FEM solution for example problem of stationary heat flow

FEM solution for example problem of stationary heat flow

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

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

This lecture was prepared within project ”The development of the didactic potential of Cracow University of Technology in the range of modern construction”, co-financed by the European Union within the European Social Fund

Determination of shape functions for 3-node element

Shape function N i (x, y)





1 x _i y _i 1 x j y j

1 x k y k







 α _1i α 2i

α 3i



 =



 1 0 0





W =      

1 x i y i

1 x j y j

1 x k y k

     

= 2P ₄

y x

i k

j e

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

x 1 i

k j

α _1i = x j y k − x k y j

2P ₄ α 2i = y j − y k

2P ₄

This lecture was prepared within project ”The development of the didactic potential of Cracow University of Technology in the range of modern construction”, co-financed by the European Union within the European Social Fund

Determination of shape functions for 3-node element

Shape function N i (x, y)





1 x _i y _i 1 x j y j

1 x k y k







 α _1i α 2i

α 3i



 =



 1 0 0





W =      

1 x i y i

1 x j y j

1 x k y k

     

= 2P ₄

y x

i k

j e

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

x 1 i

k j

α _1i = x j y k − x k y j

2P ₄ α 2i = y j − y k

2P ₄

This lecture was prepared within project ”The development of the didactic potential of Cracow University of Technology in the range of modern construction”, co-financed by the European Union within the European Social Fund

Determination of shape functions for 3-node element

Shape function N i (x, y)





1 x _i y _i 1 x _j y _j 1 x k y k







 α _1i α _2i α 3i



 =



 1 0 0





W =      

1 x i y i

1 x j y j

1 x k y k

     

= 2P ₄

W α _1i =      

1 x i y i

0 x j y j

0 x k y k

     

= x j y k − x k y j

α 1i = W _{α 1i}

W = x _j y _k − x k y _j 2P ₄

y x

i k

j e

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

x 1 i

k j

α _1i = x j y k − x k y j

2P ₄ α 2i = y j − y k

2P ₄

This lecture was prepared within project ”The development of the didactic potential of Cracow University of Technology in the range of modern construction”, co-financed by the European Union within the European Social Fund

Determination of shape functions for 3-node element

Shape function N i (x, y)





1 x _i y _i 1 x _j y _j 1 x k y k







 α _1i α _2i α 3i



 =



 1 0 0





W =      

1 x i y i

1 x j y j

1 x k y k

     

= 2P ₄

W α _1i =      

1 x i y i

0 x j y j

0 x k y k

     

= x j y k − x k y j

α 1i = W α 1i

W = x j y k − x k y j

2P ₄

y x

i k

j e

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

x 1 i

k j

α _1i = x j y k − x k y j

2P ₄ α 2i = y j − y k

2P ₄

This lecture was prepared within project ”The development of the didactic potential of Cracow University of Technology in the range of modern construction”, co-financed by the European Union within the European Social Fund

Determination of shape functions for 3-node element

Shape function N i (x, y)





1 x _i y _i 1 x _j y _j 1 x k y k







 α _1i α _2i α 3i



 =



 1 0 0





W =      

1 x i y i

1 x j y j

1 x k y k

     

= 2P ₄

W α _2i =      

1 1 y i

1 0 y j

1 0 y k

     

= y j − y k

α 2i = W _{α 2i}

W = y _j − y k

2P ₄

y x

i k

j e

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

x 1 i

k j α _1i = x j y k − x k y j

2P ₄

α 2i = y j − y k

2P ₄

This lecture was prepared within project ”The development of the didactic potential of Cracow University of Technology in the range of modern construction”, co-financed by the European Union within the European Social Fund

Determination of shape functions for 3-node element

Shape function N i (x, y)





1 x _i y _i 1 x _j y _j 1 x k y k







 α _1i α _2i α 3i



 =



 1 0 0





W =      

1 x i y i

1 x j y j

1 x k y k

     

= 2P ₄

W α _2i =      

1 1 y i

1 0 y j

1 0 y k

     

= y j − y k

α 2i = W α 2i

W = y j − y k

2P ₄

y x

i k

j e

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

x 1 i

k j α _1i = x j y k − x k y j

2P ₄

α 2i = y j − y k

2P ₄

This lecture was prepared within project ”The development of the didactic potential of Cracow University of Technology in the range of modern construction”, co-financed by the European Union within the European Social Fund

Determination of shape functions for 3-node element

Shape function N _i (x, y)





1 x i y i

1 x j y j

1 x _k y _k







 α 1i

α 2i

α _3i



 =



 1 0 0





W =      

1 x _i y _i 1 x _j y _j 1 x _k y _k

     

= 2P ₄

W _{α 3i} =      

1 x i 1 1 x j 0 1 x k 0

     

