FEM for stationary heat flow

FEM for stationary heat flow

Piotr Pluciński e-mail: pplucin@L5.pk.edu.pl

Jerzy Pamin

e-mail: jpamin@L5.pk.edu.pl

Lecture contents

1 Stationary heat flow in 3D Model - strong formulation Model - weak formulation FE equations

2 Selection of approximation functions

Shape functions for 1D problems

Shape functions for 2D problems

Shape functions for 3D problems

Stationary heat flow in 3D

q n

q _n = q ^T n

∇ =







∂

∂x

∂

∂y

∂

∂z







cold hot

q ∇T

Fundamental unknown - temperature T

Fourier’s law of heat conduction q = −D ∇T

heat flux density vector q = {q x q y q z } [W/m ² ] temperature gradient gradT = ∇T = n

∂T

∂x

∂T

∂y

∂T

∂z

o [K/m]

heat conduction matrix D = {k ij } [W/(mK)]

Flux density grows with temperature gradient.

Heat flows from higher to lower temperature.

Stationary heat flow in 3D

q n

q _n = q ^T n

cold hot

q ∇T

Fundamental unknown - temperature T

Fourier’s law of heat conduction q = −D ∇T

heat flux density vector q = {q x q y q z } [W/m ² ] temperature gradient gradT = ∇T = n

∂T

∂x

∂T

∂y

∂T

∂z

o [K/m]

heat conduction matrix D = {k ij } [W/(mK)]

Flux density grows with temperature gradient.

Heat flows from higher to lower temperature.

Heat energy balance

Heat generated equal to heat flowing out

Z

V

f dV = Z

S

q n dS ∀ V

f – heat source density – energy supplied to the body per unit volume and time [W/m ³ ]

V

S

Using Green-Gauss-Ostrogradsky theorem about integration by parts

Z

S

q _n dS = Z

S

q ^T n dS = Z

V

divq dV = Z

V

 ∂q _x

∂x + ∂q _y

∂y + ∂q _z

∂z

 dV Z

V

f dV = Z

V

∇ ^T q dV ∀ V ⇒ ∇ ^T q = f ∀x ∈ V

Heat energy balance

Heat generated equal to heat flowing out

Z

V

f dV = Z

S

q n dS ∀ V

f – heat source density – energy supplied to the body per unit volume and time [W/m ³ ]

V

S Using Green-Gauss-Ostrogradsky theorem about integration by parts

Z

S

q _n dS = Z

S

q ^T n dS = Z

V

divq dV = Z

V

 ∂q _x

∂x + ∂q _y

∂y + ∂q _z

∂z

 dV Z

V

f dV = Z

V

∇ ^T q dV ∀ V ⇒ ∇ ^T q = f ∀x ∈ V

Heat flow equations

Heat conduction equation (strong formulation)

∇ ^T (D∇T ) + f = 0 ∀x ∈ V + boundary conditions

q _n = q ^T n = b q on S _q – natural b.c. (Neumann) T = b T on S _T – essential b.c. (Dirichlet)

V

S _T S _q

Conduction matrix for isotropic materials D = kI

∂ ² T

∂x ² + ∂ ² T

∂y ² + ∂ ² T

∂z ² + f

k = 0 – Poisson equation For isotropic materials without heat source

∂ ² T

∂x ² + ∂ ² T

∂y ² + ∂ ² T

∂z ² = 0 – Laplace equation

Heat flow equations

Heat conduction equation (strong formulation)

∇ ^T (D∇T ) + f = 0 ∀x ∈ V + boundary conditions

q _n = q ^T n = b q on S _q – natural b.c. (Neumann) T = b T on S _T – essential b.c. (Dirichlet)

V

S _T S _q

Conduction matrix for isotropic materials D = kI

∂ ² T

∂x ² + ∂ ² T

∂y ² + ∂ ² T

∂z ² + f

k = 0 – Poisson equation

For isotropic materials without heat source

∂ ² T

∂x ² + ∂ ² T

∂y ² + ∂ ² T

∂z ² = 0 – Laplace equation

Heat flow equations

Heat conduction equation (strong formulation)

∇ ^T (D∇T ) + f = 0 ∀x ∈ V + boundary conditions

q _n = q ^T n = b q on S _q – natural b.c. (Neumann) T = b T on S _T – essential b.c. (Dirichlet)

