Geometry cells and surface definition by finite elements

Computer Aided Geometric North-Holland

213

Design 2 (1985) 213-222

Geometry cells and surface definition finite elements

by

Ingolf GRIEGER Depurtment

of Aero Spuce Engineenng

Presented at Oberwolfach Revised 8 March 1985

16 November

Umwrsity

OJ Stuttgcrrt. Fed. Rep. Germonr

1984

1. Introduction The approximation of the geometry is important for computer-aided design. Due to the different requirements on the representation of the geometry of technical objects in various fields of applications a certain variety of curve, surface and body representation should be permitted [Boehm et al. ‘841. This paper presents an element-wise representation of geometrical objects. Basis of this technique is the approximation of the geometry by curve, surface and body elements. In order to avoid conflicts with the expression element we call this small geometric unit a cell. Advantages of this geometry cells are: (a) the representation is general (it exists no limitation to simple bodies), (b) the representation is computer-oriented (complex geometrical objects can be generated from simple cells in the computer and interactively modified), (c) the representation is geometry-oriented (several geometrical operations can be easily performed). For various applications a set of cells with different qualities is needed. This cell-wise approximation of the geometry is particularly well-suited if the designed object should later be analysed with the finite element method [Grieger ‘781 and, if the finite element idealisation should automatically created by mesh generation program. The second part of the paper deals with a special application of the finite element method to the surface definition problem [Grieger ‘731. A thin elastic plate is moved from the initial plane position to the final position in space. This procedure can be performed with a finite element program if plate elements of the fifth order are incorporated. An extension of this technique is the use of a shell element based on shape functions of complete fifth order.

2. Definition of cells The representation and manipulation of one, two and three dimensional objects in the computer is a typical problem. Complicated forms are approximated by simpler ceils whereby many smaller cells or fewer cells with better characteristics can be used. Different criteria are available for the classification of the cells. One is the dimension of the cell: curve, surface and bodies. Another alternative is the cell form (e.g. surface cells can be divided into triangular and quadrilateral cells). The degree of continuity or the degree and type of the interpolation function can also be used to classify cells. For that purpose it is necessary to know how a cell is built. A typical cell contains a certain number of nodal points in which the coordinates and according to the type possibly derivatives are known. After a determined rule (interpolation) an 0167-8396/U/$3.30

c 1985, Elsevier Science Publishers

B.V. (North-Holland)

I. Grreger / Grometr,:

214

cells

arbitrary point within the cell can be computed. This interpolation procedure can also be used for the interactive work with the cells. For the interactive use of the cell concept two independent areas may be defined: Topology means the subdivision into cells, which is largely independent of their real dimension. For the actual shape of objects the geometry must also be defined. Interactive design means that the topology as well as the geometry can be modified.

3. Cell interpolation

For the computation of geometric properties a function must be defined in terms of the values of the function at the nodal points. This function can be the coordinates at a point in the cell in terms of the nodal point coordinates. In each cell the relationship between the value of a function f (a coordinate, thickness, displacement, temperature etc.) at an arbitrary position in the cell and the corresponding values of the function at the nodal points f, can be written

(1)

f= Ch /=I

where w, are the shape functions of that special cell type and n the number of nodal points. For the interpolation function it is very helpful to use special coordinates (Fig. l), e.g., barycentric or other dimensionless coordinates. Each nodal point of the cell has a certain number of ‘values’. If only coordinates are used, the shape functions can be set up by Lagrange polynomials, but Hermite interpolation polynomials are recommended when derivatives as well as coordinates are taken into account [Boehm et al. ‘841. The most convenient forms for surface cells are the triangle (Fig. 2) and quadrilateral (Fig. 3). For a triangular cell complete polynomials of the degree m can be adopted. Various nodal

1 0

2 D 4-t

Curve

Tetrahedron

Triangle

Pentahedron

Fig. 1. Parametric

cell prepresentation

Quadrilateral

Hexohedron

215

Ilnear

lnterpolatlon

quadratlc

lnterpolatm

1

A 9

1

B

1”

2

6

3

cubic mterpolotmn

cubic

5

mterpolotm

4

1

L (1)

(2)

Fig. 2. Triangle

v, 3

2

;I cubic

natural

mterpolotm

cubac mterpolotlon

(3)

cell interpolation

point arrangements for the cubic casE are shown in Fig. 3. However, barycentric coordinates are recommended for the interpolation over triangles. Quadrilateral cells can be obtained if Lagrange or Hermite polynomials are applied in the two directions 5 and 71. The extension of this concept leads in the three-dimensional case to body cells. Typical cell forms are tetrahedron, hexahedron and pentahedron. The interpolation of the tetrahedron cell is based on complete polynomials of the degree m. Hexahedron cells can be derived using Lagrange polynomials in the three coordinate directions 6, 17 and {. A typical interpolation polynomial of the point ijk is given by the product of the corresponding Langrange polynomials Wl,,k

=

w:,(8 $h)

0;k(l)

(2)

where 1 is the number of points in the 5 direction, m in the ( direction and n in the { direction. The pentahedron cell is based on a triangle interpolation multiplied by a Lagrange polynomial in the third coordinate direction. Besides these basic cells, several variants with different nodal point arrangements are feasible.

