Automatic triangular mesh generation in flat plates for finite elements

cmQvlrrs & .wlutlvc8 Vol. I I. pp. 43964 Pcrgama Press Ltd.. 1980. Printed in Grcac Britain

AUTOMATIC TRIANGULAR MESH GENERATION IN FLAT PLATES FOR FINITE ELEMENTS

G. D. STEFANOU School of Engineering, University of Patras, Patras, Greece

and

K. SYRMAKEZIS Civil Engineering Department, The National Technical University of Athens, Greece

(Received 3 August 1978; received for publication 28 June 1979)

Abstract-The paper describes a method of generating triangular finite elements for a two dimensional region bounded by rectangles. The work is divided into two parts: Part 1 describes the method dealing with the problem of generating graded triangular meshes for two dimensional regions. The mesh could be uniform or graded in four directions at prescribed regions where stress concentrations are expected to appear. Five typical groups of triangular elements are ,introduced and their properties and recurring formulae referring mainly to nodal and triangular numbering are given in detail. These properties are used later for developing the computer programme. Part 2 describes the process required for developing the computer programme in FORTRAN. The programme is divided into three sections (a) preparation of data (b) running of the programme and (c) printing of results.

The method is demonstrated by an illustrative example of a simply supported thin beam loaded in its plane.

1. INTRODIJCTlON The purpose of this paper is to study the problem of dividing the surface of plate or disc structures into triangular elements by a computer and hence, compute the geometrical properties of the elements. The solution of this problem will be useful for the application of numerical approximate methods, such as ,finite elements, finite difference or dynamic relaxation, to plate or shell problems [ 1,2].

The structures analysed may be continuous or discontinuous with the local co-ordinate system at right angles or at any angle 9, as shown in Fig. 1 (a) and (b). The co-ordinates of any system x’, y’ may be expressed in terms of the co-ordinates of the orthogonal system as

X’=XtyCOSQ I

y’=ysiny, 1 (1)

The investigation is based on three main criteria: (a) The mesh of the elements is not uniform between regions and can be graded in four directions within the region. (b) Mesh grading is taking place gradually from larger to finer elements and vice versa. (c) All elements are connected at the nodes.

The structure is divided by the user into regions by

1’

EL cl0 0 x

(0)

Fig. 1. Discontinuous plates.

‘Y

I

I I I

I I I I I I

:pn_ --- ---

--- A---+

I / ----t---bt---*

/ ’ I I

j j I ’ I 1

0, I 1 1

X

Fig. 2. Discontinuous plate divided into regions.

straight lines drawn parallel to the co-ordinate axes as shown in Fig. 2. The area is therefore divided into rectangular “hyper-elements” which are subdivided by the programme into “triangular elements”

2. SuBlWEXoN PROCJCDIJRE OF MAIN GROW G = I

To divide a hyper-element into triangular elements, a main group G = 1, Fig. 3, is introduced with ek (k = 1,2,. . . n) hyper-elements.

Index k indicates the manner in which the hyper- element is divided. The elements of the group are chosen to have an area, ET, of their triangles equal to

ET=$E (2)

where E = Axby is the area of hyper element, and NT = 2k the number of triangular elements of each hyper-element.

The division of hyper-elements into biangular elements is based on the following rules: (a) The t&g&u

439

G. D. STEFANOU and K. SYRMAKEZIS

e2 ,k=2

e3 , k.3

3 6 9 12 15

i Ayl4

t

elements are produced by continuously dividing their sides the numbering of the triangular elements. The number of Ax, Ay (side Ax is divided first), by vertical and horizontal nodes per column jC, and line jr. for each hyper-element is lines and by drawing the diagonals of the parallelograms so defined by formed as shown in Fig. 3(b) The diagonals have always an ascending slope from right to left. (c)The numbering of the jC=lt2’k-‘)‘2 for k=l,3,5 ,... nodes starts from bottom to top for the columns and from left to right for the lines Fig. 3. (d) The same rule holds for or j,=lt2 (k-2)‘2 for k = 2,4,6,. . (3)

Automatic triangular mesh generation in flat plates for finite elements 441

(i*l 1 Nodal he -----------

i Nodal line --~-----~--

i

_-_

(j-l)(r,*l)+i j (se+11 + i

Fig. 4.

and

jL=l+2(k-‘)‘2 for k=l 3 5 , 9 ,**- I or iL = 1 t 2”“’ for k=2,4,6,...]

(4)

Let

and

SC = jC - 1

SL = jL+r 1

(5)

be the numbers of spaces of the hyper-element in the y and x-axis respectively. Then the number of the smaller “rectangular sub-elements” so formed is N, = SC x S,_. Each hyperelement is subdivided by a diagonal into two triangular elements. Any sub-element which is formed by the inter-section of ith and jth lengths (i = 1.2,. . . SC, j = 12 t . . . . S,_) will be defined, Fig. 4, by numbers as follows

(j-l)(S,tl)ti, j(S,tl)ti, j(S,tl)titl, (j-l)(S,tl)titl.

The two triangular elements in the sub-element have element numbers

2[(j - l)S, t i)]-’ and 2[(j - l)S, t jl.

3. !WBDMSION PROCRDURE Op SRCONDARY CROUPS: c=s, c=j, c=J, c=s

The described method of dividing the elements of group G = 1 is useful only for uniform element subdivision of the plate since it is not possible to connect directly two rectangular sub-ekments of different k

numbers. To do this four secondary groups G = 2, G = 3, G = 4 and G = 5, are introduced. The subdivision procedure is described in the following.

3.1 Secondary group G = 2. (Fig. 5) Each hyper-element is defined by a two digit index

number T, the first and second numbers of this index indicating the indices k of elements of group G = 1 to which they may be connected above or below the element respectively. Therefore, elements el and e2, k = 1, k = 2) of G = 1 are connected by element el (T = 12) of G = 2, Fig. 9(a). Index T does not exist for G = I. The serial number k for G = 2 is given by the expression

k =; [r- lOAk(r/lO)] (6)

where (Ak(r/lO) = the real part of 1llOth of index r. The numbers of nodes per column jC and line jL are

jC = 1 t2’-’ jL = 1 t 2k. I

(7)

To facilitate the calculations, fictitious nodes Jo are introduced temporarily during nodal numbering, Fig. 5. The fictitious nodes will make eqn (7) valid and are given by

j,, = 2k-l. 03)

Their serial number nio is:

n~o=lth(lt2k-‘) (9)

for

A=l,3,5 ,... <jL. (10)

442 G. D. STEFANOU and K. SYRMAKEZIS

1 3

l\ 2

1 3

G=t?

l g,T ~34

Ax14 ,L AxH ,” Ax14 1’ Ax14 ,L X

Fig. 5. Four typical elements of G = 2.

The total of nodes jl the real number of node jl and the The numbering of fictitious node j0 is again given by eqn number of elements in hyper-element, become respec- (8) and their serial numbers nio by expression tively

fljO=A(l t2’-‘) (16) jt = 1 t2k-'(3+2k) (11)

j,=j,-jo=1+2k(1+2k-‘) (12) for

r, = 22k - 2’-‘. (13) h=2,4,6 ,... <jr_. (17)

The numbering of triangular elements is performed as for group G = 1, and the modifications required for mak- ing new elements are made, after subtracting the fictitious nodes. For elements el (k = 1) of group G = 2 element e2 (k = 2) of group G = 1 is firstly created. From this element, after cancelling fictitious node 3, sides 2-3 and 3-4 are also cancelled and diagonal l-4 is drawn

3.2 Secondary group G = 3 (fig. 6) Index number T and serial number k of the elements

are as described in Section 3.1, Fig. 9(b). The serial number k of group G = 3 is given by the expression

k = ; Ak(~/l0). (14)

The number of nodes per column jC and line jL is given by

The total number of real and fictitious nodes jh the number of real nodes jr and the number of triangular elements t. are given from eqns (l&(13) respectively.

3.3 .!%condary grOllp G = 4 (Fig 7) The serial number k of group element G = 4 is given

by eqn (14), whereas the number of nodes per column je and line jL is given respectively by expressions

jC=l-2k

J,=1t2k. I (18)

The number of fictitious nodes is given by eqn (8), also

Iljo = A (19)

for

jc = 1 t2k-’

1 (15) A=2,4,6,...<j, C3-N

jL = 1 t 2”. j, =(1+2k)z (21)

Automatic triangular mesh generation in tIat plates for finite elements 443

G=3

el , ~21 e2, 7=43

e3,r=65 e. ,T=67


and

j,=j,-/O=(1t2*)2-2*-’ (22)

t. = 2*-‘[2k+2 - 11. (23)

3.3 Secondary group G = 5 (Fig. 8) The serial number k is given as

k = ; [T - 10. Ak(rjlO)] (24)

and the numbers of nodes per column and line are given from eqn (15). The number of fictitious nodes is given from eqn (8), also

n,0=2k(1t2k)tA (29

for

A=2,4,6,...<j, (26)

and X J, and f. are given by eqns (21)-(23) respectively.

4.uEvxuWMENToFciMNnWlntoGMMMxFoll AUWMAYU’IG MESB GRADING AND ELXMENT

G-Y CALCUUllONS

In this section the computer programme “DIVIDE”

developed for automatic mesh grading and for element geometry calculations is described. The programme is coded in FORTRAN language and was originally written for the IBM 1620 computer of the National Technical University of Athens. Later, the programme was modified for the CDC 3u)o computer unit of “Demo- critos” Nuclear Research Centre, Athens. The programme is divided into three main parts: (a) Preparation of data,(b) Running of the programme and (c) data printing[3,4].

4.1 Data preparation To prepare the data it is necessary to divide the area of

the plate regions Figs. 2 and 10. The mesh so produced consists of hyper-elements and is known as the “original mesh”. Such a mesh for a simply supported thin beam loaded in its plane with a concentrated load, is shown in Fig. 10. This simple example was chosen in order to demonstrate the proposed method of mesh grading.

4.1.1 Data for the problem. (a) The number of columns of hyperclements along the length of the beam NELEM &); (b) The number of lines of hyperclemeds along the height of the beam MELEM (SC); (c) The lengths of hyper-elements of the various cohmms of the elements in the beam are BHMAX (I) (A& (d) The heights of hyper-elements are BHMY (1) (A,,); awl(e) The matrix of indices INDEX (IJ), (i.e. indices r), which has dimensions (MELEM x NELEM), &fines the manner in which each


el, ~=23 e2, r=46


hyper-element is divided into triangular elements. For the example considered in Fig. 10, we have: NEMEL = 8 and MELEM = 7, also

INDEX (1,J)

3 32 1 1 1 1 23 3 22111122

21 21 1 1 1 1 21 21 1 1 1 1 1 1 1 1 1 1 12 12 12 12 1 1 11222211 1 1 23 3 3 32 1 1

The data are punched on computer cards as follows: 1st card (214): includes the numbers NELEM (in

columns l-4) and MELEM (in columns 5-8). 2nd card (lOF8.3): Every 8 columns the number of

MELEM elements of matrix BHMAX is putiched. If the number of elements is MELEM > 10 the punching is continued on the next card.

