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Automatic triangular mesh generation in flat plates for finite elements
cmQvlrrs & .wlutlvc8 Vol. I I. pp. 43964 Pcrgama Press Ltd.. 1980. Printed
inGrcac Britain
AUTOMATIC TRIANGULAR MESH GENERATION IN FLAT PLATES FOR FINITE ELEMENTS G. D. STEFANOU Schoolof Engineering,Universityof Patras, Patras, Greece
and K. SYRMAKEZIS Civil Engineering Department,
The National Technical University
of Athens, Greece
(Received 3 August 1978;receivedfor publication28 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
‘Y
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 geometricalproperties of the elements. The solution of this problem will be useful for the application of numericalapproximate methods, such as ,finiteelements, 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-ordinatesof the orthogonal system as
I
I I
I
I I I
:pn_ I I
I
---
---
A---+
---
----t---bt---* I
I
0,
I
I/ j j
1
1
/
I I
’ ’
1 X
Fig. 2. Discontinuous plate divided into regions. X’=XtyCOSQ I
y’=ysiny,
1
(1)
The investigationis 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 gradingis taking place graduallyfrom 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
2.SuBlWEXoN PROCJCDIJRE OF MAINGROWG= I To divide a hyper-elementinto triangularelements, 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 hyperelement is divided.The elementsof the group are chosen to have an area, ET, of their trianglesequal to ET=$E
cl0
0
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 programmeinto “triangularelements”
(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 followingrules: (a) The t&g&u
x
(0)
Fig.1. Discontinuous plates. 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 Ax, Ay (side Ax is divided first), by vertical and horizontal lines and by drawing the diagonals of the parallelograms so formed as shown in Fig. 3(b) The diagonals have always an ascending slope from right to left. (c)The numbering of the 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
the numbering of the triangular elements. The number of nodesper column jC,and line jr. for each hyper-element is
defined by jC=lt2’k-‘)‘2 or j,=lt2
for
(k-2)‘2 for
k=l,3,5
,...
k = 2,4,6,. .
(3)
Automatictriangularmesh generationin flat plates for finite elements
441
i
(i*l 1 Nodal he -----------
i Nodal --~-----~--
line
j (se+11
_-_
(j-l)(r,*l)+i
+
i
Fig. 4.
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.
and
or
jL=l+2(k-‘)‘2
for
k=l , 395,**-
iL = 1 t 2”“’
for
k=2,4,6,...]
I
(4)
Let SC= jC - 1 (5)
and SL = jL+r 1
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, = SCx 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)titl,
j(S,tl)ti, (j-l)(S,tl)titl.
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-
3. !WBDMSION
and
2[(j - l)S, t jl.
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:
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
(6)
where (Ak(r/lO) = the real part of 1llOth of index r. The numbers of nodes per column jC and line jL are
The two triangular elements in the sub-element have element numbers 2[(j - l)S, t i)]-’
lOAk(r/lO)]
n~o=lth(lt2k-‘)
(9)
for A=l,3,5
,...
(10)
G. D. STEFANOUand
442
K.
SYRMAKEZIS
G=t? lg,T ~34
1
l\
3
2
1
3
Ax14
Fig. 5.
Four typical
,L
AxH
,”
1’
Ax14
X
The numbering of fictitious node j0 is again given by eqn (8) and their serial numbers nio by expression fljO=A(l t2’-‘)
jt = 1 t2k-'(3+2k)
(11)
j,=j,-jo=1+2k(1+2k-‘)
(12)
r, = 22k- 2’-‘.
(13)
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
,L
elements of G = 2.
The total of nodes jl the real number of node jl and the number of elements in hyper-element, become respectively
The numbering of triangular elements is performed as for group G = 1, and the modifications required for making 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
Ax14
(16)
for h=2,4,6 ,...
(17)
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 k = ; Ak(~/l0).
(14)
The number of nodes per column jC and line jL is given by jc = 1 t2k-’ jL = 1 t 2”.
1
(15)
Iljo = A
(19)
for A=2,4,6,...
C3-N
j, =(1+2k)z
(21)
Automatictriangularmesh generationin tIatplates for finiteelements
443
G=3 el , ~21
e2,
e3,r=65
7=43
e. ,T=67
Fig. 6. Four typical elements of G = 3.
j,=j,-/O=(1t2*)2-2*-’
(22)
t. = 2*-‘[2k+2 - 11.
(23)
and
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
A=2,4,6,...
(26)
for
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 FORTRANlanguageand was originallywritten for the IBM 1620computer of the National Technical University of Athens. Later, the programme was modified for the CDC 3u)o computer unit of “Democritos” NuclearResearchCentre, Athens.The programme is dividedinto three mainparts: (a) Preparationof data,(b) Runningof the programmeand (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-elementsand 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 numberof columns of hyperclements along the length of the beam NELEM &); (b) The numberof lines of hyperclemeds alongthe height of the beam MELEM (SC); (c) The lengths of hyper-elementsof the variouscohmmsof the elements in the beam are BHMAX (I) (A& (d) The heights of hyper-elementsare BHMY(1)(A,,);awl(e) The matrixof indicesINDEX(IJ), (i.e. indicesr), whichhas dimensions (MELEMx NELEM), &fines the manner in which each
G. D. STEFANOU and K.
444 el,
SYRMAKEZIS
~=23
e2, r=46
Fig. 7. Four typical elements of G = 4.
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) _ . INDEX, first, per column __ .. . of matrix and then per line, is punched. Matrix INDEX is constructed according to the desired
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
for
i=odd,
aij=i
for
i=even
for i=j: Ui,=i=j
(27)
for i > j: ai,=j
for
j=odd,
a,=lljtl
for j=even.
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
(28)
Automatic triangular meshgenerationin flat plates for finite elements
44s
G=!5
*
Ax/2
1 !x 1
Ax12
1 AX
X’ 1 L 1
e4, t=96
ea.T=76
Fig. 8. Four typical elements of G = 5.
Sector C. (a) Cij=aijt9
Cii=aj-9
for i>j for i
and j=odd 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 0 1 . ..... 0 00 . . . . . . . 0 0 [E] = 0 0 ....... 1 : 0 ...................................... 0 1 ....... 0 0 0 0 ....... 0 0 0 I . I I 0”
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 perform, 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.
D.
STEFANOU
SYRMAKEZIS
and K.
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
1st element
+
2nd element
-8
r
0 0 0 0.20 0.20 0.20 -_ + 0.40 ..............0.40 ........,.. ...0.40 ...... I + 0.80 ..............*0.80 ..............0.80 ...... I f
3rd element
-+jj
f;
........13 .......
4th element 2.00
G-l
G-l
Gm4
G=l
G=l
0
( = lYC]
kf 2.00
G.4
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 whichthefictitious Fig. 9. Typicalcolumns 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 necessarily all lying on the MNOFC columns. Consequently, the serial numbers of columns where the element under consideration has nodes, are
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 subelement (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
ZSTEP=lt(A-I)~~~_~ll~MNOFC
for
cr__________________-._-----__--_---_____+
A
=12, 9.‘.
