femlab

Transcription

femlab

How to contact COMSOL:
Germany
FEMLAB GmbH
Berliner Str. 4
D-37073 Göttingen
Phone: +49-551-99721-0
Fax: +49-551-99721-29
[email protected]
www.femlab.de
Benelux
COMSOL BV
Röntgenlaan 19
2719 DX Zoetermeer
The Netherlands
[email protected]
www.femlab.nl
Norway
COMSOL AS
Verftsgata 4
NO-7485 Trondheim
Phone: +47 73 84 24 00
Fax: +47 73 84 24 01
[email protected]
www.comsol.no
Denmark
COMSOL A/S
Rosenkæret 11C
DK-2860 Søborg
Phone: +45 39 66 56 50
Fax: +45 39 66 56 20
[email protected]
www.comsol.dk
Sweden
COMSOL AB
Tegnérgatan 23
SE-111 40 Stockholm
Phone: +46 8 412 95 00
Fax: +46 8 412 95 10
[email protected]
www.comsol.se
Finland
COMSOL OY
Lauttasaarentie 52
FIN-00200 Helsinki
Phone: +358 9 2510 400
Fax: +358 9 2510 4010
[email protected]
www.comsol.fi
France
COMSOL France
19 rue des bergers
F-38000 Grenoble
Phone: +33 (0)4 76 46 49 01
Fax: +33 (0)4 76 46 07 42
[email protected]
www.comsol.fr
Switzerland
COMSOL AB
Repräsentationsbüro Schweiz
Technoparkstrasse 1
CH-8005 Zürich
Phone: +41 (0)44 445 2140
Fax: +41 (0)44 445 2141
[email protected]
www.femlab.ch
United Kingdom
COMSOL Ltd.
Studio G8 Shepherds Building
Rockley Road
London W14 0DA
Phone:+44-(0)-20 7348 9000
Fax: +44-(0)-20 7348 9020
[email protected]
www.uk.comsol.com
United States
COMSOL, Inc.
1 New England Executive Park
Suite 350
Burlington, MA 01803
Phone: +1-781-273-3322
Fax: +1-781-273-6603
[email protected]
www.comsol.com
COMSOL, Inc.
1100 Glendon Avenue
17th Floor
Los Angeles, CA 90024
Phone: +1-310-689-7250
Fax: +1-310-689-7527
[email protected]
www.comsol.com
COMSOL, Inc.
744 Cowper Street
Palo Alto, CA 94301
Phone: +1-650-324-9935
Email: [email protected]
For more representatives, see
www.comsol.com/contact
Company home page
www.comsol.com
Product suggestions
[email protected]
FEMLAB user forums
www.comsol.com/support/forums
FEMLAB MEMS Modeling Course
 COPYRIGHT 1994–2005 by COMSOL AB. All rights reserved
Patent pending
The software described in this document is furnished under a license agreement. The software may be used
or copied only under the terms of the license agreement. No part of this manual may be photocopied or
reproduced in any form without prior written consent from COMSOL AB.
FEMLAB is a registered trademark of COMSOL AB.
Other product or brand names are trademarks or registered trademarks of their respective holders.
Version:
3.1, May 2005
FEMLAB 3.1i
C O N T E N T S
Preface
2
Tube in Current
3
Introduction to the Lesson . . . . . . . . . . . . . . . . . . .
3
Key Instructive Elements . . . . . . . . . . . . . . . . . . . .
3
Currents in Aluminum Deposit
20
Model Description . . . . . . . . . . . . . . . . . . . . . . 21
Current Balance and Heat Balances in a Two-dimensional cross section . . 22
Using the Graphical User Interface . . . . . . . . . . . . . . . . 23
Convergence analysis (optional) . . . . . . . . . . . . . . . . . 37
The Three-Dimensional Current and Heat Balances
. . . . . . . . . 39
Creating the 3D Model Geometry . . . . . . . . . . . . . . . . 40
Summary of Equations . . . . . . . . . . . . . . . . . . . . . 47
i|
CONTENTS
CONTENTS
| ii
1
FEMLAB Introductory Course
|
1
Pr ef a ce
Mathematical modeling has become a very important part of the research and
development work in engineering and science. Competitive edge requires speed on the
path between idea and prototype, and mathematical modeling and simulation provides
a valuable shortcut for understanding both qualitative and quantitative aspects of
scientific and engineering design. To your assistance, FEMLAB 3 offers state-of-the art
performance, being built from the foundation with Java3D interface and C/C++
solvers.
This course gives you an introduction to modeling in FEMLAB 3 and takes you
through all the steps of the modeling process, from drawing or geometry import, to
parametric analysis.
The exercises do not require any prior expertise in mathematical modeling or
FEMLAB use to be rewarding.
Enjoy your modeling!
2 |
CHAPTE R 1 : FEM LAB INT RO DU CT OR Y C OU RS E
T ub e i n C u rr en t
Introduction to the Lesson
This is a simple fluid-solid interaction problem intended to introduce the student to
FEMLAB. It is relatively simple to create and solves quickly on most computers. It is
an example of one of the more important classes of multiphysics: fluid-solid interaction
(FSI). It is a simplified case of FSI in that it assumes that the deflection of the solid
does not significantly change the flow - which FEMLAB can also simulate, using more
advanced techniques that shown here.
In this problem we take on the role of research marine biologist, interested in the
deflection and resulting stresses of a soft sponge deflected in a mild ocean current.
Here we study the equilibrium case, but the same setup could be used to study
dynamic deflection in varying currents, or (with a much denser mesh) the vibratory
response of the tube when the current velocity increased to the point that vortices were
forming and shedding on alternating sides of the tube-sponge. Flow-induced
vibrations of this type can be quite destructive in some very practical engineering cases.
Such an analysis requires a much denser mesh and involves longer run times than is
practical in a mini-course environment. Hence we will only do equilibrium.
Key Instructive Elements
The key elements we hope you will learn in this example are
3 |
1 How to create and manipulate simple 3D geometries in FEMLAB
2 Find geometric symmetry planes to reduce the model size.
3 Practice simulating flow and structural analyses
4 How to define materials, set boundary conditions, and link sets of physics together
5 Use of the parametric solver to plot both system and user-created functions vs.
parameter
6 Demonstrating a variety of FEMLAB’s postprocessing capabilities including
animations
PHYSICS
In this problem seawater is flowing over and around a single branch tubular sponge which is deflected by this flow. The two key physics involved are incompressible fluid
flow (for the seawater) and 3-D static structural analysis. We might also represent the
tube with structural shells, but in this case the wall thickness of the tube is relatively
thick compared to the diameter. The flow over the tube creates forces on the surface
due to pressure and shear stress. These forces load the tube structurally and deflects it.
The sponge walls are quite elastic and bend relatively easily in these mild currents. The
problem is assumed to be symmetric down the center plane - eliminating the possibility
of simulating side-to-side vibrations. Finally, the velocity profile of the ocean current
this close to the bottom has been found to typically follow a variety of relations,
depending on the up-current profile of the bottom. We will simulate the problem with
one such profile.
P ROBLEM SET UP
1 Start FEMLAB by double clicking on the FEMLAB icon on your screen. The