3rd card (lOF8.3): Every 8 columns the number of MELEM elements of matrix BHMAY is punched. If the number of elements is MELEM > 10 the punching is continued on the next card.

4th card (2014): Every 4 columns the number of elements (NELEM x MELEM) of matrix INDEX, first, per column _ _ . . . _ .

degree of mesh concentration at various neighborhoods of rectangular sub-elements. The matrix INDEX in indices T is assembled to fit four possible ways of mesh grading as shown in Fig. 9.

To make the assemblage of matrix INDEX easy an auxillary array is assembled (Fig. ll), based on the element arrangement shown in Fig. 9.

The elements of the array in Fig. 11 could be produced automatically by the computer. -

4.1.2 Elements of sector A. For

i<j: aij=lli+lO for i=odd, aij=i for i=even

for i=j: Ui,=i=j

for i > j:

ai,=j for j=odd, a,=lljtl for j=even.

(27)

4.1.3 Elements of sectors B, C and D. The elements of sectors B, C and Dare produced from elements of sector A if we consider the following:

4.1.4 SectorB. (a)

bii=aii-9fori<jandi=odd (28)

and then per line, is punched. Matrix INDEX is constructed according to the desired and (b) if the lines of sector A are transposed.

Automatic triangular mesh generation in flat plates for finite elements

G=!5

* Ax/2 Ax12 X’

1 1 1 !x AX L

1 1

ea.T=76 e4, t=96

44s


Sector C. (a)

Cij=aijt9 for i>j and j=odd Cii=aj-9 for i<j and i=even (29)

and (b) If the lines and columns of sector A are transposed. 4.13 Sector D. The elements of sector D are

produced from elements of sector A if we consider the following

(a) d,,=aii+9fori>jandj=even (30)

and (b) if the columns of sector A are transposed. For the transposition of lines and columns of matrix A

an auxiliary matrix [El is introduced

I

0” 00 . . . . . . 0 0 1 . . . . . . . 0 0

[E] = 0 0 . . . . . . . 1 : 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

0 1 . . . . . . . 0 0 0 I 0 . . . . . . . 0 0 0 I .

The product matrix [AITIEl* [Al * [El is a matrix which results from matrix [A] if all its columns and lines

are transposed. For the example considered in this paper four 3 x 3) matrices were taken from sectors A to D of Fig. I I as shown by the dotted line and on the beam in Fig. IO.

4.2 Running of the computer programme The purpose of the computer programme is to per-

form, after mesh grading is completed, the nodal point numbering and hence to calculate the geometry of the elements. Essentially, the programme works out the data for only one column of rectangular sub-elements.

In the lirst place the programme calculates the expression (61, (14) and (24) and the given matrix of indices INDEX (7) of the rectangular sub-elements, the group number NGRUP (G) and its serial number NTYPE (K) in the group. The flow chart diagrams for these operations is shown in Fig. 12.

The computation of the number of nodal columns (real and fictitious) NOFC, for each rectangular element and the maximum number of nodes MNOFC that occur in the column is performed from eqns (31, (71, (15) and (18). In the column matrix XPR (IJX), with (UK) varying from 1 to MNOFC, the x-co-ordinates for each nodal column in MNOFC are registered.

Each of the rectangular sub-elements in MELEM is then considered separately. The number of nodal lines,


G-l G-l Gm4 G=l G=l G.4

Fig. 9. Typical columns of rectangular sub-elements.

NOFL, (real or fictitious) is considered first. Based on this number the y-co-ordinates, for each nodal line are registered in the unit matrix, YPRELM (IJK), for (IJK) varying from 1 to NOEL.

Each rectangular sub-element has its nodes not neces- sarily all lying on the MNOFC columns. Consequently, the serial numbers of columns where the element under consideration has nodes, are

ZSTEP=lt(A-I)~~~_~ll~MNOFC for A

=12 , 9.‘.

P

-+.A I 2 2 1111 2 2

3 32 1 1 1 1 23 3 E

Fig. 10. Rectangular mesh of beam.

The nodal co-ordinates of the rectangular sub-elements are stored in a temporary matrix YC, having MNOFC columns and infinite number of lines. The co-ordinates of real and fictitious nodes of the element, are entered in matrix YC. If the rectangular element is the first in the column, we enter in matrix YC all its nodal co-ordinates of all nodal lines. If this is not happening, then we omit the co-ordinates of the first nodal line as they have already been entered as the co-ordinates of the last line of the previous element. The YC matrix for the first column of rectangular sub-elements is given by

r 0 0 0 1st element + 0.20 0.20 0.20

- 0.40 0.40 0.40

2nd element I _ +

f

. . . . . . . . . . . . . . . . . . . . . . , . . . . . . . . . . .

-8

I + 0.80 0.80 0.80 . . . . . . . . . . . ...* . . . . . . . . . . . . . . . . . . . .

3rd element

4th element -+jj f; . . . . . . . . 13 . . . . . . . kf ( = lYC]

2.00 0 2.00

If the rectangular element has fictitious nodes, its co-ordinates in matrix YC are located to be eliminated later, by shiiting one space upwards, all the following elements of the column in YC matrix, in which the fictitious node belongs.

The column matrix ISTAR (JF), defines the last line of matrix YC on which a co-ordinate is entered. Initialising ISTAR = 0, after considering each rectangular sub- element (i.e. cancelling of fictitious nodes, etc.), we register the new co-ordinates of the next rectangular element at each column JF of matrix YC from column ISTAR + 1. In the example, the first column in Fig. 10, of matrix ISTAR which originally is [000], su-

Y

cr__________________-._-----__--_---_____+ 1, 7 1 5 % 3 032 , 1 23 3 46 5 67 7 1

; 6 6 -6 64 3 22 I 1 23 3 46 5 6 6 1

166565 50 221

/ 4 4 4 4 3 ‘z - _l__

I 23 3 465

4 5656I 4

; 34 24 24 24 r;-- 32 1

__I__‘_)__) 4 4

I 23 31 24 24 24 34 1. , 12 2 2 2 I 2 I 1 I I

, 12

1 12 I2 I2 I2 112 II I I “’ 8

, , I

‘I

; 2l n 2l 21 ml 2, I I

I 2 , 2221221 I

1 2 212 2 2 2/

I I

( ” 4’ 4’ 43 rl___u___r__..__l_Z’__d I

‘2 42 43 43 1

.I4 4 ‘ ‘1 221 1 23 3 4 4 4 4 1

.166666 0 643 22 I ,

1 21 1 465 6666;

!6 6 5563 32I 1 23 2 46 5 6 6!

Fig. 11. Auxiliary array.


Fig. 12. Flow chart diagram. Note: D = 1st digit of INDEX; M = 2nd digit of INDEX: P = odd number: A = even number.

ccessively becomes:

(2 2 21 ]3 3 31 [4 3 41 15 3 51 16 3 61 [7 3 71 [8 3 81.

When matrix ISTAR is considered a new two-dimensional matrix LBEGG is created. This matrix has MNOFC lines and MELEM t 1 columns. Thus, in the example considered in the paper, matrix LBEGG is

Hence, matrix LBEGG of the example. after the final numbering of nodes becomes

[LBEGG] =

: 10 I2 I6 I4

4 I3 I7 5 13 18 6 I3 I9 7 I3 20

; 13 I3 ::.

For each column of matrix LBEGG we keep the number NABCO LBEGG(1, MNOFC) - I. After cal- culating the total number of nodes for column (i.e. columns 9,4,9 in the example), column matrices X and Y of the nodal co-ordinates, are formed as follows: for matrix X, each element of matrix XRR(IJK =

I , . . . MNOFC) is repeated ISTAR(IJK) t I times. For matrix Y,ISTAR(IJK) t I elements are taken successively and are placed in matrix YC in the same order. Thus for the example, matrices X and Y become

[Xl =[O.OO 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.25 0.25 0.25 0.25 0.50 0.50 0.50 0.50 0.50 0.50 0.50

0.50 0.501

[Y] =[O.OO 0.20 0.40 0.80 1.20 1.60 2.00 2.40 2.80 0.00 0.20 0.40 0.80 0.00 0.20 0.40 0.80 1.20 1.60 2.00

2.40 2.801.

Before going from the local to the global nodal numbering the necessary changes of matrix NEM are perforated for each case, as described in the following paragraphs.

4.2.1 Group G = 2. Comparing Figs. 3 and 5 we see that the element of group G = 2 results from element 2k of group G = 1, by modifying the first line of its triangular elements. The triangular elements of the first line are considered in sets of four, Fig. 13(a) and (b). Let

I, = I t 40 - l)(L - I)

12 = I, t I 1, = b t 2(j, - I)

I, = I3 t I

be the serial numbers of the four triangular elements in the set, Fig. 13(a). Where A = l/2,. . . < l/2(ji - 1) and jc and jL are taken from eqns (7). From the above four

(a)

Fig. 13.

(b)


Fig. 14.

triangular elements, three new triangular elements Zl,Z:,Z;, are created as shown in Fig. 13(b).

The nodes of these elements are numbered in an anti-clockwise direction as is also in Fig. 12, according to the following notation

Element I;: Node 1 corresponds to node I of elemtnt II Node 2 corresponds to node 2 of element Zz Node 3 corresponds to node 3 of element II

Element I;: Node 1 corresponds to node 1 of elements II Node 2 corresponds to node 1 of element Z4 Node 3 corresponds to node 3 of element I,

Element Z;: Node 1 corresponds to node 1 of element Is Node 2 corresponds to node 2 of element Z4 Node 3 corresponds to node 3 of element I,.

The above modification made in matrix NEM, corresponding to the element I,, Z2, Z, produce the new arrangement of triangular elements. Element 14 is finally eliminated by shifting upwards by one line all lines of element NEM. Thus, for element k = 2 of group G = 2, 16 lines of matrix NEM are formed in the following manner

I*-) 4 : 2 : 6 3 3

4 : 5 I 5 5 8 6 8 9 6

4+ Z4+

7 10 8

11 10 13 11 14 ,..............

10 8 II 8

11 12 z 13 I1 14 11 14 12 15 12 . . . . . . . . . . . . . . .

