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Weighted residual method

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(1)

Piotr Pluciński, Jerzy Pamin

Institute for Computational Civil Engineering

Cracow University of Technology

(2)

Weighted residual method

Strong formulation - local model

A(x)y 00 (x) + B(x)y 0 (x) + C(x)y(x) = D(x) , x ∈ (x a , x b ) + (example) boundary conditions

y(x a ) = b a – essential (Dirichlet) boundary condition y 0 (x b ) = b b – natural (Neumann) boundary condition

˜ y = X

i=0

Φ i α i = Φα

Residuum

R(x) = A(x)˜ y 00 (x) + B(x)˜ y 0 (x) + C(x)˜ y(x) − D(x)

(3)

Weighted residual method

Strong formulation - local model

A(x)y 00 (x) + B(x)y 0 (x) + C(x)y(x) = D(x) , x ∈ (x a , x b ) + (example) boundary conditions

y(x a ) = b a – essential (Dirichlet) boundary condition y 0 (x b ) = b b – natural (Neumann) boundary condition Function approximation

˜ y =

n

X

i=0

Φ i α i = Φα

R(x) = A(x)˜ y (x) + B(x)˜ y (x) + C(x)˜ y(x) − D(x)

(4)

Strong formulation - local model

A(x)y 00 (x) + B(x)y 0 (x) + C(x)y(x) = D(x) , x ∈ (x a , x b ) + (example) boundary conditions

y(x a ) = b a – essential (Dirichlet) boundary condition y 0 (x b ) = b b – natural (Neumann) boundary condition Function approximation

˜ y =

n

X

i=0

Φ i α i = Φα

Residuum

R(x) = A(x)˜ y 00 (x) + B(x)˜ y 0 (x) + C(x)˜ y(x) − D(x)

(5)

Weighted residual method

Residuum minimization Z x b

x a

w(x)R(x)dx = 0 ∀w w(x) 6= 0 – weighting function

Z x b x a

w(x) (A(x)˜ y 00 (x) + B(x)˜ y 0 (x) + C(x)˜ y(x) − D(x)) dx = 0

Different WRM variants depending on weighting function

least squares method – w i = dα i

Bubnov-Galerkin method – w i = Φ i

(6)

Weighted residual method

Residuum minimization Z x b

x a

w(x)R(x)dx = 0 ∀w w(x) 6= 0 – weighting function

Z x b x a

w(x) (A(x)˜ y 00 (x) + B(x)˜ y 0 (x) + C(x)˜ y(x) − D(x)) dx = 0

Different WRM variants depending on weighting function

least squares method – w i = dα i

Bubnov-Galerkin method – w i = Φ i

(7)

Weighted residual method

Residuum minimization Z x b

x a

w(x)R(x)dx = 0 ∀w w(x) 6= 0 – weighting function

Z x b x a

w(x) (A(x)˜ y 00 (x) + B(x)˜ y 0 (x) + C(x)˜ y(x) − D(x)) dx = 0

Different WRM variants depending on weighting function collocation method – w i = δ(x − x i )

Bubnov-Galerkin method – w i = Φ i

(8)

Weighted residual method

Residuum minimization Z x b

x a

w(x)R(x)dx = 0 ∀w w(x) 6= 0 – weighting function

Z x b x a

w(x) (A(x)˜ y 00 (x) + B(x)˜ y 0 (x) + C(x)˜ y(x) − D(x)) dx = 0

Different WRM variants depending on weighting function

collocation method – w i = δ(x − x i ) least squares method – w i = dR

dα i

(9)

Residuum minimization Z x b

x a

w(x)R(x)dx = 0 ∀w w(x) 6= 0 – weighting function

Z x b x a

w(x) (A(x)˜ y 00 (x) + B(x)˜ y 0 (x) + C(x)˜ y(x) − D(x)) dx = 0

Different WRM variants depending on weighting function

collocation method – w i = δ(x − x i ) least squares method – w i = dR

dα i

Bubnov-Galerkin method – w i = Φ i

(10)

Residuum minimization Z x b

x a

w(x)R(x)dx = 0 ∀w w(x) 6= 0 – weighting function

Z x b x a

w(x) (A(x)˜ y 00 (x) + B(x)˜ y 0 (x) + C(x)˜ y(x) − D(x)) dx = 0

Different WRM variants depending on weighting function

collocation method – w i = δ(x − x i ) least squares method – w i = dR

dα i

Bubnov-Galerkin method – w i = Φ i

(11)

