”THS

  
     

  

LIBRARY ' .
Michigan State .
University

ifmiiia? 15%

This is to certify that the

thesis entitled

A Bond-Graph Approach to Finite Element
Methods

presented by

Craig Wendel Galer

has been accepted towards fulfillment
of the requirements for

M.S. degree in Mech. Eng.

Major professor I?"

 

 

 

Date ll/l0/78

 

0-7 639

 

A BOND-GRAPH APPROACH TO FINITE

ELEMENT METHODS

By

Craig Wendel Galer

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

MASTER OF SCIENCE

Department of Mechanical Engineering

1978

ABSTRACT

A BOND-GRAPH APPROACH TO FINITE ELEMENT METHODS

by

Craig Wendel Galer

Bond-graph techniques are applied in finite-element
analysis. A procedure is developed for constructing a bond-
graph model for constant—strain triangular elements from their
stiffness matrices. Causality is proposed as a useful
tool in deriving the response of finite-element assemblies.

A comparison of bond-graph and standard finite-element
methods is also made, and some simple extensions of the bond-

graph method are proposed.

ACKNOWLEDGEMENTS

Many thanks to Professor Ronald C. Rosenberg for all
his assistance and guidance both on this thesis and through-
out my M.S. program.

Thanks also to all the friendly people in Mechanical
Engineering, who helped make my graduate studies a most
enjoyable experience.

Humble thanks to my parents, who "Trained me up in the
way I should go” (Proverbs 22:6).

And deepest gratitude to my brothers and sisters in the
Work of Christ Community, for their ongoing love and support.
"How good and pleasant it is when brothers live together in

unity." (Psalm 133:1).

ii

TABLE OF CONTENTS

Page
LIST OF FIGURES. . . . . . . . . . . . . . . . . . . . . v
Chapter

I. INTRODUCTION. . . . . . . . . . . . . . . . . . . 1

1-1 Introduction to the Thesis . . . . . . . . . 1

1-2 Introduction to Terminology. . . . . . . . . 2

II. THE BOND-GRAPH FINITE-ELEMENT METHOD. . . . . . . 5
2—1 The Desired Model - Description And

Derivation . . . . . . 5

2-2 Examples 8

2-2.1 A One-Dimensional Example . . . 8

2-2.2 A Two-Dimensional Example . . . . . .12

2-3 Applying Causality to the Model. . . . . . .13

2-“ Some Simple Extensions . . . . . . . . . . .20

2-A.l Higher-Order Elements . . . . . . . .20

2-4.2 Three-Dimensional Elements. . . . . .21

III. A COMPARISON OF STANDARD AND BOND-GRAPH METHODS
IN FINITE-ELEMENT ANALYSIS . . . . . . . . . . .29

3-1 Superposition. . . . . . . . . . . . . . .29
3-2 Boundary Conditions. . . . . . . . . . . . .32
3-3 Solution . . . . . . . . . . . . . . . . . .33

IV. CONCLUSIONS AND RECOMMENDATIONS FOR FURTHER
INVESTIGATION . . . . . . . . . . . . . . . . . .39

A—l Conclusions. . . .39
A-2 Recommendations For Further. Investigation. .40

iii

APPENDICES

A. Internal Structure Of A Constant-Strain
Triangular (6-Port) Element.

B. Supporting Mathematics

LIST OF REFERENCES

iv

Page

A2
A7

A9

Figure

1-1.

LIST OF FIGURES

Constant-strain triangular elastic element
with nodal displacements shown.

Page

A

6-port C-field representation of elastic element. A

Desired form of elastic element model

Model with six external ports, internal 3-port
C- field . .

Model with decoupled C-elements

Axisymmetric rod with exponentially-varying
cross-section . . . .

Bond-graph element model.

Bond-graph assembly model

Constant-strain triangular element for example.

Bond-graph model.

(a) Boundary conditions applied to triangular
element . . .

(b) 6- -port causal obond- -graph model .

Causality applied to the bond-graph model

The junction-structure transformation matrix.

Linear-strain triangular element.

Form of linear-strain triangle bond—graph model

Constant-strain tetrahedral element . . . .

Form of tetrahedron bond-graph model.

Mesh of four triangular elements. . .

Bond—graph model of element mesh.

V

.22

.22

.22

.23
.23
.23
.2A
.2A
.25
.25
.25
.26
.27
.28
.28
.28
.36
.36

Figure Page
3-3. Distributed load applied to element mesh. . . . .37
3-A. Discretized nodal forces. . . . . . . . . . . . .37

3-5. Port causality applied to the global bond—graph

model . . . . . . .38
A-l. 3-port pseudo-bond-graph model. . . . . . . . . .A6
A—2. (2 + l)-port pseudo-bond—graph model. . . . . . .A6

vi

CHAPTER ONE: INTRODUCTION

l-l Introduction to the Thesis

 

The Finite-Element Method has come into extensive use
in the last fifteen years on a wide range of applications
from fluid flow and heat transfer to structural analysis.
The pOpularity and utility of this method are due to its fa-
cility for converting a complicated problem with various associ-
ated nonlinearities and mathematically difficult boundary con-
ditions into a straightforward (albeit large) problem in linear
algebra. This approach also adapts the method for easy imple-
mentation on the digital computer, which has also enhanced its
popularity in industry.

Bond-Graph techniques have emerged in the last ten years
as a viable means of modeling dynamic systems. They present
a straightforward procedure for deriving system state equations
by means of simple modeling rules, which in turn allows one to
analyze the dynamic response of the system. They also contain
a certain extractable "intuition" for energy flow in a system.
Causal rules inherent in bond-graph techniques allow a reason-
ably straightforward generation of an input/output form for

the system model, which is very useful in many applications.

This study employs multiport bond-graph techniques to
gain insight into and augment existing finite-element methods.
By using a bond—graph modeling approach, an alternative means
of analyzing the behavior of a given finite-element mesh will
be derived.

