Week 9
We finished talking about Laplace’s Equation and started talking about the Heat Equation.
In VBA we looked at finite difference methods for Laplace’s Equation. This completes the examinable VBA material. The Heat Equation that we cover in Week 10 will not be examinable.
Week 10
We will look at finite differences for the Heat Equation. This completes the examinable written material.
In VBA we will implement same.
Week 11 — 2nd 20% VBA Assessment
In the 09:00 class we will have a revision session, geared towards the 20% VBA Assessment 2.
In the 12:00 class we will have a revision session, geared towards the 40% Written Assessment 2.
Formulae will be provided in the VBA 2 Assessment.
To understand how your student numbers generate constants (see below) see this VBA Test 2 from last year (do not read this as a sample – it included e.g. the Heat Equation which you will not be examined on and the Laplace’s Equation might be slightly simpler than what ye will have).
The VBA 20% Assessment 2 format will be as follows.
Q. 1. Boundary Value Problem with a Shooting Method
Specifically,
;
,
,
for some ,
, and step-size
determined by your student number.
I want Euler Shooting Method approximations to for
.
You can use:
- An Excel Worksheet, or
- Excel’s Goal Seek, or
- A VBA program
but you have to use a Shooting Method (technically Goal Seek takes loads of shots so I am happy to call it a shooting method).
It is up to you to understand which method is easiest for you.
Example
Use a shooting method to solve the following with :
,
,
.
Solution: The preliminary work is to turn this into a system of first order initial value problems. To do so introduce a new variable for the first derivative (as it happens
is the shear).
Let together with the initial value
.
If is the first derivative of
with respect to
then
so that we have
.
We have no initial value for so we just guess for the moment… say
.
Worksheet Solution
Please see the first worksheet of Shooting Method for Bending Moment (it will be emailed) for the implementation of Euler’s Method for the system:
;
,
;
,
The first shot with produced
, an undershoot (we are trying to get
).
We try again with a larger , say
. This produces an overshot of
.
Now use equation (3.38) on p. 130 of the notes to find the correct :
.
Now see the worksheet where the Euler Method is run with this value and the resulting graph (I am happy with just the values but if you can input the graph). Note this value of yields
as required.
(Usually in engineering we plot underneath the
-axis… don’t worry about this.)
Goal Seek Solution
Very similar set-up to the previous except we don’t have to take any shots and instead ask Excel to try a load of shots.
See Worksheet 2 of Shooting Method for Bending Moment.
So perhaps just put as a placeholder.
Now do Goal Seek (see p. 131). This produces and
.
VBA Solution
Again the set up is similar but we run the Euler Method via VBA.
See Worksheet 3 of Shooting Method for Bending Moment (or moreover the code behind the worksheet).
We have to take two shots and use equation (3.38) to get . Finally, we must run the program one more time.
Q. 2. Boundary Value Problem with Finite Differences
Specifically,
;
and
for some ,
,
,
and
. These constants will be determined by your student number.
Use a Finite Difference Method with a mesh size (determined by your student number) [Sample: Lab 7, p.134, Problem 2], to produce approximations to
for
.
I have actually sent ye an email on 11 April with a worked example of this.
Q. 3 Laplace’s Equation
Specifically,
for the temperature at the point
of a rectangular plate
with boundary conditions given by
, where
is the boundary/perimeter of the rectangular plate
.
The boundary temperature will be given in terms of your student number.
The above equation, Laplace’s equation, can instead by framed as the Mean Value Property which can be approximated using the ‘four adjacent gridpoint average’ once the rectangular plate is meshed using a square grid.
Sample Question: Lab 7, Problem 1, P. 133
Week 12 — 40% Written Test
I have not yet decided the format of the 40% Written Test but am toying with the idea of splitting the Test in two.
The problem with this is that I can do the first part of the test at 09:00 on the Tuesday in B242 but A183L is too small to conduct an assessment in.
I might consider putting the second part of the test into your VBA slot.
To ensure some kind of fairness, this would work as follows:
The first part of the Test would take place at 09:00 . It would be designed to be easily completed in 30-40 minutes. It would be geared more towards theoretical questions.
The second part of the Test would take place in your VBA slot. I would have to tell you in advance what questions are coming up, e.g. maybe
- Q. 1 Second Order Problem Using Heun’s Method
- Q. 2 Euler Shooting Method
- Q. 3 Heat Equation
Each group would get questions with only minor variations from the sample questions. I will confirm this next week.
Study
Study should consist of
- doing exercises from the notes
- completing VBA exercises
Student Resources
Please see the Student Resources tab on the top of this page for information on the Academic Learning Centre, etc..
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