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g = 9.8
3 1 2

1 2 3 x
0.02
0.015
0.025

FIGURE 8.1 Simple two-dimensional example.


Output Lines Comments

1-title(1:40),™ at increment:™, 1 Title line:
title(1:40): partial problem title
incrm, ™, load: ™,xlamb
incrm: increment number
xlamb: current load
parameter

2-eltyp(1:5) 1 Element type

3-npoin 1 Number of mesh nodes

4-ip,icode(ip),(x(i,ip), Coordinate and force data:
npoin
ip: node number
i=1,ndime) (force(i),
icode(ip): boundary code
i=1,ndime)
x(i,ip): x, y, z coordinates
force(i): total external force
or reaction vector
ndime: No. of dimensions

5-nelem 1 Number of elements

6-ie,matno(ie),(lnods(i,ie),
See input item 6
i=1,nnode) nelem

7-(stress(i,ig,ie),i=1,nstrs) Cauchy stresses for each Gauss point
ngaus
(ig=1, ngaus) of each mesh

element (ie=1,nelem):
nelem
(2-D) stress: σxx , σxy , σyy
(plane strain)
(2-D) stress: σxx , σxy , σyy , h
(plane stress)
(3-D) stress: σxx , σxy , σxz , σyy ,
σyz , σzz
7
8.3 OUTPUT FILE DESCRIPTION




BOX 8.1: Input ¬le for example in Figure 8.1
2-D Example
quad4
9
1 3 0.0 0.0
2 2 1.0 0.0
3 3 2.0 0.0
4 0 0.0 1.0
5 0 1.0 1.0
6 0 2.0 1.0
7 0 0.0 2.0
8 3 1.0 2.0
9 0 2.0 2.0
4
111254
226523
315874
425698
2
14
1.0 100. 100. 0.1
26
1.0 100. 0.1
1 3 3 0.0 -9.8
9 1.2 3.4
3 1 0.02
2 2 -0.025
3 2 -0.015
1 8 7 0.25
2 7 4 0.25
3 1 4 -0.25
2 10. 5. 25 1.e-10 0.0 0.0




The output ¬le produced by FLagSHyP for the simple example shown
in Figure 8.1 is listed in Box 8.2.
8 COMPUTER IMPLEMENTATION



8.4 ELEMENT TYPES

Nodes and Gauss points in a given ¬nite element can be numbered in a
variety of ways. The numbering scheme chosen in FLagSHyP is shown in
Figures 8.2 and 8.3.
In order to avoid the common repetitious use of shape function rou-
tines for each mesh element, FLagSHyP stores in memory the shape func-
tions and their nondimensional derivatives for each Gauss point of the cho-
sen element type. This information is stored in a three-dimensional array
eledb(1:ndime+1,1:nnode+1, 1:ngaus) as follows:
® 
N1 (ξi , ·i , ζi ) · · · Nn (ξi , ·i , ζi ) Wi
 ‚N1
(ξi , ·i , ζi ) · · · ‚Nn (ξi , ·i , ζi ) ξi 


 ‚ξ ‚ξ  for i = 1, . . . , ngaus,
 
 ‚N1 (ξ , · , ζ ) · · · ‚Nn (ξ , · , ζ ) ·  (n = nnode)
 ‚· i i i iii i
‚·
° »
‚N1 ‚Nn
‚ζ (ξi , ·i , ζi ) · · · ‚ζ (ξi , ·i , ζi ) ζi

Exactly the same type of array is constructed for the line or surface elements
and stored in the array variable elebdb.

BOX 8.2: Output ¬le for example in Figure 8.1

2-D Example at increment: 1, load: 5.00
quad4
9
1 3 0.0000E+00 0.0000E+00 -0.3361E+01 0.9500E+00
2 2 0.1189E+01 -0.1250E+00 0.0000E+00 -0.2195E+01
3 3 0.2100E+01 -0.7500E-01 -0.1262E+01 -0.2211E+01
4 0 0.2906E+00 0.7809E+00 0.1006E+01 -0.2481E+01
5 0 0.1283E+01 0.1062E+01 0.0000E+00 -0.4900E+01
6 0 0.2053E+01 0.1226E+01 0.0000E+00 -0.2450E+01
7 0 0.5021E-01 0.1609E+01 0.7619E+00 -0.1668E+01
8 3 0.1000E+01 0.2000E+01 -0.3877E+01 -0.4350E-01
9 0 0.2396E+01 0.3825E+01 0.6000E+01 0.1577E+02
4
1 1 1 2 5 4
2 2 6 5 2 3
3 1 5 8 7 4
4 2 5 6 9 8
31.165 16.636 -29.752 0.99858E-01
37.922 7.0235 29.804 0.92369E-01
9.8170 28.948 23.227 0.96515E-01
-9.1664 52.723 -52.341 0.10566
-31.460 9.0191 69.610 0.97692E-01
-44.255 19.009 40.029 0.10422
-10.503 14.344 58.661 0.94115E-01
9
8.4 ELEMENT TYPES



