- Email: [email protected]
Chapter 2 Interactive Computer Graphics for Finite Element, Boundary Element, & Finite Difference Methods
Unification of Finite Element Methods H. Kardestuncer (Editor) 0 Elsevier Science Publishers B.V. (North-Holland), 1984
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CHAPTER 2 INTERACTIVE COMPUTER GRAPHICS FOR FINITE ELEMENT, BOUNDARY ELEMENT, & FINITE DIFFERENCE METHODS
J.F. Abel, A.R. Ingraffea, R. Perucclzio, T - Y. Han, & J.F. Hajjar
The t r e n d toward u b i q u i t y o f i n t e r a c t i v e g r a p h i c s i n computational e n g i n e e r i n g i s h e l p i n g t o develop an atmosphere conducive t o t h e combination and u n i f i c a t i o n o f v a r i o u s numerical a n a l y s i s methods. A number of f e a t u r e s o f i n t e r a c t i v e computer a i d e d design which f o s t e r u n i f i c a t i o n a r e discussed. Examples a r e drawn from t h r e e p r i n c i p a l aspects o f computer graphics i n e n g i n e e r i n g a n a l y s i s : p r e processing, postprocessing, and i n t e r a c t i v e a d a p t i ve a n a l y s i s . INTRODUCTION The purpose o f t h i s paper i s t o present and e x p l o r e some ideas about how t h e growing use of i n t e r a c t i v e computer g r a p h i c s i s b o t h a f f e c t i n g t h e u t i l i z a t i o n o f numerical methods by e n g i n e e r i n g a n a l y s t s and i n f l u e n c i n g t h e p o s s i b l e u n i f i c a t i o n o f these methods. Recent decades have seen a dramatic development and p r o l i f e r a t i o n o f numerical methods. T h i s a c t i v i t y has i n c l u d e d t h e s t r e n g t h e n i n g and d i v e r s i f i c a t i o n o f e s t a b l i s h e d methods such as f i n i t e d i f f e r e n c e s and f i n i t e elements, t h e r a p i d expans i o n o f r e d i s c o v e r e d techniques such as b o u n d a r y - i n t e g r a l equations, t h e i n c r e a s i n g r e c o g n i t i o n of u n i f y i n g f o r m u l a t i v e n o t i o n s such as v a r i a t i o n a l approaches , and t h e coinbi n a t i o n o f methods t o r e a l i z e d i v e r s e advantages such as t h e m i x i n g o f f i n i t e elements and boundary elements. The ideas presented here r e p r e s e n t some r e c e n t c o l l e c t i v e thoughts o f a research group a t C o r n e l l U n i v e r s i t y t h a t has, f o r n e a r l y a decade, been i n v o l v e d i n developments o f i n t e r a c t i v e computer graphics f o r t h e i r a p p l i c a t i o n s t o both computational mechanics research and computer-aided design. Because most o f t h i s research has been l i m i t e d t o s t r u c t u r a l e n g i n e e r i n g and mechanics and has i n v o l v e d p r i m a r i l y f i n i t e element and boundary element approaches, t h i s can h a r d l y be considered a g l o b a l view o f d i s c i p l i n e s and methods. Nevertheless, t h e emphasis i n t h i s paper on growing t r e n d s i n g r a p h i c s and t h e i r r e l a t i o n t o u n i f i c a t i o n o f a n a l y s i s methods can be considered a manifesto, a l b e i t modest, r e g a r d i n g d e s i r a b l e d i r e c t i o n s f o r f u t u r e developments i n i n t e r a c t i v e graphics. C l e a r l y i n t e r a c t i v e graphics i n i t s e l f i s d i s t i n c t from a n a l y s i s and, t h e r e f o r e , can be viewed from one p e r s p e c t i v e as having l i t t l e t o do w i t h u n i f i c a t i o n o f a n a l y s i s techniques. However, i t has been w i d e l y demons t r a t e d t h a t i n t e r a c t i v e g r a p h i c s can be h i g h l y successful i n b r e a k i n g down b a r r i e r s and d i f f i c u l t i e s i n t h e performance o f e n g i n e e r i n g a n a l y s i s . The enhanced access t o , and c o n t r o l o f , a n a l y s i s can i n t h e same way s i g n i f i c a n t l y h e l p remove o b s t a c l e s t o u n i f i c a t i o n , as w i l l be argued
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subsequently. The p r o v i s i o n o f a proper atmosphere f o r amalgamation i s no l e s s i m p o r t a n t than t h e r e c o g n i t i o n o f t h e p o t e n t i a l f o r u n i f i c a t i o n . U l t i m a t e l y , t h e obvious goal i s t h e p r o v i s i o n o f t h e most e f f e c t i v e and a p p r o p r i a t e a n a l y s i s methodologies f o r engineers t o c a r r y out design r e s p o n s i b l y and c r e a t i v e l y . The computational environment f o r a n a l y s i s i s changing r a p i d l y . Computeraided a n a l y s i s and design i n both p r a c t i c e and i n d u s t r y a r e being t i e d t o g e t h e r wit h computerized i n f o r m a t i o n r e p o s i t o r i e s and data f l o w c a p a b i l i t i e s , w h i l e a t t h e same t i m e i n c r e a s e d computational power i s being p r o vided t o t h e i n d i v i d u a l engineer. Three c h i e f c u r r e n t developments i n hardware and software are e n g i n e e r i n g w o r k s t a t i o n s which p r o v i d e i n t i m a t e access t o computing, n e t w o r k i n g which enables t h e s h a r i n g and r a p i d t r a n s f e r o f l a r g e volumes o f data among w o r k s t a t i o n s , and i n t e r a c t i v e g r a p h i c s c a p a b i l i t i e s i n t e g r a l t o t h e w o r k s t a t i o n s . The l a s t o f these i s t h e theme here. I n t h i s environment, w i t h i n t h e next decade, n e a r l y a l l computing w i l l be a s s o c i a t e d w i t h i n t e r a c t i v e graphics. Both l i t e r a l l y and f i g u r a t i v e l y , i n t e r a c t i v e computer graphics i s becoming t h e "window" t o t h e world o f analysis. I n t e r a c t i v e computer graphics was f i r s t a p p l i e d t o p r e p r o c e s s i n g and p o s t p r o c e s s i n g only, and these a r e s t i l l t h e phases o f a n a l y s i s where e n g i n e e r i n g p r o d u c t i v i t y i s most d r a m a t i c a l l y improved by i t s use. However, although a n a l y s i s o r p r o c e s s i n g i t s e l f has t r a d i t i o n a l l y been a batch p r o cedure, t h e continued i n c r e a s e o f cheap, d i s t r i b u t e d computational capab i l i t y and t h e growing demands o f d e s i g n e r l a n a l y s t s i n d i c a t e t h a t i n t h e near f u t u r e some i n t e r a c t i v e processing w i l l be e f f e c t i v e and d e s i r a b l e even f o r such c o m p u t a t i o n a l l y i n t e n s i v e procedures as f i n i t e element a n a l y s i s . This n o t i o n has given r i s e t o i n t e r a c t i v e - a d a p t i v e a n a l y s i s [l], which i s d e f i n e d as a n a l y s i s d u r i n g which t h e engineer can c o n t i n u o u s l y m o n i t o r what i s happening and can choose t o i n t e r v e n e a t any t i m e t o change system c h a r a c t e r i s t i c s , models, a n a l y s i s parameters, and a1 gor i t h m s . I t i s most s u i t e d t o n o n l i n e a r and t i m e - v a r y i n g analyses. The engineer i s a b l e t o move backward and forward i n t h e a n a l y s i s a t w i l l and t o invoke a v a r i e t y o f p a r a l l e l analyses a t any stage. Moreover, d i f f e r e n t types o f analyses can be s t r u n g t o g e t h e r i n a sequence. T h i s a n a l y s i s c o n t r o l c a p a b i l i t y c l e a r l y must r e l y on i n t e r a c t i v e computer graphics f o r i t s success. I n t h i s paper a l l t h r e e aspects o f i n t e r a c t i v e computer graphics i n e n g i n e e r i n g a n a l y s i s are used t o i l l u s t r a t e t h e n o t i o n s discussed. UNIFYING INFLUENCES OF INTERACTIVE GRAPHICS Experience w i t h t h e development o f i n t e r a c t i v e g r a p h i c a l s o f t w a r e programs f o r a n a l y s i s a p p l i c a t i o n s i n d i c a t e s t h a t , f o r w e l l designed i n t e r a c t i v e systems, programming i s a major task t h a t exceeds i n complexity and d i f f i c u l t y t h e programming o f much a n a l y s i s s o f t w a r e [2]. In addition t o the a r a t i o n a l u n d e r l y i n g database; m o d u l a r i t y t o f o s t e r usual requirements e x p a n d a b i l i t y ; c l e a r , s t r u c t u r a l programming; a system f o r a n t i c i p a t i n g , handling, and c o r r e c t i n g e r r o r s ; t r a n s p o r t a b i l i t y t o t h e maximum e x t e n t p o s s i b l e w h i l e m a i n t a i n i n g necessary performance and response; and t h e design f o r human f a c t o r s o r "user f r i e n d 1 i thorough documentation ness" places e x t r a demands on t h e program developer. Among these a r e completeness o f f u n c t i o n , f l e x i b i l i t y i n sequence, and a humanized command language and menus [31 141. The achievement o f these l a s t o b j e c t i v e s
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tends t o r e s u l t i n a g e n e r a l i t y o f t h e i n t e r f a c e between t h e user and t h e computer t h a t i s conducive t o f l e x i b i l i t y , d i v e r s i f i c a t i o n , and u n i f ic a t i on o f a n a l y s i s methods. I n a d d i t i o n t o t h e fundamental design o f i n t e r a c t i v e software, a f a c t o r which i s a f f e c t i n g t h e way i n which p a r t i c u l a r a n a l y s i s methods f i t i n t o an o v e r a l l design process i s t h e i n c r e a s i n g use o f system models independent from a n a l y s i s models. I n t h e a p p l i c a t i o n o f , say, f i n i t e element methods, one was p r e v i o u s l y l i k e l y t o d e s c r i b e t o t h e computer t h e geometry, boundary c o n d i t i o n s , and l o a d i n g s o f t h e problem under
I n t e r a c t i v e System M o d e l l i n g ( P r e l i m i n a r y D e s i g n , Geometry)
I n t e r a c t i v e Preprocessing (Geometry, A t t r i b u t e s , A n a l y t i c a l M o d e l )
t
T r a n s l a t o r (s) Ana 1y s i s
A n a l y s i s M e t h o d (s) I n v e r s e T r a n s l a t o r (s)
t I I I n t e r a c t i v e Postprocessing
I n t e r a c t i v e Design/Re-Design (Geometry, A t t r i b u t e s )
F i g u r e 1.
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The computer-aided design process. I n t e r a c t i v e computer graphics i s t h e p r i n c i p a l medium f o r man-machine communication and c o n t r o l . The i n t e g r a t i o n o f t h e v a r i o u s stages shown i s p r e d i c t e d on computerized databases and data flow.