I. Grieger / Geometry cells

216

/----J .(yj 1

1

brImear

interpolation

blcublc

mterpolatlon

3

blquadratlc

(1)

blcubic

Fig. 3. Quadrilateral

mterpolotm

interpolation

(2)

cell interpolation

4. Cell operations

In this section a few geometrical operations are described application of cells. Due to the large variety of possible operations a few typical ones are described. A point on a cell is defined in cuvilinear coordinates: Curve

p=

MO

Surface

P=

[x(-C, 17)

Body

P=

[x(6,

y(E)

y(t,

77, s>

4‘91~ 17)

which are typical for the and the available space only

(3)

z(t,

17)1,

Y(‘5> 97 l)

z(t>

(4)

01.

713

(5)

For some operations only a single cell must be taken into account and the entire result can be obtained by summation over all cells. However, for corresponding differentials of the first order within the Cartesian and the curvilinear coordinate system are required. This relation is given by the Jacobian matrix: Curve segment

ax

J=Tg I

1

ay aZ -a.$ at

(6)

aZ

Surface

segment

11 2

aZ

aq

7

(7)

I. Grteger / Geometry cells

217

(8)

The corresponding

differentials

can now be written

with this prerequisite

in the following

form: Line element

dl=

(dx’+d$

+dz’)“‘=md&

Surface element

ds = dx dy =JdetlJJ’I

Volume element

du=dxdydz=

det

IJI

(9)

dt dq,

(10)

dtdqd{.

(11)

Length, surface or volume of a cell can now be defined necessary easily numerically integrated. Length

L= jldl=

jm

d<= jf(E)dt=

in curvilinear

coordinates

t w,f(E,),

and if

(12)

!=I

Surface s=lds=jj~~dZdn=jji(~,n)d~dl = i 2 /=1 r=l Volume

v= j,;do=

yw,f(L jjj

det

T,)>

(13)

IJI d5 dv dl=jjjf(S.

77.0 dt dq d.C (14)

For

the interactive

0)

Global

b)

Local

coordinate

coordinate

use of the cells two other

operations

system

system

Fig. 4. Mapping

of cell geometry

are important:

(a) interactive

218

I. Grieger / Geometry cells

3

b)

Cl

1

Position

on 0

Position

on 0 triangle

Position

on

line

CI quadrilateral

Fig. 5. Position

in mapped

cells.

position on cells and (b) extraction of curves from surfaces, surfaces from bodies etc. For the modification of coordinates the user must be able to define an arbitrary point in a cell. This can be achieved without.difficulties using the following technique. A cell can be considered in the local coordinate system (Fig. 4). The straight-edged parent cell can be transformed into the real cell by using the interpolation functions. For example, a curved triangular cell in space can be mapped into an equilateral triangle in the plane. This property can be used for the interactive position of a point in the cell. A reference scale is required for the input of a dimensionless coordinate along the curve cell, a unit triangle for the input of barycentric coordinates or a unit square for the input of the dimensionless coordinates of a quadrilateral (Fig. 5). For a three-dimensional hexahedron cell, a reference scale and a reference square can be combined. This technique is easy to program.

5. Surface definition

by finite elements

The definition of surfaces for engineering applications is often that of defining a surface which passes a number of given points and satifies prescribed boundary conditions. In addition, the surface should satisfy a smooth criterion. These requirements lead to minimal surfaces [Birkhoff et al. ‘601 where the integral

is a minimum. this corresponds to minimizing the strain energy of a thin element method [Zienkiewicz ‘771 is well suited for the solution of the with different types of boundary conditions. The finite element method is used for many years for the solution engineering problems. Therefore, we restrict the description to a few those parts of the theory relevant to the surface definition problem. A structure (e.g. a thin elastic plate) is divided into a number of finite are connected together at the so called ‘nodal points’. The displacements and derivatives) at the nodal points are unknown and must be computed. on a set of functions which describes the displacement u any location dependence of the discrete displacements at the nodal points: u(x.

y. z)=wp.

elastic plate. The finite plate bending problem of structural and other basic equations and to elements. The elements (translation. rotation Each element is based within the element in

(16)

The matrix w contains the shape functions and p the vector of the displacements of the element nodes. The element strain vector E is the derivative of the element displacement vector u if and only if the displacement state is kinematically consistent and the derivative exists everywhere in the element. The strains E and the stresses u are connected via the material law which is in the linear elastic case the Hooke’s law. An elastic structure is in equilibrium under given loads if for any virtual displacement 6~ the increment of external work is equal to the increment of strain energy which is the principle of virtual work or virtual displacements. This energy theorem can be used to derive the force-displacement equation at the element level: P=

kp.

