Weigted Residual Methods

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Approximate solutions, including FE solutions, can be constructed from governing differential equations. One approach is the Galerkin method. This can be applied to non-structural problems.

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Galerkin Method

Notation:

x independent variables, e.g. coordinates of a point

u u x dependent variables, e.g. displacements of a point

u u x approximate solution

f function of x (may be constant or zero)

D differential operat

or

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Problem Statement

In domain V:

Du - f 0

with appropriate B.C.

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Problem Statement

Residual in domain V:

R Du - f

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Approximate Solution

th

i

u is a linear combination

of basis functions

u is a typically a polynomial

of n terms whose

i term is muliplied

by a generalized d.o.f. a

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Solution

i

Minimize residual w.r.t. weights

Best Approximation:

W Rdv 0 i 1,2, ,n

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One-Dimensional Example

q cx P

TL

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One-Dimensional Example

x x

2

2

x

duE E

dx

d uAE q 0

dxd

A q 0dx

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One-Dimensional Example

2

2

T

d u cx0

dx AE

duAE P at x = L

dx

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Exact Solution

3

2

T

T

2

2

d u cx0

dx AEdu

AE P at x

cL

= Ld

P cu x x x

AE 2AE 6A

x

E

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Galerkin ProblemTL 2

i 20

ii

d u cxW dx 0

dx AE

duW

da

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Integration by Parts

d uv udv vdu

udv uv vdu

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Galerkin ProblemT

T T

L 2

i 20

L L2

i i20 0

i

2

2

i

d u cxW dx

dx AE

d u cxW dx W d

d u dudv dx v

dx d

dWu W

x 0dx

x

A

dux

E

d

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Galerkin Problem

T

T T

L 2

i 20

L L

ii

00

d uW dx

dx

dWdu duW dx

dx dx dx

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Galerkin Problem

T

T T

T T

L 2

i 20

L L

L L

ii i

00

2

i i20 0

dWdu du cxW W dx

d u cxW dx

dx AE

d u cxW dx W dx 0

dx AE

dx dx dx AE

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Assumed Function2

1 2

11

1

2 22

2

u a x a x

du dWW x 1da dxdu dW

W x 2xda dx

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Integrals

T

T

L

1 2 T

0

L2

1 2 T

0

cx P1 a 2a x x dx L 0

AE AE

cx P2x a 2a x x dx L 0

AE AE

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Solution

2T

1

T2

7cLPa

AE 12AE

cLa

4AE

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Solution

22T T

2T T

7cL cLPu x x x

AE 12AE 4AE

7cL cLdu PE x

dx A 12A 2A2

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Solution2

23T

2T T7cL cLPu x x x

AE 12AE 4AE

cLP cu x x x

AE 2AE 6AE

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Comparison of Results

Quantity Location Exact Galerkinu L/2 0.2292 0.2292u L 0.3333 0.3333u,x 0 0.5000 0.5833u,x L/2 0.3750 0.3333u,x L 0.0000 0.0833

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Galerkin FEM Formulation:Uniform Bar, Axial Load

2

,xx2

,x

d uAE q(x) AEu q(x) 0

dxF AEu

L

q q x

x,u

A,E

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Galerkin FEM Formulation:Uniform Bar, Axial Load

T

1 2

u N d

L x xN

L L

d u u

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Galerkin FEM Formulation:Uniform Bar, Axial Load

1 1 2 2

i ii

u N u N u

uW N

d

L x xN

L L

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Galerkin FEM Formulation:Uniform Bar, Axial Load

jels

LN

i ,xxj 1 0

N AEu q dx 0

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Galerkin FEM Formulation

j

j

L

i ,xx

0

LL

i ,x i,x ,x00

For EA constant:

N AEu dx

N AEu N AEu dx

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B.C.