= x _k − x _j

α 3i = W α _3i

W = x k − x j

2P ₄

y x

i k

j e

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

x 1 i

k j α 1i = x _j y _k − x k y _j

2P ₄ α _2i = y j − y k

2P 4

This lecture was prepared within project ”The development of the didactic potential of Cracow University of Technology in the range of modern construction”, co-financed by the European Union within the European Social Fund

Determination of shape functions for 3-node element

Shape function N _i (x, y)





1 x i y i

1 x j y j

1 x _k y _k







 α 1i

α 2i

α _3i



 =



 1 0 0





W =      

1 x _i y _i 1 x _j y _j 1 x _k y _k

     

= 2P ₄

W _{α 3i} =      

1 x i 1 1 x j 0 1 x k 0

     

= x _k − x _j

α 3i = W α _3i

W = x k − x j

2P ₄

y x

i k

j e

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

x 1 i

k j α 1i = x _j y _k − x k y _j

2P ₄ α _2i = y j − y k

2P 4

This lecture was prepared within project ”The development of the didactic potential of Cracow University of Technology in the range of modern construction”, co-financed by the European Union within the European Social Fund

2D example of heat flow – 3-node elements

q n = 0

q n = 0 q n = 5 J/m ² s

T = 20 ◦ C

4 m

3 m

k = 0.9 J/ms ^◦ C f = 2 J/m ² s h = 1 m

X Y

1 i

2 j

3 k

i

j k

4 Discretization

1 2

This lecture was prepared within project ”The development of the didactic potential of Cracow University of Technology in the range of modern construction”, co-financed by the European Union within the European Social Fund

2D example of heat flow – 3-node elements

q n = 0

q n = 0 q n = 5 J/m ² s

T = 20 ◦ C

4 m

3 m

k = 0.9 J/ms ^◦ C f = 2 J/m ² s h = 1 m

X Y

1 i

2 j

3 k

i

j k

4 Discretization

1 2

This lecture was prepared within project ”The development of the didactic potential of Cracow University of Technology in the range of modern construction”, co-financed by the European Union within the European Social Fund

Heat flow in 2D

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

T = NΘ, w = Nβ = β ^T N ^T , ∇T = BΘ

∇w = β ^T B ^T , D = kI, h = const

This lecture was prepared within project ”The development of the didactic potential of Cracow University of Technology in the range of modern construction”, co-financed by the European Union within the European Social Fund

Heat flow in 2D

Z

A

(∇w) ^T Dh∇T dA = − Z

Γ

whq _n dΓ + Z

A

whf dA + boundary condition

T = b T on Γ T

T = NΘ, w = Nβ = β ^T N ^T , ∇T = BΘ

∇w = β ^T B ^T , D = kI, h = const

This lecture was prepared within project ”The development of the didactic potential of Cracow University of Technology in the range of modern construction”, co-financed by the European Union within the European Social Fund

Heat flow in 2D

Z

A

(∇w) ^T Dh∇T dA = − Z

Γ

whq _n dΓ + Z

A

whf dA + boundary condition

T = b T on Γ T

T = NΘ, w = Nβ = β ^T N ^T , ∇T = BΘ

∇w = β ^T B ^T , D = kI, h = const

This lecture was prepared within project ”The development of the didactic potential of Cracow University of Technology in the range of modern construction”, co-financed by the European Union within the European Social Fund

Heat flow in 2D

Z

A

(∇w) ^T Dh∇T dA = − Z

Γ

whq _n dΓ + Z

A

whf dA + boundary condition

T = b T on Γ T

T = NΘ, w = Nβ = β ^T N ^T , ∇T = BΘ

∇w = β ^T B ^T , D = kI, h = const β ^T

Z

A

B ^T kBdA Θ = −β ^T Z

Γ

N ^T q n dΓ + β ^T Z

A

N ^T f dA, ∀β 6= 0

This lecture was prepared within project ”The development of the didactic potential of Cracow University of Technology in the range of modern construction”, co-financed by the European Union within the European Social Fund

Heat flow in 2D

Z

A

(∇w) ^T Dh∇T dA = − Z

Γ

whq _n dΓ + Z

A

whf dA + boundary condition

T = b T on Γ T

T = NΘ, w = Nβ = β ^T N ^T , ∇T = BΘ

∇w = β ^T B ^T , D = kI, h = const Z

A

B ^T kBdA Θ = − Z

Γ

N ^T q n dΓ + Z

A

N ^T f dA

This lecture was prepared within project ”The development of the didactic potential of Cracow University of Technology in the range of modern construction”, co-financed by the European Union within the European Social Fund