(17)

P is the element

force vector corresponding to the element displacement vector p. The next step in the finite element procedure is the assembly of the finite elements to the global structure. The relation between the element displacements p and the global structural displacements r are given by a simple connection matrix (1: p = ar.

(18)

The matrix a contains only zero and one elements if both displacements are defined in the same global coordinate system. The element displacement vector p introduced into the element equilibrium (equation (17)) and premultiplied by the at yields to a’P = a’kar = Kr.

(19)

The left hand side of equation (19) are the external loads R corresponding to the structural displacements r. The global stiffness of the structrue K is defined by the symmetric transformation of the element stiffness k with the boolean matrix a. A typical problem in computer-aided geometric design is the definition of a surface through a number of points. This can be achieved with the finite element method. The surface is idealized as a plate or a shell, which is moved from the x-r-plane to given positions in space (normally moved in the z-direction) and in addition fixed by certain boundary conditions. However, no

I. Grieger / Geometry cells

220

Fig. 6. Surface

definition

by finite elements

external loads are applied in the surface definition. The result will be the unknown geometric data (mainly derivatives) at the nodal points and furthermore the geometric data at any position via the interpolation scheme. The important equation is the equilibrium on the entire structure Kr=

R.

(20)

This equation can be partitioned into local (index L) and prescribed (index P) displacements for the surface definition problem. The prescribed displacements rp are the z-coordinates at all nodal points (Fig. 6) and the local displacements r,_ are first and second derivatives at these nodal points.

(21) or

K,,r,

+ K,pr, = R,,

(22)

K,,r,

+ K,,r,

(23)

= R,.

The vector of applied loads R, is zero in the surface definition problem. Therefore, equation (22) allows the computation of the unknown geometrical quantities at the nodal points: rL = - K,;K,,r,. The complete r=

displacement rL rp [

(24) vector r defines

the surface completely:

I

The element displacement (or geometry) vector p can be extracted ment vector r by a simple transformation with a boolean matrix a: p = ar.

(25) from the global displace(26)

221 0)

Generaltriangle A

Y F

//

E

/2

B

I

D

5

C

0

Y

x

b)

Boric

triangle

A

5” A F

6

D

E

c

I

0'

Fig. 7. Triangle

c’ cell with quintic

interpolation

For the interpolation of data within the finite element we have to define a point compute the displacement vector ~‘(6, n) by applying the shape functions:

(5, 7) and to

w([, ?j)=UOp=war. (27) The surface definition problem can be solved with existing finite element programs using quadrilateral [Throsby ‘691 or triangular [Grieger ‘731 plate bending elements. A typical triangular plate element is based on complete or incomplete fifth order interpolation with the IV,,., w,.,. at the corner points and if applied the normal geometrical properties u‘. M;, MI,.,w,,, derivative at the mid-side nodes. A further extension of this technique is the use of triangular shell elements based on fifth order interpolation [Barnhill, Farin ‘811. The triangular shell element (Fig. 7) based on complete interpolation scheme for the geometry [Zienkiewicz ‘771 needs 18 parameters at the corner points and 3 parameters at the mid-points to describe a surface without folds. Examples are carried out with the finite element program ASKA but due to lag of space not presented. An advantage of this method is the use of available finite elements programs without any change. The author

thanks

the referees for their helpful

suggestions.

References Argyris, J.H., Haase, M. and Malegiannakis, G.A. (1973), Natural geometry of surfaces with specific reference to the matrix displacement analysis of Shells. Koninkl. Nederl. Akademie van Wetenschappen, Amsterdam, Proceedings Series B. 76 (5). 361-410.

Barnhill, R.E. and Farin. G.E. (1981). C’ quintic Interpolation over triangles: Two explicit representations, Int. J. Num. Methods in Engineering 17. 1763-177X. Birkhoff. G. and Garabedian. H.L. (1960). Smooth surface interpolation. J. Math. Physics 39. 25X-268. Boehm, W.. Farin. G. and Kahmann. J. (1984). A survey of curve and surface methods in GAGD. Computer Aided Geometric Design 1. I-60. Grleger. 1. (1973). Darstellungsverfahren fuer das computergestuetzte Konstruieren. in: W. Brauer. ed. Lecture Notes in Computer Science 1. Springer. Berlin, 455-464. Grieger. I. (197X). Geometry elements in computer aided design. Computer and Structures 8. 371-381. Schrem. E. (1976). A short description of ASKA, ISD-Report No. 194. University of Stuttgart. Throsby. P.W. (1969). A finite element approach to surface definition. Computer J. 12. 3X5-387. Zienkieulcz, O.C. (1977). The F~‘,n,teElemmr ,Merhod. 3rd ed.. McGraw-Hill. New York.