# Introduction to Finite Element Methods

## Presentation on theme: "Introduction to Finite Element Methods"— Presentation transcript:

Introduction to Finite Element Methods
UNIT I Introduction to Finite Element Methods

Numerical Methods – Definition and Advantages
Definition: Methods that seek quantitative approximations to the solutions of mathematical problems Advantages:

What is a Numerical Method – An Example

What is a Numerical Method – An Example

What is a Numerical Method – An Example

What is a Numerical Method – An Example

What is a Numerical Method – An Example

What is a Finite Element Method

Discretization 1-D 2-D ?-D 3-D Hybrid

Numerical Interpolation Non-exact Boundary Conditions
Approximation Numerical Interpolation Non-exact Boundary Conditions

Applications of Finite Element Methods
Structural & Stress Analysis Thermal Analysis Dynamic Analysis Acoustic Analysis Electro-Magnetic Analysis Manufacturing Processes Fluid Dynamics

Lecture 2 Review

Matrix Algebra Row and column vectors
Addition and Subtraction – must have the same dimensions Multiplication – with scalar, with vector, with matrix Transposition – Differentiation and Integration

Matrix Algebra Determinant of a Matrix: Matrix inversion -
Important Matrices diagonal matrix identity matrix zero matrix eye matrix

Numerical Integration
Calculate: Newton – Cotes integration Trapezoidal rule – 1st order Newton-Cotes integration Trapezoidal rule – multiple application

Numerical Integration
Calculate: Newton – Cotes integration Simpson 1/3 rule – 2nd order Newton-Cotes integration

Numerical Integration
Calculate: Gaussian Quadrature Trapezoidal Rule: Gaussian Quadrature: Choose according to certain criteria

Numerical Integration

Numerical Integration - Example
Calculate: Trapezoidal rule Simpson 1/3 rule 2pt Gaussian quadrature Exact solution

Linear System Solver Solve: Gaussian Elimination: forward elimination + back substitution Example:

Linear System Solver Solve: Gaussian Elimination: forward elimination + back substitution Pseudo code: Forward elimination: Back substitution: Do k = 1, n-1 Do i = k+1,n Do j = k+1, n Do ii = 1, n-1 i = n – ii sum = 0 Do j = i+1, n sum = sum +

Finite Element Analysis (F.E.A.) of 1-D Problems
UNIT II Finite Element Analysis (F.E.A.) of 1-D Problems

Historical Background
Hrenikoff, 1941 – “frame work method” Courant, 1943 – “piecewise polynomial interpolation” Turner, 1956 – derived stiffness matrice for truss, beam, etc Clough, 1960 – coined the term “finite element” Key Ideas: - frame work method piecewise polynomial approximation

Axially Loaded Bar Review: Stress: Stress: Strain: Strain:
Deformation: Deformation:

Axially Loaded Bar Review: Stress: Strain: Deformation:

Axially Loaded Bar – Governing Equations and Boundary Conditions
Differential Equation Boundary Condition Types prescribed displacement (essential BC) prescribed force/derivative of displacement (natural BC)

Examples fixed end simple support free end

Potential Energy Elastic Potential Energy (PE) - Spring case
Unstretched spring Stretched bar x - Axially loaded bar undeformed: deformed: - Elastic body

Potential Energy Work Potential (WE) Total Potential Energy
f P f: distributed force over a line P: point force u: displacement A B Total Potential Energy Principle of Minimum Potential Energy For conservative systems, of all the kinematically admissible displacement fields, those corresponding to equilibrium extremize the total potential energy. If the extremum condition is a minimum, the equilibrium state is stable.

Potential Energy + Rayleigh-Ritz Approach
Example: f P A B Step 1: assume a displacement field f is shape function / basis function n is the order of approximation Step 2: calculate total potential energy

Potential Energy + Rayleigh-Ritz Approach
Example: f P A B Step 3:select ai so that the total potential energy is minimum

Galerkin’s Method Example: f P Seek an approximation so
B Seek an approximation so In the Galerkin’s method, the weight function is chosen to be the same as the shape function.