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c o n s i d e r a t i o n i n terms o f t h e elements, nodes, and c o n n e c t i v i t y o f t h e d e s c r i p t i o n t h a t was e n t i r e l y l i n k e d w i t h t h e a n a l y t i c a l d i s c r e t i z a t i o n . With t h e t r e n d toward computer-aided design, a n a l y s i s i s now u s u a l l y viewed as being i n t e g r a t e d i n t o an o v e r a l l design process such as d e p i c t e d i n general form i n F i g u r e 1. The l i n k s between t h e v a r i o u s steps shown a r e achieved through a computerized database and data flow. Here t h e geometric d e s c r i p t i o n o f t h e design problem i s c r e a t e d on t h e computer by a modeller or, i f a modeller i s n o t a v a i l a b l e , i t i s created as t h e f i r s t s t e p o f preprocessing. Then, i n preprocessing, a d d i t i o n a l a t t r i b u t e s o f t h e problem, such as m a t e r i a l p r o p e r t i e s , boundary c o n d i t i o n s , and l o a d i n g s , a r e a s s o c i a t e d w i t h t h e geometry. The r e s u l t i s t h e complete d e s c r i p t i o n o f an a n a l y s i s problem t h a t e x i s t s w i t h i n t h e computer p r i o r t o any meshing and, indeed, t h a t i s independent o f any p a r t i c u l a r method. I n t h e f u t u r e , t h e generation o f t h e a n a l y t i c a l model o r d i s c r e t i z a t i o n a p p r o p r i a t e f o r a p a r t i c u l a r method o f a n a l y s i s w i l l probably be automated [5]. T h i s w i l l be p a r t i c u l a r l y s u i t a b l e f o r a n a l y s i s procedures t h a t i n c l u d e s e l f - a d a p t i v e mesh a l g o r i t h m s t o achieve a s p e c i f i e d degree o f refinement o r accuracy i n each p o r t i o n o f t h e system. However, f o r a t l e a s t t h e next decade, it i s c l e a r t h a t t h e e n g i n e e r i n g a n a l y s t w i l l need t o continue t o e x e r c i s e s t r o n g i n f l u e n c e over t h e design o f t h e a n a l y t i c a l model because o f i t s d i r e c t e f f e c t on accuracy and r e l i a b i l i t y o f t h e a n a l y s i s . I n t e r a c t i v e mesh generators a r e i d e a l l y s u i t e d t o t h i s task [6]. Moreover, e f f e c t i v e i n t e r a c t i v e preprocessors p r o v i d e a f l e x i b i l i t y t h a t i s o n l y beginning t o be recognized. Not o n l y a r e computer-assisted mesh generators i n many cases a b l e t o be g e n e r a l i z e d f o r a v a r i e t y o f a n a l y t i c a l approaches, b u t t h e i n t e r a c t i v e t o o l s i n c o r p o r a t e d f o r , say, s u b s t r u c t u r i n g o r p a r t i t i o n i n g a l s o present n a t u r a l o p p o r t u n i t i e s f o r preprocessing f o r h y b r i d o r combined techniques. Another aspect o f i n t e r a c t i v e s o f t w a r e design t h a t s i g n i f i c a n t l y a l l e v i ates s p e c i f i c i t y t o a s i n g l e a n a l y s i s t y p e i s t h e humanization o f command languages. One d e t e c t a b l e t r e n d i s toward t h e use o f commands more n a t u r a l t o t h e engineer. For example, i n t h e s p e c i f i c a t i o n o f boundary c o n d i t i o n s f o r f i n i t e element a n a l y s i s , one may be asked t o s p e c i f y t h e t y p e o f c o n d i t i o n (e.g., "symmetry") r a t h e r t h a n a node-by-node l i s t o f r e s t r a i n t codes f o r each degree o f freedom. The c o r o l l a r y i s t h a t t h e preprocessing software o r t h e t r a n s l a t o r programs t h a t a r e t h e i n t e r f a c e t o t h e s p e c i f i c a n a l y s i s program ( F i g u r e 1) must i n c l u d e i n t e r n a l coding which a u t o m a t i c a l l y converts between a n a l y s i s - s p e c i f i c data and t h e engi n e e r ' s n a t u r a l vocabulary.
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Two f a c t o r s r e l a t i n g t o data o r g a n i z a t i o n f o r i n t e r a c t i v e systems potent i a l l y c o n t r i b u t e t o t h e d i v e r s i f i c a t i o n o f such software. The f i r s t o f these i s t h e o r g a n i z a t i o n o f data s t r u c t u r e s f o r e f f i c i e n t and r a p i d g r a p h i c a l d i s p l a y . This i n v o l v e s t h e e x p l o i t a t i o n o f geometric coherence which may d i f f e r from t h a t imposed by a p a r t i c u l a r a n a l y s i s method. For instance, i n d i s p l a y i n g a f i n i t e element mesh, one may n o t wish t o draw t h e mesh element by element because t h i s w i l l i n v o l v e t h e redrawing o f l i n e s f o r t h e edges o f adjacent elements. Instead, one would p r e f e r t o t a k e advantage o f t h e coherence o f mesh l i n e s by drawing each complete l i n e i n one step. The rearrangement o f element edge data i n t o complete mesh-line data i n t h i s f a s h i o n i s perhaps more reminiscent o f data o r g a n i z a t i o n s a p p r o p r i a t e t o h i g h e r o r d e r f i n i t e d i f f e r e n c e operators. Another example a r i s e s from t h e d e s i r e t o p r o v i d e d i s p l a y s i m p l i f i c a t i o n s f o r three-dimensional geometries. Here, one technique used t o p e r m i t c l e a r e r viewing o f three-dimensional f i n i t e element meshes i s t h e s e l e c t e d removal
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o f l i n e s o r t h e s e l e c t i v e d i s p l a y of some elements o r groups o f elements. One u s e f u l v e r s i o n of t h i s i s t h e d i s p l a y of o n l y t h e e x t e r i o r s u r f a c e s o f elements ['I] [&I]. The d i s t i n c t i o n here i n t h e data s t r u c t u r e between i n t e r i o r and e x t e r i o r element faces c r e a t e s an obvious 1 ink t o boundary element techniques. The second d a t a - r e l a t e d f a c t o r i s t h e o v e r a l l data t r a n s m i s s i o n scheme u n d e r l y i n g computer-aided design systems w i t h data f l o w s such as i n d i c a t e d s c h e m a t i c a l l y i n F i g u r e 1. The need not o n l y t o t r a n s m i t i n f o r m a t i o n from one stage o f design t o another b u t a l s o t o exchange data between d i f f e r e n t computer-aided design systems has g i v e n r i s e t o s t a n d a r d i z e d procedures, most n o t a b l y t h e I n i t i a l Graphics Exchange S p e c i f i c a t i o n o r I G E S [Sl. Although s t i l l under development and n o t y e t as e f f i c i e n t as d e s i r able, I G E S i s now being s u c c e s s f u l l y used t o t r a n s m i t geometrical data. Moreover, t h e s p e c i f i c a t i o n i s being expanded t o i n c l u d e a n a l y s i s i n f o r m a I G E S data f i l e s a r e r i g i d l y f o r t i o n , p a r t i c u l a r l y f i n i t e element data. matted b u t a r e so complete and f l e x i b l e i n t h e i r r e p r e s e n t a t i o n o f data t h a t t h e f i l e s can be considered e s s e n t i a l l y system independent. The f l e x i b i l i t y o f I G E S a r i s e s because i t s e n t i t i e s may be e i t h e r geometric o r nongeometric and because v a r i o u s e n t i t i t e s may be a s s o c i a t e d i n ways a p p r o p r i a t e t o any a p p l i c a t i o n . The i m p l i c a t i o n f o r a n a l y s i s o f t h e e x i s t e n c e of such schemes as IGES i s t h e n o t i o n t h a t e i t h e r problem d e s c r i p t i o n s o r a n a l y s i s models and r e s u l t s can be s t a n d a r d i z e d i n a comp l e t e , n e u t r a l form. T h i s f o s t e r s an exchange o f data t h a t p e r m i t s n o t o n l y comparisons b u t a l s o combinations o r h y b r i d i z a t i o n s o f a n a l y t i c a l
F i g u r e 2.
Assignment o f boundary c o n d i t i o n s d u r i n g f i n i t e element p r e p r o c e s s i n g f o r s h e l l s t r u c t u r e s [lo]. Note t h e n a t u r a l t e r m i n o l o g y f o r boundary c o n d i t i o n types (middle p o r t i o n o f menu a t r i g h t ).
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procedures. Some o f t h e ideas i n t r o d u c e d thus f a r are now i l l u s t r a t e d by d i s c u s s i o n s and examples o f i n t e r a c t i v e programs developed f o r research o r i n s t r u c t i o n a l purposes a t C o r n e l l . A l l o f t h e subsequent f i g u r e s are photographs taken from graphic d i s p l a y devices, and these have been reduced t o t h e degree t h a t some small t e x t i s not e a s i l y l e g i b l e . I n each o f these cases, t h e small t e x t c o n t a i n s l i t t l e p e r t i n e n t i n f o r m a t i o n and i s r e t a i n e d o n l y t o i l l u s t r a t e t h e n a t u r e o f t h e computer graphics d i s p l a y , I n a d d i t i o n , t h e photographs o f c o l o r d i s p l a y s are, by n e c e s s i t y , reproduced here i n black and white. PREPROCESSING
A s i n g l e glimpse a t an i n t e r a c t i v e preprocessor f o r s h e l l a n a l y s i s [ l o ] provides an example o f t h e humanization o f commands. I n F i g u r e 2, t h e menu page f o r t h e assignment o f boundary c o n d i t i o n s i s shown. The commands along t h e r i g h t edge are d i v i d e d i n t o groups. The upper group c o n s i s t s o f permanent menu items f o r t h e m a n i p u l a t i o n o f t h e main image, w h i l e t h e group below these are commands and m o d i f i e r s s p e c i f i c t o t h e immediate t a s k . Here t h e user has a c t i v a t e d t h e command ASSIGN w i t h t h e boundary c o n d i t i o n t y p e SYMMETRY and t h e m o d i f e r s e t t o LINE. (All active commands and m o d i f i e r s a r e i n d i c a t e d by a box drawn about them.) Simply by p o i n t i n g t o t h e m e r i d i o n a l c u t s o f t h e h y p e r b o l i c c o o l i n g tower s h e l l shown, t h e engineer assigns t h e symmetry c o n d i t i o n t o t h i s l i n e . By f a r t h e most s i g n i f i c a n t e f f o r t i n preprocessing i s t h e c r e a t i o n o f t h e
F i g u r e 3.
Complete three-dimensional f i n i t e element mesh f o r a p r o p e l l e r generated by l o f t i n g between s e c t i o n s [8].
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a n a l y t i c a l model o r mesh. Even w i t h mesh g e n e r a t i o n a l g o r i t h m s , some i n t e r v e n t i o n by t h e engineer i s u s u a l l y necessary o r d e s i r a b l e t o c o n t r o l t h e design of t h e d e s c r e t i z a t i o n . For example, i n t h e d i s c r e t e t r a n s f i n i t e mapping methods of mesh generation, t h e d e n s i t y and g r a d a t i o n o f t h e mesh i n a r e g i o n i s determined by t h e placement and spacing o f nodes around t h e boundary of t h e r e g i o n [7l, and t h i s t a s k i s b e s t performed by t h e analyst interactively. The techniques o f mesh c r e a t i o n t h a t have been found successful f o r i n t e r a c t i v e a p p l i c a t i o n s have a l s o proven t o be a p p l i c a b l e f o r a v a r i e t y of d i f f e r e n t a n a l y s i s types. A t C o r n e l l , t h e l o f t i n g s p e c i a l i z a t i o n of d i s c r e t e t r a n s f i n i t e mapping has been used f o r b o t h f i n i t e element [7] C8l and boundary element [ l l ] C121 preprocessing. F i g u r e 3 shows a complete three-dimensional f i n i t e element mesh f o r a p r o p e l l e r - l i k e s o l i d generated i n t e r a c t i v e l y i n t h i s fashion. The mesh f o r t h e hub has been c r e a t e d by l o f t i n g among p a r a l l e l p l a n a r s e c t i o n a l meshes w i t h d i f f e r e n t c i r c u l a r diameters, w h i l e t h e l o f t i n g s e c t i o n s f o r t h e blades a r e n o t p a r a l l e l and, a t t h e r o o t o f t h e blades, a r e nonplanar In s e c t i o n s i n t e r a c t i v e l y s e l e c t e d from t h e s u r f a c e o f t h e hub mesh [8]. F i g u r e 4, t h e mesh i n F i g u r e 3 i s s i m p l i f i e d f o r c l e a r e r v i e w i n g by d i s p l a y i n g o n l y t h e e x t e r i o r f a c e t s o f elements, and t h e r e s u l t i s c l e a r l y suggestive o f t h e a p p l i c a b i l i t y o f t h i s meshing procedure f o r boundary element preprocessing. An example o f a d i r e c t a p p l i c a t i o n o f t h i s meshing t e c h n i q u e f o r a boundary element problem i s shown i n F i g u r e s 5 and 6 [ll]
CW.
