Three transition interpolation schemes for use in h-or r-refinement have been analyzed in terms of accuracy, implementation ease and extendability. They include blending-function interpolation, displacement averaging, and strain matching at discrete points along the transition edge lines. The results suggest that the choice of matching depends strongly on the element formulations, (viz. displacement or assumed strain, etc.) and mesh refinement criteria employed, and to a lesser extent the choice of computer architecture (serial vs. parallel) and the equation solution procedures. A recommended pairing of some of the elements with the choice factors is suggested

Levit, Itzak

Park, K. C.

Stanley, Gary M.

English

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NASA-CR-199037
CU-CSSC-91-15
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CENTER FOR SPACE STRUCTURES AND CONTROLS
EVALUATION OF DISCRETIZATION
PROCEDURES FOR TRANSITION
ELEMENTS IN ADAPTIVE MESH
REFINEMENT
(NASA-CR-199037) EVALUATICN CF
DISCRETIZATION PROCEDURES FOR
TRANSITION ELEMENTS IN ADAPTIVE
MESH REFINEMENT (Colorado Univ.}
16 p
G3/61
N95-32190
Unclas
0060392
May, 1991
by
K. C. Park, I. Levit and G. M. Stanley
COLLEGE OF ENGINEERING
UNIVERSITY OF COLORADO
CAMPUS BOX 429
BOULDER, COLORADO 80309
https://ntrs.nasa.gov/search.jsp?R=19950025769 2020-06-16T06:13:49+00:00Z

EVALUATION OF DISCRETIZATION PROCEDURES
FOR
TRANSITION ELEMENTS IN ADAPTIVE MESH REFINEMENT
K. C. PARK
Center for Space Structures and Controls and
Department of Aerospace Engineering Sciences
University of Colorado
Boulder, CO 80309
and
ITZAK LEVIT AND GARY M. STANLEY
Computational Structural Mechanics Group
Lockheed Palo Alto Research Laboratory
3251 Hanover St., Palo Alto, CA 94304
May 1991
ABSTRACT
Three transition interpolation schemes for use in h or r-refinement have been analyzed in
terms of accuracy, implementation ease and extendability. They include blending-function
interpolation, displacement averaging, and strain matching at discrete points along the
transition edge lines. The results suggest that the choice of matching depends strongly on
the element formulations, (viz. displacement or assumed strain, etc.) and mesh refinement
criteria employed, and to a lesser extent the choice of computer architecture (serial vs.
parallel) and the equation solution procedures. A recommended paring of some of the
elements with the choice factors is suggested.

1. Introduction
The h-version and/or r-version of mesh refinements have been steadily gaining popularity
in recent years. A primary reason for this popularity has been due to the simplicity
of data handling for subsequent refinement and solution strategies. The error estimate
criteria employed for these refinements are by and large deduced from the assumption
that there is no loss of accuracy in the transitional meshes or the transitional meshes
are sufficiently remote from the regions of rapid stress or strain gradients (Babuska and
Dorr, 1981; Felippa, 1976; Shephard and Gallagher, 1979; Zienkiewicz and Zhu, 1987).
Two techniques most frequently adopted for discretizing the transitional regions are the
displacement matching and the blending function interpolations (Gordon and Hall, 1973;
Surana, 1980; Gordon, 1970; Cavendish and Hall, 1984).
From the viewpoint of element formulations, these techniques go well in hand with the
displacement-based elements as they involve only displacements at the element boundaries.
Hence, it appears that transition interpolations for other element fornmlations such as hy-
brid and assumed strain elements would require the matching of different field quantities.
A recent numerical study of mesh refinement strategies that utilize displacement matching
has shown that the displacement-based 9-node element yields a nmch better mesh refine-
ment pattern compared to the assumed natural-coordinate strain (ANS) 9-node element
(Stanley et al., 1989). This has motivated us to examine various interpolation sclmmes
used in transition elements and classify the desirable conjugacy of element formulations
vs. transition interpolation schemes.
