“WWWNUWWWI!!!”WWWl

"UNI!

 

3
U HSI

 

 

(II—L
A
WV

  

   

T-‘z’r‘h M35

 

h.
ULRARY
5 P. Q 29 '7 9/

Michigan State
University

 

 

 

This is to certify that the
thesis entitled

Modular Finite Element Modeling By Distributed Internet
Engineering Design Agents

presented by

Umar Farooq

has been accepted towards fulfillment
of the requirements for the

M. 3 degree in Mechanical En ineerin
\

\LL

 

 

PLACE IN RETURN BOX to remove this checkout from your record.
TO AVOID FINES return on or before date due.
MAY BE RECALLED with earlier due date if requested.

 

DATE DUE DATE DUE DATE DUE

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

6/01 c:/C|RC/DateDua.p65-p.15

MODULAR FINITE ELEMENT MODELING BY DISTRIBUTED INTERNET

ENGINEERING DESIGN AGENTS

By

Umar Farooq

A THESIS
Submitted to
Michigan State University

in partial fulfillment of the requirements
for the degree of

MASTER OF SCIENCE
Department of Mechanical Engineering

2002

ABSTRACT

MODULAR FINITE ELEMENT MODELING BY DISTRIBUTED INTERNET
ENGINEERING DESIGN AGENTS

By

Umar Farooq

Finite Element Analysis of structural assemblies using distributed Internet
Engineering Design Agents (i-EDA) is presented here. Finite Element modeling of
distributed models in the past has been a complicated issue. Typically, the assembly of
these structures has required global reformulation of the Finite Element equations for
solution and this reformulation consumes time and prevents efficient development of
large models. Structural models with incompatible Finite Element meshing are
assembled here using of the Modular Modeling Method (MMM). The method presented

here does not require reformulation and allows efficient distributed model assembly and

solution.

The MMM is a power-based systematic technique that eliminates global
reformulation of model equations. The method uses displacement and work constraints
to assemble the components at connections. This method is implemented through a set of
Internet Engineering Design Agents (i—EDA) to assemble the Finite Element Models
from models provided by component subsystem agents. The structural analysis of Plate
Assembly of two triangular plates is used as an example and 4 cases presented. Models

of the plate component and an assembly of these two plates are published as software

agents on the worldwide Internet.

TABLE OF CONTENTS

List of Tables ...................................................................................................................... iv
List of Figures ...................................................................................................................... v
Introduction ......................................................................................................................... 1
The Modular Modeling Method .......................................................................................... 4
Modular Assembly of Two Components ............................................................................. 6
Assembly Examples for FEM Plate Structure Models ...................................................... 10

Case 1: Two, Single Element, Nodally Compatible, Plate Models ............................... 11

Case 2: Nodally Compatible Component Plates with 4 Elements ................................. 14

Case 3: Nodally Incompatible Plate Assembly Model. ................................................. 16

Case 4: Assembly of Higher Resolution Finite Elements. ............................................ 19
Internet Agents for Plate Models and their Assembly ....................................................... 23
Conclusions ....................................................................................................................... 27
Appendix ........................................................................................................................... 30
References ......................................................................................................................... 40

iii

List of Tables
Table 1 Displacement Values At Each Node For Both Plates With 1 Element Each ....... 13
Table 2 Nodal Displacements For Both Plates With 4 Elements In Each Plate ................ 16
Table 3 Displacement Values Of Assembled Plate Of Incompatible Nodal Geometry
Using Modular Modeling ...................................................................................... 18
Table 4 The Displacement Values Of Assembled Plate Of Incompatible Nodal Geometry

Using Modular Modeling Method ......................................................................... 21

iv

List of Figures

Figure 1 The Steel Plate Example ..................................................................................... 10

Figure 2 Finite Element Meshes Of Plates and Assembly ................................................ 12

Figure 3 Component Plate FEM Meshes With 4 Elements Each And The Assembled
Model .................................................................................................................. 14

Figure 4 Assembly of Models having Incompatible Nodal Geometry. Grayed Node ls

Condensed Out .................................................................................................... 17
Figure 5 Comparisons Of The Displacement Values At Nodal Interface Boundary ........ 19
Figure 6 Higher Resolution Components with Incompatible Nodal geometry ................. 20

Figure 7 Assembled Model With Incompatible Nodal Geometry. The Gray Nodes
Indicate Condensed Nodes .................................................................................. 21
Figure 8 Comparisons Of All Cases At Nodal Interface Boundary .................................. 22
Figure 9 The i-EDA System Schematic Showing A User Client That Queries Model
Agents Registered in the Agent Registry Using Valid Queries From the Query
Ontology. ............................................................................................................. 23
Figure 10 The Framework of an Agent in the i-EDA System ........................................... 24
Figure 11 The Front Panels Showing Stiffness Matrices Of The Upper And Lower
Component Plates ............................................................................................. 25

Figure 12 The Front Panel Of Plate Agent Assembler. ..................................................... 26

Introduction

Structural analysis problems frequently involve the assembly of two or more
components to form the structural system to be analyzed. Structural design and analysis
widely incorporates the Finite Element method. Finite Element method is a systematic
modeling method that generates mathematical models in structural design [Zienkiewicz,
1977 i]. While the Finite element method (FEM) can yield accurate, detailed stress
solutions for individual subsystem components, FEM is not efficient at assembling
components into assemblies. This is because every change in a substructure finite
element model requires global reformulation of equations to ensure compatible nodal
geometry of components at their interfaces. Reforrnulation of finite element mesh
equations is the single most time consuming activity in FEM analysis. It involves
rewriting the set of equations for each component and hence prevents efficient
development of assembly models. Additionally, the need to reformulate component

nodal geometries prevents the reuse of FEM models in a distributed modeling system.

At each component connection, the finite element method typically requires
compatible node geometry across the boundary connecting components. Mesh nodal
compatibility is typically required to construct a finite element model of an assembly
from the finite element models of its components. Methods that do not require nodal
compatibility still require that at the common interface the nodes on all subdomains must
have the same number and same type of degrees of freedom. These methods either need
interface elements to connect subsystems [Aminpour et a1, 1995ii] or require additional

constraints, typically in form of Lagrange multipliers for coupling the components into

system [Farhat and Geradin, 1992i“, Farhat and Roux, 1991”]. Both cases add to the
degrees of freedom of the system that results in overall increase of the size of the
problem. The Modular Modeling Method does not require equation reformulation or
nodal compatibility and does not add additional degrees of freedom to the system by

adding extra constraints to it.

Modular Modeling is a relatively new method for solving structural analysis
applications of distributed models. Modular modeling is a systematic equation assembly
scheme well suited to multi DOF systems. While this method can be used in several
energy domains like Electrical, Hydraulics, Acoustics and Heat transfer, the method will
be applied here to structures. This method is a power-based systematic modeling method
that eliminates equation reformulation from large model design, development, and
refinement across multiple energy domains [Byam and Radcliffe, 1999"]. The Modular
Modeling Method is a systematic approach to generate mathematical models with fixed

input-output structure.

The Modular Modeling Method couples the subsystems or components having
independent finite element modeling. The components need not to be nodally
compatible. The Modular Modeling Method uses constraint equations that identify the
relationship of the components to derive a model of the assembly of those components.
These assembly models then have the same fixed input-output structure as their

components and can be used to derive more complex assemblies.

