# MEEN40050 Computational Continuum Mechanics I

Computational Continuum Mechanics I (CCM I) course is designed to introduce basics of Continuum Mechanics, and Finite Volume (FV) and Finite Element (FE) methods. By doing so, it is aimed to demonstrate how fundamental laws of Physics can be turned into powerful numerical tools.The CCM I course is designed to provide theoretical basis and knowledge required for practical applications of FV and FE methods which will be the topic of the CCM II module.

Syllabus:
1. Basic concepts and definitions: Concept of continuum; Continuity, Homogeneity and isotropy; Mathematical basis; Elements of matrix algebra.
2. Stresses: Body and surface forces; Stress tensor; Principal stresses. Stress tensor's invariants; Spherical and deviatoric stresses.
3. Deformation and flow: Material and spatial description; Deformation; Motion and flow.
4. Fundamental laws of Continuum Mechanics: Mass conservation; Conservation of linear momentum; Conservation of angular momentum; Conservation of energy; Law of entropy production.
5. Constitutive relations: Ideal materials; Classical constitutive relations and equations of state; Elastic solids; Ideal and Newtonian fluids.
6. Mathematical models: Linear elastic solids; Newtonian fluids; Initial and boundary conditions; Generic transport equation.
7. Finite volume disretisation: Main concepts and ideas of the Finite Volume method; Time and space discretisation, equation discretisation, conservativness, boundedness, stability, accuracy, consistency, convergence; Discretisation for 2D Cartesian mesh: u-momentum, v-momentum, energy; Initial and boundary conditions: Dirichlet boundary condition, Neumann boundary condition, mixed boundary condition, symmetry boundary condition; 3D problems in domains of arbitrary shapes.
8. Finite element discretisation: Principle of virtual work; Finite element discretisation; Linear elastic finite element model; Assembly of element and global stiffness matrix; Shape functions; Numerical quadrature; Mapping of elements; Solution of the finite element equations.

Tutorials:Tutorials are integral part of this module and students are strongly advised to attend all tutorial sessions. Tutorials are also the main component of the continuous assessment. There will be five tutorial assignements, and the solutions are to be handed in within one week after the corresponding tutorial session.

Recommended text:1. G. E. Mase, Theory and Problems of Continuum Mechanics, Schaum's Outline Series, McGraw-Hill, Inc. 1970.2. L. E. Malvern, Introduction to the Mechanics of a Continuous Media, Prentice-Hall, Inc., 1969.3. O. C. Zienkiewicz and R. L. Taylor, The Finite Element Method, 4th ed.,Volume 1, McGraw-Hill, Inc. 1989.4. K. J. Bathe, Finite Element Procedures, Prentice-Hall, Inc., 1996.5. S.V. Patankar, Numerical Heat Transfer and Fluid Flow, Mc Graw-Hill, Inc. 1980.6. J.H. Ferziger and M. Peric, Computational methods for Fluid Dynamics, Springer-Verlag Berlin Heidelberg, 1996.7. CCMI lecture notes and overheads.

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Curricular information is subject to change

Learning Outcomes:

At the completion of the module, the students should be able to:
1. Demonstrate knowledge and understanding of basic concepts of continuum, field theory, and conservation and constitutive laws.
2. Formulate and solve simple Continuum Mechanics problems, i.e. problems involving field theory, stresses, deformation and flow, conservation and constitutive laws.
3. Demonstrate understanding of fundamental basics of FV discretisation procedure; Define and set-up within the FV framework a variety of steady and unsteady continuum mechanics problems.
4. Demonstrate understanding of fundamental basics of FE discretisation procedure; Define and set-up within the FE framework a variety of steady and unsteady continuum mechanics problems.
5. Identify main differences between FV and FV methods, their advantages and disadvantages, and hence decide which of the method will best suit a particular problem.
6. Present and justify the main assumptions and findings in formulating and solving a CCM problem.