1, 7
1
5
%
3
032 ,
1
23
3
46
5
67
7
1
;
6
-6
64
3
22
1
23
3
46
5
6
6
1
I
23
3
465
4
4
4
3
‘z - _l__
4
4
4 5656I4
; 34 12
24 2
24 2
32 2
1 I
I 1
31 I
24
24
24
, 1
24 r;-2 I 12
I2
I2
I2
II
I
I
2l
I 21 ml
2,
I
I
6
166565 /
P
“’
, , ;
4
12 2l
I
2
2
3
32
1111 1
1
1
1
2
2
23
3
Fig. 10. Rectangular mesh of beam.
n
,I
2
2221221
(
”
4’
4’
4
‘
.I4
-+.A
50
.166666 0
E
!6
6
I
221
112 8
__I__‘_)__)
34 ,1. I
‘I
1 I I 43 rl___u___r__..__l_Z’__d ‘1
221
643
22
5563
23
32I
I
2
212 I
2
2
‘2
42
43
2/ I 43 1
4
4
4
1
23
3
4
1
21
1
465
1
23
2
46
Fig. 11. Auxiliary array.
6666; 5
6
6!
1 ,
441
Automatic triangular mesh generation in flat plates for finite elements
Fig. 12. Flow chart diagram.
Note:
D = 1st digit of INDEX; 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
M = 2nd digit of INDEX:
P = odd
number:
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.OO0.00 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.OO0.20 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)
Hence, matrix LBEGG of the example. after the final numbering of nodes becomes
[LBEGG] =
:
I2 10
I6 I4
4 5 6 7
I3 13 I3 I3
I7 18 I9 20
;
I3 13
::.
For each column of matrix LBEGG we keep the number NABCO LBEGG(1, MNOFC) - I. After calculating 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 =
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
(b)
(a) Fig. 13.
448
G.
D. STEFANOU and K.
SYRMAKEZIS
Fig. 14.
triangular elements, three new triangular Zl,Z:,Z;, are created as shown in Fig. 13(b).
elements
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 Node Node Element I;: Node Node Node Element Z;: Node Node Node
1 corresponds 2 corresponds 3 corresponds 1 corresponds 2 corresponds 3 corresponds 1 corresponds 2 corresponds 3 corresponds
to node I of elemtnt II to node 2 of element Zz to node 3 of element II to node 1 of elements II to node 1 of element Z4 to node 3 of element I, to node 1 of element Is to node 2 of element Z4 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+ Z4+
4 :
: 6
2 33
4 I 5 8
:
5 5 6 6
8 9
7 10 II 10 8 11 11 12 10 13 13 14 11 14 14 ,.............. .15 ...........
8 8 z I1 11 12 ..12 .
The above matrix is then transformed into 14line matrix as follows
.............................*.. I 5 2 2 5 3 5 6 3 1 I ‘A=1 7 8 : 6 : ! 6 7 11 8 8 11 11 12 ; 7 13 II pA=l 13 I4 II 11 14 I2 I4 I5 I2 ................................ .........................*.*.... ........,............. ........
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= ZJt 1
-m (a)
Fig. 15. Triangular elements for G = 3.
Automatic triangular mesh generation in flat plates for finite elements
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 Node Node Element I;: Node Node Node Element Z;: Node Node Node
1 corresponds 2 corresponds 3 corresponds 1 corresponds 2 corresponds 3 corresponds 1 corresponds 2 corresponds 3 corresponds
to to to to to to to to to
node node node node node node node node node
1 of 2 of 3 of 1 of 2 of 3 of 1 of 2 of 3 of
element element element element element element element element element
II I, I, Z, L I2 Z, b L.
449
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 Z4is 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
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 Node 2 corresponds Node 3 corresponds Element ii: Node 1 corresponds Node 2 corresponds Node 3 corresponds
to node 1 of element to node 2 of element to node 2 of element to node 1 of element to node 2 of element to node 3 of element
I, I, b II Z2 b
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, .ZLare 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 Node Node Element I;: Node Node Node Element Z;: Node Node Node
1 corresponds 2 corresponds 3 corresponds 1 corresponds 2 corresponds 3 corresponds 1 corresponds 2 corresponds 3 corresponds
to node 1 of element I, to node 2 of element I, to node 2 of element I, to node 1 of element Zz to node 2 of element Z4 to node 3 of element Zz to node 1 of element b to node 2 of element Z, 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
0 2
3
3
14
13
13
1
3'
3
2
/ I2
I
12
11
2’
1
(a)
(PI
Fig. 16. Triangular element for G = 4. CM
Vol. I I No. 5-F
(a>
1;
\1
fa
(PI
Fig. 17. Triangular elements for G = 5.
G.
D.
STEFANOU
and K.
SYRMAKEZIS
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.
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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.
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Acknowledgements-‘his
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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'
l
2.t:t 2.2‘7 2.9ll Z.*b'
l
l
. . . . . . l l I
l .
l . . . . .
l l l
l l l
ContinuumMechanics.McGraw-Hill, London (1%7). 2. R. W. Clough. The finite element method in plane stress analysis. Proc. ASCE 2nd Conf. Efectronic Camp~tatio~, Pit&burg, Pennsylvania (Sept. t%O). 3. D. D. McCracken and W. S. Dam, ~urn~ca~ Methods and Fortran Progmmming.Wiley, New York (I%@. 4. W. Weaver. ComouterPmemms for Stmctuml Analvsis. Van
454
G. D.
STEFANOU
and K.
SYRMAKEZIS
APPENDIX 1
Input listing-results An input listing of the computer
programme PROGRAN
s C
i
3
15
30
3,
*(i
*5
DIVIOEIINPUT.OUTPUT,
NELl.NELEN*l READ I~IllNDEXlI.J,~I=l.NELEM~~J=l~NELEM, FOWAl ILOIwl RELD 3rlBHN4VlI,rI=l.NELEN, REID ~~I~NM~XII,~~=I.NELEN, FDtiNAT llOFL).3,
C
*....*..**..*.***.******..*.....**..*.*......******.**.**.....*.** ~iii~~~i~iri~i..i~.i~.~~.~*~~~~~~~~~~..~.~*.***.*******.****.*.**.
;
CALCljiAtE’GtOMETRIcAL~DAt4 WFT~EOINIT14L WECTANGULAR s
c
LO
sample problem are included in this section (see input and tables Tl, T2. T3).