The above matrix is then transformed into 14line matrix as follows

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . *.. I 5 2 2 5 3 5 6 3 1 I 7 8 :

: ! 6 6

7 11 8 8 11

11 12 ; 7 13 II

13 I4 II 11 14 I2 I4 I5 I2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . *.*....

. . . . . . . . , . . . . . . . . . . . . . . . . . . . . .

‘A=1

pA=l

From this we eventually go to the final numbering. 4.2.2 Group G = 3. The triangular elements of the last

line are considered in sets of four Fig. 15(a) and (b). Let

I, = 2(2A - l)& - I) - I z* = I, t 1 Z,=4h(j,-I)-1 14 = ZJ t 1

-m (a)

Fig. 15. Triangular elements for G = 3.


be serial numbers of the four triangular elements in the set, Fig. 15(a). Where A = 1,2,. . . < 1/2(jL - 1) and jc and jL are taken from eqns (15). From the above four triangular elements, three new triangular elements, II, 14, I; are created as shown in Fig. 15(b). The nodes of these elements result from the nodes of the original four elements as follows

Element II: Node 1 corresponds to node 1 of element II Node 2 corresponds to node 2 of element I, Node 3 corresponds to node 3 of element I,

Element I;: Node 1 corresponds to node 1 of element Z, Node 2 corresponds to node 2 of element L Node 3 corresponds to node 3 of element I2

Element Z;: Node 1 corresponds to node 1 of element Z, Node 2 corresponds to node 2 of element b Node 3 corresponds to node 3 of element L.

The above modifications are again carried out in the corresponding lines of the elements I,, Z2, G, of matrix NEM and element I, is eliminated as described in paragraph 4.2.1.

4.2.3 Group G = 4. Comparing Figs. 3 and 7 we see that the element of group G = 4 results from the element 2k of group G = 1, by modifying the first line of its triangular elements. The triangular elements of the first column are considered in sets of four.

LA

Z,=4A-3

zz = z, t 1

I, = I, t 2

L = I, t 3

be the serial numbers of the four triangular elements in the set, Fig. 16(a). Where A = 1.2,. . . < l/2(5, - 1) and .Z, and 1‘ is taken from eqn (18). From the above four elements, three new elements, Z, Z2, 4 are created as show in Fii. 16(b). Their nodes resulting from the following:

Element Zl: Node 1 corresponds to node 1 of element I, Node 2 corresponds to node 2 of element I, Node 3 corresponds to node 2 of element b

Element ii: Node 1 corresponds to node 1 of element II Node 2 corresponds to node 2 of element Z2 Node 3 corresponds to node 3 of element b

(a) (PI Fig. 16. Triangular element for G = 4.

Element I;: Node 1 cm-responds to node 2 of element b Node 2 corresponds to node 2 of element Z4 Node 3 corresponds to node 3 of element 4

The above modifications are again carried out in the corresponding lines of elements Z,, Z2, 4, of matrix NEM and the line corresponding to element Z4 is eliminated.

4.2.4 Groupe G = 5. Comparing Figs. 3 and 8 we see that element of group G = 5 having serial number k(k = 1,2,. . .) results from the element 2k of group G = 1, by modifying the last column of its triangular elements. The triangular elements are again considered in sets of four.

Let

z* = I, t 1

Is = I, t 2

I, = I, t 3

be the serial numbers of the four triangular elements in the set, Fig. 17(a). Where A = 1,2,. . . < l/2 (jc - 1) and .L, .ZL are taken from eqn (15). From the above four elements three new elements I,, Zz, b, are created as shown in Fig. 17(b), their nodes resulting from the following:

Element Z{: Node 1 corresponds to node 1 of element I, Node 2 corresponds to node 2 of element I, Node 3 corresponds to node 2 of element I,

Element I;: Node 1 corresponds to node 1 of element Zz Node 2 corresponds to node 2 of element Z4 Node 3 corresponds to node 3 of element Zz

Element Z;: Node 1 corresponds to node 1 of element b Node 2 corresponds to node 2 of element Z, Node 3 corresponds to node 3 of element I,.

The above modifications are again carried out in the corresponding lines of the elements I,, Z,, Z, of matrix NEM and the line corresponding to element I, is eliminated.

After completing the local numbering of triangular elements, follows the transition from the local to the global numbering of the plate. This is done by column matrix NOFJO(1) of the rectangular sub-element which

3 2

14

13

1 3' 0 3 2

12

11

1 2’

(a>

3

13

/

I I2

\ 1;

1 fa

(PI

Fig. 17. Triangular elements for G = 5.

CM Vol. I I No. 5-F


6

4

Fig. 18. Triangular mesh of beam.

includes the serial numbers of its nodes from the global numbering. Matrix NOFJO(I) is formed by means of the serial numbers of the first and last node of each nodal column of the rectangular sub-element. Matrix NOFJO(1) is formed from the assembled matrix LBEGG.

Matrix NOFJO(I) is formed by means of the serial numbers of the first and last node of each nodal column of the rectangutar sub-element, defined from the pair of lines of matrix LBEGG. For the example considered, the first element of the first column, Fig. 1, from the first and second lines of matrix LBEGG:

matrix NOFJO(1) is formed in the following manner

[l 2 3 10 11 12 14 15 161.

After the observation that full correlation exists between the elements of matrix NOFJO(I) and the indices I

[l 2 3 4 5 6 7 8 91

of the elements of matrix NEM is equal to K(K = NEM(I,J), then the corresponding final element NEM(I,J) will be

NEM(I,J) = NOFJO(K).

4.3 Printing of results The third and final task of the programme relates to

the printing of data and of the results. The printing is carried out successively:

(a) The given indices, INDEX, of each original rectangular sub-element of the area considered, which is the

intersection of J column and I line, as well as the serial numbers of group G(NGRUP) and of the element K( = NTYPE) in the group.

(b) The lengths of the rectangular hyper-elements A,i (BHMAX(1)) and the heights Axi ( = BHMAY(I)).

(c) The nodal point array X and Y of the nodes of the triangular elements.

(d) The array of geometrical data of triangular elements which include the serial numbers of the elements, the three nodes l-3 together with their coordinates taken in an anti-clockwise direction, the area (AREA) of each element and the co-ordinates (XC,YC) of the centre of gravity.

In the paper table Tl of the indices INDEX, NG RUP, NTYRE of the example is also shown, together with the lengths and heights of the rectangular hyper-elements as shown in Fig. 18, in table T2 of co-ordinates X and Y of the nodes as well as in tables T3 of geometrical data of the triangular elements of the plate.

5. COMMENTS AND CONCLUSlONS

The method of dividing the area of a flat plate into triangular elements by a digital computer is undoubtedly more advantageous compared with conventional methods. This is because (a) human errors are avoided in data bonding and data preparation; (b) large numbers of card punching is avoided which has the advantage of time saving and minimises the errors and (c) it is fast, particularly, when solutions are repeated more than once and for various mesh gradings.

The method of generating a mesh of triangular elements in its final form adopted in this paper is not the only one possible. This procedure was chosen hence, because it furnishes us with more facilities in coding the whole process.


. . . . . ..*.........................+e........*..

. COLWW l LIta *CLEWEM l CLtnZMr l Ll.EIt!tll .

. . . . CklUP l 1YPL l

l d l I ’ 1lvxx l c . I: l . ..*....“.I..*...............................

. 1 l .

: l

.

.

: : : l : :

f ’ 2 ’

. 1 ’ 21 ’ 1 l 1 l

. 1 ’ .

: ’ I l .

: g 1 ’

. 1 ’ 1 l 1 l l

i’ : b ’ 1 l .

. 7 ’ 1 l : l : :

..~~..-~~-~--~~-~.-~~------~~-~~.~-I-~~~~

.

. f : : : a2 l 9 ’

r: : 1 ’ : :

. 2 ’ t l 3. 1.

. 2 ’ b l 1 ’ 1 l 1 ’

. f :

s ’ . : ’

1 . . b l 1 ’ :. f . 2 l ? l 1 l 1 l 1 ’ ~~~~~__~_~~.~__~_~~~~~~~~~~~~~~.~~~~~~~~~~~~~ .

f : .

: ’ .

: l

. . : l

1 ’ 1 l

. f ’ 3 l 1 9 1 l

. 3 l b ’ 1 ’ . : .’

. a ’ . 12 ' :. .

.

. : : 1' 1 ’ : ’ ? l 2: : c . i ’

.-ee.-----w-- _______~______________I__________

. b ’ . 1 l 1 l 1 *

. c ’ : ’ . : ’

1 l . . b ’ 1 l 1 ’ : ’ . b ’ b l 1 ’ . . b l 5’ 12. : l : : . b ’ b l 2 l 1 l 2 l

. b ’ 1 l a ’ 1 ’ I ’ ~._~_~~~_~~_~~____~~~~~~~~~~~~~~~~~~~~~~~~~~~~ . s ’ 1 l .

: l

1 ’ t l . s l 2 ’ 1 l 1 l . s l s l . 1 l 1 ’ .

; : . :. .

. :e 12’

.

: ’ 1 l

1 l . s l b ’ 2 l 1 ’ . . I l I l a ’ 1 ’ : ’ ________-_~_.___--__----~~~.-----~----.--~--- ‘ . . : ’ : :

.

: l

.

: l

1 ’ 1 l

. .

t ’ t ’ . .

. . : l

:* 12. : ’

1 ’ .

. b ’ 2 l : ’

. b l a l 2 l 1 . 2 l

. b l ?. 12. I l I ’

.___I___--___-_---___ ______(._________________

. ? l .

: ’ 23 l b ’

. ? l . . : :

. ? ’ I’ 2L : ’ 1 l

. ? l b l 1 l 1 ’ 1 ’

. r l I l 1 l . : 9

1 l . I l b l 1 ’ 1 ’ . I * 1 l 1 ’ 1 ’ 1 l . .._.~_____~~~___~.~~~~~~~-~~~~~~~~~~~~-~~~~~~ . I l .

: l

I*’ 1 ’ J l . I l 2 ’ . 2 8 * . ::

I l 21 l : * 1 l . 1. 1 l 1 l

. 8 l : . I l .

: l

1 l . 6 ’ b l 1 l 1 ’ . 8 * I ’ 1 l I l 1 l

____..___.ew. ~.~.~~_~~~~~~~~~__~~_L___________


Table 2.

JOINT COORDIFiATES

. . . ..~..~.~...~...~~...~.................~.........~....................~...,......,.~.......,

. . I . . . . . . .

. J:lMf ' I ' I l JOIN1 l I l v ’ JOIN1 l I ’ I l

. . . . . . . . . .

. . . . . . . ..*......*..*....................................*......*..............................

.

.

.

.

.

.

.

.

.

.

.

.

.

.

. . . . . . * . l

.

.

.

.

.

.

.

.

.

.

.

.