V

S _T S _q

Conduction matrix for isotropic materials D = kI

∂ ² T

∂x ² + ∂ ² T

∂y ² + ∂ ² T

∂z ² + f

k = 0 – Poisson equation For isotropic materials without heat source

∂ ² T

∂x ² + ∂ ² T

∂y ² + ∂ ² T

∂z ² = 0 – Laplace equation

Stationary heat flow in 3D

V

S T S q

Weak formulation Z

V

w ∇ ^T (D∇T ) + f
dV = 0 ∀w 6= 0

Stationary heat flow in 3D

V

S T S q

Weak formulation Z

V

w ∇ ^T (D∇T ) + f
dV = 0 ∀w 6= 0

− Z

V

(∇w) ^T D∇T dV + Z

S

w (D∇T ) ^T ndS + Z

V

wf dV = 0

Stationary heat flow in 3D

V

S T S q

Weak formulation Z

V

w ∇ ^T (D∇T ) + f
dV = 0 ∀w 6= 0

− Z

V

(∇w) ^T D∇T dV + Z

S

w  D∇T

−q

 ^T ndS +

Z

V

wf dV = 0

Stationary heat flow in 3D

V

S T S q

Weak formulation Z

V

w ∇ ^T (D∇T ) + f
dV = 0 ∀w 6= 0

− Z

V

(∇w) ^T D∇T dV − Z

S

wq ^T ndS + Z

V

wf dV = 0

Stationary heat flow in 3D

V

S T S q

Weak formulation Z

V

w ∇ ^T (D∇T ) + f
dV = 0 ∀w 6= 0

− Z

V

(∇w) ^T D∇T dV − Z

S

w q ^T n q n

dS + Z

V

wf dV = 0

Stationary heat flow in 3D

V

S T S q

Weak formulation Z

V

w ∇ ^T (D∇T ) + f
dV = 0 ∀w 6= 0

− Z

V

(∇w) ^T D∇T dV − Z

S

wq n dS + Z

V

wf dV = 0

Stationary heat flow in 3D

V

S T S q

Weak formulation Z

V

w ∇ ^T (D∇T ) + f
dV = 0 ∀w 6= 0

− Z

V

(∇w) ^T D∇T dV − Z

S

wq n dS + Z

V

wf dV = 0 Z

V

(∇w) ^T D∇T dV = − Z

S q

w qdS − b Z

S T

wq n dS + Z

V

wf dV ∀w

Stationary heat flow in 3D

V

S T S q

Weak formulation Z

V

w ∇ ^T (D∇T ) + f
dV = 0 ∀w 6= 0

− Z

V

(∇w) ^T D∇T dV − Z

S

wq n dS + Z

V

wf dV = 0 Z

V

(∇w) ^T D∇T dV = − Z

S q

w q dS − b Z

S T

w q n dS + Z

V

wf dV ∀w

natural b.c. secondary unknown

Stationary heat flow in 3D

V

S T S q

Weak formulation Z

V

w ∇ ^T (D∇T ) + f
dV = 0 ∀w 6= 0

− Z

V

(∇w) ^T D∇T dV − Z

S

wq n dS + Z

V

wf dV = 0 Z

V

(∇w) ^T D∇T dV = − Z

S q

w qdS − b Z

S T

wq n dS + Z

V

wf dV ∀w + boundary condition

T = b T on S T

Stationary heat flow in 3D

Strong formulation

∇ ^T (D∇T ) + f = 0 ∀x ∈ V + boundary conditions

q n = q ^T n = q b on S q

T = b T on S T

V

S T S q

Weak formulation

Z

V

(∇w) ^T D∇T dV = − Z

S q

w q dS − b Z

S T

wq _n dS + Z

V

wf dV ∀w 6= 0

+ boundary condition

T = b T on S _T

Stationary heat flow in 3D

Strong formulation

∇ ^T (D∇T ) + f = 0 ∀x ∈ V + boundary conditions

q n = q ^T n = q b on S q

T = T b on S T

V

S T S q

Weak formulation

Z

V

(∇w) ^T D∇T dV = − Z

S q

w q b dS − Z

S T

wq _n dS + Z

V

wf dV ∀w 6= 0

+ boundary condition

T = T b on S _T

Stationary heat flow in 3D

Set of FE equations

T = NΘ – approximated temperature function N – shape function vector

Θ – nodal temperature (dof) vector

∇T = BΘ – approximated temperature gradient function B = ∇N – shape function derivative matrix