Agent software facilitates the engineering design in a globally distributed

environment. Gosciak, [2001“] developed a component based design agent prototype

based on the i-EDA methodology [Radcliffe and Sticklen, 2001Vii] that uses a strict
communication protocol. The Design Agents are capable of representing the whole
system by providing answers to system attribute queries. In the work presented here,
individual components are modeled as design agents, their assembly is done according to
Modular Modeling Method and then the agents and the assembly are published on the

intemet using the i-EDA system.

The Modular Modeling Method

Fixed input-output structure is important to the Modular Modeling Method. Fixed
input-output structure requires standardization of input and output variables. In the
structural models discussed here, all inputs are nodal forces and all outputs are nodal
displacements. With standardized input and output variables, the mathematical model
that describes each component can remain fixed, independent of the system model into
which it is inserted. The assembled system generated by Modular Modeling has the same

standardized input and output variables [Byam and Radcliffe, 1999

Structural components defined by the FEM can be assembled using the Modular
Modeling method. In this work, the components are constructed independently and their
finite element models need not to have compatible nodal geometry. FEM models
generated without applied boundary conditions have rigid body modes and singular
stiffness matrices. Modular Modeling uses systems of constraint equations to connect
these singular component stiffness matrices by requiring conservation of power at
structural connections. This eliminates the need for both global reformulation and
compatible component nodal geometry. Because assembly models are assembled
without applied boundary conditions, they also have rigid body modes and singular

stiffness matrices.

Displacement Constraints connect two components by requiring consistent
displacements along their connecting boundaries. In this work, the components are
termed as Master and Slave components because the interface boundary displacements of

the slave component are written in terms of the master component. The idea of Master

and Slave is not new. Cook et a1 [1989“] used these terms within a subsystem. In the
substructure or component the nodes at boundary were called master and internal nodes
interior in that component were termed as the Slave nodes. Here, the whole component is
assigned the respective name. The component having fewest nodes at connection is used
as the master component. This allows nodal displacements of the slave to be written in
terms of the master component and results in removal of unconstrained nodes of slave
component at the connection. Work constraints require conservation of work along the
boundary of physical connections. By definition, work is the product of force and
displacement. The total work done across the connection boundary must sum to zero in

this static analysis.

Assembly of components using these constraints condenses out the connection
boundary nodes of slave component and retains connection boundary nodes of the master
component. The assembled model has the same fixed input output structure as its
constituent components. The model now can either be used a new subsystem to connect
to other subsystems or some boundary conditions and loads can be applied to allow for

the solution of the system displacement response.

Modular Assembly of Two Components

Static analysis of assembly of two components using the M is presented here.
The analysis assembles two components, which may or may not have nodal compatibility
at their connecting boundary. The stiffness matrices of both components are obtained

independently and are written in the linear form,

[KlIU] = [F] (1)

K is a component structure stiffness matrix, U is the vector of component nodal
displacements and F is vector of component nodal loads. Because every nodal
displacement has an associated nodal force, K is square. Additionally, K is often
symmetric [Zienkiewicz and Taylor, 1989*] and, in the MMM assembly, if all component
stiffness are symmetric, the assembly stiffness will also be symmetric. Here the

component structures have rigid body modes associated with them, thus K is singular.

The assembly of the component equations begins with the unconstrained, uncoupled,

[‘3‘ élfiiHiiil <2>

Displacement and work constraints will be used to assemble the subsystems. The

system equations,

displacement constraints are a set of equations that identify the physical relationships
between displacement output values at each connected node of the components. The
displacement values of connected nodes of one component are written in terms of

displacement values of the other component. The system equations are first rearranged,

[ml
mil

where U and E are component displacements and forces at unconnected nodes while

C1> Cl
l___l

(3)

~11» 'rll

U and E are nodal displacement and force variables included in the work constraints.
Using the master “M” and slave “S” component notation, the stiffness matrices of the

components obtained are represented in the form of master and slave subsystems.

1
FKMlI KMlz 0 -61"- “FM—
KMZI. ..... KM22K ........... KW EM = EM (4)
° Ki” .3: 1193:
_ 21 22‘ ‘ T A

 

 

 

 

 

The displacement constraints give the slave nodal displacementUS as a linear

combination of the master nodal displacements U M

[OSHSHOMI (5)

When these displacement constraints are applied, the number of columns of the system
equations is reduced by the number of slave displacements along interface boundary, the

constrained assembly equations result in a rectangular assembly matrix

 

 

 

KM“ KM12 0 W—UMF FFM-

$72.1. ..... KMzz ..... .0.... 0 = FM (6)
0 KSIZSEKS“ :1"- F5

_ K5228 K52 :US _ _FS_

 

 

Work constraints are applied to remove E M and generate a square system matrix. The

work constraints require work done across the assembled component boundaries to be
zero when no external forces are present at the connected nodes. If some external forces
are present at the connected nodes (for example reaction forces), then this relation sums

to that external force vector.

FAI'IUM +F§US = AgxtfiM (7)

where the first product is the work done on the master component by the connection and
the second term is the work done on the slave component. Applying displacement

constraints on (7) yields

in}; + 1313810,, = 13;,(1 M (8)

Because (8) must be valid for any displacement vector UM,

A TA A
FM +S F5 =Fex, (9)

The work constraints (9) applied to (6) results in a square, symmetric, assembly stiffness

model,

 

 

 

 

KMH KM12 0 UM EM
1‘ T " _ ‘
KMzi [KMzz +8 K5228] S K321 EM. — Eff. (10)
0 K s K Us Fs
512 511

 

 

This new mathematical model of the assembly has different characteristics from its
component models in terms of nodes and geometrical shape. The matrix is symmetric as

long as the constituent stiffness matrices of master and slave components are symmetric.

The assembly model completely describes the structural properties of the assembled
model, has the same form as component models (1) and can be used to assemble more

complex assemblies.

Assembly Examples for FEM Plate Structure Models

The mathematical modeling of the structural assembly of two components has been
described using the Modular Modeling Method. This method will be applied below to a
square steel plate (Figure. 1) assembled from two triangular component models and

solved to verify the modular assembly method discussed above.

    
 

l<————c

: ilmlll““""«miiillitmuiimamii“

 

Modulus of ElastICIty 20x106 N/cm?
Poisson‘ 5 Ratio: 0.27

60m

i ‘4‘
Hi“ RN” .1!“

 

' ‘ “ ' """"“”‘“" V“ "U ‘ Load=141 N

Figure 1: The Steel Plate Example

 

 

Four cases are presented for different finite element meshes of each triangular plate.
The first two cases have compatible interface geometry with different grid size and are
used to compare the M result with conventional FEM assembly. The third case
generates an assembly with incompatible nodal geometry from the elements developed in

cases one and two. This case has no conventional FEM model comparison, however, so

10

its accuracy must be evaluated against similar FEM. models that should bound this case
from both above and below. Finally, a case of higher mesh resolution is presented to
demonstrate that, the higher the FEM component model resolution, the better is the

accuracy of the resulting, assembled, MMM model.

The individual component stiffness matrices are computed using triangular finite
elements by standard finite element procedures [Segerlind, 1984’“) and assembled with
the Modular Modeling Method. Boundary conditions and applied load shown in Figure 1
are applied to enable a solution [Cook et a1 1989“] of the singular assembled stiffness
matrix. The displacement values at each node are compared with the solution generated
by ANSYS for the first two cases and the RMS difference is calculated to verify the
accuracy of the M results. Finally, a third incompatible nodal geometry case is
demonstrated whose FEM component mesh resolution are in-between the first two cases.