Indicative Module Content:

1. Basic concepts and definitions: Concept of continuum; Continuity, Homogeneity and isotropy; Mathematical basis; Elements of matrix algebra.
2. Stresses: Body and surface forces; Stress tensor; Principal stresses. Stress tensor's invariants; Spherical and deviatoric stresses.
3. Deformation and flow: Material and spatial description; Deformation; Motion and flow.
4. Fundamental laws of Continuum Mechanics: Mass conservation; Conservation of linear momentum; Conservation of angular momentum; Conservation of energy; Law of entropy production.
5. Constitutive relations: Ideal materials; Classical constitutive relations and equations of state; Elastic solids; Ideal and Newtonian fluids.
6. Mathematical models: Linear elastic solids; Newtonian fluids; Initial and boundary conditions; Generic transport equation.
7. Finite volume disretisation: Main concepts and ideas of the Finite Volume method; Time and space discretisation, equation discretisation, conservativness, boundedness, stability, accuracy, consistency, convergence; Discretisation for 2D Cartesian mesh: u-momentum, v-momentum, energy; Initial and boundary conditions: Dirichlet boundary condition, Neumann boundary condition, mixed boundary condition, symmetry boundary condition; 3D problems in domains of arbitrary shapes.
8. Finite element discretisation: Principle of virtual work; Finite element discretisation; Linear elastic finite element model; Assembly of element and global stiffness matrix; Shape functions; Numerical quadrature; Mapping of elements; Solution of the finite element equations.

Tutorials:Tutorials are integral part of this module and students are strongly advised to attend all tutorial sessions. Tutorials are also the main component of the continuous assessment. There will be five tutorial assignements, and the solutions are to be handed in within one week after the corresponding tutorial session.

Student Effort Hours:
Student Effort Type Hours
Autonomous Student Learning

70

Lectures

30

Tutorial

10

Total

110

Approaches to Teaching and Learning:
Face to face and online lectures. Lectures are designed to cover essential points of each topic given in syllabus and their relevance will be discussed and you will be encouraged and will have the opportunity to ask questions.
Face to face and online tutorial sessions to apply the theory to gain practical experience in problems solving. Examples of exam questions will be solved step by step (during lectures and tutorials).
Assignments and homework
On-line submissions/feedback. The two in-class tests, which are the part of the second and third assignments, will take place in 2022/23.

Requirements, Exclusions and Recommendations
Learning Requirements:

Mechanics of Solids I
Mechanics of Solids II
Mechanics of Fluids I
Mechanics of Fluids II

Module Requisites and Incompatibles
Not applicable to this module.

Assessment Strategy
Description Timing Open Book Exam Component Scale Must Pass Component % of Final Grade
Examination: Written exam. The default scale (Default 40%) is used. 2 hour End of Trimester Exam Yes Standard conversion grade scale 40% No

70

Continuous Assessment: Online Tutorial 1 [5%]
Online Tutorial 2 [5%]
In-class test 1 [2.5%]
Online Tutorial 3 [5%]
In-class test 2 [2.5%]
Online Tutorial 4 [5%]
Online Tutorial 5 [5%]
Varies over the Trimester n/a Standard conversion grade scale 40% No

30

Carry forward of passed components
No

Resit In Terminal Exam
Spring Yes - 2 Hour
Feedback Strategy/Strategies

• Feedback individually to students, post-assessment
• Group/class feedback, post-assessment
• Online automated feedback

How will my Feedback be Delivered?

Feedback to be given by TAs within two weeks after each assignments and in class tests.

Timetabling information is displayed only for guidance purposes, relates to the current Academic Year only and is subject to change.

Autumn

Tutorial Offering 1 Week(s) - 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 Fri 11:00 - 12:50
Lecture Offering 1 Week(s) - Autumn: All Weeks Fri 14:00 - 14:50
Lecture Offering 1 Week(s) - Autumn: All Weeks Thurs 12:00 - 12:50
Lecture Offering 1 Week(s) - Autumn: All Weeks Tues 11:00 - 11:50
Autumn