Indian Journal of Engineering & Materials Sciences Vol. 6. June 1999. pp. 135- 143

Adaptive mesh refinement in finite element analysis

S Rajasekaran & V Remeshan

p.s.G. College of Technology. Coimbatore 641 004. Indi a

Received 27 April 1998; accepled / / November / 998

Among the acceptab le numerical methods. Finite Element Analysis stands as the most acceptable one for problems characterised by partial differential equations. However. inaccuracy in Finite Element Analysis is· unavoidable since a continuum with infinite degrees of freedo m is modelled into finite degrees of freed om. In addition to the mesh generation task being tedious and error prone. the accuracy and cost of the analysis depend directly on size. shape and number of d e ments in the mesh. The procedure of refining the mesh automatically based on the error estimate and distribution of the error is known as "adap ti ve" mesh refinement. A si mplified method called "Divide and Conquer" rule based on "Fuzzy Logic" is used to refi ne the mes h by using Ir , p and Irp versions . Automatic mesh generator developed in thi s paper based on Fuzzy log ic is able to develop well shaped elements. The program for automatic mesh generation and subsequent mesh re linement is developed in "C' language and the analys is is carried out usi ng "ANSYS" I package. Automatic mesh generation i ~ app licd to problems such as dam, square pl ate with a ho le. thi ck spherical pressure vesse l and a corbel and error less than 5% is achi eved in most of the cases.

The initial necess ity for automatic mesh generation was to reduce time and money spent for creating a mes h. whi ch could amount to half of the total time required for the Finite El ement Analysis . Adaptation inc luding both mes h optimi sati on and mes h refin e­ment. was shown to be very import ant in case of problems for loca li sed stress grad ient s and point loads.

In ea rl y 1970s. Zienki ewicz and Phillips6 intro­duced the technique of iso-parametric mapp:ng and most of numeri cal mesh ge nerati on techniques still empl oy mapp ing tcchniques . An attempt has been made to aut omate the mapping technique by Blacker and Stephensonc thi'ough the ultimate decomposition of the comp lex domain into mappable sub regions. Va lliappan (' I (//. 1-1 introduced fuzzy logic for aut omati c mapping tec hnique by di viding the region into sub regions by :-.e lecting optimum splitting lines. In thi s met hod " Di vide and Conquer" ru le by using fuzzy Logic through ex pert sys tem is used for auto mat ic mesh gcnerati on. By thi s method, the whole domain is di\ 'idcd into a number of sub reg ions and quadrilatera ls or triangular shapes and the mes h is ge nerated :-.eparatcly in each of them satisfying presc rihed co nditi ons and finall y sub reg ions are attached toge ther.

Error Estimation The tcrm "e lTor" is generall y applied to the

d if Terence het wec ll res ul ts of computati onal and the

actual phenomenon observed experimentally. The Fin ite Element Analysis yields the followin g errors: (a) errors due to spatial discretization, (b) round off errors, (c) approximation in volved in mathematical model. With the use of today's computers, the round off errors can be minimised assuming that the so lution of the mathemati cal model is exact. Hence, the error due to discreti zation is considered in thi s paper. Usuall y, posteri or approach is used to est imate the error. Finite Element Anal ys is is carried out for raw distribution and refined mesh. Based on the error between them, further refinement of mesh is carried out.

The error approx imation tec hnique used tn

ANSYS ' is similar to that one proposed by Zienki ewicz and Zhu 5. ~ . The estimati on of error in stress ca ll be done by using Eq. ( I):

. .. ( I )

where L\a is the error and a " is the nodal stress Ii

calculated by simple ave ragin g of element at a node

and a:: an averaged stress at a node. Then error norm

e j in the ith element is given by

... (2)

where [C] is the constituti ve matrix. The error norm 'e ' over the entire model can be assumed to be equal to

136 INDIAN J. ENG. MATER. SCI , JUNE 1999

/I

(' = L I', / :::: 1

.. . (3)

\\ 'here ' /1' is t he nu mber of the elements in the model or part of the modc l. ThL' error energy ' E' can be normali sed aga inst strain energy ' U'. Thi s is done in POSTI of ANSYS as

{'

E = ( ) I I c x I 00 (U + c )

. .. (4)

where

V = L V , .. (5)

where V , is the internal strain energy of the element i.

