Template Development for the 5-Node, Bilinear Pyramid
ASEN 6367: Advanced Finite Element Analysis
May 8, 2013
Gunnar V. Ashton
Deparment of
Mechanical Engineering
Mark Robinson
Deparment of
Aerospace Engineering
1 Introduction
1.1 Pyramids
Pyramid elements are primarily used as transitions between tetrahedra and hexahedra during mesh generation. 1
In particular, 5-node pyramids are effective in combining meshes including 4-node tetrahedra and 8-node
hexahedra. The geometry of the 5-node pyramid is simplistic. As indicated by its name, the pyramid has 5
vertices. In addition, it features 8 edges and 5 faces. The quadrilateral face is referred to as the base and may be
warped, while the remaining 3 faces are called apex faces and are strictly planar.
The isoparametric formulation of the 5-node pyramid is well known, having been in existence since the early
1970s. The pyramid shape functions are derived directly from the shape functions of the 8-node hexahedron.
The base node shape functions of the pyramid take on the exact value of the shape functions from nodes 1-4 in
the hexahedron. The apex node shape function of the pyramid is an amalgamation of the shape functions from
nodes 5-8 in the hexahedron. The shape functions satisfy compatibility and completeness requirements, and thus
can be used to generate element stiffness and mass matrices through the standard isoparametric formulation.
The element constructed in this manner passes multiple patch tests. That being said, this formulation does not
guarantee an optimal element with respect to a specific characteristic, for example bending. An alternative
element formulation technique, known as templates, allows a user to customize an element through parameters,
so that it optimally responds within its envisioned use.
1.2 Templates
Using templates forms the element stiffness matrix in a completely different fashion as compared to the
isoparametric formulation. Within templates, the master stiffness matrix, K, is the combination of the basic
stiffness matrix, Kb, and the higher order stiffness matrix, Kh:
ࡷ ൌ ࡷ௕ ൅ ࡷ௛
The basic stiffness matrix functions to produce consistency and mixability.2 It is a result of the Free
Formulation (FF) school of thought. In contrast, the higher order stiffness matrix stabilizes with correct rank
and can be adjusted through parameters to provide desired accuracy. It descends from the Assumed Natural
Strain Formulation (ANS). The higher order stiffness matrix is orthogonal to rigid body motions and constant
strain states.
The purpose of this project is to extend 5-node pyramids beyond the isoparametric formulation into templates,
so that they may be optimized for bending. In doing so, analyzing rank is an effective method to verify results.
The rank of the master stiffness matrix is 9, assuming 15 degrees of freedom, which will be discussed below,
and 6 rigid body modes. The rank of the basic stiffness matrix is 6, as it is purely a function of the rigid body
modes. Accordingly, the correct rank of the higher order stiffness matrix must be 3. As will be described in
greater detail below, this necessitates the use of 3 parameters applied within the higher order stiffness matrix.
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2 Objectives
There were four interrelated objectives for this project. We wished to construct a template for the 5-node
pyramid transition element, conduct higher order patch tests on a grouping of these created elements, verify the
element formulation, and determine the optimal parameters for the element in bending. We were successful in
creating the template for the parameterized, 5-node pyramid although it has some limitations, discussed further
in Section 6.2.
3 Assumptions
The main assumptions for developing the 5-node pyramid template relate to its geometry and coordinate
system. The base of the pyramid must be orthogonal to and symmetric about the X and Y axes. The height of
the pyramid runs along the Z-axis, with the apex of the pyramid lying on the axis. The element geometry and
coordinate system can be seen in the figure below:
Z
Y
X
h
b
a
Figure 1: 5-Node Pyramid Element Geometry and Local Coordinate System
Poisson’s ratio was neglected (set to 0) to simplify all algebraic computation.
Each node was given 3 degrees of translational freedom in the principal coordinate directions. This means the
element had a toal of 15 DOFs.
4 Methodology
4.1 Basic Stiffness Matrix
In the FF, the basic stiffness matrix, Kb, is constructed with the element volume, V, the element material matrix,
E, and the force lumping matrix, L:
ࡷ௕ ൌ
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The basic stiffness matrix for the 5-node pyramid has dimensions 15 X 15, the element material matrix has
dimensions 6 X 6, and the force lumping matrix has dimensions 15 X 6.
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The element volume is found in a straightforward manner due to the assumed geometry. The element material
matrix is also simplified due to removal of Poisson’s ratio. It is now only a function of Young’s modulus, Em:
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The force lumping matrix requires a unique approach in its derivation. The first step involves determining the
planar equations for the pyramid faces in the assumed geometry. These being known, the direction cosines, l, m,
and n, are found for each face. The basic stiffness matrix is a constant stress hybrid within the confines of the
FF. As a result, the next step in deriving the force lumping matrix is to assume 6 constant stress fields, σXX, σYY,
σZZ, σXY, σYZ, and σXZ. Using the assumed stress fields, Cauchy’s formula is applied 6 times to form surface
tractions, tX, tY, and tZ:
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These surface tractions are resolved on a stress field-by-stress field basis into forces using face surface areas.