Z

V

(∇w) ^T D∇T dV = − Z

S _q

w qdS − b Z

S _T

wq _n dS + Z

V

wf dV ∀w 6= 0

KΘ = f b + f K =

Z

V

B ^T DBdV, f b = − Z

S _q

N ^T q dS− b Z

S _T

N ^T q n dS, f = Z

V

N ^T f dV

Selection of approximation functions

Fundamental steps in FE procedure

1 Build a strong formulation of a problem

2 Convert the formulation into a weak format

3 Select approximation of unknown function

4 Select weighting function

Selection of approximation functions

Fundamental steps in FE procedure

1 Build a strong formulation of a problem

2 Convert the formulation into a weak format

3 Select approximation of unknown function

4 Select weighting function

Selection of approximation functions

Fundamental steps in FE procedure

1 Build a strong formulation of a problem

2 Convert the formulation into a weak format

3 Select approximation of unknown function

4 Select weighting function

Selection of approximation functions

Fundamental steps in FE procedure

1 Build a strong formulation of a problem

2 Convert the formulation into a weak format

3 Select approximation of unknown function

4 Select weighting function

Selection of approximation functions in 1D

1 Strong formulation d

dx Ak dT dx

!

+ f = 0 + boundary conditions

q _x = q b at x _q ( e.g. x _q = 0) T = b T at x T ( e.g. x T = l)

x _q = 0 x T = l

2 Conversion into 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

+ b.c.

T = b T at x T ( e.g. x T = l)

Selection of approximation functions in 1D

3 Selection of approximation functions Linear function

T ^e (x) = α 1 + α 2 x = Φα ^e Φ = [1 x], α ^e =

 α 1

α 2



T ^e (x) = N _i ^e (x ^e )T i + N _j ^e (x ^e )T j = N ^e Θ ^e N ^e = [N _i ^e (x ^e ) N _j ^e (x ^e )], Θ ^e =

 T i

T j



dT ^e

dx ^e = B ^e Θ ^e , B ^e = dN ^e dx ^e =

" dN _i ^e

dx ^e dN _j ^e

dx ^e

#

x T

T i

T j

x _i x _j l ^e

x ^e

0 ^e l ^e

1

N _i ^e (x ^e )

x ^e

0 ^e l ^e

1

N _j ^e (x ^e )

Selection of approximation functions in 1D

3 Selection of approximation functions Linear function

T ^e (x) = α 1 + α 2 x = Φα ^e Φ = [1 x], α ^e =

 α 1

α 2



T ^e (x) = N _i ^e (x ^e )T i + N _j ^e (x ^e )T j = N ^e Θ ^e N ^e = [N _i ^e (x ^e ) N _j ^e (x ^e )], Θ ^e =

 T i

T j



dT ^e

dx ^e = B ^e Θ ^e , B ^e = dN ^e dx ^e =

" dN _i ^e

dx ^e dN _j ^e

dx ^e

#

x T

T i

T j

x _i x _j l ^e

x ^e

0 ^e l ^e

1

N _i ^e (x ^e )

x ^e

0 ^e l ^e

1

N _j ^e (x ^e )

Selection of approximation functions in 1D

3 Selection of approximation functions Linear function

T ^e (x) = α 1 + α 2 x = Φα ^e Φ = [1 x], α ^e =

 α 1

α 2



T ^e (x) = N _i ^e (x ^e )T i + N _j ^e (x ^e )T j = N ^e Θ ^e N ^e = [N _i ^e (x ^e ) N _j ^e (x ^e )], Θ ^e =

 T i

T j



dT ^e

dx ^e = B ^e Θ ^e , B ^e = dN ^e dx ^e =

"

dN _i ^e dx ^e

dN _j ^e dx ^e

#

x T

T i

T j

x _i x _j l ^e

x ^e

0 ^e l ^e

1

N _i ^e (x ^e )

x ^e

0 ^e l ^e

1

N _j ^e (x ^e )

