The project is based upon research of Tessler et al. (1994) on an improved variational formulation for post-processing stress predictions in Finite Element Analysis. The methodology, called Smoothing Element Analysis (SEA), employs a three-node smoothing finite element. The present effort focused on verifying the basic constant strain criterion for the three-node smoothing element subject to a set of internal penalty constraints. The convergence characteristics of the element are assessed by first deriving the constrained form of the assumed element stress and stress gradient fields, and then by verifying the validity of the constant strain criterion once the element penalty constraints are explicitly imposed. The analytical investigation is carried out with the use of the symbolic manipulation code 'Mathematica'

Sobel, Seth S.

English

NASA Technical Reports Server

/Characteristics of Three-node Smoothing Element
Under Penalty Constraints
Seth S. Sobel
LARSS Student
Integrated Science And Technology
James Madison University
Harrisonburg, VA
Dr. Alexander Tessler
Computational Structures Branch
NASA Langley Research Center
Hampton, VA
Research and Technology Group
Structures Division
Computational Structures Branch
667
https://ntrs.nasa.gov/search.jsp?R=19970003114 2020-06-18T00:11:18+00:00Z
Abstract
The project is based upon research of Tessler et al. (1994) on an improved variational
formulation for post-processing stress predictions in Finite Element Analysis. The
methodology, called Smoothing Element Analysis (SEA), employs a three-node
smoothing finite element. The present effort focused on verifying the basic constant strain
criterion for the three-node smoothing element subject to a set of internal penalty
constraints. The convergence characteristics of the element are assessed by first deriving
the constrained form of the assumed element stress and stress gradient fields, and then by
verifying the validity of the constant strain criterion once the element penalty constraints
are explicitly imposed. The analytical investigation is carried out with the use of the
symbolic manipulation code Mathematica.
Introduction
An improved variational formulation called Smoothing Element Analysis (SEA),
developed by Tessler et al., (1994), serves as a foundation for the enhancement of finite-
element obtained deformation and stress response. In the case of stress predictions, C "_ -
continuous stress field from a finite element solution is enforced into a C_-continuous
stress field with continuous stress gradients. These enhanced results are ideally suited for
error estimation since the stress gradients can be used to assess equilibrium satisfaction.
The approach is employed as a post-processing step in finite element analysis. The
variational statement combines the discrete-least squares, and penalty-constraint
functionals, thus enabling automated recovery of smooth stresses and stress gradients.
The practical issues whose adequate resolution is essential for a successful application
of the approach are:
(l) The SEA mesh and the number of the discrete stresses extracted from the Finite
Element Analysis (FEA) mesh should be properly interrelated in order to produce a
determined system of SEA equations. To fulfill this requirement, Tessler et al. (1994)
proposed specific guidelines. An automated generation of the SEA mesh may also be
necessary to make the post-processing transparent to the user.
(2) The FEA stresses need to be extracted at the discrete elemental locations that are
best suited for the recovery. Optimal (i.e., superconvergent) Barlow points and Gauss
integration points have been successfully used.
(3) The smoothing element should not exhibit locking -- a pathological stiffening
phenomenon commonly exhibited in penalty-constrained elements. In this connection, a
judicious choice of the element shape functions is key to avoiding locking.
The purpose of this effort is investigate the influence of penalty constraints on the
characteristics of the smoothing element used in Tessler et al. (1994). Particularly, the
convergence characteristics of Tessler's smoothing element are assessed by way of
deriving the constrained form of the assumed element stress and stress gradient fields, and
by verifying the validity of the constant strain criterion once the element penalty
constraints are explicitly imposed. This analytical investigation is facilitated by the use of
the symbolic manipulation code Mathematica.
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Error Functional
In this section an error functional proposed by Tessler et al. (1994) for a two-
dimensional plane formulation is reviewed. It is assumed that within a two-dimensional
region fl = {xe _2}, where x={x,-}, i=1,2, represents a position vector in Cartesian
coordinates, a finite element-derived stress field orb(x) has been obtained by means of a
discretization of fi with characteristic element size h. The smoothed stress field, a'(x),
is to be constructed from orb(x) via a variational formulation. The variational statement
involves scalar quantities only, and so each component of trh(x) is smoothed
independently. Hence, in the following reference is made only to components O_and _.
