Body-oriented coordinates applied to the finite-element method W. A. C O O K
Technical Engineering Support Group, Los Alamos National Laboratory, Los Alamos, 87545, USA
The objective of this research is to increase the accuracy of the finite-element method using coordinates intrinsic to the shape of the body being analyzed. We refer to these coordinates as body coordinates. Existing finite elements use Cartesian coordinates and are more accurate for solving rectangular shaped problems than for solving nonrectangular-shaped problems. To check the feasibility of this research, we developed finite-element codes that used both cylindrical and Cartesian coordinates to solve problems in which the bodies were cylindrical-shaped. We obtained the most accurate solutions using the code that used cylindrical coordinates. Body coordinates become Cartesian coordinates for rectangular-shaped bodies and cylindrical coordinates for circular-shaped bodies. The body coordihate's finite-dement formulation uses coordinate transformations from the body to the Cartesian coordinates. These transformations are developed using blending functions and boundary functions. Gradients of the Cartesian coordinates, with respect to body coordinates, are needed for stiffness calculations. Thus, the criterion for the blending function derivation is 'the nearest boundaries dominate,' both for coordinate transformations and for gradient of coordinate transformations. For our studies, we developed two codes, one that uses body coordinates and one that uses Cartesian coordinates. These codes have been used to solve six example problems.
1. INTRODUCTION Before the advent of computers, engineering analysis consisted of solving a few select problems for which coordinate systems were already available or could be developed and that corresponded to the geometric shape of the body being analyzed. Thus, many coordinate systems, such as cylindrical, spherical, parabolic, and elliptic were developed. Each of these orthogonal coordinate systems was especially wellsuited for analyzing problems with particular geometries. After the advent of computers, finite-difference and finiteelement numerical methods became popular. These methods generally used Cartesian coordinates to approximate solutions over many small regions in such a manner that, in the limit, the resulting solutions could solve problems in the accumulated region. This paper will address the finite-element method. Whether or not the finite-dement method has the capability to approximate a solution depends upon the shape of the element. 1 For example, if the element has a rectangular shape, the continuum stress solution of a longitudinal beam with a transverse load can be accurately approximated with only a few quadratic elements. If the same few elements have a skewed shape, however, the resulting approximate solution is poor. Our objective is to eliminate inaccurate elements (that is, elements whose approximation capabilities are reduced as a result of their geometric shapes) and thus to increase the accuracy of the finite-dement method. Inaccurate elements have a skewed shape and do not approximate solutions as accurately as elements with rectangular shapes. Cartesian coordinates are used to develop conventional finite dements. The Cartesian coordinates are very accurate for solving Accepted August 1987. Discussion doses May 1988
2 Engineering Analysis, 1988, Vol. 5, No. 1
rectangular problems (the bodies are rectangular-shaped) in which the elements are rectangular-shaped. In our research, however, we use body coordinates to develop the finite elements. These coordinates are intrinsic to the geometric shape of the body. When the finite-element method is formulated with body coordinates, there is no need for inaccurately shaped elements with respect to body coordinates. To establish the feasibility of this research, we first used cylindrical coordinates to develop a two-dimensional finiteelement formulation. This formulation, which had nine nodal points, was a Lagrangian interpolation element for solving two-dimensional continuum stress problems with the the principle of minimum potential energy. Then we used Cartesian coordinates to develop a finite-element formulation. We next compared computer programs that used cylindrical coordinates with those that used Cartesian coordinates. Each of these programs used three finite-element formulations to solve a 90°-plane-stress curved beam geometry with a transverse load; the solution calculated using cylindrical coordinates was 93 % accurate, and the solution calculated using Cartesian coordinates was 80 % accurate. The body coordinate element uses coordinate transformations from the body coordinates to the Cartesian coordinates with blending and boundary functions. To make the stiffness calculations, we need gradients of the Cartesian coordinates with respect to body coordinates. Thus, the criterion for the blending function derivations is 'the nearest boundaries dominate' for both coordinate and gradients of coordinate transformations. A comparison of the Cartesian coordinte dement with the body coordinate element, showed that the body coordinate uses approximation functions with body coordinates to approximate variables (displacements) in the body coordinate system, and the Cartesian coordinate uses approximation functions with body coordinates to
© 1988 Computational Mechanics Publications
Body-oriented coordinates: IV. A. Cook approximate variables (displacements) in the Cartesian coordinate system. Thus, body coordinates are equivalent to Cartesian coordinates when the shape of the body is rectangular and to cylindricalcoordinates when the shape of the body is cylindrical. W e used dements that had four, nine, and sixteen nodal points with shape functions developed from Lagrangian interpolation formulas. W e used the principle of minimum potential energy as a variational principle for the finiteelement method in our studies on the deformation of twodimensional linearelasticcontinuums. The calculatedpotentialenergy can also be used to measure the accuracy of the solution.