.

.

J.00 l

3.00 l

o.uo l

0.00 * a.00 ’ 1.00 ’ 0.00 l

3.06 l

0.00 ’ a.25 ’ 0.25 l

0.:5 ’ 0.25 ’ 0.50 ’ 0.50 ’ 0.50 l

0.50 ’ 0.50 ’ J.53 ’ 0.50 l

0.50 ’ 0.50 ’ 0.?5 l

0.15 ’ 0..?5 l

a.75 l

1.30 l

1.00 l

l.JO ’ 1.03 ’ l.CO ’ 1.00 ’ 1.00 ’ l.Jl l

1.i5 ’ 1.35 ’ I.25 l

o.uo l

3.29 l

o.*o l

0.60 ’ l.dO ' 1.60 ' 2.00 ’ 2.cu l

2.86 l

3.00 l

a.20 l

0.50 l

0.00 l

O.OG ’ 0.20 ’ J.Clr ’ 3.dC ’ 1.20 q 1.03 ’ 2.tAo l

2.co l

.?.oL, ’ o.uo * 0.10 l

J*CO l

0.80 l

0.00 l

O.bi l

0.80 . 1.20 l

1.00 l

2.c3 ’ Z.bO l

2.80 l

2.UJ ’ Z.%O ’ 2.bO l

38 ’ 39 9 b0 ’ Cl ' l 2 l

51 l

c4 l

*5 l

bh * L? l

56 l

69 ’ 50 l

51 * 52 l

53 l

54 l

55 l

5b ’ 57 ’ 56 l

59 l

60 . 61 . 02 l

SJ l

64 ’ 65 ' bb ' 67 ’ 68 l

b9 l

ro l

71 ' 72 ' ?J ' 74 l

1.25 l

1.50 l

1,50 l

1.50 l

1.50 l

1.50 l

1.50 ' 1.50 ' 1.53 l

1.5t ' 1.75 ' 1.15 l

1.15 ' 1.w l

2.00 ' 2.00 l

2.00 l

2.00 l

2.00 l

Z.JO l

2.00 l

2.00 ' 2.00 * 2.25 ' 2.25 l

2.25 ' 2.25 l

2.50 * 2.50 l

2.50 . 2.50 l

2.50 l

2.5c ' 2.i3 l

2.50 l

2.50 l

2.75 '

2.80 l

0.00 l

o.*o l

0.80 l

1.20 l

1.60 l

2.00 l

2.60 l

?.bO i 2.80 l

2.00 ' 2.40 * 2.60 l

2.80 ‘ 0.00 l

0.40 l

0.80 l

1.20 l

l.bO l

2.00 l

2.sa ' 2.63 l

2.80 l

2.03 9 2.co l

2.60 l

2.60 l

0.00 l

O.bO ' 0.6J l

1.20 l

1.60 l

2.00 l

?.bO l

2.bO ' 2.80 l

2.00 l

t5 l

?b ' 77 l

78 l

79 l

60 ' 61 l

82 l

83 l

8C l

b5 ’ 86 l

87 l

88 l

89 l

90 l

91 l

92 l

91 ' 9s l

95 l

9b l

97 l

98 ' 99 l

100 * 101 l

102 l

IOJ l

106 l

105 l

lOI l

107 ' 108 l

109 l

110 ' 111 l

2.75 l

2.?5 l

2.7s ' 1.00 l

3.00 '

:*:: l .

Loo ' 5.00 l

3.00 l

3.00 l

3.25 * 1.25 l

1.25 l

:'f: l .

s:so l

:*:: ' .

I:50 ' 3.50 l

S.I@ l

J.50 l

3.50 l

1.75 l

1.75 l

1.75 l

J.75 l

b.00 l

c.00 ' c.00 l

b.OE l

4.00 l

*.no l

$.O@ l

r.00 l

b.00 l

2.60 l

2.bJ * 2.10 l

8.00 l

O.bO ' 0.40 l

I.20 . l.bO l

2.00 l

2.so 9 2.10 l

0.00 ' 0.79 ' O.LO l

0.80 l

0.00 l

0.20 9 o.co l

0.80 l

1.20 l

l.bO l

2.00 l

2.bO ' 2.10 l

4.00 ' 0.20 l

O.bO l

0.80 l

e.00 l

0.20 l

o.co ' 0.10 ' 1.20 ' 1.10 l

2.00 l

2.bO l

2.60 l

Automatic triangular mesh generation in Bat plates for finite elements 453

. l 54 l :: ’ $0 . 90 l SC) . 51 l St . 98 l bT . 91 * 5J . ' Cb . :: l 5b . 9c l 57 . Y? l 95 . 54 l 59 l * 56 . t:: l 5b l lJ1 l b9 l :*,c * 57 . t:J ' bt . lJb l 0 . $05 * 'J . tab l 59 . JO? * h& . ICI l 5: l 1Sll ‘ bJ . tto l 52 . JI: l ft . $1) l ct . 1:: ' '2 . tt* ' l 5 . 11% l '9 . :Ib ’ i’) . ::I l t’t . 11) l $I . $19 l 90 . II* l 09 ‘ $21 l a: l tt2 l b5 . JdJ l b: . $4. l 92 . 125 l ?O . tab ' 'm . I)' l '5 . $29 ' I9 . tw l 'L . JJJ l 15 ‘ tJt l 72 l as2 ' I- * tJJ ' '5 . .lJb l ¶c . 115 * 'b . tJh l ?9 . I?? ' '6. . It1 * 9' . tJ4 l 55 . 140 l +I . 1*1 * 97 l lb2 ‘ 91 l :IJ l ft . 1.r l ,I . :*5 l 59 l tr5 . 4: . I*' l 9: . INI * 9: . $51 l 9‘) . 00 l 01 . l $0 ‘ :;: l ,? l J'rJ l 9% . J5C * If l $55 l ‘5 l t5b l . . . $5' l 9' . 1'1 l ,:o . 15Y l II . 153 ' II . Jbt ,* ts1 . :b2 l If . :**i l toi l J'*b l 10) l 1,s ' If- * 1~ l J.' l ts1 l 101 l t21 l $Jt . Jb9 * I!5 . t'a * 'I I . 171 l IIf . l 1)' . ::: . *; . or l 11; l J7, ‘ 1+ . I'S l tJ9 l :*7 l 9, . lf5 l I,# l $7, l w

t.75 i,?b t.oo i.79 2,CJ Z.00 8.50

:::; Z.OU 1.50 2.co 2.51 2*JJ ?.OU 2.59 2.ao 2.?5 1.25 1.54 z.00 2.Z5 C.09 2.25 2.15 2.58 I.25 ?.5: Z.5b J.OJ 2.50 J.Cu 2.50 1.00 c-52 J.CO t.:a t.iu J.OY 2.50 2.75 2.75 J.OJ 2.50 2.75 2.50 2.7% 2.75 1.0) t.75 l.OU 3.LO 1.25 1.25 J-5* J.25 J.5.r t.OJ J.29 1.2, J.511 J.,U J.?', 1.f-l J.CU 1.5& J.uJ I.54 J.dJ a.5: t.u* J.5b J.5" I.72 J.5U 1.7% I.'5 r.33 J.?5 l .". 1.41 J.?" I.?', b.JJ J-5, 1.7% 1.15 1-s: b.11 J.94 ..I!" ,.,. l .2L J.'.r:

t.a, l J3Y :.aa ‘ II0 2..J l AlJ

1.7% 2.00 2.08 2.00 2.00 t.90 t.50 2.50 2.90 2.5* 2.90 2.9a 2.90 2.25 2.50 2.5) 2.29 2.29 1.90 i.5J 2.25 2.2s 2.25 2.25 2.5J 2.5J 2.94 2.5J J.OJ J.OJ 3.00 1.90 1.4I) 3.00 1.OJ 1.01 2.79 3.00 1.04 f.75 ?.'I 3.00 3.00 2.'5 2. '5 2.75 ?.?I 1.00 J.01 3.00 3.25 s.25 3.25 J.5J l.SJ 3.50 J.50 1.25 J.2S I.50 1.50 J.25 J.5J J.50 J.IJ J.50 3.50 I.50 3.50 1.50 J-54 J.SO 3.75 J.'Z J.75 J.?9 b.00 4.OJ b.0, r.00 l.F9 J.?5 b.:J l .OJ J.'9 b.00 b.JO L. Or b.OJ b.JJ r.OJ ..YJ c.JJ b.23 _.

2.0) l b' t.r0 l 90 2.b6 l 50 2.bO l 5J 2.eo l 5t 9.ee l 5J 0.4: l 51 e.*e l 5c e.to ' II Leek l 55 I.'0 l 55 1.20 l 9b :.bO l fb 2.90 l I' \.bQ * bt 2.BO l b: ?.OO ' 9b 2.60 ' 9b ?.OO ' bt Z.\O l b2 zero l 99

:*:i * 99 ‘ b8 2::O l b0 2.rO * bJ I.60 * bJ 2.CO l b. 2.:0 l ir 0.00 l bb O.bO ' 6b O.+O l 61 0.10 * b? 0.30 l be t.20 ‘ be I.20 l b5 1.60 l bt 2.00 l 70 1.60 l Fb 2.00 l tr 2.00 l 'I 2.bO l II 2.00 l 75 2.*0 l 75 2.&O l '2 2.50 l ?2 2.60 l ?J 2.10 l 7J 2.+3 l ib 2.40 l Ib 2.&O l 71 0.00 ' 0' 0.20 l ?9 a.rn l 19 0.00 ' 0 a.*9 * 9' 0.20 l 9e o.ca l 90 n.ro l 90 e.15 l 90 O.bO l 19 o.eo l e9 0.10 l e: 1.20 l e: O.LJ l *. 1.20 l 82 :.:a l e2 t.bO l b3 2.90 l OJ 2.00 * e* 2.*1 l .c 2.*9 l e5 2.00 l b5 0.00 ' 91 0.2J ’ 9: 0.Z0 l 11 O.bO ’ *I 0.00 l :ce 0.2. - 100 0.28 l :a1 e.*J l tat O.bJ l 91 o.ea * 9s O.be ' JJ2 a.ee l IO2 o.ea ' 9* I.'0 l 9. O.LY ' 197 t.>a * +5 1.50 l 55 I.60 * SC 2.00 l :b 2.00 l *I 2..0 l 9'

1.00 $.I$ l*?% J.?5 1.75 Z.OO 1.00 t.oa t.oe 2.00 t.00 z.*a Z.00 ztao 2.25 I.25 2.90 2.00 2.25 2.25 2.oo 2.00 t.on 2.ea 2.25 2.25 2.25 2.25 2.50 2.50 2.50 2.5P 2.50 2.5s 2.5@ 2.50 2.50 2.75 2.F5 2.50 2.56 2.75 2.75 2.59 2.50 2.50 2.50 2.75 2.15 2.75 3.25 J.60 1.00 1.25 J.25 J.25 Jr25 J.OD J.00 3.25 J.25 a.00 1.60 J.50 1.00 1.00 J.OC 3.00 3.50 1.00 J.00 J.00 1.50 1.50 J.50 J.5C J.I5 J.75 l.75 J.75 3.50 1.50 J.75 J.75 11.50 I.56 r.00 1.50 J.5J J.53 3.50 J.9k 3.93 1.90 __

*..a ’ i).ttoa l ,.‘,I l 2.t:t l

2.bJ * a.1510 l 5.9:: l 2.2‘7 l

2.r0 l 99 . 1.1 - 11, ..U, <.C, - ,I, ..J4

y: : *.toJo l S.9b' l 2.9ll l

2.:, l 99 4.99 . O.tOJO l l.OJ: * Z.*b' l ,,,,.(~~~~~..,..,..,....,.~~.~.,,......,~..,~...~.................................~.~..........*..........~*.~.