FEMLAB Model Navigator will launch.
2 Select 3D as the Space Dimension
3 Then in the list of Physical Models select FEMLAB > Fluid Dynamics > Incompressible
Navier Stokes. To be sure you have the right application mode, the Application mode
TU B E I N C U R R E N T
|
4
name should be ns and the dependent variables are u v w and p. Check that your
screen looks like the image below and then click OK.
The main graphics window of FEMLAB will open. This has the usual form of a series
of pull-down commands along the top and then commonly-used commands arranged
as toolbar buttons arrayed along the top and left side of the window.
GEOMETRY
We will build the geometry assuming SI units but we could use any consistent set of
units in FEMLAB. Since our material properties are in SI, we will set the problem up
in meters. We first create a 2D cross-section of the geometry and later we extrude it to
a 3D geometry:
1 Select Draw>Workplane settings from the choices in the top horizontal command
pull-downs. Leave the default settings, which gives you a 2D work plane at z = 0
and click OK.
5 |
62 Select Draw > Specify Objects > Circle and enter 0.001 in the Radius field. Leave the
other fields at their default values and click OK to create the circle.
3 Select the Zoom Extents button (the one with the Red Cross) in the top horizontal
toolbar.
We will create the hollow center by creating another circle and then subtracting it from
the first one.
4 Select Draw > Specify Objects > Circle and enter 5e-4 in the Radius field. Leave the
other fields at their default values and select OK to create the circle.
5 To subtract the inside from the outside choose Edit > Select All and then the
Difference button on the left vertical toolbar (just above the +-/x button).
Since the problem is symmetric, we can cut the tube (so far the circle before extruding
it to 3D) in half.
6 Select Draw > Specify Objects > Rectangle and enter Width = 0.016 and Height =
0.008. Leave the Base Position to be the Corner and enter x = -0.008 . Click OK.
|
6
7 Select the Zoom Extents button (the one with the Red Cross) in the top horizontal
toolbar.
8 With the rectangle still highlighted, select Edit > Copy (or select Ctrl-c). Then, select
Edit > Select all and click the Intersection tool button on the left vertical toolbar.
You should see half of the hollow circle cut vertically on the symmetry plane.
9 Paste the rectangle back in by selecting Edit > Paste and leave the Displacements to
zero. Last of all we remove the inner semi-circle from the rectangle.
10 Select Draw > Specify Objects > Circle and enter 0.001 in the Radius edit field. Leave
the other fields at their default values and click OK to create the circle.
11 Select Draw > Create Composite Object and enter R1-C1 in the Set Formula edit field.
Click OK.
7 |
12 Select Draw > Extrude and select both CO2 and CO3. Enter 0.01 in the Distance
edit field and click OK.
13 To see the geometry in a shaded mode, select the Headlight button on the left
vertical toolbar.
Your geometry should look like what is shown below. If you had trouble, you can load
a completed geometry at this point. To do so, select File > New and load the file:
Tube_in_Current_GEOM.fl Only do this if your geometry does not look like that
below.
|
8
We have defined the geometry and created introduced the flow physics to the problem.
We next need to define the material properties and boundary conditions for the flow
problem and then introduce and do the same for the structural part of the problem.
M A TE R IA L PRO PE R TIE S - F L OW
We first assign material properties for the flow, under the Physics pull-down commands
9 |
CHAPTER 1: FEMLAB INTRODUCTOR Y COURSE
1 Choose Physics > Subdomain Settings and then select the block in the graphics
window (or select subdomain 1 in the subdomain list) enter 998 for the Density and
1.4e-3 for the Dynamic viscosity. Leave the other fields at their default values.
The flow equations only apply in the flow subdomain, they are not valid in the tubular
structure, thus we need to make them inactive where they are not appropriate. We do
this in the subdomain settings dialog box using the Active in this domain selection box.
2 Select subdomain 2 in the subdomain selection box. The hollow tube in the graphics
window should be highlighted in pink. Clear the Active in this domain checkbox
under the selection list. The number two in the list and all the material property
fields should grey out when you do this. Click OK.
BOUNDARY CONDITIONS - FLOW
We next assign boundary conditions for the flow- again, under the Physics pull-down
commands
1 Choose Physics > Boundary Settings and choose the outflow boundary surface
(number 15 in the boundary list) Change the boundary condition type to Outflow/
Pressure and leave the Pressure at 0.
2 Select the top and side boundaries (boundaries 2, 4, 5, and 14) from the list. To pick
multiple boundaries, hold the Ctrl key down when you select the boundaries.
Change the boundary condition to Slip/Symmetry. Click OK .
3 Select the outer surfaces of the hollow cylinder (boundaries 7 and 12) and set the
Boundary condition to No Slip (The bottom, surface 3 was set this way by default)
Click OK.
|
10
4 Finally, choose the inflow boundary (boundary 1). Here we specify the inflow
velocity. Change the boundary condition type to Inflow/outflow velocity.
Normally we would just enter numerical values for the velocity. In FEMLAB, anywhere
you can enter a number you can also enter an equation. For this problem the velocity
follows a known profile that is a function of the vertical depth - thus a function of z in
our coordinate system. In this case, we will let the velocity vary with the fifth-root of
the depth as Vin(z/0.01)(1/5). Additionally, we would like to change this velocity
parametrically. Thus we will write the profile as a function of z and of a (yet to be
defined) constant V_in. V_in will be the magnitude of the velocity at the top of our
domain, which we will eventually vary from 1 to 5 cm per second.
5 To finish specifying the inlet velocity boundary: Enter V_in*(z/.01)^0.2 in the u0
field and leave the other two components as zero. Click OK and close the window.
6 To define the constant: Select Options > Constants and enter V_in as the Name and
0.03 as the Expression. Click OK. Be sure you get the right number of decimal places
here! Similarly, you might want to double check the boundary expression in the
previous step. If you inadvertently enter too large a velocity, the character of the flow
may change from steady to unsteady and there simply is not a stationary solution for
the solver to find. It, of course, does not know this and continues iterating until it
reaches its maximum iteration limit and gives up.
At this point we could either introduce the structural equations and their