This final case is then compared against the displacements predicted by the two; high and

low resolution results from ANSYS.

Case 1: Two, Single Element, Nodally Compatible, Plate Models

Case one is a simple 2D problem (Fig. 2) that assembles two triangular plates each
consisting of one triangular element with three nodes. Plates are assumed to have 2 DOF
at every node and two nodes along each side. Because their meshes have the same
number and placement of boundary nodes, the plates have compatible nodal geometry at
the boundary. Plates are assigned master and slave names, the stiffness matrices of both
plates derived, and the components are assembled using MMM. The boundary
conditions and loads are then applied and the nodal displacements found at each node for

comparison with commercial finite element software ANSYS results.

11

2.

 

Master Plate

Slave Plate

 

 

 

 
 
   

 

 

 

 

 

Plate Assembly

Figure 2 Finite Element Meshes Of Plates and Assembly

The FEM models of these component plates are relatively small and are given as.

' 1.97
0.00
—1.97
1.97
0.00
_—1.97

 

' 5.39
0.00
-5.39
1.46
0.00
L—1.46

 

0.00
5.39
1.46
—5.39
—1.46
0.00

0.00
1.97
1.97
— 1.97
- 1.97
0.00

—1.97 1.97
1.46 —5.39
7.36 -3.42

—3.42 7.36

-5.39 1.46
1.97 —1.97

- 5.39 1.46
1.97 —1.97
7.36 — 3.42

- 3.42 7.36

-1.97 1.97
1.46 — 5.39

 

0.00 —1.97' in” “ ' f,,
—1.46 0.00 u., f,,.
_5.39 1.97 x106 ”2x = f2x

1.46 -1.97 “2,. f,,

5.39 0.00 us. f,,

0.00 1.97 _ _u,_,. _M _f,,, _M

0.00 —1.46' "ulx ' ' f1, '
-1.97 0.00 uly fly

1.97 1.46 u

x106 2x = f2x
1.97 0.00 :43, f3,
0.00 5.39 d —u3y-S _f3y_s

 

-

 

 

 

 

 

 

 

 

(11)

(12)

The assembly of master (11) and slave (12) models uses the displacement constraints (5),

p—

“1x
uly
“3x

u3y

—

 

- r-

 

 

.5-

OOOH

0 O Oqfulx1
1 O 0 “1y
0 l 0 “3x
0 0 1‘_u3y_

 

 

M

so that the constraint matrix S = I . The work constraints (9) are,

12

(13)

— —‘

fr.
f,,
f3.
f3, _

 

 

OOOH

 

Gov—O

OHOO

Using (10), the assembly model in case 1 is,

” 7.36
- 3.42
—1.97
1.46
— 5.39
1.97
0.00
_ 0.00

 

— 3.42

7.36
1.97

—5.39

1.46
—1.97
0.00
0.00

— 1.97 1.46 — 5.39
1.97 - 5.39 1.46
7.36 0.00 0.00
0.00 7.36 — 3.42
0.00 — 3.42 7.36

- 3.42 0.00 0.00

— 5.39 1.97 -—1.97
1.46 - 1.97 1.97

HOOO

 

1.97

—1.97

— 3.42

0.00
0.00
7.36
1.46

 

 

 

 

fr. Ff].

f f
" = " (14)

f3): f3x

Lf3yJS -f3y4
0.00 0.00 “ Fug.” "f2.-
0.00 0.00 an f2,
—5.39 1.46 u... fr.
1.97 -1.97 u

x106 1” = f” (15)
— 1.97 1.97 u3. f3.
1.46 ”5.39 “3), f3),
7.36 -3.42 u... f...
—5.39 -3.42 7.36 _. Lu“)- ..f4yd

 

 

 

 

 

The assembly process (3) — (10) requires a reordering of the nodal variables and the

above result reflects the new nodal displacement and force order.

Now boundary

conditions and Nodal forces are applied to further reduce the order of the problem (see

Appendix). The result of application of boundary conditions and load is the set of nodal

displacements shown in Table 1. Because standard element equations are used for the

finite element model, these results agree exactly with the results computed by AN SYS for

this simple model.

Table 1 Displacement Values At Each Node For Both Plates With 1 Element Each

 

 

 

 

 

 

 

 

 

 

 

Node X-Displacement X-Displacement Y-Displacement Y-Displacement
Number U x (ANSYS) (cm) Uy MMM (cm) U x (ANSYS) (cm) Uy (MMM (cm)
1 0.0000E-O4 0.0000E-O4 0.0000E-04 0.0000E-04
2 0.0000E-O4 0.0000E-O4 0.1403E-04 0.1403E-O4
3 -O.1 137E-04 -O.1 137E-O4 0.4407E-O4 0.4407E-O4
4 0.3137E-O4 0.3137E-O4 0.6350E-04 0.6350E-04
RMS DIFFERENCE I Ux=0.0000E-O4 J U, =0.0000E-04

 

 

13

Case 2: Nodally Compatible Component Plates with 4 Elements

This case involves two finite element models with 3 nodes on each side. Again this is
a case of compatible nodal geometry. This case demonstrates that M produces the
same results as a standard FEM package for multiple element models and will be used for

later verification of the nodally incompatible model.

 

 

 

 

 

 

 

 

 

 

 

 

Master Plate 5 5
4 9
6
4 .
2 . MMM
5
1 1 . A
Slave Plate 2 3 Y Plate Assembly

 

Figure 3 Component Plate FEM Meshes With 4 Elements Each And The Assembled
Model

The master plate stiffness matrix,

' 1.97 0.00 -1.97 1.97 0.00 —1.97 0.00 0.00 0.00 0.00 0.00 0.00 ‘
0.00 5.39 1.46 -5.39 —l.46 0.00 0.00 0.00 0.00 0.00 0.00 0.00
-1.97 1.46 14.72 -3.42 —10.73 3.42 -1.97 1.97 0.00 -3.42 0.00 0.00
1.97 —5.39 -3.42 14.72 3.42 —3.93 1.46 —5.39 3.42 0.00 0.00 0.00
K: 0.00 -l.46 -10.78 3.42 14.72 —3.42 0.00 0.00 3.93 3.42 0.00 -1.97 (16)
-1.97 0.00 3.42 —3.93 —3.42 14.72 0.00 0.00 3.42 —lO.78 -l.46 0.00
0.00 0.00 —1.97 1.46 0.00 0.00 7.36 —3.42 —5.39 1.97 0.00 0.00
0.00 0.00 1.97 -5.39 0.00 0.00 -3.42 7.36 1.46 -1.97 0.00 0.00
0.00 0.00 0.00 —3.42 —3.93 3.42 —5.39 1.46 14.72 —3.42 -5.39 1.97
0.00 0.00 —3.42 0.00 3.42 -lO.78 1.97 —1.97 —3.42 14.72 1.46 -1.97
0.00 0.00 0.00 0.00 0.00 -l.46 0.00 0.00 5.39 1.46 5.39 0.00
0.00 0.00 0.00 0.00 —1.97 0.00 0.00 0.00 1.97 —1.97 0.00 1.97 _

x106

 

 

14

Nodal Displacements,

U = [111x L11
and Nodal forces,

F=[f1,r fly fzx fzy f3x f3y

y

“2x

uzy

“3x

The Slave plate stiffness matrix,

' 5.39
0.00
-5.39
1.46

_ 0.00

0.00
0.00
-l.46
0.00
0.00
0.00

 