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 READ AND PRINT DATA REAO B.NELEN.NELEM z FOHN4T (214)
10
15
and results of
FOR’tiit‘SUBELENENTS MESH
OF
TNE
NE,.=o swsx=o KBiF=O 00 1100 J=irNELEN CALCULATE N(lRUP AND NTVPE FOR LAW RECT4NGULAR ELENENT OF THE E 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 t*****.***.****.****.*..*..***.*...*.****.*~*.*~*~**~*.**~*.**.**~ L 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,=. ISrARIL,=O 3r. 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 CALCULATE’JOINT CUONOIlhtES bF tiiE SWELEMENTS OF EACM INITIAL c UECTCIYGULAR ELENcNT INCLUDING-FICTITIOUS JOINTS 00 301 I=i.NELEH KSTX(I,=NOFCII,-I IF (INSEXlI.J,, 301.30i.lr01 C CALCUL4Tt NO OF LINES Ii01 KTVPE=YTVPElI,J, KGPUP=YGRUPlI,J~ KdOHsNGRUPlIrJ,/Z IF tKlOH-i, 305130Jr30* 305 NOFLll*c.~((KTVPE-l)/2, GO TO 308 303 NOFL.l*L**(KTYPE-I, ‘0 TO 308 JUI. NOFL=l*C’*KTVPE 308 AL=NOFL-I KSTVIIi=AL IF (1-i) 3i9r319r3LO 320 IF (INDEXII-r,J)) 319.31 9.1105 ii05 NOfLsYDfL-i C CALCULATE AdSCISAE VPREL OF EACH LINE OF JOINTS
.~ _. _. ..*.........**.****.***...*.**.~**.*.*********~**.*.*....*.......*
d5
90
YS
100
COLUMNS
Automatic triangular mesh generation in flat plates for finite elements
1GS
110
115
120
IL5
139
135
IWJ
l*S
150
155
109
319 DO 310 IJK=l.NOFL If (1-l) 31is311*312 311 AI=lJI(-i GO TO 310 312 IF (lNDEX(l-leJ)b 3~lr3ll*lr07 Af.LM ‘E 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 3*3 YCILI.ISTEP)=YPREL(LWN) ISlARlISlEPI~ISlAWllSTEP~*NOFL J*‘ OWIT iIClI~IOUS JOINTS C ICOYS’Z*~~KrYPE~*I GO TO I3~~,3~6,3*7r3A9,34~~,KG~~P AD IF II-.) 3*7,3*7*3*>1 IF IINDEXII-r.JbB .bSr367r3QS 3*5i 3*7 DO 3% J<=L.ICONS.C IF IKGQJP-r) 35lrJblr352 35i 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 ooc K”AQ=L bui 00 37~ LJ’KVARIHNO‘C ~ST4~=lSTARlLJ) 00 37i Ll=r+hSTAH LIJ=KE+Ef*L, AlLIJ)=XWILJ# 37; YlLIJl=VClLI,LJ) J7‘ KdCfs<~tf.IS,AH(LJ, ,QAYSfJt(M ‘YlTlAL 10 ilkA~ INOLA MAT*,X LaEGG C “0 U+.L .,fF=<.MNOrC
.._ __
IO>
*a& 7oj
ri0 783 Iff=r,HELI L~tGGIIFF.JFF)=LdrcGi(IFF,JFFk*NAbCO
179 t C C 175 1991
100
rY9 *97 492 u95
185
*Pi C
190
1% C C 290
205
501
21J C
61.
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
456
G. D. STEFANOUand K. SYRMAKEZIS
215
220
5i3 SI*
225 C
SLI 230
235
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
523 SL* 2-o
SC9
2*5 ‘
53i 250
255
933 33* 260
2b5
539 C 54.
270
215
563 5+a
280
285
5.9 37. L
,LJ ivu0 rlO0 L i b
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 NOFJO~
?EScJLI,
_._ ., jUar 3UI., iulr
730‘ 3dia 3u.5
c
L)O 30,. J=r.NELEH dU 30,~ I’rrYELE” PHIUT 3UiJrJ~I~IN~th~I~Jl~NGRU~~I~J~~NTYPE~I~JI FO~*~T(,~A,.~..~(Z*.IM.~~.,H*II WIYI 3OIS FU
3Oi7FU4YPT
(*OX*
Fi0.a)
AWAY
Automatic
x5
3039 3uro 30*r
330
3061 3062
335
3*s 3033 3036 303” 3036
34s
3032 3~8%
350
3C53 3055 355
3O5.e
3056 360 3U57
365
37u
375 3r)bu
36U
4566
365
3Obl 390 467
SYMBOLIC ENTkV 2106
6262
CAS Vol. II No. 5-C
PRIYT 3u39 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 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 CUNTINJE PHINI 303Y PHlYT 30% FOMAT (rn;/4OX,ZUnGEDMET*ICAL DATA OF tLtHENTS/lH r39X,26(lk-l//l PHI*1 3053 FOKqAT (iOx,a3(ln~)**6(i~*,, PRlhlT 3055 FOSYAT (lUX~~n*~9XtiH~rbeXr~M. .YX.,H*,2OH ELtHENl CENTER OF ‘1 PRIYT 30% 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 FORMAT ~rOX~rn~t9X170~1~~l~~OX~~~~~19x~l~~l PRIYT 3057 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 FOMAT ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ iik*rd(F7.3*2XerH*ll IkD~lND*i IF (IYD-501 306Ar3Obl.4566 PRINT 3053 PRINT 3052 PRINT 3053 PRIYT 3055 PRINT 3054 PRINT 3056 PRINT 3057 PKINT 3053 lNO=9 COYTINVE PRINT 3053 CONTINUE END REFLRENCE
HAP
(RI,)
POINTS OI’JIOE
VARlA8LES 6201 6174 10152 b157 6233 6224 6261 1;:;: 6211 6277 6222 6210 6232 6250 6263 6223 6165 6166 bZO* 6236 6170 6237 6270 6265 6274 10520 6166 blb7 6LOO 6175 1;;:;
triangular mesh generation in tlat plates for finite elements
SN
Al AN BHkAX I IFF Ill4 IKL ;gEx ISTEP 12 I* JE JFF JJ JOINT J4 KAPA KLCF KklOk KF KHELP KKF KKJ KLUR KPEK3 KSTX KTLN KUNIT KX5T L LGtGG LIJ LKKJ
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
458
G.
61% b172 6212 oi*3 61b2 bI54 6252 6255 62>7 6213 b22U 02r7 6207
NENO NLAdX
bZU5 odU2 63~4
NDfL Nl N3
f,Lt
NPHLZ 0
STATEMENT 5%7 4227 4214 4206 4225 4244 0 4342 *375 4336 0 4450 0 0 45u5 0 u u 0 " 0 5051 0 u 4540 5222 u
5550 56U7 MC4 0 5b57 5676 6003 b037 61~6 0 0
Y YCCI
INPUT
61>5 b1¶3 6161 62~2 b&5 bL~4 6207 b&4 6256 LO544 6466 b246 6644 ILId 63L2 b303 6LI)Z
INtcci*
INttbER INtEuEr? SN
SJHSX xC6
SYRMAKEZIS
IhTtGEH INTt_i.W INTEGER
NLlNl NOfCE
VAHIAdLES b16.I 63ub 7bs2 h307
and K.
STEFANOU
INTEGER INTEGER INiE6W INiEbER INttbER INktiER lNit6tH INttbER INiE6EI) INtEGEH lNiE6EH
LMN MEL1 MNOFC NdcG NHOW NLL NELEH NkLHi NtLi NkL4
o21r
D.
