVEIRA (1 98611.

The papers inc luded i n the book i n d i c a t e t h a t a p o s t e r i o r e r r o r est imates and adaptive ref inement methodologies are we1 1 understood mathematical ly and tes ted f o r a wide range o f l i n e a r e l a s t i c problems. However, the computational processes look ove r l y compl i c a t e d and extensions t o nonl i nea r analys is o f s o l i d s and s t ruc tu res w i l l need f u r t h e r research and development. App l i ca t i on o f the adaptive mesh ref inement concept has been demonstrated t o be t r a c t a b l e using a r e l a t i v e l y simple e r r o r i n d i c a t o r f o r so lu t i ons t o aerospace f l ow problems modeled by the compressible Euler equations. Note t h a t the adapt ive s o l u t i o n o f the compressible Euler equations i s the f i r s t systematic at tempt a t extending the technology t o nonl inear problems. The r e s u l t s inc luded i n the book show the p o t e n t i a l advantage o f the concept.

4

A simple a lgor i thm t o est imate e r ro rs a p o s t e r i o r i i s proposed by ZIENKIEWICZ, ZHU (1987). The l o g i c i s based on the d i f f e rence i n values f o r q u a n t i t i e s obtained d i r e c t l y w i t h i n the elements and the same in fo rmat ion pro jec ted by nodal i n t e r p o l a t i o n funct ions. The nodal p ro jec t i on techniques used i n the mixed, i t e r a t i v e process t o recover the nodal s t r a i n a l l ow a simple and e f f e c t i v e way t o ca l cu la te e r ro rs . Th is process i s r e l a t i v e l y easy t o implement, i n comparison w i t h o ther seemingly complicated mathematical ly der ived a p o s t e r i o r i e r r o r est imate algor i thms. Examples shown i n the paper i n d i c a t e the e f f i c i e n c y o f the proposed process. However, r igorous mathematical ana lys is i s s t i l l t o be done t o v e r i f y the methodology. It i s i n t e r e s t i n g t o note t h a t the res idua l vector ca lcu la ted a t the zeroth i t e r a t i o n o f the equal order mixed i t e r a t i v e s o l u t i o n i s indeed the Zienkiewicz-Zhu e r r o r i n d i c a t o r weight-averaged a t nodes.

With t r i a n g u l a r elements being employed, e f f o r t s t o combine automatic mesh-generation and adapt ive ref inement concepts have been reported. Examples inc luded i n ZIENKIEWICZ, ZHU (1987) use t h i s idea. I n the a p p l i c a t i o n o f adapt ive mesh ref inement t o compressible f l ow problems, a number o f papers have been publ ished i n which t r i a n g u l a r mesh generat ion methods are discussed i n conjunct ion w i t h l o c a l ref inement s t ra tegy t o capture shocks.

PERAIRE, MORGAN, ZIENKIEWICZ (1986) and LOHHER (1 986) descr ibe adapt ive mesh ref inement and de - re f i nement a1 g o r i thms designed f o r t r i a n g l es and specul a t e on the a p p l i c a b i l i t y o f such a concept t o three-dimensional computations. The use o f automatic mesh generat ion o f t r i a n g u l a r and te t rahedra l elements has been found t o be v iab le i n f i n i t e d i f f e rence flow computations i n v o l v i n g unst ructured gr ids. JAMESON (1 986) demonstrated poss ib le u t i l i z a t i o n o f t h i s techno1 ogy f o r geometr ica l ly complex problems.

To summarize then, i n the area o f a p o s t e r i o r i e r r o r est imates, post-processing and adapt ive mesh ref inement, usefu l technology has been devel oped f o r 1 i near s t r u c t u r a l analys i s and c e r t a i n nonl i near probl ems. However, f u r t h e r research and devel opment i nc l ud i ng extensive numerical experiments w i l l be needed t o establ i s h a methodology base f o r three-dimensional i n e l a s t i c analyses i n v o l v i n g complex l oad ing h i s t o r i e s .

2.4 NONLINEAR ANALYSIS OF SOLIDS AND STRUCTURES

App l ica t ion o f f i n i t e element methods t o nonl inear s t r u c t u r a l ana lys is invo lves i nc reas ing l y complicated geometrical and mater ia l model s and pushes c u r r e n t l y a v a i l ab1 e supercomputi ng f a c i 1 i t i e s t o t h e i r 1 i m i t s . Codes s p e c i f i c a l l y designed f o r supercomputers have demonstrated such computers ' po ten t ia l . Papers have appeared recen t l y discussing performance improvement o f f i n i t e e l ement codes on vector and para1 1 e l machines u t i 1 i z i ng a1 g o r i thms t a i l o r e d f o r a s p e c i f i c computer environment. BENSON, HALLQUIST (1986) discusses the r i g i d body a lgor i thm implemented i n the DYNA code which reduced the amount o f computations s i g n i f i c a n t l y f o r e x p l i c i t f i n i t e element t ime in tegra t ions . Implementation of an i t e r a t i v e so lu t i on a lgor i thm based on the e l ement-by-element precondi t i o n e r i n the N I K E code i s discussed by HUGHES, FERENCZ, HALLQUIST (1986). The u t i l i z a t i o n o f supercomputers i n an engineer ing envi ronment i nvol v i ng the devel opment o f these codes i s presented by GOUDREAU, BENSON, HALLQUIST, KAY, ROSINSKY, SACKETT (1 986).

5

Investigations on vectorizabil i t y of basic f i n i t e element operations continue. For instance, the de ta i l s of vector-matrix multiplication i n an element-by-element manner a re discussed, including t iming figures, for various f i n i t e element models by HAYES, DEVLOO (1986). An algorithm t o take fu l l advantage of vector machines i n solving a system of ordinary different ia l equations i s proposed by BROWN, HINDMARSH (1986). I te ra t ive algorithms based on the ORTHOMIN method and incomplete factorization a re studied by ZYVOLOSKI (1986). I t i s anticipated tha t the development of algorithms closely t i ed t o the actual data manipulations and arithmetic operations i n computing machines will eventually lead to f a s t e r f i n i t e element codes not only on supercomputers b u t a1 so on small er scal a r machi nes.

, I

Utilization of parallel computing for equation solution i s a subject investigated extensively i n the l a s t few years. A se r i e s of papers by U t k u , Me1 osh and thei r col 1 aborators 1 ooks i n t o the possi b l e para1 1 el i zation of d i rec t s t i f fnes s matrix factorization [ UTKO, MELOSH, SALAMA, CHANG (1 986) ; and UTKU, SALAMA, MELOSH (1986)l. The possible u t i 1 ization of hypercube architecture i s s tudied by NOUROMID, ORTIZ (1986) for the factorization and i t e r a t ive so lu t ion of f i n i t e element equations. I t i s observed i n these papers t ha t a simple reduction i n the total number of operations does not i t s e l f always provide the optimal algorithm for vector and parallel processing. An increas ing number o f sessions a t s c i e n t i f i c meetings are now b e i n g devoted t o d i scussi ng t h i s aspect of f i n i t e el ement computations , and extensive l i t e r a t u r e i n this area i s being accumulated. However, fur ther research and development s t i l l needs t o be done before robust a1 gor i thms become avai 1 ab1 e f o r a wide range of engineering applications i n order t o ut i l ize ful ly the potenti a1 of modern computing machinery .

,

Notable progress has been made i n the l a s t few years i n the fie1 d of computational p las t ic i ty , i n par t icular , des ign and analysis o f integration a1 gori thms for rate-independent p las t ic i ty const i tut ive equations. ORTIZ, SIMO (1986) summarizes the development of such integration algorithms and investigates their accuracy i n great de ta i l . The basic idea i s to generalize the return mapping algorithm based on the e l a s t i c predictor. Application t o v i scopl a s t i c const i tut ive equations i s a1 so incl uded i n this paper. SIMO (1986) extends the resu l t s t o f ini te deformation p las t ic i ty by developing a simple and e f f i c i en t f i n i t e element algorithm i n which e l a s t i c deformation of f i n i t e amp1 i tude i s accommodated.

global f i n i t e element computations have been developed which enable discontinuities t o be modeled a t the subelement scale. The work by HUGHES, SHAKIE (1986) extends the conventional return mapping algorithm for J2 flow theory to y ie ld surfaces w i t h corners. This extension enables the e f fec ts of localized p las t ic flow to be captured i n a global manner. ORTIZ, LEROY (1986) proposes an algorithm which allows shear bands t o generate inside a f i n i t e element. The numerical resu l t s indicate the potential of the method for capturing local fa i lure of structures without excessive mesh refinement. Methods for b r i n g i n g i n local fa i lure modes i n f i n i t e elements t o capture

I Approaches for capturing localization due t o ine las t ic material response i n

I

6

localization e f fec ts under s t ra in softening are discussed by WILLAM, SOBH, STURE (1 986 1. A survey of the f i n i t e element techno1 ogy fo r capturing localized fa i lure modes i s given by NEEDLEMAN (1986). T h i s is a re la t ively new f i e l d and further work will be needed before localized microscopic material responses due t o embedded si ngul a r i ties i n 1 arge-scal e structural systems can be cal cul ated effectively.

2.5 CONCLUDING REMARKS

In the theoretical aspects of f i n i t e element development, the u t i l ization of the mixed variational principle has been and continues t o be the main thrust of research and development work, T h i s approach not only provides tools t o improve the performance of elements b u t also leads t o a deeper i n s i g h t on the solution algorithms for l inear and nonlinear problems. The mixed i t e r a t ive methodology developed under the HOST contract takes fu l l advantage o f these modern theoretical developments. A1 so, new ideas demonstrated i n the MHOST program have at t racted s ignif icant academic research in t e re s t i n the computational mechanics community.

In the appl ication and computational aspects o f f i n i t e el ements, progress made i n the l a s t few years has not y e t been ful ly incorporated into the MHOST code development. The information available i n the l i t e r a t u r e indicates t ha t fur ther development t o u t i l ize modern algorithms and coding techno ogy would enhance the performance of the MHOST code even more s ignif icant ly .

7

3.0 GLOBAL SOLUTION STRATEGY AND ELEMENT FORMULATION

3.1 INTRODUCTION

This section suriimarizes the technology developed and implemented in the MHOST program during Task VB.

The mixed variational formulation of Hu-Washizu type [HU (1955), and WASHIZU (1955, 1974)] is used as a driving mechanism to derive the iterative solution algorithms. A system of algebraic equations is generated, including the effects of the stress stiffening and the centrifugal mass terms in conjunction with the dynamic transient effects in a semi-discrete manner. The fully-discretized version of the finite element equation system is included in this report, with the Newmark family of algorithms used for temporal discretization. As demonstrated in ZIENKIEWICZ, LI, NAKAZAWA (1986), other time integration algorithms can be utilized to generate recursive forms for transient analyses.

The implementation of iterative solution algorithms for the mixed finite element equations is briefly discussed in this section. A comparison of options made available in the MHOST code indicates that the secant version of the Davidon rank-one quasi-Newton update performs most reliably, except for a class o f ill-conditioned problems.