Z.bt l St 2.59 l %b 2.5) l St tat ’ 99 1-b) ’ b0 o.et ’ b5 o.oa l bb O.CJ l bb t.bJ l b7 t,tY ’ b' O.Ol) ’ b# I.?0 l b4 :*a0 ’ 69 t.bJ ‘ bt :.bO l b9 L.hY l 70 2.00 l 41 Z.OJ ' b2

:*:: l I0 0 l ‘1 ?.-a l b2 2,bO l bl ‘.bO l bJ Z.bJ l .bC 2.cr l 7: 8.21 l It Z*bO * '2 1.60 l '3 a.01 l '4 o.aa l '5 b.50 * I9 Y.., l 60 e.eJ l e0 0.1) l et t.ti l 4: l.ZJ l oz trSJ ‘ 'b 1.L) l 52 l.bO l e1 2.01 ' 'b l.00 l ?5 1.OJ l ex 2.e3 ' Rb 2.r3 l '5 2.CJ l ‘b 2.b., ’ 'b 2.51 l I? 8.40 ‘ "4 2.4) l .e5 2.50 l 85 0.01 ' 56 0.4) ' e7 ..2J ' 5e i.aa ' 30 z.ea l 1: e.21 l i: t.tJ ‘ eZ a.*3 l l e o.*p l 19 C.&J l 32 e..a l '93 'r.lJ l 59 r.5J ' eb *.a" . SJ I..'* l *. i.21 l . . 95 l.bJ l 15 1.61 l *b C.&J = 41,

:*ri l 5' .:- 97 p.r3 ' qn t.03 ' 99 ..OJ l IJO &.?J l :ac b.20 l 13: L.01 * IJJ *."a l lib Ir.ta l tar c*ta ‘ 145 C..J ‘ 1st (I.., l II2 "..J l If5 L.cl l IJb u.91 ' $lZ b.II l IJ' J.03 l lo* :.?I l IO7 1.2) l :>e t.irJ ‘ ate t.l.3 l 109

Table 3. Geometrical data of elements 1.00 l

t.be * 2.bf l :*:: :

0:ri l

O.&O l

9.10 l

o.oo ' 1.2J ' 1.21 l

t.bO l

i.bO l

2*09 ' 2.90 ' Z.00 l

z.ra l

2.*0 ' 2.ra ' 2.r. ' 2.hO ' 2.bO ' 2.:0 l

2.oa l

Z.b9 '

:*:: l l

t:,a l

0.48 l

o.*o l

O.bO ' S.80 l

1.10 l

t.ro l

t.CJ 9. :.bO l

2.OD l

2.J9 * z.oo l

2.ro l

?.CO ' 2.bO l

2.6) ' 2.50 l

t.bO ’ t.ea ' 2.43 l

2.ca ' 2.60 l

2.5e l

0.20 l

o..o l

o..* l

0.22 ' 3.20 l

‘... l

O.hO ' a.so l

e-es l

e.10 ' e.:o l

I.20 l

z: l . ’ t.bO l

&.bO l

2.eo l

2.00 l

2.68 l

Z.CO l

2.90 l

Z.¶O ' e.20 l

0.29 l

0.60 l

o.re l

a.20 ' e.20 l

O.CO l

o.ra l

0.90 l

0.60 ' 0.90 ' 0.10 l

1.20 ' t.2e l

1.10 l

:.bO l

1.b. l

2.J) l

2.SJ l

e.82~1 ‘

0.0250 l

o.0250 *

8.8290 l

o.e250 .

*.tooo .

0.1100 .

0.1000 .

o.to4e .

O.le80 .

ortoe l

0.10e0 l

1.1010 l

0.05@0 l

0.1000 l

0.0590 l

0.85JO .

e.0500 .

e.asea .

I.oJaa .

3.025e .

o.ozio .

0.0250 .

e.e?50 .

e.9250 l

@r025J l

O.D254 l

0.025: l

0.10e0 .

O.IOJO .

e.t4os .

1.1OJJ .

0.1OJ8 ‘

0.100f I

o.:OOo .

e.:a10 .

8.OSe-t ‘

a.tcae l

e.etao .

0.05PJ .

O.L9JO .

o.ooe .

e.O%Pe l

0.0150 .

O.P750 .

0.025J l

a.0250 l

e.0253 l

O.OIJl l

6.5<51 .

J.O?SO .

o.e350 .

0.0250 .

O.C2SO .

0.0?50 .

O.E?5C .

(I.0250 .

0.05JO *

0.053J w

e.osJo .

0.05JO 6

e.0'500 ‘

O.lOOJ .

e.05;0 .

O.lOBO .

O.i$JO .

o.s0:5 .

O.lOJO l

P.IOPO l

0.:000 ‘

O.IOJO .

o.1900 .

0.0299 .

O.J250 .

0.0250 .

e.az9J .

0.0150 .

0.029? .

0*02so l

O.J150 l

*.JfPa l

0.0900 l

0.05ilo .

o.t5ca .

I.0590 .

P.IOJO .

(l.o510 .

o.toaa .

e.tJ*o .

*.t3*e *

#.:aJa l

:rbbt I.&II Lrtr I.111 1.91) 2.1)’ 2.111 2.16) 2.1)) 2.lb' Z.JJJ 2.:47 2.JJJ 2.09J 2.25e 2.b:F 2.09J 2.1b' Z.JJJ 2.51' 2.0:s i.lb? 2.ObJ I.tb? 2.338 i.*1t 2. JSS I.*:’ 2.bb’ Z.IJJ 2.667 2.:31 2.661 2.931 2.55' 2.9:1 Z.lEl 2.'50 2.9f' 2.5JJ 2.bh7 2. BJJ 2.91’ 2.9.3 i.bSi

i*SbJ 2. bb? 2.833

I.947 2.511 3.1&F J.J#J J.lb’ LJJ3 S.blF l.JJJ f.5$? s.ats 1.$&F J.133 1.*1t J.UdJ J.250 J.kl? J.:b? 1.111 l.lb? s.3:3 J. ii? J.JlJ J.:b? 3.11% 3.591 J.bh'

1.5iI 3.bi7 J.0) l.Ol? l.lJJ 1.917 1.99J J.bb' 3.933 i9;i 1.911 J.751 1.91F J.bb7 J.bJJ S.lb? J.tJJ

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2.111 2.bV I.911 t.bb? 2.711 O.tJJ 0.20 O.IJ3 O.bb? (.9Jf l.Ob' I.111 t.bb’ l.Bb? t.733 (.0&I 2.1JJ 2.2b7 1.111 2.2b7 Z.bb? 1.533 t.bb? Z.?lf 2.bb’ 2. (33 2.bb7 2.?JJ O.IJJ 1.2b? o.9JJ O.Sb? I.CSJ :.?b? 1. JSJ :.bb? J.#67 I.‘Jl 1.9b' 2.138 2.267 2.tJl 2.2b' 2.bb' 2.5J) t.bb? 2.7s3 2.5b7 t.bOO 2.7JJ 11.11' 0.2JO o.1aa o.:b' 1.111 0.2b' O.ll3 0.93s O.bb7 9.93s e.bb' 0.9JJ I.907 0.9)s l.JJf t.bb' S.'lS t.tb?

F::: 2:9J, 2.b.L' O.Ob' O.lJJ #.ZC? e.JlJ 0.1:7 e.tas (.2&f O.JJJ 9.933 O.bL' J.9Jl l.bb' I.911 1.9b' 9.llJ L.JJJ 1.11' t.?Jl 1.9b'

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Acknowledgements-‘his work was carried out in the laboratory Continuum Mechanics. McGraw-Hill, London (1%7).

of The Institute of Structural Analysis of the National Technical 2. R. W. Clough. The finite element method in plane stress

University of Athens. Part of the work appeared by G. D. analysis. Proc. ASCE 2nd Conf. Efectronic Camp~tatio~,

St~~u and K. Symmkezis in the magazine, issue No. 12172, of Pit&burg, Pennsylvania (Sept. t%O).

The Greek Inst&e of Engineers, 3. D. D. McCracken and W. S. Dam, ~urn~ca~ Methods and

REiWsiWE§ Fortran Progmmming. Wiley, New York (I%@.