TYPc tiEAL 17EAL REAL rltAL
MELEM HNLIZ NABCO N1)OHX NCOL NELtlE NELGE NELI NEL3 NEM NGUUP
INtiGER IHtEbtR IN1Ebt.R IYtEbER IYttUEU IYtEbER IVtEGtH IYtE6tR
NLINE NOFC NOFJO YTYPE N2 5unst
INtEGtR INtt‘ER IHTEGEM INTEbER IXTEXR RcAL
x XPH
REAL PEAL REAL RECIL
INttGtR IVtEGtR INTLbiR ARRAY ARRA" ARRAY
RcLOCA!lVN 7332 b310 6626 6032
ARRAY
YC YPREL
AHRA" ARRAY AWAY ARRAY
MODt
co*,
FM1
FM,
OUTPUT
LABELS I 111 114 131 134 2~3 2Ub 3"* 310 319 342 346 351 361 364 372 452 495 501 51* 523 531 539 544 601 1000
FM,
0 ~176 4Cli uz54 4L50 453i 0 4370 0 0
STATISTICS PtiOCi?AcI LENGTH WFFEH LENGTH
*455 **I,
INACTIVE
0 0 0 *652 0 4735 0 0 0 5124
Ii05 3910 3013 3Ulb 3033 3837 3040 3053 3656 3b64 3451 7362
>51b
INACTIVE
0 0 0 0
INACTIVE
5C60 z.616 5637 5352
FMT FMT fHT FYT FM1 FM1 FM1 FM1
57% 5705 6022 0046 0 0
INACTIVE INACTIVE
bOb38 *IOdB
2 1.2 I‘, r3c LUI CO4 JO1 305 3ii 320 3*3 347 352 3C2 365 391 491 *97 51i 520 524 533 541 5*9 602 1100 1AU7 3011 3014 3017 3034
FMT
3036 3041 3654
FM1 FM1 FMT FMT
3057 3081 4567
55*3 INACTIVE
205
4336 4345 4372
INACTIVE INACTIVE
0 4523 4476 0
INACTIVE
4503 0 0 4635 0 0
INACTIVE 4774
0 0 0 5205 INACTIVE
0 0 0 0
INACTIVE FHT FHT
rltCtANGJLA3 ELEMENT ________________-____---
5627 5363 5735 5670 5771 6011 5476 5326
INACTIVE
0
OAIA
.**....*.**...*..**....*.***...*.****.*.*.**.*
l
i iINt ELE*ENi iELEM
lEiEMiNt l l
i +
“ 3c* >* i' c* C* L' 3 0 :. * ‘(1 2. * * ii ** IL* i+ . 2' 3* c* I _____________________-___-_______-_-____-___-_
l
6
1YPi I(
l
6
'3
l
i
<* l
l
* l
0
4222
3123 2AA4
* COLiPiN II
0
3 113 122 133 202
0 0
f ,*
*
i* L* I*
*
:
l
. *
303 308 312 341 345 349 353 363 371 451 492 '499 513 521 529 534 543 571 lb3 1101 1901 3012 3015 3032 3036 3039 3052 3055 3061 3082 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
__~~~~_~_~~___~__~~_~~~~~~~~_~~_~~~~~~~~_~_~~~ * 3 6 * 3. I* I * * *
b*
l
6 6
*
c.*
LO
4
3.
a
.
;.
A. iL
l
a l
I 3.
l
z
l
X-DIRECTION 1.000 1.000 i-000 1.000 i.000 1.000
STEPS
Y-DLRECTION 1.000 1.000 1.000
l
i
+
r*
l
LO I* ~~~__~~_~~__~~_~~____~_~_~~~_~__~~~~~~~~_~____ STEPS
2
1.006
1.000
l l
2
l
3 4 5 b 7 8 9
a
0.00
l
l
0.00
l
0.00 0.00 0.00
l
4 4 ;
.GJ 1.00 2.00 3.00 4.00 5.00 0.00 .SO 1.00 2.00 0.00 .50 1.00 2.00 3.00 *.oo 5.00 0.00 .so 1.00 2.00 4.00 5.00
l l
::
3c . 33 . l l 34 l l . 35 . l . . . 36 l . . 37 . .50 l . l 36 .50 . . 39 .4 .50 . . :: . 40 . 1.00 . 13 12 f . 41 A.00 . . . 42 . i 1.00 . l* l 43 . 15 1.00 l . 44 16 4 1.00 . . l 45 . i-00 . *4 :: . 46 l . 47 : . ff 19 4 :*:: .l . 46 . 1:so . . 49 . l * 50 22 . l . :*5’: . . : 23 . 51 : . :.3 . 52 24 . i 53 25 $3 . l l 54 . 26 x. l 2.00 . 55 2:oo . 28 27 ; . 56 2.00 . 3.00 . . ..*...*..**..*......*..*..****...*..*.**.*****.~..**.*..****..******.*~........****.********* l
“2,”
l
i l
4 4 l
* l
4 . 4 4 4 6 4 4 4 . 4 4 6 . 4 4 4 4 4 4
GEOWTRICAL
2.66
l
2.5.0 2.54 2.50 2.50 2.50 2.50 3.00 3.00 3.00 3.00 3.00 3.00 3.00 3.50 3.>0 3.50 3.50 3.50 3.50 4.00
. .
. t*:: 4.60 4.00 4.00
. . . .
+.oo
DATA
OF
. . . . . . . . . . . . . . l
. . l
l
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:*:a: 2:733 2.461 2.600 2.733 .067 .200 .333 .067 .I33 .267 .333 ,533 ,667 .533 .b67 .933 1.067 .933 1.333 1.467 1.733 l.ab7 2.133 2.267 2.533 2.667 .067 ,133 .267 .333 .067 .133 .267 .333 .533 .667 .533
. l
. l . II . . . . . . . l . . .
:%. 3.417 3.083 3.250 3.417 3.167 3.333 3.167 3.333 3.167 3.333 3.167 3.333 3.583 3.667 3.583 3.667 3.833 3.917 3hj3 3.917 3.583 3.667 3.833
. ELENENT + GRAVITY tLtMtN1 NODES . *...*..............*...........4...**....*......**.***..*.**.*....*... _.._.. ‘y1 -. LX2 Y3 . AREA . XC x3 l NUWER + i. Xi Y2 . 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...**................*.....*.***. ..bO. 3.Y17 3.75 .80 . .0500 . .40 l 106‘ l 102 ‘* 105 ‘a.00 169 3.7> .80 l 94 3.50 1.20 . .0500 . * 93 .L(O . iO2 3.50 170 : :q:; . 3.50 1.20 . .lOOO 4.00 A.20 . 94 . ror 3.75 .80 . 107 171 . 3.017 . 1.20 . .0500 ’ 4.00 .a0 . 101 4.00 .80 . 10b 172 . 1ui 3.75 . 3.667 . 1.60 . .lOOO 4.00 3.50 173 . Y* 1.20 l 95 3.50 I.LO . r07 . 3.833 l 95 1.60 l 4.00 3.50 *IO00 l * iO7 iad0 . rod 4.00 114 . . 3.667 . 96 3.50 2.00 . 4.00 f’Z .lOOO 175 . 9a r.60 . 108 3.50 + 3.833 . 4.00 3.50 2.00 . 2:oo . 96 .I000 r.60 . ~09 * 1vn 176 4.00 . 3.661 . 4.0” 3.50 2.40 * l r09 .I000 177 . Yb 2.00 l 97 3.50 Z.OO . 3.833 2.40 + . 4.00 2.40 l 97 3.50 .I000 ild * IO9 c.00 . hi0 u.00 * *.uu 2.80 + . 3.667 174 . Y7 L.40 l ii0 2.40 . 98 3.50 .lOOO 3.50 . . 3.633 3.50 2.80 l 4.00 *IO00 . 110 L.40 . 111 2.80 l 9b 180 *.oo .**....*.***.*....*........*...~...*....*.**.*...***............**.....*.*..*.*...............*..........*..... LLEMLNT
.
. . . .
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.
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tLEMENTS
*.............*..........*..........*.............*..*..**.*.............*............*.......*....*****....... .. . -.-. . . . .
CENTER Of COORDS
hC 2.833 2.917 2.583 2.667 2.583 2.667 2.833 2.917 2.833 3.167 3.083 3.167 3.333 3.417 3.333 3.417 3.083
.
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.0500 .0509 .lOOO .0500 .lOOO .I000 .1000 .1000 .I000 .lOOO .I000 .lOOO .0250 .0250 .0250 .0250 .0250 .0250 .0250 .0250 .0500 .0500 .0500
ELEMENT GRAVITY
CENTER Of COORDS YC . . . . . . . l
. . . .