Weak formulation - global model (∀w 6= 0)

Z x b x a

w(x) (A(x)˜ y 00 (x) + B(x)˜ y 0 (x) + C(x)˜ y(x) − D(x)) dx = 0

(12)

Weak formulation - global model (∀w 6= 0)

Z x b x a

w(x) (A(x)˜ y 00 (x) + B(x)˜ y 0 (x) + C(x)˜ y(x) − D(x)) dx = 0 Z x b

x a

w(x)A(x)˜ y 00 (x)dx+

+ Z x b

x a

w(x)B(x)˜ y 0 (x)dx + Z x b

x a

w(x)C(x)˜ y(x)dx − Z x b

x a

w(x)D(x)dx = 0

(13)

Weak formulation - global model (∀w 6= 0)

Z x b x a

w(x) (A(x)˜ y 00 (x) + B(x)˜ y 0 (x) + C(x)˜ y(x) − D(x)) dx = 0 Z x b

x a

w(x)A(x)˜ y 00 (x)dx+

+ Z x b

x a

w(x)B(x)˜ y 0 (x)dx + Z x b

x a

w(x)C(x)˜ y(x)dx − Z x b

x a

w(x)D(x)dx = 0

w(x)A(x)˜ y 0 (x)

x b

x a

− Z x b

x a

w 0 (x)A(x)˜ y 0 (x)dx − Z x b

x a

w(x)A 0 (x)˜ y 0 (x)dx+

+ Z x b

x a

w(x)B(x)˜ y 0 (x)dx + Z x b

x a

w(x)C(x)˜ y(x)dx − Z x b

x a

w(x)D(x)dx = 0

(14)

Weak formulation - global model (∀w 6= 0) w(x)A(x)˜ y 0 (x)

x b

x a

− Z x b

x a

w 0 (x)A(x)˜ y 0 (x)dx − Z x b

x a

w(x)A 0 (x)˜ y 0 (x)dx+

+ Z x b

x a

w(x)B(x)˜ y 0 (x)dx + Z x b

x a

w(x)C(x)˜ y(x)dx − Z x b

x a

w(x)D(x)dx = 0

w(x b )A(x b ) ˜ y 0 (x b ) − w(x a )A(x a )˜ y 0 (x a ) − Z x b

x a

w 0 (x)A(x)˜ y 0 (x)dx+

+ Z x b

x a

w(x)[B(x)−A 0 (x)]˜ y 0 (x)dx+

Z x b x a

w(x)C(x)˜ y(x)dx−

Z x b x a

w(x)D(x)dx = 0

(15)

Weak formulation - global model (∀w 6= 0) w(x)A(x)˜ y 0 (x)

x b

x a

− Z x b

x a

w 0 (x)A(x)˜ y 0 (x)dx − Z x b

x a

w(x)A 0 (x)˜ y 0 (x)dx+

+ Z x b

x a

w(x)B(x)˜ y 0 (x)dx + Z x b

x a

w(x)C(x)˜ y(x)dx − Z x b

x a

w(x)D(x)dx = 0

w(x b )A(x b ) ˜ y 0 (x b ) b b

− w(x a )A(x a )˜ y 0 (x a ) − Z x b

x a

w 0 (x)A(x)˜ y 0 (x)dx+

+ Z x b

x a

w(x)[B(x)−A 0 (x)]˜ y 0 (x)dx+

Z x b x a

w(x)C(x)˜ y(x)dx−

Z x b x a

w(x)D(x)dx = 0

(16)

Weak formulation - global model (∀w 6= 0) w(x)A(x)˜ y 0 (x)

x b

x a

− Z x b

x a

w 0 (x)A(x)˜ y 0 (x)dx − Z x b

x a

w(x)A 0 (x)˜ y 0 (x)dx+

+ Z x b

x a

w(x)B(x)˜ y 0 (x)dx + Z x b

x a

w(x)C(x)˜ y(x)dx − Z x b

x a

w(x)D(x)dx = 0

w(x b )A(x b )b b − w(x a )A(x a )˜ y 0 (x a ) − Z x b

x a

w 0 (x)A(x)˜ y 0 (x)dx+

+ Z x b

x a

w(x)[B(x)−A 0 (x)]˜ y 0 (x)dx+

Z x b x a

w(x)C(x)˜ y(x)dx−

Z x b x a

w(x)D(x)dx = 0 + essential boundary condition

y(x a ) = b a

(17)

