lecture 1: course introduction. - university of victoriamech420/lecture1.pdf · lecture 1: course...

MECH 420: Finite Element Applications

Lecture 1: Course Introduction.

What is the “Finite Element Method” (FEM)?“…a numerical method for solving problems of engineering and

mathematical physics.” –(Logan Pg. #1).In MECH 420 we are concerned with problems arising in applied mechanics.

Mechanics: That science, or branch of applied mathematics, which treats of the action of forces on bodies. Applied Mechanics: the practical application of the laws of matter and motion to the construction of machines and structures of all kinds.

The FEM is a numerical tool that factors into the design of machines and structures.

In the modern engineering paradigm:CAD/CAM & Solids Modelling.Computational Fluid Dynamics.Virtual prototyping.

The FEM is equally applicable to fluid flow, heat transfer, solid mechanics.



COMPUTATIONAL MECHANICS1. Solids & Structures (statics & dynamics).2. Fluid Flow3. Heat Transfer4. Coupled Systems (Mass transport – ie: Fuel cells)

There are several ways to obtain approximate solutions to the DE’s that arise in these fields of study.

Finite Element Method.Finite Difference Method.Finite Volume Method.Boundary Element Method.

Computational Fluid Dynamics



In terms of the UVic Mech Eng curriculum:Mechanics

Theoretical Applied Computational

MECH 141MECH 220MECH 320MECH 330MECH 342MECH 395

MECH 420

•A means for the “solution” of the governing equations that are developed through application of theory and idealizations.

•Applied mechanics generally includes some idealization (ie: small displacements).


Lecture 1: Course Introduction

What is the objective of MECH 420?

???

Vector calculus.Vibrations.Elasticity.Linear algebra.Object oriented programming.


Lecture 1: Course Introduction

Why is MECH 420 a part of a valuable part of a B.Eng. degree?By its very definition – the FEM can NOT give an exact solution (see comments to come on “weak form” formulations).FE modelling is a tool to establish the best approximation possible.The best approximation can only be identified by evaluation of the FEM output.Without knowledge of the FEM’s limitations and the assumptions inherent in element equations the “best guess” can be a terrible approximation.MECH 420 will show that the FEM requires the “user” to exert a great deal of creative choice.

Variational formulations – choosing a functional.Weighted Residual Formulations – choosing a weak form.Direct Stiffness Method – choosing a displacement field.


Lecture 1: FEM Introduction

§1.1 Brief History.In historical discussions, “FEM” is often replaced with “MSA”.The evolution of the current FEM/MSA occurred in distinct stages.1930’s: aeronautic engineers begin to put aeroelasticity problems in discrete, or matrix, form.

British National Physics Laboratory (NPL).Concerned with flutter and wing vibration.Divide an airscrew blade into 10 segments and define screw vibration in terms of ten DOF.Use an iterative procedure to solve a matrix set of equations.The expressions “Mass”, “Force”, & “Stiffness” matrices were coined.

WWII interrupted the development of MSA.Timelines were short and experimentation ruled the day.Publication restrictions prevented an evolution of the technique.


Lecture 1: FEM Introduction1950-1955: Boeing’s analysis of the Delta Wing.

Engineers couldn’t model the wing using 1D beams and struts.Placed restrictions on the displacement on the nodes of a triangular piece.Related the displacement field to stresses and then to nodal loads.Method was referred to as the Direct Stiffness Method (DSM).In parallel, British Engineers developed a rectangular element.

1956 – 1959: Direct Stiffness Method (DSM).Method of relating forces and displacements algebraically (matrix notation).Solution to these systems was fostered by a “growth” in computing power.UNIVAC (1952): 1000 45 bit word capacity.A lot of work conducted in the solution process.

1960’s & 1970’s – the FEM is born.Clough (Boeing) coined the phrase “Finite Element” referring to use of tangible rectangular and triangular elements to solve plane stress problems.Sparked by developments in computing power.The assembly process is mentioned in print.



