An isoparametric 3-D finite element mesh generator was developed with direct interface to an interactive geometric modeler program called POLYGON. POLYGON defines the model geometry in terms of boundaries and mesh regions for the mesh generator. The mesh generator controls the mesh flow through the 2-dimensional spans of regions by using the topological data and defines the connectivity between regions. The program is menu driven and the user has a control of element density and biasing through the spans and can also apply boundary conditions, loads interactively

Kayrak, C.

Ozsoy, T.

English

NASA Technical Reports Server

ISOPARAMETRIC 3-0 FINITE ELEMENT MESH GENERATION USING INTERACTIVE 
COMPUTE2 GRAPHICS 
Cuneyd Kayrok m a  Dr. Tulga Ozsoy 
Department o f  Mechanical Engineer i  ng and Mecharli cs 
Lehi  gh l ln i  v e r s i  t y  
SUMMARY 
An isoparametr ic  3-0 f i n i t e  element mesh generator was developed w i t h  d i r e c t  
i n t e r f a c e  t o  Lehigh developed i n t e r a c t i v e  geometric modeler program c a l l e d  
POLYGON. POLYGON defines t h e  model geometry i n  terms o f  boundaries and mesh 
r e g i ~ n s  f o r  t h e  mesh generator. The mesh g ~ n e r a t o r  con t ro l s  t he  mesh f l o w  through 
t he  2-dimensional "spans o f  reg ions"  by us ing  t h e  t opo log i ca l  data def ines t he  
connec t i v i t y  between regions. The program i s  menu d r i ven  and t h e  user  has a 
con t ro l  o f  element dens i ty  and b ias ing  through t h e  spans and can a l so  apply 
b ~ u n d a r y  condi t ions,  1 oads i n t e r a c t i v e l y .  
INTRODUCTION 
The POLYGON (1) package developed a t  t h e  CAD Laboratory o f  Lshigh Un i ve rs i t y  
has been q u i t e  successful  i n  p rov id i ng  t h e  user w i t h  t o o l s  f o r  3-D w i r e  frames t o  
B-REP type  so l  i d  model representat ions conversion process. POLYGON can accept 3-2 
w i re  frame data from o ther  CAD systems e i t h e r  d i r e c t l y  o r  through format ted f i l e s .  
P3LYGON a1 so provides some geometry generat i  on and manipu la t ion capa t i  1 i t i e s .  
Furthermore, d i f f e r e n t  f i l e s  can be w r i t t e n  from POLYGON t o  be i npu t  f o r  o ther  
s o l i d  modelers use 0-REP type  representat ions. The f i n i t e  element modeling 
package was developed as a sub module w i t h i n  POLYGON, thus i t  has a d i r e c t  access 
t o  t h e  w i r e  frame data as w e l l  as t o  t h e  bounded surfaces, "regions" created by 
POLYGON. The main purpose o f  t h i s  i n t eg ra ted  approach i s  t o  reach a h igh degree 
o f  automation dur ing  t h e  mesh generat ion by reducir lg t h e  model p repara t ion  t i t t l e  
w i t h  l ess  requ i red  user inpu t .  Mesh f l c ~  i s  au tomat ica l l y  d i r ec ted  through t he  
number o f  spans which can be def ined us ing  t h e  t opo log i ca l  data f o r  regions. 
Cu r ren t l y  on ly  f ou r  and/or three-s ided regions are a i  lowed. I n  each reg ion nodes 
are created us ing  c u r v i - l i n e a r  coord inate system and quadra t i c  shape funct ions 
2 ) .  Supported element types a re  1 i near o r  parabol i c t r i a n g u l a r  and quadri  1 a t e r a l  
elements. Loads and r e s t r a i n t s  can a l s o  be app l iod  ?n t -e rac t i ve l y  t o  t h e  model. 
F i n a l l y ,  f i l e s  can be w r i t t t . n  f o r  i npu t  t o  COSMIC NASTRRN and SDRC's SUPERB f i n i t e  
element ana lys is  programs. F u l l  c o l o r  v i s u a l i z a t i o n  o f  t h e  f i n i t e  element mod.1 
i s  poss ib le  a t  any stage w i t h  extens ive view con t ro l  c a p a b i l i t i e s  such as dynamic 
r o t a t i o n ,  sca l ing,  t r a n s l a t i o n ,  zooming, etc.  I n t e r a c t i v e  menus 1 inked  i n  a t r ee -  