The forces are accordingly lumped at nodes. The force lumping matrix is then assembled like so:
With the element volume, material matrix, and force lumping matrix known, the basic stiffness matrix is then
easily found through matrix calculations. It is important to note, however, that in our formulation of the basic
stiffness matrix is completely dependent upon the element geometry in the assumed coordinate system rather
than nodal coordinates directly.
4.2 High-order Stiffness matrix
4.2.1 Assumed Displacements
The route to find the higher order stiffness matrix, Kh, is significantly more complicated than determining the
basic stiffness matrix, Kb. There is no equation that can be referenced and no universal approach. The
formulation of the higher order stiffness matrix is not guaranteed and may require multiple iterations. Our first
attempt at finding the higher order stiffness matrix for the 5-node pyramid involves assumed displacements.
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The method first constructs a prism from 6, 5-node pyramids centered at the coordinate origin. This solid is
shown in the figure below:
Figure 2: Assumed Displacement Prism Constructed From 6, 5-Node Pyramids
Because one of the goals of this project is to improve the 5-node pyramid with respect to bending, 6 constant
curvature bending modes are applied to the prism, κXXY, κXXZ, κYYX, κYYZ, κZZX, and κZZY. Overall displacements in
the 3 principal directions, uX, uY, and uZ, are then derived from these bending modes using solid mechanics
relationships:
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Simultaneously, the basic-mode matrix, Gr, and constant-strain matrix, Gc, are developed. The basic-mode
matrix spans the element’s 6 rigid body modes. The constant-strain matrix spans the element’s 6 constant strain
states. In both matrices, the entries result from pyramid nodal displacements. As such, both matrices have
dimensions 15 X 6. They are represented below:
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It is paramount in template formulation that the higher order stiffness matrix be orthogonal to rigid body modes
and constant strain states. As the higher order stiffness matrix is presently unknown, it necessary to use another
means to ensure the required orthogonality. This matrix tool, Hh, is called the geometric projector of the
assumed displacement field. It is derived from the isolated constant curvature contributions from the assumed
displacements at each element node. It has dimension 6 X 15 and is shown below:
Since the basic-mode and constant-strain matrices represent the rigid body modes and constant strain states and
the geometric projector matrix acts on behalf of the higher order stiffness matrix, ideally, the following
relationships should hold true:
ࡴ௛ ࡳ௥ ൌ ૙
ࡴ௛ ࡳ௖ ൌ ૙
For the assumed displacement method, the team was unable to obtain orthogonality between the higher order
stiffness matrix and the rigid body modes and constant strain states. The above relationships did not hold true
for the developed Hh, Gr, and Gc. Because of this, we were forced to pursue an alternate method to formulate
the higher order stiffness matrix for the 5-node pyramid.
4.2.2 Energy Orthogonality, Stress Hybrid Method and Filtering
After the failed attempt at strong orthogonality of Hh and Grc, another method was required to prove that the
geometric projection of the displacement field under constant curvature, Hh, was orthogonal to the rigid body
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modes and constant strain states. This method was energy orthogonality and is considered a weak for of the
equations:
ࡴ௛ ࡳ௥ ൌ ૙
ࡴ௛ ࡳ௖ ൌ ૙
The proving factor of this method was that the average h-strain must be equal to 0.
With this condition was met, the formation of Kh could be tackled. Instead of creating a Kh for the element all at
once, the stress hybrid method was used. This method consisted of finding a higher order stiffness matrix for
each face as if they were each independent elements. This left us with 5 Kh matrices that must be reduced to 3 to
satisfy the ability to use 3 parameters in optimization of the element. The first step in doing so was
parameterizing the individual Kh matrices properly. Because the symmetric faces were selected to be combined,
the faces were parameterized and grouped as follows:
ࡷ௛ଵ ൌ ߚଵ ሺࡷ௛ିଵଶହ ൅ ࡷ௛ିଷସହ ሻ
ࡷ௛ଶ ൌ ߚଶ ሺࡷ௛ିଶଷହ ൅ ࡷ௛ିସଵହ ሻ
ࡷ௛ଷ ൌ ߚଷ ࡷ௛ିଵଶଷସ
Disappointingly, when these are combined to create a single Kh, the rank is an excessive 12, so a method to
reduce the rank of the matrix had to be used. This method was filtering and was derived from the rigid body
modes and constant strain states matrix, Grc.3 The filter or projector was calculated by the following equation:
ࡼ௥௖ ൌ ࡵ െ ࡳ௥௖ ሺࡳ௥௖ ் ࡳ௥௖ ሻିଵ ࡳ௥௖
By definition, this makes any matrix pre and post-multiplied by it orthogonal to Grc, something we want as Kh
should be independent of the rigid body modes and constant strain states. So, we now filter each Kh and
combine them to create the higher order stiffness matrix, which is represented below:
ࡷ௛௙ ൌ ࡼ௥௖ ࡷ௛ ࡼ௥௖
The projector/filtering matrix is as follows:
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4.3 Master Stiffness Matrix
To create the master stiffness matrix, K, only simple addition is necessary:
ࡷ ൌ ࡷ௕ ൅ ࡷ௛
Kb was found in Section 4.1 and will be true for all 5-node pyramid elements of this geometry regardless of
optimization. Kh is much more complicated as explained above, but once it has been filtered, parametrized and
the three parts combined it is simply another 15 X 15 matrix.