Example formulation - one-dimensional heat flow

Heat amount

Q [J]

amount of heat energy

H = ∂Q

∂t [J/s=W] amount of heat per unit time

Heat flux density

q n = ∂H

∂A [W/m 2 ] for 1D: H(x) = A(x)q x (x)

heat flux per unit area

(18)

Example formulation - one-dimensional heat flow

Heat amount

Q [J]

amount of heat energy Heat flux

H = ∂Q

∂t [J/s=W]

amount of heat per unit time

q n = ∂H

∂A [W/m 2 ] for 1D: H(x) = A(x)q x (x)

heat flux per unit area

(19)

Heat amount

Q [J]

amount of heat energy Heat flux

H = ∂Q

∂t [J/s=W]

amount of heat per unit time Heat flux density

q n = ∂H

∂A [W/m 2 ] for 1D: H(x) = A(x)q x (x)

heat flux per unit area

(20)

Example formulation - one-dimensional heat flow

A(x)

x T

dx

0 l

f (x) [J/ms] – heat source

0 l x

T

(21)

A(x)

x T

dx

T (x)

0 l

f (x) [J/ms] – heat source

T (x)

0 l x

T

(22)

Example formulation - one-dimensional heat flow

A(x)

x T

dx

T (x)

0 l

f (x) [J/ms]

T (x)

0 l x

T

A ( x ) A ( x + d x )

dx

H H + dH

f (x)

H + f dx = H + dH

⇒ dH dx = f

Fourier’s law (H = Aq x ) q x = −k dT

dx k – heat conductivity coefficient

for Ak = const Ak d 2 T

dx 2 + f = 0

(23)

Example formulation - one-dimensional heat flow

A(x)

x T

dx

T (x)

0 l

f (x) [J/ms]

T (x)

0 l x

T

A ( x ) A ( x + d x )

dx

H H + dH

f (x)

H + f dx = H + dH ⇒ dH dx = f

Fourier’s law (H = Aq x ) q x = −k dT

dx k – heat conductivity coefficient

for Ak = const Ak d 2 T

dx 2 + f = 0

(24)

Example formulation - one-dimensional heat flow

A(x)

x T

dx

T (x)

0 l

f (x) [J/ms]

T (x)

0 l x

T

A ( x ) A ( x + d x )

dx

H H + dH

f (x)

H + f dx = H + dH ⇒ dH dx = f

Fourier’s law (H = Aq x ) q x = −k dT

dx k – heat conductivity coefficient

for Ak = const Ak d 2 T

dx 2 + f = 0

(25)

Example formulation - one-dimensional heat flow

A(x)

x T

dx

T (x)

0 l

f (x) [J/ms]

T (x)

0 l x

T

A ( x ) A ( x + d x )

dx

H H + dH

f (x)

H + f dx = H + dH ⇒ dH dx = f

Fourier’s law (H = Aq x ) q x = −k dT

dx k – heat conductivity coefficient

− d

dx Ak dT dx

!

= f

Ak dx 2 + f = 0

(26)

A(x)

x T

dx

T (x)

0 l

f (x) [J/ms]

T (x)

0 l x

T

A ( x ) A ( x + d x )

dx

H H + dH

f (x)

H + f dx = H + dH ⇒ dH dx = f

Fourier’s law (H = Aq x ) q x = −k dT

dx k – heat conductivity coefficient

− d

dx Ak dT dx

!

= f

for Ak = const Ak d 2 T

dx 2 + f = 0

(27)

x = 0 x = l x

Local model (strong formulation)

d

dx Ak dT dx

!

+ f = 0 + boundary conditions

q(x = 0) = − k dT dx

! x=0

= q b natural (Neumann) b.c.

T (x = l) = b T essential (Dirichlet) b.c.