The updated Lagrangian forniulation is used to facilitate large displacement effects. The inclusion of finite strain effects provides an experimental capability in the latest version of the MHUST code. A subsection is devoted to t ii i s de ve 1 o pme n t . Element formulations based on the selectively reduced integration and the assumed strain approaches are investigated. In the mixed iterative solution framework, eleriient formulations applicable to anisotropic material response with rate independent deviatoric plasticity are identified as requiring further investigation.

The subelement method is extended to handle inelastic response inside subelement regions. The validation cases included in this context indicate the performance of this new approach in capturing local inelastic responses around embedded singular i t ies.

3 . 2 SEMIDISCRETE FINITE ELEMENT EQUATIONS AND TEMPORAL DISCRETIZATION

The finite element equations of dynamic equilibrium for a deformable body can be written in terms of the nodal acceleration vectora, the nodal velocity vector v and the nodal stress vector s as follows:

8

where M i s the mass matrix defined by

M = P fNTNdx

with p and N being the material density and the vector of interpolation functions respectively. The damping matrix C m a y be defined as a sum of mass and stiffness matrices,MandK, such that

C = P1M+ Pz K ,

witti P i and P 2 being preassigned constants representing the effects o f viscous and structural damping respectively. Symbolically, the stiffness rnatrix can be written by

K = /ONTO VN dx , sz

( 3 )

( 4 )

with V being the gradient operator and D t h e material modulus. The third term in Equation (1) represents the internal energy, with the B matrix being the discrete gradient operator defined in the finite element approximation subspace such that

B = h N T N dx. ( 5 1

sz Here an equal order interpolation using the same basis functions is assumed for the acceleration, velocity, displacement, and stress. The nodally interpolated strain is recovered via

e=G" STu, (6 1

where G is the diagonalized gram matrix whose Ith diagonal entry is calculated by

NP i- GI = c J NI NJ dx,

J=1 ( 7 )

with Np being the total number of nodes in a mesh. The nodal stress state is recovered by

s = D e . ( 8 )

dote that a l l the integrations indicated in the above equations are evaluated approxiiiiately using numerical quadrature. The particular choice o f numerical integration rule is discussed in the subsection on element formulation.

9

Next, an abstract iterative process is constructed for the semi-discrete Equations (1) by means of displacement preconditioning. The approximation of internal energy by the product o f stiffness matrix Kand total displacement vector u leads to the recursive expression

with subscripts denoting the iteration counter. For linear elastic problems, the displacement converges linearly given an appropriate initial condition such as uo = U.

Note that the strain projection procedure and the stress recovery operation, Equations (6) and (81, are executed every time the displacement vector i s updated.

The fully discretized systeni of equations is derived for the mixed iterative solution by introducing the temporal approximation. For the sake of simplicity and clarity, the Newmark family o f algorithms is introduced in a finite difference fashion, e.g., HUGHES (1984), such that dynamic equilibrium is satisfied at time t + A t as follows:

I M + C v ~ + ~ ~ + B s t+At = Ft+At (10)

T h i s then leads t o a time discrete displacement preconditioning

with the updates for displacement and velocity given by

Aitn the aid o f

Aut+At = u t + A t - t l i i

and

~

a recursive form is obtained as follows:

I 10

+(;- I ) c v t +($- 1 ) A t c at

'(&) vt + ( & - 1) M at,

where the incremental di spl acement Jui t+At +1 i s updated by

and the increriiental strain is recovered at each node by

Aet+At = ~ - 1 BT Aui+At i

(16)

In a general setting of history dependent inelastic constitutive integration, the stress recovery process is written as

which is fed into the equilibrium iteration defined by Equation (16).

The coinputational effort in the above iterative process for direct time integration o f the discrete equations of motion involves assembly of the element operator matrices which appear on the left-hand side of Equation (16). T h i s requires exactly the same amount o f effort as to solve the problem by the conventional displacefiient method. Additional computations required for the mixed solution involve a few back substitutions and strain and stress recovery at nodes, computations which are relatively inexpensive compared t o matrix assembly and factorization. This additional computational cost would vanish when the scheme is applied to nonlinear problem in which matrix assembly and factorization or back substitution need to be performed repeatedly no matter which finite element method is used to drive the solution. Therefore, the

1 1

seemingly complicated derivation of the mixed iterative solution scheme does liot increase the coriiputational effort in comparison with the displacement formulation. Thus, the advantages of equal order interpolation of displacewent, strain and stress can be fully utilized without fear of central processing unit (CPU) time penalties.

With particular attention to turbo machinery blades and other rotating structures, the stiffness matrix can be modified to include the effects of the stress stiffening and the centrifugal mass terms. These results are based on linearized models of the large displacement and corresponding follower force equations under a given angular velocity.

The centrifugal load vector i s generated by integrating the body force

F O = N T p u * r ( x ) d x ,

witn o being the angular speed and r the vector distance between a point and the axis of rotation. Under the centrifugal loading given above, the nixed iterative algorithm generates a certain stress field represented by the nodal stress vector so such that the discrete equilibrium equation

B so = Fo ( 2 1 )

is satisfied. Our interest here is to calculate the linearized response o f the structure under an initial state of stress given by so. If these initial stress term are taken into account, the equilibrium equation system is modified (excluding the damping term) to

(22 1 T M a + KE u + 6 6 = F ,

whereKE is the geometric stiffness term associated with the initial stress

field so and is given by

KE = VNT so VN dx .

M ( 2 3 )

Note that Equation ( 2 2 ) is a general expression for the linearized response of structural systems under a prescribed initial stress field.

0 dhen this tend, i.e., Kp, is included in the nixed iterative computations, t t i e abstract recursive forin, Equation (91, becoines:

1 2

or in terms of displacement update 0 (KE t K) Aui= F - ( B si - K G Ui),

with = ui t Aui . ui+ 1

In the residual force calculations, the initial stress matrix needs to be evaluated repeatedly with the stress field calculated at the beginning of the i ncreriieti t . Tile additional force due to the relative motion froin the equilibrium position, referred to as the centrifugal inass term, is also added to the equilibrium equation system resulting in

Ma+(K:-M,)u tBs = F ,

where Mc is the centrifugal mass matrix defined by

Mc = pu2 NT L N dx , a

with L being a matrix based upon the components of the unit vector m parallel to the axis of rotation. Lis given by

L =

2 2 2 3 -"1"2 -m3m 1

2 2 -m1m2 m3tml '?"3

2 2

m t rn

m +m '"3" 1 -"2"3 1 2

When this term is activated i.n the quasi-static and transient dynamic conputations, an additional correction of the residual vector calculation i s

lliade by replacing K o with (KE - Mc). Typically, these optional terms are activated for the vibration analysis of rotdting structures by extracting the modes o f the eigenvalue problern

u

{M- A(K;-M~+K)} x = o .

1 3

In the modal analysis, the displacement preconditioner is assumed t o represent the structural stiffness. However, the s t r e s s values used i n the evaluation of i n i t i a l s t r e s s terms are calculated by the mixed i t e r a t ive method. Thus , a more accurate s t r e s s fie1 d i s generated than via the conventional displacement method. Hence, the solution of Equation (29) is expected to be somewhat more accurate than tha t associated w i t h the displacement method approach.

For de t a i l s of derivations of i n i t i a l s t r e s s and centrifugal mass matrices and ful ly worked o u t examples, see THOMAS, MOTA SOARES (1973) and RAWTANI,

I DOKAINISH (1 971 1. In summary then, Equation (16 ) represents the recursive form of the basic time discrete displacement preconditioning as given by Equation (1 1 ; furthermore , Equation (11 ) i t s e l f i s a discretized form of the original basic equation of dynamic equilibrium as given by Equation (1 ) . The bottom l ine , however, i s tha t the sequence of computations for the basic equation s e t i s given by the order of Equations (16) through (19) . For centrifugal load e f fec ts , Equations (24) and (25) replace Equations (16) and ( 1 7 ) and Equations (20) and (23 ) are also added.

I 3.3 GLOBAL SOLUTION ALGORITHMS

In the present implementation o f the mixed i t e ra t ive so lu t ion procedures, the displacement s t i f fness equations need t o be assembled and factorized. To enhance the efficiency of the MHOST code, a new profi le solver algorithm has been devel oped to rep1 ace the constant bandwidth Crout decomposition algorithm. The implementation of this new profi le solver i s based on code published by TAYLOR (1985). Examples presented i n t ha t paper include both symnetric and nonsymmetric cases us ing Crout decomposition. Imp1 ementation o f the prof i le solver i n t o the MHOST code excludes a nonsymmetric capabili ty i n order to gain maximum efficiency. The prof i le storage of the global s t i f fnes s equations reduces memory requirements as well as time required for factorization. Additional computational overhead i s required t o prepare the elimination table fo r a s t i f fnes s matrix stored i n prof i le form. A table comparing the performance of these solvers i s included i n Section 4.4.1.

In the quasi-static computations, a se r ies of options t o accelerate the ra te of convergence for nonl inear i t e ra t ions has been made operational w i t h the new profi 1 e sol ver capabi 1 i ty . These options are as fol 1 ows : the modi f ied Newton method, the quasi-Newton method of inverse BFGS update, the secant Newton implementation of Davidon rank-one quasi-Newton update, the conjugate gradient method, and the line search method. The spherical path version of the arc length method incorporating modified Newton i te ra t ion i s a1 so now available w i t h the prof i le solver capability. For de t a i l s of the implementation of these numerical processes i n the m i xed i te ra t ive sol ution framework, see the t h i r d year progress report [ NAKAZAWA ( 1986 ) 1.

The eigenvalue extraction procedure implemented into the MHOST code is based on the subspace i te ra t ion approach documented by BATHE, WILSON (1976). In this process, the large eigenproblem of structural systems i s mapped into a subspace of f i n i t e dimensions and Jacobi i t e ra t ion i s performed t o solve the smal 1 e i genprobl em s e t i n the subspaces. T h i s eigenvalue extraction procedure

i s used t o compute the natural frequency and vibration mode of unstressed and prestressed s t ructures as well as the buckling analysis of prestressed structures undergoing e l a s t i c or ine las t ic deformation of infinitesimal or f i n i t e amplitude.

Note here tha t i n the eigenvalue extraction, the standard displacement s t i f fness matrix i s used to represent the response of structural systems. However, i n buckling and modal analyses w i t h prestress, the improvement of the s t r e s s f i e ld generally improves the quality of the solution considerably. The u t i l i za t ion of mixed s t r e s s interpolation i n the eigenvalue analysis can be viewed as equivalent t o a perturbed eigenvalue problem i n which the e f f ec t of the independent stress approximation i s regarded as the perturbation t o the displacement f i n i t e element system. I t i s anticipated tha t an algorithm proposed by SIMO (1985) and implemented for the probabilist ic f i n i t e element code a t MARC may be usable for eigenvalue extraction of mixed systems of f i n i t e element equations.

