Dr A. Cookson Department of Mechanical Engineering
20192020
ME40064: System ModellingSimulation ME50344: Engineering Systems Simulation
University of Bath
Assignment 2: Transient MATLABBased FEM Modelling
Summary
In this coursework you must extend your finite element code to be able to solve the transient diffusionreaction equation:
cD 2ccf t x2
You will then use this code to model the effects of heat damage to human skin, thereby being able to estimate the level of thermal shielding that a piece of protective clothing must provide.
Throughout this assignment it is assumed that more thorough answers to the questions will achieve a higher mark i.e. by exploring the effect of element size, time step, basis function polynomial order, and time integration scheme on the accuracy of the results and conclusions.
Note that you may reuse your existing functions from Assignment 1 to calculate the local element matricesvectors etc. Alternatively, you can rewrite your code to use Gaussian quadrature. However, you should not use MATLABs inbuilt functions for numerical or symbolic integration to calculate these matrices.
Part 1: Software DevelopmentVerification
Check that your code is working correctly by solving the following transient diffusion equation, for the domain defined from x0 to 1, and subject to the following initial and Dirichlet boundary conditions:
c2c t x2
x0,1, cx,00, c0,t0, c1,t1
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Dr A. Cookson Department of Mechanical Engineering 20192020 University of Bath
In order to achieve good accuracy in your solution, use an element size of 0.1 i.e. a 10 element mesh and a time step, t0.01.
Compare your results to the following analytical solution:
2 X1 1n n22t
cx, tx n1 n e sinnx
A MATLAB function, TransientAnalyticSoln.m, to compute this analytical solution is provided on Moodle.
Create the following two figures to demonstrate that your code is correct:
a Plot your solution cx vs. x, showing the solutions at t0.05, 0.1, 0.3, 1.0,
in the format shown in Lecture 13. 30
b Plot both the analytical solution and your numerical solution at x0.8, for
t0 to 1.0, to demonstrate that your solution is converged. 5
Extra credit is available for the following tasks if performed, be sure to include
appropriate description and figures in your report.
Important: note that Part 2 of this assignment can be carried out to a highlevel without using any of these advanced features.
Investigating accuracy and stability of forward Euler, backward Euler and CrankNicolson time stepping methods 5
Using Gaussian quadrature to evaluate element integrals 5
Implementingtesting performance of quadratic basis functions 5
Investigating errorsconvergence using the L2 norm 5
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Dr A. Cookson Department of Mechanical Engineering
20192020
University of Bath
Part 2: ModellingSimulation Results
1. UseyourcodetosolvethetissueburnmodelfromLecture1415.
Note that you may find the backward Euler much faster than CrankNicolson for achieving stable results.
Your finite element mesh and material parameters must represent the three layered structure of skin, as follows:
This is modelled using the equation:
T k2T GbcbTGbcbTb tcx2 c c
The parameter values for each layer are provided in Table 1. However, in this question we assume zero blood flow i.e. G0 everywhere. The layers of the tissue are defined at the following x coordinates: E0.00166667, D0.005, B0.01.
a Run your code for the following initial conditions and Dirichlet boundary conditions, for a maximum of 50 seconds:
Tx,0310.15K, TxB,t310.15K, Tx0,t393.15K
Create a plot showing the spatial temperature distribution in the tissue for various points in time between 0 and 50 seconds. Explain the shape of the curves as they change through time based on your knowledge of the physics, material parameters, initial conditions, and boundary conditions.
tburn Page 3 of 5
b Use these temperature profiles to determine whether tissue damage will occur or not. To do this, evaluate the following integral numerically at the mesh node for xE:
Zt 9812017
210 exp T273.15 dt
Dr A. Cookson Department of Mechanical Engineering 20192020 University of Bath
The limits of the integral are ttburn, i.e. the time value at which the temperature T, at xE, becomes greater than 317.15 Kelvin, and t50.
If 1 at xE, this produces a seconddegree burn. State the value ofyou have calculated at this node dont be alarmed if it is very large or small, and hence state whether you expect a seconddegree burn to occur.
Note that the MATLAB function, trapzx, uses the trapezium rule to integrate the vector x, assuming an interval of 1. Therefore, multiply the output of the function by your timestep, t, to obtain the final value of this integral.
2. Useyourcodetodeterminetheminimumtemperaturereductiontothe nearest 0.5 degree K that must be achieved by the protective clothing at the boundary x0, in order to prevent a seconddegree burn as defined in Q1b.
State the final Dirichlet boundary condition at x0 that achieves this, and explain how you used your code, and any other calculations, to estimate this value. Include any data or figures that you feel are relevant in order to demonstrate your approach to estimating this. 10
3. RerunyoursimulationsforQuestions12,butnowincludingtheblood flow related reactionsource terms, using the parameter values in Table 1. How much does this change your results? Is blood flow an important effect to consider for future modelling of this problem? Finally, discuss how realistic you think the initial conditions, boundary conditionsmodel assumptions are. 10
Parameter
Epidermis
Dermis
Subcutaneous
k
25
40
20
G
0
0 Q1a 0.0375 Q2
0 Q1a 0.0375 Q2
1200
1200
1200
c
3300
3300
3300
b
1060
1060
cb
3770
3770
Tb
310.15
310.15
Table 1: Parameter values for tissueblood. Note that some are realistic and others are not, having been chosen to allow you to solve your model in a reasonable time frame.
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Dr A. Cookson Department of Mechanical Engineering 20192020 University of Bath
Presentation 5
Clear graphical presentation of equations, text, and results
Quality of English, proper use of references, figures, and tables
Overall Code Quality 5
Correctness, elegance, and readability, of MATLAB code
Evidence of software verification and other good development practice
SUBMISSION GUIDELINES
You may structure your report as a set of answers to these questionsthere is no requirement to write this in a lab report format. You also do not need to re explain the entirety of the finite element methoda highlevel overview of the method is sufficient. However, your report must not assume that the reader knows the content in this assignment document.
You must include all your Matlab source code as text in the Appendicesfailure to do so will cause you to lose marks. Do not paste your code into the document as an OLE item or as an image.
Do not upload archivedzippedcompressed folders of these source files.
Do not use MATLABs symbolic algebra toolbox.
Your code should use meaningful variable names and include comments,
in line with good practice.
Word limit of 2000 words not including source code.
Submit your work using the online submission function on the units Moodle page.
Deadline: 4pm on Monday, 2nd December 2019.
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