For the purpose of this study, we will restrict ourselves
to problems in elasticity, with the understanding that the
method generated here can be later generalized to other areas
of application. We will also restrict our analysis to linear
constant-strain, two-dimensional algebra in our bond-graph
models, which will allow clearer initial understanding of the
new method without sacrificing depth of insight, as well as
saving time in unnecessarily involved computations. Again,
this is with the understanding that this restriction can be

generalized later to include a wider range of problems.

1-2 Introduction to Terminology

 

In this section, we will introduce some of the notation
and terminology which will appear regularly in this thesis.
We are concerned in this thesis with deveIOping a bond—graph
model of an element used in a finite-element analysis of two—
dimensional elasticity.

The typical element used in finite—element analysis is
a constant-strain triangular element (Figure 1-1). These ele-
ments will fit together in a mesh to approximate the shape
and elastic properties of the object being analyzed (e.g. an
airplane wing, automobile body, etc.). Typically, in elasticity

problems, the primary variables are force and displacement

(deformation) at the element nodes (vertices). These forces
and displacements form the nodal force vector, Fn’ and the no-
dal displacement vector, Xn’ which are related by the global

assembly stiffness matrix, as in

In bond-graph terminology, such a stiffness relationship
is characteristic of a C-field, which models elastic (or, in
a sense, mechanical capacitance) properties of an object (see
(1) ). We may also speak of the number of ‘ports' associated
with a given bond-graph element, and this simply refers to the
means of energy flow between an element and adjacent elements.
A 2-port element, for example, is able to transfer energy along
two "paths" (in a bond-graph sense) with adjacent elements.
In the case of the constant—strain triangle, it "communicates"
with adjacent elements through nodal forces and displacements.
In a two-dimensional problem, two linearly independent coor-
dinates are necessary to describe the loading and motion of
the node. Each node, then, has two directions (x and y) in
which to transfer energy to adjoining elements, and thus, in
regard to the elastic properties of the triangular element,
we model it as a 6-port C-field (Figure 1-2). Some prior
knowledge of bond-graph notation and methodology is assumed.

(If such is not the case, please see (1) as a reference).

 

Figure 1-1. Constant-strain triangular elastic element with
nodal displacements shown

Figure 1-2. 6-port C-field representation of elastic element

CHAPTER TWO: THE BOND—GRAPH FINITE—ELEMENT METHOD

2-1 The Desired Model - Description and Derivation

 

Our primary motivation here is to develop a means of
modeling a finite—element mesh in terms of bond-graph struc-
tures so as to take advantage of the strengths of bond-graph
techniques. For the elastic elements considered here, we
model force as the effort variable and velocity as the flow
variable.

In the case of a linear elastic element, then, we would
desire a model of the type shown in Figure 2—1, where a junc-
tion structure composed of O- and 1-junctions and TF-elements
connects the external ports (inputs, boundary values) to the
internal elastic storage field (see (1) ). The number of ex-
ternal and internal bonds are determined by the geometry of
the problem (i.e., number of element nodes, dimensionality,
etc.) and the junction structure is determined by relation-
ships between the external and internal ports. This type of
model, along with the proper assignment of causality, allows
a direct transformation from "input" to "output" for each
element, and is the most systematic type of model with which
to work.

In Appendix A, we see that the internal structure of a
constant-strain triangular element is that of an external

6-port with three internal parts; thus, we will construct

5

6

our model along the same pattern of six external and three
internal ports (Figure 2-2). Even better would be a model
such as the one shown in Figure 2—3, where the 3—port
elastic storage field is in the form of three decoupled
l-port C-elements. If such a model is possible, it would
allow for a more direct and explicit representation of the
element model, as well as being intuitively and analytically
easier to work with.

The mathematics of this decoupled model can be derived
from the constitutive and connective relationships of the bond-
graph structure. Each (linear) C-element has a constitutive

equation of the form

relating the force and displacement in the storage element;
the constant of proportionality, Kc’ is the stiffness of the
C-element. In 3-port form, the 3-port stiffness matrix would
be a diagonal matrix whose elements are the individual stiff-

nesses of each distinct C-, and the constitutive relationship

would be
F1 K1 0 0 x1
P2 = 0 K2 0 x2
F O O K x
L 3d b ii b 3-

 

 

 

 

 

 

The junction structure shown in Figure 2-2 would be repre—
sented by a 6x3 matrix whose elements would be the moduli of

the TF-elements joining the appropriate external and internal

bonds according to the position in the transformation matrix
(e.g., the element in the fifth row and second column would
indicate the modulus of a transformer connecting the fifth ex-
ternal port to the second internal port). An element whose
value is zero would indicate no connection. To model the
external 6-port constitutive relationship of the triangular
element, the junction structure (J) and stiffness (K) matri-

ces are assembled in the form

Ke = JKJT , (2-1)

since, when power is directed through the junction structure,
the effort (force) transformation is the transpose of the flow
(velocity) transformation. Thus, the 6-port element stiffness
matrix is expressible as a combination of linear transforma-
tions - from external forces through the junction structure
to the internal forces; then, by means of the internal con-
stitutive relationships, the internal displacements are trans-
formed into internal displacements, which are mapped through
the junction structure into external displacements.

So, we see that we seek a synthesis of the form of (2-1),
where K is a diagonal matrix. Such a synthesis can be shown
to exist for every Ke (see Appendix B). Any real symmetric
matrix A (a condition satisfied by all elastic element stiffness
matrices) can be decomposed into a diagonal matrix M, whose
elements are the eigenvalues of A and a transformation T, whose
columns are the corresponding eigenvectors of A, normalized
to unit magnitude, so that

A = TMTT . (2-2)

We need only consider the non-zero eigenvalues, since any
zero eigenvalues will have no effect on the matrix product.
In the case of a stiffness matrix Ke for a constant-strain
triangular element, M will be 3x3, since Ke will always have
at least three zero eigenvalues, due to its construction,
(Appendix B) and T, then, will be 6x3.