2. Formulation of Transition Discretization
Consider two quadrilateral shell elements that have one common transition boundary as
shown in Fig. 1. Although one eould avoid transition nodes in triangular elements, such
a strategy can lead to different nodal patterns or severe element distortion, which in turn
deteriorates the accuracy in the transition regions. This is illustrated in Fig. 2. Conceiv-
ably, the distortion brought about by an h-refinement can be regularized to some extent
if an r-refinement is followed up after each h-refinement. However, such an r-refinement
follow-up may not contribute to overall accuracy improvements.
During the present study of transition discretizations, we will impose one important imple-
mentation requirement: the construction of transition boundary conditions should involve
only the nodal degrees of freedom (dofs) along the transition element edges. If one views
the nodes along the transition element edges as connectors for tying element A to elements
B and C in Fig. 1, there exist three possibilities. These possibilities include blending
displacement interpolation, displacement averaging, and strain (or stress) matching. We
will describe in detail the three procedures.
() 0 ()
4-Node Element and
Transition Node(X)
) A O
) x 0 x
B
I
C
0
9-Node Element and
Transition Nodes (X)
()
Transition Node(s) Belong to Element A
Fig. 1 Transition Nodes for quadrilateral elements due to (h, r)-refinements
Use of Two Transition Nodes
Allows Accurate Triangular
Discretization:
/
/
/
t'
K = ( K + K )/2[] F1
Without Transition Nodes
Union-Jack Pattern Develops,
Hence Loss of Accuracy
in General
Fig. 2 Transition Nodes vs. Union-Jack Pattern for Triangular h-Refinements
2.1 Blending Displacement Interpolation
This is by far the simplest procedure to introduce additional nodes as connectors so that
the generalized degrees of freedom associated with the additional nodes remain linearly
independent of the remaining ones. For example, when a 4-node element possesses one
transition edge, one introduces the following blending interpolations:
where
N
u = E Nj((,r/)ui (1)
j=l
N, = Ll(q). Qx(() ]
]
N2 = L,(r/) Q2(()
N3 = L2(r/) LI(()
N4 = L2(r/) L2(()
N5 = Ll(r/) Q3(()
(2)
in which L's and Q's are linear and quadratic Lag'range interpolation functions. A similar
blending interpolation can be constructed for a 9-node element with one or more transition
edges.
q
LI(_)
L2(q) ()
3
I
L1(q) ( )
QI({)
5
X
O3(_)
L2 b
(
4
2
()
Q2(_)
Fig. 3 Blended Interpolation of Transition Nodes
It should be pointed out that while the interpolation procedure is straightforward, the
resulting element would require a (2 × 3)-integration rule instead of (2 × 2)-integration in
order to preserve rank-sufficiency.
4
2.2 Displacement Averaging
Displacement averaging is by far the simplest procedure.
at node 5 can be obtained by an average of
For example, the displacement
u5 = au2 + (1 - o)u_, E1gl + t2 (3)
where gx and g2 are the side length from node 1 to node 5, and node 5 to node 2, respectively.
If ga = g2, then we have
1¢ = u5 - _(ul + us) = 0 (4)
Hence, u5 can either be eliminated or the constraint equation (4) can be augmented via the
method of Lagrange multipliers (X) and u and .k can be solved as the system unknowns.
2.3 Strain Matching
Instead of displacement matching or interpolation, one may impose strain matching as
a connector requirement. For this case, in conjunction with the present implementation
requirement, viz., the use of dofs along the transition edges only, we are to match the
membrane, transverse shear and bending strains:
10ue
e_ - A_ O_
1 Ow_
"_ = Ae O(
1 O0
_- A_ O_
(5)
where A_ is the first fundamental coefficient of the neutral surface of the shell along the
boundary _, (u_, w_, 0_) are the tangential, normal and rotational deformation along _.