corresponding material properties and boundary conditions or we could check if the
flow part is set up as planned. In multiphysics problems it is always wise to check them
on each major step. This makes debugging the problem later easier if you made
mistakes along the way. Lets be prudent and check what we have thus far.
11 |
M ESH AND SOLVE FOR FLOW
FEMLAB has a very sophisticated mesh algorithm which meshes the problem based
on the geometry you have defined. Because this is a class problem and not a true
research problem we would like the problem to be as small as possible, so that each
solution will converge quickly and so that we do not inadvertently define a problem
that is too large for classroom computers.
Thus we will first change the default mesh parameters that FEMLAB uses to mesh the
problem.
1 Select Mesh > Mesh Parameters and change the Predefined mesh sizes pull-down field
at the top of the window to Extra Coarse. Click Remesh and OK. You should get
around 2500 elements.
FEMLAB also sets up the solver parameters based on the problem you have defined.
This is a 3D flow problem, which in general, could be quite large. Thus FEMLAB
selects the solver expecting large system matrices and large memory demands. For this
particular geometry, the problem is quite small and we can save ourselves quite a bit of
time if we select a direct solver instead of the predefined iterative one.
1 Select Solve > Solver Parameters and change the Linear System Solver (in the upper
left side of the menu) to Direct (UMFPACK) . Leave everything else at its default. Click
OK.
2 Select Solve > Solve Problem . After a short time you should see the figure on the
right: (You can toggle off the headlight to remove shading and thus brighten the
image)
|
12
1 Select Postprocessing > Plot Parameters and pick the Slice tab at the top. In the Slice
subwindow change the x-levels to 3, the y-levels to 0, and the z-levels to 1. Click
Apply.
2 While still in the postprocessing window, select the Arrow tab at the top. In the
Arrow subwindow, set the x-points to 15, the y-points to 10 and the z-points to 7.
Change to Arrow type to 3D arrow. Clear the Auto scale factor checkbox and enter the
Scale factor to be 1.3. Finally, select the Arrow plot checkbox in the upper left corner
of the window. (Don’t click Apply yet!)
Before you add the 3D arrows to the result plot, you may want to save your model.
Sometimes the graphics cards on seminar computers do not have much memory in
them. Arrow plots, particularly 3D arrow plots, call for extra memory on this card. If
the graphics card has a small memory, the computer sometimes can crash and your
model (if not previously saved) will be gone! If this is a new computer to you - I
suggest you save your model now (File > Save).
3 Select Apply to add the arrows to your plot.
4 Select the Streamline tab and change the Number of streamlines to 40. Select the
Streamline checkbox in the upper left corner. In the Streamline color box, select the
Use expression to color streamlines radio button and then select the Color Expression
button. Leave the Expression as U_ns but change the Colormap to cool. Clear the
Colorscale checkbox and click OK to close the color expression window. Then click OK
to close the Plot parameters window.
5 To turn off the coordinate system grid on the screen, select the Options > Axis/Grid
Settings and clear the Visible checkbox in the Axis window. Click OK.
You should see the plot below. You can toggle between orientations by selecting the
XY-View button and the Default View button (both located on the left vertical toolbar).
13 |
We would like to see the fluid-loading on the post. FEMLAB calculates this for us for
all surfaces in the model. Graphically we only want to see the post loading. To set this
up we can define a variable for the surface loading that only is defined on the post.
1 Pick Options > Expressions > Boundary Expressions and select the outer surface of the
post (boundaries 7 and 12). Enter P_Post as the Name and T_x_ns for the
Expression. Click OK.
2 Select Solve > Update Model to evaluate this variable without recalculating the
solution.
3 Select Postprocessing > Plot Parameters and pick the Boundary tab . Enter -P_Post
in the Expression field (remember the negative!) and select the Boundary checkbox.
Click the General tab and clear both the Arrow plot and Slice plot types. Click OK.
Reorient the view to see the x-component of the drag per unit area on the post.
SETUP STRUCTURAL PROBLEM
1 Select Multiphysics > Model Navigator. In the list of Physical Models select FEMLAB >
Structural Mechanics > Solid, Stress-Strain > Static Analysis. To be sure you have the
right application mode, the Application mode name should be solid3 and the
|
14
Dependent variables are u2 v2, and w2. Select Add (at the top right side), then click
OK to close the window.
2 Select Physics > Subdomain Settings and choose subdomain 1 (the flow volume). The
structural equations are not applied in this subdomain and thus need to be
inactivated. Clear the Active in this domain checkbox to turn them off. Click Apply.
3 Select subdomain 2 (the hollow cylinder) to define it’s material properties. Enter
2e3 for Young’s Modulus, 0.3 for the Poisson’s ratio and 1004 for the Density. Click
OK .
4 Select Physics > Boundary Settings and enter the boundary conditions as follows
(Notice the negative signs on the forces!) Once done, click OK.
BOUNDARIES
7, 12
8
6, 13
LOCATION
BOUNDARY CONDITION
VARIABLE
EXPRESSION
Click Load Tab
Fx
-T_x_ns
Fy
-T_y_ns
Fz
-T_z_ns
Rx
0
Ry
0
Rz
0
Ry
0
Constraints: Select all three: Rx, Ry, Rz
Constraints: Select only Ry
The three surface stresses, T_x_ns, T_y_ns and T_z_ns are application variables that
are internally calculated by the Navier Stokes equation mode. They are the total fluid
force per unit area on a given surface pushing back on the fluid. Using them as surface
15 |
force boundary conditions in the structural mechanics equations links the flow to the
deformation. Again, remember to set the surface force as the negative of the fluid
surface stress!
SOLVE FLOW-IND UCED D EFOR MAT ION P ROBLEM
The coupled problem is now completely set up. In this case the two physics (flow and
structures) is only linked one way: The deformation depends on the fluid drag force
and thus the flow distribution, but the flow (in our model) does not depend on the
deformation. This is an example of one-way coupling between the equations. We could
reduce the problem size and solve first for flow and then use that solution to solve a
second structural deformation solution.
By default, FEMLAB solves everything simultaneously - as if all physics depend on one
another. FEMLAB has a solver manager to enable sequential solution of some of the
equations in the system. The solver manager is used to manage overall problem size