Most general purpose structural FE packages use element equations derived in or prior to the 1970’s using DSM.However, DSM is a natural fit for only structural problems.Weighted Residuals Formulations & Variational Methods (which is called the work-energy method when applied in structural analysis) bridge the gap between structural analysis and multi-physics applications.

Multi-physics: Fluid Flow, Mass Transport, Electromagnetic fields.2000 and beyond: Bio-engineering (non-linear elasticity); large displacement or finite rotation problems (non-linear structural FEA); MEMS; two-phase flows; …



The FEM as a whole includes DSM, variational, and weighted residual formulations…

‘Grey Matter’ ‘Bits and Bytes’

Logan Ch# 2,3,4, & 5

Logan Ch# 4,6,9,13,14, & 16

1950

’s

1960

’s



Recovery & Interpretation

MECH 420 will provide knowledge of “how” an FE code or package can generate a “solution.”MECH 420 will focus on the importance of the human presence in the FE “analysis loop”.MECH 420 will begin by looking at structural analysis but will conclude by extending the FE technique to other classes of problems.



§1.2 Introduction to Matrix Notation.The FEM translates differential equations into a matrix set of algebraic equations.

ˆ and

ˆx x xdu Edx

ε σ ε= =

1 1

2 2

ˆ ˆ1 1ˆ ˆ1 1

x x

x x

d fAEL d f

⎧ ⎫ ⎧ ⎫−⎡ ⎤ ⎪ ⎪ ⎪ ⎪=⎨ ⎬ ⎨ ⎬⎢ ⎥−⎣ ⎦ ⎪ ⎪ ⎪ ⎪⎩ ⎭ ⎩ ⎭

FEM



The state of the structural system is defined by a matrix of displacements (generalized displacements), .

The external factors acting on the system are given by a force (generalized force) matrix, .

However, cartesian coordinate systems are used to define the sense of each displacement and force term.

D

F

{ }1 1 1 2

T

x y z x Ny NzF f f f f f f=

{ }1 1 1 2

T

x y z x Ny NzD d d d d d d=

Node 35{ }35 35 35 35

T

x y zf f f f=

Force is a vector with the sense of components being set by some x-y coordinate system



§1.3 Role of the Computer.The assembly process takes advantage of computing power.We will see in Chapters 2 and 3 how assembly occurs.Assembly can be automated.

It only requires knowledge of where an element sits within the finite element model.When working by hand we can use a connectivity table to help us perform assembly.Modern FE packages will handle assemblies of thousands of elements.The FE procedure has the advantage of producing a banded set of equations.Banded equations are less expensive to solve.



§1.4 General Steps to the Finite Element Method (p.6-13).1) (Discretize and) Set the Element Type.2) Set the Displacement Function.3) Define the Governing DE(s).4) Derive the Element Equations*.5) Assemble the Element Equations to Produce the Global

Equations & Apply BC’s to Reduce the Global System.6) Solve for the Unknowns.7) Recovery.8) Interpret.

*Step 4 has wildly differing looks depending on whether we apply a DSM, Variational, or Weighted Residual (WR) formulation.


Lecture 1: The Generalized FEM.

For now, we preview a WR formulation to emphasize the multi-physics nature of the FEM, and give some meaning to FEM terminology.

2

( ) 2d du xxdx dx x

⎛ ⎞ =⎜ ⎟⎝ ⎠

1.0 2.0x≤ ≤

[ ] 1.02.0

12.0 ; 2=

=

⎡ ⎤= −⎢ ⎥⎣ ⎦=x

x

duu xdx

The governing DE is the actual problem: find u(x).

The global domain of the problem

The conditions that must hold at the boundaries: the BC’s.