Galerkin’s Method Example: f P A B 1 2 3 1 2 3

Finite Element Method – Piecewise Approximation
u x u x

FEM Formulation of Axially Loaded Bar – Governing Equations
Differential Equation Weighted-Integral Formulation Weak Form

Approximation Methods – Finite Element Method
Example: Step 1: Discretization Step 2: Weak form of one element P1 P2 x1 x2

Approximation Methods – Finite Element Method
Example (cont): Step 3: Choosing shape functions - linear shape functions x x x=-1 x=0 x=1 x1 l x2

Approximation Methods – Finite Element Method
Example (cont): Step 4: Forming element equation E,A are constant Let , weak form becomes Let , weak form becomes

Approximation Methods – Finite Element Method
Example (cont): Step 5: Assembling to form system equation Approach 1: Element 1: Element 2: Element 3:

Approximation Methods – Finite Element Method
Example (cont): Step 5: Assembling to form system equation Assembled System:

Approximation Methods – Finite Element Method
Example (cont): Step 5: Assembling to form system equation Element 1 Element 2 Element 3 1 2 3 4 Approach 2: Element connectivity table local node (i,j) global node index (I,J)

Approximation Methods – Finite Element Method
Example (cont): Step 6: Imposing boundary conditions and forming condense system Condensed system:

Approximation Methods – Finite Element Method
Example (cont): Step 7: solution Step 8: post calculation

Summary - Major Steps in FEM
Discretization Derivation of element equation weak form construct form of approximation solution over one element derive finite element model Assembling – putting elements together Imposing boundary conditions Solving equations Postcomputation

Exercises – Linear Element
Example 1: E = 100 GPa, A = 1 cm2

Linear Formulation for Bar Element
x=x1 x= x2 u1 u2 f(x) L = x2-x1 u x x=x2 1 f2 f1 x=x1

Higher Order Formulation for Bar Element
1 3 u1 u3 u x u2 2 1 4 u1 u4 2 u x u2 u3 3 1 n u1 un 2 u x u2 u3 3 u4 …………… 4

Natural Coordinates and Interpolation Functions
x x=-1 x=1 x x=x1 x= x2 Natural (or Normal) Coordinate: 1 2 x x=-1 x=1 1 3 2 x x=-1 x=1 1 4 2 x x=-1 x=1 3

x=-1 x=0 x=1 f3 f1 f2

f(x) P3 P1 P2 x=-1 x=0 x=1

Example 2: E = 100 GPa, A1 = 1 cm2; A1 = 2 cm2

Some Issues Non-constant cross section: Interior load point:
Mixed boundary condition: k

Finite Element Analysis (F.E.A.) of I-D Problems – Applications

Plane Truss Problems Example 1: Find forces inside each member. All members have the same length. F

UNIT II

Arbitrarily Oriented 1-D Bar Element on 2-D Plane
Q2 , v2 q P2 , u2 Q1 , v1 P1 , u1

Relationship Between Local Coordinates and Global Coordinates

Relationship Between Local Coordinates and Global Coordinates

Stiffness Matrix of 1-D Bar Element on 2-D Plane
Q2 , v2 q P2 , u2 Q1 , v1 P1 , u1

Arbitrarily Oriented 1-D Bar Element in 3-D Space
ax x gx bx y z 2 1 - ax, bx, gx are the Direction Cosines of the bar in the x-y-z coordinate system -

Stiffness Matrix of 1-D Bar Element in 3-D Space
ax x gx bx y z 2 1 -

Matrix Assembly of Multiple Bar Elements
Element I Element I I Element I I I

Matrix Assembly of Multiple Bar Elements
Element I Element I I Element I I I

Matrix Assembly of Multiple Bar Elements
Apply known boundary conditions

Solution Procedures u2= 4FL/5AE, v1= 0

Recovery of Axial Forces
Element I Element I I Element I I I

Stresses inside members
Element I Element I I Element I I I

FEM of 1-D Problems: Applications
Lecture 5 FEM of 1-D Problems: Applications

Torsional Shaft Review Assumption: Circular cross section
Shear stress: Shear strain: Deformation:

Finite Element Equation for Torsional Shaft

Bending Beam y Review x Pure bending problems: Normal strain:
Normal stress: Normal stress with bending moment: Moment-curvature relationship: Flexure formula:

Bending Beam y Review q(x) x
Relationship between shear force, bending moment and transverse load: Deflection: Sign convention: M + M - M V + - V V

Governing Equation and Boundary Condition
0<x<L Boundary Conditions ----- { Essential BCs – if v or is specified at the boundary. Natural BCs – if or is specified at the boundary.