_ 0.00

0.00
1.97
1.97
-l.97
0.00
0.00
-1.97
0.00
0.00
0.00
0.00
0.00

-5.39
l .97
14.72
—3.42
-5.39
1.46
-3.97
3.42
0.00
—3 .42
0.00
0.00

1.46
—l .97
—3.42
14.72

1.97
—l .97

3.42

-lO.78
—3.42

0.00
0.00
0.00

0.00
0.00
-5.39
1.97
7.36
-3.42
0.00
0.00
-l .97
1 .46
0.00
0.00

£43),

“4x “4y “5x
f 4x f 4 y f 5):
Oil) 0.00 —l .46
01X) -1 .97 0.00
1.46 -3 .97 3.42
-l .97 3.42 -10.78
-3 .42 0.00 01X)
7.36 0.00 01X)
0.“) 14.72 -3.42
0.“) -3.42 14.72
1.97 -lO.78 3.42
-5.39 3.42 -3.93
0.00 0.00 -l .97
01X) - l .46 01X)

The corresponding Nodal Displacements and Forces are

T
[ulx “1y “2x “2y "3x “3y “4x “4y ”5x “5y “6x “6y] and

£45),

fSy

0.00
0.00
0.00
—3 .42
-l .97
1.97
-lO.78
3.42
14.72
-3.42
-l.97
1.46

f6x

0.00
0.00
-3.42
0.00
1.46
-5.39
3.42
-3.93
-3.42
14.72
1.97
-5.39

0.00
0.00
0.00
0.00
0.00
0.00
0.00
-1.97
-1.97
1.97
1.97
0.00

0.00 '

0.00
0.00
0.00
0.00
0.00
—1 .46
0.00
1.46
-5.39
0.00

 

5.39 -

(17)
1:10‘5

[flx fly f2x f2y f3x f3y f4x f4y fo fSy f6x fslereSpectively.

The displacement constraints (5) for this case are

p- —

“1x
“1y
“3x
1(3),

“6x

 

 

u6y_

S

OOOOOF—

 

Work Constraints using S = I are

CD CD CD CD H‘ CD

CD CD CD H‘ CD CD

CD CD —‘ CD CD CD

15

CD _. CD CD CD CD

H‘ CD CD CD CD CD

 

“1): i
uly
“4x
u4y
“6x

_ “6Y‘M

 

(18)

fr.
f,_..
f3.
f5
fa

 

 

féy

—

_M

OOOOOH
OOOOl-‘O
OOOHOO
OOHOOO
CHOOOO

 

h

HOOOOO

- p _

 

 

 

 

T "f1. f...
fly fly
f4. _ f3.
12. ‘ f.
f6. f6.

-f6)’ .13 ..f‘W _

 

.1

(19)

The assembly is done using the above constraints. By applying boundary conditions and

loads, nodal displacements are found and compared with ANSYS results. The RMS

difference calculated for this nodally compatible assembly shows no difference in results.

Table 2 Nodal Displacements For Both Plates With 4 Elements In Each Plate

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Node X-Displacement X-Displacement Y-Displacement Y-Displacement
Number (Ux ) (AN SYS) (Ux ) (Modular Uy (ANSYS) Uy (Modular
(cm) Modeling) (cm) (cm) Modeling) (cm)
1 0.0000E-04 0.0000E-04 0.0000E-04 0.0000E-04
2 0.0067E-04 0.0067E-04 0.1493E-04 0.1493E-04
3 0.0009E-04 0.0009E-04 0.3641 E-04 0.3641 E-04
4 0.0000E-04 0.0000E-04 0.241 1 E-04 0.241 1 E-04
5 -0.1621 E-04 -0.1621 E-04 0.3796E-04 0.3796E-04
6 -0.2241 E-04 -0.2241 E-04 0.7092E-04 0.7092E-04
7 0.3003E-04 0.3003E-04 0.4024E-04 0.4024E-04
8 0.5739E-04 0.5739E-O4 1 .0127E-04 1 .0127E-04
9 0.0319E-04 0.0319E-04 0.7784E-04 0.7784E-04
RMS DIFFERENCE x=0.0000E-04 l Uy =0.0000E-04

 

Case 3: Nodally Incompatible Plate Assembly Model.

Here a plate problem with incompatible interface nodes is solved. The assembly
involves a master plate having 2 nodes on each side and a slave plate with 3 nodes on
each side as shown in figure (4). This case is combination of previous two cases. The

finite element model already found in (12) is connected with finite element model already

16

 

found in (16) to make an assembly of finite element models having incompatible nodal

geometry.

 

 

 

 

 

 

 

 

 

Slave Plate
4 5 6 5 6
t O
3 3
2 4
MMM
A
1 Master Plate
0 1 2 .
1 2

Assembled Plate

Figure 4 Assembly of Models having Incompatible Nodal Geometry. Grayed Node Is
Condensed Out

The displacement constraints given for this case are

 

[‘qu

“1y
“3x
U3y
“6x

u6y

.1

 

-S

 

h

l O O 0
0 1 0 0
0.5 0 0.5 0
0 0.5 0 0.5
0 0 1 0
O 0 0 1

In

“1x
uly

“3x

 

 

u3y_

(20)

Note that the S matrix is not identity in this case as opposite to the previous two cases.

The work constraints conserving work at the boundary in this case are

17

The assembled model uses the constraint equations (20-21) for assembly.

 

 

F 1 0 0
0 1 0
0 5 0 0.5
+
0 0.5 0
0 0 1
_ 0 0 0

 

 

 

 

077,,"
O f.,

0 f3. _
0.5 f3), —
0 f6.
1) _f6y_s

fix
fry
f3x

 

 

f3y_

(21)

The

application of nodal loads and boundary conditions results in nodal displacements shown

in table 3.

Table 3 Displacement Values Of Assembled Plate Of Incompatible Nodal Geometry

 

 

 

 

 

 

 

Using Modular Modeling
Node X-Displacement Y-Displacement
Number Ux (cm) Uy (cm)
1 0.0000E-04 0.0000E-04
2 0.3236E-04 0.7543E-04
3 -0.1 884E-04 0.5699E-04
4 -0.0648E-04 0.1281 E-04
5 0.0000E-04 0.2237E-04
6 -0.1 584E-04 0.3035E-04

 

 

 

 

 

The displacement values are in accordance with the expectations as this model lies in

between the stiff model (case 1) and the relatively compliant model (case 2). The grayed

node “A” is the simple linear interpolation of nodes 1 and 3 and its displacement values

are defined by the displacement constraints in terms of nodes 1 and 3. The connection

force values can be ascertained by substituting the displacement values calculated above

(table 3) in the assembled model.

18

The comparison of displacement values for all 3 cases shown in graph 1 indicates the
behavior of displacement values at the interface. Case 3 is less stiff than case 1 and more
stiff than case 2 so the nodal displacement values obtained in this case are expected to lie
between the first two cases. Figure 5 indicates the expected results. It also indicates that
whether the nodal interface geometry is compatible or not, MMM assembles the model
and gives the expected results. Thus it can be assumed that MMM correctly assembles

the model

 

Total Displacements Along Interface Boundary

 

 

8.E-05

7.E-05 i-

6.E-05 — - -—O—Compatible 3-3
55.05 1 ., Nodes (15 DOF)
4.E-05 —~—-- ----- - I- - Incompatible 3-2

 

3.5-05 .____.-. - ' ' / Nodes (QDOF)
25.05 -_.__..-H-_.-- ‘ ,v , - . 22 .. - i- Compatible 2-2
1.5-05 :./ , . Nodes (5 DOF)
0.E+00 i
1 2 3

Nodes at Interface Boundary

 

 

 

 

 

 

 

 

Displacement Magnitude(cm)

 

 

 

Figure 5 Comparisons Of The Displacement Values At Nodal Interface Boundary
Case 4: Assembly of Higher Resolution Finite Elements.