The methods i n these examples c o u l d be s i m i l a r l y extended t o such a l t e r n a t i v e s as f i n i t e d i f f e r e n c e mesh generation. However, one o b s t a c l e remaini n g t o t h e complete g e n e r a l i z a t i o n o f i n t e r a c t i v e p r e p r o c e s s i n g techniques
F i g u r e 4.
The mesh i n F i g u r e 3 i s s i m p l i f i e d f o r c l e a r e r viewing by d i s p l a y i n g o n l y e x t e r i o r (boundary) f a c e t s o f elements.
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f o r a n a l y t i c a l model g e n e r a t i o n i s t h e nonorthogonal i t y o f t h e c o o r d i n a t e systems employed i n t h e mesh generation procedures. Many f i n i t e d i f f e r ence a l g o r i t h m s , f o r example, r e q u i r e a mesh which c o n s i s t s of i s o c o o r d i n a t e l i n e s i n an orthogonal c o o r d i n a t e system. The common f i n i t e e l e ment mesh g e n e r a t i o n techniques based on i s o p a r a m e t r i c o r t r a n s f i n i t e mappings do n o t a u t o m a t i c a l l y f u l f i l l t h i s requirement. Nevertheless, i t i s c l e a r t h a t t h e meshing methods can be extended w i t h minor adaptions t o l e s s r e s t r i c t i v e a n a l y s i s methods such as f i n i t e c e l l methods. I n many cases, boundary element a n a l y s i s r e q u i r e s n o t o n l y a d i s c r e t i z a t i o n of t h e s u r f a c e o f t h e domain b u t a l s o an i n t e r i o r mesh. The conside r a t i o n o f body e f f e c t s such as g r a v i t y loads and o f n o n l i n e a r i t i e s a r e two examples. I n a d d i t i o n , i f f u l l f i e l d o r s e l e c t e d i n t e r i o r r e s u l t s are required, t h e n i n t e r i o r p o i n t s must be chosen as l o c a t i o n s f o r t h e evaluat i o n o f responses. Although i n t e r i o r meshes f o r t h e boundary element method need n o t s a t i s f y t h e same s t r i c t t o p o l o g i c a l and c o n n e c t i v i t y requirements as a f i n i t e element mesh, t h e use o f an i n t e r i o r f i n i t e e l e ment d i s c r e t i z a t i o n i s acceptable i n most cases. The combination o f f i n i t e and boundary element preprocessing c a p a b i l i t i e s i s t h e r e f o r e f r u i t f u l . A modest two-dimensional example i s shown i n F i g u r e s 7 and 8 [13] [14]. Here t h e m o t i v a t i o n f o r use o f an i n t e r i o r mesh i s t h e i d e n t i f i c a t i o n o f b o t h i n t e r i o r p o i n t s and a n a t u r a l i n t e r p o l a t i o n scheme f o r t h e e v a l u a t i o n and v i s u a l i z a t i o n o f t h e f u l l s t r e s s f i e l d . F i r s t t h e boundary element mesh i n F i g u r e 7 i s c r e a t e d i n t e r a c t i v e l y by s e l e c t i n g t h e number, spacing, and g r a d a t i o n o f mesh p o i n t s a l o n g each edge. Then, i n F i g u r e 8, an automatic i n t e r i o r mesh generation scheme i s used t o f i n d r a p i d l y an i n t e r i o r mesh of sampling p o i n t s w i t h a n a t u r a l i n t e r p o l a t o r y b a s i s , i n t h i s case t r i a n g u l a r regions. This mesh i s determined s o l e l y by t h e
F i g u r e 5.
Planar s e c t i o n s f o r c r e a t i o n by l o f t i n g methods o f a boundary element mesh f o r t h e r o o t of a t u r b i n e blade [ll] [12].
Interactive Computer Graphics boundary nodes and by a s i n g l e i n t e r i o r d e n s i t y parameter. Although t h e r e s u l t i n g mesh c o u l d be a d j u s t e d i n t e r a c t i v e l y , t h e design c o n s i d e r a t i o n s c o n s t r a i n i n g t h e i n t e r i o r mesh a r e l e s s s t r i n g e n t t h a n those f o r a f i n i t e element a n a l y s i s . It i s s u f f i c i e n t t h a t t h e mesh i s f i n e i n t h e r e g i o n of expected s t r e s s c o n c e n t r a t i o n , and t h i s refinement i s achieved d i r e c t l y as a consequence o f t h e design o f t h e boundary element mesh w i t h s m a l l e r elements i n t h i s region.
POSTPROCESSING The combination of f i n i t e and boundary element c a p a b i l i t i e s c a r r i e s over i n t o postprocessing, p a r t i c u l a r l y f o r an example m o t i v a t e d by f u l l - f i e l d v i s u a l i z a t i o n such as t h e two-dimensional problem shown i n F i g u r e s 7 and 8. F i g u r e s 9 and 10 show how an i n t e r a c t i v e two-dimensional f i n i t e p o s t processor [151 i s used, w i t h o u t s i g n i f i c a n t m o d i f i c a t i o n , t o i n s p e c t t h e r e s u l t s o f t h e boundary element a n a l y s i s . The f i r s t o f these f i g u r e s shows t h e examination o f t h e v a l u e o f displacements a t nodes by p o i n t i n g t o a p a r t i c u l a r node; t h e deformed mesh can a l s o be d i s p l a y e d a t any s e l e c t e d m a g n i f i c a t i o n . F i g u r e 10 i s a b l a c k and w h i t e photo o f c o l o r s t r e s s contours throughout t h e domain. Here some of t h e s t r e s s ranges have been darkened i n t e r a c t i v e l y t o produce a c l e a r e r view o f how s t r e s s e s vary near t h e c e n t r a l hole. The general g r a p h i c a l problem i n p o s t p r o c e s s i n g i s t h e v i s u a l i z a t i o n o f how a s c a l a r , v e c t o r , o r t e n s o r response parameter v a r i e s over t h e geome t r y o f t h e system. P a r t i c u l a r l y f o r three-dimensional s o l i d s o r s u r faces, i t i s u s u a l l y p o s s i b l e t o d i s p l a y o n l y a s i n g l e parameter component over t h e s u r f a c e o f t h e body w h i l e u s i n g c o l o r a t i o n t o convey
F i g u r e 6.
The complete boundary element mesh r e s u l t i n g f r o m F i g u r e 5.
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simultaneously t h e t h r e e d i m e n s i o n a l i t y o f t h e geometry, F i g u r e 11 [15]. As l o n g as one has both an a p p r o p r i a t e way o f e v a l u a t i n g response parame t e r s a c c u r a t e l y and an i n h e r e n t o r assumed b a s i s f o r s p a t i a l i n t e r p o l a t i o n o f these parameters, an i n t e r a c t i v e g r a p h i c a l postprocessor need n o t be s p e c i f i c t o a p a r t i c u l a r method o f a n a l y s i s and may, i n f a c t , be used e f f e c t i v e l y even f o r combined methods,
As an i l l u s t r a t i o n , f o l l o w i n g a r e some o f t h e major design f e a t u r e s of a general -purpose i n t e r a c t i v e g r a p h i c a l postprocessor c u r r e n t l y under d e v e l opment a t C o r n e l l . The postprocessor i s being designed t o accommodate three-dimensional geometries w i t h speci a1 iz a t i ons a v a i l a b l e f o r both curved surfaces i n three-dimensional space and two-dimensonal problems. The postprocessor serves f o r f i n i t e element, f i n i t e d i f f e r e n c e , boundary element, and combined f i n i t e / b o u n d a r y element analyses. F l e x i b i l i t y f o r a v a r i e t y o f t e c h n i c a l analyses i s b u i l t i n , i n c l u d i n g s t r e s s a n a l y s i s , heat conduction, and compressible f l u i d f l o w , The b a s i c d i s p l a y c a p a b i l i t y c o n s i s t s o f c o l o r c o n t o u r i n g on t h e surfaces as shown i n F i g u r e 11. The user i s a b l e t o choose any viewing angle and may have s i n g l e o r m u l t i p l e views o f t h e system. Moreover, t h e user i s a b l e t o s e c t i o n , c u t , o r " u n f o l d " t h e geometry i n t e r a c t i v e l y i n a v a r i e t y o f c o o r d i n a t e systems t o o b t a i n views o f t h e i n t e r i o r d i s t r i b u t i o n s o f parameters. When a c u t i s made, t h e program a u t o m a t i c a l l y i n t e r p o l a t e s i n t e r i o r r e s u l t s t o o b t a i n t h e v a r i a t i o n o f t h e parameter over t h e exposed surface. When no i n t e r i o r r e s u l t s are a v a i l a b l e f o r a boundary element a n a l y s i s , t h e program can t r e a t postprocessing i n much t h e same manner as i t would handle r e s u l t d i s p l a y s on surfaces such as s h e l l and membrane s t r u c t u r e s . F i n a l l y ,
?I
B
F i g u r e 7.
I
I n t e r a c t i v e boundary element preprocessing f o r a twodimensional problem. C r e a t i o n o f a d i s c r e t i z e d o u t l i n e f o r one q u a r t e r o f a s t r e t c h e d p l a t e w i t h a c i r c u l a r h o l e [13].
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a l t h o u g h t h e program has i t s own i n t e r n a l data o r g a n i z a t i o n designed f o r e f f i c i e n t i n t e r a c t i v e g r a p h i c a l m a n i p u l a t i o n , i t i s assumed t h a t data from a l l d i f f e r e n t types o f analyses a r e r e c e i v e d by t h e program i n a standardi z e d I G E S format. This represents a f i n i t e element p r e j u d i c e c o i n c i d i n g w i t h t h e i n i t i a l d i r e c t i o n s of t h e IGES t r e a t m e n t of a n a l y s i s data, b u t f o r t h e most p a r t i t seems f e a s i b l e t o expect t h a t t h e r e s u l t s from n e a r l y a l l a n a l y s i s types can be t r a n s l a t e d i n t o a f i n i t e - e l e m e n t - l i k e format.
INTEKACTIVE-ADAPTIVE ANALYSIS I n t e r a c t i v e - a d a p t i v e analyses has been d e f i n e d e a r l i e r as continuous g r a p h i c a l m o n i t o r i n g of t h e progress o f a n a l y s i s by t h e engineer w i t h t h e o p p o r t u n i t y t o i n t e r r u p t and t o change e i t h e r t h e design o r t h e a n a l y s i s [l]. I n a sense, t h i s i s a combination o f p r e p r o c e s s i n g and p o s t p r o c e s s i n g w i t h a d d i t i o n a l f a c i l i t i e s o f user c o n t r o l over a n a l y s i s . Obviously i f a n a l y s i s r e s u l t s a r e t o be monitored i n r e a l time, t h e computation must be very r a p i d o r t h e user w i l l be w a s t i n g h i s t i m e s i t t i n g before a very s l o w l y changing graphic d i s p l a y . C u r r e n t l y , t h i s l i m i t s t h e use o f i n t e r a c t i v e - a d a p t i v e techniques t o r e l a t i v e l y small problems, t o e f f i c i e n t s p e c i a l -purpose a n a l y s i s and design systems, t o t e a c h i n g a p p l i c a t i o n s , and t o research. However, t h e t r e n d s toward i n c r e a s i n g l y c o s t - e f f e c t i v e comp u t i n g and memory and i n c r e a s i n g l y r a p i d data t r a n s f e r i n d i c a t e t h a t t h e s e c o n s t r a i n t s w i l l be a l l e v i a t e d . An example of t h e use o f t h e approach t o m o n i t o r n o n l i n e a r s t r u c t u r a l behavior as i t i s being c a l c u l a t e d i s shown i n F i g u r e 12. Here f o u r d i f f e r e n t p o r t i o n s o f t h e d i s p l a y a r e used t o
F i g u r e 8.
