Mesh Generation A Survey

W. R. BUELL

Associate Principal Research Engineer.

B. A. BUSH

Research Engineer.

Ford M o t o r C o m p a n y , D e a r b o r n , Mich.

Mesh Generation—A Survey

This report discusses pertinent schemes in finite element mesh generation. It is di-vided into two parts; node point generation and element generation. Node generation consists of automatically determining the coordinates of the finite element nodes. The basic node generation methods reviewed are straight line interpolation, sides and parts electro-mechanical devices, simplified finite difference, the equipotential method and the natural coordinate system method. Element generation consists of automatically connecting the nodes to form elements. These methods are simple increments, sides and parts pseudo I-J transformation, fully surrounded point, and I-J transformation.

T,

I HE increased popularity of the finite element ana-lytical method due to the availability of high speed, large memory computers has led to the solution of many heretofore unsolvable problems. Now the method in addition to being used in struc-tural analysis, is receiving increased emphasis in heat transfer and fluid flow analysis. In any of the computer programs for these analyses, "the preparation of the input . . . should be simple enough not to require a specialized team of experts" [1] .* Mesh generation is an attempt at simplifying input data for the finite element programs. The generation described in this report can be used also in finite difference programs.

There have been many incentives to automating the inputs, specifically the definition of the topology. Reference [2] men-tions cost and states t h a t "typically 50 percent (or more) of the total analysis time and dollars is spent generating, (and) checking the input data, . . . . " Thus, if the user can communicate with the computer in a better, faster, error-free environment, money and labor will be substantially saved. Other incentives include: simpler to perform parametric studies; reduced human error; insured regularity of the mesh; ease in using other types of ele-ments; and closer control on errors in stress values.

Zienkiewicz, reference [3], reiterates the importance of mesh generation when he writes: " . . . automatic mesh generation . . . are items on which efforts must be continued. If further break-throughs are to be expected, it is in this area t h a t they will occur."

In addition, reference [2] specified a few guidelines for data for finite element programs by recommending t h e programs receive

1 Numbers in brackets designate References at end of paper.

Contributed by the Design Engineering Division and presented at the Winter Annual Meeting, New York, N. Y., November 26-30, 1972, of T H E AMERICAN SOCIETY OF MECHANICAL ENGINEEBS. Manuscript received at ASME Headquarters, May 8, 1972. Paper No. 72-WA/DE-2.

the data by: d a t a cards, F O R T R A N statements, mesh genera-tion, and/or a combination of the above. This then gives the user maximum flexibility. This report is concerned only with mesh generation. T h e implementation of the other two meth-ods, data cards and F O R T R A N statements, is very straight-forward.

Because of the many mesh generation programs coming into existence, this report was initiated to organize and catagorize the methods being used. I t is anticipated t h a t such topics will generate more creative work.

T h e first generator for triangular mesh codes, per reference [4], was written, b u t never published, in 1958 by Mr. R. MacLean for an I B M 704 computer. Since then, much of the information on mesh generation has been embedded in the various finite element programs.

T h e report discusses pertinent schemes in mesh generation and is conveniently divided into: node point generation and ele-ment generation. Both are discussed separately. T h e table of reference is limited to the sources of information on mesh genera-tion which the authors were able to obtain.

Node Generation

In the finite element method, one replaces the continuous structural system by an assemblage of elements. T h e continuous system is divided into pieces by fictitious cuts and the intersec-tion of the cutting lines are called "gridpoints" or "nodes." The node d a t a consist of the coordinates of the node and other items (such as temperature, thickness, etc.) t h a t are point functions. This data preparation is a rigorous part of mesh generation. In the simplest form, the node d a t a is punched on computer cards (or other forms of temporary storage) for each and every node, until all the nodes of the entire structure are defined. On the other extreme, special programs such as F E D G E , reference [5,14, and 15] require special set-up procedures. F E D G E requires A overlays, 5 scratch files plus I/O files, and a 32k computer. Some-where between these two extremes are some very useful schemes t h a t free the user of the laborious, and time-consuming work. T h e following paragraphs detail some of the various schemes that

332 / F E B R U A R Y 1 9 7 3

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Node

Fig. 1 Straight line interpolation node generation

have been and are being used in node generation, beginning with the simpler schemes and progressing to the more complex and sophisticated schemes.