The assembling of higher resolution meshes is done in the same way as discussed
above. Consider the same example plate with the master and slave plates having 3 and 4
nodes along their boundaries shown in figure (6). This is a problem with relatively large
DOF and more incompatibility at the interface boundary. The displacement and work

constraints are given respectively as

19

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

2O

 

3,” 1 0 0 0 0 0
(3130 0 01a];
“3x 3 0 3 0 0 O ury
M3), = 0 %- 0 %- O 0 u4x
“6x 0 0 § 0 § 0 144,
“6y 0 0 0 g 0 31%— “6x
“10x 0 0 0 0 1 0_“6y_M
-“lOy_s _0 0 0 0 0 1-
F100000‘Tfu‘
f1,‘ 010000 f,, "f“
fly :1;- 0 €- 0 0 0 f3): fly
f4.+0%0%00 f3,__f4.
f4y 0 O '3' O 21;— 0 f6x f4y
f6x O 0 O % 0 % f6y f6x
fat” 0 0 0 0 1 0 f10x _f6,.._
-0 O 0 O O 1J_’f10y_is
SlavePlate
7 11 o .10 ,
I
ll
5 l
4 6 ,
II
I 4
2 I
l
l
I
ll /’ 1 3
MasterPlate

Figure 6 Higher Resolution Components with Incompatible Nodal geometry

(23)

(24)

11 12

 

...L
Co

 

 

 

 

 

 

 

Figure 7 Assembled Model With Incompatible Nodal Geometry. The Gray Nodes
Indicate Condensed Nodes

The displacement values obtained at each node are given in Table 4.

Table 4 The Displacement Values Of Assembled Plate Of Incompatible Nodal

Geometry Using Modular Modeling Method

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Node X-Displacement Y-Displacement
Number U, cm Uy (cm)
1 0.0000E-04 0.0000E-04
2 0.3153E-04 0.4606E-04
3 0.5899E-04 1 .1037E-04
4 -0.0075E-04 0.4265E-04
5 0.0188E-04 0.8704E-04
6 -0.2874E-04 0.7985E-04
7 -0.0005E-04 0.1397E-04
8 -0.0669E-04 0.21 14E-04
9 -0.0918E-04 0.3178E-04
10 0.0000E-04 0.2965E-04
1 1 -0.1688E-04 0.3380E-04
12 -0.2610E-04 0.5476E-04

 

21

 

The case predicts the displacement response of the system at meshes that have higher
resolutions than the previously discussed cases. It is evident from the results that the
displacement response actual solution as the number of DOF of system increase as shown
in fig 8. Again, the displacement values of gray shaded nodes “A” and “B” are the

combination of nodal values of l, 4 and 6 as defined in (23).

 

Total Displacements at Nodal Interface

 

 

a: 9.E-05

E 3:82 + Incompatible 4-3 Nodes
g, 6 505 (21 DOF)

‘5' A ' +Compatible 3—3 Nodes (15
E E 515-05 DOF)

o 3 4.E-05 _

§ 3.505 + zgcagqatlble 3-2 Nodes
.L' 2.E-05

3 1.5-05 +Compatible 2-2 Nodes (5
P 0.E+OO DOF)

 

 

Nodes at Interface Boundary

 

 

 

Figure 8 Comparisons Of All Cases At Nodal Interface Boundary

22

Internet Agents for Plate Models and their Assembly

A globally distributed market requires software to design, manufacture and market
the assemblies of components and subsystems. An intemet-based engineering design
system would reduce design cycle time and increase efficiency. These intemet-based
agents should, not only be able to effectively and rapidly assemble these components but
also protect proprietary information underlying these models. In earlier work by Gosciak
(2001), a prototype system of Internet Engineering Design Agents (i-EDA) was
organized (Radcliffe and Sticklen, 2001) to facilitate the exchange of engineering design
performance data between corporate organizations while protecting proprietary design
information. The current work incorporates the plate models and their assemblies into

the i-EDA system (Fig 9).

 

 

  
   

 

Model Model
Agent , Agent _,

14mm

 

 

[her
Client
:4 '
Que ry
Ontology

Agent
Registry

 

 

 

 

 

 

 

      

List of
Agents

 

 

 

 

  
   

  

Model
Agent 1-

Model Model
A entu A entkg

Figure 9 The i—EDA System Schematic Showing A User Client That Queries Model
Agents Registered in the Agent Registry Using Valid Queries From the Query Ontology.

 

 

 

 

 

23

Users of the i—EDA system employ client software to submit queries to agents
representing physical components and systems from which a new design is to be
assembled. The agents are registered with the i-EDA Registry. The i-EDA system
includes an Ontology, or ordered list, of valid queries. Typical queries return information
on engineering, geometric, and or economic performance of the physical objects they
represent. The model agents represent either components or assemblies of components.
The framework of an i-EDA agent is shown in Fig. 10. The individual design agent
includes a network communication protocol, a query handler, a knowledge based system
and the resource set. Queries are received via the communication protocol and parsed by
query handler into a suitable form. The knowledge base then utilizes the internal
resources to assemble the response. For agents representing system, these resources may

consult agents representing system components.

 

Communication via TCP/IP

 

Query Handler

 

Knowledge base system

 

 

 

Resources
Model Application Rule
Database Software Database

 

 

 

 

 

 

 

 

 

Figure 10 The Framework of an Agent in the i-EDA System

24

The distributed Design Agents for the plates, represented by two separate companies
“Upper Plates Inc” and “Lower plates Inc” were added to the existing i—EDA system.
These design agents are component agents that provide independent answers to the query
for stiffness and return the plate stiffness matrix for each of these components. In
addition queries for cost, color, weight, area, size and delivery time are answered. These
plate models follow the same architecture as existing agents for truss and spans system
assemblies formed from bar components. The attributes are requested as queries to the
design agents. For answering the queries, separate software routines address each query
attribute for any part number of the specific plate. The front panel, block diagram and

framework of the plate agents are similar.

The stiffness query responses for the l-element models of the upper and lower plates
are shown in Fig (1 1). For a specific plate part number as shown, the agents build and
display the FEM model of the plate. Here, the stiffness matrices displayed on the front

panels of both plate agents are same as the stiffness matrices generated in (13-14).

 

     

 

 

 

Lower Plates _|n_1£3- .333
now ‘ 1 ‘ ' 'STLEEEEELW. £51 Tm
. . 1 ‘ 1 ‘ 1 [53395 00050 5.3955 11.655 0mm ~1.46E+6 I
1.3758 0.00E+0 49796 1.97E+8 0.0.150 -1.S7E46
‘ 000M 1.97m 197m 19754; 1.97515 11005.0 1
0.0380 5.39E+6 1.46E4-5 5.3856 -1.46E¢6 0.11141 .