i
X2
xqo,1) 4 ¢........~
,o¢o.:~
I
xi(O'O)
xi(al, 1)
.
/
xi(al O) 2
xi(1,0)
2. SUMMARY This paper derives and discusses the transformations from body coordinates to Cartesian coordinates, derives the formulation of the body and Cartesian coordinate elements, and discusses the problems solved with computer programs that used body and Cartesian coordinate elements. The problems discussed are traction loads on the following geometries. 1. 2. 3. 4. 5. 6.
Longitudinal beam geometry, plane stress. Curved beam geometry, plane stress. Straight-line tapered geometry, plane stress. Circular tapered geometry, plane stress. Skewed straight-line geometry, plane strain. Skewed, curved-boundary geometry, plane strain.
The problems show that the body coordinate element is as accurate as the Cartesian coordinate element for elements with nine and sixteen nodal points. However, for elements with four nodal points, the body coordinate element is generally less accurate than the Cartesian coordinate element. Also, the elements with sixteen and nine nodal points are approximately equally efficient (accuracy per unit of time) and are significantly more efficient than the elements with four nodal points. We arrived at these conclusions by using potential energy as a measure of accuracy.
Figure 1. Cartesian and body coordinates with blending functions, boundary functions, and corners depicted
where the body coordinates are normalized coordinates (0 to 1). The transformation equations interpolate linearly between the boundary functions xt(~t, 0); xi(=t, 1); xi(0, =2); and xi(1, ~2). The criterion we used for these transformation equations was that the nearest boundaries dominate. Thus, as the body coordinate approaches a boundary, all other boundaries become negligible in comparison. However, when partial derivatives of these transformation equations are needed, this criterion is not satisfied. To maintain this criterion for both partial derivatives and coordinates, we must generalize the transformation equations as follows: Xi(=l, =2) =
fil(~XZ)xi(=1 '
0) + fi2(~xZ)xi(=1 , I)
-l-f~(~l)x/(O, =2) 4-:4(gl)xl(l,g2) --
f~(~2)Jn3(~t)X'((O,O)
-
f~(et2)f~(=l)x~(1,
3. T R A N S F O R M A T I O N S
- f~(c~2)f~(=l)x'(O,
In this section, we derive a coordinate transformation that is intrinsic to the geometric shape of the body being analyzed. Reference 2 describes the use of linearly blended interpolation formulas for generating finite-element meshes. These formulas have been used for over two decades to approximate two- and three-dimensional regions and surfaces. Reference 3 describes a physical derivation for twodimensional linearly blended interpolation formulas. The two coordinate systems that define the two-dimensional region shown in Fig. 1 are the Cartesian coordinates (x t , x 2) and the body coordinates (=t, =2). The body coordinates are generally a nonorthogonal coordinate systems. The transformations from the body to the Cartesian coordinates derived in Ref. 3 are
-- f~(o~2)f~(=l)Xi(1,
x i ( = l , =2) =
(1 -=2)Xt(=I, O) "~ =2Xi(O[1, 1) + (1 -
=l)xi(O,
=2) "1-=lxi(1,
(2)
xi(1, 1)xi(0, = z ) _ xi(0, 1)xi(1, ~z) f~(=2) = xi(O, O)x~(1, 1) - x~(1, O)xi(O, 1) ' Xi(0, 0)xi(1, 0t2) -- gi(1, 0)xi(0, ¢Xz) f~(~Z) = Xi(0, 0)xi( 1, 1) -- X~(1, 0)Xi(0, 1) ' xi(1, 1)xi(=l, 0) - xl(1, 0)xi(~1, 1
fi- 1,
i)- x'(l,0)x'(0,I))'
(3)
and