Straight Line Interpolation. T h e simplest method, and one step above t h a t of specifying t h e coordinates of each and every node is called "straight line interpolation" and is contained in the computer code of reference [6]. Instead of specifying every node, nodes may be omitted; the computer code is written to interpolate all omitted nodes. Usually straight line interpolation is used, b u t higher order interpolation could be used when more intermediate points are included or when other information is supplied.

Straight line interpolation involves specifying the end-point coordinates (and other variables as may be applicable) and grid point numbers for a straight line; such as xi2 = 1.0, y^ = 0.5 for node 1, and x-t = 4.0, y-, = 1.25 for node 7. The difference of the nodal point numbers determines the number of divisions in the line. T h e difference of the coordinates gives the length of the line. T h e length divided by the numer of divisions deter-mines the equal increments of the straight line. These incre-ments, Aa; and Ay are added to the lower nodal point coordinates to obtain the next intermediate nodal point and so on, until the last nodal point of t h e line is reached. See Fig. 1.

The scheme m a y be expanded easily to three-dimensional structures. I t is still very crude, b u t it does eliminate much d a t a if the topology can be divided easily into long straight lines and the meshes are evenly divided into equal segments. Variations may be made such as interpolation along circular and parabolic arcs.

Another version of the straight-line method (reference [7]) in-volves weighting integers, which may or may not be used. T h e coordinates of the end-points of each line, called a row, must be given along with the number of points in a row. T h e initial nodal point number and element number are given on a separate card as well as the number of rows and a switch integer to tell the computer whether weighting integers will be used. Depend-ing upon the spacDepend-ing switch, either the coordinates of the gener-ated points are determined by interpolation to give equal spacing; or the weighting integers are used to give the proportional lengths of the segments in each row. If the weighting integers are to be used, integers m u s t be applied for all spaces between points t h a t !are inputted or generated and this must be done for each row. ^Por example, if t h e weighting integers for a row of seven points are 9, 35, 16, 15, 14, 11, then in a ten-in. row, along the .r-axis (starting at the origin, points will be at 0, 0.9, 4.4, 6.0, 7.5, 8.9, 'and 10.0 in.

Sides-and-Parts. Generating nodes using the sides-and-parts method has been used exclusively for defining axisymmetric and two-dimensional structures in reference [8]. I n this method, the structure is described as a composite of four-sided figures, called parts; each side being any combination of straight-line segments I and circular arcs, as shown in Fig. 2. T h e sides of each p a r t are numbered clockwise and the parts numbered vertically (y

direc-2 The subscript on these coordinates x and y refer to the node point

number.

Fig. 2 Sides a n d regions node generation

tion) and then horizontally {x direction) starting with the p a r t closest to the x-y origin. T h e number of divisions for each side must be given along with the part number, side number, and co-ordinates. T h e number of subdivisions along two opposite sides must be the same because the mesh algorithm connects corre-sponding divisions of opposing sides, 1 to 3, and 2 to 4. T h e sub-divisions are then connected. T h e node points of each part are numbered from side 1 to 3, then side 2 to 4. Regions are joined together by being connected at node points, not mid-sides, dupli-cate nodes eliminated, and the whole structure renumbered.

Electro-mechanical Hardware. T h e electro-mechanical hardware used in nodal point generation takes on as m a n y forms as there are manufacturers. Basically, there are two broad categories of hardware: two-dimensional and three-dimensional. Nodal point coordinates are obtained using the two-dimensional devices by following preselected lines or points on a drawing with a stylus. Three-dimensional devices are used to obtain the co-ordinates of physical structures. For both categories of hard-ware, peripheral equipment, such as magnetic tape, cards, and paper t a p e are used t o store t h e coordinate information.

