Unit of study_

# BMET9960: Biomedical Engineering Mathematical Modelling

## Overview

BMET9960 is designed to equip you with the necessary tools to mathematically model and solve a range of canonical problems in engineering: conduction heat transfer, vibration, stress and deflection analysis, convection and stability. You will learn how to compute analytical and numerical solutions to these problems, and then apply this to relevant and interesting biomedical examples. By the end of this unit you will know how to derive analytical solutions via separation of variables, Fourier series and Fourier transforms and Laplace transforms. You will also know how to solve the same problems numerically using finite difference, finite element and finite volume approaches. The theoretical component of the unit is complemented by tutorials where you will learn how to use Matlab to implement and visualise your solutions. There is plenty of support in the early weeks of the unit to refresh your Matlab knowledge, or to learn Matlab for the first time if you've had no prior experience. Gaining a good working knowledge of Matlab to solve engineering problems and explore the solution space of these problems is one of the key benefits of this unit - it will set you up very well for future units requiring programming expertise! There is a strong emphasis in both the lectures and tutorials on example-based learning - you will see and attempt many different examples involving a wide range of biomedical applications. Applications include electrical, mechanical, thermal and chemical mechanisms in the human body and specific examples include heat regulation, vibrations of biological systems, and analysis of physiological signals such as ECG and EEG. This is a challenging but very rewarding unit and you'll come away feeling well-equipped with useful tools for your future engineering career. We hope you enjoy it!

### Unit details and rules

Unit code BMET9960 Biomedical Engineering 6 BMET2960 or AMME2960 None None Undergraduate mathematics (1000-level) and an appreciation of the biomedical engineering process Yes

### Teaching staff

Coordinator Andre Kyme, andre.kyme@sydney.edu.au

## Assessment

Type Description Weight Due Length
Final exam (Open book) Final exam
Type C
40% Formal exam period 2 hours
Outcomes assessed:
Small test Mid-Semester Quiz 1
In-class quiz
10% Week 04
Due date: 17 Mar 2022 at 12:00
60 min
Outcomes assessed:
Assignment Assignment 1
Assignment 1
10% Week 06
Due date: 01 Apr 2022 at 23:59
n/a
Outcomes assessed:
Small test Mid-Semester Quiz 2
In-class quiz
10% Week 10
Due date: 05 May 2022 at 12:00
60 min
Outcomes assessed:
Assignment Assignment 2
Assignment 2
15% Week 12
Due date: 20 May 2022 at 23:59
n/a
Outcomes assessed:
Weekly online quiz
5% Weekly n/a
Outcomes assessed:
Assignment Tutorials Assessment
Weekly online tutorial assessment
10% Weekly n/a
Outcomes assessed:
= Type C final exam

### Assessment summary

• Assignment 1 (10%): Analytical and numerical solution of the heat equation.
• Assignment 2 (15%): Analytical and numerical solution of the heat and/or wave and/or Laplace equations.
• Quiz 1 (10%): Students will be advised what is examinable.
• Quiz 2 (10%): Students will be advised what is examinable.
• Weekly pre-lecture quizzes (5%): A short weekly online quiz based on the pre-lecture work for the week, to be completed prior to the lectures that week. Students have unlimited attempts up until the deadline each week.
• Tutorial assessment (10%): One exercise from each tutorial must be completed online by 9 am Tuesday of the following week. The exercise associated with the Week 1 tutorial is not assessed. Each of the 12 exercises from Week 2 onwards is scored as 1 or 0 and the best 11 scores are used to compute the final score (/10%). A student successfully completing 11 or 12 of the tutorial exercises from Week 2 onwards will gain the full 10%.
• Final exam (40%): 2-hour exam.

Detailed information for each assessment can be found on Canvas.

### Assessment criteria

See Canvas for details

### Late submission

In accordance with University policy, these penalties apply when written work is submitted after 11:59pm on the due date:

• Deduction of 5% of the maximum mark for each calendar day after the due date.
• After ten calendar days late, a mark of zero will be awarded.

This unit has an exception to the standard University policy or supplementary information has been provided by the unit coordinator. This information is displayed below:

5% per day

The Current Student website  provides information on academic integrity and the resources available to all students. The University expects students and staff to act ethically and honestly and will treat all allegations of academic integrity breaches seriously.

We use similarity detection software to detect potential instances of plagiarism or other forms of academic integrity breach. If such matches indicate evidence of plagiarism or other forms of academic integrity breaches, your teacher is required to report your work for further investigation.

You may only use artificial intelligence and writing assistance tools in assessment tasks if you are permitted to by your unit coordinator, and if you do use them, you must also acknowledge this in your work, either in a footnote or an acknowledgement section.

Studiosity is permitted for postgraduate units unless otherwise indicated by the unit coordinator. The use of this service must be acknowledged in your submission.

## Learning support

### Simple extensions

If you encounter a problem submitting your work on time, you may be able to apply for an extension of five calendar days through a simple extension.  The application process will be different depending on the type of assessment and extensions cannot be granted for some assessment types like exams.

### Special consideration

If exceptional circumstances mean you can’t complete an assessment, you need consideration for a longer period of time, or if you have essential commitments which impact your performance in an assessment, you may be eligible for special consideration or special arrangements.

Special consideration applications will not be affected by a simple extension application.

### Using AI responsibly

Co-created with students, AI in Education includes lots of helpful examples of how students use generative AI tools to support their learning. It explains how generative AI works, the different tools available and how to use them responsibly and productively.