For a 4-node element having one transition edge as shown in Fig. 3, it turns out the strain
matching is the same as the displacement averaging as long as the transition node 5 bisects
nodes 1 and 2. To exasnine the strain matching for a 9-node case, we adopt a flat plate
for illustrative purposes as shown in Fig. 4.
)9 01o 11()
8()
3_)
()6 0 7
2
0
1 _b 4 12
°
nts
,._k.]= I ---[_
5 _b 3
1/gg
1/gg
Fig. 4 Strain Matching Along A 9-Node Edge
Since there are two transition nodes with (u, w,/9,)-dofs affected, a total of six constraints
are required in order to eliminate them. Hence, we must enforce the strain matching of the
three strain states given by (5) at two locations along the transition edge. Among several
candidates, it has been determined that the most desirable strain-matching locations are
the two Barlow points (_ = -t-1/x/_). Matching the strains at the two Barlow points yields
the following constraints:
I) =
8u, - (3u, + 6u2 - u3) = 0
8u5 - (-ul + 6u2 + 3u3) = 0
8v4 - (3Vl 4- 6v2 - v3) = 0
805 - (-v, + 6v2 + 3v3) = 0
804 -- (301 4- 6/92 -/93) ----0
8/95 -- (--/91 4- 6/92 4- 3/93) -- 0
8/34 - (3fl, 4- 6fl2- fl_) = 0
8fl5- (-fl, 4-6_2 4-3fl3)= 0
8w4 -(3wx +6w2 - w3) 4- {(03 - 202 4- 0,) = 0
g (03 - 2/92 +/9, ) = 08w_ - (-w, + 6w2 + 3w3) -
(6)
6
Rewriting (6), one obtains
Bsd = 0, d = [u 1 _)1 Wl 01 #1, ..-, .-., ..., u5 u5 w5 05 /35 ]T (7)
If the displacement averaging is adopted, one replaces the two strain constraint equations
on the transverse shear strains by
8w4 - (3w, + 6w2 - w3) = 081/35 --(--Wl +61/32 -_" 3w3)=0 (8)
3. Implementation
There are two distinct needs for implementing the blending function approach vs the
method of Lagrange multipliers. We shall describe the two procedures separately.
3.1 Blended Assumed Natural-Coordinate Strains
To illustrate the blended assumed natural-coordinate strain (ANS) interpolation, let us
consider the following 5-noded membrane square element.
"rl, y, v
)
3 4
1 5 2
) x (
•_ _,X,U
Figure 5 5-Node Blended-Function Square Plate Element
From the above figure one readily obtains the following strain expressions:
7
To see whether (9a) yields a rank-sufficient stiffness matrix, it is adequate to check
f
= ] B_EB_ge=z T dV
Jy
where
(9)
(10)
ezz = Bzzu, u = [Ul,
= [_(-1 +2_)g,(r/), _(1 +2_)gz(r/),B_
in which
112, U3, U4, us J T ill)
(12)
Nz(T/) = ½(1 -- r/), N2(_) = ½(1 + 17)
and E is Young's modulus. Evaluation of (10) for a constant thickness plate (h) gives
14
2
Eh
K_zz - 18 3
-3
-16
2 3 -3 -16
14 -3 3 -16
-3 3 -3 0
3 -3 3 0
-16 0 0 128
(13)
which has the required correct rank since there is only one physical rigid mode. Hence,
(2 x 2)-integration of the transition element involving node 5 will have a correct rank.
For higher order elements, e.g., 9-node element, similar ANS transition elements can be
constructed.
3.2 Augmentation of Displacement Averaging or Strain Matching Condition
For linear cases, the transition conditions given by (6) or (7) can be augmented as follows:
so that one obtains
6H = 6uTKu + 6ATBu + 6uTBTA -- 6uTf = 0 (14)
[KB BT] {u}= {f}0 X 0 (15)
Partitioning u further into the transition node ut and the original node u0, (15) can be
rearranged as
K,o K,t B T u, = ft (16)
Bo Bt 0 A 0
So that (15) can be solved directly or by
Koouo=fo (17)
where
I_oo Koo KotB71Bo T -T T -T -1= -- -BoB t Kto+BoB t K.B t Bo
T -Tf= fo-BoBt ft
(18)
4. Strain Matching for 9-ANS Transition Element
As alluded to in Introduction, the ANS family of shell elements are constructed by inter-
polating the strains along the element edges and certain interior natural-coordinate lines.