Lecture 1: The Generalized FEM

1( )xφ

0

1

1 1.0x = 2 1.25x =

2 ( )xφ

0

1

1 1.0x = 2 1.25x =

(1)1 1 2 2( ) ( ) ( )u x u x u xφ φ= ⋅ + ⋅

( )u x

x1 1.0x = 2 1.25x = 3 1.50x = 4 1.75x = 5 2.0x =

node 1 node 2 node 3 node 4 node 5element 1domain element 2

domain element 3domain element 4

domain

Step 1: (Discretize and) Set the Element Type.

We will use line segments to model the 1D function u(x).

Step 2: Set the Displacement Function.

For this problem the “displacement”is u.

u1

u2

(1) ( )u x



Step 3: Define the Governing DE(s).The problem defines the governing DE in this case.

Step 4: Derive the Element Equations.We are using a WR formulation.In Step 4 we are only concerned with satisfying the governing DEwithin the element.The WR formulation forces the approximate elemental solution to be a good one.How does one define a good approximation?

Each existing WR method uses a different criterion for the measure of goodness.

Collocation WR MethodLeast-Squares WR methodGalerkin WR Method – used exclusively in MECH 420.



Step 4: Derive the Element Equations. (cont’d…)

(1) (1) (1)1,1 1,2 1 1

(1)(1) (1)2 22,1 2,2

k k u Fu Fk k

⎡ ⎤ ⎧ ⎫⎧ ⎫=⎨ ⎬ ⎨ ⎬⎢ ⎥

⎩ ⎭ ⎩ ⎭⎣ ⎦

2

1

2

1

(1)

12

(1)

22

( ) 2 ( ) 0

( ) 2 ( ) 0

φ

φ

⎧ ⎫⎛ ⎞− =⎨ ⎬⎜ ⎟

⎝ ⎠⎩ ⎭⎧ ⎫⎛ ⎞

− =⎨ ⎬⎜ ⎟⎝ ⎠⎩ ⎭

∫

∫

x

xx

x

d du xx x dxdx dx x

d du xx x dxdx dx x

Element 1( )1 2x x x∈ →

(1) (1) (1) (1)1,1 1 2 1 1,2 1 2 2 1 1

1

(1) (1) (1) (1)2,1 1 2 1 2,2 1 2 2 2 2

2

( )( , ) ( , )

( )( , ) ( , )

⎡ ⎤+ = =⎢ ⎥⎣ ⎦

⎡ ⎤+ = − =⎢ ⎥⎣ ⎦

+

+

x

x

du xk x x u k x x u x Fdxdu xk x x u k x x u x F

dx

f

f



Step 4: Derive the Element Equations. (cont’d…).The evaluation of the residual equations for elements 1, 2, 3, and 4 produces 4 sets of element equations.

(1) (1) (1)1,1 1,2 1 1

(1)(1) (1)2 22,1 2,2

k k u Fu Fk k

⎡ ⎤ ⎧ ⎫⎧ ⎫=⎨ ⎬ ⎨ ⎬⎢ ⎥

⎩ ⎭ ⎩ ⎭⎣ ⎦Element 1

( )1 2x x x∈ →

(2) (2) (2)1,1 1,2 2 2

(2)(2) (2)3 32,1 2,2

k k u Fu Fk k

⎡ ⎤ ⎧ ⎫⎧ ⎫=⎨ ⎬ ⎨ ⎬⎢ ⎥

⎩ ⎭ ⎩ ⎭⎣ ⎦Element 2

( )2 3x x x∈ →

(3) (3) (3)1,1 1,2 3 3

(3)(3) (3)4 42,1 2,2

k k u Fu Fk k

⎡ ⎤ ⎧ ⎫⎧ ⎫=⎨ ⎬ ⎨ ⎬⎢ ⎥

⎩ ⎭ ⎩ ⎭⎣ ⎦Element 3

( )3 4x x x∈ →

(4) (4) (4)1,1 1,2 4 4

(4)(4) (4)5 52,1 2,2

k k u Fu Fk k

⎡ ⎤ ⎧ ⎫⎧ ⎫=⎨ ⎬ ⎨ ⎬⎢ ⎥

⎩ ⎭ ⎩ ⎭⎣ ⎦Element 4

( )4 5x x x∈ →



Step 5: Assemble the Element Equations to form the Global System.The assembly process (or superposition process as in Logan §2.4) produces a set of global equations.The global equations form a boundary value problem…