Selection of approximation functions in 1D

3 Selection of approximation functions Quadratic function

T ^e (x) = α 1 + α 2 x + α 3 x ² = Φα ^e Φ = [1 x x ² ], α ^e =



 α 1

α 2

α 3





T ^e (x) = N _i ^e (x ^e )T _i + N _j ^e (x ^e )T _j + N _k ^e (x ^e )T _k

= N ^e Θ ^e

N ^e = [N _i ^e (x ^e ) N _j ^e (x ^e ) N _k ^e (x ^e )], Θ ^e =



 T _i T j

T k





dT ^e

dx ^e = B ^e Θ ^e , B ^e = dN ^e dx ^e =

" dN _i ^e

dx ^e dN _j ^e

dx ^e

#

x T

T i

T j

T k

x i x j x k

l ^e

x ^e 0 ^e x ^e _j l ^e 1 N _i ^e (x ^e )

x ^e 0 ^e x ^e _j l ^e

1 N _j ^e (x ^e )

x ^e 0 ^e x ^e j l ^e

1

N _k ^e (x ^e )

Selection of approximation functions in 1D

3 Selection of approximation functions Quadratic function

T ^e (x) = α 1 + α 2 x + α 3 x ² = Φα ^e Φ = [1 x x ² ], α ^e =



 α 1

α 2

α 3



 T ^e (x) = N _i ^e (x ^e )T _i + N _j ^e (x ^e )T _j + N _k ^e (x ^e )T _k

= N ^e Θ ^e

N ^e = [N _i ^e (x ^e ) N _j ^e (x ^e ) N _k ^e (x ^e )], Θ ^e =



 T _i T _j T k





dT ^e

dx ^e = B ^e Θ ^e , B ^e = dN ^e dx ^e =

"

dN _i ^e dx ^e

dN _j ^e dx ^e

dN _k ^e dx ^e

#

x T

T i

T j

T k

x i x j x k

l ^e

x ^e 0 ^e x ^e _j l ^e 1 N _i ^e (x ^e )

x ^e 0 ^e x ^e _j l ^e

1 N _j ^e (x ^e )

x ^e

1

N _k ^e (x ^e )

Selection of approximation functions in 2D

1 Strong formulation

∇ ^T (D∇T ) + f = 0 ∀x ∈ A + boundary conditions

q n = q ^T n = q b on Γ q

T = b T on Γ T Γ T

Γ q

2 Conversion into weak formulation (h - configuration thickness)

Z

A

(∇w) ^T Dh∇T dA = − Z

Γ _q

wh qdΓ − b Z

Γ _T

whq _n dΓ + Z

A

whf dA

+ boundary condition

T = b T on Γ T

Selection of approximation functions in 2D

3 Selection of approximation functions Three-node element

T ^e (x, y) = α ₁ + α ₂ x + α ₃ y = Φα ^e

Φ = [1 x y], α ^e =



 α ₁ α ₂ α ₃





T ^e (x, y) = N _i ^e (x ^e , y ^e )T i + N _j ^e (x ^e , y ^e )T j + + N _k ^e (x ^e , y ^e )T k = N ^e Θ ^e

N ^e = [N _i ^e (x ^e , y ^e ) N _j ^e (x ^e , y ^e ) N _k ^e (x ^e , y ^e )], Θ ^e =



 T i

T j

T k





y T (x, y)

x T i

T k

T j

i

k

j

y ^e x ^e

i k

j

e

Selection of approximation functions in 2D

Pascal triangle

– element

1

x y

x ² xy y ²

x ³ x ² y xy ² y ³

x ⁴ x ³ y x ² y ² xy ³ y ⁴

Selection of approximation functions in 2D

Pascal triangle – three-node element 1

x y

x ² xy y ²

x ³ x ² y xy ² y ³

x ⁴ x ³ y x ² y ² xy ³ y ⁴

Selection of approximation functions in 2D

Pascal triangle – six-node element 1

x y

x ² xy y ²

x ³ x ² y xy ² y ³

x ⁴ x ³ y x ² y ² xy ³ y ⁴

Selection of approximation functions in 2D

Convergence conditions - requirements for FE approximation

completeness – approximation must be able to represent a constant field and a constant field gradient