and allow sequential solutions of loosely coupled physics. Again, we will simply solve
everything simultaneously here, as if there was bidirectional coupling.
As an aside, this problem could be set up with bidirectional coupling. If the
deformation is large the flow clearly depends on the deformation. We can do this in
FEMLAB but it involves an advanced technique of setting up and using a governable
mesh in the flow subdomain that moves with the bending of the hollow tube. Such a
problem has three sets of “physics” that are solved simultaneously: The flow, the
structural deformation, and the deformation of the mesh.
1 Click the Solve button (the one with the equals sign) on the top horizontal toolbar.
2 Select Postprocessing > Plot Parameters and select the Arrow tab. Change the Plot
Arrows on selection list (top right) to Boundaries. Change the Predefined quantities
to Global Force (solid3). This is the total face load on the cylindrical structure. Select
the Arrow plot checkbox in the upper left corner. Click Apply.
3 Click the Slice tab and change the x-slices from 3 to 0. Select the Slice plot checkbox
in the upper left corner. Click Apply
4 Select the Boundary tab and change the Predefined quantities to Total displacement
(solid3) and change the Colormap to Pink. Select the Boundary plot checkbox in the
upper left corner. Click Apply
5 Finally, select the Deform tab. Clear the Subdomain checkbox under Domain types to
deform. Click the Boundary tab in the Deformation Data area and change the
Predefined quantities to Displacement (solid3). Clear the Auto scale checkbox and
|
16
change the scale factor to 1. Finally, select the Deformed shape plot checkbox in the
upper left and click OK.
PA RA M E T RIC S O L VER - EF F E CT OF VA R YIN G F L OW VE L O C ITY
1 We first define a new variable which is the integral of the x-component of fluid
pressure on the cylinder. Select Options > Integration Coupling Variables > Boundary
Integration and select boundaries 7 and 12 (the outer surface of the cylinder). Enter
Drag as the Name and enter -T_x_ns as the Expression. (Remember the negative
sign!) Click OK.
2 Next, we set up the solver to automatically sweep through a series of parametric
values. Go to Solver > Solver Parameters and select Parametric nonlinear in the Solver
17 |
list. Type V_in as the Name of parameter and 0.01:0.01:0.05 in the Parameter list
field. Click OK.
3 Select the Solve button (the one with the equals sign) on the top horizontal toolbar.
The solver solves sequentially for V_in = 0.01, 0.02, 0.03, 0.04 and 0.05 m/sec.
Depending on your computer this can take 5-6 minutes. Once solved, we can for
example plot parametric results of maximum displacement or our integration-defined
variable of Drag vs. inlet velocity. Finally we can animate the flow and deformation vs.
inlet velocity.
1 Select Postprocessing > Domain Plot Parameters. Select the Point Plot radio button
and then the Point tab. Enter Drag as the Expression and select any point from the
list. Click OK. Close this figure by selecting its red “x”in the upper right corner of
the figure window. Be careful not to close the FEMLAB main window!
2 Again select Postprocessing > Domain Plot Parameters. Pick Total displacement
(solid3) from the Predefined quantities list and select point 6 (The top front point on
the cylinder) from the list. Click OK. Again close this new figure with it’s red “x”.
|
18
3 Finally select Postprocessing > Plot Parameters and select the Animate tab. Change
the frames per second field to 1 and the Number of frames to 50. Leave everything
else at its default value. (Don’t click the start animation button until you read the
following!)
Before you click the animate button, you may want to save your model. Sometimes the
graphics cards on seminar computers do not have much memory in them. Creating
animations call for extra memory on this card. If the graphics card has a small memory,
the computer sometimes can crash and your model (if not previously saved) will be
gone! If this is a new computer to you - I suggest you save your model now (File >
Save)
Select the Start Animation button on the lower right corner of the screen. Once the
animation is complete, slide the slider bar back and forth at the base of the animation
window and/or reset and replay the animation in this window. Click OK to close.
19 |
C u r r e n t s i n A l u m i n um D e p os i t
This model treats current conduction and heat generation in an aluminum film
deposited on a silicon substrate. The purpose of the model is to investigate the
temperature in the device after the current is applied. The model was submitted by
M.S. Cohen at Aegis Semiconductors.
The model exemplifies several steps that can be important in the modeling process. It
also gives a quick review of the features available for advanced modeling in FEMLAB’s
graphical user interface. We will address the following important FEMLAB features:
• The use of predefined physics.
• The definition of a multiphysics problem.
• The expression feature to define physical properties that depend on the solution
itself.
• The extraction of design numbers using the postprocessing tools.
• The parametric solver feature that provides fast and efficient parameter screening as
well as smooth convergence for highly nonlinear problems.
In addition to the FEMLAB features, several modeling methods are exemplified in the
exercise, among them:
• Reduction of the dimension of parts of the problem from three to two dimensions.
• Scaling of the mesh due to large differences in the dimensions of the geometry.
• Control of the obtained results to estimate the validity of the solution.
Apart from theses highlights, we will go through the typical modeling steps, which
include:
1 Definition of the geometry.
2 Definition of the physics in the volume and at the boundaries.
3 Meshing.
4 Solving.
5 Postprocessing.
6 Parametric studies.
CURRENTS IN ALUMINUM DEPOSIT
|
20
Model Description
Deposit
Tair=293 K
500 mA
Substrate
A film of aluminum is deposited on a silicon substrate. The aluminum is about 4 µm
thick, with the pattern shown above. The width of the device is about 1000 µm, with
a hole in the center about 250 µm in diameter. A constant current is passed through
the deposit starting at time t = 0.
We wish to study the temperature distribution in the aluminum pattern as a function
of time. We also want to see the electrical power distribution as well as the total
resistance as a function of time.
The problem couples two physical phenomena, a current balance and a heat balance.
The electrical resistance increases with temperature while the current density itself
serves as a heat source through ohmic losses in the current conduction. This implies
that the coupling between the two physics is a two-way coupling.