Hence, displacement matching or blended function approach can destroy much of the good
ANS element attributes. A preferred construction of transition strategies for ANS elements
is to match the strains along the transition edges at some discrete points. To this end, let
us consider Fig. 6 in which the element edge A is to match with the element edges B and
C as shown below.
T
_- 2 _ Edge A
"b JStrain Mat_hin9 Points_5 "b
Figure 6 Strain Matching Along Three ANS Curved Element Edges
9
The position vector of the particle point P (Fig. 7) on the deformed shell is given by
r = r ° + _b3 (19)
where
r° = xe, + re2 + ze3, (x V z) = {(X + u) (V + v) (Z + w)} (20)
in which x, rand z are the deformed neutral shell surface position coordinates, (u, rand w)
are the displacements measured in the inertial e-system, _ is the distance of the material
point P from the shell neutral surface measured in the b-system attached on the deformed
cross-section of the shell, and the vector b is related to the vector e by
b = Re (21)
The linearized (or incremental) displacement vector, U, is obtained from:
tl = uTe + _£1 (22)
where the pseudo-rotation vector, fi, is related to the shell-surface, pseudo-rotation quan-
tities, _ according to
{/31} /)= T/s&Tb. (23)fl = (--tTg(2) tTg(1)) /32 '
in which tTg(i) represents the i-th row of the transformation matrix, Tsg, defined by
s = Tsge (24)
that are attached to the deformed shell surface; and, Tb, relates the shell-surface basis
vectors s, to the inertial basis vectors according to
b = Tb,s (25)
The covariant natural-coordinate unit vectors are then obtained by:
1 Or ° 1
-- (x,( e, + y,(e2 + z(e3)= tie (26)
a_ = _- 0_ A_
1 Or ° 1
-- (X,_ el + y,,e2 + znea)= tTe (27)
a'7- A n Oq A n
a( x a n _(z,¢el +y,(e2+z,(e3)=t_'e
a<- [a(x an[
(2s)
10
£'* xe!÷ ye2÷ze 3
1 (,. ,. ,l.{Ix+ul. _x+,l. _z+,,)}
!2
r !1
Figure 7 Geometry of Shell Surface
where the two fundamental shell surface quantities, Ae and A_, are given by:
vOr° 0r ° Or ° Or °
AI= 0_ _-' A_= 07 07 (29)
For subsequent applications, we express the above relation in a compact form:
{<}a=T.ge= t T e (30)
Finally, the covariant partial derivatives axe given by:
0 1 0 0 1 0 0 1 0
OS_ - A e O_ OS, 7 - A,_ OrI 0S¢ = -A¢ O( A¢ = h((, rl)/2 (31)
where h(_, 7}) is the shell thickness.
First, the membrane strains, eee, at the two Barlow points along the edge A are given (
see Park et al.(1989) for details in the natural-coordinate strain formulation), respectively,
by:
_(_)= ! _ •1½(-_--,)+_(u_- 2._+ u,)
A¢, t¢,
(32)
11
1_l_(_b ) ---- A__bt_. [½(u 3 -- Ul)-}-_b(U3 -- 2u 2 -}- Ul) (33)
where _b and _b are the two right and left Barlow points along the edge A.
On the other hand, the assumed strains at the same location along the edge B and C can
be derived, respectively, as
C (_h) = '(1+ _}' C , ½(1 _}_e c f_'_ (34)
e_,_,(5_)= ½(1+ --5}_)e ,_,(_,)+ ½(1- ____b)e¢,¢,(_b)5'LB -,g (35)
in which c , ce_,_,(_b) and e_,_,(_) are the strains at the two Baxlow points along the edge C,
and B , Be_,_,(%) and (_;) axee_,_, the strains at the two Barlow points along the edge B.
