CE768 Lecture 1 2012 Post

The finite element method Formulation of the problems Derivation of a model problem

Introduction to the finite element method

CIVIL EN / MECH ENG 768: Lecture 1

Introduction

and

Formulation of the basic problems

March 26, 2012

1 / 50Introduction to the finite element method


Today’s lecture

1 The finite element method

2 Formulation of the problems

3 Derivation of a model problem



The finite element method

1 WHAT IS IT?

Thefinite element methodis a general technique for constructingapproximate solutions toinitial boundary value problems.

2 HOW IS IT APPLIED?

The method involves dividing the domain of the problem into afinitenumber of simple subdomains — thefinite elements— and usingvariational conceptsto construct approximate solutions over thecollection of elements.

3 WHAT IS IT USED FOR?

Because of its generality, it has been used to successfully solveproblems in virtually all branches of engineering and physicalsciences.



The finite element method

Some finite element show and tell



Differential equations

Many problems in engineering and science lead to equations involvingderivativesof unknown functions.

Such an equation is called adifferential equation (DE).

Examples:

dudx

= 5x+ 3

∂u∂t

+∂u∂x

= 0

∂2u∂x2

+∂2u∂y2

+∂2u∂z2

= 0




The unknown function,u, is called thedependent variable.

It is a function of one or moreindependent variables.

Independent variables are typically coordinates of a pointin space (x,y, andz) and/or timet.

Dependence ofu on the independent variables is expressed as:

u = u(x, y, z, t)




When there is one independent variable the DE is called anordinarydifferential equation (ODE), e.g.,

dudx

= 5x+ 3

When there are two or more independent variables the DE is called apartial differential equation (PDE) , e.g.,

∂2u∂x2

+∂2u∂y2

+∂2u∂z2

= 0

Theorder of the DE is defined as the order of the highest derivativeappearing in the equation.




In addition tou there may be other given functions that appear in theDE, e.g.,

a(x, t)∂u∂t

− b(x)∂2u∂x2

+ c(x, t)u = f (x, t)

These functions, together with any other information givenbeforehand,constitute thedata of the problem.

If the functionu and its derivatives appear only linearly in the DE, andthere are no products ofu and its derivatives, then the equation is calleda linear DE.

We will only be concerned with linear DEs in this class.



Specification of a domain

DEs are often used to describe physical processes over some,generallylimited, extent of space and/or time.

This extent is called thedomain of the problem.

To indicate that a point in space or time is in our domain, we write

x ∈ (0, l) or t ∈ (0, l) or (x, y) ∈ Ω,

where “x ∈ Ω” is read “x belongs toΩ”.

Ω

Ω = (0, l)

0 l y

x



Boundary conditions

For problems involving spatial variables, we need to specify the valueof u and/or one or more of its derivatives along the boundary∂Ω of thedomain., e.g.,

u = g or∂u∂x

= g on ∂Ω

This is called aboundary condition (BC).

∂Ω

Ω = (0, l)

x = 0 x = l

y

x

Ωu(0) = g1

u|∂Ω = g

u′(l) = g2



Initial conditions

For problems involving time, we must specify the value ofu andpossibly some of its derivatives at the initial timet = 0, e.g.,

u = g at t = 0

This is called aninitial condition (IC) .

∂Ω

Ω = (0, T ]

t = 0 t = T y

x

Ω

u(0) = g

u(x, y, 0) = g



Complete specification of a problem

So complete specification of a problem, or mathematical model,consists of the following:

Specification of a problem

1 A domain of interest,Ω, over which we wish to solve our problem.

2 DE(s) describing the process(es) of interest overΩ.

3 Boundary and/or initial conditions on∂Ω.



Types of problems

1 When only spatial variables are involved the problem is called a

boundary value problem

2 When only time is involved it is called an

initial value problem

3 When both time and space are involved it is an

initial boundary value problem



Steps involved in the formulation and analysis of aproblem

1

CONSTRUCTION OF A MATHEMATICAL MODEL

The DE-based model is one such model.

There are other, more or less equivalent, models that can be constructed.

Thefinite element methoduses avariational or weak form of theproblem.




1 CONSTRUCTION OF A MATHEMATICAL MODEL

2

CONSTRUCTION OF A SOLUTION(IF POSSIBLE)

Find a functionu that satisfies the DE overΩ and the boundary and/orinitial conditions on∂Ω.

In general, it isnot possible to findexact solutionsto these problems.

Numerical methods, such as thefinite element method, provide a meansto computeapproximate solutionsto these problems.





2 CONSTRUCTION OF A SOLUTION(IF POSSIBLE)

3

CONSIDER WELL-POSEDNESS OF THE PROBLEM

Exact solutions cannot generally be found, but we can still try to obtainsomequalitative informationabout the solution:

1 Does a solution exist?

2 If so, is the solution unique?

3 Does the solution depend continuously on the data?

If the answer to all three questions is yes, then the problem is said to bewell-posed.

In this class, most of our problems will be well-posed.






3 WELL-POSEDNESS OF THE PROBLEM

4

CONSTRUCTION OF AN APPROXIMATE SOLUTION

Given a well-posed problem for which an exact solution cannot be found,at this stage we attempt to construct anapproximate solution.

