Automatic mesh generation with tetrahedron elements

INTEiRNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEFRING, VOL. 18, 273 289 (1982)

AUTOMATIC MESH GENERATION WITH TETRAHEDRON ELEMENTS

NGUYEN-VAN-PHAI

Knorr-Rremse GrnhH, Miinchen, W. Gennaizv

SUMMARY

To reduce the manual work involved in the application of the FEM in practice, preprocessors can be applied for the construction of network structures, which are complicated in generation strategy and do not form any optimum discrete structure. The time necessary for generation can be minimized even more by the application of only one element type within the whole network structure. A technique for the automatic generation of 3D-network structures with tetrahedron elements is presented in this paper. In this proposed technique, the nodal points of the network structure must be defined manually before the generation procedure, since a random positioning of points is usually undesirable for FEM calculation. The nodal points are connected by a program to a network structure consisting of tetrahedron elements which have optimum form for the numerical computation of the element matrices. After the generation, the element side.; forming any part of the boundary surface of the network structure can be automatically identified. If necessary, the network structure can be automatically refined.

INTRODUCTION

The choice of the type of element for the application of the FEM has a decisive influence on the obtainable accuracy of the results. This resulted from the efficiency examinations on plane and axisymmetric stress concentration problems by Schnack.' If these examination results are applied to generaliy three-dimensional problems, the tetrahedron element with linearly varying strain, developed by Argyris,* has the optimum qualities. This element, however, can hardly be used in practice because of great difficulties concerning the manual construction of a network.

Techniques exist for automatic mesh generation which make possible the application of the tetrahedron element. But they are too complicated in generation strategy and their application necessitates great manual work.

According to the technique of Zienkiewicz and Phillips,? the three-dimensional solid is divided into several subregions by the application of isoparametric hexahedron elements, which are defined by 20 nodes. These subregions are split up into smaller hexahedrons by indicating the number of the divisions along the local co-ordinates. These are divided finally into tetrahedron elements. The disadvantages of this technique are the large amount of manual work involved and the regularity of the network structure.

After Kame1 and E i~ens t e in ,~ a network structure consisting of triangular elements is tirst constructed for the surface of a three-dimensional solid. Then a central point is defined inside the solid. The nodal points on the surface are connected with this point to form narrow tetrahedrons. The tetrahedrons are divided into pentahedron segments and finally the pentahe- drons into tetrahedron elements. By an iteration procedure the final number of nodal points and their co-ordinates are then automatically determined. The disadvantage here is that many elongated elements with small angles are produced on the boundary surface of the network structure.

OO29-S98 1/82/O20273-17$01.70 @ 1982 by John Wiley & Sons, Ltd.

Recciwd December 1980 Recised Apri l 1981

274 NGUYEN-VAN-PHAI

A new technique for automatic mesh generation is presented in this paper, which in contrast to the above-mentioned procedure performs more efficiently and has a larger applicability. Here the number and the co-ordinates of the nodal points are needed as program input data. The points have to be determined before the mesh generation and can be located arbitrarily in the solid and on its surface, according to the problems to be handled. They are mutually connected to a continuous 3D-network structure, consisting of tetrahedron elements, by a calculating algorithm, in such a manner that the elements do not penetrate each other and have an optimum form for the numerical calculation of the element matrices.

When the mesh generation procedure is finished, the parts of the surface of the solid, approximated by triangles as tetrahedron sides, can be registered automatically for the consideration of the boundary conditions.

In addition, with this proposed technique a possibility exists to refine automatically the constructed network structure. Thereby, according to a particular principle, every tetrahedron element is split up into eight smaller ones.

GENERATION STRATEGY

The procedure described in the paper of three-dimensional mesh generation consists in forming tetrahedron elements from a particular number of given points in a defined 3D-domain by laying straight lines and plane triangles.s Thereby the elements ought to form a continuous network structure and have an optimum form.

The automatic 2D- and 3D-mesh generation is based on two elementary laws of analytical geometry: (1) every polygon is divisible into triangles; (2) every polyhedron is divisible into tetrahedrons. This means that every plane can be represented by a network structure consisting of arbitrary triangles without any blank spaces. Thereby the given points on this plane are connected by lines with no intersections. The boundary contour of the plane is approximated by the sides of the outer triangles (Figure 1).