,xF=AEu

For ends of the element

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Galerkin FEM Formulation:Uniform Bar, Axial Load

jels

jels els

LN

i ,xxj 1 0

LN NL

i,x ,x i i ,x 0j 1 j 10

N AEu q dx 0

N AEu N q dx N AEu 0

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,x

1,x

2

B N

u1 1u B d

uL L

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jels

els

els

els els

LN

i,x ,x ij 1 0

NL

i ,x 0j 1

LNT

jj 1 0

LN N LT T

0j 1 j 10

N AEu N q dx

N AEu 0

B AE B dx d

N qdx N F 0

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els

els

els

LNT

jj 1 0

LNT

e jj 1 0

N LT

0j 1

k B AE B dx

r N qdx

P N F 0

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Beam Dynamics

4 2

L4 2

,xxxx L

d v d vEI (x) 0

dx dtEIv v 0

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B.C.

,xx B

,xxx B

EIv -M =0

EIv -V =0

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Shape Functions

3 2 31 3

3 2 2 32 3

3 23 3

3 2 24 3

1ˆ ˆN 2x 3x L L

L1

ˆ ˆ ˆN x L 2x L xLL1

ˆ ˆN 2x 3x LL1

ˆ ˆN x L x LL

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Shape Functions for Beam Element

-0.500

0.000

0.500

1.000

0

N1 N3

N2

N4

L

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1 1 2 1 3 2 4 2

1

1

2

2

i ii

v N v N N v N

v

dv

vW N

d

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jL

T

,xxxx L

0

N EIv v dx 0

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LT

,xxxx

0

L LT TT

,xx ,xx ,xxx ,x ,xx00

EI constant Integration

(by parts twice!)

N v dx

N v dx N v N v

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jLT

,xxxx L

0

LT T

,xx ,xx L

0

LTT

,xxx ,x ,xx0

N EIv v dx 0

N EI v N v dx

N v N v 0

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L

T T

,xx ,xx L

0

LTT

,xxx ,x ,xx0

N EI v N v dx

N v N v

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,xx B

,xxx B

LT T

,xx ,xx L

0

LTT B B

,x0

EIv -M =0

EIv -V =0

N EI v N v dx

V MN N

EI EI

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,xx

,xx

v N d

v B d

B N

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LT

0

LT

L

0

LTT B B

,x0

k B EI B dx

m N N dx

V MR N N

EI EI

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Heat Flow in a Bar

,x

dAkT

dx

Af Af d Af

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T

eT N T

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1 1 2 2

i ii

T N T N T

TW N

d

L x xN

L L

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L

T

,x

0

0dN AkT dx

dx 0

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L

T T

,x ,x

0

0N AkT dx N Af

0

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,x ,x e

LT 1 1

,x ,x e2 20

T N T

A FN Ak N T dx

A F

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LT

T ,x ,x e

0

T

k N Ak N T dx

1 1k kA

1 1

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Two Dimensional Problems

x ,x y ,y

x ,x y ,y B

In volume V: k k Q 0x y

is knownOn boundary S: either

lk mk f 0

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Poisson’s Equation

x ,x y ,y

x y

2

k k Q 0x y

If k k k constant:

k Q 0

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Shape Functions

eN

Shape functions depend on 2D element:L

CST, LST, Quad, Quadratic, etc.

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Galerkin Residuals

T

x ,x y ,y

0

0N k k Q dxdy

x y

0

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Integration by Parts

T

x ,x

T

,x x ,x

T

x ,x

N k dxdyx

N k dxdy

N k ldS

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Integration by Parts

T

y ,y

T

,y y ,y

T

y ,y

N k dxdyy

N k dxdy

N k mdS

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,x ,x e

,y ,y e

N

N

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Galerkin Residuals

TT

,x x ,x ,y y ,y e

T T

B

N k N N k N dxdy

N Qdxdy N f dS

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Galerkin Residuals

TT

,x x ,x ,y y ,y

T T

B

k N k N N k N dxdy

r N Qdxdy N f dS