The two most well known approximation methods are thefinitedifferenceandfinite elementmethods.

This stage of the process using finite element methods will be the mainfocus of this class.






3 WELL-POSEDNESS OF THE PROBLEM

4 CONSTRUCTION OF AN APPROXIMATE SOLUTION

5

QUALITY OF THE APPROXIMATE SOLUTION

How good is the approximate solution?

Does the approximation procedure give a family of solutions thatconvergeto the exact solution?

If so, what is therate of convergence?



Basic Conservation Law

x = 0 x = l

A Ω

Time rate of change of

amount of quantity

in the domain

︸︷︷︸

1

=

Net rate at which quantity

flows across the

boundary

︸︷︷︸

2

+

Rate at which quantity

is created, or destroyed,

in the domain

︸︷︷︸

3



Basic Conservation Law: Term 1

Time rate of change ofamount of quantity

in the domain

=ddt

∫ b

aρ(x, t)︸︷︷︸

Amount of quantityVolume

Volume︷︸︸︷

Adx

x = a x = b

x = 0 x = l

A Ω′




Netrate at which quantity

flows across theboundary

= σ(a, t)︸︷︷︸

Amount of quantity(Area a)·time

A − σ(b, t)︸︷︷︸

Amount of quantity(Area b)·time

A

x = a x = b

x = 0 x = l

A Ω′

σ(a, t)




Rate at which quantityis created, or destroyed,

in the domain

=

∫ b

af (x, t)︸︷︷︸

Amount of quantity createdVolume·time

Volume︷︸︸︷

Adx

x = a x = b

x = 0 x = l

A Ω′



Fundamental integral conservation law

Putting it all together...

Time rate of change of

amount of quantity

in the domain

︸︷︷︸

=

Net rate at which quantity

flows across the

boundary

︸︷︷︸

+

Rate at which quantity

is created, or destroyed,

in the domain

︸︷︷︸

1 2 3

ddt

∫ b

aρ(x, t)Adx = Aσ(a, t)− Aσ(b, t) +

∫ b

af (x, t)Adx



Fundamental integral conservation law

Fundamentalintegral form of the conservation law:

ddt

∫ b

aρ(x, t)dx = σ(a, t) − σ(b, t) +

∫ b

af (x, t)dx

Must assume some “smoothness” of the functions to derive PDE.



Digression on the “smoothness” of functions

The degree of “smoothness” of a function can be characterized by howmany of its derivatives are continuous.

A function that is continuous is called aC0 function.

Or we say the function isC0 continuous.

Or we say that the function “belongs to” the set ofC0 functions,f ∈ C0.

x

f(x)

f(x) is C0 continuous

x

g(x)

g(x) is not C0 continuous!



Digression on the “smoothness” of functions

A function with continuous 1st derivatives is called aC1 function.

A function with continuous 2nd derivatives is called aC2 function.

...

A function with continuousm-th derivatives is called aCm function.

x

f(x)

f(x) is C0 continuous

x

f ′(x)

....but not C1 continuous!

0



Converting to a PDE

1 If∂ρ

∂tis continuous over the domain, then

ddt

∫ b

aρ(x, t)dx =

∫ b

a

∂ρ

∂tdx (1)

Special case ofLeibniz’s integral rule .

2 If∂σ

∂xis continuous over the domain, then

σ(b, t) − σ(a, t) =

∫ b

a

∂σ

∂xdx (2)

From theFundamental Theorem of Calculus.



Converting to a PDE

Substituting (1) and (2) into the integral form of conservation law

∫ b

a

[∂ρ

∂t+

∂σ

∂x− f

]

dx = 0

Only true for arbitrarya andb if [ · ] = 0. Thus, thePDE form of theconservation law is

∂ρ

∂t+

∂σ

∂x= f



Some specific examples

Specific problems can be derived by looking at a given conservationprinciple:

1 Conservation of energy

2 Conservation of mass

3 Conservation of linear momentum

Conserved quantity

ρ = ρ(u) =Amount of quantity

Volume,

whereu is some physical property (displacement, velocity,temperature, etc).

Flux σ = σ(u) is given (in part) by aconstitutive law.



Example: Heat transfer

Conservation principle: Conservation of energy

ρ = Energy Density= c(x)︸︷︷︸

specific heat

mass density︷︸︸︷

ρm(x) u(x, t)︸︷︷︸

temperature

Constitutive law: Fourier’s law

σ = − k(x)︸︷︷︸

Thermal conductivity

temperature gradient︷︸︸︷

∂u∂x



Example: Heat transfer

ρ = c(x)ρm(x)u(x, t) σ = − k(x)∂u

∂x

∂ρ

∂t+

∂σ

∂x= f

cρm∂u∂t

−∂

∂x

(

k∂u∂x

)

= f



Example: Mass transfer

Conservation principle: Conservation of mass

ρ = Mass Density or concentration= u(x, t)