-- (1 -- =2X1 - ¢ l ) x i ( 0 , O) -- (1 - =2)=txi(1, O)
- =2(1 - =l)xl(0, 1) - ~2~1xil,
1) 1),
where f~(CX2), j02(=2), f~(~l), and f~(=t) (see Fig. 1) are blending functions that will be determined using the criterion that the nearest boundaries dominate both for the derivatives and for the coordinates. For this criterion to be maintained for coordinate transformation,
3t= ~= ~-~.~
=2)
O)
1)
Xi(0, 0)Xl(0t 1, 1 ) - X~(0, 1)XI(= 1, 0)
(1)
f,(=l)
=
x'(0, 0)x% 1)-x'(1, 0)x'(0, 1) Engineering Analysis, 1988, Vol. 5, No. 1
3
Body-oriented coordinates: W. A. Cook and
However, 02x~l, ~2) Og~ 0~z
D
d~d d~t I
d~ 2
xi(1, O) dxi(o?, 1) dxi(0, ~t2) D -q
d~ 1
(plane strain)
1
(Ovl
Ovj'~
~'J = ~ \UxJ + ~x')
d~ 2
xi(0, 0) dxi(~ 1, 1) dxi(1, ~2) D d~ 1 d~ 2
(4)
I11)
for the Cartesian coordinates and 1
% = ~ (uij + uj.i)
where the determinant D is D = xi(0, 0)xi(1, 1) - x i ( l , 0)xi(0, 1). Equations (4) do not satisfy the criterion that the nearest boundaries dominate. An examination of Eqs. (3) shows that when all boundaries are linear functions, the general transformation equations, Eqs. (2), become identical to Eqs. (1), and ~ 2 x ~ ' ~2)
(12)
for the body coordinates, where the comma (,) represents covariant differentiation (see Appendix). Because the material tensors are symmetric, DJikl= D jikl = D ijlk--- D jilk
(13)
for the Cartesian coordinates and (14)
E ijkt = E jikt-~ E ijtk = E jilk
for the body coordinates. The virtual work principle can be written as
Oa 1 00~2
are constants. Thus, Eqs. (4) are the most satisfactory for making the blending functions simple, low-ordered polynomials. Note that Eqs. (4) equally represent all boundary functions and corner points. Detailed derivations of these transformations are presented in Ref. 4.
fv D'iu Ov' 6[~-(vt~ {ovk' d V - fs Tirvi \
/
dS=0
for the Cartesian coordinates and
In this section, the stiffness and loads are derived for Cartesian and body coordinate elements. In these two formulations, the displacements are approximated differently. Equations (5) and (6) are the principles of virtual work that are used to derive the stiffness tensor. These principles of virtual work are limited to traction loads. For example, (5)
T
for the Cartesian coordinates, and
fvEiJkteijrek, d V - ~ tirui dS=O
(6)
dSr
for the body coordinates. Here the material tensors for plane stress, plane strain, and isotropic material are 22, ~ c~06kt+/~(cSikaj,+
6it~kXplane stress)
- \2 + 2#/
(plane strain)
\,t + zt~/
vl = p,(o~1, ~2)v7
(17)
for the Cartesian coordinates and
ui = p.(gl, ~2)u7
(18)
for the body coordinates, where p,(al, ~2) are the shape functions for the Lagrangian element. Ref. 5 describes how these shape functions are derived. Note the nodal point displacements ~ and ~. Each of these has a covariant tensor component of i, where i is 1 or 2 for this two-dimensional geometry, and a contravariant discrete tensor (see Ref. 6) component of n, where n is 1, 2, ... number of nodal points. In the following development, note that B, b, K, and K have both tensor and discrete tensor components. Thus, repeated indexes i, L k, l, r, and s sum 1 and 2, and repeated indexes m and n sum 1, 2. . . . number of nodal points. We define the B tensors for the Cartesian coordinates as
OVi Op, Ox--7 = Ox--5
(8)
= ~j.~
for the Cartesian coordinates and /' 22# "~ o u + ~g~kg~Z+ gUgjk) E "k'= 1,--7--~--/0 0
