The University of Manchester

School of Mechanical, Aerospace & Civil Engineering

Modelling & Simulation 3 - (3rd Year) [Unit Code: MACE 30051]

Semester 1 – Dr M A Sheikh - 12 Lectures

Dr M Cotton - 12 Lectures

Lectures Schedule & Venue

Please consult your timetable

Assessment

Written Examination (2 hrs): 80% ; Coursework: 20%

Examination Paper Format:

Two Sections A & B: Section A (Dr Sheikh) – 2 Qs; Section B (Dr Cotton) – 2 Qs
Attempt THREE Qs.

Computational Laboratory Exercises

1. FE Modelling of a Structural Design Problem (Dr M Sheikh) – 10%

2. Flow Simulation (Dr M Cotton) – 10%

Contact

Dr. M. A. Sheikh
Office: P/G20 – Ext. 63802
E-mail: m.sheikh@manchester.ac.uk

Topics: [Modelling of Linear Elastic Structural Problems (Dr M Sheikh)]

Numerical Techniques & their applications in Engineering

Basic Theory of the Finite Element Method

FE Modelling Techniques for Linear Elastic Analysis

Basic Elements & Shapes

Interpolation Functions for Simplex Elements

Element Equations for one-, two- and three-dimensional elasticity equations

Higher Order Elements

Numerical Integration & Convergence of Solutions

Mesh Design – Modelling Procedures & Model Validity

Main Text:

Segerlind, Larry J., Applied finite element analysis, 2nd ed. Wiley, New York,
Chichester, 1984.

Secondary Texts:

Rao, S. S., The Finite element method in engineering, Pergamon Press, Oxford, 1982.

Hinton, E. and Owen, D. R. J., An introduction to finite element computations,

Pineridge Press Ltd, Swansea, 1979.

Hinton, E. and Owen, D. R. J., A simple guide to finite elements, Pineridge Press Limited,
Swansea, 1980.

Problem Definition:
Γ1 (φ = φ')

Ω
∇2φ = 0

Γ = Γ1 + Γ2
Γ2 [ (∂φ/∂n) = (∂φ/∂n)' ]

1. Governing Equations

∇2φ = 0 (Laplace Equation: In 1-D)

2. Boundary Conditions

φ = φ' (Essential Condition) [Direchlet]

(∂φ/∂n) = (∂φ/∂n)' (Natural Condition) [Neumann]

Approximate Solution

∇2φ ≠ 0 in Ω ⇒ε

φ - φ' ≠ 0 on Γ1 ⇒ ε1

(∂φ/∂n) - (∂φ/∂n)' ≠ 0 on Γ2 ⇒ ε2

Aim: Make the error as small as possible. How? Distribute the error
 The way in which this error is distributed ⇒ Different Weighted Residual Techniques

Weighted Residual Statement

∫Ω ε w dΩ = ∫Γ1 ε1 w dΓ + ∫Γ2 ε2 w dΓ [w: Weighting Functions]

Finite Element Method (FEM):

Satisfy ‘Essential Condition’ on Γ1 identically, and approximate ‘Natural Condition’ on Γ2 , so

that:
∫Ω ε w dΩ = ∫Γ2 ε2 w dΓ

or ∫Ω (∇2φ) w dΩ = ∫Γ2 (q - q') w dΓ [q = ∂φ/∂n]

Integrate by parts:

∫Ω ∂φ/∂xk ∂w/∂xk dΩ = ∫Γ2 q' w dΓ - Statement of FEM

[k = 1, 2, 3 depending upon the dimensionality of the problem]
Three Steps in FE Analysis

I. Pre-processing:

Problem Discretization

Geometry, Loads, Boundary Conditions, Material Properties etc

Selection of Approximating Function

Linear, Quadratic, Cubic

II. Analysis:

Form the System of Equations

Element Equations; Assembly

Incorporate Boundary Conditions

Apply Constraints

Solve the System of Equations

Element Equations
Gaussian Elimination ,
Wavefront Methods etc.

III. Post-processing
System Equations
Evaluation of Results
{F} = [k] {u}
At Nodal Points, Gauss Points etc

Presentation of Results

Plots etc.
Conclusions:

Geometric Modelling:

2-D Plane Strain Representation of 3-D Problems

2-D Plane Stress Representation of 3-D Problems

 Types of Symmetry

Planar Symmetry

Plate with a hole – Stress Concentration Problem

Quarter of the plate required for analysis– due

to symmetry

Cyclic Symmetry

Repetitive Symmetry
 Basic Element Shapes

Finite Elements with curved boundaries

Example - FE Models of a Thin-Walled Cylinder

More than ONE models are possible

 Cooling Fin – ‘Mesh Size’ effect on the resulting solutions

Cooling Fin Problem - Solution Convergence Study

Changing the Mesh Size

 Aspect Ratios – allowable element distortions

Curved Sides

Location of Nodes:
 Bandwidth:

= (D + 1) f

D: Largest Difference between the node numbers

in an element
f: Number of D.O.F per node

B = (D + 1) f

Model (a) ⇒ B = 20

Model (b) ⇒ B = 12

Solution by the Wavefront Method

Polynomial Approximation in 1-D

1-D Elements and their Interpolation Functions

Two-dimensional Elements and their Interpolation Functions

Simplex, Complex and Multiplex Elements

Let's consider:

which is a linear Interpolation Polynomial for a Simplex Element in 1-D

Apply nodal conditions:

@ x = xi ; φ = φi

@ x = xj ; φ = φj

Ni and Nj are called ‘SHAPE FUNCTIONS’

Each Shape Function is associated with a

‘Particular Node’

Property: = 1 at its own assigned node

= 0 at all other nodes

Quadratic Element
Variation of Shape Functions

Linear Element
2-D Simplex Element

Interpolation function:

Shape Functions:

In general,

3-D Simplex Element

Natural Coordinates

The natural coordinates can be used

to describe the geometry

Line Integration:

(2-D Local Coordinates)

[Area Coordinates]

L1, L2, L3 vary from 0 to 1 – as before, so that

Area Integration Formula

It is now possible to calculate the ‘Global’

coordinates of the point ‘P’ as:
 Natural Coordinates of a three-dimensional element

Volume Integration Formula

Vector Problems

Degrees of Freedom for a 2-D Stress Problem

Displacement components

Shape Functions

Storage of Equations:

Alternative definitions for the Degrees of Freedom

of a two-dimensional element

Equations for a three-dimensional element with four nodes