We will divide the study into two parts. In the first part, we will look at the current and
heat balances in a two-dimensional approximation of the real geometry. This will give
us an estimation of the time scale of the problem and of the magnitude of the studied
fields, potential and temperature.
In a second and more advanced step, we will introduce the real three dimensional
geometry. We will study the current balance first, to check this against the
two-dimensional approximation and we will then add the heat balance.
21 |
Current Balance and Heat Balances in a Two-dimensional cross
section
Let us first look at the current distribution. Current is only conducted in the
aluminum, since the silicon substrate is an electrical insulator. Assuming isothermal
conditions, the current density along the thickness should be perfectly uniform, since
no variations are introduced by the geometry, current conduction, or boundary
conditions. This means that we can reduce the problem down to two dimensions to
study the current balance. The current passes through the device from right to left, in
the figure above, and all the boundaries except the left and right boundary are assumed
to be insulated. We will study the current density distribution, and the influence of
temperature on the total ohmic losses in the device.
The heat balance in the device is clearly a three dimensional phenomenon. However,
we can get an estimate of the production of heat in the device by performing a
two-dimensional analysis. We can introduce the convective cooling perpendicular to
the surface as a heat sink, in order to investigate the convective cooling’s influence on
the time scale of the phenomena. This will give us a good start as we take the next step
to three-dimensional modeling.
Below follows a detailed description of the modeling procedure in this exercise. The
step by step instructions will guide us through the current balance at steady state and
the introduction of the multiphysics content in the time-dependent heat transfer
problem.
|
22
Using the Graphical User Interface
1 Start FEMLAB.
2 In the Model Navigator, set Space dimension to 2D and then select
FEMLAB>Electromagnetics>Conductive Media DC in the list of available application
modes.
Note: Before proceeding, make sure that you have not selected the Electromagnetics
Module branch in the model navigator tree. It should be the FEMLAB branch like
described in item 2 above.
3 Click OK.
GEOMETRY MODELING
The second step in the modeling process is to create or import the model geometry.
FEMLAB is able to read and repair DXF and IGES files from other solid modeling
packages. However, the geometry of the aluminum deposit is very easily created in
FEMLAB’s built-in solid modeling tool.
1 Select Axes/Grid Settings in the Options menu.
2 Clear the Axis equal checkbox.
23 |
3 Type -5.5e-5 in the x min and 1.05e-3 in the x max edit field,
Type -3.5e-4 in the y min and 3.5e-4 in the y max edit field.
4 Click on the Grid tab.
5 Clear the Auto checkbox.
6 Type 1e-4 in the x spacing and y spacing edit fields.
7 Insert extra grid lines by typing -2.5e-5 2.5e-5 5e-5 1.25e-4 1.75e-4 in the
Extra y edit field and click OK.
This gives a proper grid spacing for the drawing of the device. You can now start
drawing the geometry. The geometry can be created using solid primitive objects or
line and curve tools. We will use the line tool to create the main part of the device.
8 Click the Line tool button.
Note: In the following process, you can see the location of the pointer
in the bottom left corner of the FEMLAB window.
9 Click on the coordinates tabulated below to create the thick part of the device.
X COORDINATES
Y COORDINATES
2e-4
0
2e-4
0.5e-4
4e-4
1.75e-4
|
24
X COORDINATES
Y COORDINATES
6e-4
1.75e-4
8e-4
0.5e-4
8e-4
0
10 Click the right mouse button to create a solid object denoted CO1, composite object
1.
11 Click the Mirror button and specify the reflection line as shown below. Click OK.
25 |
You have created the geometry objects CO1 and CO2 and can continue with FEMLAB’s
Create Composite Object button to unify these objects.
12 Select CO1 and CO2 by using the key combination Ctrl+A and click the Create
Composite Object button.
13 Clear the Keep interior boundaries box and click OK.
We will now draw the contact strips to the main body of the device.
14 Click the Rectangle/Square button (top button on the vertical toolbar).
15 Hold down the left mouse button and drag the pointer between the coordinates (0,
-0.25e-4) and (1e-3, 0.25e-4) to create a rectangle.
16 Select the rectangle R1 and the composite CO1 (Ctrl+A) and unify them by using the
Create Composite Object tool. The Keep interior boundaries box should be cleared.
17 Click OK.
To complete the geometry, create the circle in the middle and drill the hole in the
structure.
18 Click the Ellipse/Circle (Centered) button.
19 Using the right mouse button, create a circle by clicking in the center of the
geometry (5e-4,0) and dragging upwards until the top of the circle snaps to
y=1.25⋅10-4.
|
26
20 Select all the objects by using the key combination Ctrl+A and click the Difference
button (seventh button from the bottom on the vertical toolbar).
21 The geometry of the aluminum deposit is completed. This is a good time to go to
File, Save As to save your work.
OPTIONS - CONSTANTS
1 Select Constants in the Options menu.
2 Define the constants tabulated below in the corresponding edit fields. sig is a
constant that you will use later on to define the electrical conductivity of the
material.
NAME
EXPRESSION
DESCRIPTION
iin
0.5
Current (A)
thick
4e-6
Thickness (m)
side
50e-6
Width (m)
sig
3.54e7
Conductivity (Ω-1 m-1 )
3 Click OK.
27 |
BOUNDARY SETTINGS
1 Click the Boundary Mode button on the main toolbar.
2 Select all boundaries by using the key combination Ctrl+A.
3 Select Boundary Settings in the Physics menu.
4 Set Electric insulation conditions at all boundaries. Click OK.
5 Double click the rightmost side boundary, boundary 16, and select the Inward
current flow condition from the drop-down list. Type the expression
iin/thick/side in the Jn edit field.
6 Select the left hand side boundary, number 1, and set the Ground boundary
condition.
7 Click OK.
This defines the boundary conditions for the current balance. Proceed to the
subdomain mode.
SUBDOMAIN SETTINGS
1 Click the Subdomain Mode button in the main toolbar.
2 Double-click on the geometry or open Subdomain Settings from the Physics menu
and select 1 from the Subdomain Selection list.
3 Type sig in the σ (isotropic) Conductivity edit field.
4 Leave the other fields at the default values (0).
5 Click OK.
The constant sig corresponds to the conductivity, in Ω-1 m-1, at 293 K.
MESH
|
28
Click the Mesh Mode button in the main toolbar to create the mesh.
COMPUTING THE SOLUTIONS
Solve the problem by clicking the Solve button (=) in the main toolbar.
