COMMUNICATIONS IN NUMERICAL METHODS IN ENGINEERING, VOl. 1 1,839-847 (1 995)

PLOTTING OF FRINGE CONTOURS FROM FINITE ELEMENT RESULTS

K. RAMESH, A. K. YADAV AND W A Y A. PANKHAWALLA Department of Mechanical Engineering

Indian Institute of Technology, Kanpur, India - 208016

SUMMARY A novel and simple approach to plot fringe contours from finite element results that simulates the fringe thickness variation as observed in experiments is presented. The procedure does not require solution of any non-linear equation. Shape functions are used as interpolation functions to plot fringe contours from the nodal values of fringe orders. The various fringe contours one usually comes across in stress analysis are also reviewed. The program code in Turbo-Pascal employing the new algorithm is given. Stress and displacement contours at the tip of a crack in an SEN (single-edge-notched) specimen are plotted using the code developed. The fringe contours plotted correspond to those observed in experiments such as photoelasticity, holography and Moire.

KEY WORDS validation of FE modelling; optical techniques; fringe contours; shape functions; stress analysis; displacement analysis

1. INTRODUCTION

The finite element method (FEM) is a numerical method for solving a given boundary value problem. ' Depending on the problem under consideration a suitable element or a combination of elements is chosen and the domain is discretized. Ideally, one would like to use a large number of elements to discretize the domain, but due to constraints on computer resources one is normally expected to optimize the discretization. Elements of smaller size may be used in zones of stress concentration, and elsewhere the discretization could be coarser. Manual discretization of the domain is tedious and time-consuming. The developments in automatic mesh generation' have helped in solving problems with complicated geometry more easily using E M . The advancements on adaptive mesh has addressed the problem of optimal discretization. The use of the Frontal technique4 for solution has made it easier to add or remove material with ease and hence has made FE analysis suitable for the design process.

Although the FEM is advantageous for a detailed parametric study of a problem, the adequacy of finite element approximation in terms of the choice of element and discretization needs to be verified at least for one configuration. This is best done by comparing the results of FEM with that of experiments.

Experiments can be of two types. One is experiments such as a strain gauge technique which yields point-by-point information and the other which includes optical techniques such as photoelasticity, Moire, holography, etc., that give the whole field information. The whole field

CCC 1069-8299/95/100839-09 0 1995 by John Wiley & Sons, Ltd.

Received 25 November I994 Revised 3 February I995

840 K. RAMESH ET AL.

information is obtained in the form of fringes, and they in general appear as broad bands. Whenever possible, the use of whole field techniques is the ideal choice for verifying FE results.

was confined to plotting line contours. The contours are generally plotted from the nodal values of FE results using appropriate interpolation functions. One of the simplest approaches is to use a linear interpolation function. The later research’ brought out the advantages of using shape functions themselves as interpolation functions, and this has come to stay as an accepted practice as the contours are smoothly bent and their exactness matches that of the finite element analysis.

In this paper, a novel approach is proposed to plot contours that appear like fringes. This makes the comparison easier and is also, in a sense, dramatic. The method proposed is simple to implement and uses shape functions as interpolation functions. Further, the approach does not require solving any non-linear equation. The efficacy of the method is shown by plotting fringes observed near the tip of a crack in a single-edge-notched (SEN) specimen subjected to uniform tensile load. Fringes corresponding to the technique of photoelasticity, holography and Moire are plotted.

The earlier research for post-processing of F%

2. FRINGE CONTOURS

In principle, one can plot contours such as u displacement, v displacement, aA, uy, t,,,, al, u2, etc. from the FE results. However, if the results are to be compared with experiments, it is desirable to plot contours that are observed in an experiment. The technique of Moire provides the contours of a, v displacements or r , 6 displacements depending on the type of gratings used. One of the most widely used techniques to visualize stress fields is the technique of photoelasticity. Photoelasticity provides the contours of difference in principal stresses, namely contours of (a, - a2). The technique of holography provides the contours of sums of principal stresses, i.e. (a, + az) contours. The contours are generally observed as fringes, and fringes are usually numbered as 0, 1,2,3, ..., etc., or 0-5,1.5,2.5, .. . , etc., depending on the specific optical arrangements employed.’ The fringes in general appear as broad bands. The thickness of the fringe is indicative of the gradient of the variable. The fringes are very broad when the gradient is small, and vice versa. Further, a zone of high density of fringes indicates a zone of stress concentration. Thus, a mere qualitative observation of the fringes can yield a wealth of useful information. In the case of Moire, the displacement is related to fringe order N as

where p is the pitch of the Moire grating. In the case of photoelasticity, the stress-optic law states