't L_,

Figure 1. 2D-network structure consisting of triangle elements

Analogically to this, every three-dimensional solid can be represented by a 3D-network structure consisting of tetrahedron elements. The solid surface is reproduced by the sides of the outer tetrahedrons (Figure 2).

In the well-known automatic mesh generation technique in the plane by Frederick et al.,' all given points within one 2D-domain are surrounded in order to form triangles from these points. The already formed triangles are taken into consideration so that no overlapping of the triangle elements may occur (Figure 3).

AUTOMATIC MESH GENERATION 275

Figure 2. 3D-network structure consisting of tetrahedron elements

k I

Figure 3 . Generation strategy for 2D-network structures by Frederick rr ill.'

The strategy of the automatic mesh generation technique in space proposed here is constructed analogically to that in the plane described above in such a manner that all connecting lines between two nodal points within the 3D-domain are Surrounded in order to form tetrahedron elements from these lines (Figure 4).

A difference to mesh generation in the plane is that all points to be surrounded are defined there as nodal points of the network structure before the generation procedure. In space, however, the nodal points are also given, but the lines which come gradually into existence during the generation procedure, that is during the laying of the tetrahedron elements, must be surrounded. For this reason, all lines and triangular planes which come into existence in the course of the generation procedure are to be recorded. In order to obtain a complete network structure all lines must be surrounded.

Figure 5 clarifies the generation procedure. First, one line (1-2) to be surrounded is fixed. The basic triangle (1-2-3) is supposed. An optimum point (4) is sought in order to form the tetrahedron element (1-2-3-4) for the basic triangle (1-2-3). The newly formed connection

0

P

r

0

9 S

Figure 4. Generation strategy for 3D-network structures by the authoi

276 NGUYEN-VAN-PHAI

L -3

(1 -2 1 l i ne to be s u r r o u n d e d

(1 -2 -3 ) basic t r iangle 3

1 - 3

Figure 5. Surrounding procedure around a line

lines and triangles are registered. The triangle (1-2-4) is now used as a new basic triangle for surrounding the line (1-2), while point (4) is fixed as a new point ( 3 ) . After the line (1-2) has been surrounded, the next line (1-2) is handled. In order that no penetration of the elements occurs the already present tetrahedron elements are considered during the surrounding of a line by jumping over them. When all existing connection lines are surrounded, the 3D-network structure is complete.

In summary, the generation strategy has the algorithm, shown in Figure 6, in which 1 9 1 - 2 is called the surrounding angle of line (1-2). If 81-2 equal 360", the line (1-2) is totally surrounded.

Determin ing the l ine to be s u r r o u n d e d

( 1 - 2 ) ~~

_-

D e t e r m i n i n g the bas i c t r i a n g l e

( 1 - 2 - 3 )

I Looking for t h e op t imum point ( L l a n d forming the t e t r a h e d r o n

( 1 - 2 - 3 - L ) 3

I ,

51-2 = s u r r o u n d i n g ang le

Figure 6. Flow chart of the generation strategy


L " L

1 spherical t r long le

Figure 7. Optimum tetrahedron element (1-2-3-4")

As the tetrahedron elements of the network structure should have a favourable form for calculating the element matrices-they should not have too acute angles-the optimum point (4) must always be found from the total number of the given points for the basic triangle (1-2-3). The point forming the greatest spherical excess is the most favourable point (Figure 7). The spherical excess is:

a = A + B + C - n (1)

where A, B and C are the three angles of the spherical triangle formed by the penetration points of the connection lines between (4) and (1-2-3) and the sphere having a radius of uni t length and point (4) as its centre.

In order that the connection lines on the surface of the solid can also be surrounded, auxiliary points must be specified outside the solid. The elements formed by the auxiliary points will be ignored afterwards. A boundary point may only be connected to a boundary triangle if the tetrahedron so formed lies within the solid. To avoid the formation of tetrahedron outside the solid, there must always be an auxiliary point available which forms a greater spherical excess with this boundary triangle than any real point at the boundary (Figure 8).

The auxiliary points must be located in such a way that the spherical excess to a boundary triangle must be greater from a real point within the solid than from an auxiliary point. This condition guarantees that for a line on the surface of the solid to be surrounded the direction of the surrounding remains constant during the whole procedure (Figure 9). In addition, it must be possible to connect all the auxiliary points together to form an enveloping surface, which does not intersect the solid. This condition guarantees that every line on the surface can be surrounded (Figure 10).