POSTPROCESSING
The graphical user interface automatically selects the Postprocessing Mode once the
problem is solved. The default plot shows the electric potential in the aluminum
deposit.
Postprocessing mode
quick buttons
You can toggle between a number of visualization options by clicking the quick
buttons in the Post Mode.
1 Click the Plot Parameters button in the main toolbar.
2 Clear the Surface checkbox and check the Contour checkbox.
3 Select the Contour tab.
29 |
4 Type 31 in the Contour levels, Number of levels edit field. This will render 31 contour
lines with constant potential increment.
5 Click OK.
Note that the iso-potential lines above and below the hole are vertical, see the figure
below. This line represents an antisymmetry plane, meaning that the potential field to
the left of this line is a mirror of that to the right of the line, with opposed sign. True
symmetry is present between the upper and lower part of the geometry. The presence
of symmetry and antisymmetry allows for the modeling of only one fourth of the
device. You will use this knowledge in the three-dimensional modeling exercise later
in this course.
Antisymmetry plane
Symmetry plane
The next step introduces the concept of multiphysics. FEMLAB allows for arbitrary
coupling of several physics phenomena. In this case, you will add the heat balance to
the existing current balance.
|
30
M O DEL N AVI G AT O R
1 Select Model Navigator from the Multiphysics menu.
2 Select FEMLAB>Heat Transfer>Conduction.
3 Click the Add button.
4 Click OK.
Define a few constants, in the temporary data base, for later use.
5 From the Options menu, select Constants.
6 Add to the existing list the constants below.
CONSTANT
EXPRESSION
DESCRIPTION
ro1
2.7e3
Density (kg m -3)
cp1
900
Heat capacity (J kg-1 K-1)
k1
240
Thermal conductivity (W m-1 K-1)
h1
5
Heat transfer coefficient (W m-2 K-1)
Tinf
293
Reference temperature (K)
7 Click OK.
In addition to constants, you can define expressions that are functions of the
dependent variables in the model, in this case voltage, V, and temperature T. You can
also make expressions of the independent variables x and y, as well as other geometric
variables. In the following secti on we will define the expressions described in
“Summary of Equations” on page 31.
31 |
1 Select Expressions/Scalar Expressions in the Options menu.
2 Define the following expression for the resistivity (Ω m).
NAME
EXPRESSION
res
2.824e-8*(1+3.9e-5*(T-Tinf))
Make sure that you do not leave spaces when you enter the expression. FEMLAB
interprets spaces as separation between the components of a vector.
3 Click OK.
BOUNDARY SETTINGS
1 Click the Boundary Mode button.
2 Select all boundaries by pressing Ctrl+A.
4 Select Heat flux from the Boundary condition drop down list.
5 Type h1 in the Heat transfer coefficient edit field.
6 Type Tinf in the External temperature edit field.
7 In the boundary selection list, make sure that only boundaries 1 and 16 are
highlighted. You can do this by holding down the Ctrl key and pointing directly on
the boundaries in the drawing of the device.
8 Select Temperature from the Boundary condition drop down list and type Tinf in the
Temperature edit field.
9 Click OK.
You will now continue by defining the domain physics for the heat transfer mode.
|
32
SUBDOMAIN SETTINGS
1 Click the Subdomain Mode button.
2 Double-click on the domain drawing.
3 Define the Subdomain Settings according to the table below.
DESCRIPTION
VALUE
Thermal conductivity
k1
Density
ro1
Heat capacity
cp1
Heat source
Q_dc
Convective heat transfer coefficient
h1/thick
External temperature
Tinf
Q_dc is called a postprocessing variable and is predefined in the conductive media
application mode as the resistive heating (W m-3). You can check the definition of
Q_dc by selecting the menu Physics/Equation System/Subdomain Settings, Variables
tab. This way you can verify that it represents σ|∇V|2, see “Summary of Equations”
on page 31.
4 Click the Init tab and type Tinf in the Initial value edit field.
5 Click OK.
6 Select Conductive Media DC (dc) in the Multiphysics menu.
33 |
7 Select Subdomain Settings in the Physics menu.
8 Replace sig with 1/res in the Conductivity edit field.
9 Click OK.
Note that the heat transfer through convection on the surface of the aluminum deposit
is introduced as a source or sink. Since symmetry along the thickness of the device
implies that the model is defined per unit length perpendicular to the paper, the heat
transfer coefficient has to be divided by the thickness of the deposit. This gives the
correct value of the cooling per unit volume of aluminum. The coupling between the
heat balance and the current balance is here present in the expression for the resistivity,
denoted res, and the source term, which depends on the gradients of the potential,
terms denoted Vx and Vy.
In order to make a two-way coupling between the heat and current balances, you have
to introduce the expression for the resistivity, res, in the conductive media application.
Heat transfer is a phenomenon that takes place in a much greater time scale than
current transfer. In comparing the two phenomena, current density can be assumed to
respond immediately to a change in potential. You can therefore assume that the
current balance is always a steady-state problem, while the heat balance remains a
transient problem.
COMPUTING THE SOLUTION
1 Select Solver Parameters in the Solve menu.
2 Select Solver: Time dependent in the left list on the General tab.
|
34
3 Type 0:0.001:0.03 in the Times edit field.
4 Select Solution form: General and click OK.
5 Click the Solve button.
The vector expression for the output times gives the solution from 0 to 0.03 seconds
with 0.001 second increments. The time stepping algorithm has built-in step length
control and the output time only defines the steps that should be saved for post
processing.
You have already solved the potential field at 293 K, which corresponds to the initial
condition on the temperature. If we now assign this calculated potential field as the
initial condition for the time dependent problem, there is perfect consistency in the
initial conditions, which gives an excellent start for the time dependent solver. Well
posed initial conditions are very important in coupled systems of stationary and time
dependent problems (DAE-systems). FEMLAB provides a fully automatic procedure
to get consistent initial conditions for a DAE systems. This is accomplished by setting
Consistent initialization of DAE systems: On on the Timestepping page of the Solver
Parameters dialog box. This will generate perturbed initial solution (t=0) of the
stationary equation (potential field), but the problem will be well-posed.
35 |
POSTP ROCESSING
The solution shows the potential distribution in the aluminum deposit for the resulting