For t h e problem o f F i g u r e 7, c r e a t i o n o f a mesh o f i n t e r i o r p o i n t s used t o o b t a i n f u l l - f i e l d i n f o r m a t i o n from a boundary element a n a l y s i s C131 C141.
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t r a c e a n a l y s i s progress. The main viewport shows t h e deformation behavior o f t h e s t r u c t u r e , w h i l e t h e upper l e f t and upper c e n t e r viewports i n c l u d e graphs o f response s e l e c t e d i n t e r a c t i v e l y by t h e user; l a s t l y , t h e upper r i g h t m o n i t o r summarizes a n a l y s i s i n f o r m a t i o n such as l o a d l e v e l , l o a d step, number o f i t e r a t i o n s , etc. The a n a l y s i s d i v e r s i f i c a t i o n o p p o r t u n i t i e s presented by i n t e r a c t i v e a d a p t i v e a n a l y s i s a r e several. F i r s t i s t h e a b i l i t y t o perform analyses which a i d i n e s t a b l i s h i n g a n a l y s i s parameters; f o r example, t h e a b i l i t y t o perform r a p i d n a t u r a l p e r i o d c a l c u l a t i o n s a s s i s t s i n d e t e r m i n i n g a p p r o p r i a t e t i m e increments f o r marching a l g o r i t h m s . Second, one r o l e f o r i n t e r a c t i ve-adapti ve approaches i s t o change a n a l y s i s s t r a t e g i e s a f t e r a n a l y s i s has begun. For example, i n F i g u r e 13 t h e a n a l y s t has, p a r t l y on t h e b a s i s o f an i n i t i a l a n a l y s i s , i d e n t i f i e d a group o f elements t o c o n s t i t u t e an e l a s t o p l a s t i c n o n l i n e a r s u b s t r u c t u r e , w h i l e t h e remaining elements a r e I n a d d i t i o n , a growing number o f grouped i n t o a l i n e a r s u b s t r u c t u r e [8]. s e l f - a d a p t i v e a l g o r i t h m s a r e becoming a v a i l a b l e f o r such aspects as f r a c t u r e propagation, s u b s t r u c t u r i n g , marching schemes, and mesh improvements. A1 though many of these s e l f -adapti ve a1 g o r i thms a r e s t a b l e , some have h e u r i s t i c bases and a r e best monitored t o ensure t h e continued v a l i d i t y o f computations. For example, t h e f i n i t e element mesh changes t h a t are necessary t o t r a c e an a r b i t r a r i l y p r o p a g a t i n g crack may l e a d t o meshes w i t h e x c e s s i v e l y d i s t o r t e d elements; such d i s t o r t i o n s can be c o r r e c t e d by t h e user under an i n t e r a c t i v e - a d a p t i v e approach. F i n a l l y , t h e implementat i o ? of i n t e r a c t i v e - a d a p t i v e procedures presents two o t h e r p o s s i b l e comput i n g strategies t h a t are p a r t i c u l a r l y s i g n i f i c a n t for nonlinear analysis.
7 i g u r e 9.
I n t e r a c t i v e boundary element p o s t p r o c e s s i n g f o r t h e displacements o f t h e problem i n Figures 7 and 8 u s i n g a twodimensional f i n i t e postprocessor [14].
Interactive Computer Graphics The f i r s t o f these i s p a r a l l e l a n a l y s i s i n which an a l t e r n a t i v e a n a l y s i s t y p e i s invoked t o diagnose system behavior. Examples i n c l u d e b u c k l i n g and f r e e - v i b r a t i o n t e s t s on n o n l i n e a r s t r u c t u r e s a t an a r b i t r a r y stage of l o a d i n g . The second i s s e q u e n t i a l a n a l y s i s where d i f f e r e n t analyses a r e performed t o i n f e r t h e e f f e c t of t h e o r d e r o f a p p l i c a t i o n o f i n f l u e n c e s . For example, a s t a t i c p r e l o a d i n g may precede a n o n l i n e a r s t r u c t u r a l dynamic a n a l y s i s . It i s apparent t h a t i n t e r a c t i v e - a d a p t i v e a n a l y s i s approaches h o l d s t r o n g promise f o r t h e e x p l o r a t i o n , development, and implementation o f u n i f i e d o r combined numerical methods. For instance, t h e same c a p a b i l i t y f o r adapt i v e s u b s t r u c t u r i n g i l l u s t r a t e d above c o u l d be used t o determine i n t e r a c t i v e l y t h e p a r t i t i o n i n g o f a geometric e n t i t y i n t o "married" f i n i t e element and boundary element zones. One o f t h e most e x c i t i n g uses o f i n t e r a c t i v e - a d a p t i v e a n a l y s i s procedures a t Cornel 1 t o date has, i n f a c t , been as a " t e s t bed" f o r research on numerical methods [l].
CONCLUSIONS The i n t e n t i o n o f t h i s paper has been t o demonstrate how i n t e r a c t i v e graphi c s i s a f f e c t i n g t h e environment i n which a n a l y s i s i s performed and t h e r e by i s i n f l u e n c i n g p o s s i b l e u n i f i c a t i o n o r h y b r i d i z a t i o n o f numerical methods. Although t h i s impact i s o n l y b e g i n n i n g t o be f e l t i n t h e comput a t i o n a l mechanics community, t h e r a p i d growth i n a v a i l a b i l i t y o f i t e r a c t i v e g r a p h i c a l c a p a b i l i t i e s w i l l d e f i n i t e l y a f f e c t such developments i n t h e near f u t u r e . It takes o n l y a small leap o f i m a g i n a t i o n t o see how i n t e r a c t i v e techniques a l r e a d y i n use f o r s p e c i f i c numerical methods such
F i g u r e 10.
F u l l - f i e l d s t r e s s v i s u a l i z a t i o n from a boundary element a n a l y s i s u s i n g t h e same f i n i t e element postprocessor as i n F i g u r e 9 [14] C151.
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as f i n i t e elements can be g e n e r a l i z e d f o r a v a r i e t y o f a n a l y t i c a l approaches. Future developments o f i n t e r a c t i v e g r a p h i c a l s o f t w a r e systems f o r e n g i n e e r i n g a n a l y s i s and design should be designed t o account f o r p o s s i b l e d i v e r s i f i c a t i o n o f a n a l y t i c a l procedures. One obvious recommendation i s t h a t s t r i c t s e p a r a t i o n be maintained between system models (geometry and a t t r i b u t e s ) and a n a l y t i c a l models (meshes o r o t h e r d i s c r e t i z a t i o n s ) . Under such c i rcumstances , developers o f i n t e r a c t i v e s o f t w a r e should be urged t o "go t h e e x t r a m i l e " t o p r o v i d e t h e a n a l y s i s f l e x i b i l i t y t h a t may be needed f o r e f f e c t i ve engi n e e r i ng design
.
ACKNOWLEDGMENTS P o r t i o n s o f t h e work used as examples i n t h i s paper have been sponsored by t h e N a t i o n a l Science Foundation, t h e N a t i o n a l Aeronautics and Space A d m i n i s t r a t i o n , and by v a r i o u s p r i v a t e companies; t h e w r i t e r s a r e g r a t e f u l f o r t h i s support. However, t h e views expressed here a r e t h e w r i t e r s ' and a r e n o t those o f any sponsor. The w r i t e r s would a l s o l i k e t o express t h e i r a p p r e c i a t i o n t o t h e i r present and former colleagues a t t h e Program o f Computer Graphics a t C o r n e l l U n i v e r s i t y , p a r t i c u l a r l y Donald P. Greenberg, D i r e c t o r o f t h e Program, and W i 11iam McGui re, a f a c u l t y c o - i n v e s t i g a t o r f o r much o f t h e research c i t e d .
F i g u r e 11.
Postprocessing u s i n g c o l o r contours on t h e surface o f a s o l i d i n c o n j u n c t i o n w i t h a d i f f u s e r e f l e c t i o n model t o convey t h r e e - d i mensi onal it y [ 151.
Interactive Computer Graphics REFERENCES and Abel , J . F., I n t e r a c t i v e - a d a p t i v e l a r g e displacement a n a l y s i s w i t h r e a l -time computer graphics, Computers and S t r u c t u r e s
c11 Gattass, M.,
16 (1983) 141-152.
C2l Kame1 , H. A.,
Design and implementation o f i n t e r a c t i v e e n g i n e e r i n g software, i n : Abel, J. F., e t a l . (eds.), I n t e r d i s c i p l i n a r y F i n i t e Element A n a l y s i s ( C o l l e g e o f Engineering, C o r n e l l U n i v e r s i t y , I t h a c a , NY, 1981) 773-803.
c31 Pesquera, C. I., McGuire, W.,
and Abel, J. F., I n t e r a c t i v e g r a p h i c a l p r e p r o c e s s i n g o f three-dimensional framed s t r u c t u r e s , Computers and S t r u c t u r e s 17 (1983) 97-104.
c41 Abel, J. F.,
I n t e r a c t i v e computer qraphics i n aDDlied mechanics. i n : Proc. 9 t h U. S. Nat. Cong. App. Mechs’., (ASME,’ New York, 1983) .
97-104.
c51 Yerry, M. A.,
and Shephard, M. S . , Authomatic three-dimensional mesh g e n e r a t i o n by t h e m o d i f i e d - o c t r e e technique, I n t l . J . Num. Meth. Engrg. ( t o appear).
C6l Shephard, M. S., and Abel, J. F.,
The i n t e g r a t i o n o f f i n i t e element methods i n t o CAD/CAM, i n : Kardestuncer, H. (ed.), F i n i t e Element Handbook (McGraw H i l l , New York, t o appear 1984).
F i g u r e 12.
Use o f m u l t i p l e d i s p l a y windows t o m o n i t o r n o n l i n e a r s t r u c t u r a l behavior as i t i s being c a l c u l a t e d d u r i n g i n t e r a c t i v e - a d a p t i v e a n a l y s i s C13.
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J.F. A b e l e t al. Perucchio, R., I n g r a f f e a , A. R., and Abel, J . F., I n t e r a c t i v e computer g r a p h i c preprocessing f o r three-dimensional f i n i t e element a n a l y s i s , I n t . J . Num. Meths. Engrg. 18 (1982) 909-926. Han, T. Y., Adaptive s u b s t r u c t u r i n g and i n t e r a c t i v e graphics f o r three-dimensional f i n i t e element a n a l y s i s , Ph.D. Thesis, Department o f S t r u c t u r a l Engineering, Cornel 1 U n i v e r s i t y (1984). Smith, B. M., e t al., I n i t i a l graphics exchange s p e c i f i c a t i o n (IGES), Version 2.0 (NBSIR 82-2531 (AF), N a t i o n a l Bureau o f Standards, 1982). Chang, S . C., An i n t e g r a t e d f i n i t e element n o n l i n e a r s h e l l a n a l y s i s system w i t h i n t e r a c t i v e computer graphics, Ph.D. Thesis, Department o f S t r u c t u r a l Engineering, Cornel 1 U n i v e r s i t y (1981). Perucchio, R., and I n g r a f f e a , A. R., I n t e r a c t i v e computer g r a p h i c s p r e p r o c e s s i n g f o r three-dimensional boundary i n t e g r a l element a n a l y s i s , Computers and S t r u c t u r e s 16 (1983) 153-166. Perucchio, R., An i n t e g r a t e d boundary element a n a l y s i s system w i t h i n t e r a c t i v e computer graphics f o r three-dimensional l i n e a r - e l a s t i c f r a c t u r e mechanics, Ph.D. Thesis, Department o f S t r u c t u r a l Engineering, Cornel 1 Uni v e r s i t y ( 1984).
F i g u r e 13.
I n t e r a c t i ve-adapti ve substr.uctu r i ng d u r i n g e l a s t o p l a s t i c The a n a l y s t has s e l e c t e d those elements (shown a n a l y s i s [S]. d a r k e r ) which should c o n s t i t u t e a n o n l i n e a r s u b s t r u c t u r e d u r i n g t h e n e x t stage o f a n a l y s i s .