Heat flow in 2D

Z

A

(∇w) ^T Dh∇T dA = − Z

Γ

whq _n dΓ + Z

A

whf dA + boundary condition

T = b T on Γ T

T = NΘ, w = Nβ = β ^T N ^T , ∇T = BΘ

∇w = β ^T B ^T , D = kI, h = const Z

A

B ^T kBdA Θ = − Z

Γ

N ^T q n dΓ + Z

A

N ^T f dA

K = Z

A

B ^T kBdA, F = Z

A

N ^T f dA, F b = − Z

Γ

N ^T q n dΓ

This lecture was prepared within project ”The development of the didactic potential of Cracow University of Technology in the range of modern construction”, co-financed by the European Union within the European Social Fund

Heat flow in 2D

Z

A

(∇w) ^T Dh∇T dA = − Z

Γ

whq _n dΓ + Z

A

whf dA + boundary condition

T = b T on Γ T

T = NΘ, w = Nβ = β ^T N ^T , ∇T = BΘ

∇w = β ^T B ^T , D = kI, h = const Z

A

B ^T kBdA Θ = − Z

Γ

N ^T q n dΓ + Z

A

N ^T f dA

K = Z

A

B ^T kBdA, F = Z

A

N ^T f dA, F b = − Z

Γ

N ^T q n dΓ

KΘ = F + F b

This lecture was prepared within project ”The development of the didactic potential of Cracow University of Technology in the range of modern construction”, co-financed by the European Union within the European Social Fund

2D example of heat flow – 3-node elements

qn = 0 q n = 0 qn = 5 J/m2 s

T = 20 ◦ C

k = 0.9 J/ms ◦ C f = 2 J/m 2 s h = 1 m

X Y

Discretization

1 i

2 j

3 k

i 1 j k 4

2

K matrix – element 1 N ¹ = 

1 − ¹ ₄ x ¹ ₄ x − ¹ ₃ y ¹ ₃ y 

B ¹ = ∇N =

 −0.250 0.250 0.000 0.000 −0.333 0.333



K ¹ = Z

A ¹

B ^T kBdA = A ¹ B ^T kB

=





0.338 −0.338 0.000

−0.388 0.938 −0.600 0.000 −0.600 0.600





K ² =





0.600 0.000 −0.600 0.000 0.338 −0.338

−0.600 −0.338 0.938





F ¹ = F ² =



 4 4 4





F ¹ _b =



 0 0 0



 ,

F ² _b =



 f _{b 1}

10 f _{b 4} + 10





K =









0.938 −0.338 0.000 −0.600

−0.388 0.938 −0.600 0.000 0.000 −0.600 0.938 −0.338

−0.600 0.000 −0.338 0.938









F =







 8 4 8 4







 ,

Fb =







 fb1

0 10 fb4 + 10









Θ =







 20 48.040 57.145 20









This lecture was prepared within project ”The development of the didactic potential of Cracow University of Technology in the range of modern construction”, co-financed by the European Union within the European Social Fund

2D example of heat flow – 3-node elements

qn = 0 q n = 0 qn = 5 J/m2 s

T = 20 ◦ C

k = 0.9 J/ms ◦ C f = 2 J/m 2 s h = 1 m

X Y

Discretization

1 i

2 j

3 k

i 1 j k 4

2

K matrix – element 1 N ¹ = 

1 − ¹ ₄ x ¹ ₄ x − ¹ ₃ y ¹ ₃ y  B ¹ = ∇N =

 −0.250 0.250 0.000 0.000 −0.333 0.333



K ¹ = Z

A ¹

B ^T kBdA = A ¹ B ^T kB

=





0.338 −0.338 0.000

−0.388 0.938 −0.600 0.000 −0.600 0.600





K ² =





0.600 0.000 −0.600 0.000 0.338 −0.338

−0.600 −0.338 0.938





F ¹ = F ² =



 4 4 4





F ¹ _b =



 0 0 0



 ,

F ² _b =



 f _{b 1}

10 f _{b 4} + 10





K =









0.938 −0.338 0.000 −0.600

−0.388 0.938 −0.600 0.000 0.000 −0.600 0.938 −0.338

−0.600 0.000 −0.338 0.938









F =







 8 4 8 4







 ,

Fb =







 fb1

0 10 fb4 + 10









Θ =







 20 48.040 57.145 20









This lecture was prepared within project ”The development of the didactic potential of Cracow University of Technology in the range of modern construction”, co-financed by the European Union within the European Social Fund