Currently on the m a r k e t are m a n y two-dimensional devices, such as the electromagnetic graph tracing table (reference [9]) and a Gerber digitizing system (reference [10]). T h e Gerber digitizing system consists of a camera and a display tube ( C R T ) . A blueprint is laid on the board and prescribed points set with crosshairs on the tube by the operator, where upon the operator pushes a button and the coordinates of these points, in this x-y plane view, are stored. This process is then repeated for a three-dimensional structure, to obtain values in t h e y-z plane. These two decks are then merged to form all the nodal point coordinates, x, y, and z. Such a system per reference [10] " . . . reduced t h e preparation time . . . from several weeks, to a few d a y s " for a vehicle body. These devices are advantageous when one has blueprints or elementary drawings of the structure. Thus, analysis of the structure may be initiated in the early de-sign process.

T h e three-dimensional devices, such as Fig. 3 depicts, use the physical structure t o obtain the coordinate data. T h e hardware must, therefore, be fabricated. This is not always advantageous when one is investigating various design concepts. However, this scheme is useful and fast when the structure is extremely complex, such as a complete automobile. Only the exposed surface structure can be digitized; hidden subassemblies must be scanned as subassemblies, and not as a composite structure.

Simplified Finite Difference. This method of node generation uses Laplace's equation in finite difference form. I t has been applied only to two dimensional structures, which also includes axisym-metric structures. The computer program and documentation for this method is given in reference [11],

T h e perimeter of the structure to be analyzed is defined by line segments. Line segments are defined as circular arcs,

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Fig, 3 A scanner

Xi+l.;

+

Xi_l';

+

Xi.Hl

+

Xi,;_l - 4Xi';

0

and for they coordinate,

Vi+l';

+

Yi-l.;

+

Vi.;+l

+

Vi';-l - 4Vi.;

=

0

These two equations are then solved for Xi andVi by iteration llsiug the

x-v

coordinates on the boundary and working to th~ interior. Each nodeiis then the average of the coordinate of its four neighboring nodes. An over-relaxation factor is used to speed the convergence of the solution.

Equipotential Method. This method described in references [4]

and [12] regards the mesh lines (and hence the nodal points)

as

two intersecting sets of equipotentials, with each set satisfYing Laplace's equation in the interior with adequate boundary condi. tious. This section will concern itself only with a brief overview of those references. In the equipotential method, the coordi. nates x, Vof an internal point must satisfy the difference equa-tions;

whereXi,Yi are the coordinates of the closest neighbors andWi

a

weighting function (a function ofa, (3, and 'Y) is shown in Fig. 6.

a, (3,and 'Yare defined as follows: a = x,,}

+

y",2

where p is the over-relaxation factor usually 1

<

P

<

2. Refer-ences [4] and [12] continue with extensive manipuln.t.ion for optimum cyclic value ofp. Assoon as "new" coordinate values are obtained, they are used in the equations. To have triangles, the smnmations used in equations (1) and (3) are taken over the six closest neighbors, as shown in Fig. 6. For quadrilateral meshes, the summations are taken over the eight neighbors.

(2) (1) (3) 2: Wi(Vl - y)

=

0 2:Wi(Xi-X)

=0

i X?~+l

=

and (3 = x,px",

+

V,pY", 'Y

=

X,p2

+

V,p2

wherex'"andX,p (and similarlyV'" andY,p)are defined in Fig. 6, together with the weights Wi for three different triangles, lIud one quaclrilateral mesh.

Equations (1) above are solved for all internal points by itera-tion, using an over-relaxation factor to speed convergeuce. Writing equation (1) above forn

+

1 iteration cycle gives:

02f 1

op

=

h2

[f(~

+

h,'I)) - 2f(~, 'I))

+

f(~ - h, 'I))]

straight lines, or points in the two-dimensional coordinate sys-tem. Ifperimeter points between the end points of the line seg-ments have been omitted, linear interpolation along these line segments is used to evaluate all missing nodes. The two-dimensional structme in the

x-v

plane is then transformed with a one-to-one correspondence of the points into the i-j plane.

Referring to Fig.4(a),a semicircular diaphragm with a central hole is shown in the

x-v

plane. The boundary points, AB, specify a circular arc line segment; BC a straight line segment; CD, a circular arc line segment; andDA,a straight line segment. This stl'Ucture is transformed into the i-j plane shown in Fig.