for the body coordinates. The finite-element approximation for each element is defined as follows:
(7) and O ijkl = ,~,t~iJt~kl + fl(t~ikt~ il + t~ilt~jk)
(16)
T
4. DERIVATIONS O F STIFFNESS A N D LOADS
fvD~%i~CSekldV- fs T~rv~dS=O
(15)
r
fvEO%,jr(u,,t) d V - fs tiru, dS=O
DOkt- (
(10)
for the body coordinates. Repeated indexes i, j, k, and 1 sum 1 and 2 for dimensions of geometry. The strains are defined as
dot 2
xi(O, 1) dxi(~d, 0) dx'(1, ~2) D
gUgjk)
Eijkt = 2gljgkt + it(gikgjl +
xi(1, 1) dxt(o?, 0) dx~(0, ~2)
(19)
thus, (plane stress)
l~on = ~x Op.j 6,k
(9)
4 Engineering Analysis, 1988, Vol. 5, No. 1
We define the tensors for the body coordinates (see Appen-
Body-oriented coordinates: I4I. A. Cook dix) as
Cartesian coordinates
u,,j =
;s T' rv, dS= ;s T'p. dS rv7
u, -
T k
T
n
= bijnu k
--
thus,
and body coordinates
bi~,= Op.
-
(20)
fs t' ru, dS= fs t'p. dS ru ~ T
From these definitions and from the virtual work principle, the stiffness tensors for Cartesian coordinates are -- ~ F~ijkllTt r
K~'~"- Jv ~
I~s
ui~muu, d V
(21).
and the stiffness tensors for body coordinates are f E i j k l b r b~ k'~, = Jv 0,, u. dV
(22)
The consistent load tensors are derived by determining the tractions on the boundary surface for Cartesian and body coordinates. For Cartesian coordinates, we can find the tractions intuitively. Figure 2 shows pressure (p) and shear (s) loads on a surface with body coordinate ~k. The tractions for these loads in Cartesian coordinates are T 1 -
p 0x 2 s c~x1 -I L 0~k L 0~k
and p
T2
(25)
~X 1
L ~k
S
q
~X 2
(23)
L c3~k
When we use the basic transformation property for a contravariant tensor, the tractions for these loads in body coordinates are c3~i ti = ~ T k (24)
(26)
T
= £ flu7
(26)
Using the theory in this section, we developed three Lagrangian elements with 4, 9, and 16 nodal points, respectively, for the Cartesian and body coordinate element formulations. These elements are illustrated in Fig. 3. Figure 4 illustrates the elements with both the Cartesian and the body coordinates. Because these elements are based on the virtual work principle for linear problems, the potential energy serves as a measure of the accuracy of the solution, that is, the minimum potential energy theorem. The Cartesian and body coordinate element formulations of Lagrangian elements are integrated 2 × 2 Gauss points for elements with 4 nodal points, 3 × 3 Gauss points for elements with 9 nodal points, and 4 × 4 Gauss points for elements with 16 nodal point elements. These were the minimum integrations required to achieve decreasing potential energies as the meshes were refined. When a reduced integration was used, the potential energies increased as the meshes were refined. For each Gauss integration and for each Cartesian and body coordinate element formulation, two potential energies were calculated: one with applied forces and displacements and one with integrated stresses and strains. In all instances, the potential energies were indentieal; thus, for the Lagrangian elements, the calculated stresses and strains are as accurate as the displacements. Ref. 7 discusses the possibility of achieving this same accuracy for other elements.
When we use the basic displacement assumptions in Eqs. (17) and (18), the force tensors are derived from the second integral in the virtual work principles (Eqs. [5] and [6]).