3.4 LARGE DISPLACEMENT ANALYSIS

In this section, the large deformation algorithm implemented i n the mixed i t e r a t ive method i s described. The formulation u t i 1 izes an updated Lagrangian mesh approach. The choice of deformation and s t r e s s measures i n f i n i t e deformation analysis depends primarily on the class of material involved, along w i t h the necessity that the measures are objective ( invariant) for r i g i d body motions. For e las t ic -p las t ic behavior where the response depends primarily on the current s t a t e of s t r e s s , a Cauchy s t r e s s and rate-of-deformation ( i .e., s t ra in r a t e ) consti tutive formulation i s frequently most convenient.

Beyond the choice of a Lagrangian mesh description, the formulation can be further simplified by evaluating the equations o f motion a t the current (deformed) configuration. T h i s i s achieved by continually updating the mesh geometry t o the current s ta te . Utilizing an updated geometry resu l t s i n simplification because a l l matrix expressions take the same form as i n small deformation theory. The only additional quanti t i e s needed are deformation g r a d i e n t s and r o t a t i o n tensors eval uated a t nodes, a1 ong w i t h f o l l ower force matrices. T h i s formulation is referred t o as the updated Lagrangian approach i n the l i t e r a tu re .

The updated Lagrangian algorithm for large deformation problems i s essent ia l ly the same as tha t for nonlinear small deformation problems, w i t h the exception tha t a l l tensors and integrals are evaluated w i t h respect t o the current configuration. The updated Lagrangian a1 gori t h m presented i n this section involves nodal coordinates being updated continuously and deformation gradients being evaluated a t nodes.

Table 1 presents a summary of the algorithm fo r an increment. All new terms i n this table are defined i n the Nomenclature section. Following in i t i a l i za t ion , the process i s repeated i te ra t ive ly u n t i l equilibrium has been sa t i s f ied .

1 5

Table 1 Updated Lagrangian Algorithm

Initialization:

i i-1 Lu]i = cup u = u

Equation Forrnulation and Displacement Solution:

Aui = ( Ki)-l Ri

update Geometry:

,i+l = ,i + ,i Aui

Fi+1/2 - i+1/2 Fi - Frel

Fi+l - - Fi+l Fi re 1

Strain Projection:

Large Strain Small Strain

Table 1 Updated Lagrangian Algorithm (continued)

Kotate Back to Global System:

i+l T [,~i+' = ROT"+^ [oji+'( ROT i+l = ROTi+l [E]~+'(ROT i+l 1 T CE1PL pLR

Forni Res idua 1 :

Ri+l = (N i+l ) T fi+l Ni+l dx i+l ni+l

I f converged, exit; otherwise update displacements via line search or displacement solution and repeat.

17

In the finite displacement calculations, the stiffness array is assembled in the following manner and repeatedly updated in the iteration loop unless another iteration process (modif ied-Newton, quasi-Newton, or secant-Newton option) is specified. Note that the stiffness matrix does not correspond to the classical Newton method in the mixed iterative solution framework because of the independent Co-continuous stress interpolation functions. Here, the conventional strain displacement matrix.

is

K i = s. -T B (x) i DT i - B(xi) dxi 52’

Tangent stiffness matrix

Initial stress matrix

T i - N (x fit’ N(xi) Ji ( Ji)-l dt ’ 5

Body force follower force due to volume change

- N T i (X hi+’ N(xi) Jf, ( J i ) -1 d, a t Surface traction follower

force due to surface area change

T i T i - [vN (x ) fit’ N(xi) t N (x ) fitlVN(xi)] dxi 52’

Body force follower force due t o rotation

- s . [vN T i (x ) hitlN(xi) t N T i (x ) hitlVN(xi)] dS a

Surface traction follower I force due to rotation

Whenever the stiffness matrix is reformulated in an updated Lagrangian algorithm, the shape functions, derivatives of the shape functions and determinants of Jacobians are evaluated at the nodal coordinates of the current configuration. In addition, the material and stress matrices used are those computed in the immediately preceding iteration.

I 18

4.0 COMPUTER PROGRAM DEVELOPMENT AND VALIDATION

4.1 INTRODUCTION

Version 4 of the MHOST code consists of over 47,000 lines of FORTRAN source statements including extensive comments. The concept of multiple-analysis drivers has been used to maintain the quality and efficiency of the code. A brief discussion of the architecture of the code is presented in this report.

Note that the multiple driver strategy has kept the code usable for well tested capabilities, while new options and enhancements of existing features were being developed and tested. Also, such a strategy has helped to keep the code readable since the number o f conditional statements is minimized. In order to use this code effectively for production purposes in engineering environments, further streamlining of the code would be needed to improve the performance of this finite element package.

The user interface has been enhanced to deal with numerous algorithmic options added to Version 4 of this code. A subsection briefly summarizes the program features and user commands.

All the development work has been carried out on a PRIME 9955 at MARC Analysis Research Corporation. The PRIME FORTRAN 77 compiler is employed without extensive uti1 iration of extensions to ANSI standard. Versions for VAX/VMS, IBM/CMS and CRAYKOS environments have been created and tested using their FORTKAN 77 compilers. The utilization of FORTRAN 77 features which are not supported by the previous FORTRAN standard have made the new code incompatible with the old FORTRAN compilers.

A profile solver approach has replaced the constant bandwidth Crout decomposition routines. The new solution package improves the efficiency of the code significantly in terms of both core storage and computing speed. A large portion of the MHOST code has been rewritten to take advantage of this new solver capability, including the quasi-static analysis, eigenvalue extraction and transient time integration subsystems. The out-of-core frontal solution subsystem remains unchanged and may be employed for large-scale small deformation quasi-static calculations.

4.2 COMPUTER CODE ARCHITECTURE

The architecture of the MHOST code is schematically shown in Figure 1. The execution supervisor routine (SUBROUTINE HOST) controls a multiple number of analysis modules in a consistent manner. I n Version 4 of the FlHOST code, a pair of subroutines has been added to check the consistency o f the Parameter Data and to generate internal flags for the selection of analysis modules. The objective here is to avoid combining features o f the program package not intended by the developers to be combined.

19

i

Bureaucrats SUBROUTINE LETCMD 8 SUBROUTINE RUNCMD

Data Input Modules SUBROUTINE DATlNl

and SUBROUTINE BULKIN

Incremental Data Input and

Initialization

Element Library

Incremental Report Generation

Utilities for Linear Constitutive Equation Algebra and Array

Library Manipulation

Figure 1 Architecture o f the MHOST Code Version 4.2

20

The analysis modules provide the control structure for the incremental iterative algorithms implemented in the MHOST code. The source program with comments is designed to serve as the schematic flow chart of the computational process.

The schematic flow of the element and nodal data manipulations is coded in the element assembly submodules which are entered from the analysis modules. The element assembly submodules perform operations independent of element type and constitutive model. The assumptions introduced in the mechanics aspects of the formulations are explicitly coded at this level. These submodules call the librarian routines for elements and constitutive models.

The element 1 ibrarian subprogram (SUBROUTINE DEHIV) generates quantities unique to the element used in an analysis. The MHOST code uses for the most part the standard finite element matrix notation, as in ZIENKIEWICZ (1977). The element specific information returned from the librarian subprogram is the strain-displacement array referred to as the matrix in previous subsections.

The constitutive equation 1 ibrarian subprogram (SUBROUTINE STRESS and BMSTRS) is designed to accommodate the nodal storage of stress and strain values. The STRESS subroutine sets u p the loop over the points at which constitutive equations are evaluated. The current implementation is a nested double loop with the outer loop being over the nodes and the inner loop over the integration layers through the thickness. Note that the conventional displacement method can be recovered by restructuring this loop in conjunction with a few minor modifications in the core allocation for stresses and strains.

The 1 ibrarian subprogram calls the constitutive equation package (SUBROUTINE NODSTR) from which individual subprograms for initial strains, stress recovery and material tangent are accessed. The librarian subprogram also controls the pre-integration of stresses, strains and material tangent over thickness for the shell element.

The evaluation of the constitutive equations is one o f the most costly operations in the nonlinear finite element computations. An attempt has been made to minimize the execution o f this process during t h e incremental iterative analysis, typically once every iteration during the recovery of the residual vector. At the beginning of each increment, this process may be executed to evaluate the material tangent as well as the contribution of initial strain terms which are often necessary for proper displacement pre-conditioning.

Except for a small amount of information related to the convergence of the iterative solution, all report generation is performed at the end of an increment. Optionally, post-processing and restart files are written at the end of user-specified increments. Generic reporting subprograms are called from analysis modules inside the loop over the increments.

21

A brief description o f major subprograms is provided below.

Execution Supervisor

MAIN PROGKAbl - declares the work space in blank common as an integer array. Also defines system parameters which are machine independent. Then passes control to the actual execution supervisor SUBROUTINE HOST.

SUBROUTINE HOST - controls the sequence of execution of analysis modules. First, this routine executes the control parameter data reader SUBROUTINE DATINl and checks for consistency by entering SUBROUTINE LETCMD and RUNCMD. The current structure of the code allows certain combinations of two analysis modules to be executed sequentially. Analysis modules called from the execution supervisor are as follows:

SUBROUTINE STATIC for a quasi-static incremental iterative solution.

SUBROUTINE DYNAMT for a transient time integration o f the dynamic equilibrium equation system i n an incremental- iterative manner.

SUBROUTINE MODAL for eigenvalue extraction in vibration mode analysis. This subsystem may be executed after a quasi-static analysis for modal analysis of prestressed structures.

SUBROUTINE BUCKLE for eigenvalue extraction in buckling load calculations. This subsystem is executable only after a quasi-static analysis.

SUBKOUTINE FRONTS for a quasi-static incremental iterative solution by the out-of-core frontal solution procedure. Note that certain options are not available in this subsystem.

SUBROUTINE SUPER for a linear dynamic response calculation by the method of mode superposition.

InDut Data Reader

There are three major subprograms:

SUBKOUTIiIE UATINl - Reads and interprets the parameter data input called by the execution supervisor. All the default values for control variables are set in this routine.

I 22

SUBROUTINE BULKIN - Reads and i n t e r p r e t s the f i n i t e element model d e f i n i t i o n data and p r i n t s the mesh and load ing data f o r the i n i t i a l increment (number 0). The bulk data reader i s entered before the execut ion o f ana lys is modules. This subrout ine u t i l i z e s the fo l l ow ing lower l e v e l rout ines:

SUBROUTINE INIT11 f o r the memory a l l o c a t i o n o f i n tege r work space i n the blank common t o s to re nodal and element data.

SUBROUTINE DATIN2 f o r the model data input .

SUBROUTINE DATOUl f o r repo r t i ng the model d e f i n i t i o n data.