At the narrow corners of a solid it is difficult to find the exact positions of the auxiliary points which fulfill the above-mentioned conditions. At such critical positions of the surface, the latter is divided into surface subdomains. The surface subdomains are defined by the nodal

o real point

aux i l i a ry p o i n t 6' > 6"

Figure 8. The position of an auxiliary point

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Figure 9. Positions of the auxiliary points

Figure 10. Positions of the auxiliary points

points and their connection lines. Only the points which are located in the same subdomain may be connected to form tetrahedron sides (Figure 11).

During the search for an optimum point (4) for the basic triangle (1-2-31, the amount of calculation can be reduced by dividing the solid into several volume subdomains before the mesh generation. This division is achieved by numbering the points into different groups. Only the points located in one volume subdomain may be connected to form tetrahedron elements (Figure 12).

Figure 11. Arrangement of the subdomains within a partial surface of the mlid


* 6 11

SiJbdornain 1 node 1 t o node 6 S u b d o m a i n 2 . node i, to node 11 S u b d o m a i n 3 node 0 t o node 15

Figure 12. Division of the solid into volume domains

The conditions (I) to (V) describe all the instructions for the selection of the optimum point (4) and the formation of the tetrahedron element (1-2-3-4). The sets are defined as follows:

NORG = Set of the real points. KRA =Set of the points located inside a surface subdomain.

IBE = Set of the points located inside a volume subdomain. VO = Volume set of a solid already filled up with tetrahedron elements.

V01-2-3-4 = Volume set of a solid occupied by the tetrahedron element (1-2-3-4). IEL = Set of the points forming tetrahedron elements according to particular principles.

Condition (I) describes that the point (4) and the basic triangle (1-2-3) must be located in the same partial volume. One of the following conditions must be fulfilled: the point set (4) may not be a subset of the set of the real points NORG; or both of the point sets (4) and (1, 2, 3} must be a subset of IBE at the same time.

NGUY EN-VAN-PHAI 280

Condition (IIj must exist in order that no tetrahedron elements outside the solid can be formed at the small angles of a solid. If the point set {1,2, 3 } describing the nodes of the basic triangle is a subset o f KRA, (4) must not be a subset of KRA.

Condition (111) guarantees a constant sense of rotation of the surrounding angle 8 around the line (1-2) and thereby a total surrounding of this line. The volume of a tetrahedron (1-2-3-4) in Figure 13 can be calculated for the Cartesian co-ordinate system from the following determinant:

( 2 ) 2 4 - 2 1 ~

l -y4- -x i y4-4‘1

v01-2-.+~-4=: I . Y 4 - X 2 y4--yz 24--zz 1 / x 4 - x 3 y 4 - y 3 z 4 - z 3 i

<“O, 2 . 2 L ’C LO1.2.3-i <G

Figure 13. Different mangements of node? of a tetrahedron element

x. y and I are co-oidinates of the corners of the tetrahedron. The volume VO, : has a different sign depending on the arrangement of the nodes as in Figure 13. If the sign remains the same during the surrounding of the line (1-21~ the sense of rotation will remain the same. too, for the total surrounding process.

The relation in condition (IV) indicates that the newly existing tetrahedron element does not penetrate an already existing one. The volume set VO,_i 34 is not a subset of VO. In order to he able to meet this demand, it must be calculated and questioned whether or not the lines newly formed as tetrahedron edges pierce already existing triangles in space. The same demand is valid conversely for the newly formed triangles and the exisring lines (Figure 14).

Figure 14. Penetration of n tetrahedron element and an existing triangle

AUTOMATIC MESH GENERATION 28 1

The conditions (I), (II), (111) and (IV) are applied for all nodal points during the search for the suitable point (4) for the basic triangle i 1-2-3). Finally the optimum tetrahedron I 1-2--3-4) is chosen as a new element, condition (V). The best tetrahedron element has the greatest spherical excess 6 from the angle point 14j to the basic triangle (1-2-3) compared with all tetrahedron elements which were considered during the search procedure.