NF, (a, -a*) = - t

where F , is the material fringe value and ‘ t ’ is the model thickness. In general, the above set of equations can be expressed as

@ = N K (3) where rp is the field variable, N is the fringe order and K is the appropriate factor to convert fringe orders to field variables which is a function of the parameters governing a particular experiment. The fringe order at a point can thus be evaluated as $ / K . Essentially, in plotting

PLOTTING OF FRINGE CONTOURS 84 1

fringes using FE results, the nodal values of @/K have to be calculated initially, and using these, contours of N = 0, 1,2,3, . . ., etc., or 0.5,1.5,2.5, . . ., etc., as the case may be, are then to be plotted.

3. ENSURING CONTINUITY ACROSS ELEMENT BOUNDARIES

Finite element calculations directly provide the displacement information which are continuous across element borders. Unlike displacements, the strains and stresses are related to the first derivatives of displacements with respect to space variables. In order to plot stresses some precautions are needed to avoid ugly jumps or breaks. In every element, the stresses at gauss points (Figure 1) are calculated using

(0) = [DI(EI = [DI[Bl(dl

{ Q I "odes = [T 1 { Q 1 gauss pts

(4)

(5 ) Stresses at the mid-side nodes are calculated by averaging the stress values at the two adjacent corner nodes. For an eight-noded quadrilateral, with the gauss points shown in Figure 1, the transformation matrix [TI using the shape functions takes the following form:

The stresses at the four comer nodes are calculated using the following equation:

- 1 c12 c13 c14

[TI = [ !:i ~ 2 2 c23 c24]

c32 c33 c34

c41 c42 c43 c44

where

i = 1,2 ,3 ,4 I C; = N I + N8 ,/2.0 + N, J2.0 ~~2 = N2; + N5;/2* 0 + N6;/2. 0 ci3 = N3; + N,;/2* 0 + N,;/2* 0 ci4 = N4; + N, ;/2- 0 + N, ;/2- 0

Nii is the shape function for node j calculated for the co-ordinates of gauss point i.

Figure 1.

fringe

't Global L

X

Fringe contour in the natural element and its co-ordinates also

8

r=-l

1

transformation into global co-ordinates: the element shows the gauss points

in local

842 K. RAMESH ETAL..

If a node is common to n elements, the stress value a; assigned to the node is calculated as

1 "

n j = l

fJ;= - c a; (7)

If contours corresponding to (a, - cr2) have to be plotted, one has to find ax, a,, and t, at the node using the above procedure and then the difference of ( a1 - az) has to be calculated using these. Finally, the fringe order at the node is calculated as N; = (a, - oz)ir/Fo. Typically, the values of N for all nodes are calculated and stored in a file for plotting contours.

4. THE NEW PLOTTING SCHEME

The earlier algorithms5-* have concentrated on finding the points within an element for which the field variable corresponds to a given contour. This necessitates the use of solving non-linear equations in determining the co-ordinates of the points forming the contour. Further, one

Mesh

t s 3 6 2 4 7

t i

3

--r

( b) ( C )

Figure 2. Information regarding the graphics module for plotting fringe patterns: (a) a typical mesh along with the window chosen; elements outside the window are not scannet!; (b) a typical element enclosed in a rectangle to

calculate pixel increments in x and y directions; (c) element in its natural co-ordinates in which scanning is done

PLOTTING OF FRINGE CONTOURS 843

requires to take special care if a particular contour has several branches or if the contour closes itself within an element.

The new plotting scheme employs an inverse approach. Rather than finding points corresponding to a specified contour, each element in the domain is scanned discretely and the field variable, for which a plot is to be made, at each of these points is calculated. A check is then made to see whether the field variable calculated is equal to a contour value. If so, then the point is plotted. The quality of the contour depends on the scanning interval. The finer the scanning interval, the better is the resulting contour. In graphical displays, the screen is divided into an assembly of pixels. If the scanning interval within an element is equal to the pixel size or finer than the pixel size then one obtains a contour that is visually smooth.