The finite element stress field is sampled at Xq, q = 1, 2 .... , N, to obtain the set of
stresses {_}, i.e., O_q_ trh(xq). The sampled stresses are those extracted at the Gauss
integration points, Barlow points, or other element locations in the finite element
analysis. To minimize the error functional, we adopt the finite element methodology and
therefore discretize fl with net "recovery" or "smoothing" finite elements such that
f2 =, ,"_ fl" where fY is the domain of smoothing element e Within our recovery element
_e=l
model, we use C°-continuous interpolation functions for the stress, _, and the
independent quantities 07, I=1, 2, whose mathematical interpretation will be readily
established. The error functional to be minimized can be written as
N i'le/
•: w,[o, + - o:;-+ - o,)Jan
e=l _,
(1)
where wq and p(x) are the appropriate weight functions; r is a normalization factor; ,;I. is
a dimensionless parameter; and a comma denotes partial differentiation. Because the
highest partial derivative in equation (1) is of order one, the field variables need only be
approximated with C°-continuous shape functions.
The first term in equation (1) represents a discrete least-squares functional in which
the squared 'error' between the smoothed stress field and the sample data is computed for
all sampled stresses. The term can be normalized in several different ways; presently, the
normalization factor equals the total number of the sampled stresses, i.e., y= N. The
discrete weights wq are introduced so that sample data known to be of higher accuracy
can be assigned more weight than less accurate data.
The second term in equation (1) represents a penalty functional which, for ,;t
sufficiently large, enforces the derivatives of the smoothed stress field 4, to approach the
corresponding _ variable pointwise, i.e.,
---) _ (i=x,y) in f2 e (2)
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Theoretically, the greater the value of 2, the closer the correlation between cry, and _,
where C I continuity of O"_is achieved as A. --->oo. In practice, $ needs to be sufficiently
large in order to enforce conditions (2), yet it should not be excessively large to cause ill-
conditioning of the smooth solution. Because _ are interpolated with continuous
functions, the smoothed stress field, for all practical purposes, is C ! continuous. The
weight function p(x) is introduced in the functional to allow the enforcement of C 1
continuity to be somewhat relaxed in certain regions of f_ and more strictly enforced in
others, if so desired. Also note that by specifying the weights Wq and p(x) to vanish in
regions outside a given domain of interest, a local orpatch analysis is admitted. Presently,
we only consider the special case where w = 1 and p(x) = 1, that is all stress data are
treated equally and the C t continuity is enforced throughout the f_ domain.
Assumed Element Fields
Although the functional (1) admits C°-continuous shape functions for the field
variables o", _, and _, the constraints (2) impose certain restrictions on the suitable
choice of shape functions. In a similar plate theory formulation, constraints of this type
are known to cause locking (i.e., severe stiffening) when conventional isoparametric
interpolations are used. (In the present context, locking would manifest itself in a
smoothed stress field, o", that grossly underestimates the 'true' stress distribution.)
When _ is interpolated with a polynomial one degree higher than those for the _ and
variables, using anisoparametric interpolations, the locMng effect is alleviated or
completely eliminated (Tessler, 1985).
Another important consideration is the nodal configuration that is best suited for the
smoothing element. It turns out that a three-node triangle is well-suited for this purpose
because (a) from a modeling standpoint, it represents the most versatile element topology,
and (b) it permits a one-to-one linear mapping between the global and element local (area-
parametric) coordinates, thus allowing a straightforward identification of the sampled
stress data within the smoothing element.
The anisoparametric interpolations for a three-node element involve quadratic
approximation of o" and linear approximations of Cx and _ which can be expressed in
matrix form as
• $(ys =zcr • +m0_ + 0._, 0 i = z0[ (i=1,2) (3)
where _, _ are 3xl vectors of nodal degrees-of-freedom (dof), z is selected as a row-
vector of a linear shape function, and m and 1 are selected as row-vectors of quadratic
shape functions. Their explicit forms given in terms of area-parametric coordinates are
where
z={zl,z__,z3}, m={m,,m2,m3}, !={lt,12,13} (3.1)
670
z, = 2-_(c, +bix +a,.y), m, = ½(akz,z i -aiz,z k ), l, = __(bjzizk -bkzizj)
ai=xk-x _, bi=yj-y k, ci=xjy_-xky _ (i=1,2,3, j=2,3,1, k=3,1,2)
and A denotes the area of a triangular element.
Note that these interpolations are consistent with a three-node element which has only
three dof per node, even though O_ is quadratic and O/ are linear functions. Moreover,
equations (3) ensure that the gradient of the smoothed stress, o_.,, is the same degree
polynomial as that representing _, i.e., they are both linearly distributed across _e. This
naturally leads to a reasonable expectation that penalty constraints (2) can be adequately
fulfilled without over constraining (locla'ng) the element.
Edge Penalty Constraints
A straightforward manipulation of the two constraint equations in (2), in which
equations (3) are introduced, produces three edge-wise constraints per element. For the
element edge defined by nodes i and j, the edge constraint equation has a simple form in
terms of the nodal dof corresponding to the edge (Pomeranz, 1995; Tessler, 1985):
! • • I • • (4)
where x, and Yk (k = i,j) are the nodal coordinates.