Interactive Computer Graphics [13] Han, T. Y., A general two-dimensional , i n t e r a c t i v e g r a p h i c a l f i n i t e / b o u n d a r y element proprocessor f o r a v i r t u a l stage environment, M.S. Thesis, Department o f S t r u c t u r a l Engineering, Cornel 1 U n i v e r s i t y (1981). [14] Han, T. Y.,
unpublished student p r o j e c t ( C o r n e l l U n i v e r s i t y , 1982).
[15] Schulman, M. A., The i n t e r a c t i v e d i s p l a y o f parameters on two- and t h r e e - dimensional surfaces, M.S. Thesis, Department o f A r c h i t e c t u r e , Cornel 1 U n i v e r s i t y (1981).
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CHAPTER 2 INTERACTIVE COMPUTER GRAPHICS FOR FINITE ELEMENT, BOUNDARY ELEMENT, & FINITE DIFFERENCE METHODS
J.F. Abel, A.R. Ingraffea, R. Perucclzio, T - Y. Han, & J.F. Hajjar
The t r e n d toward u b i q u i t y o f i n t e r a c t i v e g r a p h i c s i n computational e n g i n e e r i n g i s h e l p i n g t o develop an atmosphere conducive t o t h e combination and u n i f i c a t i o n o f v a r i o u s numerical a n a l y s i s methods. A number of f e a t u r e s o f i n t e r a c t i v e computer a i d e d design which f o s t e r u n i f i c a t i o n a r e discussed. Examples a r e drawn from t h r e e p r i n c i p a l aspects o f computer graphics i n e n g i n e e r i n g a n a l y s i s : p r e processing, postprocessing, and i n t e r a c t i v e a d a p t i ve a n a l y s i s . INTRODUCTION The purpose o f t h i s paper i s t o present and e x p l o r e some ideas about how t h e growing use of i n t e r a c t i v e computer g r a p h i c s i s b o t h a f f e c t i n g t h e u t i l i z a t i o n o f numerical methods by e n g i n e e r i n g a n a l y s t s and i n f l u e n c i n g t h e p o s s i b l e u n i f i c a t i o n o f these methods. Recent decades have seen a dramatic development and p r o l i f e r a t i o n o f numerical methods. T h i s a c t i v i t y has i n c l u d e d t h e s t r e n g t h e n i n g and d i v e r s i f i c a t i o n o f e s t a b l i s h e d methods such as f i n i t e d i f f e r e n c e s and f i n i t e elements, t h e r a p i d expans i o n o f r e d i s c o v e r e d techniques such as b o u n d a r y - i n t e g r a l equations, t h e i n c r e a s i n g r e c o g n i t i o n of u n i f y i n g f o r m u l a t i v e n o t i o n s such as v a r i a t i o n a l approaches , and t h e coinbi n a t i o n o f methods t o r e a l i z e d i v e r s e advantages such as t h e m i x i n g o f f i n i t e elements and boundary elements. The ideas presented here r e p r e s e n t some r e c e n t c o l l e c t i v e thoughts o f a research group a t C o r n e l l U n i v e r s i t y t h a t has, f o r n e a r l y a decade, been i n v o l v e d i n developments o f i n t e r a c t i v e computer graphics f o r t h e i r a p p l i c a t i o n s t o both computational mechanics research and computer-aided design. Because most o f t h i s research has been l i m i t e d t o s t r u c t u r a l e n g i n e e r i n g and mechanics and has i n v o l v e d p r i m a r i l y f i n i t e element and boundary element approaches, t h i s can h a r d l y be considered a g l o b a l view o f d i s c i p l i n e s and methods. Nevertheless, t h e emphasis i n t h i s paper on growing t r e n d s i n g r a p h i c s and t h e i r r e l a t i o n t o u n i f i c a t i o n o f a n a l y s i s methods can be considered a manifesto, a l b e i t modest, r e g a r d i n g d e s i r a b l e d i r e c t i o n s f o r f u t u r e developments i n i n t e r a c t i v e graphics. C l e a r l y i n t e r a c t i v e graphics i n i t s e l f i s d i s t i n c t from a n a l y s i s and, t h e r e f o r e , can be viewed from one p e r s p e c t i v e as having l i t t l e t o do w i t h u n i f i c a t i o n o f a n a l y s i s techniques. However, i t has been w i d e l y demons t r a t e d t h a t i n t e r a c t i v e g r a p h i c s can be h i g h l y successful i n b r e a k i n g down b a r r i e r s and d i f f i c u l t i e s i n t h e performance o f e n g i n e e r i n g a n a l y s i s . The enhanced access t o , and c o n t r o l o f , a n a l y s i s can i n t h e same way s i g n i f i c a n t l y h e l p remove o b s t a c l e s t o u n i f i c a t i o n , as w i l l be argued
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subsequently. The p r o v i s i o n o f a proper atmosphere f o r amalgamation i s no l e s s i m p o r t a n t than t h e r e c o g n i t i o n o f t h e p o t e n t i a l f o r u n i f i c a t i o n . U l t i m a t e l y , t h e obvious goal i s t h e p r o v i s i o n o f t h e most e f f e c t i v e and a p p r o p r i a t e a n a l y s i s methodologies f o r engineers t o c a r r y out design r e s p o n s i b l y and c r e a t i v e l y . The computational environment f o r a n a l y s i s i s changing r a p i d l y . Computeraided a n a l y s i s and design i n both p r a c t i c e and i n d u s t r y a r e being t i e d t o g e t h e r wit h computerized i n f o r m a t i o n r e p o s i t o r i e s and data f l o w c a p a b i l i t i e s , w h i l e a t t h e same t i m e i n c r e a s e d computational power i s being p r o vided t o t h e i n d i v i d u a l engineer. Three c h i e f c u r r e n t developments i n hardware and software are e n g i n e e r i n g w o r k s t a t i o n s which p r o v i d e i n t i m a t e access t o computing, n e t w o r k i n g which enables t h e s h a r i n g and r a p i d t r a n s f e r o f l a r g e volumes o f data among w o r k s t a t i o n s , and i n t e r a c t i v e g r a p h i c s c a p a b i l i t i e s i n t e g r a l t o t h e w o r k s t a t i o n s . The l a s t o f these i s t h e theme here. I n t h i s environment, w i t h i n t h e next decade, n e a r l y a l l computing w i l l be a s s o c i a t e d w i t h i n t e r a c t i v e graphics. Both l i t e r a l l y and f i g u r a t i v e l y , i n t e r a c t i v e computer graphics i s becoming t h e "window" t o t h e world o f analysis. I n t e r a c t i v e computer graphics was f i r s t a p p l i e d t o p r e p r o c e s s i n g and p o s t p r o c e s s i n g only, and these a r e s t i l l t h e phases o f a n a l y s i s where e n g i n e e r i n g p r o d u c t i v i t y i s most d r a m a t i c a l l y improved by i t s use. However, although a n a l y s i s o r p r o c e s s i n g i t s e l f has t r a d i t i o n a l l y been a batch p r o cedure, t h e continued i n c r e a s e o f cheap, d i s t r i b u t e d computational capab i l i t y and t h e growing demands o f d e s i g n e r l a n a l y s t s i n d i c a t e t h a t i n t h e near f u t u r e some i n t e r a c t i v e processing w i l l be e f f e c t i v e and d e s i r a b l e even f o r such c o m p u t a t i o n a l l y i n t e n s i v e procedures as f i n i t e element a n a l y s i s . This n o t i o n has given r i s e t o i n t e r a c t i v e - a d a p t i v e a n a l y s i s [l], which i s d e f i n e d as a n a l y s i s d u r i n g which t h e engineer can c o n t i n u o u s l y m o n i t o r what i s happening and can choose t o i n t e r v e n e a t any t i m e t o change system c h a r a c t e r i s t i c s , models, a n a l y s i s parameters, and a1 gor i t h m s . I t i s most s u i t e d t o n o n l i n e a r and t i m e - v a r y i n g analyses. The engineer i s a b l e t o move backward and forward i n t h e a n a l y s i s a t w i l l and t o invoke a v a r i e t y o f p a r a l l e l analyses a t any stage. Moreover, d i f f e r e n t types o f analyses can be s t r u n g t o g e t h e r i n a sequence. T h i s a n a l y s i s c o n t r o l c a p a b i l i t y c l e a r l y must r e l y on i n t e r a c t i v e computer graphics f o r i t s success. I n t h i s paper a l l t h r e e aspects o f i n t e r a c t i v e computer graphics i n e n g i n e e r i n g a n a l y s i s are used t o i l l u s t r a t e t h e n o t i o n s discussed. UNIFYING INFLUENCES OF INTERACTIVE GRAPHICS Experience w i t h t h e development o f i n t e r a c t i v e g r a p h i c a l s o f t w a r e programs f o r a n a l y s i s a p p l i c a t i o n s i n d i c a t e s t h a t , f o r w e l l designed i n t e r a c t i v e systems, programming i s a major task t h a t exceeds i n complexity and d i f f i c u l t y t h e programming o f much a n a l y s i s s o f t w a r e [2]. In addition t o the a r a t i o n a l u n d e r l y i n g database; m o d u l a r i t y t o f o s t e r usual requirements e x p a n d a b i l i t y ; c l e a r , s t r u c t u r a l programming; a system f o r a n t i c i p a t i n g , handling, and c o r r e c t i n g e r r o r s ; t r a n s p o r t a b i l i t y t o t h e maximum e x t e n t p o s s i b l e w h i l e m a i n t a i n i n g necessary performance and response; and t h e design f o r human f a c t o r s o r "user f r i e n d 1 i thorough documentation ness" places e x t r a demands on t h e program developer. Among these a r e completeness o f f u n c t i o n , f l e x i b i l i t y i n sequence, and a humanized command language and menus [31 141. The achievement o f these l a s t o b j e c t i v e s
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tends t o r e s u l t i n a g e n e r a l i t y o f t h e i n t e r f a c e between t h e user and t h e computer t h a t i s conducive t o f l e x i b i l i t y , d i v e r s i f i c a t i o n , and u n i f ic a t i on o f a n a l y s i s methods. I n a d d i t i o n t o t h e fundamental design o f i n t e r a c t i v e software, a f a c t o r which i s a f f e c t i n g t h e way i n which p a r t i c u l a r a n a l y s i s methods f i t i n t o an o v e r a l l design process i s t h e i n c r e a s i n g use o f system models independent from a n a l y s i s models. I n t h e a p p l i c a t i o n o f , say, f i n i t e element methods, one was p r e v i o u s l y l i k e l y t o d e s c r i b e t o t h e computer t h e geometry, boundary c o n d i t i o n s , and l o a d i n g s o f t h e problem under
I n t e r a c t i v e System M o d e l l i n g ( P r e l i m i n a r y D e s i g n , Geometry)
I n t e r a c t i v e Preprocessing (Geometry, A t t r i b u t e s , A n a l y t i c a l M o d e l )
t
T r a n s l a t o r (s) Ana 1y s i s
A n a l y s i s M e t h o d (s) I n v e r s e T r a n s l a t o r (s)
t I I I n t e r a c t i v e Postprocessing
I n t e r a c t i v e Design/Re-Design (Geometry, A t t r i b u t e s )
F i g u r e 1.
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The computer-aided design process. I n t e r a c t i v e computer graphics i s t h e p r i n c i p a l medium f o r man-machine communication and c o n t r o l . The i n t e g r a t i o n o f t h e v a r i o u s stages shown i s p r e d i c t e d on computerized databases and data flow.