4(b). The boundary points and type of line segments are all thA,t is needed for the program to generate all missing boundary and iut.ernal nodes.

The internal points are found by satisfying the finite difference form of Laplace's equation over the transformed grid for each of the coordinates,xandV. Laplace's equation in two-dimensional form is:

02X 02X

oi

2

+0j2=O

The central finite difference approximation to the second deriva-tive is:

where~, 'YJisthe coordinate system and h represents an increment

in the~direction.

By substituting i,j for the coordinate system~,'I) and by set-t.ing botht::.iandt::.jincrements in the Tespective directions equal to one, the finite difference form of Laplace's equationisobtained for the x coordinate:

y

Fig.4(h) I-Jgrid transformation from Laplacian grid in Fig. 4(a)

I--.---_+_x

Fig. 4(a) Laplacian grid with equally spaced boundary prints

A D

i,j+l

i - l . j i»3 i+1,3

i>j-l

Fig. 5 I-J transformed grid

0-/3 H

Equilateral Trian^k';;

1+1 I T 1-1

RigU Triangles ( 1 even)

/ 2 y

•V - \ [<*2 * 2xl+XG> " (X3 f 2 x4 + x5>]

x* ° f [C-'I ' 2 x 2' - <x5 * 2*4>] x , = as above

A> special case is obtained when the weights, w%, are taken as unity. For the case of triangles, one obtains "fairly" adequate equilateral triangles. x = - > xt 6< - i and (4) V = r E W u i = i

When the weighting functions for quadrilaterals are unity, and only neighbors 2, 4, 6, and 8 (as illustrated in Fig. 6(e)) are used, the equations become:

x =

I E

*••''

*' =

2

>

4

>

6

>

8

a+j9

Right Triangles (1 odd)

<Mff -0 ^Sh 4 s"'^ 2

/ ^

jSb d. Right Triangles i [<*2 1- 2Xl) - <x4 * zx6)] :T [( xl - 2x2 ' x3> - (X4 ' 2 x5 * and (5) V = 4 Z ) 2"' * = 2' 4' 6>

Equation (5), is Laplace's equation in finite difference form, as described in the previous section.

Included in this overall method is the concept of using a re-flecting plane (i.e., imaginary mesh points) near the boundaries to obtain near regular-shaped elements at the boundary. This is performed by modification of the weighting functions. The use of the weights near indented boundaries is preferred, since the simplified equations of (4) or (5) can cause mesh points to fall outside the region and can produce very skinny elements.

Fedge. This is a very large data generation computer program capable of providing a means of discretization of continuous domains in one, two and three dimensions. It is a three-link program for an IBM 7094 computer. It will also handle field quantities associated with different regions in the domain. The algorithm uses a natural coordinate system to define the bounda-ries of the regions.

The natural coordinate system is a very simple scheme. It is represented by an integer and a fraction for describing a point on a continuous line, two integers and two fractions for a plane, etc. For example, in Fig. 7, the line m, is defined by the points 1, 2, 3, ... L in the x, y coordinate system. Then point P, given by xp and yv can also be represented by the integer 5, and the fraction of the distance between point 5 and 6, 5P/56 = C,-, i = 5. Thus, the continuous line may be represented in this natural coordinate system. All coordinate points are calculated from an equation which is inputted by the specification of coefficients for a second order polynomial.

When the region is thus defined in this coordinate system by toesh lines of rectangles, i.e., lines going to opposite sides in a plane, the mesh nodal points may then be determined. Using

3 4 r 2 - 1 8 «! 7 Quailnlalerals Y , , "W <*,-*«> Fig. 6

Fig. 7 Natural coordinate system

the i-j transformations for two dimensions and the i, j , k trans-formations for three dimensions, member incidence tables may also be generated.

The available references [5, 14, and 15] do not give specific details, but some of the general highlights are:

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(a) accurate boundary definition (6) easy mesh refinement

(c) renumbering—bandwidth minimization (d) a fine mesh at a selected point of interest

(e) only lines, rectangles, and cubes have been used in the transformations to obtain the line, quadrilateral and 3-D plate elements. Meshes for triangles and other elements are men- tioned b u t no details are given.