In ANSY S. the \',due ' c' is obtained as EN RM and the va lue of V-..IS SENE.

Refinement Methods Mes h refi nement tec hn iques can be categorised as:

a ) /r-\'(~ rs i o n in which refinement is done by :--ubdiv is ion of element , where ' fl ' stands for typica l geometric parameter li ke element length or diamcter. If thc refinement is being done throughout the lomain then it is uniform /r-refinement : else it is carried out locall y, and ca lled adapt ive /r-refinement.

h ) p-refi nclllent in whi ch refinement is done by increasing the in terpolating function order. If {I-ve rsion is app li ed th roughout the domain it is ca lled fJ -ve r:-, ion, else it is ca ll ed adaptive p­ve rsion.

c ) /rf)-ve rsion. whi ch combines both of the schemes desc ri bed abovc , i.e , simultaneous reducti on in the size of the e lement and increase in the po lynomial order. In the h­ve rsion finite element convergence study, it has bee n fo und that if the sequence of mes hes is designed in such a way that the error is approx imate ly equall y di stributed on each clement. the mes hes are called optimal meshes and the influcnce of singul arities is eliminated

... (6)

where

(' = posi ti vc co nstant depending upon the problem (' = error In energy norm N = the num ber of degrees of freedom {J = the polynomi al degree of used shape functions

p-version on the other hand converges exponenti ally if the exact solution is smooth and converges always faster than the fi xed pol ynomi al degree. The error estimate is then given by : e=e- {3cN . .. (7)

where f3 is a positi ve constant which depends on the

smoothness of the exact solution. If the mesh is not des igned properl y, then the p-version does not perfo rm so well for problems with singularities although its perfo rmance is always better than the h­version with uniform mesh with respect to the nu mber of degrees of freedom.

Finall y, the mesh size ' h' is refin ed in combinati on with increase in the degrees of the polynomial in a proper manner. hp-version converges exponenti all y even in the presence of singul arities in the exact solution. The error es timate for hp-version is given by

- aN8 e= ce . . . (8)

where a and e are pos iti ve constants depend ing

upon the smoothness of the exact solution of the problem and the finite element mesh.

Types of Elements Used Adapti ve mesh refin ement of the finite element is

carried out with or without the help of automatic mesh generati on. Finite Element Model requires that the domain of interes t be di vided into a mesh of di screte elements, and the accuracy and expense of the ca lcul ati ons are strongly affected by goodness of underlying mesh. An ideal mesh is charac teri sed by small element when stress gradients are hi gh and larger elements when stress gradients are small , elements being regul ar, or undi storted in shape. When quadrilateral elements are used, these cha racteri stics can sometimes be di fficult to achieve. In thi s paper four and eight node iso-parametric quadrilaterals are used. In case of eight node quadrilateral e l ment s, the triangul ar elements can be obtained by merging 4, 7 and 3 on one edge into one node.

Mesh generation algorithm

At the beginning of the mesh generat ion process, the domain boundary is di vided into a number of divi sions such that eac h divi sion has a size less than or equal to the spec ified element size, The re-entrant comer must be detected as given below.

A comer is said to be re-entrant comer if the included angle is > 180°. If any re-entrant comer is present, this polygon is call ed concave pol ygon. This

) ..

RAJASEKARAN & REMES HAN: ADAPTIVE MESH REFINEMENT IN FINITE ELEMENT ANALYSIS 137

corner is detected initially and the subdivi sion of the concave polygon is done along the splitting line from the re-entrant corner to make it convex . The ai m is to create co nvex polygon (a ll inc luded angles ~ 180°) so th<lt mes h genera ti on in each sub domain can easi ly be done. Consider two lines meeting at a comer. The precedin g line is denoted by " p" and the foll ow ing line as " p ' . There are twenty poss ibiliti es ex isting wi th respect to vari ous conditions.

After finding the re-entran t corner, the next step is to find the optimum sp litting line from re-entrant corner for dividing the concave polygon into convex polygo ll s. The selec ti on of optimum splitting line is carri ed out by using Fuzzy Logic. The basic idea beh illd the selecti on of sp litting line is that it should make all angles as c lose [ 0 90° as poss ible.

