gel-ia: MECHANIKA z. 108 Nr kol. 1161
International Conference on
COMPUTER INTEGRATED MANUFACTURING
Internationale Konferenz über
RECHNERINTEGRIERTE FERTIGUNGSSYSTEME
Zakopane, March 24-27 1992
M arek K R Z A C Z E K
The Institute o f Technology and Building M aterials Technical University o f G dansk, G dansk, Poland
NEW M E T H O D O F DATA E D IT IN G F O R F E AN D FD M E T H O D S
Summary: In particular, this is the case when the F E num erical m ethods are concerned.
This paper presents G raphic D a ta E ditor (GDE) - a new m ethod for d ata editing for 2D problem s solved by the F E (F D ) num erical m ethods. G D E m akes it possible to inp u t d a ta in graphic form. It is also possible to m ake use o f selected fragm ents o f the technical draw ing which describe the construction element under analysis. G D E recognizes the draw ing, and generates a file containing numerical data which unequivocally describe th e com putational task. The problem of recognition and in terpretation has been solved by m eans o f the theory o f fuzzy sets.
1. In troduction
T his pap er presents a new m ethod for d a ta editing, called G raphical D a ta E ditor (GDE).
It should be m entioned, th a t d ata editing is interpreted here as a m ethod for generating, correcting and m odifying the description o f a com putational task.
G D E enables detim ng a com putational task by m eans o f a natural language of graphic symbols, unequivocal and easy to understand by a designer. A n additional advantage is the opportunity for creating a model o f the analyzed construction element directly out o f the technical draw ing.
T he G D E m ethod can be em ployed to describe various engineering problem s, and it can be used along with various com putational program s. O riginally, it has been developed to cooperate with the F E m ethods in 2D space.
2. Problem F orm ulation
G D E is a program m ing system providing the following functions:
- creating draw ings (or im porting drawings from a draw ing editor) which are the graphical description o f a task (geometry, boundary conditions, loadings),
- recognizing the draw ing o f a construction model (e.g. static structure in 2D m em ber systems, o r m ulticonnected regions in 2D plate statics problem s) and its understanding, - recognizing the logical description o f working conditions o f a construction element, - checking whether the logical construction description is correct.
- generating a num erical description o f the com putational task, - invoking a com putational program .
The last four functions are realized by a T ask Identification M odule (TIM ).
The basic assum ption taken here is, th a t G D E should enable generating both the com putational model o f the construction based on a technical draw ing, and th e logical description o f the construction with the help o f a set o f predefined graphic symbols supplied by G D E (e.g. symbols o f loading types, b o u ndary conditions, etc.). I t should b e as easy and natural to define a com putational task as ordinary d raft designing.
G D E analyzes by oneself the draw ing describing the m odel o f th e geom etry o f the analyzed element, and working conditions. Such assum ptions considerably com plicate the problem . A technical draw ing is essentially a set o f graphic symbols which are unequivocal to a substantial degree, and which are draw n with precision adapted to m a n ’s perceptive abilities. This m eans th a t a technical draw ing, although unequivocal and clear for a m an, is n o t always a precise m athem atical description o f the construction (from a num erical m ethod point o f view).
Fig. 1. G eneral A rchitecture o f G D E
GDE m ust generate an unequivocal and m athem atically precise description o f a com putational task which m ust be clear for num erical program s. It seems quite n atural th a t the problem should be solved based on procedures for description conversion, similar to image (technical draw ing in this case) perception and recognition processes going on in human brain.
3. D raw ing Recognition
A technical draw ing m ade with the help o f a draw ing editor (DE) is n o t free from inaccuracies which arc inessential from the poin t o f view o f technical docum entation. Such a draw ing is an unequivocal m athem atical description o f a real construction, and consequently, it is worthless. In the m ethod for d ata editing suggested here, the technical draw ing is recognized and interpreted by an artificial intelligence m odule D R M which has been developed based on an indeterm inistic algorithm for interpreting draw ing inaccuracies
and determ ining intersection points o f graphic symbols. D R M im itates the drawing recognition processes going on in hum an brain.