X2~
LAak/ ~
/
C
/
I ,xx2
^.~
I'
\
\-.\
II
//
"2t 5 Figure 2. A pressure (load/area) shear (load/area) applied to a boundary
,o,_ x1
Figure 3. Lagrangian elements with element coordinate system and Cartesian coordinate system Engineering Analysis, 1988, Vol. 5, No. 1 5
Body-oriented coordinates." W. A. Cook cz2
\
<
1.0
- -
(m)
<
,,q X2
(11
6.3661977
(m)
x1
(11
Figure 4. Body coordinate element with the body coordinates, element coordinates, and Cartesian coordinates p = 10.0
(N/m 2 )
ELASTIC MODULUS= 106 (N/m 2 ) 5. RESULTS AND C O N C L U S I O N S The body coordinate formulation reduces to the Cartesian coordinate element formulation for rectangular problems, as confirmed by the problem in Fig. 5 and the approximate solutions in Table 1. Table I also shows that with only three Lagrangian 9-nodal-point elements, we calculated a 98 % accurate solution to a longitudinal beam problem with transverse loads. Similarly, the body coordinate element formulation reduces to a cylindrical coordinate formulation for cylindrical problems, as shown in Fig. 6 and Table 2. Again, with only three of the Lagrangian 9-nodal-point elements, we calculated a 93 % accurate solution using the body or cylindrical coordinate element formulation.
POISSON'S RATIO= 0.3 Figure 6. Curved beam problem modeled with three Lagrangian elements (plane stress).
Table 2. Potentialenergy (J) for curvedbeamproblenP Mesh Nodal Points (cd x ct2) 9-Nodal-Point Elements Cartesian coordinate Body coordinate Cylindrical coordinate
7x3 -0.0.9726 -0.11271 - 0.11271
39 x 3 -0.12155 -0.12157 - 0.12157
"See geometry in Fig. 6. 1.0 (m)
:
10.0 (m) p= 10.0 (Nlm 2 ) ELASTIC MODULUS=IO 6 (Nlm 2 ) POISSON'S RATIO= 0.3
Figure 5. Longitudinal beam problem with three rectangular Lagrangian elements (plane stress) Table 1.
Potential energy (J) for longitudinal beam problem"
Mesh Nodal Points (~d x ~t2) 9-Nodal-Point Elements Cartesian coordinate Body coordinate
7x 3 -0.19685 -0.19685
"See geometry in Fig. 5.
6
Engineering Analysis, 1988, Vol. 5, No. 1
39x3 -0.2~77 -0.2~77
For rectangular- and cylindrical-shaped problems, the body coordinates are orthogonal. Figures 7 through 10 show some problems in which the body coordinates are not orthogonal. The approximate solutions for these problems are evaluated in Tables 3 through 6. The accuracy parameter is the normalized potential energies, which are normalized to the minimum potential energy of all approximate solutions. From Tables 3 through 6, the following conclusions can be made. • When comparing elements with the same number of nodal points, those with 16 nodal points are generally more accurate than those with 9 nodal points, and those with 9 nodal points are generally more accurate than those with 4 nodal points. • The Cartesian coordinate element formulation is generally more accurate than the body coordinate element formulation for elements with 4 nodal points. • The body and Cartesian coordinate dement formulations are similar in accuracy for elements with 9 nodal points.
Body-oriented coordinates." W. A. Cook
P
121 p = I O O . O ( N I m 2) ELASTIC MODULUS = 10 7 (N/m 2 ) POISSON'S RATIO = 0.3 f
p=lO.O
(NIm 2)
Figure 9. Skewed straight-line geometry (plane strain).
ELASTIC MODULUS =10 6 (N/m 2) POISSON'S RATIO= 0.3
Figure 7. Straight-line tapered geometry (plane stress)
• The body and Cartesian coordinate element formulations are similar in accuracy for elements with 16 nodal points. A more meaningful comparison of elements with 4, 9, and 16 nodal points can be made when computational time is considered. Tables 7 and 8 tabulate the computational times required for the four problems shown in Figs. 7 through 10. On the basis of the results shown in Tables 7 and 8, the following conclusions are offered. • The difference in computational time between body and Cartesian coordinate element formulations is generally small, approximately 2 % or less. • For elements with equal numbers of nodal points, the elements with 4 nodal points require approximately one half the computational time required by the elements with 9 nodal points, and the dement with 9 nodal points require approximately one-half the computational time required by the dement with 16 nodal points.
~0 ×
XI X Xi
>i N × N
P
From the computational times shown in Tables 7 and 8 and from the normalized potential energies shown in Tables 3 through 6, accuracy vs computational time plots can be made for the problems shown in Figs, 7 through 10. The most accurate formulation was made for each dement and the resulting plots are shown in Figs. 11 and 14. The following are our general conclusions. • The higher order elements (those with 9 and 16 nodal points) are significantly more efficient than those with 4 nodal points. • The body and Cartesian coordinate element formulations are similar in accuracy and efficiency.