Fuzzy inference a l):orithm

Fuzzy al go ri thms are di vided into four basic categori es: Fuzzy definiti onal algorithm, Fuzzy ge nerat ional algor ithms, Fuzzy relat ional and be hav ioural algorithms and Fuzzy dec isional algo rithms. Fuzzy dec isional algorithm is adopted here to dec ide the optimum sp litting line.

n al .CXc./3I,/32are the four angles of the splitting

line, then the optimum sp litting line is calculated by maximising the fo ll owing functi on. Maximise:

I (a l • a 2' /31' /31 ) = I (a l ) +l (a 1 ) + f ([31 ) + I (/31)

= R(a l )+ I« a 2 )+ I«/3I )+ R( /3 2) ... (9)

If " r " is a scalar va ri abl e which takes the values of four cut angles (a l '(;(2' /31' /32) in the interva l [0 ,27r] ,

thell the criteria for the cut angles as close as poss ible to 90" call be ex pressed by the following fF-THEN rul es. (see Fig. I a and I b).

Rule I I I' . 1" is ve ry clo,c III l)0" then R ( I' ) is very large Rule 2 II' ' 1" i, c lose 1(1 ()O" thell R ( I' ) is large Rule 3 II' ' 1" is medium close tll <)0" then R (I') is med ium Rule 4 I I' . 1" is ra r rrom l)()" then R ( r) is small Rule 5 II' 'r' is very 1',11' rrom <)()" then R ( r ) is very small Rule (, If ' r ' is ve ry very rar rrullll)O" then R (r ) is very very sm311 Rule 7 If 'r ' is lOu f~lr rrom <)0" then R ( r ) is too small

Computation of fuzzy control output or de-fuzzilication

The possible splitting lines are the imaginary lines joining the re-entrant comer and the division points along the domai n boundary. Let PQ be one of the splitting lines as shown in Fig.2. Now to calculate the va lue of (/3t ' /32), the angles at "Q" consider the

adjoining di vis ion of d I. d2 along the boundary and so lve the two tri angles thus formed by applying cos ine rule to get (/31' /32)' Similarly following the same procedure one gets the value of angles (al. (X2)

by consideri ng two adjacent di vis ion points at the re­entrant corer. Any splitting line which is having any of its angles as zeroes, more than zero and less than

TF WF VF F MC C VC C MC F VF W F TF

o 15 30 45 60 75 90 105 120 135150165 180

Degrees

TF - Too Far; VVF - Very Very Far; VF - Very Far;

F - Fair; MC - Mediwn close; C - Close; VC - Very dose;

Fi g. 1 a - Member function for angles

VL

Rating Variable

TS - Too Small; VVS - Very Very Small; VS .- Very Small;

S - Small; M - Medium; L - Large; YL - Very Large

Fig.1 b - Membership function fur rating variablt:

138 INDIAN 1. ENG. MATER. SCl.,lUNE 1999

or equal to 180°, will be a candidate for competition for selection of optimum splitting line. After getting the four angles, we are normali sing it by subtracting it from 180°, if the angle obtained is >90°. In general, the input (angles) for getting Fuzzy rating value

should be < 90°.

Procedure Consider the angle (XI which is between 70° and

90°. Now, produce a vertical line to strike two Fuzzy sets very close and closes as shown in Fig.3 . From these intersecting points, produce a horizontal line to meet the corresponding Fuzzy sets of membership grade and draw triangles with its base equal to the base of Fuzzy sets in membership function and height equal to the height of the hori zontal line with vertex lying at the midpoint of the Fuzzy set. Rating value is calcul ated by taking moment about origin to find rating va lue. Similarly the rating of other three angles is calculated and added to get the rating va lue of the splitting line. The procedure is repeated for all other poss ible splitting lines and that havi ng max imum va lue is selec ted. If there are more than one re-entrant corners (hen (he procedure is repeated for the concave polygons till sub polygons are convex in nature as shown in Fig. 4.