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c o n s i d e r a t i o n i n terms o f t h e elements, nodes, and c o n n e c t i v i t y o f t h e d e s c r i p t i o n t h a t was e n t i r e l y l i n k e d w i t h t h e a n a l y t i c a l d i s c r e t i z a t i o n . With t h e t r e n d toward computer-aided design, a n a l y s i s i s now u s u a l l y viewed as being i n t e g r a t e d i n t o an o v e r a l l design process such as d e p i c t e d i n general form i n F i g u r e 1. The l i n k s between t h e v a r i o u s steps shown a r e achieved through a computerized database and data flow. Here t h e geometric d e s c r i p t i o n o f t h e design problem i s c r e a t e d on t h e computer by a modeller or, i f a modeller i s n o t a v a i l a b l e , i t i s created as t h e f i r s t s t e p o f preprocessing. Then, i n preprocessing, a d d i t i o n a l a t t r i b u t e s o f t h e problem, such as m a t e r i a l p r o p e r t i e s , boundary c o n d i t i o n s , and l o a d i n g s , a r e a s s o c i a t e d w i t h t h e geometry. The r e s u l t i s t h e complete d e s c r i p t i o n o f an a n a l y s i s problem t h a t e x i s t s w i t h i n t h e computer p r i o r t o any meshing and, indeed, t h a t i s independent o f any p a r t i c u l a r method. I n t h e f u t u r e , t h e generation o f t h e a n a l y t i c a l model o r d i s c r e t i z a t i o n a p p r o p r i a t e f o r a p a r t i c u l a r method o f a n a l y s i s w i l l probably be automated [5]. T h i s w i l l be p a r t i c u l a r l y s u i t a b l e f o r a n a l y s i s procedures t h a t i n c l u d e s e l f - a d a p t i v e mesh a l g o r i t h m s t o achieve a s p e c i f i e d degree o f refinement o r accuracy i n each p o r t i o n o f t h e system. However, f o r a t l e a s t t h e next decade, it i s c l e a r t h a t t h e e n g i n e e r i n g a n a l y s t w i l l need t o continue t o e x e r c i s e s t r o n g i n f l u e n c e over t h e design o f t h e a n a l y t i c a l model because o f i t s d i r e c t e f f e c t on accuracy and r e l i a b i l i t y o f t h e a n a l y s i s . I n t e r a c t i v e mesh generators a r e i d e a l l y s u i t e d t o t h i s task [6]. Moreover, e f f e c t i v e i n t e r a c t i v e preprocessors p r o v i d e a f l e x i b i l i t y t h a t i s o n l y beginning t o be recognized. Not o n l y a r e computer-assisted mesh generators i n many cases a b l e t o be g e n e r a l i z e d f o r a v a r i e t y o f a n a l y t i c a l approaches, b u t t h e i n t e r a c t i v e t o o l s i n c o r p o r a t e d f o r , say, s u b s t r u c t u r i n g o r p a r t i t i o n i n g a l s o present n a t u r a l o p p o r t u n i t i e s f o r preprocessing f o r h y b r i d o r combined techniques. Another aspect o f i n t e r a c t i v e s o f t w a r e design t h a t s i g n i f i c a n t l y a l l e v i ates s p e c i f i c i t y t o a s i n g l e a n a l y s i s t y p e i s t h e humanization o f command languages. One d e t e c t a b l e t r e n d i s toward t h e use o f commands more n a t u r a l t o t h e engineer. For example, i n t h e s p e c i f i c a t i o n o f boundary c o n d i t i o n s f o r f i n i t e element a n a l y s i s , one may be asked t o s p e c i f y t h e t y p e o f c o n d i t i o n (e.g., "symmetry") r a t h e r t h a n a node-by-node l i s t o f r e s t r a i n t codes f o r each degree o f freedom. The c o r o l l a r y i s t h a t t h e preprocessing software o r t h e t r a n s l a t o r programs t h a t a r e t h e i n t e r f a c e t o t h e s p e c i f i c a n a l y s i s program ( F i g u r e 1) must i n c l u d e i n t e r n a l coding which a u t o m a t i c a l l y converts between a n a l y s i s - s p e c i f i c data and t h e engi n e e r ' s n a t u r a l vocabulary.
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Two f a c t o r s r e l a t i n g t o data o r g a n i z a t i o n f o r i n t e r a c t i v e systems potent i a l l y c o n t r i b u t e t o t h e d i v e r s i f i c a t i o n o f such software. The f i r s t o f these i s t h e o r g a n i z a t i o n o f data s t r u c t u r e s f o r e f f i c i e n t and r a p i d g r a p h i c a l d i s p l a y . This i n v o l v e s t h e e x p l o i t a t i o n o f geometric coherence which may d i f f e r from t h a t imposed by a p a r t i c u l a r a n a l y s i s method. For instance, i n d i s p l a y i n g a f i n i t e element mesh, one may n o t wish t o draw t h e mesh element by element because t h i s w i l l i n v o l v e t h e redrawing o f l i n e s f o r t h e edges o f adjacent elements. Instead, one would p r e f e r t o t a k e advantage o f t h e coherence o f mesh l i n e s by drawing each complete l i n e i n one step. The rearrangement o f element edge data i n t o complete mesh-line data i n t h i s f a s h i o n i s perhaps more reminiscent o f data o r g a n i z a t i o n s a p p r o p r i a t e t o h i g h e r o r d e r f i n i t e d i f f e r e n c e operators. Another example a r i s e s from t h e d e s i r e t o p r o v i d e d i s p l a y s i m p l i f i c a t i o n s f o r three-dimensional geometries. Here, one technique used t o p e r m i t c l e a r e r viewing o f three-dimensional f i n i t e element meshes i s t h e s e l e c t e d removal
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o f l i n e s o r t h e s e l e c t i v e d i s p l a y of some elements o r groups o f elements. One u s e f u l v e r s i o n of t h i s i s t h e d i s p l a y of o n l y t h e e x t e r i o r s u r f a c e s o f elements ['I] [&I]. The d i s t i n c t i o n here i n t h e data s t r u c t u r e between i n t e r i o r and e x t e r i o r element faces c r e a t e s an obvious 1 ink t o boundary element techniques. The second d a t a - r e l a t e d f a c t o r i s t h e o v e r a l l data t r a n s m i s s i o n scheme u n d e r l y i n g computer-aided design systems w i t h data f l o w s such as i n d i c a t e d s c h e m a t i c a l l y i n F i g u r e 1. The need not o n l y t o t r a n s m i t i n f o r m a t i o n from one stage o f design t o another b u t a l s o t o exchange data between d i f f e r e n t computer-aided design systems has g i v e n r i s e t o s t a n d a r d i z e d procedures, most n o t a b l y t h e I n i t i a l Graphics Exchange S p e c i f i c a t i o n o r I G E S [Sl. Although s t i l l under development and n o t y e t as e f f i c i e n t as d e s i r able, I G E S i s now being s u c c e s s f u l l y used t o t r a n s m i t geometrical data. Moreover, t h e s p e c i f i c a t i o n i s being expanded t o i n c l u d e a n a l y s i s i n f o r m a I G E S data f i l e s a r e r i g i d l y f o r t i o n , p a r t i c u l a r l y f i n i t e element data. matted b u t a r e so complete and f l e x i b l e i n t h e i r r e p r e s e n t a t i o n o f data t h a t t h e f i l e s can be considered e s s e n t i a l l y system independent. The f l e x i b i l i t y o f I G E S a r i s e s because i t s e n t i t i e s may be e i t h e r geometric o r nongeometric and because v a r i o u s e n t i t i t e s may be a s s o c i a t e d i n ways a p p r o p r i a t e t o any a p p l i c a t i o n . The i m p l i c a t i o n f o r a n a l y s i s o f t h e e x i s t e n c e of such schemes as IGES i s t h e n o t i o n t h a t e i t h e r problem d e s c r i p t i o n s o r a n a l y s i s models and r e s u l t s can be s t a n d a r d i z e d i n a comp l e t e , n e u t r a l form. T h i s f o s t e r s an exchange o f data t h a t p e r m i t s n o t o n l y comparisons b u t a l s o combinations o r h y b r i d i z a t i o n s o f a n a l y t i c a l
F i g u r e 2.
Assignment o f boundary c o n d i t i o n s d u r i n g f i n i t e element p r e p r o c e s s i n g f o r s h e l l s t r u c t u r e s [lo]. Note t h e n a t u r a l t e r m i n o l o g y f o r boundary c o n d i t i o n types (middle p o r t i o n o f menu a t r i g h t ).
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procedures. Some o f t h e ideas i n t r o d u c e d thus f a r are now i l l u s t r a t e d by d i s c u s s i o n s and examples o f i n t e r a c t i v e programs developed f o r research o r i n s t r u c t i o n a l purposes a t C o r n e l l . A l l o f t h e subsequent f i g u r e s are photographs taken from graphic d i s p l a y devices, and these have been reduced t o t h e degree t h a t some small t e x t i s not e a s i l y l e g i b l e . I n each o f these cases, t h e small t e x t c o n t a i n s l i t t l e p e r t i n e n t i n f o r m a t i o n and i s r e t a i n e d o n l y t o i l l u s t r a t e t h e n a t u r e o f t h e computer graphics d i s p l a y , I n a d d i t i o n , t h e photographs o f c o l o r d i s p l a y s are, by n e c e s s i t y , reproduced here i n black and white. PREPROCESSING
A s i n g l e glimpse a t an i n t e r a c t i v e preprocessor f o r s h e l l a n a l y s i s [ l o ] provides an example o f t h e humanization o f commands. I n F i g u r e 2, t h e menu page f o r t h e assignment o f boundary c o n d i t i o n s i s shown. The commands along t h e r i g h t edge are d i v i d e d i n t o groups. The upper group c o n s i s t s o f permanent menu items f o r t h e m a n i p u l a t i o n o f t h e main image, w h i l e t h e group below these are commands and m o d i f i e r s s p e c i f i c t o t h e immediate t a s k . Here t h e user has a c t i v a t e d t h e command ASSIGN w i t h t h e boundary c o n d i t i o n t y p e SYMMETRY and t h e m o d i f e r s e t t o LINE. (All active commands and m o d i f i e r s a r e i n d i c a t e d by a box drawn about them.) Simply by p o i n t i n g t o t h e m e r i d i o n a l c u t s o f t h e h y p e r b o l i c c o o l i n g tower s h e l l shown, t h e engineer assigns t h e symmetry c o n d i t i o n t o t h i s l i n e . By f a r t h e most s i g n i f i c a n t e f f o r t i n preprocessing i s t h e c r e a t i o n o f t h e
F i g u r e 3.
Complete three-dimensional f i n i t e element mesh f o r a p r o p e l l e r generated by l o f t i n g between s e c t i o n s [8].
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a n a l y t i c a l model o r mesh. Even w i t h mesh g e n e r a t i o n a l g o r i t h m s , some i n t e r v e n t i o n by t h e engineer i s u s u a l l y necessary o r d e s i r a b l e t o c o n t r o l t h e design of t h e d e s c r e t i z a t i o n . For example, i n t h e d i s c r e t e t r a n s f i n i t e mapping methods of mesh generation, t h e d e n s i t y and g r a d a t i o n o f t h e mesh i n a r e g i o n i s determined by t h e placement and spacing o f nodes around t h e boundary of t h e r e g i o n [7l, and t h i s t a s k i s b e s t performed by t h e analyst interactively. The techniques o f mesh c r e a t i o n t h a t have been found successful f o r i n t e r a c t i v e a p p l i c a t i o n s have a l s o proven t o be a p p l i c a b l e f o r a v a r i e t y of d i f f e r e n t a n a l y s i s types. A t C o r n e l l , t h e l o f t i n g s p e c i a l i z a t i o n of d i s c r e t e t r a n s f i n i t e mapping has been used f o r b o t h f i n i t e element [7] C8l and boundary element [ l l ] C121 preprocessing. F i g u r e 3 shows a complete three-dimensional f i n i t e element mesh f o r a p r o p e l l e r - l i k e s o l i d generated i n t e r a c t i v e l y i n t h i s fashion. The mesh f o r t h e hub has been c r e a t e d by l o f t i n g among p a r a l l e l p l a n a r s e c t i o n a l meshes w i t h d i f f e r e n t c i r c u l a r diameters, w h i l e t h e l o f t i n g s e c t i o n s f o r t h e blades a r e n o t p a r a l l e l and, a t t h e r o o t o f t h e blades, a r e nonplanar In s e c t i o n s i n t e r a c t i v e l y s e l e c t e d from t h e s u r f a c e o f t h e hub mesh [8]. F i g u r e 4, t h e mesh i n F i g u r e 3 i s s i m p l i f i e d f o r c l e a r e r v i e w i n g by d i s p l a y i n g o n l y t h e e x t e r i o r f a c e t s o f elements, and t h e r e s u l t i s c l e a r l y suggestive o f t h e a p p l i c a b i l i t y o f t h i s meshing procedure f o r boundary element preprocessing. An example o f a d i r e c t a p p l i c a t i o n o f t h i s meshing t e c h n i q u e f o r a boundary element problem i s shown i n F i g u r e s 5 and 6 [ll]
CW.