## Weekly schedule

WK Topic Learning activity Learning outcomes
Week 01 1. Introduction to the UoS; 2. Introduction to numerical methods; 3. Discretisation; 4. Interpolation; 5. Least squares; 6. Cubic Splines; 7. Taylor series; 8. Finite differences Lecture (2 hr)
Introduction to Matlab Tutorial (2 hr)
Week 02 1. What is a PDE?; 2. Generic PDE introduction inc. classification; 3. Derivation of the heat diffusion equation; 4. Exact solution of the heat diffusion equation (Fourier series); 5. Solution of heat equation via separation of variables; 6. Heat equation with non-homogeneous boundary conditions Lecture (2 hr)
Numerical methods, Taylor series, PDEs and interpolation Tutorial (2 hr)
Week 03 1. Initial value problems, boundary value problems, initial conditions, boundary conditions, well posed problems; 2. Accuracy, stability, consistency; 3. Linear algebra; Lecture (2 hr)
Analytical solution to heat equation Tutorial (2 hr)
Week 04 1. Forward-in-time centred-in-space solution of the heat diffusion equation. Lecture (2 hr)
Numerical solution to heat equation Tutorial (2 hr)
Week 05 1. Heat equation with more complex initial and boundary conditions; 2. Introduction to and derivation of the wave equation; 3. Classification of wavelike equations; 4. Approximate solution using Fourier series. Lecture (2 hr)
Solution to heat equation with more complex BCs Tutorial (2 hr)
Week 06 1. Wave equation with complex initial conditions; 2. Numerical solution of the wave equation. Lecture (2 hr)
Analytical solution to wave equation Tutorial (2 hr)
Week 07 1. Introduction and derivation of the Laplace and Poisson equation; 2. Applications; 3. Exact solution based on Fourier series. 4. Numerical discretization of the 2D Laplace equation; 5. Solution using iterative methods. Lecture (2 hr)
Numerical solution to wave equation Tutorial (2 hr)
Week 08 1. Understanding PDEs - method to determine behaviour. 2. Fourier integrals and transforms. Lecture (2 hr)
Analytical solution to Laplace equation Tutorial (2 hr)
Week 09 1. Fourier integral solutions to infinite problems; 2. FFT and Signal Processing; 3. Fourier Transform solutions to PDEs. Lecture (2 hr)
Fourier integral solution to the heat equation Tutorial (2 hr)
Week 10 1. Laplace transforms; 2. Solution of the semi-infinite wave equation using Laplace transforms. Lecture (2 hr)
Fourier transform Tutorial (2 hr)
Week 11 1. Laplace Transform solution to the heat equation; 2. Introduction to finite elements. Lecture (2 hr)
Laplace transform solution to heat and wave equation Tutorial (2 hr)
Week 12 1. Piecewise linear basis functions; 2. Method of weighted residuals; 3. Weak formulation of the PDE and solution. Lecture (2 hr)
Implicit numerical methods Tutorial (2 hr)
Week 13 1. Foundations of stress analysis; 2. FEA solution for an axially loaded bar. Lecture (2 hr)
Finite element analysis Tutorial (2 hr)
Weekly Individual learning and problem solving Independent study (4 hr)

### Study commitment

Typically, there is a minimum expectation of 1.5-2 hours of student effort per week per credit point for units of study offered over a full semester. For a 6 credit point unit, this equates to roughly 120-150 hours of student effort in total.

All readings for this unit can be accessed on the Library eReserve link available on Canvas.

• Advanced Engineering Mathematics, E. Kreyszig, 10th Edition, Wiley, 2011.

## Learning outcomes

Learning outcomes are what students know, understand and are able to do on completion of a unit of study. They are aligned with the University's graduate qualities and are assessed as part of the curriculum.

At the completion of this unit, you should be able to:

• LO1. Understand and apply the physical relations and mathematical modelling of fundamental problems in engineering structures, fluid mechanics and heat and mass transfer.
• LO2. Creatively solve assignment problems which focus on real-life engineering applications
• LO3. Develop proficiency in a structured approach to engineering problem identification, modelling and solution; develop proficiency in translating a written problem into a set of algorithmic steps, and then into computer code to obtain a solution

The graduate qualities are the qualities and skills that all University of Sydney graduates must demonstrate on successful completion of an award course. As a future Sydney graduate, the set of qualities have been designed to equip you for the contemporary world.

 GQ1 Depth of disciplinary expertise Deep disciplinary expertise is the ability to integrate and rigorously apply knowledge, understanding and skills of a recognised discipline defined by scholarly activity, as well as familiarity with evolving practice of the discipline. GQ2 Critical thinking and problem solving Critical thinking and problem solving are the questioning of ideas, evidence and assumptions in order to propose and evaluate hypotheses or alternative arguments before formulating a conclusion or a solution to an identified problem. GQ3 Oral and written communication Effective communication, in both oral and written form, is the clear exchange of meaning in a manner that is appropriate to audience and context. GQ4 Information and digital literacy Information and digital literacy is the ability to locate, interpret, evaluate, manage, adapt, integrate, create and convey information using appropriate resources, tools and strategies. GQ5 Inventiveness Generating novel ideas and solutions. GQ6 Cultural competence Cultural Competence is the ability to actively, ethically, respectfully, and successfully engage across and between cultures. In the Australian context, this includes and celebrates Aboriginal and Torres Strait Islander cultures, knowledge systems, and a mature understanding of contemporary issues. GQ7 Interdisciplinary effectiveness Interdisciplinary effectiveness is the integration and synthesis of multiple viewpoints and practices, working effectively across disciplinary boundaries. GQ8 Integrated professional, ethical, and personal identity An integrated professional, ethical and personal identity is understanding the interaction between one’s personal and professional selves in an ethical context. GQ9 Influence Engaging others in a process, idea or vision.