?
N >.
p =
10.0 (NIm 2 )
ELASTIC MODULUS=IO 6 (N/m 2 ) POISSON'S RATIO =0.3
Figure 8. Curved-boundaries tapered geometry (plane stress)
In summary, the body coordinate element formulation is as accurate and as efficient computationaUy as the Cartesian coordinate element formulation. A careful study of Tables 3 through 6 shows that the differences in these two formulations are considerably smaller than we had expected them to be on the basis of the results of the curved beam problem (Table 2 and Fig. 6).
Engineering Analysis, 1988, Vol. 5, No. 1 7
Body-oriented coordinates." IV..4. Cook
X
\
C2 1
p=IO0.O (N/m 2) ELASTIC MODULUS =10 7(N/m 2) POISSON'$ RATIO = 0.3 Figure 10.
Skewed curved-boundaries geometry
(plane strain).
Table 3. Normalized potential energy. Straight-line tapered geometry~ Mesh Nodal Points (cd x a 2) 4-Nodal-Point Elements Cartesian coordinate Body coordinate
7x 7
13 x 13
19 x 19
25 x 25
37 x 37
0.98673 0.99533
0.99592 0.99817
0.99799 0.99901
0.99880 0.99938
0.99944 0.99970
9-Nodal-Point Elements Cartesian coordinate Body coordinate
0.99729 0.99859
0.99936 0.99947
~99974 0.99976
0.99987 ~99988
0.99997 0.99997
16-Nodal-Point Elements Cartesian coordinate Body coordinate
0.99853 0.99633
0.99965 0.99892
0.99987 0.99949
0.99994 0.99971
0.99988
1.0 b
=See Fig. 7. b M i n i m u m P o t e n t i a l Energy ( C a r t e ~ a n C o o r d i n a t e , 37 x 37) = - 2 . 1 5 0 6 1 5 9 10 - a J
Table 4. Normalized potential energy. Curved-boundaries tapered geometry a Mesh Nodal Points (~tI x ~t2) 4-Nodal-Point Elements Cartesian coordinate Body coordinate
7x 7
13 x 13
19 x 19
25 x 25
37 x 37
0.96042 0.90131
0.98461 0.96133
0.99142 0.97933
0.99442 0.98711
0.99705 0.99359
9-Nodal-Point Elements Cartesian coordinate Body coordinate
0.98548 0.98586
0.99612 0.99640
0.99822 0.99839
0.99903 0.99914
0.99968 0.99974
16-Nodal-Point Elements Cartesian coordinate Body coordinate
0.99202 0.99524
0.99784 0.99851
0.99909 0.99934
0.99957 0.99970
0.99995 1.0b
=See Fig. 8. b M i n i m u m P o t e n t i a l Energy (Body Coordinate, 37 x 37) =--2.0743773 10 -4 J
8
Engineering Analysis, 1988, Vol. 5, No. 1
Body-oriented coordinates." W. A. Cook Table 5. Normalized potential energy. Skewed straight-line geometr~ Mesh Nodal Points (cd x ~t2) 4-Nodal-Point Elements Cartesian coordinate Body coordinate
7x 7
13 x 13
19 x 19
25 × 25
37 x 37
0.98756 0.97514
0.99598 0.99193
0.99795 0.99603
0.99875 0.99763
0.99939 0.99763
9-Nodal-Point Elements Cartesian coordinate Body coordinate
0.99703 0.99682
0.99922 0.99918
0.99966 0.99965
0.99982 0.99982
0.99995 0.99995
16-Nodal-Point Elements Cartesian coordinate Body coordinate
0.99831 0.99845
0.99957 0.99960
0.99983 0.99984
0.99992 0.99993
1.0 1.0b
aSee Fig. 9. bMinimumPotential Energy (Body Coordinate, 37 x 37)=-5.0103266
10 - 3
J
Table 6. Normalized potential energy. Skewed curved-boundaries geometry a Mesh Nodal Points (cd x u2) 4-Nodal-Point Elements Cartesian coordinate Body coordinate
7x 7
13 x 13
19×19
25x25
37x37
0.97897 0.90271
0.99327 0.96889
0.99653 0.98483
0.99784 0.99104
0.99891 0.99580
9-Nodal-Point Elements Cartesian coordinate Body coordinate
0.99477 0.99208
0.99857 0.99888
0.99939 0.99951