By correlating equations 2-1 and 2-2 above, we see then
that we may construct a bond graph representation of Ke by
means of three l-port C-elements whose stiffnesses are the
non-zero eigenvalues of Ke (corresponding to the diagonalized
matrix K in equation 2-1) and a junction structure of trans-
formers whose moduli are determined by the corresponding nor-
malized eigenvectors of Ke (corresponding to the matrix J in

equation 2-1). This is the core of the modeling method.

2-2 Examples

 

Some examples will be helpful in demonstrating the method

described above.

2—2.l A One-Dimensional Example

 

For the sake of computational simplicity, let us begin
with a one-dimensional linear 2-port example. This will pro—
vide an easy first case, and will allow us easily to demonstrate
the junction-structure transformation matrices, as well as
to demonstrate how the l-junction connectors naturally use
the principle of superposition for nodal forces. (This ex-

ample problem is build upon an example given in (2), p. 59).

We consider the case of an axisymmetric rod whose area
varies exponentially with length (Figure 2-A). We subdivide

the rod into two elements, each of length L, and define

A(x) = A e_PX and 2LP = 1

so that

A = .6O6AO, A3 = .368AO.

Choosing the cross-sectional area for each element as the

average of the two nodal areas for that element, we obtain

A3 = .803Al, Ab = .A87Al

We now find the element stiffness matrices (by standard

methods) to be

.803 -.803 AOE .A87 -.A87 ADE
Ka = ——— and KD = ———
-.803 .803 L -.A87 .A87 L

At this point the bond-graph analysis begins.

We first solve for the eigenvalues and eigenvectors of
Ka and Kb' Ka has eigenvalues 0 and 1.606. Kb has eigenvalues
(not surprisingly) O and 0.97A. In each case, the normalized

eigenvector corresponding to the non-zero eigenvalue is

.707
-.707

So, in the case of element a (and similarly for element b)

we have

10
a .707 [;.60é] [:T07 -.70{] = .803 -.803

-.707 —.803 .803

174
II

from which we construct our bond—graph model. By form we can
see it will be an external 2-port with one internal port.

The stiffness of the internal l-port C-element will be the
eigenvalue (ka = 1.606), and this internal l-port capacitance
is connected to the two external ports by transformers whose
moduli are the elements of the eigenvector, giving us the
bond-graph model shown in Figure 2-5.

Now, let us analyze the bond-graph model which we have
constructed to show how the model engenders the mathematical
construction described above. We have three basic sets of
equations; relating the external displacements to internal

displacements, we have

Xa = .707xl - .707x2 ;

relating internal forces to external forces,

~11
ll

.707Fn

F = —.707Fa ;

and the internal stiffness, relating force to displacement,

F = 1.606x ;
a a

which, in a succession of transformations, lead to

1

l
I! "l '
Fl i ”- .707 I E..606 ] [.707 -.707-] Fxl

F2 -.707
uni r— c—d L- cl

 

 

 

 

 

and we see that the product of these transformations is
exactly the element stiffness matrix.

Now, having the element models, we wish to assembly them
into a global model. At the node, the elements have the same
displacement, and forces are additive, which is precisely the
definition of a l-junction in bond-graph modeling. If we con-
nect the separate element models with l -junction (thus arriv-
ing at the assembly model shown in Figure 2-6), we obtain an
assembly stiffness relationship from the l-junction constitu-

tive law at node 2

F2 = F2a + F2b
from which
FF; F .803 - 803 o -« ”-qu
F2 = -.803 1.290 -.A87 X2
i“F34 _ 0 -.A87 .A87”d _.X%Jl

 

 

 

 

 

 

which is exactly the assembly stiffness matrix generated in
finite-element methods by node-wise superposition of element
stiffness matrices. We see here, then, that superposition

is inherent to the constitutive laws of the 0- and l-junctions

in bond-graph methods.

2-2.2

A Two—Dimensional Example

12

 

element, as shown in Figure 2-7 (this example was adapted

from (3), pp. 109, 115).

(E

71
II

which would be the stiffness representation for a 6-port

model

2.9'

1593

of th

107 psi,

-'9.Ao
A.875
—12.5
.u —5.25
3.10

A. 0.375

 

e element.

U =

A.875
11.19
-A.50
-A.375
-9.375
-6.81

-12.5
-A.50
25.0
0
-l2.5
A.50

-5.25
-A.375
O
8.75
5-25
-A.375

0.3) the stiffness matrix is

3.10
-0-375
—l2.5
5.25
9.A0
-A.875

Here we will consider the case of a single triangular

By standard finite—element methods,

0.375.1
—6.81
u.50
-u.375
-u.875
11.19J

 

In order to obtain our more detailed

model, we first employ a digital computer to obtain the eigen-

values of Ke, which are 0, 0, 0, 2A300, 30960 and 6A100,

the normalized eigenvectors corresponding to the non-zero

eigenvalues are

 

b

F1,135

1

.667
.270
O
--l35
-.667 ,

 

-.A03
-.336
0
.671
.A03
-.336

 

 

-3851
—.23u
.771
o
-.385

 

 

L .23A _

so we can construct the (3x1)-port-plus-junction—structure

model shown in Figure 2-8 (an implied—TF convention has been

employed, with the moduli shown in parentheses).
our work in terms of the element model is complete.

further analysis would take place at the assembly level,

With this,

Any

13
connecting a mesh of elements using l-junctions as shown in
section 2-2.l, or applying causality to the model, which is
discussed in section 2-3.

It should also be noted that in formulating the model
here, we have sacrificed insight into the internal structure of
the element (at least in any real, physical sense) for simpli-
city and directness at a mathematical level. The relationships
illustrated in Figure 2-8 don't necessarily have any connection
with the internal interactions in any "real" material (see
Appendix A). The model shown here was formulated because it is
direct, adapts easily to bond-graph methods, and generates
the same 6-port stiffness relationship as the standard finite-
element technique. As we shall see, however, this may be per-

fectly adequate.