Equating e_(_b) and c ,e_,_,((L) one obtains the following constraint at (b = 1/V_ on the
edges A and C:
cy.u, + 4.u + cy.u2+c1 .Ul= o (36)
where ci are given by
c5 = (1 - V_)_-¢ t:; + (1 - l"f_' 1 t C
1 C
.f4_1 t c 5/v_) _-___c3 = -(0.5 + 1/V_)A--_t_, + }(1 + v oj_-_ _ + }(--3 + t_
1 C 1/V_) 1 C
= 2 1 tf, + }(-5 + 3x/_ _--c-t,, ¼(1 + t_;
cl = (0.5- 1/V_)A--_t_,
(37)
where the superscript C denotes the quantities pertaining to the edge C.
Similarly, the membrane strain matching constraint at _b = --1/X/_ along the edges A and
B is obtained by:
eT.u4 + eT.ua + eT-u2 +eT'ul : 0 (38)
where c:i are given by
12
c4 = (1 - l/v/3) ' c (1 v_)2-_-.t_; + - ' t C
e3 = (0.5 - 1/x/3) '-Lt -
A(b _b
2 I
(:2 -- v_ A<bt& - ¼(1 + 1/x/g) _-qt_;' " + I(-5 + 3v_l..ql -,B
_:, = -(0.5 + 1/V_)A--_t & + ¼(--3 + 5/V/3) A--_-t,; + ¼(1 + V_>A--__ t_B
(39)
It should be noted that the two bending strain constraints at the two Barlow points
axe obtained by replacing u by fi in the preceding membrane matching conditions. The
remaining two strain matching conditions, viz., the transverse shear strains,
T 1 0u
lead to the following two constraints:
----+tga (40)
dsT.us+dT-ua+dT-u2+dT.u,+eT-fi5+eT-fia+eT-fi2+eT.fil =0 (41)
-T -T -T -
d4 .u 4 q_ d3 .u 3 nc d2 .u 2 q_ dlT . Ul + @T. 1714 .it_ @3T " £1 3 _it_ _2 T " 1712 q_ _1T " 1711 : 0 (42)
where di and di are obtained by replacing t_ with to, etc., into ci and _:i, respectively,
and ei and ei axe given by
e5 = (1 - 1/v_)t_ + ½(V_- 1)t_
ea = -_(1 + X/3)t_b + 6-_t_ + _(Vf3 - 2)t_
=2 '(3- 2v_)t_ ___1 Ce2 _t_, + g gt_
e, = -_(1- X/_)t_b
1)t + (1- 1/v%t 
2 1 B
e2 = _t& + gt_ + _(3- 2x/_)t_
_, = -k(1 + v_)t& + _(V/3 - 2)tg; + 4t_
(43)
(44)
Finally, the displacement u. and fin, namely, those perpendicular to t( and t( ( t, =
t( x t(), can be obtained by
gT'u5 -k-gT'ua + gT'u2 -{'-gT'ul = 0
13
(45)
gT.u, + gT.u, + g[.u_ + gT.u, = o
where
g3 -----tn(_b) g3 = tn(_b)
g2 - ' g2 -
gl +3 gl -3
The interpolation functions for ft, at the two Barlow points are the same as for u,.