0.99968 0.99976
0.99989 0.99993
16-Nodal-Point Elements Cartesian coordinate Body coordinate
0.99646 0.99803
0.99914 0.99925
0.99967 0.99974
0.99985 0.99990
0.99998
~See Fig. 10. bMinimumPotential Energy (Body Coordinate, 37 x 37) =-2.9748175
10 - 3
1.0 b
J
Table 7. Computational time for the two tapered geometries (Cray computer seconds) Mesh Nodal Points (ul x ~t2) 4-Nodal-Point Elements Cartesian coordinate Body coordinate
7x7
13 x 13
19 x 19
25 x 25
37 x 37
0.7 0.8
2.5 2.8
5.8 6.4
11.0 12.1
29.5 31.8
9-Nodal-Point Elements Cartesian coordinate Body coordinate
1.1 1.2
4.4 4.7
10.7 11.2
21.0 21.9
59.3 61.3
16-Nodal-Point Elements Cartesian coordinate Body coordinate
2.3 2.3
9.0 9.3
21.7 22.3
42.1 43.2
116.7 119.3
APPENDIX COVARIANT DIFFERENTIATION COORDINATES
FOR BODY
This A p p e n d i x describes the c o v a r i a n t differentiation used to derive the stiffness t e n s o r using b o d y coordinates. T h e t r a n s f o r m a t i o n f r o m b o d y c o o r d i n a t e s (ct1, ~t2) to C a r t e s i a n c o o r d i n a t e s is defined by Eqs. (1). U s i n g this t r a n s f o r m a t i o n , we can define the base vector, the metric tensor, the Christoffel s y m b o l of the first kind, a n d the
Christoffel s y m b o l of the second k i n d as follows (see Ref. 8). W e define the base vector as
Oxk(~l' ~t2)~"
(A-l)
W e define the metric t e n s o r as
Ox~ ~x k - t%ti ~cd
(A-2)
Engineering Analysis, 1988, Vol. 5, No. 1
9
Body-oriented coordinates." IV. A. Cook Table 8. Computationaltimefor the two skewedgeometries (Cray computer seconds) Mesh Nodal Points (=1 x =2) 4-Nodal-Point Elements Cartesian coordinate Body coordinate
19x19
25x25
37x37
7x7
13 x 13
0.8 0.8
2.5 2.6
5.8 6.0
l 1.0 11.4
29.4 30.1
9-Nodal-Point Elements Cartesian coordinate Body coordinate
1.2 1.2
4.4 4.5
10.7 10.8
20.9 21.0
59.2 59.2
16-Nodal-Point Elements Cartesian coordinate Body coordinate
2.3 2.3
9.0 9.1
21.7 22.3
42.0 43.1
117.0 119.1
$
~ 099~
•
16-NODAL-POINT ELEMENT (CARTESIAN COORDINATE)-
&
g-NODAL-POINT ELEMENT (BODY COORDINATE)
a
4-NODAL-POINT ELEMENT (BODY COORDINATE)
..... N
. ,BTog;~%%.~ ~ ; . ; •
9-NODAL-POINT ELEMENT (CARTESIAN COORDINATE)
•
4-NODAL-POINT ELEMENT (CARTEBIAN COORDINATE)
~ i
70 _ 099(I 0
2O
3O
4O
60
8O
100
2OO
30.0
7O 0
500
ComPutational Time (s)
0,990
Figure 11. Accuracy (normalized potential energy) vs cornputational time for straight-line tapered geometry (see Fig. 7) 1ooo
;
I
Computational
' T i m e (s)
Figure 13. Accuracy (normalized potential energy) vs cornputational time for skewed straight-line geometry (Fig. 9)
: i
i
i
~
i
i
i
I
i
i
$ ,=, ~ 0.995 g ~_
0.995 •
1 6 - N O D A L - P O I N T ELEMENT
(BODY COORDINATE)
o
(BODY C O O R D I N A T E )
0990
20
30
40
610
I 80
100
20.0
3(30
5
0
700
I
900
•
4 - N O D A L - P O I N T ELEMENT (CARTESIAN COORDINATE)
Computational Time (s) 0 990
Figure 12. Accuracy "normalized potential energy) vs computational time for circular-boundaries tapered geometry (see Fig. 8)
__
lO
2.0
3.'0
4.0
60
8,0
10.0
2 .0 = ' 3 0 0
4O0
Computational Time (s)
Figure 14. Accuracy (normalized potential energy) vs cornputational time for skewed curved-boundaries geometry (Fig. 10).