The methods i n these examples c o u l d be s i m i l a r l y extended t o such a l t e r n a t i v e s as f i n i t e d i f f e r e n c e mesh generation. However, one o b s t a c l e remaini n g t o t h e complete g e n e r a l i z a t i o n o f i n t e r a c t i v e p r e p r o c e s s i n g techniques
F i g u r e 4.
The mesh i n F i g u r e 3 i s s i m p l i f i e d f o r c l e a r e r viewing by d i s p l a y i n g o n l y e x t e r i o r (boundary) f a c e t s o f elements.
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f o r a n a l y t i c a l model g e n e r a t i o n i s t h e nonorthogonal i t y o f t h e c o o r d i n a t e systems employed i n t h e mesh generation procedures. Many f i n i t e d i f f e r ence a l g o r i t h m s , f o r example, r e q u i r e a mesh which c o n s i s t s of i s o c o o r d i n a t e l i n e s i n an orthogonal c o o r d i n a t e system. The common f i n i t e e l e ment mesh g e n e r a t i o n techniques based on i s o p a r a m e t r i c o r t r a n s f i n i t e mappings do n o t a u t o m a t i c a l l y f u l f i l l t h i s requirement. Nevertheless, i t i s c l e a r t h a t t h e meshing methods can be extended w i t h minor adaptions t o l e s s r e s t r i c t i v e a n a l y s i s methods such as f i n i t e c e l l methods. I n many cases, boundary element a n a l y s i s r e q u i r e s n o t o n l y a d i s c r e t i z a t i o n of t h e s u r f a c e o f t h e domain b u t a l s o an i n t e r i o r mesh. The conside r a t i o n o f body e f f e c t s such as g r a v i t y loads and o f n o n l i n e a r i t i e s a r e two examples. I n a d d i t i o n , i f f u l l f i e l d o r s e l e c t e d i n t e r i o r r e s u l t s are required, t h e n i n t e r i o r p o i n t s must be chosen as l o c a t i o n s f o r t h e evaluat i o n o f responses. Although i n t e r i o r meshes f o r t h e boundary element method need n o t s a t i s f y t h e same s t r i c t t o p o l o g i c a l and c o n n e c t i v i t y requirements as a f i n i t e element mesh, t h e use o f an i n t e r i o r f i n i t e e l e ment d i s c r e t i z a t i o n i s acceptable i n most cases. The combination o f f i n i t e and boundary element preprocessing c a p a b i l i t i e s i s t h e r e f o r e f r u i t f u l . A modest two-dimensional example i s shown i n F i g u r e s 7 and 8 [13] [14]. Here t h e m o t i v a t i o n f o r use o f an i n t e r i o r mesh i s t h e i d e n t i f i c a t i o n o f b o t h i n t e r i o r p o i n t s and a n a t u r a l i n t e r p o l a t i o n scheme f o r t h e e v a l u a t i o n and v i s u a l i z a t i o n o f t h e f u l l s t r e s s f i e l d . F i r s t t h e boundary element mesh i n F i g u r e 7 i s c r e a t e d i n t e r a c t i v e l y by s e l e c t i n g t h e number, spacing, and g r a d a t i o n o f mesh p o i n t s a l o n g each edge. Then, i n F i g u r e 8, an automatic i n t e r i o r mesh generation scheme i s used t o f i n d r a p i d l y an i n t e r i o r mesh of sampling p o i n t s w i t h a n a t u r a l i n t e r p o l a t o r y b a s i s , i n t h i s case t r i a n g u l a r regions. This mesh i s determined s o l e l y by t h e
F i g u r e 5.
Planar s e c t i o n s f o r c r e a t i o n by l o f t i n g methods o f a boundary element mesh f o r t h e r o o t of a t u r b i n e blade [ll] [12].
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POSTPROCESSING The combination of f i n i t e and boundary element c a p a b i l i t i e s c a r r i e s over i n t o postprocessing, p a r t i c u l a r l y f o r an example m o t i v a t e d by f u l l - f i e l d v i s u a l i z a t i o n such as t h e two-dimensional problem shown i n F i g u r e s 7 and 8. F i g u r e s 9 and 10 show how an i n t e r a c t i v e two-dimensional f i n i t e p o s t processor [151 i s used, w i t h o u t s i g n i f i c a n t m o d i f i c a t i o n , t o i n s p e c t t h e r e s u l t s o f t h e boundary element a n a l y s i s . The f i r s t o f these f i g u r e s shows t h e examination o f t h e v a l u e o f displacements a t nodes by p o i n t i n g t o a p a r t i c u l a r node; t h e deformed mesh can a l s o be d i s p l a y e d a t any s e l e c t e d m a g n i f i c a t i o n . F i g u r e 10 i s a b l a c k and w h i t e photo o f c o l o r s t r e s s contours throughout t h e domain. Here some of t h e s t r e s s ranges have been darkened i n t e r a c t i v e l y t o produce a c l e a r e r view o f how s t r e s s e s vary near t h e c e n t r a l hole. The general g r a p h i c a l problem i n p o s t p r o c e s s i n g i s t h e v i s u a l i z a t i o n o f how a s c a l a r , v e c t o r , o r t e n s o r response parameter v a r i e s over t h e geome t r y o f t h e system. P a r t i c u l a r l y f o r three-dimensional s o l i d s o r s u r faces, i t i s u s u a l l y p o s s i b l e t o d i s p l a y o n l y a s i n g l e parameter component over t h e s u r f a c e o f t h e body w h i l e u s i n g c o l o r a t i o n t o convey
F i g u r e 6.
The complete boundary element mesh r e s u l t i n g f r o m F i g u r e 5.
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simultaneously t h e t h r e e d i m e n s i o n a l i t y o f t h e geometry, F i g u r e 11 [15]. As l o n g as one has both an a p p r o p r i a t e way o f e v a l u a t i n g response parame t e r s a c c u r a t e l y and an i n h e r e n t o r assumed b a s i s f o r s p a t i a l i n t e r p o l a t i o n o f these parameters, an i n t e r a c t i v e g r a p h i c a l postprocessor need n o t be s p e c i f i c t o a p a r t i c u l a r method o f a n a l y s i s and may, i n f a c t , be used e f f e c t i v e l y even f o r combined methods,
As an i l l u s t r a t i o n , f o l l o w i n g a r e some o f t h e major design f e a t u r e s of a general -purpose i n t e r a c t i v e g r a p h i c a l postprocessor c u r r e n t l y under d e v e l opment a t C o r n e l l . The postprocessor i s being designed t o accommodate three-dimensional geometries w i t h speci a1 iz a t i ons a v a i l a b l e f o r both curved surfaces i n three-dimensional space and two-dimensonal problems. The postprocessor serves f o r f i n i t e element, f i n i t e d i f f e r e n c e , boundary element, and combined f i n i t e / b o u n d a r y element analyses. F l e x i b i l i t y f o r a v a r i e t y o f t e c h n i c a l analyses i s b u i l t i n , i n c l u d i n g s t r e s s a n a l y s i s , heat conduction, and compressible f l u i d f l o w , The b a s i c d i s p l a y c a p a b i l i t y c o n s i s t s o f c o l o r c o n t o u r i n g on t h e surfaces as shown i n F i g u r e 11. The user i s a b l e t o choose any viewing angle and may have s i n g l e o r m u l t i p l e views o f t h e system. Moreover, t h e user i s a b l e t o s e c t i o n , c u t , o r " u n f o l d " t h e geometry i n t e r a c t i v e l y i n a v a r i e t y o f c o o r d i n a t e systems t o o b t a i n views o f t h e i n t e r i o r d i s t r i b u t i o n s o f parameters. When a c u t i s made, t h e program a u t o m a t i c a l l y i n t e r p o l a t e s i n t e r i o r r e s u l t s t o o b t a i n t h e v a r i a t i o n o f t h e parameter over t h e exposed surface. When no i n t e r i o r r e s u l t s are a v a i l a b l e f o r a boundary element a n a l y s i s , t h e program can t r e a t postprocessing i n much t h e same manner as i t would handle r e s u l t d i s p l a y s on surfaces such as s h e l l and membrane s t r u c t u r e s . F i n a l l y ,
?I
B
F i g u r e 7.
I
I n t e r a c t i v e boundary element preprocessing f o r a twodimensional problem. C r e a t i o n o f a d i s c r e t i z e d o u t l i n e f o r one q u a r t e r o f a s t r e t c h e d p l a t e w i t h a c i r c u l a r h o l e [13].
In teructive Computer Graphics
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a l t h o u g h t h e program has i t s own i n t e r n a l data o r g a n i z a t i o n designed f o r e f f i c i e n t i n t e r a c t i v e g r a p h i c a l m a n i p u l a t i o n , i t i s assumed t h a t data from a l l d i f f e r e n t types o f analyses a r e r e c e i v e d by t h e program i n a standardi z e d I G E S format. This represents a f i n i t e element p r e j u d i c e c o i n c i d i n g w i t h t h e i n i t i a l d i r e c t i o n s of t h e IGES t r e a t m e n t of a n a l y s i s data, b u t f o r t h e most p a r t i t seems f e a s i b l e t o expect t h a t t h e r e s u l t s from n e a r l y a l l a n a l y s i s types can be t r a n s l a t e d i n t o a f i n i t e - e l e m e n t - l i k e format.
INTEKACTIVE-ADAPTIVE ANALYSIS I n t e r a c t i v e - a d a p t i v e analyses has been d e f i n e d e a r l i e r as continuous g r a p h i c a l m o n i t o r i n g of t h e progress o f a n a l y s i s by t h e engineer w i t h t h e o p p o r t u n i t y t o i n t e r r u p t and t o change e i t h e r t h e design o r t h e a n a l y s i s [l]. I n a sense, t h i s i s a combination o f p r e p r o c e s s i n g and p o s t p r o c e s s i n g w i t h a d d i t i o n a l f a c i l i t i e s o f user c o n t r o l over a n a l y s i s . Obviously i f a n a l y s i s r e s u l t s a r e t o be monitored i n r e a l time, t h e computation must be very r a p i d o r t h e user w i l l be w a s t i n g h i s t i m e s i t t i n g before a very s l o w l y changing graphic d i s p l a y . C u r r e n t l y , t h i s l i m i t s t h e use o f i n t e r a c t i v e - a d a p t i v e techniques t o r e l a t i v e l y small problems, t o e f f i c i e n t s p e c i a l -purpose a n a l y s i s and design systems, t o t e a c h i n g a p p l i c a t i o n s , and t o research. However, t h e t r e n d s toward i n c r e a s i n g l y c o s t - e f f e c t i v e comp u t i n g and memory and i n c r e a s i n g l y r a p i d data t r a n s f e r i n d i c a t e t h a t t h e s e c o n s t r a i n t s w i l l be a l l e v i a t e d . An example of t h e use o f t h e approach t o m o n i t o r n o n l i n e a r s t r u c t u r a l behavior as i t i s being c a l c u l a t e d i s shown i n F i g u r e 12. Here f o u r d i f f e r e n t p o r t i o n s o f t h e d i s p l a y a r e used t o
F i g u r e 8.
For t h e problem o f F i g u r e 7, c r e a t i o n o f a mesh o f i n t e r i o r p o i n t s used t o o b t a i n f u l l - f i e l d i n f o r m a t i o n from a boundary element a n a l y s i s C131 C141.