2D example of heat flow – 3-node elements

qn = 0 q n = 0 qn = 5 J/m2 s

T = 20 ◦ C

k = 0.9 J/ms ◦ C f = 2 J/m 2 s h = 1 m

X Y

Discretization

1 i

2 j

3 k

i 1 j k 4

2

K matrix – element 1 N ¹ = 

1 − ¹ ₄ x ¹ ₄ x − ¹ ₃ y ¹ ₃ y  B ¹ = ∇N =

 −0.250 0.250 0.000 0.000 −0.333 0.333



K ¹ = Z

A ¹

B ^T kBdA = A ¹ B ^T kB

=





0.338 −0.338 0.000

−0.388 0.938 −0.600 0.000 −0.600 0.600





K ² =





0.600 0.000 −0.600 0.000 0.338 −0.338

−0.600 −0.338 0.938





F ¹ = F ² =



 4 4 4





F ¹ _b =



 0 0 0



 ,

F ² _b =



 f _{b 1}

10 f _{b 4} + 10





K =









0.938 −0.338 0.000 −0.600

−0.388 0.938 −0.600 0.000 0.000 −0.600 0.938 −0.338

−0.600 0.000 −0.338 0.938









F =







 8 4 8 4







 ,

Fb =







 fb1

0 10 fb4 + 10









Θ =







 20 48.040 57.145 20









This lecture was prepared within project ”The development of the didactic potential of Cracow University of Technology in the range of modern construction”, co-financed by the European Union within the European Social Fund

2D example of heat flow – 3-node elements

qn = 0 q n = 0 qn = 5 J/m2 s

T = 20 ◦ C

k = 0.9 J/ms ◦ C f = 2 J/m 2 s h = 1 m

X Y

Discretization

i

2 j k

1 1

i

3 j k 4

2

K matrix – element 2 N ² = 

1 − ¹ ₃ y ¹ ₄ x ¹ ₃ y − ¹ ₄ x 

B ² = ∇N =

 0.000 0.250 −0.250

−0.333 0.000 0.333



K ² = Z

A ²

B ^T kBdA = A ² B ^T kB

=





0.600 0.000 −0.600 0.000 0.338 −0.338

−0.600 −0.338 0.938





K ¹ =





0.338 −0.338 0.000

−0.388 0.938 −0.600 0.000 −0.600 0.600





K ² =





0.600 0.000 −0.600 0.000 0.338 −0.338

−0.600 −0.338 0.938





F ¹ = F ² =



 4 4 4





F ¹ _b =



 0 0 0



 ,

F ² _b =



 f _{b 1}

10 f _{b 4} + 10





K =









0.938 −0.338 0.000 −0.600

−0.388 0.938 −0.600 0.000 0.000 −0.600 0.938 −0.338

−0.600 0.000 −0.338 0.938









F =







 8 4 8 4







 ,

Fb =







 fb1

0 10 fb4 + 10









Θ =







 20 48.040 57.145 20









This lecture was prepared within project ”The development of the didactic potential of Cracow University of Technology in the range of modern construction”, co-financed by the European Union within the European Social Fund

2D example of heat flow – 3-node elements

qn = 0 q n = 0 qn = 5 J/m2 s

T = 20 ◦ C

k = 0.9 J/ms ◦ C f = 2 J/m 2 s h = 1 m

X Y

Discretization

i

2 j k

1 1

i

3 j k 4

2

K matrix – element 2 N ² = 

1 − ¹ ₃ y ¹ ₄ x ¹ ₃ y − ¹ ₄ x  B ² = ∇N =

 0.000 0.250 −0.250

−0.333 0.000 0.333



K ² = Z

A ²

B ^T kBdA = A ² B ^T kB

=





0.600 0.000 −0.600 0.000 0.338 −0.338

−0.600 −0.338 0.938





K ¹ =





0.338 −0.338 0.000

−0.388 0.938 −0.600 0.000 −0.600 0.600





K ² =





0.600 0.000 −0.600 0.000 0.338 −0.338

−0.600 −0.338 0.938





F ¹ = F ² =



 4 4 4





F ¹ _b =



 0 0 0



 ,

F ² _b =



 f _{b 1}

10 f _{b 4} + 10





K =









0.938 −0.338 0.000 −0.600

−0.388 0.938 −0.600 0.000 0.000 −0.600 0.938 −0.338

−0.600 0.000 −0.338 0.938









F =







 8 4 8 4







 ,

Fb =







 fb1

0 10 fb4 + 10









Θ =







 20 48.040 57.145 20









This lecture was prepared within project ”The development of the didactic potential of Cracow University of Technology in the range of modern construction”, co-financed by the European Union within the European Social Fund