W e define t h e Christoffel s y m b o l o f t h e first k i n d as
[ij, k ] = ~
O~~
W i t h t h e s e t e n s o r definitions, c o v a r i a n t d i f f e r e n t i a t i o n of t h e v e c t o r ui is
630tkJ
(~Ui k u,,; = ~ - {u}uk
where ~g/j__
a~ k
~2Xl
~X l
actk ~ t i & t ;
~2xl ~X l -&t k & d & d
(A-3)
W e define t h e Christoffel s y m b o l o f t h e s e c o n d k i n d as {~j} ----where
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Engineering Analysis, 1988, Vol. 5, No. 1
NOMENCLATURE
Description
gktrij, I]
gikgkj = ~
(A-5)
(A-4)
Coordinates Displacements Material tensor B tensor
Cartesian Coordinates
Body Coordinates
x~ vi
=i ui
Dukt
E Ukl
~,
b~j.
Body-oriented coordinates: W. A. Cook Stiffness tensor Strain tensor Traction tensor (v-') Force tensor fi~,
ACKNOWLEDGEMENTS ~ij T
eo t
[] {}
Kron~ker delta and variation operator Lames constants (material properties) Normal vector to boundary surface Element coordinates Shape function for a finite element Pressure Shear Length of a boundary surface Boundary surface Boundary surface with tractions Volume of a region Base vector, metric tensor Cartesian base vector Christoffel symbol of first kind Christoffel symbol of second kind
Comma ui.j
Covariant differentiation
Pn P s
L S Sr V g~, go
7~
Subscripts and Superscripts i, j, i, l, r, a n d s
m, n
Indexes for covariant and contravariant tensors. Repeated indexes imply summation over geometry (1, 2, for two-dimensions). Indexes for covariant and contravariant discrete tensors (see Ref. 6). Repeated indexes imply summation over all nodal points.
This research has been s u p p o r t e d by the D e p a r t m e n t of Energy Advanced Weapons Technology. T h e following associates h a v e been very helpful: J. D. Allen, W. D. Bitchier, J. W. Bolstad, R. F. D a v i d s o n , J. Genin, R. E. Holt, N. L. J o h n s o n , M. R. M a r t i n e z , D. A. Rabern, L. L. Shelley, J. W. Straight, and S. C. Woolley.
REFERENCES
1 Cook, W. A. The effect of geometric shape on two-dimensional finite elements, Nuclear engineering design, 1982, 70, 13-26 2 Cook, W. A. Body oriented (natural) co-ordinates for generating threedimensional meshes, InternationalJournalfor NumericalMethods, 1974, 8, 27-43 3 Cook, W. A. and Oakes, W. R. Mapping methods for generating threedimensional meshes, A. Siereg, Ed., Computers in me~,haaic-.lengineering (CIME), August 1982, ASME, 1983,1, 67-72 4 Cook, W. A. Body-oriented coordinates applied to the finite-element method, Los Alamos national laboratory report LA-10858-MS, October 1986 5 Cook, R. D. Conceptsand applications offinite element analysis, 2nd ed., John Wiley and Sons, Inc., New York, 1981 6 Kardestuncer, H. Finite-element method via tensors, The matrix and tensor Quarter),, (UK) 1976, 26 (14), 129-140 and 27 (1), 1-14 7 Barlow, J. Optimal stress locations in finite element models, International Journalfor Numerical Methods, 1976, 10, 243-251 8 Green, A. E. and Zerna, W. Theoretical elasticity, 2nd ed. (Oxford University Press, London, 1968), Chap. 1, pp. 1-52
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