The three edge constraint equations ensure that there are only six independent dof per
element, thus properly describing the complete parabolic field of_. They also facilitate a
simple calculation of the total number of independent dof in the mesh. The key aspect of
these constraints is that they control the mechanisms of locMng. Their assessment in the
context of assembly of elements can provide proper insight into preferable discretization
patterns for such elements. For example, a fully non-locking behavior is achieved by
producing SEA meshes made of quadrilateral macro-elements that are formed with four
triangles in a cross-diagonal pattern.
Constant Strain Criterion
Let us consider an arbitrary triangular element as shown in the diagram below.
/_ x3,y3)
1 (xI,yl)__2 (x2,Y2)
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The satisfaction of the constant strain criterion in the finite element method ensures
convergence of the method as the mesh is refined. In this case, it is expected that each
individual finite element accommodates constant strains. Mathematically, the criterion is
verified by summing up on all shape functions for each field that is approximated, and the
resulting sum should add up to unity. This can be readily verified for the unconstrained
element fields in (3).
The constraint equations for the edges of this element can be written as follows.
Edge 1-2
Edge 2-3
Edge 3-1
0-1-or2 =_'((xt-x_)(O,, +O,,z)+(yt-y,,,XOy , +Oyz) )
a2-0-3 = ½((xz-x3)(Ox2 +Ox3)+(Y,.-Y3XOy2 +0,3))
0-3-a, = ½((x 3 -x, Xo_ 3 +ox,)+(y 3 -y,)(o,3 +o_,,))
(5)
Using Mathematica, the three constraint equations are solved for 1_1"1' 0"2, and 0x3.
When these solutions are substituted into the original definitions for o'*, 0_, and 0_, the
following expressions of the three element fields are derived
0-" = giG3 + g2(x)O,i + g3(x)O:_ + g4(x,y)Oyl + gs(x, y)Oy2 + g6(x,Y)Oy3
O_ = d I (x)O,_ + d 2 (x)Ox2 + d 3(x, y)Oy I + d, (x, y)Oy 2 + d 5(x, y)Ov3
$
Oy = e I (x, y)O: + e I (x, y)Oy z + e 3(x, y)Ov3
(6)
where g, d, and e are shape functions whose expressions are summarized in the Appendix.
To verify the constant strain criterion for the resulting element fields, the summation of
the g, d, and e shape functions is carried out with the use of Mathematica. The resulting
equations are as follows
6 5 3
g-Zgi=l+x-x3+Y-Y3, d=Zdi=l, e=-Zei=l (7)
i=1 i=1 i=1
t $Note that bo h 0_ and 0_ fulfill the constant strain criterion for a finite size element since d
=1 and e=l. On the other hand, g only approaches unity in the limit as the element size
diminishes to zero, i.e.,
x--') x3, Y"-) Y3 (8)
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giving rise to g--_ 1. Thus, the element convergence is ensured as the smoothing mesh is
refined.
Conclusions
The project has been a continuation of the research of Tessler et al. (1994) on an
improved variational formulation for post-processing stress predictions in Finite Element
Analysis. The effort focused on verifying the basic constant strain criterion for the three-
node smoothing element subject to a set of internal penalty constraints. The convergence
characteristics of the element were assessed by first deriving the constrained form of the
assumed element stress and stress gradient fields through a process of simplifications
using Mathematica. The element penalty constraints were explicitly imposed using the
formulas set out by Tessler et al. (1994). Then the validity of the constant strain criterion
for the element was verified. As the smoothing mesh is refined, the constant strain
criterion is satisfied.
References
Tessler, A., Riggs, H. R., Macy, S. C., 1994, "A variational Method for Finite Element
Stress Recovery and Error Estimation," Computer Methods in Applied Mechanics and
Engineering, Vol. 111 pp. 369-382.
Tessler, A., 1985, "A Priori Identification of Shear Locking and Stiffening in Triangular
Mindlin Elements, Computer Methods in Applied Mechanics and Engineering, Vol. 53,
pp. 183-200.
Pomeranz, S., 1995, Personal Communication.
Wolfram, S., 1988, Mathematica TM A System for Doing Mathematics by Computer,
Addison-Wesley Publishing Company, Inc., Redwood City, CA.