For the guarantee of the generation procedure, also at the rough transition points from a fine to a coarse network structure, the edges of the tetrahedron elements ma?; not run too close to one another. This means that the penetration point of a line and the plane of a tetrahedron side must not only be located outside the triangle but must also not be located too close. Figure 15 describes the postition of a penetration point. The total of the surface

\ I D

A DIJ

Figure 15 Position of penetration point of a line and the plane of a tetrahedron side

areas of the triangles formed by the penetration point D and the three sides of the triangle (i, j , k) must be greater than the product of the surface area of the triangle and one constant factor. This factor indicates how close a penetration point may be located to the triangle and is estimated generally to be 0.1 :

A D i ; + A D , k +AD, ) O * l A i l k ( 3 )

After the new element (1 -2-3-4) has been formed it must be decided if it is a real element, being located inside the solid, or an auxiliary element, having been formed by the auxiliar) points and being located outside the solid. If (1-2-3-4) is an auxiiiary element, it will be ignored. This process is described by the following flow chart, consisting of the conditions (VI), (VII), (VIII! and (IX).

Conditions (VI) and (VII) describe whether the newly-formed element is a real one or an auxiliary one. If one of the point sets {3} or (4) of the tetrahedron is not an element of the set NORG, it is an auxiliary point and (1-2-3-4) is located outside the solid (Figure 16).

If (1-2-3-4) is a real element, IEL becomes E L + 1. At the same time the set VO becomes VO + V O I - ~ - ~ .4. The newly-formed triangles and connection lines are recorded.

It is questioned by condition (VIII) whether the line (1-2) has been already totally surrounded. If this is the case, the line (1-2) will be marked with ICIRC = ICIRC+ 1. If the node (4) of the newly-formed tetrahedron element is equal to the node (3) of the first hasic triangle (1-2-3) during the surrounding of the line (1-2), the latter has been totally surrounded.

After (1-2) has been surrounded, it is examined if all already existing lines inside the network structure are surrounded. If this is not the case, a new line (1-2) will be supposed. The surrounding procedure for this line takes place once again. When the number of surrounded

282 NGUYEN-VAN-PHAI

~ Determining t n e l ine , to be s u r r o u n d e d i

.- , Determining the I

i I basic t r i a n g l e i

' I f l - 7 - 3 1 - - - - - . - A

1 ;-Looking for the I i j I op t imum point I L I 7 - - c- - - - -. - I :

no-- IRC= LINE2 +3 lines is equal to the number of connection lines present inside the network structure, that is when the equation in condition (IX) is fulfilled, the total generation procedure is finished.

L I EII N O R G N O R G ( 3 1 9 NORG , 3 : E N O R G

Figure 16. Real element and auxiliary element

AUTOMATIC REGISTRATION OF THE BOUNDARY SURFACE

After the mesh generation is finished, any part of the boundary surface of the network structure, approximated by tetrahedron sides, can be automatically recorded for the consideration of the boundary conditions.


As the network structure is constructed automatically and the procedure cannot be directly influenced, one does not know its construction and thus the approximated form of the surface of the solid by element sides before its completion. Now all those triangles are recorded which are located on the surface of the solid. The whole surface of the solid is defined by these boundary triangles.

All triangles existing as tetrahedron sides in the network structure are then examined to see how many tetrahedrons they belong to. The triangles which are part of only one tetrahedron element are located o n the surface of the solid. Condition (X) describes this fact. {FAIl, 2, 31 is the point set consisting of three points 1, 2 and 3 which form a particular triangle on the surface of the solid:

(XI

Only one point set IEL exists, so that {FAIl, 2,3} is a subset of IEL. As the boundary conditions can only act on whole elements at the surface of the solid,

‘boundary limitation lines’ must be defined for these conditions before the mesh generation procedure. By this it is guaranteed that tetrahedron sides are formed completely inside or outside the ‘boundary limitation lines’.

The ‘boundary limitation lines’ enclose the boundary triangles which are subject to the direct influence of the boundary conditions. They are defined by the nodes which lie upon them. The nodes are joined as initial point and final point of several lines in a counter-clockwise direction at one part of the surface of the solid (Figure 17). Any part of the boundary surface

3 ! IEL: {FA/ 1 ,2 ,3 ) c IEL

Bmundary l imitat ion line B o u n d a r y l imi t a t ion l i n e 1 - 7 - 3 - 5 - 8 1 - 8 - 5 - 3 - 7

Figure 17. Different parts of the boundary surface

within one boundary limitation line may have any shape in space, but must be without any gap. Now the boundary triangles are sought for any part of the boundary surface enclosed by

the boundary limitation lines. For one defined part of the boundary surface, those boundary triangles ar are sought first which are directly located at the boundary limitation line successively in a counter-clockwise direction (Figure 18). Conditions (XI) and (XII) are applied here.