continuity (conforming FE) – approximation must be continuous

along interelement edges

Selection of approximation functions in 2D

Convergence conditions - requirements for FE approximation

completeness – approximation must be able to represent a constant field and a constant field gradient

continuity (conforming FE) – approximation must be continuous

along interelement edges

Selection of approximation functions in 2D

3 Selection of approximation functions Three-node element

T ^e (x, y) = α ₁ + α ₂ x + α ₃ y = Φα ^e

Φ = [1 x y], α ^e =



 α ₁ α ₂ α ₃





T ^e (x, y) = N _i ^e (x ^e , y ^e )T i + N _j ^e (x ^e , y ^e )T j + + N _k ^e (x ^e , y ^e )T _k = N ^e Θ ^e

N ^e = [N _i ^e (x ^e , y ^e ) N _j ^e (x ^e , y ^e ) N _k ^e (x ^e , y ^e )], Θ ^e =



 T i

T j





y T (x, y)

x T i

T k

T j

i

k

j

y ^e x ^e

i k

j

e

Selection of approximation functions in 2D

3 Selection of approximation functions Three-node element

N ^e = [N _i ^e (x ^e , y ^e ) N _j ^e (x ^e , y ^e ) N _k ^e (x ^e , y ^e )]

y ^e N i (x ^e , y ^e )

x ^e 1 i

k

j

y ^e N j (x ^e , y ^e )

x ^e 1

i k

j y ^e

N k (x ^e , y ^e )

x ^e

1 i

k

j

Determination of shape functions N _i (x ^e , y ^e ) = α _1i + α _2i x ^e + α _3i y ^e





1 x i y i

1 x j y j

1 x k y k







 α 1i

α 2i

α 3i



 =



 1 0 0





=⇒

α 1i = ^{x j y k} _2P ^{−x k y j}

4

α _2i = ^y _2P ^{j −y k}

4

α _3i = ^x _2P ^{k −x j}

4

Selection of approximation functions in 2D

3 Selection of approximation functions Three-node element

N ^e = [N _i ^e (x ^e , y ^e ) N _j ^e (x ^e , y ^e ) N _k ^e (x ^e , y ^e )]

e.g. for N _i (x ^e , y ^e ) N _i (x ^e _i , y _i ^e ) = 1 N i (x ^e _j , y _j ^e ) = 0 N i (x ^e _k , y ^e _k ) = 0

y ^e N i (x ^e , y ^e )

x ^e 1 i

k

j

y ^e N j (x ^e , y ^e )

x ^e 1

i k

j y ^e

N k (x ^e , y ^e )

x ^e

1 i

k

j

Determination of shape functions N _i (x ^e , y ^e ) = α _1i + α _2i x ^e + α _3i y ^e





1 x i y i

1 x j y j

1 x k y k







 α 1i

α 2i

α 3i



 =



 1 0 0





=⇒

α 1i = ^{x j y k} _2P ^{−x k y j}

4

α _2i = ^y _2P ^{j −y k}

4

α _3i = ^x _2P ^{k −x j}

4

Selection of approximation functions in 2D

3 Selection of approximation functions Three-node element

N ^e = [N _i ^e (x ^e , y ^e ) N _j ^e (x ^e , y ^e ) N _k ^e (x ^e , y ^e )]

e.g. for N _i (x ^e , y ^e ) N _i (x ^e _i , y _i ^e ) = 1 N i (x ^e _j , y _j ^e ) = 0 N i (x ^e _k , y ^e _k ) = 0

y ^e N i (x ^e , y ^e )

x ^e 1 i

k

j

y ^e N j (x ^e , y ^e )

x ^e 1

i k

j y ^e

N k (x ^e , y ^e )

x ^e

1 i

k

j

Determination of shape functions N _i (x ^e , y ^e ) = α _1i + α _2i x ^e + α _3i y ^e