Constitutive law: Fick’s law

σ = − D(x)︸︷︷︸

Diffusivity

concentration gradient︷︸︸︷

∂u∂x



Example: Mass transfer

ρ = u(x, t) σ = − D(x)∂u

∂x

∂ρ

∂t+

∂σ

∂x= f

∂u∂t

−∂

∂x

(

D∂u∂x

)

= f



Example: Axial displacement of an elastic bar

Conservation principle: Conservation of linear momentum

ρ = Momentum Density= ρm(x)︸︷︷︸

mass density

velocity︷︸︸︷

v = ρm(x)∂u∂t

Constitutive law: Hooke’s law

σ = − E(x)︸︷︷︸

elastic modulus

axial strain︷︸︸︷

∂u∂x



Example: Axial displacement of an elastic bar

ρ = ρm(x)∂u

∂tσ = − E(x)

∂u

∂x

∂ρ

∂t+

∂σ

∂x= f

ρm∂2u∂t2

−∂

∂x

(

E∂u∂x

)

= f



Many Examples

Physical Conservation Variable Flux Constitutive Material SourcesProblem Principle u σ Law Modulus Sinks

Heat Conservation of Temperature Heat Fourier’s law Thermal Heattransfer energy flux conductivity sources

Mass Conservation of Concen- Diffusive Fick’s law Diffusivity Masstransfer mass tration flux sources

Axial deformation Conservation of Axial Stress Hooke’s law Elastic Bodyof elastic bar of momentum displacement Modulus forces

Groundwater Conservation of Hydraulic Flow Darcy’s law Hydraulic Fluidflow mass head rate conductivity sources

Electrostatics Conservation of Electric Electric Coulomb’s law Dialectric Chargeelectric flux potential flux permittivity

Fluid flow Conservation of Velocity Shear Newton’s law Viscosity Bodymomentum stress of viscosity forces



PDE Model

General form of equations:

a(x)∂nu∂tn

−∂

∂x

(

k(x)∂u∂x

)

= f (x, t)

wherea(x) > 0 andk(x) > 0 andn = 1 or 2.

In this class, we will not consider the case wheren = 2.



PDE Model

Two important considerations:

1 Flux σ may have a component due to advection (or convection) of thequantity:

σ = c(x, t)u︸︷︷︸

Advective flux

−

Diffusive flux︷︸︸︷

k(x)∂u∂x

.

2 Source termf may have a component that is proportional tou, i.e.,

f = f (x, t)− b(x, t)u.



PDE Model

With these considerations, our one-dimensional PDE model becomes:

a(x)∂u∂t

+∂

∂x

(

c(x, t)u− k(x)∂u∂x

︸︷︷︸

σ

)

+ b(x, t)u = f (x, t)



Stages of development

1 One-dimensional, time-independent, no advection

a(x)∂u∂t

+∂

∂x

(


)

+ b(x, t)u = f (x, t)

−ddx

(

k(x)dudx

)

+ b(x)u = f (x)

2 Two-dimensional, time-independent, no advection

∇ ·

(

k(x, y)∇u(x, y)

)

+ b(x, y)u(x, y) = f (x, y)



Stages of development

3 One-dimensional, time-independent, advection included (c >> k)

−ddx

(

k(x)dudx

)

+ b(x)u = f (x)

ddx

(

c(x)u− k(x)dudx

)

+ b(x)u = f (x)

4 One-dimensional, advection included, time-dependent

a(x)∂u∂t

+∂

∂x

(


)

+ b(x, t)u = f (x, t)



First model DE

Our beginning model DE

−ddx

(

k(x)dudx

)

+ b(x)u = f (x) 0 < x < l

Need boundary conditions atx = 0 andx = l.

Called atwo-point boundary value problem.



Boundary conditions

Two types of boundary conditions:

1 Dirichlet or essentialboundary conditions: u = g on ∂Ω

2 Neumannor natural boundary conditions:dudx

= g on ∂Ω

Ω = (0, l)

x = 0 x = l

u(0) = g1 u′(l) = g2

Dirichlet BC Neumann BC



Boundary conditions

Calledhomogeneousboundary conditions wheng1 = g2 = 0.

Calledinhomogeneousboundary conditions wheng1 6= 0 andg2 6= 0.

We will use homogeneous, Dirichlet BCs for our first model problem.

Ω = (0, l)

x = 0 x = l

u(0) = 0 u′(l) = g2 6= 0

Homogeneous, Inhomogeneous,

Dirichlet BC Neumann BC



Model problem

Components of our model problem

1 Domain:0 < x < l or x ∈ (0, l)

2 DE:

−ddx

(

k(x)dudx

)

+ b(x)u = f (x)

3 BCs:u(0) = u(l) = 0



Model problem

Model two-point boundary value problem

Findu ∈ C2 such that

−ddx

(

k(x)dudx

)

+ b(x)u = f (x) 0 < x < l

u(0) = u(l) = 0

This is called theclassicalor strong form of the problem.



Next time

The finite element method makes use of an alternative form of theproblem.

This alternative formulation is called avariational or weak form.

In the next lecture, we will look at thevariational form of our modeltwo-point BVP.


Documents

CE768 Lecture 1 2012 Post