(/) mesh of a three-dimensional frame element is not possible.

Element Generation

All nodal points of the structure found by any of the methods so far discussed must have unique names. These nodal points must then be connected to form elements, and the assemblage of the elements forms the continuous structure. The connectivity of the nodes, t h a t is, member incidence, to form an element de-pends on the type of finite element t h a t will be used in the sub-sequent analysis. Typical types of elements t h a t have appeared in the literature are: beams (2 nodes), triangles (3 to 7 nodes), quadrilaterals and rectangles (4 nodes and up), and three-dimensional elements such as the tetrahedron (4 nodes), prism (6 nodes) and the hexahedron (20 nodes). For a two-dimensional structure shown in Fig. 8, the member incidence for the quadri-lateral element number 2 is: node 2, node 3, node 8, and lastly node 7.

Analogous to the previous section, there have been many schemes for determining the nodes to form the desired finite element. In the simplest case, the user transcribes to the com-puter every element by hand, using punched cards, paper or magnetic tape, etc. This scheme is generally too cumbersome, hence the following paragraphs give some other methods, be-ginning with simple schemes and progressing to more complex ones.

Simple Increments. I n this method (reference [6] and [28]) the current member incidence data are generated by adding " 1 " to the preceding member incidence values. T h e member incidence for the first and last element of a row is given. All intermediate elements are obtained by adding " o n e " to the preceding member incidence, until the last element as initially given is obtained. Referring to Fig. 8, the member incidence for element 1 is 1, 2, 7, and for element 4 is 4, 5, 10, 9.

Then adding 1 to each node in the member incidence of element 1 gives the member incidence for element 2. This scheme is continued until element 4 is reached; then the scheme can be repeated for all the "rows" in the structure. This lends itself to very simple coding, b u t it supposes t h a t all node numbers and elements be monotonically increasing, t h a t no numbers are omitted, and t h a t all numbers from 1 to the n t h element number and from 1 to the mth number must be used in the structure.

Sides and Parts. T h e pseudo I-J transformation numbering scheme is used very effectively by the sides-and-parts node

gener-v..

Element

Numbei-15-4 Node Number

ation method (reference [8]). Since the sides-and-parts method is limited to four-sided figures in the x-y (two-dimensional) co-ordinate system, it in effect has a rectangular grid similar to an I-J transformed grid. T h e numbering of t h e nodal points and elements proceeds through t h e mesh first from side 1 to side a then from side 2 to side 4, as shown in Figs. 9(a) and 9(b). Xhfi element numbering is begun by counting in the direction of in. creasing / (side 1 to side 3) for the minimum value of J, using the grid points whose names are a t : (i, j) node 1; (i, j _|_ jv node 2; (i + 1, j 4- 1), node 3 and (i + 1, j) node 4. When the maximum J is reached for t h a t I, the next point is numbered bv increasing / by one (thus working from side 2 to side 4). Then J is increased again from its minimum. The procedure is repeated until the maximum value of J (at side 4) is reached. Note that it is not necessary t h a t incrementing proceed in the J direction-the incrementation direction is a matter of choice.

Fully Surrounded Point, T h e fully surrounded point method pre-supposes t h a t the nodes have been obtained by the use of any of the aforementioned methods of generating the nodes. Because this method is very general, it was used in reference [9] in con-junction with the electromagnetic graph-tracing table and a dig-itizer for obtaining the node point coordinates. The fully surrounded point method generates as near to equilateral tri-angles as the system of points allows. As suggested by the name of this method, each node is surrounded in turn with triangular elements.

T h e method ."begins by selecting the nearest point to node 1 and establishing a side, lj, where j is the node number of the selected point.

T h e side, ij, is then used to find a node k, such t h a t the angle ikj is a maximum and ijk is in a counterclockwise sequence, as shown in Fig. 10. T h e new point, k, then becomes j , and the process is repeated until the node is fully surrounded—when a side is obtained which is the side from which the process started. A new i node is selected and the process is repeated.