(1) (1) (1)1,1 1,2 1 1(1) (1) (2) (2) (1) (2)2,1 2,2 1,1 1,2 2 2 2

(2) (2) (3) (3) (2) (3)2,1 2,2 1,1 1,2 3 3 3

(3) (3) (4) (4)42,1 2,2 1,1 1,2

(4) (4)52,1 2,2

0 0 00 0

0 00 00 0 0

k k u fk k k k u f f

k k k k u f fu fk k k kuk k

⎡ ⎤ ⎧ ⎫⎢ ⎥ ⎪ ⎪+ +⎢ ⎥ ⎪ ⎪⎪ ⎪⎢ ⎥+ = +⎨ ⎬⎢ ⎥ ⎪ ⎪+⎢ ⎥ ⎪ ⎪⎢ ⎥ ⎪ ⎪⎩ ⎭⎣ ⎦

(1)

1

(3) (4)4 4

(4) (4)5

5

000

duxdx x

ff dux

dx x

⎧ ⎫⎡ ⎤⎪ ⎪⎢ ⎥⎣ ⎦⎧ ⎫ ⎪ ⎪

⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪+⎨ ⎬ ⎨ ⎬⎪ ⎪ ⎪ ⎪+⎪ ⎪ ⎪ ⎪

⎡ ⎤⎪ ⎪ ⎪ ⎪⎩ ⎭ −⎢ ⎥⎪ ⎪⎣ ⎦⎪ ⎪⎩ ⎭

Banded matrix structure makes numerical solution inexpensive. The structure is a result of the choice of node numbering scheme.



Step 5: Applying the Boundary Conditions to Reduce the Global System. (cont’d…)

Recall the conditions that existed at the boundaries of the global domain (the BCs):

1

2.0

2.0

12x

u

duxdx =

=

⎡ ⎤−⎢ ⎥⎣ ⎦=

This is a condition on a flux term. A flux is a rate of change of the desired function, u(x). This natural boundary condition is satisfied by the actual/true function.


(1) (1) (1)1,1 1,2 1 1(1) (1) (2) (2) (1) (2)2,1 2,2 1,1 1,2 2 2 2

(2) (2) (3) (3) (2) (3)2,1 2,2 1,1 1,2 3 3 3

(3) (3) (4) (4)42,1 2,2 1,1 1,2

(4) (4)52,1 2,2

0 0 00 0

0 00 00 0 0

k k u fk k k k u f f

k k k k u f fu fk k k kuk k

⎡ ⎤ ⎧ ⎫⎢ ⎥ ⎪ ⎪+ +⎢ ⎥ ⎪ ⎪⎪ ⎪⎢ ⎥+ = +⎨ ⎬⎢ ⎥ ⎪ ⎪+⎢ ⎥ ⎪ ⎪⎢ ⎥ ⎪ ⎪⎩ ⎭⎣ ⎦

(1)

1

(3) (4)4 4

(4) (4)5

5

000

duxdx x

ff dux

dx x

⎧ ⎫⎡ ⎤⎪ ⎪⎢ ⎥⎣ ⎦⎧ ⎫ ⎪ ⎪

⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪+⎨ ⎬ ⎨ ⎬⎪ ⎪ ⎪ ⎪+⎪ ⎪ ⎪ ⎪

⎡ ⎤⎪ ⎪ ⎪ ⎪⎩ ⎭ −⎢ ⎥⎪ ⎪⎣ ⎦⎪ ⎪⎩ ⎭


Step 5: Applying the Boundary Conditions to Reduce the Global System. (cont’d…)