4. W. Weaver. Comouter Pmemms for Stmctuml Analvsis. Van

454

Input listing-results


APPENDIX 1

An input listing of the computer programme and results of sample problem are included in this section (see input and tables Tl, T2. T3).

s

10

15

LO

15

30

3,

*(i

*5

d5

90

YS

100

PROGRAN DIVIOEIINPUT.OUTPUT,

OIYLNSION XPRIiO~~NtVP~ilO;lO~1NCIYUPtlO~~O,.VPREL~lO~,NOFCIlO~, ~V~I~~~~~~~X~~OO~~VI~OO~.~~N~XIAO,~~HN~VIIO,~IST~R~~O~~INOEXI~O,~O~

DIYENSION Lt)LtGIAO~lo~rKSTXl10~.KSTVI10,,NENl250r3,rNOFJ0I35~ DO 667 NNL.L=I,L

C READ AND PRINT DATA REAO B.NELEN.NELEM

z FOHN4T (214)

NELl.NELEN*l READ I~IllNDEXlI.J,~I=l.NELEM~~J=l~NELEM,

i FOWAl ILOIwl RELD 3rlBHN4VlI,rI=l.NELEN, REID ~~I~NM~XII,~~=I.NELEN,

3 FDtiNAT llOFL).3, C *....*..**..*.***.******..*.....**..*.*......******.**.**.....*.**

; ~iii~~~i~iri~i..i~.i~.~~.~*~~~~~~~~~~..~.~*.***.*******.****.*.**. CALCljiAtE’GtOMETRIcAL~DAt4 FOR’tiit‘SUBELENENTS OF TNE COLUMNS

c WFT~EOINIT14L WECTANGULAR MESH s

NE,.=o swsx=o KBiF=O 00 1100 J=irNELEN

E CALCULATE N(lRUP AND NTVPE FOR LAW RECT4NGULAR ELENENT OF THE INITIAL MEW DO 111 I=I.NELEN NGWPII.J,=r UA?A=IYDEXIIIJ) NTYPE~lrJ,=KAPA IF IKAD4-.0) ~l,.l~l.llL

11.2 KTf.N=KPPA/,U KUYIT=KAPA-,O’KTEN KHCLP=KTEN/L KHtLP=KHELP*c INTEN=, IF iKTiN-KHELP) iLirA&?~rLi

ik INlENs i2i IF IKTEN-KUNlT) rlJril3ri~r ii3 GU TO ll3i.r32)rINTtN i3i NGWUPlIrJ,=L

NTYPEll.J,=KUNIT/~ GO TO hii

13‘ NGdUPlI.J,=r NTYPElI*Jl=KTEWZ GO TO irl

AI* GO TO (.33,r3u,rINTEN ,J3 NGWPllrJ,=>

NTVPEllrJ,=KUNIT/L GO TO rir

139 NG.tuPII,J,=3 NTVPE(IrJ)=KTEN/~

rir iOYT1Y;lc L t*****.***.****.****.*..*..***.*...*.****.*~*.*~*~**~*.**~*.**.**~

C CALCULPTE‘N~MBERS UF COLUANS ~~‘>U~NTS NDFCIII OF EACH ELEMENT C UF THE r(PlN COLUMN CDNSIOWEO 45 IltiLL IS THE NAXINUW NUNELK OF

C THEY YVOFC HNJFC=O L)D cOi I=irNLLEM KTVP~=YTYPE(I,J)

IF (NG+UP(IvJI-I, LO.?*LOL~~U~ .zI)d NOFC(Il=I*Z+*(KTVP~/2)

b0 TO 2Or LO3 NOFC(ll=r*L**KTYPE c”* IF lNOFC(I,-HNOFC) cO,,20,,205 ~05 HNOFC=VDFC(I, cOi CONTINUE

AN=HNOFC-r C INITIALIZE AUXILIAHV ARRAY5

00 3~. L=l.NNOFC LdiGGlr.L,=.

3r. ISrARIL,=O IPkNNOFC’NELEN

00 39, lJK=,.IPR 00 39i JIK=I.NNOFC

3Y. VC(IJK.JIK,=O. c CALCULATE DHDlNATtS XPR OF EACH COLUMN OF JOINTS

00 206 LJK=~.NNOFC AI=IJK-A

,?“a XPHlIJK~=AI~BNNAX(J,/*NISJnSX

SUYSX=SUWSX+dNNAXlJl S”*ST=o.

‘ c c

C

Ii01

305

303

JUI. 308

320 ii05

.~ _. _. ..*.........**.****.***...*.**.~**.*.*********~**.*.*....*.......* CALCULATE’JOINT CUONOIlhtES bF tiiE SWELEMENTS OF EACM INITIAL UECTCIYGULAR ELENcNT INCLUDING-FICTITIOUS JOINTS 00 301 I=i.NELEH KSTX(I,=NOFCII,-I IF (INSEXlI.J,, 301.30i.lr01 CALCUL4Tt NO OF LINES KTVPE=YTVPElI,J, KGPUP=YGRUPlI,J~ KdOHsNGRUPlIrJ,/Z IF tKlOH-i, 305130Jr30* NOFLll*c.~((KTVPE-l)/2, GO TO 308 NOFL.l*L**(KTYPE-I, ‘0 TO 308 NOFL=l*C’*KTVPE AL=NOFL-I KSTVIIi=AL IF (1-i) 3i9r319r3LO IF (INDEXII-r,J)) 319.31 NOfLsYDfL-i

9.1105

C CALCULATE AdSCISAE VPREL OF EACH LINE OF JOINTS

Automatic triangular mesh generation in flat plates for finite elements

1GS

110

115

120

IL5

139

135

IWJ

l*S

150

155

109

IO>

179

175

100

185

190

1%

290

205

21J

319

311

312

‘E

3*3 J*‘

C

AD 3*5i

3*7

35i

ooc bui

37; J7‘

C

*a&

7oj

t

C C

1991

rY9 *97 492

u95

*Pi C

C C

501

C 61.

DO 310 IJK=l.NOFL If (1-l) 31is311*312 AI=lJI(-i GO TO 310 IF (lNDEX(l-leJ)b 3~lr3ll*lr07 Af.LM .._ __ YPRELIIm,=rI~enHrrIII/AL*swsr SU*ST=SUYsl*nmMA*Il~ INrROOUCE AGSCISAL VPREL INTO 1HE GENERAL MATRIX YC OF TKE MAIN COLUMN N6FCE=YOFC(I) 00 3+L JEsirNOFCL ISTEPai*(JE-I)~(MNOFC-I)/(NOFCt’r) Nd~G=lSTARIISTEPl*r NEND=lSTAKIISTEPl*NOFL 00 3C3 LMN=rrNOFL LI=YGEG*LMN-r YCILI.ISTEP)=YPREL(LWN) ISlARlISlEPI~ISlAWllSTEP~*NOFL OWIT iIClI~IOUS JOINTS ICOYS’Z*~~KrYPE~*I GO TO I3~~,3~6,3*7r3A9,34~~,KG~~P IF II-.) 3*7,3*7*3*>1 IF IINDEXII-r.JbB .bSr367r3QS DO 3% J<=L.ICONS.C IF IKGQJP-r) 35lrJblr352 NEYO=ISTARtJd) NLARX=ISTAR~JL~-KS~Y~I~ UO 3% KLJ=NLAYX.NCNG YCIKLJ.J~)=VC~KLJ*..JZ) ISlAP(JL~=ISTAU(JL~-1 GO TO 3-S 00 361 I*=i.lCONS*d IF IKG?UP--1 36~..10~.363 JUZi b0 TO 3ou Ju*ICDYS NEW=ISIAU(JU) 111~=151A~~J~~-IKSTlorilrlu JO 3b3 fiLJ=IIIurNtW YCIKLJ.J*~=YL(KLJ*I.J*) lSIA~IJCl=ISTA~~J4,-r FOQWLATE INDEX HAlnIX LLcGG OU *S, JF=ivMNUFC LdLGGII*,rJi)iISTA~(J,) CJYTIYJE *...*...*.......***...**....~...*.*.*.~.*....**~*..~...~*.....*... FOQYJLITE~FINAL 1-Y CULUMi4 ‘4AtlilCLb OF tHct MAIN CVLUHY CONSIDERLO KV4Q1. IF (J-r) aO,,bO~ra’Jd K”AQ=L 00 37~ LJ’KVARIHNO‘C ~ST4~=lSTARlLJ) 00 37i Ll=r+hSTAH LIJ=KE+Ef*L, AlLIJ)=XWILJ# YlLIJl=VClLI,LJ) KdCfs<~tf.IS,AH(LJ, ,QAYSfJt(M ‘YlTlAL 10 ilkA~ INOLA MAT*,X LaEGG “0 U+.L .,fF=<.MNOrC

ri0 783 Iff=r,HELI L~tGGIIFF.JFF)=LdrcGi(IFF,JFFk*NAbCO

FOQMULATE WIANGtiLAI( eLRM:kT hi&Y 00 1000 I=lrKELEM FOdNlJLATE COLUMN MATRIX NOFJO GF THE NUMBERS OF JOINTS OF THE SQUARE ELEMENT CONSIDLREO IF IINDEXiI;J)) lYOAriOOO1l90l KaEFE=O. DO 491 JF=l.MNOFC KNUWG=LGEGGlI*1,JFl-LBEGG(lrJF)rl IF II-11 *9Ct*92.*99 IF IKNUMG-11 *91,W7,492 IF lLBEGGIlrJF~-LGLGGII-l,JFII ~Ylr~91,692 00 *95 KF=ivKNUW KKF=KBEFEiKF NOFJOIKKFl=LaEGG~IrJFI*KF-i KB~FE=KBE~E+KtWJMG CONTINUE AUXILIARY VARIAGLES KTYPE=YTYPElI.J) KGRUP=YGRUPIl,J) KXST=KSTXlI) KVSTrKSTYII) NBOHXaKXSl/L NBOHY=KYSt/L NELGE=NEL NCOL=NOFCIIl NLINE=KSlY~i~~l NLINl=NLINE-r FOllMULATE TRIANGULAU ELEMENT AWAY FOR EACH RECTANGULAR ELEMENT NGRUP=I 00 SOi JJEi.KXST 00 SOi II~irKYST NEL=Z~llJJ-r~~KYS~*II~-I*N~L6E NE~INEL,,~~IJJ-AI~~KY5~*ll*II NL~INEL.cI’JJ~IKYSI*II*II NE~INEL.3J.IJJ-I)~IKYSTli)rllll NEL=NEL+, NEM~YEL~~l~JJ~o(YS~rl)rlI kE~lN~L,2l=JJ~IKYSi*l)rII*A NE~INEL~3l=lJJ-i~~~KYS7~ll*Il*~ NELYl=YEL-1 GO TO la7r.~ll~521.531r54Al~KG~UP NGWP=Z 00 S,* LLL=i,NGO”X


215

220

225

230

235

2-o

2*5

250

255

260

2b5

270

215

280

285

5i3 SI*

C SLI

523 SL*

SC9

‘ 53i

933 33*

NELi=r*4.(LLL_i).KYSl*NELUE NELZ=,*NELi NEL31iri.(~.LLL-AJrKYSl*N~LUE NEL%=,+NEL3 NE~~HEL~,L~=NEMINCL~~~) NEY(YELL.~I=NEWIN~L,.I) NEY(NELL.c)=NEHIN~L~.~) NE*IN~Lc,~I=NEM(N~L~~~) 00 5i3 JKLzi.3 00 513 IhL=NEL3 ,N~ZLHI NE*(IKL.JKL)=NEM(lKL*,.JKL~ NEL=NiL-r GO TO 57, NGRUP=3 0” 52. LLL=,.N8OHl NELi=Z.(C*LLL-r)TKYSl-~*N~LnE NELZ=NELA*l riL3=r.LLL.KYST-l*NELL(E NELs=YiL3*; NEY(NELc,L~=NEN(N~L..~I NEY(NEL3,~i=NEHINtLr.~J 00 523 JKL-r.3

NE*IVEL*,,JKL)=O. DO 5,3 IKL=NELZ,NtL

NEY~IKL,JKL~=NEM~IKLII,JLLI NEL=NEL-r DO 529 LKKJ=,t,NCOL,Z JOIYT=LKKJ.~~*~.*~KTYPE-I~I*~LKKJ-I~/~ KOIF.KaEFE-JOINT’I O” 529 KLOH=r,KDIF KIYO=KBEFE+L-KLOR NOFJO~KINDl=NOFJO~KIND-1) GO TO 57, NGIIUP=*

00 534 LLLcivNBOHY NELl-u*LLL-3*NtL& NEL<=NELA*, NEL3=NEL,*Z NEL*=YEL,+3 NE~INEL,,J)=NENIN~LL,~) NE~~YEL~,AI=NE~~N~L~~~I NE*(YELL.~)=NEWINLL~~)) 00 533 JKL=,r3 UO 533 IKL=NEL3rNtLMI NE~(IKL.JKLI=NEM(IKL*I,JKLI NEL=NEL-r UO 533 LKKJ’rrNLINArZ JOIYT=l*LKKJ*~LKKJ-,)/Z KOIF=KBEFE-JOINT*1 00 S39 KLOR=rrKDIF KIW=K3EFE+c-KLOH

539 NOFJO~<INDI=NOFJO~KIND-i) GO TO 57i

C

54.