SUBROUTINE CHKELM f o r the de tec t ion o f c lockwise element connec t i v i t y which r e s u l t s i n a negat ive Jacobian matr ix . This rou t i ne automatical l y cor rec ts the connec t i v i t y tab1 e t o a counterclockwise d i rec t ion .

SUBROUTINE SUBDIV f o r memory a l l o c a t i o n o f subelement mesh data and automatic mesh generation o f subelements.

SUBROUTINE I N C R I N - Reads and i n t e r p r e t s the load ing and c o n s t r a i n t data fo r each increment. This rou t i ne i s invoked by i nd i v idua l analys is modules from i n s i d e the loop over the increments.

SUBROUTINE DATIN3 - A small subset o f the bu lk data reader SUBROUTINE DATINl, i s used f o r actua l operat ions i ncl ud i ng i n i t i a l i z a t i o n o f arrays a t the beginning o f an increment.

A1 gebraic Operations Subsystem

There are th ree packages o f rou t ines f o r the p r o f i l e solver, f r o n t a l so lver and eigenval ue ex t rac t ion . The p r o f i l e sol u t i o n package cons is ts o f :

SUBROUTINE COMPRO - Sets up the in teger array f o r t h e p r o f i l e o f t h e global s t i f f n e s s equations t o be s tored i n a p r o f i l e form.

SUBROUTINE ASSEMS - Assembles the element s t i f f n e s s equations i n t o the g lobal equation system stored i n a p r o f i l e form.

SUBROUTINE SOLUTl - Control s the i t e r a t i v e so lu t i on processes i n c l u d i n g the vector update requ i red f o r the quasi- and secant-Newton i t e r a t i o n s .

SUBROUTINE DECOMP - Factor izes the global s t i f f n e s s equations s tored i n t o a

SUBROUTINE SOLVER - Performs the back s u b s t i t u t i o n and generates the update

I

p r o f i l e form.

vector f o r the incremental d i spl acement.

23

The frontal solution package consists of:

SUBROUTINE FRONTW - Estimates the front matrix s ize t o be accommodated i n the core memory.

SUBROUTINE INTFR - Allocates memory fo r the work space required for the frontal solution.

I SUBROUTINE PRFRNT - Sets u p the elimination table for the frontal solution.

SUBROUTINE FRONTF - Assembles and factor izes the global s t i f fness equations simultaneously.

SUBROUTINE FRONTB - Performs the back substi tution and generates the updates for the displacement vector.

SUBROUTINE FRONTR - Controls the i t e r a t ive solution processes including the vector update required f o r the quasi- and secant-Newton i terat ions. Calls SUBROUTINE FRONTB for the incremental displacement update vector.

and t h e actual out-of-core storage devices.

I

SUBROUTINE VDSKIO - Controls t h e d a t a stream stored i n the in-core buffer area

The eigenvalue analysis package consists o f :

SUBROUTINE EIGENV - Controls execution of the eigenvalue extraction subsystem.

SUBROUTINE INIMOP - In i t i a l i ze s the array for eigenvectors.

SUBROUTINE SUBSPC - Performs the subspace i te ra t ion and generates a specified number o f e i genval ues and e i genvectors.

SUBROUTINE JACOB1 - Solves the eigenvalue problem i n the subspace by Jacobi i terat ion.

There a re a number of subprograms used commonly by the Algebraic Operations Su bsy s tern :

SUBROUTINE STRUCT - Controls memory allocation for global algebraic manipulations a t the beginning of every increment.

SUBROUTINE INITI2 - Allocates memory required for storage of the g loba l s t i f fness matrix and other vectors necessary i n the l inear

SUBROUTINE LINESR - Calculates the search distance when the l ine search option

I algebraic manipulation of the f i n i t e element equations.

i s turned on.

Element Assembly Submodul es

These a re subprograms constructing vectors and matrices appearing i n the a1 gori t h m i c descri p t i on of the m i xed i terati ve process d i scussed i n previous subsections.

SUBROUTINE ASSEM1 - Assembles the d i spl acement s t i f fnes s matrix for preconditioning purposes. All the options for kinematic and consti tutive assumptions are tested i n this module.

SUBROUTINE ASSEM2 - Assembles the coeff ic ient matrix for the t ransient time in t eg ra t ion by the Newmark family o f algorithms. T h i s routine i s evolved from SUBROUTINE ASSEMl and contains a l l the options.

SUBROUTINE ASSEM3 - Assembles the coeff ic ient matrix f o r the quasi-static analysis u s i n g the frontal solution subsystem. Large displacement, s t r e s s s t i f fening and centrifugal mass terms are n o t available i n this package.

SUBROUTINE ASSEM4 - Calculates the nodal s t r a in and recovers the residual vector i n a mixed form. The subelement solution package i s entered from t h i s subprogram.

Element Loop Structure and Library Routines

In the element assembly submodules, element arrays a re generated i n loops over the elements. The protocol for accessing the element l ib rary involves a sequence o f subroutine c a l l s as follows:

SUBROUTINE ELVULV - Sets u p the current element parameters (see Table 2 for variables and values) from the element l ibrary table.

SUBROUTINE CNODEL - Pulls o u t quantit ies for the current element from the global nodal array and restores them i n the element work space. Coordinate t ransformat ions necessary f o r beam and she1 1 elements are performed i n t h i s subprogram.

SUBROUTINE DERIV - Sets up the displacement-strain matrix fo r the current element by cal l ing the element l ib rary subroutines. Those are :

SUBROUTINE BPSTRS f o r plane s t r e s s elements, types 3 and 101.

SUBROUTINE BPSTRN f o r plane strain elements, types 11 and 102.

25

SUBROUTINE BSOLID for three-dimensional solid element, type 7.

SUBROUTINE BSHELL f o r three-dimensional shell element, type 75.

SUBROUTINE BAXSYM for ax i symmetric sol i d-of-revol u t ion elements, types 1 0 and 103.

SUBROUTINE BTBEAM f o r linear Timoshenko beam element, type

SUBROUTINE BASPST f o r the assumed stress plane stress element, type 151 . SUBROUTINE BASPSN for the assumed stress pl ane strain element, type 152.

SUBROUTINE BASSOL for the assumed stress three-dimensional sol id element, type 154.

98.

SUBROUTINE UDERIV - A s l o t f o r a user coded element B matrix routine.

The following subprograms are used t o calculate terms appearing in the finite el ement equations :

SUBROUTINE LMPMAS - Calculates nodal weight fac tors for the strain projection.

SUBROUTINE STIFF - Performs matrix tr iple products t o assemble the element stiffness matrix and the element load vector associated w i t h the i n i t i a l strain terms.

SUBROUTINE STRAIN - Calculates element strains a t specified sampling points

SUBROUTINE CNSMAS - Assembles the consistent mass matrix for modal and

SUBROUTINE INITST - Generates init ial stress terms for quasi-static, b u c k l i n g

SUBROUTINE CENMAS - Evaluates the centrifugal mass terms for r o t a t i n g

and projects t o nodes.

transient analysis.

and modal analysis of prestressed structures.

structures a t speed.

element.

element residual vector for the subelement mesh.

I SUBROUTINE RESID - Calculates the element residual vector for the global

SUBROUTINE SUBFEM - Performs the subelement solution and calculates the

I 26

SUBROUTINE RELDFG - Calculates the re la t ive deformation g rad ien t a t the element sampling point and projects t o a node.

SUBROUTINE RESDYN - Calculates the contribution of mass and damping terms i n the element residual vector when the t ransient dynamics option i s used.

Material Library

A system of subroutines i s included i n the VlHOST code which covers a wide range of material models and i n i t i a l s t ra in assumptions:

SUBROUTINE SIMPLE - Integrates the s t r e s s over the increment assuming the elastic-pl a s t i c response of the material i s represented by a total nonl inear el a s t i c secant modul us. A1 so generates the material modulus matrix.

SUBROUTINE PLASTS - Integrates the s t r e s s over the increment and calculates the p las t ic s t ra in by u s i n g the radial return algorithm.

SUBROUTINE PLASTD - Calculates a consistent e las t ic -p las t ic modulus i f the incremental equivalent plast ic strain i s positive.

SUBROUTINE WALKEQ - Integrates the Mal ker unified creep p l a s t i c i ty consti tutive equation and a1 so generates a material modulus based on the temperature dependent el a s t i c i ty assumpti on.

SUBROUTINE LELAST - Calculates the s t r e s s fo r the constant material modulus given as data.

SUBROUTINE THRSTN - Calculates the thermal s t ra in .

SUBROUTINE CRPSTN - Calculates the creep strain.

4 . 3 USER INTERFACE

A large number of o p t i o n s a r e made available i n the MHOST code. A user selects options by specifying cer ta in keywords i n the parameter data section of the i n p u t data f i l e . Some of the infrequently used features available i n the previous versions a re deleted for the sake of c l a r i t y . Table 2 summarizes the analysis capabili ty of MHOST Version 4.2.

27

Table 2 MHOST Analys is Capabil i ty

E l ement Three- Three- D e f i n i t i o n Plane P1 ane Axisymmetric Dimensional Dimensional Options - Beam Stress S t r a i n Sol i d Sol i d She1 1

L inear I s o t r o p i c E l a s t i c i t y X X X X

Ani so t rop i c*l E l a s t i c i t y X X X

Composi t e * l Laminate

S i m p l i f i e d P1 a s t i c i t y

EI asto-*3 P1 a s t i c i t y

U n i f i e d Creep-P1 a s t i c i ty

Thermal *2 S t r a i n

Creep*2 S t r a i n

Large D i spl acement x

F i n i t e S t r a i n

X

Notes:

"1 Appl icable on l y t o l i n e a r e l a s t i c i t y .

*2 Not app l i cab le t o the u n i f i e d c r e e p - p l a s t i c i t y model i n which t h e quant i t i e s a re the i n t e g r a t e d p a r t o f t he model . An iso t rop ic y i e l d sur face can be de f ined f o r a continuum element. *3

I 28

Analysis Modul e Option

Quasi-static Analysis

Buck1 i ng Anal y s i s

Modal Analysis

Transient Dynamics

Tab1 e 2 MHOST Analysi s Capabi 1 i ty (continued)

Beam -

X

X

X

X

P1 ane Stress

Pl ane Strain

In the model data section of the i n p u t

Three- Axisymmetric Dimensional

Sol i d Sol i d

Three- Dimensional

She1 1

data f i l e , certain nonstandard features are made avai 1 ab1 e. T h i s subsection' summarizes program features avai 1 ab1 e i n Version 4.2 of the MHOST code by the keywords. For f u l l detail s , see the MHOST User's Manual.

*ANISOTROPY

The anisotropic material property option i s flagged by this parameter. The orientation vector for the material axes must be added i n the model data section. The material properties a long the material axes must be given by e i ther the "DMATRIX option ( f o r continuum elements, type 3, 7, 1 0 and 1 1 ) or the *LAMINATE o p t i o n ( for shell element type 75) . The anisotropic p las t ic response of a material i s described by the user subroutine ANPLAS as documented i n the MHOST User's Manual.