This method features ghost points which are points beyond the structure boundary whose use eliminates the need to identify external boundary nodes. These ghost points are not sur-rounded. Triangles are generated to them and when the mesh is completely generated, the triangles containing ghost points are deleted and then the elements are numbered. More than one material may be taken into account.

I-J Transformation. The element generation using the I-J trans-formation is very similar to the node generation scheme using

a. X - Y P l a n e

1

1

Side 1

_ j i i _

Fig. 8 Structure depicting quadrilateral finite element w i t h node and element numbers

0 1 2 :l 4 5 (i 7 8 0 10 11 12 13 14 b. P s e u d o I - J P l a n e

Fig. 9 Sides and parts member incidence generation

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Y u

- • X "C" Section Three Dimensional Beam with a Triangular Mesh Fig. 10 Fully surrounded poinf method

a. Mesh on a Semicircle with a Central Hole

Element Number Opened C Section Showing Node :md Klomont Numbers of Figure 12a above Fig. 12 M e m b e r incidence generation using the slab method

w 1 A' i — •—

—

—i JJ • • R

b. I-J Transformation of Figure 11a above

Fig. 11 M e m b e r incidence generation using the i-J transformation method

A representative section, or slab, is defined in detail, and then all the other slabs are interpolated and incremented until t h e last portion of the structure is modeled. T h e entire "C" section, shown in Fig. 12 was required to be modeled by triangular ele-ments (three nodes per element). I n this method, t h e first slab is defined completely by: (1) the coordinates of all the node points of the first slab and any other variables t h a t are position-oriented, such as thickness, temperature, etc.; and (2) all element num-bers and their associated member incidence for the first slab. T h e procedure is then to repeat this slab, "n" number of times; each time the corresponding element numbers and nodal points are " b u m p e d " a n d / o r interpolated (as in the case of the co-ordinates) for each new slab.

the I-J transformation along with the Laplace's equation for the internal node generation. An I-J transformed grid as shown in Pig. 11(b), simple counting is used. T h e procedure for element generation starts after all nodal points have been determined. The procedure for element generation starts on the lower left corner of the transformed grid on the line represented by the least value of J. T h e point on the J line with the least value of / is designated as the first nodal point for element 1. See Pig. 11(6). T h e grid points whose names are a t : (i, j) node 1; (i + l,j) node 2; {i + l,j + 1) node 3; and (i, j + 1) node 4 form t h e member incidence for each quadrilateral element. Incrementa-tion of / is continued in the direcIncrementa-tion of increasing I. When the maximum value of I corresponding to the present value of / , is reached, the next element and member incidence is found by in-creasing J by one and resetting I. Then I is increased again from its minimum for t h a t J. Note here t h a t the incrementa-tion was chosen to proceed in the I direcincrementa-tion. This procedure continues until the entire mesh has been traversed. This scheme adapts itself readily to using two-dimensional variables in F O R T R A N ; hawever, the problem size is immediately limited by memory size and hence, the scheme is usually applicable to prblems of small size, around 1000 nodes or less.

Slab Generation. This method is extremely useful when struc-tures with constant meshes can be used, as shown in Fig. 12(a).

Conclusions

T h e report has collected a number of different schemes for automating input d a t a to finite element programs. E a c h of the schemes is usually applicable to a special set of topologies; hence the reason for so m a n y currently in use today. There is no "one best" scheme t h a t would satisfy all users of finite element meth-ods. I t would be desirable to have a "library" of schemes from which one could pick the method to best satisfy his modelling problem.

All of these mesh generation methods are for two-dimensional elements; i.e., they handle general three-dimensional plate ele-ments and/or axisymmetric eleele-ments. Little emphasis has been made thus far, on developing algorithms for generation of three-dimensional solid elements. Two of the references [4, 24] indi-cate the capability of expanding their methods to generate three-dimensional elements, b u t the expansion has not yet been done. Progress should be made in this area and efforts are necessary in simplifying the input preparation for these programs.