-533E16 1.97155 738E+6 -3 421516 497E413 148E443 ‘
4.971545 1.48546 7.3896 {3.4258 -5.3$46 1.97EoS

1.4E1E46 187EvE 3-42E+5 736E05197E06 53358 I

01201540 19755-19755 1.97E06 197E45 011E111

1.97E¢S 5.508 -3.42E+8 7.3856 1.48E46 4.97596 '1
OIIJEoO 14656 -5.3$+6 1.4856 5.39E~6 [100540 4

19796 0.00540 1.9755 1975.6 0.0090 renew/cm ”5515 000543 145M 5.3356 0.00510 533E+6]N/cm 1

 

 

 

 

 

 

 

 

Figure 11 The Front Panels Showing Stiffness Matrices Of The Upper And Lower
Component Plates

25

The plate agent assembler “Miller Smith Plates”(fig 12) is a system class of design
agents that uses the information provided by the component plate agents to answer the
queries. As an example, this design agent takes the FEM models provided by the other
plate agents as input, assembles the combined Plate stiffness matrix using Modular
Modeling Method for a plate part number and represents the stiffness of the assembled
plate as a new attribute. Different part numbers for different plates that may or may not
have compatible nodal geometry are visible on the front panel. Future class of agents can
further use the information provided by this agent in the similar fashion. This is the
advantage of i-EDA that generalized architecture allows for use of the resources by other

design agents. The stiffness matrix obtained here is same as obtained in (15).

 

MLLLCY Smith Ptates

Q'JEQLJ
[ [3141111 + Stiffness

Design Component Data

 

 

' 4- .‘\. .24.;Jfiu-M

PartNO “cm {11 “N {6114114 .... 1NCN9114'
Plate‘l {Ker
Plate2 173L411

' manna-mu

1111434

«Iii-1L3:- - ‘

.. 111-11439,“
3W.“ {31114 '

 

 

 

 

[ 7.36E +6

 

 

-3.42E +6
-1 .97E +6
1 .46E +6
-5.39E+6
1.97E +6
0.00E +0
0. 00E +0

3.426 +5
7. 366 +6
1 .976 +6

25.3955

1.46E +6
-1 .S7E +6
0.00E +0
0.00E +0

-1 .97E +6

1 .97E +6
7.3613 +6
0.00E +0
0.00E +0

3. 42E +6
-5.39E +6

1.46E+6

1.46E+6 -5.39E+6 1.87E+6 0.006+0 0.00E+0

~5.39E +6

0. 00E +0
7. 36E +6

3.426 +6

0.006 +0
1.97E +6
.1 .976 +6

1.46E+6 -1.97E+6 0.00E+0 0.00E+0
0.006+0 -3.4ZE+6 -5.39E+6 1.46E+6
-3.42E+6 0.00E+0 1.97E+6 -1.97E+6
7.36E+6
0.00E+0

-1 .976 +6

0.00E+0 -1.97E+6 1.97E+6
7.36646 1.4SE+6 -5.39£+6
1466+6 7.36E+6 -3.42E+6

1.97E+6 ~5.39E+6 -3.4ZE+6 7.3SE+6]N/cm

 

 

Figure 12 The Front Panel Of Plate Agent Assembler.

26

 

 

Conclusions

The new method (MMM) of solving structural problems is both efficient and time
saving. It avoids the reformulation of system equations and allows easy assembling for
problems that involve Incompatible Finite Element meshing without adding constraints to
them. Also, it condenses the order of problem to a level that is far more convenient than

some of the methods currently used.

Modular modeling of structures using independent, Finite Element formulations for
components has been explained. This method does not require components to have
compatible finite element grids, assembles the components without reformulating

equations or requiring any interface element or additional constraints.

For assembling the two components the displacement and work constraints were
applied after the partitioning of components into connected and unconnected nodes.
These equations conserved the work done at connections and eliminated the linearly
dependent equations of the system thus reducing the order of the system. The assembly
model formed from its component models had different shape and nodal geometry but the
same input output structure as its components. This model can now be used
independently as a new finite element model of plate assembly. This model can also be

used as a component for further assemblies.

Four test cases were presented. Each test case represents differing resolution models
of identical physical plate geometry, boundary conditions and nodal loads. The first two

cases are of compatible nodal interface geometry, first case having course mesh

27

resolution resulting in a stiff model, and the second having relatively fine mesh resulting
in a compliant problem. Solving these cases demonstrated that MMM produces the same
results as any standard FEM used for the solution of such problems. The third case is an
incompatible nodal interface geometry case that uses the FEM models obtained by the
first two cases. This case demonstrated that the MMM produced the expected results that
are less stiff than the first case and less compliant than the second case. The fourth case
completed the examples by demonstrating that as resolution of models increases there is
an improvement in the results, and that MMM accurately predicts the displacement

response along the boundary of models.

These test cases show

0 Modular Modeling Method derived for assembling components agrees exactly with
the standard FEM package hence it is valid.

0 Modular Assembly has the same fixed input output structure as its components. The
assembly of components does not require reformulation of system equations and the
order of the system equations is reduced.

0 Modular Modeling Method accurately predicts the behavior of models having
incompatible nodal interface geometry.

0 Method uses simple constraints to assemble any two components. The methodology
is generic for all cases and no special element or additional constraints are required to

assemble the models with incompatible interface boundary.

Internet Design agents for component plate models and their assemblies have been

built for global distribution of engineering design database. It is shown that the plate

28

agent models generate the same FEM models as other commercial FEM packages. The
plate assembly agent assembles the FEM models of different plates provided by plate
agent models using the Modular Modeling Method. The plate assembly agents maintain
the structural properties of assembly by maintaining the fixed input output structure thus
enabling the assembly to be used as a component for further assemblies. The system of i-

EDA allows publishing of these models over the global internet market.

Structural assemblies of models involving incompatible nodal geometry require
special treatment either in the form of global reformulation or additional constraints. The
Modular Modeling Method presented here avoids the global reformulation of equations
or additional constraints and efficiently assembles the systems giving accurate results.
Thus it avoids complicated designs and saving computation time. The modeling methods
developed here form the analytical foundation for new i-EDA agents permitting

distributed modeling of structural systems.

29

Appendix

The Finite Element Model Of Plate

The triangular plates used in analysis are equal in dimensions (6 x 6 x 0.5) and are
assembled to form a square plate shown in (figure 1). Plates are made of steel and have a
Young’s Modulus of 30 x 106 N/cmz. Each plate uses one or more Triangular Elements
to form the stiffness matrix of the plate. The stiffness matrix obtained for each plate is

according to standard FEM procedure [Segerlind, 1984] for triangular elements.

For case 1, after the model is assembled in (15), the rigid body modes are removed by
the application of Boundary conditions and Nodal forces according to (figure 1). The
applied forces can be resolved into 100 N in x and y directions. Also, since node 1 has
zero displacement in both x and y directions and node 3 has zero displacement in x
direction only, hence the equations associated with these nodal displacements are deleted

and the model is reduced to the form

 

”7.36 1.45 -1.97 0.00 0.00" ’11,,“ '0‘
1.45 7.36 0.00 -1.97 1.97 14,, 0
-1.97 0.00 7.36 1.45 .5.39 x106 14,, = 0
0.00 -1.97 1.45 7.36 -342 11,, 100
_0.00 1.97 -539 -342 736‘ or,“ 1100‘

 

 

 

 

 

This square symmetric matrix is inverted and nodal displacements are ascertained.