.667 .933 1.067 .933 1.333 1.467 1.733 1.867 2.133 2.267 2.533 2.667
. l
.
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. . . . . . . . l l
.
inGrcac Britain
AUTOMATIC TRIANGULAR MESH GENERATION IN FLAT PLATES FOR FINITE ELEMENTS G. D. STEFANOU Schoolof Engineering,Universityof Patras, Patras, Greece
and K. SYRMAKEZIS Civil Engineering Department,
The National Technical University
of Athens, Greece
(Received 3 August 1978;receivedfor publication28 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
‘Y
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 geometricalproperties of the elements. The solution of this problem will be useful for the application of numericalapproximate methods, such as ,finiteelements, 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-ordinatesof the orthogonal system as
I
I I
I
I I I
:pn_ I I
I
---
---
A---+
---
----t---bt---* I
I
0,
I
I/ j j
1
1
/
I I
’ ’
1 X
Fig. 2. Discontinuous plate divided into regions. X’=XtyCOSQ I
y’=ysiny,
1
(1)
The investigationis 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 gradingis taking place graduallyfrom 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
2.SuBlWEXoN PROCJCDIJRE OF MAINGROWG= I To divide a hyper-elementinto triangularelements, 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 hyperelement is divided.The elementsof the group are chosen to have an area, ET, of their trianglesequal to ET=$E
cl0
0
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 programmeinto “triangularelements”
(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 followingrules: (a) The t&g&u
x
(0)
Fig.1. Discontinuous plates. 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 Ax, Ay (side Ax is divided first), by vertical and horizontal lines and by drawing the diagonals of the parallelograms so formed as shown in Fig. 3(b) The diagonals have always an ascending slope from right to left. (c)The numbering of the 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
the numbering of the triangular elements. The number of nodesper column jC,and line jr. for each hyper-element is
defined by jC=lt2’k-‘)‘2 or j,=lt2
for
(k-2)‘2 for
k=l,3,5
,...
k = 2,4,6,. .
(3)
Automatictriangularmesh generationin flat plates for finite elements
441
i
(i*l 1 Nodal he -----------
i Nodal --~-----~--
line
j (se+11
_-_
(j-l)(r,*l)+i
+
i
Fig. 4.
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.
and
or
jL=l+2(k-‘)‘2
for
k=l , 395,**-
iL = 1 t 2”“’
for
k=2,4,6,...]
I
(4)
Let SC= jC - 1 (5)
and SL = jL+r 1
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, = SCx 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)titl,
j(S,tl)ti, (j-l)(S,tl)titl.
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-
3. !WBDMSION
and
2[(j - l)S, t jl.
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:
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
(6)
where (Ak(r/lO) = the real part of 1llOth of index r. The numbers of nodes per column jC and line jL are
The two triangular elements in the sub-element have element numbers 2[(j - l)S, t i)]-’
lOAk(r/lO)]
n~o=lth(lt2k-‘)
(9)
for A=l,3,5
,...
(10)
G. D. STEFANOUand
442
K.
SYRMAKEZIS
G=t? lg,T ~34
1
l\
3
2
1
3
Ax14
Fig. 5.
Four typical
,L
AxH
,”
1’
Ax14
X
The numbering of fictitious node j0 is again given by eqn (8) and their serial numbers nio by expression fljO=A(l t2’-‘)
jt = 1 t2k-'(3+2k)
(11)
j,=j,-jo=1+2k(1+2k-‘)
(12)
r, = 22k- 2’-‘.
(13)
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
,L
elements of G = 2.
The total of nodes jl the real number of node jl and the number of elements in hyper-element, become respectively
The numbering of triangular elements is performed as for group G = 1, and the modifications required for making 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
Ax14
(16)
for h=2,4,6 ,...
(17)
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 k = ; Ak(~/l0).
(14)
The number of nodes per column jC and line jL is given by jc = 1 t2k-’ jL = 1 t 2”.
1
(15)
Iljo = A
(19)
for A=2,4,6,...
C3-N
j, =(1+2k)z
(21)
Automatictriangularmesh generationin tIatplates for finiteelements
443
G=3 el , ~21
e2,
e3,r=65
7=43
e. ,T=67
Fig. 6. Four typical elements of G = 3.
j,=j,-/O=(1t2*)2-2*-’
(22)
t. = 2*-‘[2k+2 - 11.
(23)
and
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
A=2,4,6,...
(26)
for
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 FORTRANlanguageand was originallywritten for the IBM 1620computer of the National Technical University of Athens. Later, the programme was modified for the CDC 3u)o computer unit of “Democritos” NuclearResearchCentre, Athens.The programme is dividedinto three mainparts: (a) Preparationof data,(b) Runningof the programmeand (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-elementsand 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 numberof columns of hyperclements along the length of the beam NELEM &); (b) The numberof lines of hyperclemeds alongthe height of the beam MELEM (SC); (c) The lengths of hyper-elementsof the variouscohmmsof the elements in the beam are BHMAX (I) (A& (d) The heights of hyper-elementsare BHMY(1)(A,,);awl(e) The matrixof indicesINDEX(IJ), (i.e. indicesr), whichhas dimensions (MELEMx NELEM), &fines the manner in which each
G. D. STEFANOU and K.
444 el,
SYRMAKEZIS
~=23
e2, r=46
Fig. 7. Four typical elements of G = 4.
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) _ . INDEX, first, per column __ .. . of matrix and then per line, is punched. Matrix INDEX is constructed according to the desired
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
for
i=odd,
aij=i
for
i=even
for i=j: Ui,=i=j
(27)
for i > j: ai,=j
for
j=odd,
a,=lljtl
for j=even.
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
(28)
Automatic triangular meshgenerationin flat plates for finite elements
44s
G=!5
*
Ax/2
1 !x 1
Ax12
1 AX
X’ 1 L 1
e4, t=96
ea.T=76
Fig. 8. Four typical elements of G = 5.
Sector C. (a) Cij=aijt9
Cii=aj-9
for i>j for i
and j=odd 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 0 1 . ..... 0 00 . . . . . . . 0 0 [E] = 0 0 ....... 1 : 0 ...................................... 0 1 ....... 0 0 0 0 ....... 0 0 0 I . I I 0”
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 perform, 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.
D.
STEFANOU
SYRMAKEZIS
and K.
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
1st element
+
2nd element
-8
r
0 0 0 0.20 0.20 0.20 -_ + 0.40 ..............0.40 ........,.. ...0.40 ...... I + 0.80 ..............*0.80 ..............0.80 ...... I f
3rd element
-+jj
f;
........13 .......
4th element 2.00
G-l
G-l
Gm4
G=l
G=l
0
( = lYC]
kf 2.00
G.4
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 whichthefictitious Fig. 9. Typicalcolumns 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 necessarily all lying on the MNOFC columns. Consequently, the serial numbers of columns where the element under consideration has nodes, are
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 subelement (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
ZSTEP=lt(A-I)~~~_~ll~MNOFC
for
cr__________________-._-----__--_---_____+
A
=12, 9.‘.