NGRUP=S DO 5.6 LLL=ivNBOHY NLLI=~.KYST .(I(.%57 -i)*r*LLL-3*NELdE NELZ=NELl*l NEL3=NELi+2 NELrzNELi*3 NEM(NELL,~~=NEH(NEL*.~~ NEM(NEL3,c)=NEM(NELr.2) UO 5.3 JKLzi.3 NEWINEL*irJKLl=O. UO 5.3 IKLsNEL4rNtL

563 NE~~IKL~rJKLl=NEII~IKLllrJ*L) 5+a NELZNEL-i

DO 569 LKKJ.ivNLINIeZ JOINT=Ii.*KlYPE~.tr*2..KTYPEi.LKKJ*I KOIF=KEEFe-~OINTil*~LaicJ;ll/2 00 5~9 KLOR=,.KOIF KIYD=KBEFE+L-KLPRIILKKJ-AI/2

5.9 NOFJO~KINDl=NOFJO~KIND-I) 37. 1NtLcJE=YtLm’.

L F”‘MU).PT~ “cuEdaL I~IANGuLP* tiCHtNT AWAY J” s,, eKJ=r.3 ~‘0 5,0 %KI=kLGt.NcL KA<I=vc~IKK,.KK.J,

,LJ “EY(<~,.KKJ,=NOFJU(K~~5, ivu0 LU\ITI\1dt rlO0 cour1\rllt

L “.Y...*.*Y..*...Y.VC.~**.****.**.**.***~.*.~.**.**.**......*.*.***

i .uu..s....u.....i*.u**...~.*.~~~~~~..*.....****.....~.***..*..~..*

b PI(,VT ?EScJLI,

_._ ., L)O 30,. J=r.NELEH dU 30,~ I’rrYELE”

jUar PHIUT 3UiJrJ~I~IN~th~I~Jl~NGRU~~I~J~~NTYPE~I~JI 3UI., FO~*~T(,~A,.~..~(Z*.IM.~~.,H*II

WIYI 3OIS iulr FU<YPT(iOXe*b(IH-),

Ihd=*Y3*Ht_LL*., Ii (IVS.YtLtH-sOl JOI~.30,Ir73bc

730‘ ,‘HIYT 30,” ,tiuv=i

3dia CO47IYJc PH,YT 30,s

3u.5 F”+MA, (rHI.rOXt,ltiSTLPS I-OIt4kCTI”N, PHINT 30~7~(dn~AX(I,.I=irNtLcMr

c 3Oi7 FU4YPT (*OX* Fi0.a)

x5

330

335

3*s

34s

350

355

360

365

37u

375

36U

365

390

Automatic triangular mesh generation in tlat plates for finite elements

3039

3uro

30*r

3061 3062

3033 3036

FOR~AlIrOX~SOliH~~r4*(ln+r~ PRINT 30*0 FOR~AT~IOX.~H~,~~~X.~~~.~OX.~~~.IOX~~~+~~ PRINT 30-t FORMA, lrOX,,n~13tzX,7~JO1NT l .5X.l~X~4X.l~~.5XtlHY.4X,l~~~~ PUIYT 30-o PRIM1 3039 KPcR3=KmEF/3 LBiFL=K6EF-3AKPER3 IF lLBEFL1 3001r306<.3081 KPER3*KPER3*, DO 3032 l=I,KPER3 ,i=l ii=il*KPER3 13=lC*KPER3 IF (13-KGEF 1 3033*30331303* PRIYT 3036,1~~Xi111.Y~111;12.X~121,V~I2~.13.X113~.V1131 FORMA1 ~IOX~An~r3lLX~I~r2X,l~~~2X,Fb.2,2X.l~~.2X.Fb.2,2X,l~~~l “0 TO 3032

303” 3036

WIN1 3036.1,rX~llJ.Y~lI,~l2.X~l~~rY~l2I F”kkAT ~rOX.~~~.L~LX,14r2X.l~~.~X.Fb.2,~X~l~~.2X.Fb.2.2X.l~o~,

i6X~IH*~r~rOX~Itl*)l 3032 CUNTINJE

PHINI 303Y

PRIYT 3u39

PHlYT 30% 3~8% FOMAT (rn;/4OX,ZUnGEDMET*ICAL DATA OF tLtHENTS/lH r39X,26(lk-l//l

PHI*1 3053 3C53 FOKqAT (iOx,a3(ln~)**6(i~*,,

PRlhlT 3055 3055 FOSYAT (lUX~~n*~9XtiH~rbeXr~M. .YX.,H*,2OH ELtHENl CENTER OF ‘1

PRIYT 30% 3O5.e FOL(Wl (rOX.,l”* ct.EMENl *.P~X.CWE L E ‘4 E N T N D 0 E 5.221.

1i~~~lOX~r~~~~Xtl~~GRAVlTY CODUVS13X~lWl PRINT 3JSb

3056 FORMAT ~rOX~rn~t9X170~1~~l~~OX~~~~~19x~l~~l PRIYT 3057

3U57 FOMAT ~iOX~iH*~2X~6HNUl4UiR ~rdX~~~lrSX~2~XltbX~2HVl~3X~lH~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~3X~*HAREAt3X~iH~r3X~2HXC~6X~~~VC~UX~l~~l

PRINT 3053 lNJ=V DO 306i K=lrNEL NI=YEM(K.II N<=YEMIK.Ll N3=YE*(K.3i ARSA=O.~‘~X~k2l.~Y~N3~-Y~Nlll*X~Nll.~Y~N2l-Y~N3lI*X~N3l’~Y~Nl~-

*YIYZ7ll XCSI(X(Y,)*X(NL)*X(N~~I/~. YCG=~Y~Nri~YlNZ~*V~N3ll/3. PL)lNT 30b*.KrN~tX(NilrY(Nl~~N2~X(N2)~Y~N2~~N3~XlN3t~Y~~3l~AkEA~

rXCGrYC3 3r)bu FOMAT ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

iik*rd(F7.3*2XerH*ll IkD~lND*i IF (IYD-501 306Ar3Obl.4566

4566 PRINT 3053 PRINT 3052 PRINT 3053 PRIYT 3055 PRINT 3054 PRINT 3056 PRINT 3057 PKINT 3053 lNO=9

3Obl COYTINVE PRINT 3053

467 CONTINUE END

SYMBOLIC REFLRENCE HAP (RI,)

ENTkV POINTS 2106 OI’JIOE

VARlA8LES 6201 Al 6174 AN

10152 BHkAX b157 I 6233 IFF 6224 Ill4 6261 IKL

1;:;: ;gEx

6211 ISTEP 6277 12 6222 I* 6210 JE 6232 JFF 6250 JJ 6263 JOINT 6223 J4 6165 KAPA 6166 KLCF bZO* KklOk 6236 KF 6170 KHELP 6237 KKF 6270 KKJ 6265 KLUR 6274 KPEK3

10520 KSTX 6166 KTLN blb7 KUNIT 6LOO KX5T 6175 L

1;;:; LGtGG LIJ

6262 LKKJ

SN TYPi REAL REAL REAL INItGER INTEGER INTEGER INTEGER INTEGER INlEGEH INTEGER INttGER

:s:::: INTEGER INTEGER 1NItGEH INTEGER INTEGER INiEGtY INTEL&R INTEGER INtE6ER INTEGER INTEGER INtC6ER INTtGER INTEGER lNti6EH INtEGLR INTtGtR INiiGER INTEGER INlECER INlEGER

RELOCATION

ARRAY

4RRAY

ARRAY

AWAY

6206 6305

IO164 6216 6251 6,177 6273 6171

10176 6276 6300 6160 6225 62OO 6260 6217 6301 bLt2 6232 6264 b2U.l 6266 6611 6221 6235 6230

1053L cl173 b2Lb 6241 6275 6.211 62d7 6253

AL AREA WIMAY ICONS II IJK IND INTEN ISTAR II 13 J JF JIK JKL J2 K KARS K&FE KOIF

KIN0 KKI KLJ KNUMB KSTAR KSTY KTYPE KVAR KIST LOEFL Ll LJ LLL

REAL REAL REAL INTEGER INTEGER INTEGER INTEGER INTEGER IYTEGER IYTEGEY INTEGER INTEGER INTEGER INlEGEk INTEGLR INTEGER IYTEGEk INTEGER INTEGEk INTEGER IYTEGER INTEGER IWEGER IYTEGEk IkTEGER INTEGER IYTEGEK INTEGER INTEGER IYTEGER INTEGER IYTEGER INTEGER IYTEGLR

ARRAY

ARRAY

ARRAV

CAS Vol. II No. 5-C


o21r LMN 61% MEL1 b172 MNOFC 6212 NdcG oi*3 NHOW 61b2 NLL bI54 NELEH 6252 NkLHi 6255 NtLi 62>7 NkL4 6213 NENO b22U NLAdX 02r7 NLlNl 6207 NOfCE bZU5 NDfL odU2 Nl 63~4 N3

VAHIAdLES b16.I SJHSX 63ub xC6 7bs2 Y h307 YCCI

f,Lt NPHLZ

0 INPUT

STATEMENT LABELS

INTEGER INTEGER INiE6W INiEbER INttbER INktiER lNit6tH INttbER INiE6EI) INtEGEH lNiE6EH IhTtGEH INTt_i.W INTEGER INtcci* INttbER INtEuEr?