2D example of heat flow – 3-node elements

qn = 0 q n = 0 qn = 5 J/m2 s

T = 20 ◦ C

k = 0.9 J/ms ◦ C f = 2 J/m 2 s h = 1 m

X Y

Discretization

i

2 j k

1 1

i

3 j k 4

2

K matrix – element 2 N ² = 

1 − ¹ ₃ y ¹ ₄ x ¹ ₃ y − ¹ ₄ x  B ² = ∇N =

 0.000 0.250 −0.250

−0.333 0.000 0.333



K ² = Z

A ²

B ^T kBdA = A ² B ^T kB

=





0.600 0.000 −0.600 0.000 0.338 −0.338

−0.600 −0.338 0.938





K ¹ =





0.338 −0.338 0.000

−0.388 0.938 −0.600 0.000 −0.600 0.600





K ² =





0.600 0.000 −0.600 0.000 0.338 −0.338

−0.600 −0.338 0.938





F ¹ = F ² =



 4 4 4





F ¹ _b =



 0 0 0



 ,

F ² _b =



 f _{b 1}

10 f _{b 4} + 10





K =









0.938 −0.338 0.000 −0.600

−0.388 0.938 −0.600 0.000 0.000 −0.600 0.938 −0.338

−0.600 0.000 −0.338 0.938









F =







 8 4 8 4







 ,

Fb =







 fb1

0 10 fb4 + 10









Θ =







 20 48.040 57.145 20









This lecture was prepared within project ”The development of the didactic potential of Cracow University of Technology in the range of modern construction”, co-financed by the European Union within the European Social Fund

2D example of heat flow – 3-node elements

qn = 0 q n = 0 qn = 5 J/m2 s

T = 20 ◦ C

k = 0.9 J/ms ◦ C f = 2 J/m 2 s h = 1 m

X Y

Discretization

1 i

2 j

3 k

1 1

i

3 j k 4

2

F vector – element 1 and 2 - A ¹ = A ²

F ^e = Z

A ^e

N ^T f dA = f 3 A ^e



 1 1 1



 =



 4 4 4





K ¹ =





0.338 −0.338 0.000

−0.388 0.938 −0.600 0.000 −0.600 0.600





K ² =





0.600 0.000 −0.600 0.000 0.338 −0.338

−0.600 −0.338 0.938





F ¹ = F ² =



 4 4 4





F ¹ _b =



 0 0 0



 ,

F ² _b =



 f _{b 1}

10 f _{b 4} + 10





K =









0.938 −0.338 0.000 −0.600

−0.388 0.938 −0.600 0.000 0.000 −0.600 0.938 −0.338

−0.600 0.000 −0.338 0.938









F =







 8 4 8 4







 ,

Fb =







 fb1

0 10 fb4 + 10









Θ =







 20 48.040 57.145 20









This lecture was prepared within project ”The development of the didactic potential of Cracow University of Technology in the range of modern construction”, co-financed by the European Union within the European Social Fund

2D example of heat flow – 3-node elements

qn = 0 q n = 0 qn = 5 J/m2 s

T = 20 ◦ C

k = 0.9 J/ms ◦ C f = 2 J/m 2 s h = 1 m

X Y

Discretization

1 i

2 j

3 k

i 1 j k 4

2

F _b vector – element 1 F ¹ _b = −

Z

Γ ¹ _ij

(N ¹ ) ^T q _n dΓ − Z

Γ ¹ _jk

(N ¹ ) ^T q _n dΓ

− Z

Γ ¹ _ki

(N ¹ ) ^T q _n dΓ

F ¹ _b =



 0 0 0





K ¹ =





0.338 −0.338 0.000

−0.388 0.938 −0.600 0.000 −0.600 0.600





K ² =





0.600 0.000 −0.600 0.000 0.338 −0.338

−0.600 −0.338 0.938





F ¹ = F ² =



 4 4 4





F ¹ _b =



 0 0 0



 ,

F ² _b =



 f _{b 1}

10 f _{b 4} + 10





K =









0.938 −0.338 0.000 −0.600

−0.388 0.938 −0.600 0.000 0.000 −0.600 0.938 −0.338

−0.600 0.000 −0.338 0.938









F =







 8 4 8 4







 ,

Fb =







 fb1

0 10 fb4 + 10









Θ =







 20 48.040 57.145 20









This lecture was prepared within project ”The development of the didactic potential of Cracow University of Technology in the range of modern construction”, co-financed by the European Union within the European Social Fund