1, 7
1
5
%
3
032 ,
1
23
3
46
5
67
7
1
;
6
-6
64
3
22
1
23
3
46
5
6
6
1
I
23
3
465
4
4
4
3
‘z - _l__
4
4
4 5656I4
; 34 12
24 2
24 2
32 2
1 I
I 1
31 I
24
24
24
, 1
24 r;-2 I 12
I2
I2
I2
II
I
I
2l
I 21 ml
2,
I
I
6
166565 /
P
“’
, , ;
4
12 2l
I
2
2
3
32
1111 1
1
1
1
2
2
23
3
Fig. 10. Rectangular mesh of beam.
n
,I
2
2221221
(
”
4’
4’
4
‘
.I4
-+.A
50
.166666 0
E
!6
6
I
221
112 8
__I__‘_)__)
34 ,1. I
‘I
1 I I 43 rl___u___r__..__l_Z’__d ‘1
221
643
22
5563
23
32I
I
2
212 I
2
2
‘2
42
43
2/ I 43 1
4
4
4
1
23
3
4
1
21
1
465
1
23
2
46
Fig. 11. Auxiliary array.
6666; 5
6
6!
1 ,
441
Automatic triangular mesh generation in flat plates for finite elements
Fig. 12. Flow chart diagram.
Note:
D = 1st digit of INDEX; 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
M = 2nd digit of INDEX:
P = odd
number:
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.OO0.00 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.OO0.20 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)
Hence, matrix LBEGG of the example. after the final numbering of nodes becomes
[LBEGG] =
:
I2 10
I6 I4
4 5 6 7
I3 13 I3 I3
I7 18 I9 20
;
I3 13
::.
For each column of matrix LBEGG we keep the number NABCO LBEGG(1, MNOFC) - I. After calculating 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 =
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
(b)
(a) Fig. 13.
448
G.
D. STEFANOU and K.
SYRMAKEZIS
Fig. 14.
triangular elements, three new triangular Zl,Z:,Z;, are created as shown in Fig. 13(b).
elements
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 Node Node Element I;: Node Node Node Element Z;: Node Node Node
1 corresponds 2 corresponds 3 corresponds 1 corresponds 2 corresponds 3 corresponds 1 corresponds 2 corresponds 3 corresponds
to node I of elemtnt II to node 2 of element Zz to node 3 of element II to node 1 of elements II to node 1 of element Z4 to node 3 of element I, to node 1 of element Is to node 2 of element Z4 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+ Z4+
4 :
: 6
2 33
4 I 5 8
:
5 5 6 6
8 9
7 10 II 10 8 11 11 12 10 13 13 14 11 14 14 ,.............. .15 ...........
8 8 z I1 11 12 ..12 .
The above matrix is then transformed into 14line matrix as follows
.............................*.. I 5 2 2 5 3 5 6 3 1 I ‘A=1 7 8 : 6 : ! 6 7 11 8 8 11 11 12 ; 7 13 II pA=l 13 I4 II 11 14 I2 I4 I5 I2 ................................ .........................*.*.... ........,............. ........
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= ZJt 1
-m (a)
Fig. 15. Triangular elements for G = 3.
Automatic triangular mesh generation in flat plates for finite elements
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 Node Node Element I;: Node Node Node Element Z;: Node Node Node
1 corresponds 2 corresponds 3 corresponds 1 corresponds 2 corresponds 3 corresponds 1 corresponds 2 corresponds 3 corresponds
to to to to to to to to to
node node node node node node node node node
1 of 2 of 3 of 1 of 2 of 3 of 1 of 2 of 3 of
element element element element element element element element element
II I, I, Z, L I2 Z, b L.
449
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 Z4is 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
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 Node 2 corresponds Node 3 corresponds Element ii: Node 1 corresponds Node 2 corresponds Node 3 corresponds
to node 1 of element to node 2 of element to node 2 of element to node 1 of element to node 2 of element to node 3 of element
I, I, b II Z2 b
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, .ZLare 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 Node Node Element I;: Node Node Node Element Z;: Node Node Node
1 corresponds 2 corresponds 3 corresponds 1 corresponds 2 corresponds 3 corresponds 1 corresponds 2 corresponds 3 corresponds
to node 1 of element I, to node 2 of element I, to node 2 of element I, to node 1 of element Zz to node 2 of element Z4 to node 3 of element Zz to node 1 of element b to node 2 of element Z, 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
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Fig. 16. Triangular element for G = 4. CM
Vol. I I No. 5-F
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Fig. 17. Triangular elements for G = 5.
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STEFANOU
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SYRMAKEZIS
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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.
Automatic
triangular mesh generation in flat plates for finite elements
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work was carried out in the laboratory of The Institute of Structural Analysis of the National Technical University of Athens. Part of the work appeared by G. D. St~~u and K. Symmkezis in the magazine, issue No. 12172,of The Greek Inst&e of Engineers,
Acknowledgements-‘his
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ContinuumMechanics.McGraw-Hill, London (1%7). 2. R. W. Clough. The finite element method in plane stress analysis. Proc. ASCE 2nd Conf. Efectronic Camp~tatio~, Pit&burg, Pennsylvania (Sept. t%O). 3. D. D. McCracken and W. S. Dam, ~urn~ca~ Methods and Fortran Progmmming.Wiley, New York (I%@. 4. W. Weaver. ComouterPmemms for Stmctuml Analvsis. Van
454
G. D.
STEFANOU
and K.
SYRMAKEZIS
APPENDIX 1
Input listing-results An input listing of the computer
programme PROGRAN
s C
i
3
15
30
3,
*(i
*5
DIVIOEIINPUT.OUTPUT,
NELl.NELEN*l READ I~IllNDEXlI.J,~I=l.NELEM~~J=l~NELEM, FOWAl ILOIwl RELD 3rlBHN4VlI,rI=l.NELEN, REID ~~I~NM~XII,~~=I.NELEN, FDtiNAT llOFL).3,
C
*....*..**..*.***.******..*.....**..*.*......******.**.**.....*.** ~iii~~~i~iri~i..i~.i~.~~.~*~~~~~~~~~~..~.~*.***.*******.****.*.**.
;
CALCljiAtE’GtOMETRIcAL~DAt4 WFT~EOINIT14L WECTANGULAR s
c
LO
sample problem are included in this section (see input and tables Tl, T2. T3).