2-3 Applying Causality to the Model

 

Causal rules implicit to bond-graph theory can be applied
to the method described here to great advantage. Causality
greatly facilitates derivation of system state equations and
multiport field relationships (see (1) pp. lAl-l62, 233-258) and
will be directly applicable here.

Let us consider the two-dimensional example of section
2-2.2, only let us fix nodes 1 and 3, and apply a 10,000 pound
load in the positive x-direction at node 2 (Figure 2-9a) which,
in 6—port causal bond-graph form, is as shown in Figure 2-9b.
From this causal form, we are able to identify "inputs" to the

elastic field as velocities al, v Q3, and v3 (all zero)

1,

and forces X2 (=10K) and Y (=0), and the "outputs" (to

2

1A

v . This is a mixed-

1’ 1’ 2’ 2

be solved for) as X Y X3, Y3, a
causal form, as opposed to a strict stiffness form (where all
inputs are displacements). We would then desire some direct

method for deriving the 6-port field relationship in this

mixed-causal "input/output" (I/O) form, i.e.,

 

 

P -v P a r
X1 ”11
Y1 V1
u2 X2

= K'
v2 Y2
X3 u3
Y v
- 3 - h J «L 3‘»

 

 

 

 

Causality allows us to do this directly. If we assign this
causality to the expanded model shown in Figure 2-8, as in
Figure 2-10, we are able to derive the input/output form from
constitutive and connective relationships in the bond-graph,
aided by the applied causality.

Immediately we notice that we have a choice in extending
the causality inward from the external ports. The derivative
causality at y2 determines derivative causality at 02, but

the derivative causality at x2 may be extended to either C1

or C3, according to the analyst's choice (in general, a force
input would allow a choice between three bonds along which
to extend the resulting derivative causality. Also, we see
that our model will not allow for more than three force inputs

on any given element, as this would result in being unable

15
to extend the causality inward; physically, it is an under—
constrained problem).

If we extend derivative causality to Cl’ 03 will have
integral causality. At this point, we may employ the ENPORT-S
program (A) to obtain a junction-structure matrix, relating
inputs and outputs of the junction structure to one another.
This matrix is shown in Figure 2-11.

We may then rearrange the information from this matrix

into equations of the form

Fop = AiiFip + A12Fic
Fee = A21Fip + A22Fic
Xop = BllXip + B12Xic
Xoc = B21Xip + B22Xic

(F and X denote force or displacement vectors; the subscripts
'o' and 'i' denote inputs or outputs to the junction structure;
'0' and 'p' distinguish between the external ports or the
internal C-field; i.e. FOp is the vector of forces which are
outputs at the external ports). The A and B transformation
matrices are all taken directly from the JS matrix in

Figure 2-11. To these we may add the C-field constitutive

relationships

ic i oc

ic d oc

16

We may further rearrange these equations into the form

Fop = AllFip + A12Kixoc
Xop = BllXip + 812Kd-1Foc
Foc ‘ A22K1Xoc = A21Fip
Xoc ' B22Kd-1Foc = B21Xip

Since, in general, none of the A's or B's are square, we now

arrange the last two equations directly above into

 

 

 

F- "l' F '1' 1- '1 r- -
I -A22Ki Foc A21 0 Fip
-1 X 0 B X.
-B22Kd I ocJ 21— 1. 1pJ
L .al ' "

 

 

 

 

 

The large matrix on the left will be square, and, in general,
invertible, so we are able to solve for FOC amui XOC in terms
of Fip and Xip’ which can in turn be substituted into the
first two equations above to complete the solution, ending

in the form

 

 

 

 

"' "‘ 1- - 1- ‘1'
F X
op ip
= K'
Xop Fip
b - b - h ‘

 

 

(Future development in ENPORT—S and subsequent versions

should condense this procedure).

17

In our case,

(x1) (yl) (x2) (y2) (x3) (y3)
_10007 —3.05 -.A99 -.601 .0007 3.05 -1
- 3.0M 13030 -.180 -.501 -3.0u -13030
' .A99 .180 (2.u9.10'5) 0 .A99 -.180
K = .601 .501 o (7.17 10'5) -.601 .501
.0007 -3.05 —.u99 .601 .0007 3.05
A. 3.011 -13030 .180 -.501 3.011 13030.;

 

 

(The ports are designated in parentheses above the appropri-
ate column). We may offer a preliminary check on this result
by applying the input vector described earlier, with a 10,000
pound force in the x-direction, zero force in the y-direction,
and the other four ports fixed, and notice the reaction forces
at the fixed vertices. In the x—direction, the reactions
are symmetric with respect to the applied force, of opposite
direction to the applied force, and very nearly one half as
large (we may expect some round-off error). In the y—direc-
tions, we note that the forces are symmetric and add to zero
also. Further, if a force in the y-direction is applied,
the y-reactions add to the negative of the applied force,
and the x-reactions add to zero. So, the model is "physically"
consistent, and passes the most basic requirement for an
acceptable model.

We might also investigate the other possible causal

form, with derivative causality extended to C3, forcing

18

integral causality at C Using the same procedure as

1'
demonstrated above, we obtain

 

'70007 —3.05 —.u99 -.601 .0007 3.05 ..,
-3.0A 13050 -.180 -.501 -3.0u —13050
" .A99 . .175 (2.52 10‘5) 0 .499 -.175
K -. .601 .501 0 (7.17 10‘5) —.601 .501
.0007 —3.05 —.499 .601 .0007 3.05
43.09 -13050 .180 -.501 3.0M 13050 .

 

which is, within roundoff error, identical to K'. This is
reasonable, since alternative internal causalities do not
alter the structure of the model, only (ultimately) the

form in which the equations occur. So, it should make no
difference how causality is extended within an element, for
a given externally-applied causality. This will give some
liberty in the actual solution of such problems. (By stan-
dard finite-element methods, as described in chapter 3, this

matrix would be

 

p q
8.0 8.0 -.500 -.599 8.0 8.0

8.0 13070 -.180 —.A99 -8.0 -13070
.500 .180 (2.5-10‘5) 0 .500 - 180
.599 .u99 0 (7.17 10'5) -.599 .499
8.0 -8.0 -.500 .599 8.0 —8.0
1.?‘0 -13070 .180 -.L199 —8.0 13070 .1

 

We notice a high degree of correlation here, which would

l9

tend to confirm the accuracy of the bond—graph model.