{LIIl, 2) is the point set consisting of two points 1 and 2 which form a particular line located on the boundary limitation line:

3 !FA: {LIIl, 2 ) c F A

AFAA>O

There is only one single boundary triangle, described by the point set FA, which has a side which is identical to one section {LIIl, 2) of the boundary limitation line, and is located within the defined part of the boundary surface.

284 NGUYEN-VAN-PHAI

1

4 - b

Figure 18. Search procedure for a defined part of the boundary surface

If the nodes at the boundary limitation line are considered in a counter-clockwise direction the surface area FAA of a boundary triangle will be positive, if i t lies within the domain defined by the boundary limitation line, condition (XII).

The outer sides of the newly-found triangles lie on the boundary limitation line. The inner sides of these triangles form a new line which is now treated as a new limitation line, to which only condition (XI) is now applied. This procedure is carried out as long as no more new limitation lines come into existence.

AUTOMATIC MESH REFINEMENT

This mesh generation procedure allows the possibility of automatically refining the constructed network structure. This is done by dividing each original tetrahedron element into eight smaller elements by a fixed principle. First the additional corner points are fixed for the new elements. They are located precisely in the centre of the edges of the old tetrahedron element. These and the nodal points of the old element form the corner points of the eight new tetrahedron elements, after they have been connected to each other by straight lines (Figure 19). So that the refined network structure corresponds better to the surface of the solid, the co-ordinates of the new tetrahedron points, located at the boundary, must be manually corrected assigning them new values.

old element

1 - 2 - 3-1

n e w e lemen ts

8 - 9 - 1 0 - L 5-9-6-7 1 - 5 - 7 - 8 8- 9-5- 7 2-6-5-9 6 - 9 ~ 10 - 7 3 - 7- 6 - 10 9 -8 - 10 - 7

$ 3 ...- . .--

I

Figure 19. Refinement specification for a tetrahedron element

EX AMPLE S

The network structure for the crankshaft of a single cylinder motor serves as a first example for mesh generation. In order to reduce the amount of calculation, the substructures of the


78 nodal points 27 nodal points 66 nodal points 92 nodal points 1L7 elements L8 elements 1LO elements 175 elements

Figure 20. Network structure of the crankshaft of a single cylinder motor

i 1

X

Figure 21. Infinite solid with a spherical cavity under uniform stress (Leon problem)

286 NGUYEN-VAN-PHAI

network are constructed first, which will finally be combined to form the whole structure (Figure 20).

The following three-dimensional notch p r ~ b l e m , ~ by means of which the stress distribution of an infinite solid with a spherical cavity under uniform stress is calculated, shows the advantages of applying this automatic mesh generation procedure in elastostatics. Here the stress fields are calculated by the algorithm for the determination of the element stiffness matrices which was developed by Schnack* and Waif'.'" for plane and axially symmetrical triangle elements and extended for the tetrahedron element by the author.'."

Figures 21-23 show the problem and the network structures for the solution of the problem. The first structure has 43 nodal points and the second one 85 nodal points. Figure 2.5 shows the third network structure consisting of 133 nodal points and 464 tetrahedron elements. In this case, a basic network structure consisting of 27 nodal points was constructed first, according to Figure 24. Then it was automatically refined.

The calculated notch factors ( C T J C T ) , , , ~ ~ for this problem, compared to the known solutions, are shown as results in Figure 26. With the first network structure, consisting of 43 nodal points, a value for (cT~/cT~J,, ,~~ = 1.99 was obtained. That means a deviation from the solution indicated by NeuberI2 of only 2.45 per cent. ( C T ~ / C T ~ , ) , ~ ~ ~ = 2.08 or 2.00 with the same deviation of 1.96 per cent was calculated for the two finer network structures consisting of 8.5 or 133 nodal points.

Figure 22. Network structure for the Leon prohlern (43 nodal points, 107 elements; [ K ] is a 681 x 315 band-matrix)


Figure 23. Network structure for the Leon problem (85 nodal points, 276 elements; [K] is a 1515 x 4 8 3 band-matrixl

Figure 2*. Network structure for the Leon problem (27 nodal points, 58 elements)

288 NGUYFN-LAN-PHAI

Figurc 25. Refined network structure of the network shown in Figure 24 ( I33 nodal points, 464 elements; [ K ] is a 2451 ~ 6 9 3 band-matrix)