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 READ AND PRINT DATA REAO B.NELEN.NELEM z FOHN4T (214)
10
15
and results of
FOR’tiit‘SUBELENENTS MESH
OF
TNE
NE,.=o swsx=o KBiF=O 00 1100 J=irNELEN CALCULATE N(lRUP AND NTVPE FOR LAW RECT4NGULAR ELENENT OF THE E 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 t*****.***.****.****.*..*..***.*...*.****.*~*.*~*~**~*.**~*.**.**~ L 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,=. ISrARIL,=O 3r. 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 CALCULATE’JOINT CUONOIlhtES bF tiiE SWELEMENTS OF EACM INITIAL c UECTCIYGULAR ELENcNT INCLUDING-FICTITIOUS JOINTS 00 301 I=i.NELEH KSTX(I,=NOFCII,-I IF (INSEXlI.J,, 301.30i.lr01 C CALCUL4Tt NO OF LINES Ii01 KTVPE=YTVPElI,J, KGPUP=YGRUPlI,J~ KdOHsNGRUPlIrJ,/Z IF tKlOH-i, 305130Jr30* 305 NOFLll*c.~((KTVPE-l)/2, GO TO 308 303 NOFL.l*L**(KTYPE-I, ‘0 TO 308 JUI. NOFL=l*C’*KTVPE 308 AL=NOFL-I KSTVIIi=AL IF (1-i) 3i9r319r3LO 320 IF (INDEXII-r,J)) 319.31 9.1105 ii05 NOfLsYDfL-i C CALCULATE AdSCISAE VPREL OF EACH LINE OF JOINTS
.~ _. _. ..*.........**.****.***...*.**.~**.*.*********~**.*.*....*.......*
d5
90
YS
100
COLUMNS
Automatic triangular mesh generation in flat plates for finite elements
1GS
110
115
120
IL5
139
135
IWJ
l*S
150
155
109
319 DO 310 IJK=l.NOFL If (1-l) 31is311*312 311 AI=lJI(-i GO TO 310 312 IF (lNDEX(l-leJ)b 3~lr3ll*lr07 Af.LM ‘E 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 3*3 YCILI.ISTEP)=YPREL(LWN) ISlARlISlEPI~ISlAWllSTEP~*NOFL J*‘ OWIT iIClI~IOUS JOINTS C ICOYS’Z*~~KrYPE~*I GO TO I3~~,3~6,3*7r3A9,34~~,KG~~P AD IF II-.) 3*7,3*7*3*>1 IF IINDEXII-r.JbB .bSr367r3QS 3*5i 3*7 DO 3% J<=L.ICONS.C IF IKGQJP-r) 35lrJblr352 35i 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 ooc K”AQ=L bui 00 37~ LJ’KVARIHNO‘C ~ST4~=lSTARlLJ) 00 37i Ll=r+hSTAH LIJ=KE+Ef*L, AlLIJ)=XWILJ# 37; YlLIJl=VClLI,LJ) J7‘ KdCfs<~tf.IS,AH(LJ, ,QAYSfJt(M ‘YlTlAL 10 ilkA~ INOLA MAT*,X LaEGG C “0 U+.L .,fF=<.MNOrC
.._ __
IO>
*a& 7oj
ri0 783 Iff=r,HELI L~tGGIIFF.JFF)=LdrcGi(IFF,JFFk*NAbCO
179 t C C 175 1991
100
rY9 *97 492 u95
185
*Pi C
190
1% C C 290
205
501
21J C
61.
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
456
G. D. STEFANOUand K. SYRMAKEZIS
215
220
5i3 SI*
225 C
SLI 230
235
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
523 SL* 2-o
SC9
2*5 ‘
53i 250
255
933 33* 260
2b5
539 C 54.
270
215
563 5+a
280
285
5.9 37. L
,LJ ivu0 rlO0 L i b
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 NOFJO~
?EScJLI,
_._ ., jUar 3UI., iulr
730‘ 3dia 3u.5
c
L)O 30,. J=r.NELEH dU 30,~ I’rrYELE” PHIUT 3UiJrJ~I~IN~th~I~Jl~NGRU~~I~J~~NTYPE~I~JI FO~*~T(,~A,.~..~(Z*.IM.~~.,H*II WIYI 3OIS FU
3Oi7FU4YPT
(*OX*
Fi0.a)
AWAY
Automatic
x5
3039 3uro 30*r
330
3061 3062
335
3*s 3033 3036 303” 3036
34s
3032 3~8%
350
3C53 3055 355
3O5.e
3056 360 3U57
365
37u
375 3r)bu
36U
4566
365
3Obl 390 467
SYMBOLIC ENTkV 2106
6262
CAS Vol. II No. 5-C
PRIYT 3u39 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 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 CUNTINJE PHINI 303Y PHlYT 30% FOMAT (rn;/4OX,ZUnGEDMET*ICAL DATA OF tLtHENTS/lH r39X,26(lk-l//l PHI*1 3053 FOKqAT (iOx,a3(ln~)**6(i~*,, PRlhlT 3055 FOSYAT (lUX~~n*~9XtiH~rbeXr~M. .YX.,H*,2OH ELtHENl CENTER OF ‘1 PRIYT 30% 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 FORMAT ~rOX~rn~t9X170~1~~l~~OX~~~~~19x~l~~l PRIYT 3057 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 FOMAT ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ iik*rd(F7.3*2XerH*ll IkD~lND*i IF (IYD-501 306Ar3Obl.4566 PRINT 3053 PRINT 3052 PRINT 3053 PRIYT 3055 PRINT 3054 PRINT 3056 PRINT 3057 PKINT 3053 lNO=9 COYTINVE PRINT 3053 CONTINUE END REFLRENCE
HAP
(RI,)
POINTS OI’JIOE
VARlA8LES 6201 6174 10152 b157 6233 6224 6261 1;:;: 6211 6277 6222 6210 6232 6250 6263 6223 6165 6166 bZO* 6236 6170 6237 6270 6265 6274 10520 6166 blb7 6LOO 6175 1;;:;
triangular mesh generation in tlat plates for finite elements
SN
Al AN BHkAX I IFF Ill4 IKL ;gEx ISTEP 12 I* JE JFF JJ JOINT J4 KAPA KLCF KklOk KF KHELP KKF KKJ KLUR KPEK3 KSTX KTLN KUNIT KX5T L LGtGG LIJ LKKJ
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
458
G.
61% b172 6212 oi*3 61b2 bI54 6252 6255 62>7 6213 b22U 02r7 6207
NENO NLAdX
bZU5 odU2 63~4
NDfL Nl N3
f,Lt
NPHLZ 0
STATEMENT 5%7 4227 4214 4206 4225 4244 0 4342 *375 4336 0 4450 0 0 45u5 0 u u 0 " 0 5051 0 u 4540 5222 u
5550 56U7 MC4 0 5b57 5676 6003 b037 61~6 0 0
Y YCCI
INPUT
61>5 b1¶3 6161 62~2 b&5 bL~4 6207 b&4 6256 LO544 6466 b246 6644 ILId 63L2 b303 6LI)Z
INtcci*
INttbER INtEuEr? SN
SJHSX xC6
SYRMAKEZIS
IhTtGEH INTt_i.W INTEGER
NLlNl NOfCE
VAHIAdLES b16.I 63ub 7bs2 h307
and K.
STEFANOU
INTEGER INTEGER INiE6W INiEbER INttbER INktiER lNit6tH INttbER INiE6EI) INtEGEH lNiE6EH
LMN MEL1 MNOFC NdcG NHOW NLL NELEH NkLHi NtLi NkL4
o21r
D.