We see here that we have a relatively straightforward
means of deriving a mixed-causal input/output form for our
6-port elastic element, by a natural application of simple
causal rules and some linear algebra. This may give the de-
signer a degree of freedom (no pun intended), as will be dis-
cussed in chapter three.

A note is worthwhile here concerning finite—element meshes
of several elements. The principle described here was pre-
sented, for the sake of time and space, for the case of a
single 6-port element. The principal of superposition (by
means of l-junction connectors, as in 2—2.1) would provide
for the means of joining individual elements into a mesh, and
for easy derivation of the global matrix. It would also pro;
vide the means of extending externally—applied causality in—
ward. The l-junctions provide one constraint on how this is
done, in that a given l—junction must have one and only one
flow (velocity) input. Another constraint on allowable ex-
tensions of causality comes from the element models themselves,
which require that no more than three effort (force) inputs
exist for a given 6-port element. These two conditions will
not, in general, completely determine the causality of the
entire mesh, however. The engineer will, in general, have
several choices in assigning causality to the global mesh.
This would suggest that we should seek an optimal causality-
assignment strategy, which is beyond the scope or intent of

this thesis, but a worthwhile topic for further investigation.

20

2—A Some Simple Extensions

 

The Finite—Element Method takes on a wide range of forms,
even if one never leaves the field of elasticity. It would
behoove us to apply our bond-graph technique to as many of
these as possible, and in this section, a few simple extensions

will be discussed.

2-A.l Higher—Order Elements

 

In some instances, the constant-strain triangular element
with which we have been dealing is not the best choice for
design purposes. In some instances, a linear-strain, or
even higher order, element is preferred. (When higher-order
elements are employed, a smaller number of elements are
required to achieve similar accuracy.) Linear-strain elements
have also been referred to as quadratic elements, since the
linear strains result in the edges of the element being de-
formed parabolically (see Figure 2-12). In order for these
quadratic displacement functions to be determined, however,
three nodes, rather than two, must exist along each edge.

The third node is usually chosen as the "mid-point" of the
edge. This gives us a total of six nodes for each element,
each capable of motion in two directions, so, externally,

the linear-strain triangular element is a l2-port. Inter-
nally, however, the structure is determined by Hooke's Law,
which in two dimensions is a 3x3 relationship. We would thus
expect the linear-strain element to be an internal 3-port

(as in Figure 2-13), which is the case. So for our model,

we would begin with the 12x12 stiffness matrix, and solve

21

for three non-zero eigenvalues and the corresponding eigen-
vectors, and proceed as before. The extension is similar to
constant—strain rectangular elements, which are external 8-ports,
but are also internal 3-ports, or to other higher-order elements,
such as linear-strain rectangles, etc. In general, for a two
dimensional element having n nodes, the element will be an

external 2n-port, but will always be an internal 3-port.

2-A.2 Three-Dimensional Elements

 

The basic three-dimensional element is s constant-strain
tetrahedron (see Figure 2-1A). Having four nodes each capable
of moving in three linearly-independent directions, it is an
external l2-port. Its internal response is determined, as
always, by Hooke's Law, which, in three dimensions, is a 6x6
relationship. Thus, we would expect an internal 6-port,
or, in terms of our model, six non-zero eigenvalues of the
stiffness matrix which determine the stiffness of six internal
C-elements, and the corresponding eigenvectors which determine
the junction structure connecting the external ports to the
internal C-field, as in Figure 2-15.

Higher-order elements may also be used in three dimen-
sions such as constant-strain octahedra or cubes, or linear
or quadratic-strain elements, etc. In general, a three-dimen-
sional element having n nodes will be an external 3n-port,
while remaining an internal 6-port, due to Hooke's Law, and
the modeling procedure would be similar to that delineated

above.

22

 

 

_____a- ----3- .
Junction , Elastic
External '
Ports . Structure . Storage
. (0,1,1?) . Field (C)
.____A. ____A5

 

 

 

 

 

 

Figure 2—1. Desired form of elastic element model

 

_*
__,.
Junction \
——.h ___~
Structure C
.....a. Ill/’.
——-7

 

 

 

Figure 2-2. Model with six external ports, internal 3-port

 

C-field
—-_
Junction
_'_'"__ --. --a- C
Str ct e
u ur
fl

 

 

 

Figure 2-3. Model with decoupled C-elements

23

“-.-“‘~‘~“_‘_ 7
a 1. j

2 3
'le 2L J

Figure 2-A. Axisymmetric rod with exponentially-varying
cross-section

 

 

 

—;-TF——r O v—TF~—-
(.707) L (1707)

Cam)

Figure 2-5. Bond-graph element model

—7TF—7 0 v—TFw— 1 —-7TF—r 0 s—TFv—
(.707) l! (1707) (.707) A (7.707)

C (1.505) C (0.774)

Figure 2-6. Bond-graph assembly model

 

 

ement

el

gular

2-7. Constant-strain trian

Figure

C (64:00) '

amal.nvrllbl
.91.. 1....