*BFGS

The inverse BFGS update procedure i s invoked by flagging this option parameter w i t h the default i t e r a t ive a l g o r i t h m being the straightforward Newton-Raphson scheme.

*BOUNDARY

The nodal displacement constraints are imposed by vir tue of penalization when the frontal solution subsystem i s invoked.

29

*BUCKLE

The i n i t i a l s t r e s s matrix i s generated and the eigenvalue extraction i s performed t o obtain buckling modes. T h i s option can be invoked a t an a rb i t ra ry s tep of the incremental nonlinear solution process so a s t o detect the change i n the buckling load due t o the ine l a s t i c response of the s t ructure .

I "COMPOS I TE

T h i s parameter data l i n e invokes the composite laminate analysis option for shell element type 75. Added t o the MHOST code upon request from the Contract Monitor a t WASA-Lewis Research Center. T h i s option rese ts the element parameters. The number of integration layers i s reduced t o 1 and a l l the components of s t r e s s resul tant are used as nodal variables. In the model data section, the user has t o supply a l l the components of the s t ress -s t ra in matrix by invoking the laminate option.

I *CONJUGATE-GRADIENT I The conjugate-gradient method i s invoked by adding this parameter. The l i n e

I with other i t era t ive algorithms such a s t h e BFGS or secant-Newton methods. I search option i s automatically turned on. Note t h a t this o p t i o n cannot be used

*CONSTITUTIVE

Three cons t i tu t ive approaches a re avai 1 ab1 e f o r describing materi a1 behavior, including the secant e l a s t i c i t y procedure (simplified p l a s t i c i t y ) i n which the material tangent i s generated fo r Newton-Raphson type i t e r a t i v e a1 gori t h m s , the von Mises p l a s t i c i ty procedure w i t h the associated flow ru le t reated by the radial return algorithm, and the Walker nonlinear viscoplast ic model i n which an i n i t i a l s t r e s s i t e r a t ion using e l a s t i c s t i f f n e s s i s u t i l i zed . For experimental purposes, the l i nea r e l a s t i c i t y o p t i o n can be flagged, w i t h the defaul t option being the conventional p l a s t i c i ty model.

*CREEP

The creep e f f e c t i s taken i n t o account via the time history being integrated

s ize a1 gor i t h m . I in an e x p l i c i t manner under control of an optional s e l f adaptive time-step

*DISPLACEMENTMETHOD

This option invokes the conventional displacement model i n which the residual loads a re evaluated d i rec t ly a t integration po in t s . For l i nea r e l a s t i c s t r e s s analyses, no i t e r a t ion would take place when this parameter card i s flagged. In ine l a s t i c analyses, the material tangent i s interpolated and m u l t i p l i e d by integration p o i n t s t r a ins d i rec t ly a t each quadrature point. T h i s o p t i o n cannot be used for the creep-plast ic i ty model which does not generate correct material tangent information a t integration points.

*DISTRIBUTELOAD

The body force and surface traction loadings are referred t o as the distributed loads i n the MHOST program. The body force option includes gravity accel eration defi nab1 e i n any d i rection, as we1 1 as centr i fuga1 1 oadi ng w i t h the center1 ine and angular velocity being user-specified.

*DUPLICATENODE

The nodal points fo r s t r e s s recovery can be disconnected by defining two nodal points a t the same geometrical location and connecting them by th i s option which assumes the compatibility of displacement a t these points. T h i s option i s used to define generic modeling regions and the i r interconnections.

*DYNAMIC

The general ized Newmark solution a1 gori t h m i s entered by us ing this parameter card. A primitive version of the adaptive time-stepping algorithm i s incorporated i f requested by the user.

*EL EM E NTS

The elements available i n this version of the code are summarized i n Table 3. Core a1 1 ocation i s performed for both nodal - and el ement-quanti t i e s based on the maximum storage space requirements among the types of element specified i n this option. All the element types must be specified here, including those only associated w i t h the subelement regions.

*EMBED

The subelement i t e ra t ion technology i s flagged by this option t o signal the code to a l locate working storage f o r subelement data i n a hierarchical manner.

The actual subelement mesh definit ion and the nodal- and element-data storage allocation take place when the individual subelement i s defined.

*FINITE - STRAIN

T h i s keyword invokes the f in i te -s t ra in o p t i o n a t a specified load/time increment.

*FOLLOWER - FORCE

The loadina data i s assumed a s the follower force when this keyword i s included i i

*FORCES

Concentrated core a l loca t

he parameter data section.

nodal forces are defined and stored i n an incremental manner. The on takes place only when this option i s invoked.

31

*FRONTALSOLUTION

The frontal solution option fo r quasi-s ta t ic analysis i s available i n this version of the code and uti l izes out-of-core storage devices, t h u s increasing the capacity of the program signif icant ly .

*HARDENING - SLOPE

The maximum number of break points fo r a piecewise linear approximation of uniaxial e las t ic -p las t ic behavior i s specified w i t h this parameter keyword.

*LARGE - DISPLACEMENT

T h i s keyword invokes the large displacement o p t i o n a t the specified loadhime increment.

*LINE-SEARCH

The l ine search o p t i o n requires posit ive action by the user t o be turned on. T h i s algorithm i s usable i n conjunction w i t h a l l i t e r a t ion algorithms available i n the MHOST program. When the conjugate-gradient i t e r a t ion i s used, the program automatically turns on the line search option.

*LOUBIGNAC

Parameters f o r numerical quadrature used i n mixed i terative processes a re definable i n a very precise way. To construct the stiffness matrix the f u l l , se lect ive, o r select ive w i t h f i l t e r i n g scheme can be chosen. For the residual vector integrat ion, fu l l or reduced integration can be chosen. The s t r a in integration can be performed by using either uniformly reduced integration, trapezoidal integration w i t h a reduced shear s t r a i n approximation, or trapezoidal integration w i t h a f i l t e r i n g option.

*MODAL

Free vibration modes a re extracted by invoking this option f o r l i nea r e l a s t i c structures only. The subspace i t e r a t ion technique i s uti1 ized together w i t h the power shi f t option.

*NODES

A l l the variables a re defined and reported a t nodal points. In the incremental processes, the deformation and s t r e s s history a re stored only a t the nodal po in t s . Note t h a t this archi tecture economizes storage substant ia l ly as compared t o the fu l ly integrated f i n i t e element displacement methods.

*NOECHO

This option suppresses the echo p r i n t of the model data.

32

*OPTIMIZE

Bandwidth optimization based on the C u t h i l l -McGee algori thm i s available i n the MHOST code, whereas no optimization for the frontal solver i s available.

*PERIODICLOADING

For transient analyses, nodal displacements and concentrated forces can be given as s inusoida l functions v ia this o p t i o n ; this reduces the effort needed t o prescribe the load h is tory for a class of simple tes t problems.

*POST

The post-processing f i l e , w h i c h contains a l l the information supplied t o and produced by the code, i s generated on the f i l e connected t o FORTRAN u n i t number 19. This f i l e i s formatted and can easily be manipulated by commercially available post-processing packages w i t h minor modifications. A version o f MENTAT supports the MHOST post-processing f i le . The header record of the f i l e i s designed t o be compatible w i t h the MARC pos t f i l e , details of w h i c h are published and processed by many finite element graphics packages.

*PRINTSETS

Report generation i s carried o u t on a nodal p o i n t basis, w i t h the element integration p o i n t op t ion supported by interpolation processes using appropriate shape funct ions.

*REPORT

The frequency of the line printer report generation i s now controlled by this o p t i o n w i t h the default being t o p r i n t a t every increment.

*RESTART

This parameter card invokes the program t o read the restart tape and t o set u p the system t o resume the analysis from the point where the tape i s written.

*SCHEME

Parameters for defining characteristics of the time integration operator can be specified by the user. The default i s the average acceleration algorithm commonly used i n nonlinear dynamic f inite element analyses.

*SECANT-NEWTON

This parameter activates the secant-Newton implementation of the Davidon rank-one quasi-Newton a1 gor i thm. The core-storage requirements of this procedure are significantly less than for the BFGS update and the execution i s re1 a t i vely f a s t . Th i s parameter i s recommended fo r i ne1 a s t i c analyses o f sol id-continua.

33

*STIFFENING

T h i s keyword i s used t o invoke i n i t i a l s t r e s s terms fo r quasi-s ta t ic , t ransient dynamic and modal analyses.

*STRESS

As an option, the boundary condition for s t r e s s can be specified by the user, although no mathematical j u s t i f i ca t ion i s ye t available for this type of boundary loading. Any s t r e s s component can be prescribed a t any nodal point. Simple numerical t e s t s have shown tha t inconsistent imposition of boundary s t r e s s values indeed lead t o rapid divergence i n the i t e r a t ive process.

*TANGENT

In the context of modified Newton i te ra t ions , the tangent stiffness matrix fo r subsequent i t e ra t ions i s user-definable. When this option i s invoked, the default i s the f i r s t tangent matrix used d u r i n g the current increment, not the e l a s t i c s t i f fness matrix. This type of implementation i s referred to as the K T ~ method of modified Newton i te ra t ion . In the BFGS process, the same approach i s used and no opt ions are made available. In the modified Newton i t e r a t i o n , t h e t a n g e n t a r r a y i s f i x e d a f t e r a s p e c i f i e d number o f u p d a t e s .

*TEMPERATURE

Nodal temperatures are read as i n p u t and used to generate thermal s t ra ins . These temperatures are used also d u r i n g the evaluation of creep strains and the integration of the coupled creep p las t ic i ty model (Walker's model ).

*THERMAL

Temperature dependent material properties a re evaluated when t h i s option i s invoked, and an appropriate user subroutine has to be provided to the system prior t o execution. T h i s operation i s - not necessary f o r the creep p las t ic i ty model i n which the temperature dependency i s readily incorporated.

*TRANSFORMATIONS

Coordinate transformations a t nodal points are specified by this o p t i o n i n which the angle of rotation i s provided by the user so tha t the code can generate necessary transformation matrices. The post-processing f i l e does not support the coordinate transformations.

*TYING

Mu1 t i p l e degree-of-freedom constraint equations are specified by the user through this option. Under certain c i rcumstances, this option i s sl ightly more f lexible than the duplicate node option; for instance, the user can disconnect three elements a t a p o i n t . Note tha t the constraint equations are generated only for displacement degrees-of-freedom, which are calculated only i n an imp1 i c i t manner.

34

The following parameters are used to signal t o the MHOST program the presence of specific user subroutines:

*UBOUN *UCOEF *UDERIV "UFORCE "UHOOK "UPRESS *UTEMP "UTHERM *WKSLP

As summarized above, the analysis capabil i t i e s incl uded i n the MHOST code cover most requirements for calculating ine las t ic deformations of turbine engine h o t section components. The MHOST code has evolved into a General Purpose Structural Analysis package w i t h strong emphasis on nonl inear material behavior usable fo r a wide range of problems. The free format data i n p u t routines and report generation packages have been improved i n the Task I1 program development e f fo r t so as t o create a comfortable environment for code users.