- - - - N E E B E H 1121 * D A R I O . B R A E L E Y I T H I A X 51 / 1 3 /

S C Y N A C K l H Y B A X ) / l . T L /

0 A R G Y R I S i T E 7 L ) 115,'

./,. N G U ' / E N ( T E T 131 /S/

5 S C H N A C K I I ? l A X C 3 ) /1,1L /

0.0 L- --.--- -- * 0 500 1000 1500 25'30 25CC 35:s

Unknowns

Figure 26. Different calculation results for the Leon problem

AIJTOMATIC MESH GENERATJON 289

CONCLUSION

The author presents in this paper a technique for the automatic generation of 3D-network structures with tetrahedron elements. Here the nodal points must be determined before the mesh generation process, but in practice this manual expenditure can be reduced by the application of digitalizers to determine the co-ordinates of the nodal points. In addition. the element sides forming the boundary surface of the network structure can be identified by the program and the structure can be automatically refined. Examples given in the paper demon- strate the efficient application of this mesh generation technique.

AC‘KNOWI.EDGEMENIS

The author wishes to thank Prof. E. Schnack of the Institute for Technical Mechanics, University of Karlsruhe, for his valuable support in the development of this work.

REFFRFNCES

1. E. Schnack, ‘Effectivitatsuntersuchung fur numerische Verfahren hei Festigkeitsberechnunyen‘. L7II-Z I I Y. i 1 /?-I .

2. J. i l . Argyris, ‘Tetrahedron elements with linearly varying strain for the matrix displacement method‘. .I. R o ) . Aero. Sor. 877-880 (1965).

3. 0. C. Zienkiewicz and D. V. Phillips, ‘An automatic rnesh generation scheme fa r plane :tnd curved surface? h y “Isoparametric” co-ordinates’, Int. J. num. Mefh. Engng. 3, 519-528 (1971 1.

4. H. 4 . Kame1 and K. Eisenstein, ‘Automatic mesh generation in two- and threz-dimensional inter-connected domains’, Les congrks et colloques de I’universitC de Liege 11971).

5. V. Ph. Nguyen, ‘Automatische Netzgenerierung fur dreidimensionale Festigkeitsberechnungen mit der Methodc der finiten Elemente’, Fortschriti-Berrcht der VDI-Z, Reihe 1, Nr. 65 Dusseldorf 11980).

6. C. 0. Frederick, Y. C. Wong and F. W. Edge, Two-dimensional iiutorndtic mesh genei-ation for structural analysis’, Int,, J. num. Meth. Engng, 2, 133- 144 (1970).

7. A. Leon, ’ Uher Sforungm der Spaniiungsverteiluizg, die in elustischen Kijrpern durch Bohritngoi i i n t l B2icc.ht.n entstehen, Wien, 1908.

8. E. Schnack, Ein Iterationsorrfuhren z u r Optimirrung Don Spartnunh..jkorIzrntrationf,n. Habilitationsschrift, U n i w r ~ i - tat Kaiserslautern, 1977.

9. M. Wolf, ‘L,iisunp von ebenen Kerb- und Rifiproblernen mit der Methode der finiten Elemente‘, Dr . - InR Distertation, ‘1U Munchen (1977).

10. E. Schnack and M. Wolf, ‘Application of displacement and hybrid stress methods to plane notch and irdck problems’, Int. J. num. Meth. Engng, 12, 963-975 (1978).

1 1. V. Ph. Nguyen, ‘Fin Algorithmus zur Berechnung der Elenientsteifigkeitsmatrix fur das Tetraederelement‘.

12. I-I. Neuber, Kerhspannungslehre, Grundlage f i i r ganaue Fectigkeitr6erechnunX, 2nd edn, Springer-Verlag, Berlin-

13. N. P. Dario and W. A. Bradley, ‘A comparison of first and second order axially symmetric finite elements’, Int.

14. E. Schnack, ‘Zur Berechnung rotationssymmetrischer Kerbprobleme mit der Methode der finiten Elernente‘,

15. J . H. Argyris, ‘Continua and discontinua’, Proc. Conf. on Matrix Methods in Structural Mechanics, Wright-

S. 43-50 (1977).

Z A A M h l , Heft 4, 118-120 (1981).

G~ttingen-Heidelberg, 1958.

J. ni4m. Meth. Engng, 5, 573-583 (1973).

Forsch. 1ng.-Wes., Bd. 42, Nr. 3, 73-81 (19761.

Patterson Air Force Base, Ohio (1965).

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Automatic mesh generation with tetrahedron elements