temperature distribution. You can plot the conductivity as a function of temperature
in the geometry:
1 Click the Plot Parameters button in the main toolbar.
2 Click the General tab.
3 Clear the Contour dialog box and check the Surface dialog box.
4 Select the Surface tab.
5 Type 1/res in the Expression edit field. As you may notice, you can type arbitrary
expressions to plot, in addition to the handy list of Predefined quantities.
6 Click OK.
You can study the development of the temperature profile by plotting the temperature
as a function of time in the hottest part of the device.
7 Select the menu Postprocessing/Domain Plot Parameters and click the Point plot radio
button.
8 Go to the Point tab and Select Temperature (ht) in the Predefined quantities drop
down list.
|
36
9 Select point 11 from the Point selection list and Click OK .
Figure 1-1: Time evolution of the temperature at a point.
The resulting plot shows that the problem has reached steady-state after 0.02 seconds.
You can continue visualizing the current density, potential, and current density
distribution by combining cross sectional, surface, contours, and flow plots.
10 You can save that data in a text file by clicking the ASCII button in the separate figure
window.
More advanced data export is possible to do in the main GUI as well as by selecting
the File/Export/Postprocessing Data menu item.
Convergence analysis (optional)
To check the numerical accuracy of the solutions, we need to examine the sensitivity
of the values to the mesh density. We will look at a key number in the solution at the
37 |
end time and compare them at different mesh densities. The integral mean
temperature can be defined as
〈 T〉
= ∫ T dA ⁄ A
and we will look at this quantity during the convergence analysis.
1 Change to a stationary problem by selecting the Solve/Solver Parameters menu and
select Solver to Stationary nonlinear. Click OK to close the dialog box.
2 Create two scalar coupling variables by selecting the menu Options/Integration
Coupling Variables/Subdomain Variables. Select subdomain 1 and define the list
according to the table below. Click OK.
NAME
EXPRESSION
INTEGRATION ORDER
DESTINATION
Tint
T
4
global
A1
1
4
global
3 Solve the problem by clicking the = button.
4 To evaluate the mean temperature, click the Plot Parameters button and enter Tint/
A1 in the expression field of the Surface tab. Click OK. You can see the numerical
value in the status field by clicking inside the domain on the drawing board.
5 Repeat the above point for different mesh sizes. For each mesh size in the table
below, open the menu item Mesh/Mesh Parameters. Go to the Subdomain tab and
enter the mesh size in the Maximum element size dialog box. Click OK and the
Initialize Mesh button to see the new mesh. Solve and plot. Fill out the mean
temperature values in the table.
MAXIMUM MESH ELEMENT SIZE
<T>
3e-5
326.138
1.5e-5
326.219
0.75e-5
326.254
You will see that the mean value of the temperature will eventually converge toward a
certain value. When you are satisfied with the relative deviation between two
refinements, you have a mesh that is sufficiently fine. Bear in mind that there might be
other quantities you will want to check, for example a total flux or heat balance.
|
38
The Three-Dimensional Current and Heat Balances
The real device is actually three-dimensional where the heating introduces
three-dimensional effects. Current heats the aluminum where heat is then dissipated
to the adjacent air and to the silicon substrate. For this reason, there will be a
temperature gradient along the thickness of the aluminum plate, which impedes us
from reducing the problem to two space dimensions. In this study, we will make use
of the symmetry and antisymmetry in the problem to reduce the geometry to one
fourth of the original description. We will only solve the stationary problem.
The boundary conditions and the expression for the conductivity are identical in the
2D and 3D problems for current conduction. This implies electrical insulation
everywhere except for the edges on the right and left.
The heat balance is a little more difficult to define. Temperature is fixed on the right
and left edges of the device. On the base of the silicon substrate and at the vertical
boundaries of the substrate, we assume thermal insulation. At all other surfaces, we
define convective heat dissipation using the film theory, where tabulated values for the
heat transfer coefficient are used. At the boundary between the aluminum film and the
substrate, FEMLAB automatically gives the continuity in temperature and heat flux.
The next step is to define the heat balance in the device. In the silicon substrate, only
conduction and the accumulation of heat takes place. This is also the case for the
aluminum film, where heat generation is also introduced by the conduction of current,
which is proportional to the square of the current density.
39 |
The thickness and length of the device geometry differ by several orders of magnitude.
This implies that a large number of elements would be required if we treated the
problem without any type of scaling, since the smallest length would determine the
edge size of the elements. Therefore, we will use an anisotropic mesh by scaling the
geometry in the thickness direction prior to meshing. FEMLAB will automatically
scale it back for viewing.
Creating the 3D Model Geometry
Save the 2D model in a file such as, intro2d.fl. Now, do Save as...intro3d.fl.
Let us start by creating the three dimensional geometry by using the existing geometry
as starting point.
1 Remove the left and the lower half of the geometry by drawing suitable rectangles
and subtracting them from the geometry by using the Create Composite Object
button. Start by clicking the Draw Mode button. To draw the rectangles, press Ctrl+A
and click the Create Composite Object to perform the suitable operations.
2 Click the Zoom Extents button.
The cross-sectional geometry is now one fourth of the original. Create the rectangular
cross-section of the substrate.
3 Click the Rectangle/Square button.
|
40
4 Draw a rectangle by snapping to the upper left vertex of the existing geometry and
drag it until the rectangle snaps to the lower right vertex of the deposit
cross-section.
Extrude this to make it three-dimensional.
5 Select Extrude from the Draw menu.
6 Select Objects to extrude: R1. Type -4e-6 in the Distance edit field to create the
three-dimensional substrate.
7 Click OK and then click the Zoom Extents button. Click the Headlight button on the
left toolbar.
8 Go back to the Geom1 tab and select Extrude from the Draw menu.
41 |
9 Select Objects to extrude: CO1. Type 3e-6 in the Distance edit field, click OK.
OPTIONS
1 Select Constants from the Options menu and add additional input data to the existing
table. See the table below:
CONSTANT NAME
EXPRESSION
L
0.05