Division of convex polygons into 4 sided quadrilaterals

The generated convex polygon (from previous

Fig .2 - Calcu lati on or rour ang les

Fig .3 - Calculation or cont ro l output

step) may contain more than 4 sides . Our aim is to get convex polygons of 4 sides or 3 sides so that mesh generation can be done in these domains easily . The Fuzzy logic based optimum splitting line is agai n calcul ated to divide convex polygons of more than 4 sides and the procedure is continued till we get all the convex polygons of sides three or four . A quadr ilatera l is accepted for generati on if the ratio between oppos ite sides is between 0.5 and 2.0 as: O.) ~ AB/CD~2.0 ... ( 10)

Sl11ootheninJ.: alonJ.: the splilli'ng line

Mesh generati on is carried out indi vidua ll y in eac h convex sub po lygon obtained by followin g procedure mentioned in pre vious paragraph . But continuity of the element s a long the boundary should be maintained. In order to achieve this, smoothening of th e element s al ong the sp litting line is carried ou t.

Algorithm Consider a sp litt ing line "AB" as shown in Fig. Sa.

The smoothening tec hnique works as fo ll ows.

I . Ca lculate the length of "AB" and divide the le ngth hy size of e lements specified in the input such that length of eac h di vision is equal to or less than that spec ifi ed size. and these points are called pi vot po int s. 2. Consider first pivot point "p" and let the node N I, N2 and N~ lying within the circle of influ ence of radi us equal to half of (A P). The smoothening is carried out by connec ting a ll nodes to the mid poi nt of the ex treme points N I and N3, i.e. co-ordinates of point N2 after smoothen ing are (XCNI)+X(N3» /2 and (Y (N 1)+ Y(N~ ))/2.

Q

p - -- --- --- -

~ A

Fig.4 - Di vision or concave pol ygon

-

RAJASEKARAN & REMESHAN: ADAPTIVE MESH REFINEMENT IN FINITE ELEMENT ANALYSIS 139

Thi s meth od avoids the large shifting of any node coming within the circle of influence if we are connective to pivot points.(see Fig.5b). Fig.6 shows the splitting line QI and Q2 consisting of uncommon nodes and the smoothened mesh.

The extra nodes which are formed during smoothening operation is removed by eliminating the larger values of nodes and renumbering the following nodes. Nodes having equal "x" and "y" co-ordinates are se lec ted and then node renumbering is done by following the continuation of the previous highest correc ted node .

Adaptive mesh refinement

The data obtained from the automatic mesh generation is g ive n for ANSYS solution and the result of this solution is analysed to find out whether refinement is requi red or not. The output from the ANSYS solution is obtained in the form of error in strain energy in eac h e lement (EN RM) and actual strain energy of the element (SENE). The total error percentage is given by the equation

ERROR = 100 x ~IENRM/(IENRM+ ISENE)

. . . (I I)

and the element to be refined IS found out by calculating

e =k~(IENRM+ ISENE)/ M ... (12)

where M is the total number of elements, k is a constant (acceptable percentage of error) ENRM=error in strain energy in element SENE=actual strain energy in element which gives the permissible percentage or error in element. If the error in strain energy in an element exceeds this value then that element is to be refined.

The element thus stored is then subjected to h/p/hp type of refinement. In this paper, the uniform h

~B

A

A

Fig .) - Silloothening along the '[1li lling line

version is adopted till one gets an error percentage around 15 %, after which the uniform p-refinement is applied after the adaptive p, adaptive h or adaptive hp type refinements are applied.

Uniform II-refinement

The coarse mesh generated initially is subjected to uniform h-refinement and the procedure is continued till one gets the error percentage around 15%.

The insertion of nodes in each element to be refined is done in counter clockwise direction . The co-ordinates of these nodes are calculated by averaging the co-ordinates of the end nodes of the s ide in which the middle node is inserted .

Centre node location is calculated by averaging the mid-point co-ordinates of the. line joining mid nodes in case of quadrilateral and by averaging the co­ordinates of the vertices in case of triangular element. During the insertion of the new node, node numbering is done continuously and there exist common nodes . The existence of the common node is found out by searching the nodes having equal x and y co-ordinates. These nodes are renumbered In

continuation with the highest corrected node .

Uniform p-relinement

In case of uniform p-refinement, the middle side nodes are inserted in each element by following the procedure described as before and finally node renumbering is done .