Table 3 summarizes the elements currently available i n the MHOST Version 4.2 code and l i s ts parameters internally used to define element character is t ics . Further detail i s available i n the MHOST User's Manual.

35

Table 3 Element Parameters

3 0 1 0 1

ITYPE NELCRD NELNFR NELNOD NELSTR NELCHR NELINT NELLV NELLAY ND I NKHEAR J LAW

NELCRD

. . . . . . 3 3 3 3 4 1 1 1 1 5 2 3 3 3 2 1 1 1 3 1 2 3 4 5 6

I NELNFR

NELNOD

NELSTR

NELCHR

NELINT

NELLV

NELLAY

ND I

NSHEAR

I J LAW

Beam P. Stress P.Strain Axsym. Brick Shell

Number of coordinate p o i n t data values per node.

Number of degrees-of-freedom per node.

Number of nodes per element.

Number of stress and strain components per node.

Number of material property data values f o r the element.

Number of 'full I integration points per element.

Number of distributed load types per element.

Number of layers of integration through the thickness of the she1 1 element.

Number of direct stress components.

Number of shear stress components.

Type o f the constitutive equation.

I 36

4.4 VALIDATION

4.4.1 Performance Improvement by the P r o f i l e Solver

Table 4 summarizes computing time and memory requirement reduc t ions a f te r implementation o f the p r o f i l e s o l v e r s o l u t i o n package i n t o the MHOST code. The performance figures a r e a p p l i c a b l e f o r the MARC Corp. PRIME 9955 computer us ing a Primos FORTRAN 77 compiler w i t h the fu l l op t imiza t ion op t ion turned on. The number o f words appear ing i n the t a b l e apply t o s i t u a t i o n s involv ing single p r e c i s i o n (32 bits) words i n the blank common work space. For 64 bi ts /word machines such as CRAY, the memory requirement i n terms of number o f words will be cons iderably smal le r . Even f o r a system o f uniformly banded equa t ions such a s occurs w i t h the composite lamina te fan b lade shown i n F igure 4, a s u b s t a n t i a l reduct ion i n computing time and some improvement i n core-s torage requirements a r e obta ined w i t h the p r o f i l e s o l v e r package.

For l a r g e problems such a s the turbine b lade model w i t h p la t form a s shown i n Figure 2 , the speed ga in and r educ t ion i n memory needed make an in-core s o l u t i o n p o s s i b l e on a small computer.

In a b u c k l i n g a n a l y s i s o f the cylinder whose mesh i s shown i n Figure 3, the reduct ion i n s o l u t i o n time i s an o r d e r of magnitude more s i g n i f i c a n t than the saving i n s t o r a g e requirements.

Note t h a t the ty ing opt ion f o r the p r o f i l e s o l v e r i s implemented i n a global manner. The a1 gebra i c c o n s t r a i n t equat ions a r e embedded i n the prof i le - form stiffness ma t r ix a f t e r a l l the element ma t r i ces a r e assembled. On the o t h e r hand, the coord ina te t ransformat ion op t ion f o r the nodal degrees-of-freedom i s handled on an el ement-by-el ement bas i s. Immediately a f t e r an element mat r ix i s assembled, the entries subjected t o coord ina te t r ans fo rma t ions are manipulated before substi t u t i ng i n t o the global a r r ay .

37

Table 4 Performance Tests f o r the Profile So lve r Solu t ion Package i n the MHOST Code

Problem 1 SSME HPFTP Blade Model w i t h 1025 S o l i d Elements (Type 7 ) and 1575 Nodes (Mesh Shown i n F igure 2 )

Profile Solver Band Solver

Working Space Required 3,483,811 4,459,647

Sol u t i on Time ( seconds ) 1 466.594 NA*

*TOO l a r g e t o be a b l e t o r u n on PRIME 9955 a t MARC.

I

Problem 2 Buck1 ing Analysis o f a Cy1 i n d e r w i t h 160 Shel l Elements (Type 75) and 176 Nodes (Mesh Shown i n F igure 3 )

Prof i 1 e Sol v e r Band Solver I Working Space - S t a t i c - Eigenvalue

So lu t ion Time (seconds ) - S t a t i c - Eigenvalue

498,873 1,044,773

62.057 66.788

596,703 1,340,375

171.430 187.551

Problem 3 Modal Analysis o f Composite Laminate Fan Blade w i t h 240 Shel l Elements (Type 75) and 279 Nodes (Mesh Shown i n F igure 4 )

Prof i 1 e Sol v e r Band Solver

Working Space 824,341 1,054,259

So lu t ion Time (seconds) 24.794 93.764

38

Figure 2

TEST CASE USING MHOST - PROFILE SOLVER; SSME HPFTP BLADE A Y

F i n i t e Element Mesh f o r Problem 1 (Turbine Blade Model w i t h P1 atform)

ANALYSIS

Figure 3 F i n i t e Element Mesh f o r Problem 2 (Buckling Analysis o f a Cy1 i nder )

33

Figure 4 Fini te Element Mesh f o r Problem 3 (Composite Laminate Fan Blade)

4.4 .2

The s t r e s s s t i f fen ing and centrifugal mass o p t i o n s were validated by analyzing a cant i lever beam rotat ing a t a given angular veloci ty , w i t h the geometry and loading conditions i l l u s t r a t e d i n Figure 5. Two f i n i t e element models were constructed on MHOST, one u s i n g the shel l element (type 7 5 ) and one using the three-dimensional sol i d element (type 7 ) . A reference sol ution was a1 so established: a solution on NASTRAN u s i n g i t s shell element (QUAD4) was compared w i t h a solution produced by MARC v i a i t s shel l element (type 7 5 ) . Both commercial codes produce almost identical v i b r a t i o n frequencies which establ ish the reference solution. As the values i n Table 5 indicate, both MHOST solutions a re almost identical t o the reference s o l u t i o n . This a lso indicates t h a t the se l ec t iv i ty integrated l i nea r so l id element i s capable of accurately representing bending mode response.

Vibration Analysis of a Rotating Beam

10000 APM

Y i

€ 1 3 ~ 1 0 7

p 7.764 x 10 -4 Y 0.3

I

F igure 5 Val i da t ion Model f o r Centr i fugal S t i f f e n i ng

Tab1 e 5 V i b r a t i o n Frequency-Canti 1 ever Beam Under Cen t r i fugal Loading (Column A: Without S t i f f e n i n g ; Column B: With S t i f f e n i n g and Centr i fugal Mass )

NASTRAN MHOST Mode She1 1 Element** S he1 1 E l emen t Sol i d E l ement

A B A 0 A B

1* 145.4 227.6 145.4 228.7 145.8 229.9

2 618.9 662.3 628.6 668.2 595.8 669.3

3 894.8 1042.8 924.5 1077.0 329.9 1083.0

4 1115.0 1199.0 131 6.7 1375.8 1205.1 1229.2

* MARC s h e l l element type 75 gives 146.2 Hz wi thout cen t r i f uga l s t i f f e n i n g and 228.3 Hz w i t h the s t i f f e n i n g and c e n t r i f u g a l mass e f f e c t s included.

** E l ement type QUAD4.

41

4.4.3 Large D i spl acement Analysi s

A plane stress test problem involving stretching an e las t ic-plast ic plate i s analyzed using the 1 arge d i spl acement analyst s capabi 1 i t ies of the MHOST code. Listing 1 shows the input da ta used for the calculations. The original f i n i t e element mesh, as well as a deformed configuration, are plotted i n Figure 6. The solution obta ined i s smooth w i t h o u t any i n d i c a t i o n of numerical i n s t a b i l i t i e s . Equivalent plast ic strain and Mises equivalent stress plots are shown i n Figures 7 and 8 respectively.

L i s t i n g 1 Input Data for a Large Displacement Analysis by Plane Stress Elements

ELASTIC PLASTIC ALUMINUM PLATE-8x8 PLANE STRESS-L 313 *POST *NODES 81 *ELEMENTS 64

3 *CONSTITUTIVE 2 *HARD 2 *BOUNDARY 40 *LARGE - DISPLACEMENT *FINITE - STRAIN *STIFFENING *LOUBIGNAC 3 1 3 *END *WORKHARD 2

45000.0 0.0000 0.0000 850000.0 10.000 0.0000

*I TERAT I ONS 20 0.050

*COORDINATES 1 0.0000 0.0000 0.0000 0.05000

2 0.0000 1 .oooo 0.0000 I 3 0.0000 2.0000 0.0000

4 0.0000 3.0000 0.0000 I 5 0.0000 4.0000 0.0000

I

42

~

6

7 8 9

10

11

12

13 14

15

16

17

18

19

20

21

22 23

24

25

26

27

28

29 30

31

32 33

34

35

36

37

38

39

40

0.0000 0.0000 0.0000

0.0000 1 .oooo 1 .oooo 1 .oooo 1 .oooo 1 .oooo 1 .oooo 1 .oooo 1 .oooo 1 .oooo 2.0000 2.0000

2.0000 2.0000

2.0000

2.0000

2.0000

2.0000

2.0000

3.0000 3.0000 3.0000 3.0000

3.0000 3.0000

3.0000

3.0000 3.0000

4.0000

4.0000

4.0000

4.0000

5.0000

6.0000 7.0000

8.0000 0.0000

1 .oooo 2.0000

3.0000

4.0000

5.0000 6.0000

7.0000 8.0000

0.0000

1 .oooo 2.0000

3.0000

4.0000 5.0000

6.0000

7.0000

8.0000

0.0000

1 .oooo 2.0000

3.0000

4.0000

5.0000 6.0000

7.0000 8.0000

0.0000

1 .oooo 2.0000

3.0000

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

0.0000 0.0000

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

0.0000 0.0000

0.0000

0.0000 0.0000 0.0000

43

41 42 43 44

~ 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68

I 69 I

70

4.0000 4.0000 4.0000 4.0000 4.0000 5.0000 5.0000 5.0000 5.0000 5.0000 5.0000 5.0000 5.0000 5.0000 6.0000

6.0000 6.0000 6.0000 6.0000 6.0000 6.0000 6.0000 6.0000 7.0000 7.0000 7.0000 7.0000 7.0000 7.0000 7.0000

4.0000 5.0000 6.0000 7.0000 8.0000 0.0000 1 .oooo 2.0000 3.0000 4.0000 5.0000 6.0000 7.0000 8.0000 0.0000

1 .oooo 2.0000 3.0000 4.0000 5.0000 6.0000 7.0000 8.0000 0.0000 1 .oooo 2.0000 3.0000 4.0000 5.0000 6.0000

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 .

