Introduction to FEM

Session 2 (01/17/2013)

Variational Methods of ApproximationNumerical Methods which utilize approximate solution

to the differential equations in an attempt to minimize the residual error. Techniques that attempt to minimize the weighted residual error over the domain are called WEIGHTED RESIDUAL METHODS.

The various methods differ from each other in the choice of the weight function, approximate function, and integral formulation used.

Assume an approximate solution for the dependent variable

N

jjjN xCuxu

10)()(

Classic Variational Methods

• RAYLEIGH-RITZ (based on weak form)• PETROV-GALERKIN (based on integral form)• GALERKIN (based on integral form)• LEAST SQUARES (based on integral form)• COLLOCATION

Rayleigh-Ritz Method (RRM)• Uses weak form of the problem

– Equivalent to the original differential equation and includes the NBCs

– Places less restrictive continuity requirements on the dependent variables

• For a symbolic development of the method, the weak form of a BVP may be written as

• In the RRM we choose the weight function to be the same family as the approximate functions

0)(, wluwB

Niw i ,...,1

RRM (cont.) On substitution of the weight and approximate function

If the functional B(0,0) is bilinear we may separate the arguments of the approximate solution and factor out the Ritz coefficient

On substitution and rearranging

B is typically symmetric

01

, ( ) ( )N

i j j ij

B C x l

1,...,i N

0 01 1

, ( ) , ,N N

i j j i j j ij j

B C x B C B

01

, ,N

i j j i ij

B C l B

FCBFCB ij

N

jij

1

RRM (continued)1. φ0 must satisfy the specified EBCs2. φi must satisfy the homogenous form of

specified EBCs3. φi must be sufficiently differentiable as

required by the weak form4. φi must be a linearly independent set

Rows/columns of B must be linearly indepnednt for a solution to exist (necessary for [B]-1 to exist)

5. φi must be a complete setContains all terms of the lowest order admissible up to

highest order desired

RRM (continued)

Satisfies EBCs

Does not satisfy homogenous form of EBCs

not complete set. can’t generate linear terms

not linearly independent

satisfies all conditions

xL

100

, , : 0, , : (0) 0, ( ) 0jii j

ddB a dx L EBC u u L

dx dx

12

2

x

x

21

32

x x L

x x L

1

2 10

x x L

x x L

1

22

x x L

x x L

Methods of Weighted ResidualsMethods based on the weighted-integral form of the

differential equationsMethods maybe generally described by considering

the operator equation A(u)=f in ΩA() : Linear or Nonlinear operator, often a differential

operator acting on the dependent variable Linear operator satisfies A(αu+βv)=αA(u)+βA(v)

Linear in dependent variable uNonlinear

f: Function of independent variables

( ) d duA u a cu

dx dx

( ) d duA u u

dx dx

MWR (continued)

• Once again, we will seek the approximate solution

• we define the residual of the approximation as

• The parameters Cj are determined by requiring the residual to vanish in a weighted-integral sense

• Ψi: weight function

N

jjjN xCuxu

10)()(

01

( ) ( )N

N j jj

R A u f A C x f

0Rdi

MWR (continued)• The requirements on Φ0 and Φj for the weighted-residual

methods are different from those for Rayleigh-Ritz• Differentiability requirements are dictated by the

weighted-integral statement. Thus, Φj must have nonzero derivative up to the order appearing in the operator A().

• Unlike the Rayleigh-Ritz method (based on weak form), the approximate solution uN must satisfy both EBCs and NBCs

• Φ0 satisfies all specified boundary conditions• Φj satisfies homogenous form of all specified boundary

conditions

Petrov-Galerkin MethodThe approximate function Ψi are not equal to the

approximation functions Φ.

When A() is linear in its arguments

[A] is not symmetric

01

0 0N

i i j jj

Rd A C f d

1

N

ij j ij

A C F A C F

01

N

i j j ij

A d C f A d

Galerkin MethodThe approximate function Ψi are equal to the

approximation functions Φ.With a linear operator A()

[A] is not symmetricAlthough both the Rayleigh-Ritz and the Galerkin

methods both assume the same form for the weight function, the two methods are different due to the former using the weak form and later, the weighted integral.

The two methods yield the same solution when the same approximation function are used and all BCs are EBCs.

1

N

ij j ij

A C F

0,ij i j i iA A d F f A d

Least Square Method Parameter Cj of the approximate solution are

determined by minimizing the integral of the square of the residual

The minimal is found asDenoting this is a weighted-residual method If A() is a linear operator

[A] is symmetric

2R d

2 0 0

i i

RR d R d

C C

i

i

R

C

i ii

RA

C

0,ij i j i iA A A d F A f A d

1

N

ij j ij

A C F

Collocation MethodUnknown parameters Cj are found by requiring the

residual to be identically zero at N selected points Xi, i=1,…N in the domain Ω

Choosing N distinct points will yield N equations for the N unknowns R(Xi,Cj)=0

This method is a weighted-residual method with the weight function chosen as the Dirac delta function. Ψi=δ(X-Xi)

, , 0i i j i jRd X X R X C d R X C

fdXfX )(

Variational Methods of ApproximationConsider the following 2nd order BVP

With algebraic polynomials find the two-parameter approximation and compare with exact solution using:

Rayleigh-RitzGalerkin Petrov-GalerkinLeast-SquaresCollocation

2

0 1

0 0 (1) 0 10

d dua x q x

dx dx

u q x u a

41120exactu x x

RAYLEIGH-RITZ METHOD

Uses the weak form of the DEUses approximation solutionWe previously found

Recall that the weight functions must vanish at boundary points where EBCs are specified. That is, w satisfies the homogenous form of the specified EBCs. u(0)=u(1)=0; w(0)=w(1)=0

01

( )N

N j jj

u u C x

00

0 0L

x L x

dw du du dua wq dx w L a w a

dx dx dx dx

RRM (continued)

The weak form becomesUtilizing the example given in previous lecture,

the admissible set for a 2-parameetr Ritz approximation was

The weak form may be cast in functional form

0

0L dw du

a wq dxdx dx

21 20, ,x x L x x L

1

0

1

0

, ( ) 0

, ,

B w u l w

dw duB w u a dx Bilinear Symmetric Functional

dx dx

l w wqdx Linear Functional

RRM (continued)The weight function is chosen to be the same family as

the approximation functions w=Φi.On substitution and rearranging

1

001

1 1 000 0

, , ,

, , 0

Nji

ij j i ij i i ij

ii i i

ddB C F B a dx F l B

dx dx

d dl qdx B a dx

dx dx

21 2

2 21 2

0, ,

2 1, 3 2 , 10,

x x L x x L

x x x a q x