Adaptive II-relinement

Result the previous analysis is verified and if the error percentage is greater, adaptive h-refinement is

followed. Refi nement procedure is same as that Illcntioncd before.

AdaptiH' p-relineJ11ent

If the error percentage is still above acceptable Ii mi t after adapt i ve II-refinement then adaptive p­refinement is followed.

Fig.() - Example fur sillootheni ng along splilling linc

I-Hl INDIAN J. ENG. MATER SCI.. JUNE 1999

..\daptin lip-refinement

T he co mbinati on of II and p type refinement can be carri eu out i I' I' und necessary and the program works as fo ll ows . First it searches for the e lement surroundin g the e lements actuall y to be refined. Adap ti ve II- refin ement of these elements is carried out and th en adapti ve II-refinement of the actual c lement s to be refined is carried out. Subsequently, the new c lement s generated from the second step are sub jec ted to uniform !,-versi on. Finally , nodes renu mbering is ca rri ed out.

Numerical Exa mples 1

Va ri ous proble ms ha ve bee n so lved and the results arc found to be we ll be low the acceptab le percentage. A~ ~ln l' ngi nee ring measure . acceptable percentage of nror is taken as I 09'r . In each problem, only two types of refinement s (un iform h and uniform p ) are c; lrried out and with thi s it se lf acceptable results are obtain cd .

Exmpll' I-L-shapcd domain

This is a plane stress problem. Geometry and load in g condit ions are given in Fi g. 7a. Initial coarse lllcs h consists of 4 noded 48 quadrilateral elements ha\ 'in g 120" of freedom in tota l. Then, uniform h­re finement is performed to ge t 192 e lements having total D.O .F eq ual to 428. Further, uniform p­refin ement is done to increase the D.O.F to 1240 and with thi s anal ys is an error percentage of 8.928% is obtained. The results are given in Table I and adapti ve /i -refinement is shown in Fig .7 .b.

5 2 T E = 2.1 XlO N / mm t = 8mm \J = 0.30 -

40mm

t 40mm

1 .... 1·----- 80 mm -----i1

Fi g. 7a - L-shaped domain

Example 2-Dam

This is a plane strain problem. Geometry and loading conditi ons are shown in Fig. 8a and the initial coarse mes h consists of 112 elements with small number of triangular elements and mesh totall y hav ing 228 D.O .F. Then adapti ve h-refin ement is carri ed out and the error percentage of 9.765 % is obtained. Then adapti ve p-refin ement is carri ed out and error percentage of 3.365 % is obtained with 2654

D.O.F. The results are given in Table and the uniform h-refinement is shown in Fig. 8b.

Example 3-Square plate with circular hoI I'

Geometry and loading conditi ons are shown in Fig. 9 a and initi al coarse mesh of 92 e lements and 2 12 nodes is subjec ted to uniform h-refin ement and then uniform p-refin ement is app lied , With thi s, an error of 1.94% is obtained . The results are gI ven tI1

Table I and the adaptive h-refinement is shown in Fig,9b.

Example 4-Thick spherical pressure vessel

Geometry and loading conditi ons are as shown tI1

Fig. lOa. Initial coarse mesh consists of 72 e lements and 1780 of freedom and after uniform h-refinement and subsequent uniform p-refinement, error percentage of 1.788% has been reached as shown in

Fig.7b - Adaptive ii -refinement for L domain

RAJASEKARAN & REMESHAN : ADAPTIVE MESH REFINEMENT IN FINITE ELEMENT ANALYSIS 141

Table I-Error in adapti ve mesh refinement

Prohl em rcli nemcn t Elements nodes

L-domain ( m 48 65

lIh 192 225

li p 192 64 1

Dam cm 112 128

lIh 431 437

lip 43 1 1388

Sq plate em 92 11 3

lIh 367 409

lip 367 11 84

Th ick vese l em 72 93

lI h 284 329

lip 284 94 1

Corhcl CIll 133 162

lIh 524 589

lip 524 170 1

cill- coa rse rm:sh: IIIr- liniforlll h: IIp- liniform fI

I---- 25 m --I

L_ 2 25 m ,

E ~ 25500 x 10 kN /m 3

p = 25 kNI m \J = 0.15

75m

~I' ------- 100 m ------1-1

Fig.8a - Dam

Table I and the meshes at different stages are shown in Fig. lOb.