F ro m the D E point o f view the graphic symbols m ake up a list o f symbols, called the list of predefined objects O. A predefined object O, is defined as follows:
Def. 1. O bject O , is a finite subset o f cartesian product o f a pair <x,., y(> in set theory meanir. g, on which the relation q> is im posed. I t can be written:
{<x,y> : <p(x,y)},
where: q>{x,y) belongs to a family R o f relations accepted in a particular solution.
The list o f predefined objects O describes a draw ing created with the help o f DE.
T he aim o f the recognition is to recreate the draw ing, to m ake it as consistent with the user’s intentions as possible, and, a t the same tim e, free from inaccuracies and equivocations.
As a result, the recognition process creates a set o f nonintersected objects, and points o f intersection. T he draw ing inaccuracy results from certain psycho-physical features o f the user and the quality o f his com puter equipm ent (e.g. graphical adapters).
T o simplify further considerations, the list O is reconstructed, and thus a new list Opf of simple objects is created. T he simple objects are defined as follows:
Def. 2. O bject Opf, is a subset o f cartesian p ro d u ct F (X x Y ), in the sense o f set theory, where for each xeX there is one and only one y, such th a t <x,y> e F(X x Y) T o solve the problem o f draw ing recognition, the theory o f fuzzy sets is used to generate the draw ing description. T o do this, fuzzy objects are introduced according to the following definition:
Def. 3. T he simple object Op, is a fuzzy relation R, which is a fuzzy subset o f a Cartesian pro d u ct o f sets X,- and
Y
¡. R e F(Z, x Y,).To each subset X and Y a m em bership function is assigned:
/ix,(x ), for subset X, /ixI(x), for subset Y.
F u rth e r idea o f draw ing recognition is based on the assum ption th a t the graphic objects included in th e Opf list will be understood according to Def. 3. Consequently, a list o f fuzzy objects can be generated out o f the Opf list.
It is assum ed, th at if X and Y are spaces, X,- is a fuzzy set in X and Y, is a fuzzy set in Y, then all fuzzy sets X, and Y, are norm al, i.e.:
sup fix(x) = 1, and sup fi (y ) = 1.
x « X y e Y
T o be able to interpret graphic objects by m eans o f fuzzy sets, and to condition the interpretation by the degree o f the draw ing precision, the concept o f a-cross-section o f fuzzy sets X, and Y, is introduced as:
X , = [xeXj | «=[0,1], Xa€=P(x), (1)
Y . - [yeY, | **& )>«]. mQ U ], Y.eP(y), (2) F ro m now on, fuzzy sets X, and Y , will be identified with their a-cross-sections.
R elation Ris characterized by m em bership function /i„(x,y). T he value o f the m em bership function determ ines how dosely th e elements o f set X { are related to the dem ents
o f set Y,. R elation R is chosen individually for each type o f simple object Opj, e.g. the following m em bership function o f fuzzy relation R,(X x Y ) (for line segment) is assumed to be:
p A/= e x p [ - w k ( y - c - x - d ) 2], f o r x , < x < x k
e x p { - k [ ( y - y , ) 2 + ( x - x , ) 2]}, f o r x ^ x , (3)
H RJ= exp {— k •[(y — y*)2 + (x — x*)2]}, forx*<x where: w — cos2/S,
k = f(a), coefficient related to a-cross-section value.
As the next step in solving the problem , th e intersection points o f simple objects Op, are determ ined and interpreted, and the Opd list o f nonintersected objects is created. T he process o f draw ing recognition presented here is sim ilar to th a t going on in hum an brain (P.H.
Lindsay and D .A . N orm an, 1972), [4],
The determ ination o f intersection points o f simple objects results in the Opd list of nonintersected objects, such that:
R, n Rj = 0 , for j *
j j
, (4)f i — 1
R, n R, = A. for j = •( . , (5)
[i+l
where A is a fuzzy set, which is the com m on p a rt o f tw o fuzzy relations.