1 x i y i

1 x j y j

1 x k y k







 α 1i

α 2i

α 3i



 =



 1 0 0





=⇒

α 1i = ^{x j y} _2P ^{k −x k y j}

4

α _2i = ^y _2P ^{j −y k}

4

α _3i = ^x _2P ^{k −x j}

4

Selection of approximation functions in 2D

3 Selection of approximation functions Three-node element

N ^e = [N _i ^e (x ^e , y ^e ) N _j ^e (x ^e , y ^e ) N _k ^e (x ^e , y ^e )]

e.g. for N _i (x ^e , y ^e ) N _i (x ^e _i , y _i ^e ) = 1 N i (x ^e _j , y _j ^e ) = 0 N i (x ^e _k , y ^e _k ) = 0

y ^e N i (x ^e , y ^e )

x ^e 1 i

k

j

y ^e N j (x ^e , y ^e )

x ^e 1

i k

j y ^e

N k (x ^e , y ^e )

x ^e

1 i

k

j

Determination of shape functions N _i (x ^e , y ^e ) = α _1i + α _2i x ^e + α _3i y ^e





1 x i y i

1 x j y j

1 x k y k







 α 1i

α 2i

α 3i



 =



 1 0 0



 =⇒

α 1i = ^{x j y} _2P ^{k −x k y j}

4

α _2i = ^y _2P ^{j −y k}

4

α _3i = ^x _2P ^{k −x j}

Selection of approximation functions in 2D

3 Selection of approximation functions Four-node element

T ^e (x, y) = α ^e ₁ + α ^e ₂ x + α ^e ₃ y + α ^e ₄ xy = Φα ^e

Φ = [1 x y xy], α ^e =







 α ^e ₁ α ^e ₂ α ^e ₃ α ^e ₄









T ^e (x, y) = N _i ^e (x ^e , y ^e )T i + N _j ^e (x ^e , y ^e )T j + + N _k ^e (x ^e , y ^e )T k + N _l ^e (x ^e , y ^e )T l = N ^e Θ ^e N ^e = [N _i ^e (x ^e , y ^e ) N _j ^e (x ^e , y ^e ) N _k ^e (x ^e , y ^e ) N _l ^e (x ^e , y ^e )] Θ ^e = {T i T j T k T l }

y T (x, y)

x T j

T k

T i

T l

j

i

k l

y ^e x ^e

j

i

k

e l

Selection of approximation functions in 2D

Pascal triangle – four-node element 1

x y

x ² xy y ²

x ³ x ² y xy ² y ³

x ⁴ x ³ y x ² y ² xy ³ y ⁴

Selection of approximation functions in 2D

Pascal triangle – eight-node element 1

x y

x ² xy y ²

x ³ x ² y xy ² y ³

x ⁴ x ³ y x ² y ² xy ³ y ⁴

Selection of approximation functions in 2D

Pascal triangle – nine-node element 1

x y

x ² xy y ²

x ³ x ² y xy ² y ³

x ⁴ x ³ y x ² y ² xy ³ y ⁴

Selection of approximation functions in 2D

3 Selection of approximation functions Four-node element

T ^e (x, y) = α ^e ₁ + α ^e ₂ x + α ^e ₃ y + α ^e ₄ xy = Φα ^e

Φ = [1 x y xy], α ^e =







 α ^e ₁ α ^e ₂ α ^e ₃ α ^e ₄







 T ^e (x, y) = N _i ^e (x ^e , y ^e )T i + N _j ^e (x ^e , y ^e )T j +

+ N _k ^e (x ^e , y ^e )T _k + N _l ^e (x ^e , y ^e )T _l = N ^e Θ ^e N ^e = [N _i ^e (x ^e , y ^e ) N _j ^e (x ^e , y ^e ) N _k ^e (x ^e , y ^e ) N _l ^e (x ^e , y ^e )]

Θ ^e = {T i T j T k T l }

y T (x, y)

x T j

T k

T i

T l

j

i

k l

y ^e x ^e

j

i

k

e l

Selection of approximation functions in 2D

3 Selection of approximation functions Four-node element (rectangular)

N ^e = [N _i ^e (x ^e , y ^e ) N _j ^e (x ^e , y ^e ) N _k ^e (x ^e , y ^e ) N _l ^e (x ^e , y ^e )]