30

 

Script CJR
This script executes the Modular Modeling examples for Structural analysis
Problem The user provides the Plates’ dimensions and their connectivity

One Element Example

XM = [0 0 6]; XS = [0 6 6]; Input of dimensions For both plates
YM=[O66]; YS= [006];
SM =[1 0 0 0 0 0; Connectivity Matrix for Master Plate
0 1 0 0 0 0;
0 0 0 0 1 0;
0 0 0 0 0 1];
SS = SM; Connectivity Matrix for Slave Plate

For Both Plates With 4 Elements
XM= [00303 6]; YM=[03 3666] 4Elements in Both Plates
XS=[036366];YS=[000336]
SM=[100000000000;
010000000000;
000010000000;
000001000000;
000000000010;
000000000001];

SS=[10000000000Q
01000000000Q
00000010000m
00000001000m
00000000001m
000000000001L

For Master Plate With 9 Elements And Slave Plate With 4 Elements
XM = [0,0,2,0,2,4,0,2,4,6]; YM = [O,2,2,4,4,4,6,6,6,6];
XS = [O,3,6,3,6,6]; YS = [0,0,0,3,3,6] ;
SM=[10000 000000000000000;
01000 000000000000000;
00001/20 00001/20 00000000;
0000 01/2000001/200000000;
00000 000000000000010;
00000 00000 0 000000001];

SS=[100000000000;

010000000000;
000000100000;

31

000000010000;

000000000010;

000000000001];
KM = Build (XM, YM,'M'); Finds Stiffness Matrices from function Build
KS = Build (XS, YS,'S');

[K] = Assemble (KM,KS,SM,SS) Stiffness of Assembled Components from Assemble

Boundary Conditions and Solution of equations.
Compatible Plates With lOne Element In Each Plate

Kmodular = K([[2:2],[5:8]],[[2:2],[5:8]]); Application of Boundary Conditions
Fmodular = [0;0;0; 100; 100]; Application of Nodal Forces.
Umodular = Kmodular\Fmodular Nodal Displacements

Compatible Plates With Each having Four elements :-
Kmodular = ktransform([[l:2],[4:6],[9:18]],[[1:2],[4:6],[9:18]]);
Fmodular = zeros (15,1);

Fmodular (12,1) = 100;

Fmodular (13,1) = 100;

Umodular = Kmodular\Fmodular;

Finally Master Plate with 9 elements and slave with 4 elements
Kmodular: K ([[1:6], [8:12], [15:26]], [[1:6], [8:12], [15:26]]);
Fmodular = zeros (23,1);

Fmodular (20,1) = 100;
Fmodular (21,1) = 100;
Umodular = kmodular\fmodular

32

The Function ASSEMBLE

function [KC] = Assemble( KM, KS,SM,SS)
[rowa cola] = size(KM);
[rowb colb] = size(KS);

TM = 1:cola;
TS = 1:colb;

[SM,KM,TM,count] = Kparse(SM,KM,TM);
[SS,KS,TS,countS] = Kparse(SS,KS,TS);

katrans = KM ;

kbtrans = KS;

kal l = katrans([l:count],[1:count]);

ka12 = katrans([lzcount],[count+1:rowa]);

ka21 = katrans([count+ l :rowa],[ 1 :count]);

ka22 = katrans([count+ l :rowa],[count+ l :rowa]);

kbll = kbtrans([l :countS],[1:countS]);

kb12 = kbtrans([l:countS],[countS+l:rowb]);

kb21 = kbtrans([countS+1:rowb],[ 1 :countS]);

kb22 = kbtrans([countS+l:rowb],[countS+1:rowb]);

[rowkall colkal l] = size(kal l);
[rowkbll colkbl 1] = size(kbl l);

ktransforrn = [ kall ka12 zeros(rowkall,colkb11);
ka21 [ka22+(SM')*(kb22)*(SM)] (SM')*(kb21);
zeros (rowkbl 1, colkal 1) kb12*SM kbl l];

KC=ktransform;
*****************************************************************
The Function BUILD

function [k]: Build(X,Y,plate)

% This function computes stiffness matrices of Triangular Plates with Plates'
% dimensions given as input.

% k = Output in the form of stiffness matrix

% X = Input of X co-ordinate of any Triangular Plate

% Y = Y co-ordinate of the triangular plate.

% Plate = Master or Slave Plate

[numelements, nodesperside] = CalcElements(X);
N = connect (nodesperside, plate);
N1 = N(1,:); NJ = N(2,:); NK = N(3,:); elem = length(NI);

*******************************************************************

33

The Function CalcElements
function [numelements, nodesperside] = CalcElements (X)
% function CalcElements (X)

% This function computes the number of elements and nodes

% in each side of plate, given the nodal Positions/total nodes.

m = length(X); Calculates length of input vector

dummy=3; A dummy variable introduced to map the nodal elements and dimensions.
n=1; A counter

m = m - dummy;.

while In > 0 For more than one elements procedure is straightforward.

dummy = dummy+ 1;

m = m-dummy;

n = n+1;
end
nodesperside = n + 1; Gives nodes on each side of plate.
numelements = n"2 Gives the total number of elements

*************************** ***************************************

The Function Kparse

function [S,K,T,count]=Kparse(S,K,T)

% Kparse parses a stiffness matrix K

% based on a supplied constraint matrix S

% and modifies the mapping transform vector T

[N ,M]=size(K); find the number of DOF N
[M,P]=size(T) Is T a column vector?

if P~=1; T=T'; M=P; end If not, make it a column vector

Test entry data
if size(T) ~= N
disp('size(T) not equal to DOF‘)

return

end

if N ~=M
disp('stiffness matrix and T do not agree')
return

end

Parse stiffness matrix
j=1 ; Point to end

count=0 ;
for i=1 :N
n=N-i+j; start at last column

if S(:,n)==0 test for no involvement in constraints S
K=[ K(:,n) K(:,lzn-l) K(:,n+1:N)]; If not move nth column left to top
K=[ K(n,:); K(l:n—1,:); K(n+l:N,:)]; and move nth row up to top

34

T=[T (n);T(l:n-1);T(n+1:N)]; Also reorder transformation map T
S=[S (: ,n) S(:, 1 :n— 1) S(:,n+1 :N)]; also move nth column of S left to top
j=j+1 ; found a constraint
count=count+1 ;
d: N-count
end

end

S=S (:,count+ 1 :M);

************************** **************************************

The Function Stiffnessmat

function K = stiffnessmat(X,Y,NI,NJ,NK,elem)

% Stiffness Matrix Computation Function for a 2D Plate

% X = X Coordinate of the Plate

% Y = Y Coordinate of the Plate

% N n= Node Numbering for each element, n=I,J,K

% elem = Total number of elements in the Plate

Defining System Constants

EE = 20* 10"6 Modulus of Elasticity in N/cm2

u = 0.27; Poisson's Ratio

t = 0.5; Thickness in cm

Kall= zeros(2*length(X),2*length(Y)); Allocation of space for Variables

for n=1:elem This Loop Computes 'B' used
Bi(n)=Y(NJ(n)) - Y(NK(n)); in Triangular elements

Bi(n)=Y(NK(n)) - Y(N1(n));
Bk(n)=Y(NI(n)) - Y(NJ (11));
Ci(n)=X(NK(n)) - X(NJ (11));
Cj(n)=X(N1(n)) - X(NK(n));
Ck(n)=X(NJ (11)) - X(NI(n));