Example S-Corbel

Geometry and loading conditions are shown in Fig. II a and the initial coarse mesh consists of 133 elements hav ing 304 D.O.F. An error percentage of 4.89% is obtained after the uniform p-refinement, which is done in continuation with uniform h­refinement (see Table I) with 3250 D.O.F. The meshes generated lI sing adaptive h-refinement are shown in Fig. 11 b.

D.O.F SENE ENRM Error (%)

120 0.364574 0.025248 24.45

432 0.374306 0.009229 15.57

1248 0.37857 1 0.003035 8.928

224 67.7849 1.58 185 15.32

8 12 66.266 0.638026 9.765

2654 66.66 0.07555 3.365

2 12 0. 11 5784 0.002702 15. 11

792 0.116968 0.0009986 9. 14

23 18 0. 117457 0.0000442 1.94

178 14.8 11 0.24785 12.83

644 14 .9 1 0.085 1544 7.534

1854 14.96 1 0.004787 1.788

3 12 0.02987 0.00 19866 24.85

11 38 0.03083 0.000774 15.65

3250 0.03 1246 0.0000751 4.89

Fig.8 b - Uniform ii-refinem ent for a dam

Conclusions Automatic mesh generator deve loped based on the

Fuzzy logic is ab le to develop well shaped e lements. For various examples solved, an error percentage of less than 5% is achieved except in case of L-shaped domain where singularity affects further refinement. The deve loped automatic mesh generator has proved

142

t-. -l....;

INDIAN J. ENG. MATER. SCI. . JUNE 1999

80mm----_

2 E = 200k N / mm U = 0.20 t =l mm

Fig.9a - Square plate wit h a circul ar hole

--

.' -\;... ,,1\ :" ....... , t-... " "

t~ ~'~ ~tl: -I

i. I

~ \ \\\\\\\ \\\\\\1 \ t l \ , I I. T I \ t , \ I

Fig.9b - Adaptive h- refi nement for plate with a ho le

E = 25500 N/mm1

= 0. 15 = 0.000025 N/mm)

ALL DIMENSIONS IN mm ~ 30 - ---1

Fig. 1 Oa - Thick spherical pressure vessel

Fig. 1 Ob - Uniform h-refinement for vessel

RAJASEKARAN & REMESHAN: ADAPTIVE MESH REFINEMENT IN FINITE ELEMENT ANALYSIS 143

T 20rrm

t 20rrm

t lOrrm

1

UNIT

~ 20rrm

1

5 2 E~2xl0 N/rrm ., = 0.30 f=lrrm

-1-20 rrm~'_---- 40 rrm ---+

Fig.11 a - Corbel

suitable for multi-connected domain provided there are no re-entrant corner at the outer boundary of the given domain. Refinement strategy followed proves to be very effective in achieving the error percentage less than the acceptable limit.

Acknowledgements The authors thank the management and the

Principal, Dr. P. Radhakrishnan, P.S.G College of Technology for providing necessary facilities to carry out the work.

References I Anonymous, Users ManuaL (USA), (1989) 15342. 2 Blacker T D & Stephenson, M B, Int J Numer Meth Eng , 32

( 199 1) 811-847. 3 Valliappan S, Ph am, T D, Murthy V & Franklin J, Micro

Fig. llh - Adaptive h-refinement for corbel

Camplil Civil El1 g, 3( 1993) 1-8. 4 Valliappan S & Pham, T D, Micro Camp tI I CiviL El1 g, 8

( 1993) 75-82. 5 Zienkiewicz 0 C, Zhu J Z, Hinton E & Wu J, Int J I'Iumer

Meth Ellg, 32 (1991) 849-866. 6 Zienkiewicz 0 C & Zhu J Z, Int J Numer Meth Eng, 28

(1989) 2839-2853 . 7 Zienkiewicz 0 C, Zhu J Z & Gong N G, Int J Numer Meth

Eng, 28 (1989) 879-891. 8 Zienkiewicz 0 C & Zhu J Z, Inl J Numer Meth Eng, 32

(1991) 783-810; 879-891.