Weak Formulation for Beam Element
Governing Equation Weighted-Integral Formulation for one element Weak Form from Integration-by-Parts (1st time)

Weak Form from Integration-by-Parts ----- (2nd time)
Weak Formulation Weak Form from Integration-by-Parts (2nd time) V(x2) x = x1 M(x2) q(x) y x x = x2 V(x1) M(x1) L = x2-x1

Weak Formulation Weak Form y(v) x Q1 q(x) Q3 Q2 Q4 x = x1 L = x2-x1

Ritz Method for Approximation
q(x) y(v) Q1 Q3 Q2 Q4 x x = x1 L = x2-x1 x = x2 where Let w(x)= fi (x), i = 1, 2, 3, 4

Ritz Method for Approximation
Q3 x = x1 y(v) x x = x2 Q1 Q2 L = x2-x1 Q4

Ritz Method for Approximation
Q3 x = x1 y(v) x x = x2 Q1 Q2 L = x2-x1 Q4

Selection of Shape Function
The best situation is ----- Interpolation Properties

Derivation of Shape Function for Beam Element – Local Coordinates
How to select fi??? and where Let Find coefficients to satisfy the interpolation properties.

Derivation of Shape Function for Beam Element
How to select fi??? e.g. Let Similarly

Derivation of Shape Function for Beam Element
In the global coordinates:

Element Equations of 4th Order 1-D Model
y(v) u1 q(x) u3 u2 u4 x x = x1 L = x2-x1 x = x2 f4 1 f1 1 f2 f3 x=x2 x=x1

Element Equations of 4th Order 1-D Model
y(v) u1 q(x) u3 u2 u4 x x = x1 L = x2-x1 x = x2

Finite Element Analysis of 1-D Problems - Applications
Example 1. F L Governing equation: Weak form for one element where

Finite Element Analysis of 1-D Problems
Example 1. Approximation function: f1 f4 x=x1 f2 f3 x=x2

Finite Element Analysis of 1-D Problems
Example 1. Finite element model: Discretization: P2 , v2 P3 , v3 P1 , v1 II P4 , v4 I III M1 , q1 M2 , q2 M3 , q3 M4 , q4

Matrix Assembly of Multiple Beam Elements
Element I Element I I

Matrix Assembly of Multiple Beam Elements
Element I I I

Apply known boundary conditions
Solution Procedures Apply known boundary conditions

Solution Procedures

Shear Resultant & Bending Moment Diagram

Plane Flame Frame: combination of bar and beam Q1 , v1 E, A, I, L
P1 , u1 P2 , u2 Q2 , q1 Q4 , q2

Finite Element Model of an Arbitrarily Oriented Frame
q x y q x

Finite Element Model of an Arbitrarily Oriented Frame
local global

Plane Frame Analysis - Example
Rigid Joint Hinge Joint F F F F Beam II Bar Beam I Beam

Plane Frame Analysis Q3 , v2 Q4 , q2 P2 , u2 P1 , u1 Q2 , q1 Q1 , v1

Plane Frame Analysis Q1 , v2 Q3 , v3 P1 , u2 P2 , u3 Q2 , q2 Q4 , q3

Plane Frame Analysis

Plane Frame Analysis

Finite Element Analysis (F.E.A.) of 1-D Problems – Heat Conduction
UNIT IV Finite Element Analysis (F.E.A.) of 1-D Problems – Heat Conduction

Heat Transfer Mechanisms
Conduction – heat transfer by molecular agitation within a material without any motion of the material as a whole. Convection – heat transfer by motion of a fluid. Radiation – the exchange of thermal radiation between two or more bodies. Thermal radiation is the energy emitted from hot surfaces as electromagnetic waves.