0.0000 0.0000 0.0000 0.0000

44

~

71

72

73

74

75 76

77 78

79 80

81

7.0000

7.0000

8.0000 8.0000

8.0000 8.0000 8.0000 8.0000

8.0000

8.0000

8.0000 *ELEMENTS 3

1 1

2 2

3 3

4 4

5 5 6 6

7 7 8 8 9 10

10 11

11 12 1 2 13 13 14

14 15

15 16

16 17 17 19

18 20

19 21

20 22

10

11

12

13

14

15

16

17

19

20

21 22

23

24

25

26 28

29

30

31

7.0000

8.0000

0.0000

1 .oooo 2.0000

3.0000

4.0000

5.0000

6.0000

7.0000

8.0000

11 2 12 3

13 4

14 5 15 6 16 7 17 8

18 9

20 11

21 12

22 13 23 14

24 15

25 16

26 17

27 18 29 20

30 21

31 22 32 23

0.0000 0.0000

0.0000

0.0000

0.0000 0.0000 0.0000 0.0000

0.0000

0.0000 0.0000

45

- -

21 23 22 24 23 25 24 26 25 28 26 29 27 30 28 31 29 32 30 33 31 34 32 35 33 37 34 38 35 39

36 40 37 41 38 42 39 43 40 44 41 46 42 47 43 48 44 49 45 50 46 51 47 52 48 53 49 55 50 56

32 33 34 35 37 38 39 40 41 42 43 44 46 47 48

49 50 51 52 53 55 56 57 58 59 60 61 62 64 65

33 24 34 25 35 26 36 27 38 29 39 30 40 31 41 32 42 33 43 34 44 35 45 36 47 38 48 39 49 40

50 41 51 42 52 43 53 44 54 45 56 47 57 48 58 49 59 50 60 51 61 52 62 53 63 54 65 56 66 57

46

51 57 52 58 53 59 54 60 55 61 56 62 57 64 58 65 59 66 60 67 61 68 62 69 63 70 64 71

*BOUNDARY 1 1 1 2 2 1 3 1 ' 4 1 5 1 6 1 7 1 8 1 9 1

10 2 19 2 28 2 37 2 46 2 55 2 64 2

66 67 68 69 70 71 73 74 75 76 77 78 79 80

73 1 0.8000 73 2

~

67 58 68 59 69 60 70 61 71 62 72 63 74 65 75 66 76 67 77 68 78 69 79 70 80 71 81 72

47

74 1 0.8000 74 2 75 1 0.8000 75 2 76 1 0.8000 76 2 77 1 0.8000 77 2 78 1 0.8000 78 2 79 1 0.8000 79 2 80 1 0.8000 80 2 81 1 0.8000

81 2 *PROPERTIES 3 C STANDARD ALUMINUM MATERIAL

1 81 0.05 10.OE6 0.330

*PRI NTOPTION INCREMENTALDISPLACEMENT NODE TOTALDISPLACEMENT NODE REACTION NODE PLASTIC STRESS

NODE NODE

STRAIN NODE FORCE NODE *END *AUTO1 NCREMENT

9 I

*END *STOP

Y

Figure 6 Elastic-Plastic Membrane; Deformed Shape

i

PREZEDING PAGE BLANK NOT

> J FILMED

51

ORIGINAL PAGE COLOR PHOTOGRAPH

LL 0 v) a 2 0

0 I- v)

a 4 , 0 I- v) a i w co

LL

A disc upsetting problem involving bo th large displacement and large strain behavior i s used as a second example. This i s a standard problem solved repeatedly by MARC, and the reference solution i s readily available. As the initial mesh and deformed configuration plots (Figure 9 ) indicate, a large ampl i tude hourgl assing i s observed. Thi s i s due t o the nearly i ncompressi b l e response of the material and the same phenomenon w i t h less significant ampl i tude i s observabl e in a s tandard d i spl acement sol u t ion . Further investigation on stabilization o f the displacement field i s necessary t o f i l t e r o u t this spurious mode. However, the stress field obtained with the present capabil i t y i s stab1 e and quantitatively agrees we1 1 w i t h the so lu t ion obtained by MARC via the conventional displacement method (Figure 10) .

For comparison, the same disc upsetting problem i s studied using the three-dimensional solid element (Figure 11 1. As indicated in the stress contour p l o t (Figure 11 model i s capable of reproducing an axisymmetric stress fie1 d quantitatively similar t o t h a t generated using the axisymmetric element.

a t an early stage of the deformation process, the

As an example of MHOST's large displacement analysis capability w i t h shell elements, a one-quarter model of a clamped square plate under uniform pressure loading i s studied. The results are plotted i n Figure 1 2 , together w i t h solutions of the same problem available i n the literature [WAY (1938); KAWAI, YOSHIMURA (1 961 ) ; HUGHES, LIU (1 981 ) ; and NAGTEGAAL, SLATER (1 981 )I . The comparison indicates t h a t the mixed iterative implementation of shell elements results in more flexible behavior compared w i t h the compatible model (Kawai -Yoshimura so lu t ion ) and w i t h model s based on the Rei ssner-Mi ndl i n theory (Hughes-Liu solution by uniform reduced integration and heterosis elements). Qualitatively, the behavior of the MHOST shell element i s similar t o the Nagtegaal-Slater so lu t ion v i a the SLICK element derived from the discrete Kirchhoff assumption (MARC element type 72). A visible difference i s shown between the solution obtained with 'full I Newton iteration (in which the displacement preconditioner i s repeatedly updated) and t h a t obtained v i a the secant-Newton update with line search. This difference may be due t o i l l -condi t ioning of the precondi tioner caused by transverse shear constraint terms.

Y

Figure 9 Compression o f S o l i d Axisymmetric Cyl inder; Deformed Shape ( Hourgl assi ng )

56

v) >

* t t t t * t t t t * t + + + + + + + + + + r r

v) W

I- v)

a

..

I- L

57

w

0 z n

t

2 t v, X a

0.d

0.6

0.2

0.0 1 .o 2.0 3.0 4.0 Center D i sp lacemen t (mn)

0 .8 r - - -

4 SLICK elements 0 9 SLICK e l m t s 0 16 SLICK eluents

.+ 4 Nine node heterosis c I c I n t s -L- 16 Four nod. u n i f o n reduced

in tegra t ion eluents Kawai and Yoshlmri -- Y.Y - - - - Linear solution

-

-

! ~

i

U.0 1 .o 2 .0 3.0 4.0 Center Displacement (mn)

E = 2 x lo4 k g / d v = 0.3 L = 1000 mn h = 2 m

I L 1

4.4.4 Application of the Subelement Method for Inelast ic Analysis

A plane s t r e s s problem of a square plate w i t h a c i rcu lar hole i s used t o demonstrate the i ne1 a s t i c sol u t i o n capabi 1 i ty of the MHOST subel ement procedures. The global f i n i t e element mesh and the subelement mesh representing a hole embedded i n the element a t the center of a plate i s i l l u s t r a t ed in Figure 13. A global f i n i t e element model expl ic i t ly including the hole in the same way as the subelement g r i d i s shown i n Figure 14.

Y

SUBELEMENT MESH K)R AN EMBEDDED HOLE

Y

1 , Figure 13 Global and Subelement Meshes for an Embedded Hole

TEST PROBLEM FOR A PLATE WITH A HOLE IN PLANE STRESS

Figure 14 Global F i n i t e Element Mesh

The i n p u t data f i l e s for both the global element mesh and the subelement mesh models are given i n Listings 2 and 3 . Note t h a t the amount of data t o represent a hole i s significantly less for the subelement mesh; this represents a major improvement i n user friendliness.

L i s t i n g 2 Global F i n i t e Element Model

TEST PROBLEM FOR A PLATE WITH A HOLE ( PLANE STRESS *POST "NOECHO

*CONSTITUTIVE 0 *ELEMENTS

3

*NODES

*BOUNDARY - C

*HARDENING *FORCE *END *ITERATION

20 *ELEMENT

1 1

2 2 3 17 4 3 5 5 6 6 7 6 8 21 9 37

1 0 7 11 25 12 25 13 39 14 38 15 18 16 22 17 29

18 31 19 33

64

28

40

19 5

1 0

. 01

5 6

21

7 18 22 25 39 35 24 27

26 40

36 9

10 10

23 11

3 10.0 . 01

6 2 21 17

7 3 8 4

22 6

27 25 39 21 37 7 24 7 20 8 28 26 40 39 38 37 35 37 10 2 2 . 29 27 23 31 11 33

24 35

20 24 21 27

22 30 23 32 24 34 25 9 26 10 27 23 28 11

*COORDINATE 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17

18 19 20 21 22 23

24

11 29 29 31 33 13 14 19 15

0.0 0.0 0.0 0.0

1 .o 1 .o 1 .o 1 .o 2.0

2.0 2.0 2.0 3.0 3.0

3.0

3.0 0.0

1.5 3.0 1.5 1 .o 1.5 2.0

1.5

12 20 30 28 31 32 33 34 35 36 14 10

19 23 15 11 16 12

0.0

1 .o 2.0

3.0 0.0

1 .o 2.0 3.0 0.0

1 .o 2.0 3.0 0.0 1 .o 2.0 3.0

1.5 0.0

1.5 3.0 1.5 1 .o 1.5 2.0

65

25 1.2146 26 1.4293 27 1 .5000 28 1.5000 29 1.7854 30 1 .5707 31 1.8000 32 1 .6000 33 1.7854 34 1 .5707 35 1.5000 36 1 .5000 37 1.21 46 38 1.4293 39 1.2000 40 1.4000

I

*PROPERTY 1 40

*BOUNDARY - C 1 1 1 2 2 1 3 1 4 1 17 1

C 13 1 C 14 1 C 15 1 C 16 1

1.2146 1.4293 1 .zoo0 1.4000 1.21 46 1.4293 1 .5000 1.5000 1.7854 1 .5707 1.8000 1.6000 1.7854 1.5707 1 .5000 1.5000

1 .o 1 .OE6 0.3

0.01 0.01 0.01 0.01

66

*FORCE

13 14 19 15 16

*PRINT

*END

*STOP

1 1 0000.0 1 1 5000.0 1 1 0000.0 1 15000.0 1 1 0000.0

TOTAL

STRESS

REACTION

L i s t i n g 3

TEST PROBLEM FOR AN EMBEDDED HOLE

*SECA

*LOUB 3 1 3

*NOECHO

*EMBED

*CONSTITUTIVE 0 *ELEMENTS 9

3 C 101 *NODES 16 *BOUNDARY - C 9 *FORCE 4 *END

* ITERATION

20 .1 10.0 1 .oo

Subel ement Model

PLANE STRESS )

67

*ELEMENT 1 1

2 2 3 3

4 5 5 6 6 7 7 9

8 10 9 11

*COORDINATE

1 2 3 4

5 6 7 8

9 10 11 12 13 14 15 16

*PROP ERTY 1 16

*BOUNDARY - C 1 1 1 2

2 1

3 1 4 1

68

3 5 6 7 9

10 11 13 14 15

0.0

0.0 0.0 0.0

1 .o 1 .o 1 .o 1 .o 2.0 2.0 2.0

2.0 3.0 3.0 3.0

3.0

6 7 a

10 11 12 14 15 16

2 3 4

6 7 8

10 11 12

0.0

1 .o 2.0 3.0

0.0 1 .o 2.0

3.0 0.0 1 .o 2.0

3.0 0.0

1 .o 2.0

3.0

1 .o 1 .OE6 0.3

*FORCE 13 14 15 16

*HOLE 5

*PRINT

*END *STOP

C 13

C 14 C 1 5 C 16

1 1 1 1

3

1 0.01 1 0.01 1 0.01 1 0.01

1 0000.0 20000.0 20000.0 1 0000.0

0.2

TOTAL STRESS REACTION

The e las , ic stress f , s l d obtained by the global solut ion i s shown i n Figure 1 5 and qual i t a t i vely agrees w i t h the classical sol ution found i n many textbooks, e.g., SAVIN (1951 1. The e l a s t i c solut ion a f t e r one i t e r a t i o n , as summarized i n Table 6, ind ica tes t h a t the quadratic subelement produces the most accurate r e s u l t s f o r the stress concentration factor .