Again, it should be noted that the master stiffness matrix is dependent on the pyramid geometry described
previously and not on nodal coordinates. Therefore, it cannot be simplistically manipulated to perform patch
tests.
5 Results
To verify our results, we evaluated our matrices numerically, checking their ranks and properties. Some of these
are shown below, however, the full numerical checks are too cumbersome to present in full here. The remainder
can be found in the Mathematica notebook Pyra5TemplateMatrices_Complete.nb.
Figure 3: Rank of master, basic, and higher order stiffness matrices
As shown, all matrices have the sufficient rank. The basic stiffness matrix has a rank of 6, representing the rigid
body modes. The master stiffness matrix has a rank of 9, which is the difference between the number of degrees
of freedom (15) and the rigid body modes. Finally, the higher order stiffness matrix has a rank of 3, which is
both a correction term and the number of parameters required to optimize. Additionally, it should be noted that
when evaluated numerically, the diagonal in all cases is greater than or equal to 0, and the matrices are
symmetric.
6 Conclusions
6.1 Future Work
There are future tasks for this project that can be continued based on our progress. First of all, the Mathematica
modules could be changed to incorporate arbitrary geometry that depend on nodal coodinates rather than
specified geometry parameters that limit the shape and orientation of the element. Our element was coded only
in the element’s local coordinate system. This should be changed by revising the modules to be coded in the
global coordinate system of the FEM or by adding supplementary coordinate transformation modules.
Secondly, the higher order patch tests should be performed. We were restricted in performing this due to the
inability to re-orient our element and assemble the 6 elements to create a testable patch.
Page 7 of 10
Furthermore, through the higher order patch tests and energy balance, the element formulation must be verified.
Lastly, using the bending patch test, the element’s three higher order stiffness parameters must be determined.
While we got to the point of finding the higher order stiffness matrix, we did not find the optimal values of
these parameters, again, as we were not able to perform the patch tests necessary.
6.2 Limitations and Lessons Learned
As mentioned above, the greatest limitation the team experienced in this study was the assumed pyramid
geometry and coordinate system. All coding completed in Mathematica was dependent upon the chosen
geometry and coordinate system rather than global nodal coordinates. Making the basic and higher stiffness
matrix accept global nodal coordinates instead of geometry parameters would facilitate easy re-orientation of
the element. This would lend itself to patch tests, element verification, and eventually, use of the element in
code.
An additional limitation the team encountered was our relative inexperience with templates. Before starting on
the project, we had only developed elements through the isoparametric formulation. Templates were a
completely new topic. To make any progress on the project, near constant consultation was required with
Professor Felippa. The team also struggled with the lack of definition in templates. There is no clear procedure
in developing a template, so some trial and error is necessary. This can be taxing because there is no guarantee
an approach will work within the constrained time period of the project. This was exemplified in forming the
higher order stiffness matrix. We attempted to use assumed displacements but was not sure when to abandon
this method if orthogonality could not be achieved. Eventually, the team used energy orthogonality and the
stress hybrid method with the assistance of Professor Felippa. However, at this time, the schedule was pushed
far enough to make accomplishing patch tests and optimization impossible.
Despite the obstacles the team confronted, this project was a success as it provided insights into 5-node pyramid
template formulation. Foremost amongst these are the techniques used to develop the basic and higher order
stiffness matrices. With our progress in creating a parameterized template for the 5-node pyramid, hopefully, a
future team can expand our work by improving our coding, conducting patch tests, verifying element
formulation, and optimizing the element for bending.
Page 8 of 10
Software List
1. Pyra5TemplateMatrices_Complete.nb
Page 9 of 10
References
1
C. A. Felippa. Pyramid Solid Elements [Online]. Available:
http://www.colorado.edu/engineering/CAS/courses.d/AFEM.d/AFEM.Ch12.d/AFEM.Ch12.pdf
2
C. A. Felippa. A Template Tutorial [Online]. Available:
http://www.colorado.edu/engineering/CAS/courses.d/AFEM.d/AFEM.Ch28.d/AFEM.Ch34.pdf
3
C. A. Felippa. Linear Algebra: Matrices [Online]. Available:
http://www.colorado.edu/engineering/cas/courses.d/IFEM.d/IFEM.AppB.d/IFEM.AppB.pdf
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