y ^e N i (x, y) = ^{(x−x j} _ab ^{)(y−y l )}

x ^e 1

i

j

k l b a

y ^e N l (x, y) = − ^{(x−x k} _ab ^{)(y−y i )}

x ^e i 1 j

k l

b a

y ^e N j (x, y) = − ^{(x−x i} _ab ^{)(y−y k )}

x ^e 1 j i

l b a

y ^e N k (x, y) = ^{(x−x l} _ab ^{)(y−y j )}

x ^e 1

i j

k l

b a

∇T ^e = B ^e Θ ^e

B ^e =





∂N ^e

∂x ^e

∂N ^e

∂y ^e





Selection of approximation functions in 2D

3 Selection of approximation functions Four-node element (rectangular)

N ^e = [N _i ^e (x ^e , y ^e ) N _j ^e (x ^e , y ^e ) N _k ^e (x ^e , y ^e ) N _l ^e (x ^e , y ^e )]

e.g. for N i (x ^e , y ^e ) N i (x ^e i , y i ^e ) = 1 N i (x ^e j , y j ^e ) = 0 N i (x ^e k , y ^e k ) = 0 N i (x ^e l , y l ^e ) = 0

y ^e N i (x, y) = ^{(x−x j} _ab ^{)(y−y l )}

x ^e 1

i

j

k l b a

y ^e N l (x, y) = − ^{(x−x k} _ab ^{)(y−y i )}

x ^e i 1 j

k l

b a

y ^e N j (x, y) = − ^{(x−x i} _ab ^{)(y−y k )}

x ^e 1 j i

k l b a

y ^e N k (x, y) = ^{(x−x l )(y−y j )}

ab

x ^e 1

i j

k l

b a

∇T ^e = B ^e Θ ^e

B ^e =





∂N ^e

∂x ^e

∂N ^e

∂y ^e





Selection of approximation functions in 2D

3 Selection of approximation functions Four-node element (rectangular)

N ^e = [N _i ^e (x ^e , y ^e ) N _j ^e (x ^e , y ^e ) N _k ^e (x ^e , y ^e ) N _l ^e (x ^e , y ^e )]

e.g. for N i (x ^e , y ^e ) N i (x ^e i , y i ^e ) = 1 N i (x ^e j , y j ^e ) = 0 N i (x ^e k , y ^e k ) = 0 N i (x ^e l , y l ^e ) = 0

y ^e N i (x, y) = ^{(x−x j} _ab ^{)(y−y l )}

x ^e 1

i

j

k l b a

y ^e N l (x, y) = − ^{(x−x k} _ab ^{)(y−y i )}

x ^e i 1 j

k l

b a

y ^e N j (x, y) = − ^{(x−x i} _ab ^{)(y−y k )}

x ^e 1 j i

l b a

y ^e N k (x, y) = ^{(x−x l )(y−y j )}

ab

x ^e 1

i j

k l

b a

∇T ^e = B ^e Θ ^e

B ^e =





∂N ^e

∂x ^e

∂N ^e

∂y ^e





Selection of approximation functions in 3D

1 Strong formulation

∇ ^T (D∇T ) + f = 0 ∀x ∈ V + boundary conditions

q n = q ^T n = q b on S q

T = b T on S T

V

S _T S _q

2 Conversion to weak formulation

Z

V

(∇w) ^T D∇T dV = − Z

S q

w b qdS − Z

S T

wq n dS + Z

V

wf dV + boundary condition

T = b T on S _T

Selection of approximation functions in 3D

3 Selection of approximation functions Tetrahedral element

T ^e (x, y, z) = α ^e ₁ + α ^e ₂ x + α ^e ₃ y + α ^e ₄ z

Hexahedral element

T ^e (x, y, z) = α ₁ ^e + α ^e ₂ x + α ^e ₃ y + α ₄ ^e z+ + α ^e ₅ xy + α ^e ₆ yz + α ^e ₇ xz + α ^e ₈ xyz

y z

x i

j

k l

y z

x i

j k

l m

n o

p

Selection of approximation functions in 3D

3 Selection of approximation functions Tetrahedral element

T ^e (x, y, z) = α ^e ₁ + α ^e ₂ x + α ^e ₃ y + α ^e ₄ z

Hexahedral element

T ^e (x, y, z) = α ₁ ^e + α ^e ₂ x + α ^e ₃ y + α ^e ₄ z+

+ α ^e ₅ xy + α ^e ₆ yz + α ^e ₇ xz + α ^e ₈ xyz

y z

x i

j

k l

y z

x i

j k

l m

n o

p