2D example of heat flow – 3-node elements

qn = 0 q n = 0 qn = 5 J/m2 s

T = 20 ◦ C

k = 0.9 J/ms ◦ C f = 2 J/m 2 s h = 1 m

X Y

Discretization

1 i

2 j

3 k

i 1 j k 4

2

F _b vector – element 1 F ¹ _b = −

Z

Γ ¹ _ij

(N ¹ ) ^T q _n b.c. = 0

dΓ−

Z

Γ ¹ _jk

(N ¹ ) ^T q _n b.c. = 0

dΓ

flow continuity along edge 1-3 q n 1

ki = −q n 2 ij

− Z

Γ ¹ _ki

(N ¹ ) ^T q _n dΓ

F ¹ _b =



 0 0 0





K ¹ =





0.338 −0.338 0.000

−0.388 0.938 −0.600 0.000 −0.600 0.600





K ² =





0.600 0.000 −0.600 0.000 0.338 −0.338

−0.600 −0.338 0.938





F ¹ = F ² =



 4 4 4





F ¹ _b =



 0 0 0



 ,

F ² _b =



 f _{b 1}

10 f _{b 4} + 10





K =









0.938 −0.338 0.000 −0.600

−0.388 0.938 −0.600 0.000 0.000 −0.600 0.938 −0.338

−0.600 0.000 −0.338 0.938









F =







 8 4 8 4







 ,

Fb =







 fb1

0 10 fb4 + 10









Θ =







 20 48.040 57.145 20









This lecture was prepared within project ”The development of the didactic potential of Cracow University of Technology in the range of modern construction”, co-financed by the European Union within the European Social Fund

2D example of heat flow – 3-node elements

qn = 0 q n = 0 qn = 5 J/m2 s

T = 20 ◦ C

k = 0.9 J/ms ◦ C f = 2 J/m 2 s h = 1 m

X Y

Discretization

1 i

2 j

3 k

i 1 j k 4

2

F _b vector – element 1 F ¹ _b = −

Z

Γ ¹ _ij

(N ¹ ) ^T q _n b.c. = 0

dΓ−

Z

Γ ¹ _jk

(N ¹ ) ^T q _n b.c. = 0

dΓ

flow continuity along edge 1-3 q n 1

ki = −q n 2 ij

− Z

Γ ¹ _ki

(N ¹ ) ^T q _n dΓ

F ¹ _b =



 0 0 0





K ¹ =





0.338 −0.338 0.000

−0.388 0.938 −0.600 0.000 −0.600 0.600





K ² =





0.600 0.000 −0.600 0.000 0.338 −0.338

−0.600 −0.338 0.938





F ¹ = F ² =



 4 4 4





F ¹ _b =



 0 0 0



 ,

F ² _b =



 f _{b 1}

10 f _{b 4} + 10





K =









0.938 −0.338 0.000 −0.600

−0.388 0.938 −0.600 0.000 0.000 −0.600 0.938 −0.338

−0.600 0.000 −0.338 0.938









F =







 8 4 8 4







 ,

Fb =







 fb1

0 10 fb4 + 10









Θ =







 20 48.040 57.145 20









This lecture was prepared within project ”The development of the didactic potential of Cracow University of Technology in the range of modern construction”, co-financed by the European Union within the European Social Fund

2D example of heat flow – 3-node elements

qn = 0 q n = 0 qn = 5 J/m2 s

T = 20 ◦ C

k = 0.9 J/ms ◦ C f = 2 J/m 2 s h = 1 m

X Y

Discretization

i

2 j k

1 1

i

3 j k 4

2

F _b vector – element 2

F ² _b = − Z

Γ ² _ij

(N ² ) ^T q n dΓ

− Z

Γ ² _jk

(N ² ) ^T q n dΓ − Z

Γ ² _ki

(N ² ) ^T q n dΓ

K ¹ =





0.338 −0.338 0.000

−0.388 0.938 −0.600 0.000 −0.600 0.600





K ² =





0.600 0.000 −0.600 0.000 0.338 −0.338

−0.600 −0.338 0.938





F ¹ = F ² =



 4 4 4





F ¹ _b =



 0 0 0



 ,

F ² _b =



 f _{b 1}

10 f _{b 4} + 10





K =









0.938 −0.338 0.000 −0.600

−0.388 0.938 −0.600 0.000 0.000 −0.600 0.938 −0.338

−0.600 0.000 −0.338 0.938









F =







 8 4 8 4







 ,

Fb =







 fb1

0 10 fb4 + 10









Θ =







 20 48.040 57.145 20









This lecture was prepared within project ”The development of the didactic potential of Cracow University of Technology in the range of modern construction”, co-financed by the European Union within the European Social Fund