In the implementation of the scheme, the user is asked to specify a window. Only the elements within the window (Figure 2) are scanned to plot for the fringe contours. For each of the elements, the lengths xext and yext are calculated. Depending on the display unit (VGA, CGA or HGC monitor-in the program shown below, the variables XScreenMaxGlb and YMaxGlb returns the number of pixels in the x and y directions of various monitors) the number of pixels representing the lengths xext and yext are then determined. These are labelled as npixx and npixy in the program. The element is scanned in its local co-ordinates using the scanning increments ri = 2/npixx and si = 2/npixy as shown in Figure 2 . For each of the points scanned, the global co-ordinates and the fringe order (variable f rn in the program) at the point are calculated, using the shapefunctions as interpolation functions, as

where x i , yi are co-ordinates of the nodes of the element and x g , y , are global co-ordinates of the point, N , , ..., N 8 are the shapefunctions, f m , , f m 2 , ..., frn8 are nodal values of fringe orders, and f m , is the fringe order corresponding to the present point being scanned.

program fem-fringe-contours; uses dos,crt,gkernel,gdriver,gshell,gwindow;

const

label

type

va r

nodmax=1900;

200;

coorarry = array [ I ..nodmaxl of real; shfar = array 11 ..81 of real; kelar= array 11 ..91 of integer;

i,j,ii,jj,iii, ia,ja,nel,nnod,gpixx,gpixy,npixx,npixy,tem:integer; xwinmax,xwinmin,ywinmin,ywinmax,al ,a2,a3,a4,a5,a6:real; s, r,si, ri,xx 1 ,xy 1 , f rn. xext 1 ,xext2, yext 1 ,yext2,xext,yext:reaI; x.y,fringe:coorarry; frngl ,tx,ty,shf:shfar; ke1,ncon:kelar; pcoor:plotarray; fin.fin1 :text; name,pic-file:string; ch:char;

clrscr; writeln1' Copyright (c) 1994 K.Ramesh, IIT Kanpur'); writeln('Enter input file name of FEM data:'); readln (name); assign(fin,name); reset (fin); readln(fin,nnod,nel); Innod - total number of nodes. 1

begin

844 K. RAMESH ETAL.

for i:=l to nnod do {nel -total number of elements}

writeln(’Enter input file name containing nodal values’); :’);

readlnhame); assign(fin1 ,name); reset(fin1); for i:=l to nnod do

read(fin1 ,fringelil); {fringe - nodal values of fringe orders} writeln(’ENTER WINDOW SIZE (Xmin,Ymax,Xmax,Ymin)’);

readln (xwinmin, ywinmax,xwinmax, ywinmin);

I nitg ra ph ic; Definewindow(1,O. 11 ,x+maxglb,ymaxglb); Defineworld(1 ,xwinmin 1.35,ywinmax,xwinmax*l.35,ywinmin); selectworld(1); selectwindow(l1; Drawborder; SetWindowModeOn; SetClippingOn;

gpixx: =round (XScreenMaxGlb/ ((xwinmax-xwinmin)) );{pixellunit length in 1 gpixy:=round(YMaxGlb/((ywinmax-ywinmin))); {global x and y directions}

kel I1 1: =I ; ke1121: =5; kelI3 1: =2; ke1141: =6; kelI5 1: =3; ke116 1: =7; ke117 I: =4; keU81: =8; ke119 1: = 1 ; for i:= 1 to nel do begin Ii loop}

for j:= 1 to 8 do

for ia := 1 to 9 do

read In (fin, x li 1, y I i 1 1;

writeln(’of contour variable to be plotted

read(fin,ncon[ jl); {reads nodal connectivity data}

ja:=nconIkelliaIl; pcoorIia.11 := xljal; pcoorlia.21 := y[jal;

for j:=1 to 8 do {draws the element} drawline(pcoor[ j, 1 I,pcoor[ j,2 1, pcoorI j + l ,I 1,pcoorI J+ 1,2 1);

{Determination of increment for scanning in local co-ordinates} xextl:=pcoor[l, l l ; yextl:=pcoor[l,21; xext2:=pcoorIl,l1; yext2:=pcoorII ,21;

begin

end;

for ia := 1 to 8 do begin

if(pcoorIia,l]> xextl) then xextl:=pcoorIia,l I; if(pcoor[ia,l J c xext2) then xext2:=pcoorIia,ll; if (pcoor[ia,2] > yextl) then yextl:=pcoortia,21; if (pcoor[ia,21 < yext2) then yext2:=pcoorIia.21; f rngl [ia 1 : = fringelnconlia 11; tx[ia I: = xIncon[ia 11; tyIia 1: = yincon lia 11;

end if (xextl < xwinmin:l.35) then goto 200; if (xext2 > xwinmax 1.35) then goto 200; if (yextl < ywinmin) then goto 200; if (yext2 > ywinmax) then goto 200; xext: =xextl -xext2; yext: =yextl -yext2;{extent of ele,ment in global co-ordl npixx:=round(gpixx xext+l); npixy:=round(gpixy yext+l); ri: =2.0/npixx; si: =2.0/npixy;{scanning increments in local co-ord}