Heat Conduction in 1-D Governing equation: Steady state equation:
Heat flux q: heat transferred per unit area per unit time (W/m2) Governing equation: Q: heat generated per unit volume per unit time C: mass heat capacity k: thermal conductivity Steady state equation:

Thermal Convection Newton’s Law of Cooling

Thermal Conduction in 1-D
Boundary conditions: Dirichlet BC: Natural BC: Mixed BC:

Weak Formulation of 1-D Heat Conduction (Steady State Analysis)
Governing Equation of 1-D Heat Conduction ----- 0<x<L Weighted Integral Formulation ----- Weak Form from Integration-by-Parts -----

Formulation for 1-D Linear Element
x1 x2 1 2 T1 x T2 f1 Let x2 x1 f1T1 f2T2

Formulation for 1-D Linear Element
Let w(x)= fi (x), i = 1, 2

Element Equations of 1-D Linear Element
x1 x2 1 2 T1 x T2 f1

1-D Heat Conduction - Example
A composite wall consists of three materials, as shown in the figure below. The inside wall temperature is 200oC and the outside air temperature is 50oC with a convection coefficient of h = 10 W(m2.K). Find the temperature along the composite wall. t1 t2 t3 x

Thermal Conduction and Convection- Fin
Objective: to enhance heat transfer Governing equation for 1-D heat transfer in thin fin w t x dx where

Fin - Weak Formulation (Steady State Analysis)
Governing Equation of 1-D Heat Conduction ----- 0<x<L Weighted Integral Formulation ----- Weak Form from Integration-by-Parts -----

Formulation for 1-D Linear Element
Let w(x)= fi (x), i = 1, 2

Element Equations of 1-D Linear Element
x=0 x=L 1 2 T1 x T2 f1

Finite Element Analysis of 2-D Problems
Lecture 7 Finite Element Analysis of 2-D Problems

2-D Discretization Common 2-D elements:

2-D Model Problem with Scalar Function - Heat Conduction
Governing Equation in W Boundary Conditions Dirichlet BC: Natural BC: Mixed BC:

Weak Formulation of 2-D Model Problem
Weighted - Integral of 2-D Problem ----- Weak Form from Integration-by-Parts -----

Weak Formulation of 2-D Model Problem
Green-Gauss Theorem ----- where nx and ny are the components of a unit vector, which is normal to the boundary G.

Weak Formulation of 2-D Model Problem
Weak Form of 2-D Model Problem ----- EBC: Specify T(x,y) on G NBC: Specify on G where is the normal outward flux on the boundary G at the segment ds.

FEM Implementation of 2-D Heat Conduction – Shape Functions
Step 1: Discretization – linear triangular element T1 Derivation of linear triangular shape functions: T3 Let T2 Interpolation properties Same

FEM Implementation of 2-D Heat Conduction – Shape Functions
linear triangular element – area coordinates T1 A2 A3 A1 T3 T2 f1 f2 f3

Interpolation Function - Requirements
Interpolation condition Take a unit value at node i, and is zero at all other nodes Local support condition fi is zero at an edge that doesn’t contain node i. Interelement compatibility condition Satisfies continuity condition between adjacent elements over any element boundary that includes node i Completeness condition The interpolation is able to represent exactly any displacement field which is polynomial in x and y with the order of the interpolation function

Formulation of 2-D 4-Node Rectangular Element – Bi-linear Element
Let Note: The local node numbers should be arranged in a counter-clockwise sense. Otherwise, the area Of the element would be negative and the stiffness matrix can not be formed. f2 f1 f4 f3

FEM Implementation of 2-D Heat Conduction – Element Equation
Weak Form of 2-D Model Problem ----- Assume approximation: and let w(x,y)=fi(x,y) as before, then where

FEM Implementation of 2-D Heat Conduction – Element Equation

Assembly of Stiffness Matrices

Imposing Boundary Conditions
The meaning of qi: 3 3 1 1 1 2 2 3 3 1 1 1 2 2

Imposing Boundary Conditions
Consider Equilibrium of flux: FEM implementation:

Calculating the q Vector
Example:

2-D Steady-State Heat Conduction - Example
AB and BC: CD: convection DA: 0.6 m C B 0.4 m y x

Finite Element Analysis of Plane Elasticity

Review of Linear Elasticity
Linear Elasticity: A theory to predict mechanical response of an elastic body under a general loading condition. Stress: measurement of force intensity with 2-D