 

 

2-8. Bond-graph model

Figure

25

.i,=o

F :4)
s§c1_‘/ ”

. g 4
70%. “j/ 6‘: ’0
&,20

(a) (b)
Figure 2-9. (a) Boundary conditions applied to triangular
element

(b) 6—port causal bond-graph model

I" l" l“ l" l" l”

1 1 1 1 1 1

 

 

r5t~r-—-C>‘-(
r3I‘r-WC)
(T5*<-i<:) 1:

Figure 2-10. Causality applied to the bond-graph model

26

 

xfihpme coameLOMmchp omsuospmeCOHpoQSw one .HHIN oeswfim

 

o o o o ow.m :H.m mooo. :H.m1 mooo.l
o o o m:.H o Hem. Hom.1 Hem. How.
0 o o o o~.m 5:.m oom. >:.ml com.
o m:.H o o o o o o o
mm.ml o ow.m o o o o o o u mh
:H.m Hom.l N:.ml o o o o o o
mooo. How. oom.l o o o o o o
:H.m1 Hom.l 5:.m o o o o o o
mooo. Hom.l oom.1 o o o o o o

27

Figure 2-12. Linear—strain triangular element

 

3 ___;C

Juncfion
'3 ’ __.xc

 

y —_'7"' Structure 2 C

 

 

 

Figure 2-13. Form of linear-strain triangle bond-graph
model

28

Figure 2-lA. Constant-strain tetrahedral element

 

WI [’- “H'lwz “s '3 W3 '4 “'4
Junction Structure

 

 

 

.11.

Figure 2-15. Form of tetrahedron bond-graph model

CCC

CHAPTER THREE: A COMPARISON OF STANDARD AND BOND-GRAPH

METHODS IN FINITE—ELEMENT ANALYSIS

In this chapter we will proceed through the solution
process of a typical finite-element problem using both stan-
dard finite element methods and the bond-graph method
developed in chapter two in a side-by-side fashion, in order
to offer a step-by-step comparison. This will demonstrate
how various processes "translate" from one method to another,
and also how the bond-graph method may be applied to advantage.

For this analysis, we assume prior knowledge of the ele-
ment mesh for a given problem, and of the individual element
stiffness matrices. Knowing the element stiffness matrices,
we are able then to formulate bond-graph models for each of
the elements, as delineated in section 2-1 and demonstrated
in section 2—2.2. From this point, we will proceed through
three steps to final solution of the problem: superposition
of element stiffness matrices to form the assembly stiffness
matrix, application of boundary conditions, and the final

steps to the solution of the problem.

3-1 Superposition

 

Let us consider the element mesh shown in Figure 3-1,
with known element stiffness matrices. (This is taken from

(3), pp. 109-12A). Its bond-graph model is shown in

29

30

Figure 3-2, with element models shown in basic 6-port form

(to show the full model would result in an extremely cluttered
drawing; visualize these basic 6-ports as being detailed
models as shown in chapter two). We note that each node in
the element mesh is represented by two l-junctions in the
bond-graph model. Once again, this is because each l-junction
represents a common flow (velocity), and each node is free to
move in two linearly independent directions. At each node,
the elements meeting at that node will have velocity and
displacement in common, hence l—junctions are used to connect
the elements in a mesh.

Now, by standard finite-element methods, when the assembly
stiffness matrix is desired, a node-wise superposition of
element stiffness matrices is performed. The stiffness matrices
are subdivided into 2x2 submatrices describing the effect of

a displacement at one node on the force at another node, as in

 

 

 

 

II- '1
K11 K12 K131 rk22 K23 K2u

K1 = K21 K22 K23 ’ K2 = K32 K33 K3A , etc.
K31 K32 K33 qu2 K13 Kuu

These 2x2 matrices are then added according to subscript to

produce the global stiffness matrix,

31

 

rZKll 2K12 2K13 2K1,4 21(15-
2K21 2K22 EK23 2K2“ 2K25
Kg = 2K31 2K32 2K33 8K3“ 2K35
ZKAl EKA2 EKA3 EKAA 2KA5
ZKSI EK52 2K53 2K5“ ZKSS

 

In the bond-graph method, this process is inherent to
the l-junction connectors. As well as connoting common flow
(velocity), l-junctions connote a summation of forces. For

example, at node 2

 

 

 

 

 

1- 1 r- 1
x, x2 x2
= +
Y Y Y
2 L. 2 .1 1 L. 2 J 2
= K21-1 ”1 + K22-1 ”2 + K23-1 ”3
v1 v2 v3
U2 L13 L14
+K22-2 +K23-2 +K211—2
V2 V3 V14
L
= K21—1 ”11 + (K22-1 + 22-2) ”2 +
V1 V2
(K23-1 + K23—2) ”3 + K211—2 ”u

32
This occurs similarly for each node in the mesh. So we see
that the very definition of a l-junction as a common-velocity,
force-summing connector results in the node-wise superposition
of element stiffness matrices. This is a nice feature, in that,
in a computational sense, it becomes relatively easy to "book-
keep" the superposition of element stiffness matrices directly

from the bond-graph.

3-2 Boundary Conditions

 

The next step, following construction of the assembly
stiffness matrix, is to establish proper boundary conditions
in the problem formulation. What this amounts to in either
standard or bond—graph methods is a proper interpretation
of the boundary conditions into suitable terms for solution.

In the standard finite-element method, this simply involves
translating forces applied to an object into nodal forces
applied on the element mesh. In the case of discrete loads
applied to the object, the element mesh may be chosen so that
the load occurs at a node. In the case of distributed loads
(as in Figure 3—3), the loads are integrated between each
node and the midpoints of the edges of the adjacent elements
on which the load is applied to derive the appropriate "nodal
forces" (Figure 3-A). These nodal forces (not all of which
are necessarily known) are then arranged into the "nodal
force vector", Fn' This vector appears in the assembly stiff-

ness equation

33

which is the step immediately proceding solution.

In the bond-graph method, the same discretizing process
must take place, since the bond-graph is by nature a lumped-
parameter rather than a continuous-distribution model, and
each node is treated as a l-junction pair. It is at this
point, however, that the bond-graph contains an added flexi-
bility in assigning external port (boundary) conditions.
Causality will permit either force or velocity inputs to the
bond—graph (in this static model, we may just as easily speak
of displacement, rather than velocity, as an input), which
are represented in opposite causal forms. Thus, a load applied
at a node may be shown as an effort (force) source to the
bond-graph, with the accordant causality, while a displacement
input (a pinned joint would be an example of this) may be
shown as a flow source, with the opposite causality. Causality
thus applied at the external ports may then be extended through
the entire assembly of elements (this extension need not be
unique, as shown in Figure 3-5). The element constitutive
relationships may then be re-evaluated in the appropriate mixed-
causal forms and constructed by means of the l-junction connec-
tors into the mixed-causal assembly constitutive matrix, setting

the stage for the final solution.