l i k e  s t r u c t u r e  a l lows users a na tu ra l  way o f  communication w i t h  t he  system. The 
program c u r r e n t l y  has been implemented on DEC ' s  l!S11 r a s t e r  te rmina ls  w i t h  16 
co lo rs  at tached t o  a VAX 11/780, and i s  supported by Lehigh developed VS11-3D 
Graphics Package (3).  
https://ntrs.nasa.gov/search.jsp?R=19850017562 2020-03-22T19:18:07+00:00Z
CURREYT CAPABILITIES OF THE FEM SYSTEM 
F i n i t e  element model c r e a t i o n  cons is ts  o f  s i x  bas ic  steps: 
1. Geonietry c rea t ion :  POLYGON prov ides user w i t h  some geometry c rea t i on  and 
manipu la t ion c a p a b i l i t i e s .  Points,  l i n e s ,  arcs, conics and cub ic  sp l ines  a re  
c u r r e n t l y  supported w i r e  frame e n t i t y  type;. Thus, 3-0 w i r e  frame geometry 
can be e i t h e r  created from sc ra tch  w i t h i n  POLYGON o r  on a ~ o t h e r  CAD system 
and then can be t r a n s f e r r e d  t o  t h e  POLYGON program. F igure  i i s  an example 
o f  part-geometry us ing  3-0 w i  r e  frame representat ion.  
Region c rea t ion :  Bounded surfaces o f  p lanar,  r u l e d  and sur face o f  r e v o l u t i o n  
t ype  can be def ined au tomat i ca l l y  and/or manually a t  t h i s  step. The same 
geometry g iven i n  F igure  1 i s  shown i n  F igure  2 w i t h  a l l  reg ions defined. 
These bounded surfaced (polygons), " reg ions"  must be e i t h e r  f o u r  o r  three-  
s ided  because o f  t h e  mesh generat ion scheme used. Once reg ions a re  def ined 
they can be ordered e i t h e r  c lockwise o r  counter-c lockwise automat ica l ly .  The 
edges o f  t h e  reg ions a re  converted i n t o  pa rabo l i c  t ype  and t h e  reg ion  data i s  
format ted i n  a way so i t  can be used d i r e c t l y  by t h e  FEM module. 
3. Span generat ion:  Span def ines t h e  f l ow  o f  mesh generat ion through t he  
regions connected by oppos i te  edges. Using t h e  edge and reg ion  data t he  
program can c rea te  spans automat ica l ly .  Spans can be h i g h l i g h t e d  f o r  user 
checking (F igure 3 and F igure  4 ) .  Spans can a l c g  be created and/or extended 
mandally t o  inc lude  more regions (Figure 5). Number o f  elements across t h e  
span and b i a s i n g  can be entered by se l ec t i ng  i n d i v i d u a l  spans o r  system 
prov ided d e f a u l t  values can be used. 
Search f o r  t h e  spans i s  b a s i c a l l y  d e f i n i n g  t h e  oppos i te  edge o f  
t he  cur rent  reg ion and t h e  o ther  reg ion shares t he  same edge. Region data 
conta ins t h e  edge loop and t h e  edge data conta ins t h e  reg ion  numbers a t tached 
t o  it. A span ends when t he re  i s  no more reg ion at tached t o  t h e  cu r ren t  edge 
( f r e e  edge) o r  i f  t h e  f i r s t  edge o f  t h e  span i s  reached. Each reg ion  w i  11 be 
crossed by two spans. A f t e r  t he  f i r s t  span has been def ined, search 
cont inues t c  de f ine  a l l  o ther  spans c ross ing  a l l  t h e  regions i n s i d e  t h e  
f i r s t  span. Automatic search ends when a l l  t h e  regions i n  t h e  model are 
crassed by two d i f f e r e n t  spans. 
4. Yesh generat ion:  Once t h e  spans a re  def ined, me3b i s  generated throughout 
the s t r u c t u r e  a t  once (F igure  6). F i r s t  nodes a re  gener..ted us i ng  t he  number 
o f  d i . i s i o n s  and b i a s i n g  in fo rmat ion  g iven across t h e  spans. Node gener3t ion 
i- performed i n  parametr ic  space us ing  c u r v i - l i n e a r  coora inate system and 
quadrat ic  shape func t ions  w i t h  m i  4-s i  de nodes p l  aced ha 1 f-way between corner-  
nodes, then they a re  mapped i n t o  t h e  cu r ren t  coord inate sys s. A f t e r  a1 1 