2D example of heat flow – 3-node elements

qn = 0 q n = 0 qn = 5 J/m2 s

T = 20 ◦ C

k = 0.9 J/ms ◦ C f = 2 J/m 2 s h = 1 m

X Y

Discretization

i

2 j k

1 1

i

3 j k 4

2

F _b vector – element 2

F ² _b = − Z

Γ ² _ij

(N ² ) ^T q n dΓ

flow continuity along edge 1-3 q n 1

ki = −q n 2 ij

− Z

Γ ² _jk

(N ² ) ^T q _n dΓ − Z

Γ ² _ki

(N ² ) ^T q _n dΓ

K ¹ =





0.338 −0.338 0.000

−0.388 0.938 −0.600 0.000 −0.600 0.600





K ² =





0.600 0.000 −0.600 0.000 0.338 −0.338

−0.600 −0.338 0.938





F ¹ = F ² =



 4 4 4





F ¹ _b =



 0 0 0



 ,

F ² _b =



 f _{b 1}

10 f _{b 4} + 10





K =









0.938 −0.338 0.000 −0.600

−0.388 0.938 −0.600 0.000 0.000 −0.600 0.938 −0.338

−0.600 0.000 −0.338 0.938









F =







 8 4 8 4







 ,

Fb =







 fb1

0 10 fb4 + 10









Θ =







 20 48.040 57.145 20









This lecture was prepared within project ”The development of the didactic potential of Cracow University of Technology in the range of modern construction”, co-financed by the European Union within the European Social Fund

2D example of heat flow – 3-node elements

qn = 0 q n = 0 qn = 5 J/m2 s

T = 20 ◦ C

k = 0.9 J/ms ◦ C f = 2 J/m 2 s h = 1 m

X Y

Discretization

i

2 j k

1 1

i

3 j k 4

2

F _b vector – element 2 F ² _b = −

Z

Γ ² _jk

(N ² ) ^T q n dΓ − Z

Γ ² _ki

(N ² ) ^T q n dΓ

− Z

Γ ² _jk

(N ² ) ^T q n dΓ = − Z 4

0

N ² (x, y = 3)
^T (−5)dx

=



 0 10 10





K ¹ =





0.338 −0.338 0.000

−0.388 0.938 −0.600 0.000 −0.600 0.600





K ² =





0.600 0.000 −0.600 0.000 0.338 −0.338

−0.600 −0.338 0.938





F ¹ = F ² =



 4 4 4





F ¹ _b =



 0 0 0



 ,

F ² _b =



 f _{b 1}

10 f _{b 4} + 10





K =









0.938 −0.338 0.000 −0.600

−0.388 0.938 −0.600 0.000 0.000 −0.600 0.938 −0.338

−0.600 0.000 −0.338 0.938









F =







 8 4 8 4







 ,

Fb =







 fb1

0 10 fb4 + 10









Θ =







 20 48.040 57.145 20









This lecture was prepared within project ”The development of the didactic potential of Cracow University of Technology in the range of modern construction”, co-financed by the European Union within the European Social Fund

2D example of heat flow – 3-node elements

qn = 0 q n = 0 qn = 5 J/m2 s

T = 20 ◦ C

k = 0.9 J/ms ◦ C f = 2 J/m 2 s h = 1 m

X Y

Discretization

i

2 j k

1 1

i

3 j k 4

2

F _b vector – element 2 F ² _b =



 0 10 10



 − Z

Γ ² _ki

(N ² ) ^T q _n dΓ

− Z

Γ ² _ki

(N ² ) ^T q n dΓ = − Z 3

0

N ² (x = 0, y)
^T q n dx

=



 f _b1

0 f _b4





K ¹ =





0.338 −0.338 0.000

−0.388 0.938 −0.600 0.000 −0.600 0.600





K ² =





0.600 0.000 −0.600 0.000 0.338 −0.338

−0.600 −0.338 0.938





F ¹ = F ² =



 4 4 4





F ¹ _b =



 0 0 0



 ,

F ² _b =



 f _{b 1}

10 f _{b 4} + 10





K =









0.938 −0.338 0.000 −0.600

−0.388 0.938 −0.600 0.000 0.000 −0.600 0.938 −0.338

−0.600 0.000 −0.338 0.938









F =







 8 4 8 4







 ,

Fb =







 fb1

0 10 fb4 + 10









Θ =







 20 48.040 57.145 20









This lecture was prepared within project ”The development of the didactic potential of Cracow University of Technology in the range of modern construction”, co-financed by the European Union within the European Social Fund

2D example of heat flow – 3-node elements

qn = 0 q n = 0 qn = 5 J/m2 s

T = 20 ◦ C

k = 0.9 J/ms ◦ C f = 2 J/m 2 s h = 1 m

X Y

Discretization

1 i

2 j

3 k

i 1 j k 4

2

Assembly

K ¹ =





0.338 −0.338 0.000

−0.388 0.938 −0.600 0.000 −0.600 0.600





K =









0.338 −0.338 0.000 0.000

−0.388 0.938 −0.600 0.000 0.000 −0.600 0.600 0.000 0.000 0.000 0.000 0.000









K ¹ =





0.338 −0.338 0.000

−0.388 0.938 −0.600 0.000 −0.600 0.600





K ² =





0.600 0.000 −0.600 0.000 0.338 −0.338

−0.600 −0.338 0.938





F ¹ = F ² =



 4 4 4





F ¹ _b =



 0 0 0



 , F ² _b =



 f _{b 1}

10 f _{b 4} + 10





K =









0.938 −0.338 0.000 −0.600

−0.388 0.938 −0.600 0.000 0.000 −0.600 0.938 −0.338

−0.600 0.000 −0.338 0.938









F =







 8 4 8 4







 ,

Fb =







 fb1

0 10 fb4 + 10









Θ =







 20 48.040 57.145 20









This lecture was prepared within project ”The development of the didactic potential of Cracow University of Technology in the range of modern construction”, co-financed by the European Union within the European Social Fund