3-3 Solution

 

Having constructed the assembly stiffness (or constitutive)
matrix and applied the proper boundary conditions, it remains
only to finally solve the system for the information which we

desire concerning forces and displacements in the assembly.

3A

In standard finite-element methods this begins with the

global stiffness equation

which is reorganized into the form ( (2), p. 37),

l' " F“
Fa Kaa Kab Xa

T
b LKab Kbb .1 XbJ

 

 

 

 

where Xa is the vector of known nodal displacements and Fa

is the corresponding vector of (unknown) nodal forces, and
Xb is the vector of unknown nodal displacements, with Fb the
(known) force vector paired with it. Now (if the boundary
conditions are such that the assembly is incapable of rigid-
body motion, assuring invertibility of Kbb) we may solve for
Xb:

-1 T

(F-K x)

X = K b ab a

b bb

which, in turn, allows us to solve for Fa

- -1 T -1
Fa _ (Kaa - Kabeb Kab ) Xa + Kabeb Fb

Thus, we have solved for the unknown nodal displacements and
the reaction forces at the constraints by essentially an
algebraic manipulation of the stiffness equation. By bond-
graph methods, we have, in the previous step (by assigning
and extending port causality), achieved an assembly consti-

tutive equation of the form (using the previous notation)

35

i- a- F 1
Fa X8.
K t
X F

b _ J _' bJ

 

 

 

 

where we may make a single-pass solution. The matrix K'
directly transforms the known (input) vectors Xa and Fb
into the output vectors Fa and Xb, concluding the solution
process in one step.

A few comments are worthwhile here concerning this com-
parison. Standard finite-element methods and bond-graph
methods have certain analogous procedures in bringing a pro—
blem to solution: l-junction connectors correspond to super-
position of element stiffness matrices, and port causality
corresponds to the boundary conditions of the element mesh.
The bond-graph technique, though, allows a compression of
the solution process. The element models may be constructed
and connected to form the assembly model, and causality ap-
plied, without any concern for derivation of the assembly

constitutive matrix. The assembly matrix may then be derived

in mixed-causal form directly from the bond-graph model.

 

 

 

 

 

36

 

 

 

 

Figure 3-1. Mesh of four triangular elements

 

 

 

 

Figure 3-2. Bond-graph model of element mesh

 

l“47"”) We.

 

 

Figure 3-3. Distributed load applied to element mesh

 

5=zsk

 

 

E=L3k

Figure 3-A. Discretized nodal forces

 

 

 

 

u 01 A L 1 p .7:
no / \ I ‘1‘"
1 1,1 92:18 i7

 

 

 

Figure 3—5. Port causality applied to the global bond-
graph model

CHAPTER FOUR: CONCLUSIONS AND RECOMMENDATIONS FOR FURTHER
INVESTIGATION

A-l Conclusions

 

The main result here is to demonstrate that bond-graph
techniques can be advantageously applied to problems in finite-
element analysis. The model shown here has the property that
the 6-port stiffness matrix it generates is identical to that
generated by standard finite-element methods, and produces
good results for the examples shown. It also has the property
that the internal 3-port C-field is represented inthe form of
three decoupled C-elements, which provides for greater ease
in mathematical construction.

Causality applied to the model proved to be a useful tool.
It allowed for a direct derivation of element matrices in mixed-
causal form, according to the boundary conditions. This made
the derivation of the mixed-causal assembly matrix a much more
direct problem.

One perhaps hidden advantage of the bond-graph model is
that it is "self—contained", so to speak. The elements of
the bond-graph are well—defined in mathematical terms, and the
generation of the bond-graph model leads fairly directly to the
derivation of the relevant equations - in other words, the

model "contains" the equations, and the equations are a direct

39

A0

product of the model. So, a model such as this, the form of

which is well known, could be used to advantage.

A-2 Recommendations for Further Investigation

 

This thesis represents only the first attempt at modeling
finite-element methods with bond—graphs; there are many
possible directions which could be taken from this point.
Certainly effort should be made to improve and streamline the
method prOposed here, as well as test its accuracy and useful-
ness in several situations. One particular manner in which
this method may be improved would be to achieve a more direct
method of deriving the bond-graph model from the finite-element
mesh. That is, construct the bond-graph directly from the
element mesh, rather than from the element stiffness matrices
(which are themselves constructed by standard finite-element
techniques), eliminating a step in the modeling process, there-
by improving the efficiency of the method. This may involve
construction of a different model - the one shown here need
not be considered a unique representation.

The model shown here could potentially be used in con-
junction with the ENPORT-S program now being developed. (5)
presents the MACRO-element approach to bond-graph modeling
incorporated into ENPORT-S. With each 6-port elastic element
modeled in this MACRO-element fashion, ENPORT-S could perform
finite-element analysis in a very direct fashion. Further
investigation should be made in this direction, as well as

how to improve and streamline the procedure.

Al

Bond—graph methods are at their best when modeling
dynamic systems, which was not the case here. However,
finite-element methods are already being applied to problems
in vibration analysis, (see (6), chapter 11) and it is in
this direction that the bond-graph techniques hold perhaps
their greatest potential. This would involve developing a
model for the 6-port I-field necessary for such problems, as
well as other steps in assembling a system model and deriving
and solving state equations.

A strategy for extending causality into the element mesh
might also be considered. For a given mesh of elements, a
number of causal forms are possible, and it would be worthwhile
to consider which, if any, are preferred.

Finite-element analysis is applied to many different
fields of investigation, including heat transfer and fluid
flow, as well as various types of elasticity problems, includ»
ing plate bending and torsion. By proper definition of vari-
ables and construction of a suitable model (which may or may
not be analogous to the model shown here), bond-graph tech-

niques could be extended to these various problems.