(46)
(47)
Combining the six strain matching constraints and the four displacement interpolation
conditions together, one obtains the following equation:
_S = BANS{q} (48)
where qi = (ui, I[li) T and BANS is given by
BANS =
_ o c_ o cT o o o c_ o
o cT o c_ o c_ o o o c_
d T e T d T e T d T e T 0 0 d T c T
gT 0 gT 0 gT 0 0 0 gT 0
0 gT 0 gT 0 gT 0 0 0 gT
eT o eT o _T o eT o o o
o _1 o _ o _ o e_ o o
- -T -T -T
d T _T d2 _T d3 _T d4 6T 0 0
gT o g_ o gT o e£ o o o
o g_ o gT o g_ o g_ o o
(49)
It can be shown that the above expression reduces to the flat, straight edge case given by
(6). It should also be noted that fi introduced in (23) must be replaced by (/3a, /32) in (48)
in actual implementation in order to avoid singularity.
5. Discussions
The present study has examined three possible ways for discretizing the mesh transi-
tion regions. It has been shown that for four-noded elements the displacement averaging
and the strain matching yield the same transition discretization expressions. For plates
with straight edges discretized by nine-noded elements, the Barlow-point displacement and
strain matching becomes the same except for the transverse shear strains.
The transition interpolation schemes for curved shell boundaries for the nine-noded ANS-
shell element (Park and Stanley, 1895; Stanley, 1985; Park et hi., 1989) has been carried
in detail for implementation in h-version mesh refinement. It is noted that the transition-
boundary strain matching directly affects the resulting assumed natural-coordinate strains
14
employed in the formulation of ANS-shell element generation. Suchexplicit enforcement
of strain-matching conditions along the element edgesis considereda distinct advantage
of the ANS-shell element.
References
1. Babuska, I. and Dorr, M., Error estimates for the combined h and p versions of the
finite element method, Numerische Mathematik, Vol. 37, 1981, pp. 257-277
2. Cavendish, J. C. and Hall, C. A., A new class of transitional blended finite elements
for the analysis of solid structures, Int. J. Numer. Methods Engrg., 20, No. 2, 1984,
pp. 241-254
3. Felippa, C.A., Optimization of finite element grids by direct energy search, Applied
Mathematical Modelling, Vol 1, 1976, pp. 93-96
4. Gordon, W. J., Blending function methods of bivariate and multivariate interpolation
and approximations, SIAM J. Numer. Anal., 8, 1977, pp. 158-177
5. Gordon, W. J. and Hall, C. A., Transfinite element methods - blending function inter-
polation over arbitrary curved element domains, Numer. Math., 21, 1973, pp. 109-129
6. Park, K. C. and Stanley, G. M., "A Curved C o Shell Element Based on Assumed
Natural-Coordinate Strains," Journal of Applied Mechanics, Vol. 108, 1986, pp.
278-290.
7. Park, K. C. and Stanley, G. M. and Cabiness, H., "A Family of C ° Shell Ele-
ments Based on Generalized Hrennikoff's Method and Assumed Natural-Coordinate
Strains," in Finite Element Methods for Nonlinear Problems, Bergan P. et al, eds.,
Springer-Verlag, Berlin, 1986, pp. 265-282.
8. Sander, G. and Beckers, S. P., The influence of the choice of connectors in the finite
element method, Int. J. Numer. Method_ Engrg., 11, No. 10, 1977, pp. 1491-1505
9. Shephard, M.S., and Gallagher, R.H., Finite Element Grid Optimization, PVP-38,
American Society of Mechanical Engineers, New York, 1979
10. Stanley, G. M., Park, K. C. and Hughes, T. J. R., "Continuum-Based Resultant Shell
Elements," Finite Element Method for Plate and Shell Structures, Volume 1: Element
Technology, Hughes, T. J. R. and Hinton, E., (eds.) Pineridge Press, Swansea, U. K.,
1986, pp. 1-45.
11. Surana, K. S., Transition finite elements for three-dimensional stress analysis, Int. J.
Numer. Methods Engrg., 15, No. 7, 1980, pp. 991-1020
12. Zienkiewicz, O.C. and Zhu, J.Z., A simple error estimator and adaptive procedure for
practical engineering analysis, Int. J. Num. Meth. Eng., Vol 24, 1987, pp. 337-357.
15


Evaluation of discretization procedures for transition elements in adaptive mesh refinement

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