(28)

Global model (weak formulation)

d

dx Ak dT dx

! + f = 0

· w

Z l

0

(29)

Global model (weak formulation)

d

dx Ak dT dx

! + f = 0

· w

Z l 0

Z l 0

w d

dx Ak dT dx

! + f

!

dx = 0, ∀w 6= 0

(30)

Global model (weak formulation)

d

dx Ak dT dx

! + f = 0

· w

Z l 0

Z l 0

w d

dx Ak dT dx

! + f

!

dx = 0, ∀w 6= 0

Z l 0

w d

dx Ak dT dx

! dx +

Z l 0

wf dx = 0

(31)

Global model (weak formulation)

d

dx Ak dT dx

! + f = 0

· w

Z l 0

Z l 0

w d

dx Ak dT dx

! + f

!

dx = 0, ∀w 6= 0

Z l 0

w d

dx Ak dT dx

! dx +

Z l 0

wf dx = 0

"

wAk dT dx

#

l

0

− Z l

0

dw dx Ak dT

dx

! dx +

Z l 0

wf dx = 0

(32)

Global model (weak formulation)

d

dx Ak dT dx

! + f = 0

· w

Z l 0

Z l 0

w d

dx Ak dT dx

! + f

!

dx = 0, ∀w 6= 0

Z l 0

w d

dx Ak dT dx

! dx +

Z l 0

wf dx = 0

"

wAk dT dx

#

l

0

− Z l

0

dw dx Ak dT

dx

! dx +

Z l 0

wf dx = 0

Z l 0

dw dx Ak dT

dx

!

dx = −(wAq x ) x=l

+ (wAq x ) x=0

+ Z l

0

wf dx = 0

(33)

Local model (strong formulation) d

dx Ak dT dx

!

+ f = 0

q x (x = 0) = − k dT dx

! x=0

= q b natural (Neumann) b.c.

T (x = l) = b T essential (Dirichlet) b.c.

Global model (weak formulation)

Z l 0

dw

dx Ak dT dx

!

dx = −(wAq x ) x=l

+ (wA) x=0

q + b Z l

0

wf dx = 0

T (x = l) = b T essential (Dirichlet) b.c.

(34)

Example

Derivation of weak format equation for 1D problem

f (x) = −x 2

x T

T 0 (0) = 1 2 T (3) = 1

A = 2 k = 1 2

q = − b 1 4 T = 1 b

Strong format d

dx Ak dT dx

!

+ f = 0 + boundary conditions

q x (x = 0) = −kT 0 (x = 0) = q b T (x = l) = b T

Weak format

Z 3 0

w T 00 − x 2  dx = 0

0 0

+ boundary condition T (3) = 1

(35)

Example

Derivation of weak format equation for 1D problem

f (x) = −x 2

x T

T 0 (0) = 1 2 T (3) = 1

A = 2 k = 1 2

q = − b 1 4 T = 1 b

Strong format

T 00 − x 2 = 0 + boundary conditions

q x (0) = − 1

2 T 0 (0) = − 1 4 T (3) = b T = 1

Weak format

Z 3 0

w T 00 − x 2  dx = 0

0 0

+ boundary condition T (3) = 1

(36)

Example

Derivation of weak format equation for 1D problem

f (x) = −x 2

x T

T 0 (0) = 1 2 T (3) = 1

A = 2 k = 1 2

q = − b 1 4 T = 1 b

Strong format

T 00 − x 2 = 0 + boundary conditions

q x (0) = − 1

2 T 0 (0) = − 1 4 T (3) = b T = 1

Weak format

Z 3 0

w T 00 − x 2  dx = 0

0 0

+ boundary condition T (3) = 1

(37)

f (x) = −x 2

x T

T 0 (0) = 1 2 T (3) = 1

A = 2 k = 1 2

q = − b 1 4 T = 1 b

Strong format

T 00 − x 2 = 0 + boundary conditions

q x (0) = − 1

2 T 0 (0) = − 1 4 T (3) = b T = 1

Weak format

Z 3 0

w T 00 − x 2  dx = 0

− Z 3

0

w 0 T 0 dx + w(3)T 0 (3) − w(0)T 0 (0) − Z 3

0

wx 2 dx = 0

+ boundary condition T (3) = 1

(38)

f (x) = −x 2

x T

T 0 (0) = 1 2 T (3) = 1

A = 2 k = 1 2

q = − b 1 4 T = 1 b

Strong format

T 00 − x 2 = 0 + boundary conditions

q x (0) = − 1

2 T 0 (0) = − 1 4 T (3) = b T = 1

Weak format

Z 3 0

w T 00 − x 2  dx = 0 Z 3

0

w 0 T 0 dx = w(3)T 0 (3) − w(0) · 1 2 −

Z 3 0

wx 2 dx

+ boundary condition T (3) = 1

Cytaty

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