APPENDICES

APPENDIX A

APPENDIX A. INTERNAL STRUCTURE OF A CONSTANT—STRAIN TRIANGULAR

(6-PORT) ELEMENT

A constant-strain triangular element could be modeled
in bond-graph terms as a general 6-port C-field (Figure l—2)
for a first approximation, since it has three vertices (nodes)
each capable of motion in two linearly independent directions
(as in Figure l-l). Each nodal displacement in either the
x- or y-direction corresponds to an external port on the
6-port C-field, and the constitutive relationship would take

the form

where Fr1 is the vector of nodal forces,
Ke is the element stiffness matrix
and Xn is the vector of nodal displacements

Given the element stiffness matrix Ke’ this could be
an adequate model of the element at one level. Since elements
are connected to each other by means of common nodal displace-
ments, we could connect the elements to each other in the
bond-graph model by means of (common-velocity, force-summing)
l-junctions, and thus arrive at a bond-graph model for the
element mesh which perfectly describes the finite-element
mesh. The l—junction connections implicityly describe the
superpositon of element stiffness matrices into the assembly
(global) stiffness matrix, which demonstrates some of the

power of bond-graph techniques.

A2

A3

This general 6-port model, however, gives no direct
information concerning internal structure of the element,
or of the effect of alternate causality on the element, or
even what causal forms are allowable. For this we need a
more detailed model, and we turn to the finite-element con-
struction of the element stiffness matrix to gain our insight.

In standard finite-element techniques, the element stiff-

ness matrix is constructed from

Ke = I BTDBdV
V

where B is the strain-displacement matrix which, given nodal
displacements, maps them into element strains and D is the
matrix mapping strain into stress according to Hooke's Law.
A closer look at these matrices will yield insight into the

internal structure of the element. We see that B is of the

form
a1 0 a2 0 a3 0
B = 0 b1 0 b2 0 b3
b1 al b2 a2 b3 a3
- -

 

 

which relates strain to displacement according to

8 = BX

T l I TI [:
e s = r . I
where 8 x y ny , and X ul ‘1 u2 v2 u3 J3

so we see that normal strains in the x- and y-directions de—

pend only on displacements in the x- and y-directions, while

AA
shear strains are related to both x and y displacements.
Thus, we can model the element in pseudo-bond-graph1 form
as shown in Figure A-l. So we see that we have now reduced
our general 6—port model to an internal 3—port C-field model
with six external ports. When we consider the constitutive

relationship of our 3-port C-field, we recall that

o = D s
and from Hooke's Taw,
1
F-l u 0
E E = Young's modulus
D = U- 1 O

Poisson's ratio

H
I
1:
1:
ll

 

 

Here we see that shear stress is dependent only on shear
strain, while normal stresses are related to normal strains
in a 2x2 relationship due to Poisson's effect. This allows
a further refinement of our 3-port model into a (2 + 1) port
model, as in Figure A—2, which is a reasonable model of a
constant-strain triangular element, insofar as it allows the

extension of externally applied causality into the model,

 

1This model does not have the property that the product of

effort and flow variables on each bond is power, since stress
and strain are "point" quantities rather than "global" quan-
tities such as force and velocity, and must be volume-integrated
to yield prOper energy quantities. Such an integration is
impossible to represent in bond-graph terms, hence we use

a pseudo—bond-graph which endeavors to represent relationships
between quantities in bond-graph style to illustrate

structure, but is not a true bond—graph.

A5

and gives some degree of information regarding the internal
energy structure of the element. This appendix, however, seeks
not to specify a model, but to provide information pertinent
to the formation of a model, the most important of which is

the internal 3-port structure of the 6-port element.

A6

 

Figure A-2. (2 + l)-port pseudo—bond-graph model

 

APPENDIX B

APPENDIX B. SUPPORTING MATHEMATICS

The key mathematical tool here is a modification of a
method for diagonalizing matrices. Any square symmetric real

matrix A can be expressed in the form

A = M-lLM

where L is a diagonal matrix whose elements are the eigenvalues
of A, and M is the nodal matrix of A, whose columns are the
corresponding normalized eigenvectors of A ( (7), pp. 179-183;
(8), p. 5A2). Since this model matrix is orthogonal, we can

express

SO

A = M LM

We can further abridge this expression for the case of an el-
ement stiffness matrix Ke' It can be shown ( (8), pp. 511-512)

that since

K = B DB dV
V 9

where B and D are both constant matrices having (in the 2 di-
mensional case) rank 3, that the rank of Ke is 3. (For n-

n(n+l).)

dimensions, Ke will have rank r = 2 From this we

can show that Ke has at least three vanishing eigenvalues ( (8),

p. 5A9) that Ke can be expressed in the form

K = M'TL'M'
8

A7

 

A8

where L' is the reduced diagonal matrix whose elements are the
non-zero eigenvalues of Ke’ and M' is the "semi-modal" matrix
whose columns are the eigenvectors corresponding to the eigen-

values in L'.

LIST OF REFERENCES

LIST OF REFERENCES

Karnopp, D.C. and Rosenberg, R.C. System Dynamics:
A Unified Approach. New York: Wiley and Sons, 1975

 

Martin, H.C. and Carey, G.F. Introduction to Finite
Element Analysis. New York: McGraw-Hill, 1973.

 

 

Ural, 0. Finite Element Method. New York: Intext,
1973

 

Rosenberg, R.C. Internal Communication, Department of
Mechanical Engineering, Michigan State University, 1978

Devlin, J.E. "Simulation of Engineering Systems Using
MACRO Bond-Graph Models", Department of Mechanical
Engineering, Michigan State University, M.S. thesis, 1978

Zienkiewicz, O.C. The Finite Element Method in Structural
and Continuum Mechanics. London: McGraw-Hill, 1967.

 

Bronson, R. Matrix Methods. New York: Academic Press,
1970

 

Wylie, C.R. Advanced Engineering Mathematics. New York:
McGraw—Hill, 1975

 

Segerlind, L.J. Applied Finite Element Analysis.
New York: Wiley and Sons, 1976

A9