Review of Linear Elasticity
Traction (surface force) : Equilibrium – Newton’s Law

Review of Linear Elasticity
Strain: measurement of intensity of deformation Generalized Hooke’s Law

Plane Stress and Plane Strain
Plane Stress - Thin Plate:

Plane Stress and Plane Strain
Plane Strain - Thick Plate: Plane Stress: Plane Strain: Replace E by and by

Equations of Plane Elasticity
Governing Equations (Static Equilibrium) Strain-Deformation (Small Deformation) Constitutive Relation (Linear Elasticity)

Specification of Boundary Conditions
EBC: Specify u(x,y) and/or v(x,y) on G NBC: Specify tx and/or ty on G where is the traction on the boundary G at the segment ds.

UNIT V

Weak Formulation for Plane Elasticity
are components of traction on the boundary G where

Finite Element Formulation for Plane Elasticity
Let where and

Constant-Strain Triangular (CST) Element for Plane Stress Analysis
Let

Constant-Strain Triangular (CST) Element for Plane Stress Analysis

4-Node Rectangular Element for Plane Stress Analysis
Let

4-Node Rectangular Element for Plane Stress Analysis
For Plane Strain Analysis: and

Evaluation of Applied Nodal Forces

Evaluation of Applied Nodal Forces

Element Assembly for Plane Elasticity
5 6 B 3 4 3 4 A 1 2

Element Assembly for Plane Elasticity
1 2 3 4 6 5 A B

Comparison of Applied Nodal Forces

Discussion on Boundary Conditions
Must have sufficient EBCs to suppress rigid body translation and rotation For higher order elements, the mid side nodes cannot be skipped while applying EBCs/NBCs

Plane Stress – Example 2

Plane Stress – Example 3

Evaluation of Strains

Evaluation of Stresses
Plane Stress Analysis Plane Strain Analysis

Finite Element Analysis of 2-D Problems – Axi-symmetric Problems

Axi-symmetric Problems
Definition: A problem in which geometry, loadings, boundary conditions and materials are symmetric about one axis. Examples:

Axi-symmetric Analysis
Cylindrical coordinates: quantities depend on r and z only 3-D problem D problem

Axi-symmetric Analysis

Axi-symmetric Analysis – Single-Variable Problem
Weak form: where

Finite Element Model – Single-Variable Problem
where Ritz method: Weak form where

Single-Variable Problem – Heat Transfer
Weak form where

3-Node Axi-symmetric Element
1 2

4-Node Axi-symmetric Element
h 4 3 b 1 2 x a z r

Single-Variable Problem – Example
Step 1: Discretization Step 2: Element equation

Time-Dependent Problems

Time-Dependent Problems
In general, Key question: How to choose approximate functions? Two approaches:

Model Problem I – Transient Heat Conduction
Weak form:

Transient Heat Conduction
and let: ODE!

Time Approximation – First Order ODE
Forward difference approximation - explicit Backward difference approximation - implicit

Time Approximation – First Order ODE
a - family formula: Equation

Time Approximation – First Order ODE
Finite Element Approximation

Stability of – Family Approximation
Example Stability

FEA of Transient Heat Conduction
a - family formula for vector:

Stability Requirment where
Note: One must use the same discretization for solving the eigenvalue problem.

Transient Heat Conduction - Example

Transient Heat Conduction - Example

Transient Heat Conduction - Example

Transient Heat Conduction - Example

Transient Heat Conduction - Example

Transient Heat Conduction - Example

Transient Heat Conduction - Example

Model Problem II – Transverse Motion of Euler-Bernoulli Beam
Weak form: Where:

Transverse Motion of Euler-Bernoulli Beam
and let:

Transverse Motion of Euler-Bernoulli Beam

ODE Solver – Newmark’s Scheme
where Stability requirement: where

ODE Solver – Newmark’s Scheme
Constant-average acceleration method (stable) Linear acceleration method (conditional stable) Central difference method (conditional stable) Galerkin method (stable) Backward difference method (stable)

Fully Discretized Finite Element Equations

Transverse Motion of Euler-Bernoulli Beam

Transverse Motion of Euler-Bernoulli Beam

Transverse Motion of Euler-Bernoulli Beam

Transverse Motion of Euler-Bernoulli Beam