673
Appendix
The following are the shape functions in (6) as solved by Mathematica TM.
gl = 1
g2 = ((-x + x3)*(x - 2"x2 + x3))/(2*(-xl + x2))
g3= ((x - x3)*(-x + 2*xl - x3))/(2*(xl - x2))
g4= (xl*x2*y ^2 " x2^2*Y ^2 -xl*x3*y^2 + x2*x3*y^2-2*x*xl*y*y2 + 2*x*x2*y*y2 +
2*xl*x3*y*y2 - 2*x2*x3*y*y2 + xA2*yl*y2 - 2*x*x2*yl*y2 + 2*x2*x3*yl*y2
x3A2*yI*y2- x^2*Y 2^2 + 2*x*xI*y 2^2 -2*xl*x3*y2^2 + x3^2*y2a2 + 2*x*xl*y*y3 -
2*x*x2*y*y3 2*xl*x2*y*y3 + 2*x2^2*y*y3 - xA2*yl*y3 + 2*x*x2*yl*y3 -
2*x2*x3*yl*y3 + x3^2*yl*y3 + xa2*y2*y3 _ 2*x*xl*y2*y3 + 2*xl*x3*y2*y3 -
X3^2*y2*y3 + xl*x2*y 3^2 - X2^2*Y 3^2 - XI*x3*y3 ^2 + X2*x3*y3^2)/(2*(-xl +X2)*(-
(x2*yl) + x3*yl + xl*y2-x3*y2-xl*y3 + x2*y3))
gS=(-(xlA2*y ^2) + xl*x2*y ^2 + xl*x3*y^2 "X2*X3*y^2 + 2*x*xl*y*yl- 2*X*X2*y*yl
- 2*xl*x3*y*yl + 2*X2*X3*y*yl - xA2*ylA2 + 2*X*X2*yl^2 _ 2*X2*X3*yI^2 +
X3^2*yl^2 + X^2*yl*y2 - 2*x*xl*yl*y2 + 2*xl*x3*yl*y2 - X3^2*yl*y2 -
2*x*xl*y*y3 + 2*xl^2*y*y3 + 2*X*X2*y*y3 - 2*xl*x2*y*y3 + xA2*yl*y3 -
2*X*X2*yl*y3 + 2*X2*X3*yl*y3 - X3^2*yl*y3 " X^2*y2*y3 + 2*x*xl*y2*y3 -
2*xl*x3*y2*y3 + X3^2*y2*y3 - xlA2*y3^ 2 + XI*x2*y3^2 + xl,x3,y3^2 -
X2*x3*y3^2)_2*(-xl + X2)*(-(X2*yl)+ x3*yl + xl*y2-x3*y2-xl*y3 + x2*y3))
g6 = (-(xl*y) + x3*y + x*yl - x3*yl - x'y3 + xI*y3)/(2*(-xl + X2)) + (x2*y - x3*y -
x'y2 + x3*y2 + x'y3 - x2*y3)/(2*(-xl + X2)) + ((-X + X3)*(yl - y2)*(-(xl*y) + x2*y +
x*yl - x2*yl - x'y2 + XI*y2))/(2*(-xl + X2)*(x2*yl - x3*yl - xl*y2 + x3*y2 + xl *y3 -
x2*y3)) + ((xl *y - x2*y - x*yl + x2*yl + x'y2 - xl *y2)*(y - y3))/(2*(x2*yl - x3*yl -
xl*y2 + x3*y2 + xl *y3 - x2*y3))
dl= (X - X2)/(xl - X2)
d:= (X - xl)/(-xl + X2)
d3= ((-(X l'y) + x2*y + x*yl - x2*yl - x'y2 + x l*y2)*(y2 - y3))/((-xl + X2)*(-(X2*yl) +
X3*yl + xl*y2 - x3*y2 - xl *y3 + x2*y3))
d4= ((xl*y- x2*y- x*yl + x2*yl + x'y2 - xl*y2)*(-yl + y3))/((-xl + X2)*(x2*yl -
X3*yl - xl *y2 + x3*y2 + xl*y3 - x2*y3))
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d5= ((yl - y2)*(-(xl*y) + x2*y + x*yl - x2*yl - x'y2 + xl*y2))/((-xl + x2)*(-(x2*yl) +
x3*yl + xl*y2 - x3*y2 - xl*y3 + x2*y3))
el = (x2*y -x3*y -x'y2 + x3*y2 + x'y3 - x2*y3)/(x2*yl- x3*yl - xl*y2 + x3*y2 +
xl*y3-x2*y3)
e2 = (xl*y-x3*y-x*yl + x3*yl + x'y3 - xl*y3)/(-(x2*yl) + x3*yl + xl*y2 - x3*y2 -
xl*y3 + x2*y3)
e3= (-(xl*y)+ x2*y+ x*yl-x2*yl-x'y2 + xl*y2)ff-(x2*yl)+ x3*yl + xl*y2 -x3*y2
-xl*y3 + x2*y3)
Notation
In the above expressions, xl=xb yl=yl, xl^2=xl 2, and the asterisk (*) denotes
multiplication.
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Characteristics of Three-Node Smoothing Element under Penalty Constraints

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