SN TYPc tiEAL 17EAL REAL rltAL

5%7 I 4227 111 4214 114 4206 131 4225 134 4244 2~3

0 2Ub 4342 3"* *375 310 4336 319

0 342 4450 346

0 351 0 361

45u5 364 0 372 u 452 u 495 0 501 " 51* 0 523

5051 531 0 539 u 544

4540 601 5222 1000

u Ii05 5550 3910 56U7 3013 MC4 3Ulb

0 3033 5b57 3837 5676 3040 6003 3053 b037 3656 61~6 3b64

0 3451 0 7362

FM,

INACTIVE

INACTIVE FMT FMT fHT

INACTIVE FYT FM1 FM1 FM1 FM1

INACTIVE INACTIVE

MODt

FM1

61>5 MELEM b1¶3 HNLIZ 6161 NABCO 62~2 N1)OHX b&5 NCOL bL~4 NELtlE 6207 NELGE b&4 NELI 6256 NEL3

LO544 NEM 6466 NGUUP b246 NLINE 6644 NOFC

ILId NOFJO 63L2 YTYPE b303 N2 6LI)Z 5unst

RcLOCA!lVN 7332 b310

ARRAY 6626 6032

co*, OUTPUT FM,

>51b 2 0 1.2

~176 I‘, 4Cli r3c uz54 LUI 4L50 CO4 453i JO1

0 305 4370 3ii

0 320 0 3*3

*455 347 **I, 352

0 3C2 0 365 0 391

*652 491 0 *97

4735 51i 0 520 0 524 0 533

5124 541 0 5*9 0 602 0 1100 0 1AU7

5C60 3011 z.616 3014 5637 3017 5352 3034 57% 3036 5705 3041 6022 3654 0046 3057

0 3081 0 4567

FMT INACTIVE

INACTIVE

INACTIVE

INACTIVE

INACTIVE

INACTIVE

INACTIVE

FHT FHT

FM1 FM1 FMT FMT

INACTIVE

x XPH YC YPREL

STATISTICS PtiOCi?AcI LENGTH bOb38 3123 WFFEH LENGTH *IOdB 2AA4

rltCtANGJLA3 ELEMENT OAIA ________________-____---

.**....*.**...*..**....*.***...*.****.*.*.**.*

* COLiPiN i iINt l ELE*ENi iELEM<NI l EiEMiNt 6 II . * * GROW l 1YPi l

* J * I * INDEX * G l I( 6 ..*.......**..*.****.*.*.**.....**.*.*..*.**** . i'A' 3 * i + '3 i * i' L* *

l <* ~ i* 3' c:* ; l l

. i* ~I) ,* I l f l

* ;* 3. ,* ** ,* ______~_~__~__~~~~~__~~__~_~____~~~~~~__~~__~~

* “ i' 3c* >* i* l c* C* L' * L* * ‘(1 3 0 2. * :. I* * ii ** IL* i+ . . 2' 3* c* I * : * _____________________-___-_______-_-____-___-_

INtiGER IHtEbtR IN1Ebt.R IYtEbER IYttUEU IYtEbER IVtEGtH IYtE6tR INttGtR IVtEGtR INTLbiR INtEGtR INtt‘ER IHTEGEM INTEbER IXTEXR RcAL

REAL PEAL REAL RECIL

ARRAY ARRA" ARRAY

AHRA" ARRAY AWAY ARRAY

55*3 3 0 113 0 122

4222 133 0 202 0 205

4336 303 4345 308 4372 312

0 341 4523 345 4476 349

0 353 4503 363

0 371 0 451

4635 492 0 '499 0 513

4774 521 0 529 0 534 0 543

5205 571 0 lb3 0 1101 0 1901 0 3012

5627 3015 5363 3032 5735 3036 5670 3039 5771 3052 6011 3055 5476 3061 5326 3082

0 4568

FM1 INACTIVE INACTIVE

INACTIVE INACTIVE

INACTIVE

INACTIVE INACTIVE

ff4T

FWT FM1 FMT FM1

INACTIVE

Automatic triangular mesh generation in tlat plates for finite elements 459

__~~~~_~_~~___~__~~_~~~~~~~~_~~_~~~~~~~~_~_~~~ * 6 * I * 3. I* 3 l

* b* LO a * 6 4 3. A. I l 2 l

3. r* l 6 a . iL l z l l

* c.* ;. LO I* i + ~~~__~~_~~__~~_~~____~_~_~~~_~__~~~~~~~~_~____

STEPS X-DIRECTION 1.000 1.000 i-000 1.000 i.000 1.000

STEPS Y-DLRECTION 1.000 1.000 1.000 1.006 1.000

l

l

l

l

.

l

.

.

.

.

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1.20 1.20 1.60 1.4.6 2.00 2.00 2.00 2.40 2.40 2.40 2.40 2.60 2.60 2.00 2.80 2.60 2.60 2.80 2.80

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2.083 . 2.250 l

2.4~7 l

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1.067 l

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119 l 60 3.00 .80 l 81 3.00 1.20 l b6 2.50 1.20 l .lOOO l 2.633 . 1.067 l

120 l w 2.50 I.20 l 81 3.00 l.LO l b9 2.50 1.60 l .lOOO l 2.667 l 1.333 l

121 * L)i 3.00 I.20 l 62 3.00 1.60 * 69 2.50 1.60 l .I000 l 2.033 l 1.467 l

122 l bY d.50 1.60 . 74 2.75 2.00 l 70 2.50 2.00 l

123 l b9 2.50 A.bO l 8L 3.00 1.60 l 74 2.75 2.00 l

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GEOMtTRlCAL DATA OF tLEMENTS ___~~__~~~_~~~_~~__~~~~~_~~~

. . . . . ..*.*****.....*...**.****.**.*...*..**.*......***.****.*...*..*.*.**...**..*.*.**..**....*.*.**..****.*..* l -‘i--

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. 127 ‘* -7.

. 128 . n3

. 129 l 71

. 130 l 75

. 131 . 7L

. 132 l 7b

. 133 l 75

. 134 . n*

. 135 * 7b

. 134 * 78

. 131 * 78

. 13b l 87 l 139 . 80 . 140 * 90 . 141 . 07 . 162 * 91 l 143 l 79 . 144 * 81 . 145 l 88 . 146 * 92 . 147 . 80 . 14b . 89 l 149 * 8Y

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L.75 c.00’+ ‘83 3.00 2.00 + 84 2.50 2.*0 . 75 2.75 2.40 l 76 2.50 d.60 l 76 2.75 d.60 l 77 i.75 d.40 l B4

3.00 2.*0 . MS 2.15 L.60 . 65

3.00 0.00 . 86 3.00 0.00 . a7 3.25 .20 . aa 3.25 0.00 . 90 3.50 0.00 . 91 3.25 .LO l 91 3.50 .20 * 92 3.00 .*o l en 3.25 .co l OY 3.25 .*o l 92 3.50 .*o l 93

3.00 .80 l 10 3.25 .80 l 9* 3.25 .80 . 93

3.09 r.20 l 94 3.50 r.10 l 95 3.00 A.60 l 95 3.50 i-60 l 96 3.00 2.00 l 90 3.50 z.00 . 91 3.09 L.40 l 97 3.50 d.40 . 98 3e5V 0.00 l 99 3.15 0.00 l 100 3.50 .LO l 100 3.75 .20 . A01 3.75 0.00 l 103 *.oo 0.00 . ,o* 3.75 .20 l i04

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3;lifJ 3.00 2.75 2.15 2.7> 2.7) 3.0-o 3.00 3.00 3.25

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2.00 . 75 2.40 . 75 2.40 l 72 2.60 . 12 2.60 l 13 2.80 . 73 2.40 l 76

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1.20 l bl .80 . 94

1.20 . a2 i.60 * 82 1.60 . 13 2.00 * b3 2.00 . 84 2.40 . 84 2.40 . n5 2.80 * 85 0.00 . 91

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.40 . Y2 0.00 . 100

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.20 . 101 ,*o l 101 .40 . 93 .60 . 93

2.75 2.75 2.50 2.50 2.50 2.50 2.75 2.15 2.75 3.25 3.00 3.00 3.25 3.25 3.25 3.25 3.00 3.00 3.25 3.25 3.00 3.00 3.50 3.00 3.00 3.00 3.00 3.00 3.00 3.00 3.00 3.50 3.50 3.50 3.50 3.75 3.75 3.75 3.75

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2.40 . 2.40 . 2.60 . 2.60 . 2.80 . 2.80 . 2.60 . 2.60 . 2.80 .

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GtOMtTRlCAL OAIA OF tLEMENTS ________________---_-____-_-

*.............*..........*..........*.............*..*..**.*.............*............*.......*....*****....... . -.-. . . . . ELENENT CENTER Of .

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Y2 . 3 x3 Y3 . AREA . XC YC l

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. 169 ‘* 105 ‘a.00 .40 l 106‘ ..bO- .no l 102 3.75 .80 . .0500 . 3.Y17 . .667 l

. 170 * 93 3.50 .L(O . iO2 3.7> .80 l 94 3.50 1.20 . .0500 .933 .

. 171 . ror 3.75 .80 . 107 4.00 A.20 . 94 3.50 1.20 . .lOOO : :q:; .

. 1.067 . . 172 . 1ui 3.75 .80 . 10b 4.00 .a0 . 101 4.00 1.20 . .0500 ’ . 3.017 . .933 . . 173 . Y* 3.50 I.LO . r07 4.00 1.20 l 95 3.50 1.60 . .lOOO . 3.667 . 1.333 . l 114 * iO7 4.00 iad0 . rod 4.00

f’Z l 95 3.50 1.60 l *IO00 . 3.833 . 1.467 .

. 175 . 9a 3.50 r.60 . 108 4.00 . 96 3.50 2.00 . .lOOO . 3.667 . 1.733 . . 176 * 1vn 4.00 r.60 . ~09 4.00 2:oo . 96 3.50 2.00 . .I000 + 3.833 l 1.867 . . 177 . Yb 3.50 Z.OO l r09 4.0” 2.00 l 97 3.50 2.40 * .I000 . 3.661 . 2.133 . . ild * IO9 u.00 c.00 . hi0 4.00 2.40 l 97 3.50 2.40 + .I000 . 3.833 . 2.267 l

* 174 . Y7 3.50 L.40 l ii0 *.uu 2.40 . 98 3.50 2.80 + .lOOO . 3.667 . 2.533 l

. 180 . 110 *.oo L.40 . 111 4.00 2.80 l 9b 3.50 2.80 l *IO00 . 3.633 . 2.667 .

.**....*.***.*....*........*...~...*....*.**.*...***............**.....*.*..*.*...............*..........*.....

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Automatic triangular mesh generation in flat plates for finite elements