M = [ 1 X(NI(1)) Y(NI(1));
1 X(NI(1)) Y(NJ(1));
1 X(NK(1)) Y(NK(1))];

A = 0.5*det(M) Area of each ELEMENT
DD = (EE/(l-u*u))*[ l u 0; Since areas are the same for all elements
U 1 0; Compute Just ANY One of them

0 0 (l-u)/2];

ij=l

35

forn=lzelem
BB(:,ij:ij+5)=(0.5*[Bi(n) 0 Bj(n) 0 Bk(n) 0;

0 Ci(n) 0 Cj(n) 0 Ck(n);
Ci(n) Bi(n) Cj(n) Bj(n) Ck(n) Bk(n)])/A;

K(:,ij:ij+5) = t*A*((BB(:,ij:ij+5)')*DD)*BB(:,ij:ij+5);

ij=ij+6;

end

ijk=zeros(l,6);muj = 1;

for n = lzelem
ijk(1)= 2*NI(n) -1;
ijk(2) = 2*NI(n) ;
ijk(3) = 2*NI(n) -1;
ijk(4) = 2*NJ(n) ;
ijk(5) = 2*NK(n) -1;
ijk(6) = 2*NK(n);
Ktemp = K(:,muj:muj+5);

for mj=l :6
for mi = 1:6
Kall(ijk(mi),ijk(mj)) = Kall(ijk(mi),ijk(mj)) + Ktemp(mi,mj);
end
end
muj = muj +6;
end
K=Kall;

************************ *******************************************

The Function Connect
function [nodalcords] = connect(nodesperside,plate)

% function nodalcords = connect(nodesperside,plate)
% This function computes nodal coordinates " N "
% if the input is in the form of nodes on any side,

% for both Master and Slave Plates.

switch plate
case 'M'
nodesperside = (nodesperside-1)*(nodesperside- 1);
maxrow = sqrt(nodesperside) + 1;
maxcol = sqrt(nodesperside) + 1;
row = maxrow;
col = 1;
count = 0;
rownum = l;
coltest = 1;
structmatM = zeros(maxrow,maxcol);
nodalcords = zeros(3,nodesperside);
while coltest <= maxcol

36

for traverse = 1 : rownum
count = count + 1;
structmatM(row,col) = count ;
col = col + 1;
end
rownum = rownum + 1;
if row ~= 1
row = row -1;
end
col = 1;
coltest = coltest + 1;
end
structmatM;
nodecol = l;
noderow = 1;
flag = 1;
row 2 maxrow;
col = 1;
nodetest = 1;
while nodetest <= nodesperside
if flag ==1
nodalcords (noderow,nodecol) = structmatM(row,col);
row = row - 1;
col = col + 1;
noderow = noderow + 1
nodalcords(noderow,nodecol) = structmatM(row ,col);
noderow = noderow + 1
col = col - l;
nodalcords(noderow,nodecol) = structmatM(row,col);
if structmatM(row+ l ,col+ 1)::

flag: 1; col = 1;
else

flag = 2;

end

elseif flag == 2

row = row +1;

nodalcords(noderow,nodecol) = structmatM(row,col);
col = col + 1;

noderow = noderow + 1;
nodalcords(noderow,nodecol) = structmatM(row,col);
noderow = noderow + 1;

row = row - l;

nodalcords(noderow,nodecol) = structmatM(row,col);
row = row+1;

flag = 1;

end

37

end

nodetest = nodetest + l;
nodecol = nodecol + 1;
noderow = 1;

otherwise

maxrow = nodesperside;
maxcol = nodesperside;
row = maxrow;

col = 1;

count = 0;
rownum = maxrow;
coltest = l;

structmat = zeros(maxrow, maxcol);
elem = (nodesperside-1 )"2;
nodalcords = zeros(3,elem);
while coltest <= maxcol
for traverse = 1:rownum
count = count + 1;
structmat (row, col) = count ;
col = col + 1;
end
rownum = rownum - 1;
if row ~= 1
row = row - 1;
end
coltest = coltest + 1;
col = coltest;

end

structmat; row = maxrow; col = 1;
noderow = 1; nodecol=l; nodetest = 1;
flag = 1;

while nodetest <= elem

if flag ==
nodalcords(noderow,nodetest) = structmat(row,col);
col = col + 1;
noderow = noderow + l;
nodalcords(noderow,nodetest) = structmat(row,col);
row 2 row -1;
oderow = noderow + 1;
nodalcords(noderow,nodetest) = structmat(row,col);
if col == maxcol
col = (maxrow - row) + 1;
flag = 1;
else
flag = 2;

38

end
else
row 2 row + l;
nodalcords(noderow,nodetest) = structmat(row,col);
col = col + 1;
row = row -1;
noderow = noderow + l;
nodalcords(noderow,nodetest) = structmat(row,col);
col = col - l;
noderow = noderow + l;
nodalcords(noderow,nodetest) = structmat(row,col);
row = row + 1;

flag = 1;
end
noderow = l;
nodetest = nodetest + 1;
end
nodalcords;

end
********************* **************************************************

39

References

 

vi

vii

viii

Zienkiewicz, O. C., 1977, The Finite Element Method, McGraw Hill Inc., New York

Aminpour, M A , Ransom, LB and McCleary, S.L .1995, “A Coupled Analysis Method For Structures
With Independently Modelled Finite Element Subdomains” Vol 38 International Journal For

Numerical Methods In Engineering

Farhat C and Geradin M, 1992 “Using a reduced number of Lagrange Multipliers For Assembling
Parallel Incomplete Field Finite Element Approximations” Vol 97 pp 333-354 Computational Methods

in Applied Mechanical Engineering

Farhat C and Roux, F.X, 1991 “A Method Of Tearing And lnterconnecting And Its Parallel solution

Algorithm” Vol 32 pp 1205- International Journal For Numerical Methods In Engineering

Byam, RB and Radcliffe, C.J., 1999 “ Modular Modeling of Engineering Systems Using Fixed Input-

Output Structure” IMECE, ASME, New York.

Gosciak,G. J, 2001 “Internet Engineering Design Agents”, MS thesis, Michigan State University, East

Lansing, Michigan.

Radcliffe, OJ. and Sticklen, J ., 2001, “Method and System for Creating Designs Using Internet-based

Agents", US. Patent 6,295,535, Sept. 25.

BR Byam and Cl. Radcliffe, 1999, "The Impact of Causality on Dynamic System Modeling", ASME
No. G01128, ISBN: 0-7918-1634-6 ISSN: 1522-2799, DSC-Vol. 67, ASME Internat'l Mech.

Engineering Congress and Exhibition, Nashville, Nov. 14-19.

Cook, R. D, Malkus, D. S. and Plesha, M. C., 1989, Concepts and Applications of Finite Element

Analysis, pp 259, 3rd Ed, John Wiley & Sons, New York.

40

 

xi

Zienkiewicz 0C, and Taylor, R.L, 1989, The Finite Element Method, 4‘h Ed, pp 479, McGraw-Hill,

UK
Segerlind, L] 1984, Applied Finite Element Analysis, 2"(1 Ed, John Wiley& Sons, New York

Cook, R. D, Malkus, D. S. and Plesha, M. C., 1989, Concepts and Applications of Finite Element

Analysis, p 48 3rd Ed, John Wiley & Sons, New York.

41

 

111311111111111111111111

 

/ <;‘-A—.—_-.- ____