2.0

0.5


(1) (1) (1)1,1 1,2 1(1) (1) (2) (2) (1) (2)2,1 2,2 1,1 1,2 2 2 2

(2) (2) (3) (3) (2) (3)2,1 2,2 1,1 1,2 3 3 3

(3) (3) (4) (4)42,1 2,2 1,1 1,2

(4) (4)52,1 2,2

0 0 0 2.00 0

0 00 00 0 0

k k fk k k k u f f

k k k k u f fuk k k kuk k

⎡ ⎤ ⎧ ⎫⎢ ⎥ ⎪ ⎪+ +⎢ ⎥ ⎪ ⎪⎪ ⎪⎢ ⎥+ = +⎨ ⎬⎢ ⎥ ⎪ ⎪+⎢ ⎥ ⎪ ⎪⎢ ⎥ ⎪ ⎪⎩ ⎭⎣ ⎦

(1)

1

(3) (4)4 4

(4)5

0000.5

duxdx x

f ff

⎧ ⎫⎡ ⎤−⎧ ⎫ ⎪ ⎪⎢ ⎥⎣ ⎦⎪ ⎪ ⎪ ⎪

⎪ ⎪ ⎪ ⎪⎪ ⎪+⎨ ⎬ ⎨ ⎬⎪ ⎪ ⎪ ⎪+⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪⎩ ⎭

−⎩ ⎭


Step 6: Solve for the remaining unknowns.

“Reduction”

(1) (2) (2) (1) (2)2,2 1,1 1,2 2 2 2

(2) (2) (3) (3) (2) (3)2,1 2,2 1,1 1,2 3 3 3

(3) (4)(3) (3) (4) (4)4 4 42,1 2,2 1,1 1,2

(4)(4) (4)5 52,1 2,2

0 0 00 0

000.50 0

k k k u f fk k k k u f f

u f fk k k ku fk k

⎡ + ⎤ ⎧ + ⎫⎧ ⎫ ⎧⎢ ⎥ ⎪ ⎪⎪ ⎪ ⎪+ +⎪ ⎪ ⎪ ⎪ ⎪⎢ ⎥ = +⎨ ⎬ ⎨ ⎬ ⎨⎢ ⎥ ++ ⎪ ⎪ ⎪ ⎪⎢ ⎥ ⎪ ⎪ ⎪ ⎪⎩ ⎭ ⎩ ⎭⎢ ⎥⎣ ⎦

(1)2,1

02.000

k⎧ ⎫⎫⎪ ⎪⎪⎪ ⎪ ⎪−⎬ ⎨ ⎬

⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪⎩ ⎭ ⎩ ⎭



1( )xφ

0

1

1 1.0x = 2 1.25x =

2 ( )xφ

0

1

1 1.0x = 2 1.25x =

(1)1 1 2 2( ) ( ) ( )u x u x u xφ φ= ⋅ + ⋅

( )1 1 2 2

2 1

1.11.1

1.10.25

x

du dx u udx dx

u u

φ φ=

⎡ ⎤ ⎡ ⎤≈ +⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦⎡ − ⎤⎛ ⎞≈ ⎢ ⎥⎜ ⎟

⎝ ⎠⎣ ⎦

Step 7: Recovery.Knowing the values of the function at the 5 key points, we can recover some information about other values of interest.For example, what is the approximate value of the flux term at the point x=1.1?The point x=1.1 lies inside the first element…



Step 8: Interpret the results.This case study illustrates the importance of the FEM user in the solution process:Interpretation includes looking at:

Convergence.Continuity and completeness conditions

What if you had wanted to know something about ?2

2

( )d u xdx

u(x)

flux

lecture 1: course introduction. - university of victoriamech420/lecture1.pdf · lecture 1: course...

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