T h e computer codes for F E D G E (reference [5]), G E O M E T R Y P R O C E S S O R (reference [13]), and the automated input d a t a preparation programs for N A S T R A N (reference [18]) may be purchased from COSMIC, Barrow Hall, University of Georgia, Athens, Georgia, 30601. For most of the other programs, it is probably necessary to contact t h e authors, the companies, or government agencies through which the reports were issued, to

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obtain the program code. Most of these mesh generation methods are "stand alone" programs. Straight line interpola-tion does appear in E. L. Wilson's original code and the simplified finite difference method is in SAASII, SAASIII, and ASAAS, (reference [11], [29], and [30]).

References

1 Chacour, S., "A High Precision Axisymmetric Triangular Element Used in the Analysis of Hydraulic Turbine Components," Paper No. 70-FE-19 presented at ASME meeting May 24-27, 1970, Detroit, Michigan.

2 Yates, D. N., et al., "The Development of Large Scale Digital Computer Codes for Production Structural Analysis," Lockheed Structures Conference, 1970.

3 Zienkiewicz, 0 . C , "The Finite Element Method; From Intu-ition to Generality," Applied Mechanics Review, Vol. 23, No. 3, Mar. 1970.

4 Winslow, A. M., "An Irregular Triangle Mesh Generator," Lawrence Radiation Laboratory, University of California, Liver-more, California, August 25', 1964. Available from NTIS, (National Technical Information Service) Springfield, Virginia, 22151, Report No. UCRL-7880, Aug. 1964.

5 Akyuz, F. A., "FEDGE—A General Purpose Computer gram for Finite Element Data Generation User's Manual," Jet Pro-pulsion Laboratory, California Institute of Technology, Pasadena, California, September 15, 1969, NASA Tech. Memorandum 33-431, Vol 1. Available from COSMIC, Barrow Hall, University of Georgia, Athens, Georgia 30601.

6 Wilson, E. L., "Structural Analysis of Axisymmetric Solids," AIAA Journal, Vol. 3, No. 12, Dec. 1965.

7 Zienkiewicz, O. C , "Triangular Mesh Generation Program," University of Wales, Swansea, Wales, Report No. 2, Sept. 1967.

8 Egeberg, J. L. "MESHGEN—A Computer Code for Auto-matic Finite Element Mesh Generation," Sandia Laboratories, Livermore, California, July, 1969. Availagle from NTIS, Spring-field, Virginia, 22151, Report No. SCL-DR-69-49.

9 Frederick, C. O., et al., "Two-Dimensional Automatic Mesh Generation for Structural Analysis," International Journal for Nu-merical Methods in Engineering, Vol. 2, No. 1, 133-144. 1970.

10 Petersen, Willy, "Application of Finite Element Method to Predict Static Response of Automotive Body Structures," Paper 710263 presented at SAE Automative Engineering Congress, Detroit, Michigan, Jan. 11-15, 1971.

11 Jones, R. M., and Crose, J. G., "SAAS II, Finite Element Stress Analysis of Axisymmetric Solids with Orthotropic, Tempera-ture-Dependent Material Properties," Aerospace Corporation No. TR-0200 (4980)-l, Sept. 1968.

12 Winslow, A. M., "Equipotential Zoning of Two-Dimensional Meshes," Lawrence Radiation Laboratory, University of California, Livermore, California, June, 1963. Available from NTIS, Spring-field, Virginia, 22151, Report No. UCRL-7312, July, 1964.

13 Kitagawa, M., "Geometry Processor," Space Division of North American Rockwell Corporation, 12214 Lakewood Blvd., Downey, California 90241. Available from COSMIC, Barrow Hall, University of Georgia, Athens, Georgia, 30601, Aug. 27, 1969.

14 Akyuz, F. A., "Natural Coordinate Systems: An Automatic Input Data Generation Scheme for a Finite-Element Method," Nuclear Engineering and Design, Vol. 11, No. 2, Mar. 1970, pp. 195-207, also appears as JPL Technical Report 32-1486.

15 Akyuz, F. A., "FEDGE—A General-Purpose Computer Program for Finite Element Data Generation Program Manual," JPL Technical Memorandum pp. 33-431, Vol. 2, Sep. 15, 1969.