It can be shown th a t for tw o simple objects Op, and Op^, th e m em bershipfunction o f fuzzy set A is given by the following formula:
Mx(x ,y) = m in[pRl<x,y), p*,(x,y)]. (6) T aking into account the above assum ptions and form ulae (4) and (5), an intersection point can be defined as follows:
Def. 4. An intersection point o f two simple objects is a p air (xp,yp), such that:
= si»p ¡>a(*.y)]- (7)
ye A
The n A(x,y) m em bership function is continuous and n o t sm ooth. In a real com putational algorithm a stochastic algorithm for determ ining local extrem um o f a n o t sm ooth function is applied. T he mechanism o f interpretation o f intersection points o f simple objects Op, is presented in Fig. 2.
The Opd list is the result o f draw ing recognition, and is an unequivocal m athem atical description o f the draw ing th at is seen by the recognition system. T he draw ing recognition aim s at realizing the intentions o f the user who creates the draw ing. H ow precise the drawing analysis is, and how much the draw ing meets the user’s expectations is determ ined by the values o f a. It can be said th at the values o f a indicates how inaccurate th e draw ing is and determ ines its final form . T his way the system learns and ad a p ts itself to the specific features
o f its w orking environm ent (e.g. the level o f precision o f the technical draw ing created by the user, o r the quality o f the equipm ent used to present graphically the drawing).
\ I
' / V
I ;
a) b) c)
Fig. 2. M echanism o f interpretation o f Intersection P oint o f three Simple Objects.
Practice has proved th a t it is extremely essential to ado p t the optim al strategy for choosing the order in which simple objects are analyzed. Therefore the strategy for analyzing the Op,- objects can be m odified depending on how far the analysis results are consistent with the user’s intentions, and thus inreaching the knowledge possessed by the system.
T he draw ing after recognition still requires further analysis (e.g. additionally, in the case o f static analysis o f 2D plate systems, it is necessary to generate unequivocal description o f m ulticonnected regions th a t would enable th e F E mesh generator to carry out the In /O u t tests). D epending on the engineering and com putational problem , D R M can cooperate, for example, w ith a 2D m odeler as a tool for generating a description of m ulticonnected regions.
4. T ask Identification
The recognized draw ing m ust be com pleted with description o f boundary conditions, loadings, etc. T he U ser’s Interface enables to create such a description. F u rth er, a task description should be identified and interpreted. In G D E, T ask Identification M odule (TIM ) enables to d o it.
T he basic functions o f TTM are as follows:
- logical identification o f graphic symbols, - geom etrical localization o f graphic symbols,
- test o f the physical sense o f the description o f objects,
- tran sfo rm atio n o f the graphic description (base o f facts) into a num erical description according to rules accessible in TIM .
T he functions listed above have determ ined the architecture o f TTM. It has been assumed th at the m ost effective solution is a m odule based on expert system techniques [3]. T he knowledge in T IM is represented in tw o ways:
- fundam ental know ledge base consisting o f rules o f inference,
- dynam ic base o f facts (called hereafter context), generated by U I individually for every com putational example, and m odified by the Inference Engine during the inference process.
T o achieve the optim al organization o f the know ledge contained in facts, it is assumed th a t th e context is com posed o f objects belonging to several classes in the sense o f Object Oriented
Program m ing. T he knowledge contained in the context is dynam ically processed during the inference, but according to the rules of inference. Two following groups of rules can be distinguished in the base o f inference ruies:
- classical rules o f the type: IF smth T H E N action,
- fuzzy rules, where a specific action is taken after a fuzzy condition has been satisfied.
Fuzzy rules are the element o f fuzzy knowledge. T here is, however, a specific relation between fuzzy rules and the mechanism o f draw ing recognition. T he level o f confidence necessary to satisfy a fuzzy rule is specified by the values o f fuzzy a-cross-section, understood and determ ined like for the DRM m odule.