s:= -1 .O; {Plotting of contours} for ii:=1 to npixy do

begin {ii loop} r:= -1.0; for jj:=1 to npixx do

begin { jj loop}

PLOTTING OF FRINGE CONTOURS 845

a1 : = I .O-r; a2:=1 .O+r; a3: = I .O-s; a4:=1 .O+s; a5:= (1 .O-s+qr(Q)/2.0; a6:=(1 .O-sqr(s))/2.;; shf [ I ]:=a1 *a3* (-r-s-I )/4.0;shf [2 ]:=a? a?* (r-s-1 V4.0; shfI31: =a2*a4 ( r+s- I )/4.Cl;shf [4l:=al a4 (-r+s-I 114.0; shf[51:=a5*a3; shf 161:=a6*a2; {shape functions for eight 1 shf[71:=a5 a4; shf [81:=a6 a1 ; {noded quadrilateral element) xxl : =O.O; xyl := 0.0; frn: =O.O; for iii:=l to 8 do

xxl : =xx1 +shf [ii i l:tx[iii 1; {Transformation from local} xyl : =xyl +shf [iiiJ ty[iii 1; {to global co-ordinates} frn:=frn+shfIiiil frngl Iiiil;{contour value at the current 1

end; {point using shape functions a s interpolation function}

begin

{Checks whether a fringe passes through the current point] if(abs((frn)-round(frn)) <= 0.1) then drawpoint (xxl ,xyl); { if 'yes' the point is plotted} {one could elaborate the above two statements to plot} {different contours with different colourslsymbols 1

r: =r + ri; if (keypressed) then goto 200;

end; { j j loop} s: =s+si; end; Iii loop}

200: end; {i loopl

ch:=readkey; if((ch = 'p') or (ch = 'P')) then hardcopy(false,l); pic-file: ='plot. pic'; savescreen (pic-file); Leaveg ra ph ic; end.

It has been mentioned earlier that fringes in general appear as broad bands. This effect is simulated by plotting the points that lie in the range N* e rather than just N . Usually the value of e is of the order of 0.1. For plotting fringe contours 0, 1,2,3, . . . , etc., when the fringe order calculated (variable frn in the programme) satisfies the following condition:

(9) abs(( fm) - round(frn)) ~ 0 . 1

the corresponding point is plotted. For plotting contours 0.5,1-5,2.5, . . . , etc., the condition changes to

abs ((fm + 0.5) - round(frn + 0.5)) s 0.1 (10) The program code shown above is written in Turbo-Pascal using the Graphix tool box. The

program has the provision for printing the contour plotted using a dot-matrix printer and also has a facility to store the plotted image as a file.

To illustrate the use of the above algorithm, the stress field and displacement field in the neighbourhood of a crack in an SEN specimen is plotted using FE results. The crack tip is modelled using quarter-point elements, and in all 452 eight-noded quadrilateral elements with 1439 nodes were used to discretize one-half of the specimen. The recommendations of Saouma and Schwemmer" are used as guidelines in modelling the crack tip. The size of the crack-tip element used was 1.33% of the crack length, and a reduced 2 x 2 integration scheme was employed to evaluate the element stiffness matrices. The non-dimensional stress intensity factor (K,/a,d(na)) calculated using FEM is 1-6737, the theoretical value" for the same is 1-6628

846 K. RAMESH ETAL.

Figure 3. Mode-I crack-tip stress field contours for isotropic material (SEN specimen, w = 25 mm, a / w = 0.3): (a) isochromatics-contours of (ol - u2); (b) isopachics-contours of (ul + 02)

Figure 4. Mode-I crack-tip displacement field contours for isotropic material (SEN specimen, w = 25 mm, a / w = 0.3); (a) u-displacement field; (b) v-displacement field

and the error is only 0.66%. Figure 3(a) shows the isochromatics, namely (al - a*) contours, and Figure 3(b) shows the isopachics, namely (a, + u2) contours. Figures 4(a) and 4(b) show the u displacement and v displacement fields. These results compare well qualitatively with those reported in Reference 9. On a PC386 machine with a VGA monitor each of the plots took less than 3 minutes to draw.

5. CONCLUSIONS

The new plotting scheme is very simple and the program code is incredibly compact. The main advantage is that the post-processing of FE results can be done on PC-based equipment and high-quality plots can be obtained using a dot matrix printer, which is cost-effective. Although

PLOTTING OF FRINGE CONTOURS 847

the scheme has been used to plot stress and displacement field contours using only eight-noded quadrilateral elements, the scheme can be extended for plotting temperature and flow fields and also to cover the entire family of isoparametric elements.

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