16 Coons, S. A., and Herzog, B., "Surface for Computer-Aid Aircraft Design," presented at the AIAA 4th Annual Meeting a i Technical Display, Anaheim, California, Oct. 23-27, 1967 AIAA

Paper No. 67-895. ' 4

17 Jensen, Paul S., "Finite Difference Techniques for VarlnW Grids," paper presented at a conference on "Computer Oriental Analysis of Shell Structures," Lockheed Palo Alto Research Labom tory, Palo Alto, California, Aug. 10-14, 1970.

18 Cook, W. L., "Automated Input Data Preparation for NAS TRAN," CGS-11039, Goddard Space Flight Center, Greenbelt Maryland, April, 1969. Available through COSMIC, Barrow JI'il]' University of Georgia, Athens, Georgia, 30601, COSMIC No. X-32i' 69-237.

19 Ice, Malcolm W., "NASTRAN User Interfaces-Automated Input Innovations," part on SAIL Data Generation Techniques NASTRAN User's Colloquium, Langley Research Center, Sent' 13-15, 1971, NASA Tech. Memorandum, NASA TMX-2378, Vol. 2.'

20 Giles, G. L., and Blackburn, C. L., "Procedure for Efficiently Generating, Checking and Displaying NASTRAN Input and Output Data for Analysis of Aerospace Vehicle Structures," Section of Fuse-lage and Wing Generation, NASTRAN User's Colloquium, Langley Research Center, Sept. 13-15, 1971, NASA T X X-2378, Vol. 2.

21 Winslow, A. M., "Numerical Solution of the Quasilinear Poisson Equation in a Nonuniform Triangle Mesh," (Appendix. Numerical Construction of Topologically Regular Nonuniform Triangle Meshes), Journal of Computational Physics, Vol. 2, 1967 pp

149-172.

22 Wang, F. D., Sun, M. C , and Wolgamott, J., "A Computer Program for Generating Finite Element Models of Mine Structures," U. S. Bureau of Mines Information circular 8510, TN23.U71, No 8510 622.06173, May, 1971.

23 Simpson, F. M., "Compressor Blade Mesh Generator for SAMIS Analysis," unpublished Ford report, Aero-Thermo Depart-ment, Turbine Operations, Mar. 19, 1971.

24 Zienkiewicz, O. C , and Phillips, D. V., "An Automatic Mesh Generation Scheme for Plane and Curved Surfaces by 'Isoparametric' Coordinates," International Journal for Numerical Methods in Engi-neering, Vol. 3, 1971, pp. 519-528.

25 Tocher, J. L., and Felippa, C. A., "Computer Graphics Ap-plied to Production Structural Analysis," A paper presented at a symposium on "High Speed Computing of Elastic Structures" in Liege, Belgium, Aug. 24-28, 1970, sponsored by the International Union of Theoretical and Applied Mechanics.

26 George, J. Alan, "Computer Implementation of the Finite Element Method," Stanford University, Stanford, California, STAN-CS-71-208, Feb. 1971. Available from NTIS, Springfield, Virginia, 22151, Report No. AD 726 171.

27 Kamel, H. A., and Eisenstein, H. K., "Automatic Mesh Gen-eration in Two and Three Dimensional Inter-Connected Domains," a paper presented at a symposium on "High Speed Computing of Elastic Structures," sponsored by The International Union of Theo-retical and Applied Mechanics, Liege, Belgium, Aug. 23-28, 1970.

28 John A. Swanson, "ANSYS-Engineering Analysis System User's Manual," Swanson Analysis Systems, Inc., Elizabeth, Penn-sylvania, Dec. 1, 1971.

29 J. C. Crose and R. M. Jones, "SAAS III, Finite Element Stress Analysis of Axisymmetric and Plane Solids with Different Orthotropic, Temperature-Dependent Material Properties in Tension and Compression," TR-0059 (S6816-53)-l, The Aerospace Corpora-tion, San Bernardino, California, June 22, 1971. Available from the Defense Documentation Center.

30 Crose, J. G., "ASAAS, Asymmetric Stress Analysis of Axisym-metric Solids with Orthotropic, Temperature-Dependent Materiiil Properties that can Vary Circumferentially," TR-0172 (S2816-15)-!, The Aerospace Corporation, San Bernardino, California Dec. 29, 1971.

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