T o sum up, the inference am ounts to processing the facts stored in th e context according to the rules contained in the knowledge base. A fter the inference process has been completed, the context contains a full num erical description o f the analyzed com putational task. The results o f inference depend on the knowledge consisting o f facts, rules, values of a-cross-section and recognition strategy.
a) b) c)
Fig. 3. C om putational task identification: a) technical draw ing, b) task definition, c) results o f calculation.
5. Con cl udin g-Rem arks
The G raphical D ata E ditor presented in this w ork is a new tool for editing d a ta for the FE numerical m ethods and, after adaptation, for m any other num erical m ethods (e.g FD m ethods) o f construction analysis.
GDE enables the user to generate the description o f the analyzed problem in the form of a task draw ing and a minimal num ber o f num erical quantities supplem enting the description.
The geom etry can be entered directly with the help o f the graphic processor which cooperates with GDE, or it can be a selected p art o f the technical draw ing.
GDE can ad ap t itself to the psycho-physical features o f the user and th e com puter equipm ent used through learning. However, if differences occur during interpretation of successive drawings, it is possible to supplem ent the GDE knowledge. T herefore, GDE is a convenient tool th a t aids designing, gives engineer access to greater n um ber o f m odem com putational m ethods, and provides integrated designing environm ent.
REFERENCES
[1] C zogala.E , Pedrycz,W.: Elementy i metody teorii zbiorow rozmytych, PW N, War
szawa 1985.
[2] Hintcm, E., Owen, D . R. J.: An introduction to F inite Elem ent C om putations, Pineridge Press Lim ited, Swansea, U .K , 1979.
[3] K rzaczek, M .: System expertowy generacji opisu zadania obliczeniowego zastosowany w systemie A R T IF , V III K onferencja M etody i Środki Projektow ania W spom aganego K om puterow o, W arszawa 1991.
[4] Lindsay, P .H ., N orm an, D . A.: H um an Inform ation Processing. A n Introduction to psychology, A cadem ic Press Inc., N ew Y o rk 1972.
[5] Zadeh, L. A.: Fuzzy sets and systems, Proc. Symp. System Theory, Polytechn. Inst, o f Brooklyn 1965.
D IE N E U E M E T H O D E E IN E S G R A FIS C H E N D A TEN V ERA RB EITU NG SSY STEM S NACH E IN E M F E (FD) -VER FA H REN
Zusam m enfassung
V orführung eines neuen grafischen D atenverarbeitungssystem s (GDB) nach einem FE (FD )-V erfahren, in dem 2D-Aufgaben rechnerisch gelösst werden. Im G D B-Verfahren können grafische D aten in das System eingeführt werden. Es besteht auch die Möglichkeit, Fragm ente technischer Zeichnungen in die Beschreibung des Berechnungsmodells einzu
setzen. G D B erkennt die Zeichnung des M odells der K onstruktionsgeom etrie und generiert eine D atensam m lung (file) m it num erischer Beschreibung der Berechnungsaufgabe, welche eindeutig diese A ufgabe form uliert. D as E rkennungs- und Interpretierungsproblem wurde mit Hilfe der T heorie der unscharfen M engen gelösst.
NOWA M E T O D A E D Y C JI DA NY CH DLA M E T O D M E S I M R S Streszczenie
W pracy om ów iono G raficzny E dytor D anych (GDE) - now ą m etodę edycji danych dla zagadnień 2D rowiązywanych za pom ocą m etod M ES (M RS). M etoda G D E pozwala na graficzne w prow adzanie danych. Umożliwia również przenoszenie do opisu m odelu obliczeniowego fragm entów rysunku technicznego. G D E rozpoznaje rysunek modelu geometrii konstrukcji i generuje plik danych zawierający opis num eryczny zadania ob
liczeniowego, k tó ry jednoznacznie opisuje zadanie obliczeniowe. Zagadnienie rozpoznaw a
nia zostało rozw iązane w oparciu o teorię zbiorów rozmytych.
Wpłynęło do redakcji w styczniu 1992 r. Recenzent: Ryszard Rnosala