t he  nodes have been d e f i n e i  t he  elements a re  generatad. Since t h e  nodes a re  
generatnd i n  an ordered manner t he  g e n e r a t i x  o f  e l ~ ~ e n t s  i s  a t r i v i a l  
process. Fade- and elements a r e  l abe led  au tomat i ca l l y  du-;nq t h e i r  
generat ion ( F i g l ~ r e  7 and F igure 81. Cur ren t l y  on ly  l i n e a r  o r  parabo l i c  
t r i ang t i l  a r  and quadr i  l a t e r -  1 isopararnstr i  c element types a re  supported. 
5. F i n i t e  element mods1 preparat ion:  For each reg ion  d i f f e r e n t  mate r ia i  t ype  
numbers can be assigned t o  t h e  elements. Ma te r i a l  p roper ty  values can be 
inpu t  by t h e  user and s to red  i n  mate-ial t ab les  which then can be accessed by 
mater ia l  type numbers. De fau l t  r e s t r a i n t  and load  d e f i n i t i o n s  can be e d i t e d  
and they can be app l ied  t o  t h e  se lected nodes o r  along t h e  edges o f  t h e  
region. D i f f e r e n t  c o l o r  codings and arrow heads a re  used f o r  d i f f e r e n t  load 
types and r e s t r a i n t s  t o  enhance t h e  use r ' s  v i sua l  i za t i on .  
6. Creat ing i npu t  data f o r  f i n i t e  element analys is :  Once t h e  f i n i t e  element 
model in fo rmat ion  i s  generated, i npu t  f i l e s  can be w r i t t e n  f o r  e i t h e r  COSMIC 
NASTRAN o r  SDRC's SUPERB f i n i t e  element ana lys is  packages. 
CONCLUDiNG REMARKS 
The cur ren t  approach uses t h e  extended concept o f  regions w i t h  span 
d e f i n i t i o n .  Region re fers  t o  a bounded area i n s i d e  a span and i t s  p o s i t i o n  i s  
t o p o l o g i c a l l y  defined. Spans can be referenced as o ther  t ype  o f  f i n i t e  elemznt 
e n t i t i e s  but a t  h igher  leve! than regions. Thus a data se t  assigned t~ a span can 
be app l ied  t o  a l l  t h e  o ther  lower l e v e l  r e l a t e d  e n t i t i e s ,  i.e. regions, elements, 
nodes, etc., such as grouping f o r  app l i ca t i on  o f  boundary condi t ions,  o r  b lank ing 
f o r  b e t t e r  v i sua l i za t i on .  By us ing  span d e f i n i t i o n  remarkable savings can be 
reached e a s i l y  by reducing t h e  number o f  user i n t e r a c t i o n s  requ i red  a t  model 
preparat ion stage. T r a d i t i o n a l l y  a l l  t he  regions a re  def ined oqe ~y one, then t h e  
mesh i s  generated i n  each of them i n d i v i d u a l l y .  I n  F igure  2 t he  p a r t  contains 47 
regions but t he re  a re  on ly  15 spans. Since t h e  in fo rmat ion  r e l a t e d  t o  t h e  number 
o f  d i v i s i o n s  and b ias ing  across t h e  span i s  entered a t  span l e v e l  but  not a t  
region l e v e l  considerable tinle savings can be reached a+ t h i s  stage o f  t h e  model 
preparat ion process. 
REFERENCES 
1. Ozsoy, T. and Ochs, J.B., "An I n t e r a c t i v e  Modeling Program f o r  t h e  
generation o f  Planar Polygons f o r  Boundarl Type s o l i d s  Representations o f  
Wire Frame Models", Proceedings from the  Symposium o f  Computer-Aided 
Geometry Model i ny ,  NASA Langley, Hampton, VA, Apr i  1 21, 1983. 
2. Zienkiewich, O.C. and P h i l  1 ips ,  D.V., "An A u t ~ m a t i c  Mesh Generation Scheme 
f o r  Dlane and Curved Surfaces by Isoparametr ic Co-ordinates, " 
I n te rna t i ona l  J o i r n a l  f o r  Numerical Methods i n  Engineering, Vol. 3, No. 
4, 1971, pp. 519-518. 
3. Orsoy. T. afid Ochs, J.B., "Lehigh Un i ve r i s t y  ' s  VSll-39 Graphics Package", 
Proceedinqj from 1983 DEiUS Meeting, S t .  Louis, MO, May 23, 1983. 




Isoparametric 3-D Finite Element Mesh Generation Using Interactive Computer Graphics

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