TYPc tiEAL 17EAL REAL rltAL
MELEM HNLIZ NABCO N1)OHX NCOL NELtlE NELGE NELI NEL3 NEM NGUUP
INtiGER IHtEbtR IN1Ebt.R IYtEbER IYttUEU IYtEbER IVtEGtH IYtE6tR
NLINE NOFC NOFJO YTYPE N2 5unst
INtEGtR INtt‘ER IHTEGEM INTEbER IXTEXR RcAL
x XPH
REAL PEAL REAL RECIL
INttGtR IVtEGtR INTLbiR ARRAY ARRA" ARRAY
RcLOCA!lVN 7332 b310 6626 6032
ARRAY
YC YPREL
AHRA" ARRAY AWAY ARRAY
MODt
co*,
FM1
FM,
OUTPUT
LABELS I 111 114 131 134 2~3 2Ub 3"* 310 319 342 346 351 361 364 372 452 495 501 51* 523 531 539 544 601 1000
FM,
0 ~176 4Cli uz54 4L50 453i 0 4370 0 0
STATISTICS PtiOCi?AcI LENGTH WFFEH LENGTH
*455 **I,
INACTIVE
0 0 0 *652 0 4735 0 0 0 5124
Ii05 3910 3013 3Ulb 3033 3837 3040 3053 3656 3b64 3451 7362
>51b
INACTIVE
0 0 0 0
INACTIVE
5C60 z.616 5637 5352
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3;lifJ L.75 c.00’+ ‘83 3.00 2.00 + 84 3.00 2.*0 . 75 2.75 2.50 . 129 l 71 2.40 l 76 . 130 l 75 2.15 2.75 d.60 l 76 . 131 . 7L 2.7> 2.50 . 132 l 7b d.60 l 77 2.7) 2.75 d.40 l B4 . 133 l 75 i.75 3.0-o . 134 . n* 2.*0 . MS 3.00 3.00 L.60 . 65 . 135 * 7b 3.00 2.15 . 134 * 78 0.00 . 86 3.25 3.00 . 131 * 78 0.00 . a7 3.00 ;-g . 13b l 87 .20 . aa . 3.25 l 139 . 80 3.50 0.00 . 90 3.25 . 140 * 90 3.50 3.50 0.00 . 91 . 141 . 07 .LO l 91 3.50 3.25 .20 * 92 3.50 3.50 . 162 * 91 l 143 l 79 .*o l en 3.23 3.00 .co l OY . 144 * 81 3.25 3.25 . 145 l 88 .*o l 92 3.5u 3.25 .*o l 93 . 146 * 92 3.50 3.50 .80 l 10 3.25 . 147 . 80 3.00 .80 l 9* . 14b . 89 3.5lJ 3.25 l 149 * 8Y .80 . 93 3.50 3.25 3.50 r.20 l 94 . 150 * ei 3.09 . 151 * Y* 3.5u r.10 l 95 3.50 A.60 l 95 . 152 . 8d 3.50 3.00 i-60 l 96 . 153 . Y5 3.50 3.50 . iS* . d3 3.50 2.00 l 90 3.00 . 155 . YD 3.50 3.50 z.00 . 91 . 150 . &I 3.50 L.40 l 97 3.09 . 157 * 97 3.fJu 3.50 d.40 . 98 * 3.7) 0.00 l 99 . YO 3e5V 15n * 159 * 99 3.15 3.7, 0.00 l 100 l 3.15 160 * Y. .LO l 100 3.50 l 3.73 Ibi * 100 3.75 .20 . A01 . 16~ . YY 4.00 3.75 0.00 l 103 l 4.00 i6A * 103 0.00 . ,o* *.oo l .20 l i04 4.06 l iO0 3.75 164 . 165 . lo* 4.00 .20 l 105 ..00 l 166 * 9.d 3.15 .40 l A01 3.50 0 3.73 * *Oi .ro + 102 167 3.75 . A05 4.00 * 101 3.75 166 .40 :.*.......*..**.*.*.*..*..***.....****......*.....**....*.*******.******.***..***.***....*.....**....*......... 127 128
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12
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2.00 2.40 2.40 2.60 2.60 2.80 2.40 :‘z . 0.00 .20 .*o 0.00 .LO .zo .40 .40 .80 .40 .no .I30 1.20 .80 1.20 i.60 1.60 2.00 2.00 2.40 2.40 2.80 0.00 .20 .20 .40 0.00 .20 .20 ,*o .40 .60 .40
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75 75 72 12 13 73 76 77 76 61 79 79 07 67 88 80 60 80 69 89 81 bl 94 a2 82 13 b3 84 84 n5 85 91 91 92 Y2 100 100 101 101 93 93 102
GtOMtTRlCAL OAIA OF ________________---_-____-_-
x3 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 .3.50 3.50 3.15
Y3
.
AREA
2.40 2.40 2.60 2.60 2.80 2.80 2.60 2.60 2.80 .20 .*o .40 .20 .PO .*o .40 .80 .80 .80 .a0 1.20 1.20 1.20 1.60 1.60 2.00 2.00
. . . . . . . . . . . . .
.0500 .0500 .o.?so .0250 .02so .0250 .0250 .0500 .0250 .0250 .osoo .0250 .0250 .OPSO .0250 .0250 .0500 .osoo
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2.40
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2.40 2.80 2.80 .20 .20 .40 .40 .20 .20 .CO .40 .80 .80 .80
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YC 2.133 2.267 2.467
. ..
:*:a: 2:733 2.461 2.600 2.733 .067 .200 .333 .067 .I33 .267 .333 ,533 ,667 .533 .b67 .933 1.067 .933 1.333 1.467 1.733 l.ab7 2.133 2.267 2.533 2.667 .067 ,133 .267 .333 .067 .133 .267 .333 .533 .667 .533
. l
. l . II . . . . . . . l . . .
:%. 3.417 3.083 3.250 3.417 3.167 3.333 3.167 3.333 3.167 3.333 3.167 3.333 3.583 3.667 3.583 3.667 3.833 3.917 3hj3 3.917 3.583 3.667 3.833
. ELENENT + GRAVITY tLtMtN1 NODES . *...*..............*...........4...**....*......**.***..*.**.*....*... _.._.. ‘y1 -. LX2 Y3 . AREA . XC x3 l NUWER + i. Xi Y2 . 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...**................*.....*.***. ..bO. 3.Y17 3.75 .80 . .0500 . .40 l 106‘ l 102 ‘* 105 ‘a.00 169 3.7> .80 l 94 3.50 1.20 . .0500 . * 93 .L(O . iO2 3.50 170 : :q:; . 3.50 1.20 . .lOOO 4.00 A.20 . 94 . ror 3.75 .80 . 107 171 . 3.017 . 1.20 . .0500 ’ 4.00 .a0 . 101 4.00 .80 . 10b 172 . 1ui 3.75 . 3.667 . 1.60 . .lOOO 4.00 3.50 173 . Y* 1.20 l 95 3.50 I.LO . r07 . 3.833 l 95 1.60 l 4.00 3.50 *IO00 l * iO7 iad0 . rod 4.00 114 . . 3.667 . 96 3.50 2.00 . 4.00 f’Z .lOOO 175 . 9a r.60 . 108 3.50 + 3.833 . 4.00 3.50 2.00 . 2:oo . 96 .I000 r.60 . ~09 * 1vn 176 4.00 . 3.661 . 4.0” 3.50 2.40 * l r09 .I000 177 . Yb 2.00 l 97 3.50 Z.OO . 3.833 2.40 + . 4.00 2.40 l 97 3.50 .I000 ild * IO9 c.00 . hi0 u.00 * *.uu 2.80 + . 3.667 174 . Y7 L.40 l ii0 2.40 . 98 3.50 .lOOO 3.50 . . 3.633 3.50 2.80 l 4.00 *IO00 . 110 L.40 . 111 2.80 l 9b 180 *.oo .**....*.***.*....*........*...~...*....*.**.*...***............**.....*.*..*.*...............*..........*..... LLEMLNT
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tLEMENTS
*.............*..........*..........*.............*..*..**.*.............*............*.......*....*****....... .. . -.-. . . . .
CENTER Of COORDS
hC 2.833 2.917 2.583 2.667 2.583 2.667 2.833 2.917 2.833 3.167 3.083 3.167 3.333 3.417 3.333 3.417 3.083
.
.osoo
.0500 .0509 .lOOO .0500 .lOOO .I000 .1000 .1000 .I000 .lOOO .I000 .lOOO .0250 .0250 .0250 .0250 .0250 .0250 .0250 .0250 .0500 .0500 .0500
ELEMENT GRAVITY
CENTER Of COORDS YC . . . . . . . l
. . . .
.667 .933 1.067 .933 1.333 1.467 1.733 1.867 2.133 2.267 2.533 2.667
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