-1.0937 4.3534 84.855 0.88759E-01
2.9733 4.9849 -8.6633 0.10056
-2.5993 10.535 -4.9380 0.10075
-10.028 16.380 -24.223 0.10326
-3.7416 10.076 -28.318 0.10306
18.711 27.033 127.70 0.80604E-01
58.710 93.889 504.64 0.52100E-01
148.61 233.72 706.89 0.39520E-01
132.88 166.87 354.22 0.54008E-01
2-D Example at increment: 2, load: 10.0
quad4
9
1 3 0.0000E+00 0.0000E+00 -0.6085E+01 0.2563E+01
2 2 0.1352E+01 -0.2500E+00 0.0000E+00 -0.3919E+01




BOX 8.3:


3 3 0.2200E+01 -0.1500E+00 -0.2444E+01 -0.2920E+01
4 0 0.5401E+00 0.6699E+00 0.1632E+01 -0.5139E+01
5 0 0.1559E+01 0.1144E+01 0.0000E+00 -0.9800E+01
6 0 0.2224E+01 0.1288E+01 0.0000E+00 -0.4900E+01
7 0 0.1912E+00 0.1305E+01 0.1663E+01 -0.3025E+01
8 3 0.1000E+01 0.2000E+01 -0.8471E+01 -0.2723E+01
9 0 0.3399E+01 0.6151E+01 0.1200E+02 0.3155E+02
4
1 1 1 2 5 4
2 2 6 5 2 3
3 1 5 8 7 4
4 2 5 6 9 8
62.596 21.249 -32.758 0.96870E-01
61.948 9.8381 54.321 0.85200E-01
21.019 44.812 45.486 0.92526E-01
-15.069 104.27 -103.93 0.11028
-50.536 18.529 105.46 0.10025
-54.947 33.161 91.325 0.10362
-11.718 32.504 117.97 0.89494E-01
-9.7154 17.872 129.69 0.86976E-01
21.962 8.2142 -4.1974 0.98174E-01
-0.20453E-01 13.036 7.8808 0.99204E-01
-33.571 37.568 -33.248 0.10611
-2.7830 29.571 -48.372 0.10477
10 COMPUTER IMPLEMENTATION



83.822 69.453 361.96 0.51329E-01
162.78 426.78 1702.4 0.29913E-01
475.04 996.13 2701.8 0.18205E-01
410.84 638.81 1376.1 0.24400E-01




8.5 SOLVER DETAILS

FLagSHyP uses a standard symmetric solver based on the LDLT decom-
position that is described in detail in Zienkiewicz and Taylor, The Finite
Element Method, 4th edition, Volume 1. In this procedure the symmetric
assembled tangent matrix is stored in two vector arrays: stifd containing
the diagonal part and stifp containing the upper or lower o¬-diagonal co-
e¬cients. The way in which the entries in the array stifp relate to the
columns of the sti¬ness matrix is determined by the pointer vector kprof
as shown in Figure 8.4.
In order to reduce the cost of the linear solution process it is necessary
to minimize the length of the o¬-diagonal array stifp. This minimization is
performed in FLagSHyP while allocating degrees of freedom to nodes using


5
3 4 3
4 3
6 3 2 4
1
1 2 1
1 2 1 2 3
1
2

FIGURE 8.2 Numbering of two-dimensional elements.


8 7 4
4
6
8 7
5
10
5
5 6
8
3 9 3
4
1
4 3
1
4 7
3 6
1 2 3
2
2
2 1
1 1 2 5

FIGURE 8.3 Numbering of three-dimensional elements.
11
8.6 CONSTITUTIVE EQUATION SUMMARY

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