cp2
778
k2
84
ro2
2330
2 Click OK.
SUBDOMAIN SETTINGS
1 Select the Model Navigator item from the Multiphysics menu. Highlight Geom2 in the
right pane. Select FEMLAB>Electromagnetics>Conductive Media DC in the left pane and
click Add.
2 Select FEMLAB>Heat Transfer>Conduction and click Add. Click OK.
3 Select Geom2: Conductive Media DC (dc2) in the Multiphysics Menu.
4 Select the menu Options>Constants and remove the entry sig and its expression
in the constant list. Click OK. The empty row should now have disappeared.
|
42
5 Select Expressions/Scalar Expressions from the Options menu and add the following
expressions.
VARIABLES
DEFINITION
res
2.824e-8*(1+3.9e-5*(T2-293))
sig
1/res
6 Click OK.
You can now define the current balance.
7 Select Multiphysics/ 3 Geom2: Conductive Media DC (dc2). Select Subdomain Settings in
the Physics menu.
8 Select subdomain 1, corresponding to the substrate, and clear the Active in this
domain checkbox.
This deactivates the current balance in the substrate, since the substrate does not
conduct current.
9 Select subdomain 2 and type sig in σ (isotropic) Conductivity edit field. Use the
default 0 in the Current source and External current density edit fields.
10 Click OK.
Toggle to the Heat Transfer by Conduction (ht2) application mode to define the
subdomain settings.
11 Select Geom2: Heat Transfer by Conduction (ht2) in the Multiphysics menu.
12 Select subdomain 1 and define the Subdomain Settings according to following table.
DESCRIPTION
VALUE
Thermal conductivity (isotropic)
k2
Density
ro2
Heat capacity
cp2
Heat source
0
13 Under the Init tab, set the initial condition to Tinf.
14 Select subdomain 2 and define the Subdomain Settings according to the table below:
43 |
DESCRIPTION
VALUE
Thermal conductivity (isotropic)
k1
Density
ro1
DESCRIPTION
VALUE
Heat capacity
cp1
Heat source
Q_dc2
15 Under the Init tab, set the initial condition to Tinf.
16 Click OK.
BOUNDARY SETTINGS
Note: You can experiment with the viewing of the thin structure by double clicking
the EQUAL field in the bottom status bar, and then clicking the Zoom Extents button.
This disables the true aspect ration view of the geometry, and can ease up the
selection of boundaries sometimes. We will not do it below, though.
Boundary 5
Boundary 17
1 Select Geom2: Conductive Media DC (dc2) in the Multiphysics Menu.
3 Press Ctrl+A to select all boundaries.
4 Set Electric insulation for all boundaries. Click Apply.
5 Select boundary 17 and click the Inward current flow condition.
6 Type iin/side/thick in the Normal current density edit field.
7 Select boundary 5 and click the Ground condition. Click OK.
|
44
8 Select Geom2: Heat Transfer by Conduction (ht2) in the Multiphysics Menu.
10 Click Ctrl+A to select all boundaries.
11 Set Thermal insulation for all boundaries.
12 Select boundaries 4, 6, 8, 10, 11, 12, 14, and 15. You can select the boundaries by
clicking once on a boundary to color it red. Then right-click to lock the selection
(the boundary turns blue). Then select the next boundary.
13 Set Heat flux for the boundaries and type h1 in the Heat transfer coefficient edit field
and Tinf in the External temperature edit field.
14 Select boundary 17 (far end of coating).
15 Select the Temperature condition and type Tinf in the corresponding edit field.
16 Click OK.
CREATING THE MESH
1 Open the Mesh Parameters window from the Mesh menu.
2 Click the Advanced tab.
3 Set z-direction scale factor to 7. Let the scale factors in the x and y directions remain
default 1.
4 Press the Remesh button. Close the window by pressing OK.
COMPUTING THE SOLUTION
1 Click the Solver Manager button.
45 |
2 Go to the Solve For tab and select only the Geom2 equations (Ctrl- click). See figure
below. Click OK.
3 Click the Solver Parameters button. On the General tab, select the Solver: Stationary
nonlinear, as well as the Solution form: General option. Click OK.
TIME DEPENDENT SIMULATION
5 Click the Solver Parameters button. On the General tab, select the Solver: Time
dependent. Click OK.
POST PROCESSING
1 Click the Plot Parameters button.
2 Check the Isosurface and Boundary and Geometry edges checkboxes and select the
Isosurface tab.
3 Select Temperature (ht2) from the Predefined quantities list. Enter 20 in the Number
of levels field. Finally, select Colormap: jet.
4 Go to the Boundary tab and select Temperature (ht2) as the Predefined quantity. Select
Colormap: gray. Click OK.
|
46
5 Select menu Options/Suppress/Suppress Boundaries. Ctrl-click boundaries 4, 8 and 12.
Click OK.
6 Click the Postprocessing Mode button.
7 Click the Headlight button to switch of the lighting.
Continue experimenting with the postprocessing variables to study he distribution of
the source, conductivity, current distribution with time, etc. Work with the Domain
plot parameters and cross section plots.
Summary of Equations
The equations in this example would be very useful in completely defining our
problem mathematically. But by using FEMLAB they are not required in our exercise
and they are definitely not required to understand the physics in our problem. The
mathematical equations express the problem in a precise and compact way. It would be
tedious to have to explain the equations in words but in the mathematical language
they can be written in a few lines.
The current balance in the aluminum deposit is given by:
47 |
∇ ⋅ ( – σ ∇V ) = 0
where σ is a function of temperature according to:
1
σ = ------------------------------------------a(1 + b( T – T0 ) )
The boundary conditions for the current balance are insulating conditions except for
the right and left boundaries of the aluminum deposit. The insulating boundaries are
expressed by:
( – σ ∇V ) ⋅ n = 0
At the inlet of the current, the boundary condition reads:
( – σ ∇V ) ⋅ n = j 0
while the outlet is defined by:
V = V0
The heat transfer equations are also based on flux balances but this time the time
dependence is introduced:
∂T
ρC p ------- + ∇ ⋅ ( – k ∇T ) = σ ∇V
∂t
2
In the silicon substrate, the following equation is valid:
∂T
ρC p ------- + ∇ ⋅ ( – k ∇T ) = 0
∂t
The convective boundary conditions are given by the expression below for both
aluminum and substrate:
( – k ∇T ) ⋅ n = h ( T – T0 )
The insulating conditions are obtained through
( – k ∇T ) ⋅ n = 0
and the temperature conditions are set by
T = T0
|
48
The thermal conductivity, k, density, ρ, and heat capacity, Cp, have different values in
the deposit and the substrate.
49 |

femlab

Transcription

Similar documents

Femlab tutorial

Page 1 Proving a triangle is a right triangle Method 1: Show two

Quinsam Coal EXISTING CITY BOUNDARY PROPOSED

Report 114-2016 Changes to Oakham North West and Barleythorpe

Wire Diagram

Zonare instructions - University of Florida

Untitled - Brookstone

Flatbed

Online Academic Record Change

PBTV Instructions.indd - Paul Bunyan Communications