2D example of heat flow – 3-node elements

qn = 0 q n = 0 qn = 5 J/m2 s

T = 20 ◦ C

k = 0.9 J/ms ◦ C f = 2 J/m 2 s h = 1 m

X Y

Discretization

i

2 j k

1 1

i

3 j k 4

2

Assembly

K ² =





0.600 0.000 −0.600 0.000 0.338 −0.338

−0.600 −0.338 0.938





K =









0.938 −0.338 0.000 −0.600

−0.388 0.938 −0.600 0.000 0.000 −0.600 0.938 −0.338

−0.600 0.000 −0.338 0.938









K ¹ =





0.338 −0.338 0.000

−0.388 0.938 −0.600 0.000 −0.600 0.600





K ² =





0.600 0.000 −0.600 0.000 0.338 −0.338

−0.600 −0.338 0.938





F ¹ = F ² =



 4 4 4





F ¹ _b =



 0 0 0



 , F ² _b =



 f _{b 1}

10 f _{b 4} + 10





K =









0.938 −0.338 0.000 −0.600

−0.388 0.938 −0.600 0.000 0.000 −0.600 0.938 −0.338

−0.600 0.000 −0.338 0.938









F =







 8 4 8 4







 ,

Fb =







 fb1

0 10 fb4 + 10









Θ =







 20 48.040 57.145 20









This lecture was prepared within project ”The development of the didactic potential of Cracow University of Technology in the range of modern construction”, co-financed by the European Union within the European Social Fund

2D example of heat flow – 3-node elements

qn = 0 q n = 0 qn = 5 J/m2 s

T = 20 ◦ C

k = 0.9 J/ms ◦ C f = 2 J/m 2 s h = 1 m

X Y

Discretization

1 i

2 j

3 k

i 1 j k 4

2

Assembly

F ¹ =



 4 4 4





F =







 4 4 4 0









K ¹ =





0.338 −0.338 0.000

−0.388 0.938 −0.600 0.000 −0.600 0.600





K ² =





0.600 0.000 −0.600 0.000 0.338 −0.338

−0.600 −0.338 0.938





F ¹ = F ² =



 4 4 4





F ¹ _b =



 0 0 0



 , F ² _b =



 f _{b 1}

10 f _{b 4} + 10





K =









0.938 −0.338 0.000 −0.600

−0.388 0.938 −0.600 0.000 0.000 −0.600 0.938 −0.338

−0.600 0.000 −0.338 0.938









F =







 8 4 8 4







 ,

Fb =







 fb1

0 10 fb4 + 10









Θ =







 20 48.040 57.145 20









This lecture was prepared within project ”The development of the didactic potential of Cracow University of Technology in the range of modern construction”, co-financed by the European Union within the European Social Fund

2D example of heat flow – 3-node elements

qn = 0 q n = 0 qn = 5 J/m2 s

T = 20 ◦ C

k = 0.9 J/ms ◦ C f = 2 J/m 2 s h = 1 m

X Y

Discretization

i

2 j k

1 1

i

3 j k 4

2

Assembly

F ¹ =



 4 4 4



 F ² =



 4 4 4





F =







 8 4 8 4









K ¹ =





0.338 −0.338 0.000

−0.388 0.938 −0.600 0.000 −0.600 0.600





K ² =





0.600 0.000 −0.600 0.000 0.338 −0.338

−0.600 −0.338 0.938





F ¹ = F ² =



 4 4 4





F ¹ _b =



 0 0 0



 , F ² _b =



 f _{b 1}

10 f _{b 4} + 10





K =









0.938 −0.338 0.000 −0.600

−0.388 0.938 −0.600 0.000 0.000 −0.600 0.938 −0.338

−0.600 0.000 −0.338 0.938









F =







 8 4 8 4







 ,

Fb =







 fb1

0 10 fb4 + 10









Θ =







 20 48.040 57.145 20









This lecture was prepared within project ”The development of the didactic potential of Cracow University of Technology in the range of modern construction”, co-financed by the European Union within the European Social Fund