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t r a c e a n a l y s i s progress. The main viewport shows t h e deformation behavior o f t h e s t r u c t u r e , w h i l e t h e upper l e f t and upper c e n t e r viewports i n c l u d e graphs o f response s e l e c t e d i n t e r a c t i v e l y by t h e user; l a s t l y , t h e upper r i g h t m o n i t o r summarizes a n a l y s i s i n f o r m a t i o n such as l o a d l e v e l , l o a d step, number o f i t e r a t i o n s , etc. The a n a l y s i s d i v e r s i f i c a t i o n o p p o r t u n i t i e s presented by i n t e r a c t i v e a d a p t i v e a n a l y s i s a r e several. F i r s t i s t h e a b i l i t y t o perform analyses which a i d i n e s t a b l i s h i n g a n a l y s i s parameters; f o r example, t h e a b i l i t y t o perform r a p i d n a t u r a l p e r i o d c a l c u l a t i o n s a s s i s t s i n d e t e r m i n i n g a p p r o p r i a t e t i m e increments f o r marching a l g o r i t h m s . Second, one r o l e f o r i n t e r a c t i ve-adapti ve approaches i s t o change a n a l y s i s s t r a t e g i e s a f t e r a n a l y s i s has begun. For example, i n F i g u r e 13 t h e a n a l y s t has, p a r t l y on t h e b a s i s o f an i n i t i a l a n a l y s i s , i d e n t i f i e d a group o f elements t o c o n s t i t u t e an e l a s t o p l a s t i c n o n l i n e a r s u b s t r u c t u r e , w h i l e t h e remaining elements a r e I n a d d i t i o n , a growing number o f grouped i n t o a l i n e a r s u b s t r u c t u r e [8]. s e l f - a d a p t i v e a l g o r i t h m s a r e becoming a v a i l a b l e f o r such aspects as f r a c t u r e propagation, s u b s t r u c t u r i n g , marching schemes, and mesh improvements. A1 though many of these s e l f -adapti ve a1 g o r i thms a r e s t a b l e , some have h e u r i s t i c bases and a r e best monitored t o ensure t h e continued v a l i d i t y o f computations. For example, t h e f i n i t e element mesh changes t h a t are necessary t o t r a c e an a r b i t r a r i l y p r o p a g a t i n g crack may l e a d t o meshes w i t h e x c e s s i v e l y d i s t o r t e d elements; such d i s t o r t i o n s can be c o r r e c t e d by t h e user under an i n t e r a c t i v e - a d a p t i v e approach. F i n a l l y , t h e implementat i o ? of i n t e r a c t i v e - a d a p t i v e procedures presents two o t h e r p o s s i b l e comput i n g strategies t h a t are p a r t i c u l a r l y s i g n i f i c a n t for nonlinear analysis.
7 i g u r e 9.
I n t e r a c t i v e boundary element p o s t p r o c e s s i n g f o r t h e displacements o f t h e problem i n Figures 7 and 8 u s i n g a twodimensional f i n i t e postprocessor [14].
Interactive Computer Graphics The f i r s t o f these i s p a r a l l e l a n a l y s i s i n which an a l t e r n a t i v e a n a l y s i s t y p e i s invoked t o diagnose system behavior. Examples i n c l u d e b u c k l i n g and f r e e - v i b r a t i o n t e s t s on n o n l i n e a r s t r u c t u r e s a t an a r b i t r a r y stage of l o a d i n g . The second i s s e q u e n t i a l a n a l y s i s where d i f f e r e n t analyses a r e performed t o i n f e r t h e e f f e c t of t h e o r d e r o f a p p l i c a t i o n o f i n f l u e n c e s . For example, a s t a t i c p r e l o a d i n g may precede a n o n l i n e a r s t r u c t u r a l dynamic a n a l y s i s . It i s apparent t h a t i n t e r a c t i v e - a d a p t i v e a n a l y s i s approaches h o l d s t r o n g promise f o r t h e e x p l o r a t i o n , development, and implementation o f u n i f i e d o r combined numerical methods. For instance, t h e same c a p a b i l i t y f o r adapt i v e s u b s t r u c t u r i n g i l l u s t r a t e d above c o u l d be used t o determine i n t e r a c t i v e l y t h e p a r t i t i o n i n g o f a geometric e n t i t y i n t o "married" f i n i t e element and boundary element zones. One o f t h e most e x c i t i n g uses o f i n t e r a c t i v e - a d a p t i v e a n a l y s i s procedures a t Cornel 1 t o date has, i n f a c t , been as a " t e s t bed" f o r research on numerical methods [l].
CONCLUSIONS The i n t e n t i o n o f t h i s paper has been t o demonstrate how i n t e r a c t i v e graphi c s i s a f f e c t i n g t h e environment i n which a n a l y s i s i s performed and t h e r e by i s i n f l u e n c i n g p o s s i b l e u n i f i c a t i o n o r h y b r i d i z a t i o n o f numerical methods. Although t h i s impact i s o n l y b e g i n n i n g t o be f e l t i n t h e comput a t i o n a l mechanics community, t h e r a p i d growth i n a v a i l a b i l i t y o f i t e r a c t i v e g r a p h i c a l c a p a b i l i t i e s w i l l d e f i n i t e l y a f f e c t such developments i n t h e near f u t u r e . It takes o n l y a small leap o f i m a g i n a t i o n t o see how i n t e r a c t i v e techniques a l r e a d y i n use f o r s p e c i f i c numerical methods such
F i g u r e 10.
F u l l - f i e l d s t r e s s v i s u a l i z a t i o n from a boundary element a n a l y s i s u s i n g t h e same f i n i t e element postprocessor as i n F i g u r e 9 [14] C151.
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as f i n i t e elements can be g e n e r a l i z e d f o r a v a r i e t y o f a n a l y t i c a l approaches. Future developments o f i n t e r a c t i v e g r a p h i c a l s o f t w a r e systems f o r e n g i n e e r i n g a n a l y s i s and design should be designed t o account f o r p o s s i b l e d i v e r s i f i c a t i o n o f a n a l y t i c a l procedures. One obvious recommendation i s t h a t s t r i c t s e p a r a t i o n be maintained between system models (geometry and a t t r i b u t e s ) and a n a l y t i c a l models (meshes o r o t h e r d i s c r e t i z a t i o n s ) . Under such c i rcumstances , developers o f i n t e r a c t i v e s o f t w a r e should be urged t o "go t h e e x t r a m i l e " t o p r o v i d e t h e a n a l y s i s f l e x i b i l i t y t h a t may be needed f o r e f f e c t i ve engi n e e r i ng design
.
ACKNOWLEDGMENTS P o r t i o n s o f t h e work used as examples i n t h i s paper have been sponsored by t h e N a t i o n a l Science Foundation, t h e N a t i o n a l Aeronautics and Space A d m i n i s t r a t i o n , and by v a r i o u s p r i v a t e companies; t h e w r i t e r s a r e g r a t e f u l f o r t h i s support. However, t h e views expressed here a r e t h e w r i t e r s ' and a r e n o t those o f any sponsor. The w r i t e r s would a l s o l i k e t o express t h e i r a p p r e c i a t i o n t o t h e i r present and former colleagues a t t h e Program o f Computer Graphics a t C o r n e l l U n i v e r s i t y , p a r t i c u l a r l y Donald P. Greenberg, D i r e c t o r o f t h e Program, and W i 11iam McGui re, a f a c u l t y c o - i n v e s t i g a t o r f o r much o f t h e research c i t e d .
F i g u r e 11.
Postprocessing u s i n g c o l o r contours on t h e surface o f a s o l i d i n c o n j u n c t i o n w i t h a d i f f u s e r e f l e c t i o n model t o convey t h r e e - d i mensi onal it y [ 151.
Interactive Computer Graphics REFERENCES and Abel , J . F., I n t e r a c t i v e - a d a p t i v e l a r g e displacement a n a l y s i s w i t h r e a l -time computer graphics, Computers and S t r u c t u r e s
c11 Gattass, M.,
16 (1983) 141-152.
C2l Kame1 , H. A.,
Design and implementation o f i n t e r a c t i v e e n g i n e e r i n g software, i n : Abel, J. F., e t a l . (eds.), I n t e r d i s c i p l i n a r y F i n i t e Element A n a l y s i s ( C o l l e g e o f Engineering, C o r n e l l U n i v e r s i t y , I t h a c a , NY, 1981) 773-803.
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and Abel, J. F., I n t e r a c t i v e g r a p h i c a l p r e p r o c e s s i n g o f three-dimensional framed s t r u c t u r e s , Computers and S t r u c t u r e s 17 (1983) 97-104.
c41 Abel, J. F.,
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c51 Yerry, M. A.,
and Shephard, M. S . , Authomatic three-dimensional mesh g e n e r a t i o n by t h e m o d i f i e d - o c t r e e technique, I n t l . J . Num. Meth. Engrg. ( t o appear).
C6l Shephard, M. S., and Abel, J. F.,
The i n t e g r a t i o n o f f i n i t e element methods i n t o CAD/CAM, i n : Kardestuncer, H. (ed.), F i n i t e Element Handbook (McGraw H i l l , New York, t o appear 1984).
F i g u r e 12.
Use o f m u l t i p l e d i s p l a y windows t o m o n i t o r n o n l i n e a r s t r u c t u r a l behavior as i t i s being c a l c u l a t e d d u r i n g i n t e r a c t i v e - a d a p t i v e a n a l y s i s C13.
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J.F. A b e l e t al. Perucchio, R., I n g r a f f e a , A. R., and Abel, J . F., I n t e r a c t i v e computer g r a p h i c preprocessing f o r three-dimensional f i n i t e element a n a l y s i s , I n t . J . Num. Meths. Engrg. 18 (1982) 909-926. Han, T. Y., Adaptive s u b s t r u c t u r i n g and i n t e r a c t i v e graphics f o r three-dimensional f i n i t e element a n a l y s i s , Ph.D. Thesis, Department o f S t r u c t u r a l Engineering, Cornel 1 U n i v e r s i t y (1984). Smith, B. M., e t al., I n i t i a l graphics exchange s p e c i f i c a t i o n (IGES), Version 2.0 (NBSIR 82-2531 (AF), N a t i o n a l Bureau o f Standards, 1982). Chang, S . C., An i n t e g r a t e d f i n i t e element n o n l i n e a r s h e l l a n a l y s i s system w i t h i n t e r a c t i v e computer graphics, Ph.D. Thesis, Department o f S t r u c t u r a l Engineering, Cornel 1 U n i v e r s i t y (1981). Perucchio, R., and I n g r a f f e a , A. R., I n t e r a c t i v e computer g r a p h i c s p r e p r o c e s s i n g f o r three-dimensional boundary i n t e g r a l element a n a l y s i s , Computers and S t r u c t u r e s 16 (1983) 153-166. Perucchio, R., An i n t e g r a t e d boundary element a n a l y s i s system w i t h i n t e r a c t i v e computer graphics f o r three-dimensional l i n e a r - e l a s t i c f r a c t u r e mechanics, Ph.D. Thesis, Department o f S t r u c t u r a l Engineering, Cornel 1 Uni v e r s i t y ( 1984).
F i g u r e 13.
I n t e r a c t i ve-adapti ve substr.uctu r i ng d u r i n g e l a s t o p l a s t i c The a n a l y s t has s e l e c t e d those elements (shown a n a l y s i s [S]. d a r k e r ) which should c o n s t i t u t e a n o n l i n e a r s u b s t r u c t u r e d u r i n g t h e n e x t stage o f a n a l y s i s .
Interactive Computer Graphics [13] Han, T. Y., A general two-dimensional , i n t e r a c t i v e g r a p h i c a l f i n i t e / b o u n d a r y element proprocessor f o r a v i r t u a l stage environment, M.S. Thesis, Department o f S t r u c t u r a l Engineering, Cornel 1 U n i v e r s i t y (1981). [14] Han, T. Y.,
unpublished student p r o j e c t ( C o r n e l l U n i v e r s i t y , 1982).
[15] Schulman, M. A., The i n t e r a c t i v e d i s p l a y o f parameters on two- and t h r e e - dimensional surfaces, M.S. Thesis, Department o f A r c h i t e c t u r e , Cornel 1 U n i v e r s i t y (1981).
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