69

ORIGINAL PAGE COLOR PHOTOGRAPff

W -1 0 I a I

5 W I- 4 a a a

z 0

Table 6 Nodal S t ress Concentration Factor a t the Edge of a Circular Hole After One I t e ra t ion

G1 obal Sol ution Subel ement Solution

Point A Linear Quadratic

2.261 5 2.31 98 2.7087

“Y 0.591 6 0.5977 0.3929

rxY 0.0002 0.0000 0.0003

Point B

UX 0.1752 0.4339 -0.2702

“Y -0.081 3 -0.2995 -0.5958

rXY 0.0000 0.0000 0.0000

For the e l a s t i c -p l a s t i c analyses, the material i s assumed t o be b i l i nea r and i s defined by *WORKLOAD data as:

S t ress P las t ic S t ra in

2.0 4.0

0.0 0.4

Note t h a t the stress here i s normalized by the uniform horizontal stress fo r the solution of the square p la te without a hole. As i s well-known from standard e l a s t i c i t y , amplification of the horizontal stress value by a fac tor of 3.0 i s expected a t Point A under pure e l a s t i c behavior.

Both l i n e a r and quadratic subelement r e su l t s , a s well as global f i n i t e element results, w i t h the same mesh, a re compared. The monotonic convergence cha rac t e r i s t i c s of the subelement method can be seen i n these comparisons, a s summarized i n Table 7.

PRECEDING PAGE BLANK NOT FILMED

Table 7 Elast ic-Plast ic Analysis of a Square Plate w i t h a Circular Hole; Nodal S t ress a t the Edge o f the Circular Hole a t Maximum SCF Location ( P o i n t A )

Linear Quadratic G1 obal Solution Subelement Solution Subelement Solution

Elas t ic Elast ic- Elas t ic Elast ic- Elas t ic Elastic- P1 as t ic Plastic P1 a s t i c

Equi Val ent P la s t i c S t ra in 0.0 0.01 91 0.0 0.0320 0.0 0.051 3

S t ress uxx 2.3492 2.2786 2.5064 2.41 51 2.7087 2.4624 UYY 0.31 60 0.4339 0.4327 0.6682 0.3929 0.4929 @XY 0.0034 0.0049 0.001 0 0.0009 0.0000 0.0000

I Note here t h a t i terat ions were carried o u t u n t i l a convergence tolerance of 10% in terms o f r e l a t ive residual was met. T h i s solution procedure improved the e l a s t i c stress r e s u l t s compared t o e l a s t i c results obtained e a r l i e r . The same convergence cri teria were used f o r a l l calculat ions shown i n Table 7. In comparison w i t h solut ions reported previously, the quadratic subelement method e x h i b i t e d superior performance, a t least fo r this pa r t i cu la r example of an embedded hol e.

I 74

5.0 REFERENCES

Babuska, I . , 0. C. Zienkiewicz, J . Gag0 and E. R. de A. Oliveira (eds.) , Accuracy Estimates and Adaptive Refinements i n Finite Element Computations, Wiley, Chichester, 1986.

Babuska, I . , "The Fi n i t e E l ement Method w i t h Lagrange Mu1 t i p l iers, I' Numer. Math., 20, 1973, pp. 179-192.

Bathe, K. J . and E. L. Wilson, Numerical Methods i n Fini te Element Analysis, Prentice Hall, Englewood Cl i f f s , New Jersey, 19/6.

Bazeley, G. P . , Y . K. Cheung, B. M. Irons and 0. C. Zienkiewicz, "Tr iangular Elements in Plate Bending. Conforming and Non-Conforming Solutions," Proc. 1 s t Conf. Matrix Methods in Structural Mechanics, AFFDL-TR-CC-80, 1966, pp. - Belytschko, T. , "A Review of Recent Developments i n Plate and Shell Elements," Computational Mechanics - Advances and Trends (ed. A. K. Noor), ASME AMD - - Vol . / 5 , 1986 3 P P * 211 232.

Benson, D. J. and J . 0. Hallquist , "A Simple Rigid Body Algorithm f o r Structural Dynamics Programs," Int . J . Num. Meth. Eng., 22, 1986, pp. 723-749.

Brezzi , F., "On the Existence, Uniqueness and Approximation of Saddle-Point Probl ems Ari si ng from Lagrange Mu1 t i pl i ers, I' RAIRO, 22, 1 974, pp. 1 29-1 51 . Brown, P. N . and A. C. Hindmarsh, "Matrix-Free Methods f o r S t i f f System o f O D E ' S , " SIAM J. Numer. Anal., 23, 1986, pp. 610-638.

Fyhre, D. and T. J. R. Hughes, "A Li terature Survey on Finite Element Methodology f o r Turbine Engine Hot Section Components," MARC Analysis Research Corporation, Palo A1 t o , Cal i fornia , 1983.

Gaudreau, G. L . , D. J. Benson, J . 0 . H a l l q u i s t , G . J . Kay, R. W . Rosinsky and S. J. Sachett , "Supercomputers for Engineering Analysis," Com u ta t iona l Mechanics - Advances and Trends (ed. A. K. Noor), ASME AMD h 9 8 6 , - PP. 117 131-

Hayes, L. 3. and P. Devloo, "A Vectorized Version of a Spare Matrix-Vector Multiply," Int . J . Num. Meth. Eng., 23, 1986, pp. 1043-1056.

Hu, H. C . , "On Some Variational Pr inciples i n the Theory of E la s t i c i ty and P la s t i c i ty , " Scint ia Sinica, 4, 1955, pp. 33-54.

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Hughes, T. J . R . , "Analysis of Transient Algorithms w i t h Particular Reference t o S t a b i l i t y Behavior," Computational Methods f o r Transient Analysis (eds. T. Belytschko and T. J. R. Hughes), Chap. 2, North-Holland, Amsterdam, 1983.

Hughes, T. J. R. and F. Shakie, "Pseudo-Corner Theory: A Simple Enhancement of J 2 - Flow Theory f o r Applications Involving Non-Proportional Loading," - Eng. Comp., 3, 1986, pp. 116-120.

Hughes, T. J . R . , R. M. Ferencz and J . 0. Hallquist , "Large-Scale Vectorized Implici t Calculations i n Solid Mechanics on a CRAY X-MP/48 Ut i l iz ing EBE Preconditioned Conjugate Gradients," Computational Mechanics - Advances and - Trends (ed. A. K. Noor), ASME AMD - Vol. /5, 1986 9 PP* 233 2/90

Jameson, A., "Current Status and Future Directions of Computational Transonics," Computational Mechanics - Advances and Trends (ed. A. K. Noor), ASME AMD - VOJ . 75, 1 986 ¶ PP- 329 368- - Kawai, T. and N. Yoshimura, "Analysis of Large Deflection of Plates by Finite Element Method," Int . J . Num. Meth. Eng., 1 , 1969, pp. 123-133.

Le Tallec, P. , "Mixed Numerical Methods i n Viscoplast ic i ty ," Proc. 1 s t WCCM, Vol. 1 , Texas I n s t i t u t e f o r Computational Mechanics, 1986.

Lohner, R . , "FEM-FCT and Adaptive Refinement Schemes f o r Strongly Unsteady F1 ows, I' Numerical Methods f o r Compressi bl e F1 ows - Finite D i f ference, E l ement and Volume Techniques (eds. T. E. Tezduyar and T. J . R. Hughes), ASME AMD - Nagtegaal , J. C. and J. G. S l a t e r , "A Simple Non-Compatible T h i n Shell Element Based on Discrete Kirchhoff Theory," Nonlinear Finite Element Analysis of Plates and Shel ls (ed. T. J . R. Hughes, A. Pifko and A. J a y ) , ASME AMD - Vol. 48, 1 981 s PP- 1 6 i -1 92.

Vol 78, 1986 , pp. 93-114.

Nagtegaal, J . C. , S. Nakazawa and 14. Tateishi, ''On the Construction of Optimal Mind l in Type Shell Elements," Finite Element Methods f o r Plate and Shell Structures, Vol. 1 : Element Technology (eds. T. J . R . Hughes and E. Hinton), P i neri dge Press, Swansea, 1 986 , pp. 348-364.

Nakazawa, S., 3-D I n e l a s t i c Analysis Methods f o r Hot Section Components, T h i r d Annual S ta tus Report; Vol ume 1 : Special Finite Element Model s , NASA CR-1/9494, P ra t t & Whitney PWA-5940 - 46, 1986.

Nakazawa, S. and J . C. Nagtegaal , ''An Eff ic ien t Numerical Procedure f o r Nonlinear Analysis of Turb ine Eng ine Hot Section Components," presented a t - 3rd Symp. on Nonlinear Const i tut ive Relations f o r High Temperature Applications, Akron, Ohio, 1986.

Nakazawa, S. and M. S. Spiege l , "Mixed Finite Elements and I t e r a t i v e Solution Processes," presented a t 1 s t WCCM, A u s t i n , Texas, 1986.

Needleman, A., "Finite Element Analysis o f Failure Modes i n Ductile Sol ids ," Proc. 1 s t WCCM, Vol. 1 , Texas I n s t i t u t e f o r Computational Mechanics, 1986.

76

Nour-Omid, B. and M. O r t i z , "Concurrent So lu t i on Methods f o r F i n i t e Element Equations," Proc. 1 s t WCCM, Vol. 1, Texas I n s t i t u t e f o r Computational Mechanics , 1986.

O r t i z , M. and Y. Leroy, "A F i n i t e Element Method f o r Loca l i zed F a i l u r e Analysis," Proc. 1 s t WCCM, Vol. 1, Texas I n s t i t u t e f o r Computational Mechanics, 1986.

Or t i z , M. and J. C. Simo. ''An Analys is o f a New Class o f I n t e a r a t i o n Algori thms f o r E l a s t o p l a s t i c C o n s 6 t u t i v e Relat ions," I n t . J.-Num. Meth. Eng., 23, 1986, pp. 355-366.

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