Galerkin Finite Element Model for Heat Transfer Introductory Course - - PowerPoint PPT Presentation

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Galerkin Finite Element Model for Heat Transfer Introductory Course - - PowerPoint PPT Presentation

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Notation remarks Local differential formulation Global integral formulations Matrix formulations Galerkin Finite Element Model for Heat Transfer Introductory Course on Multiphysics Modelling T OMASZ G. Z IELI NSKI

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Notation remarks Local differential formulation Global integral formulations Matrix formulations

Galerkin Finite Element Model for Heat Transfer

Introductory Course on Multiphysics Modelling

TOMASZ G. ZIELI ´

NSKI bluebox.ippt.pan.pl/˜tzielins/

Institute of Fundamental Technological Research

  • f the Polish Academy of Sciences

Warsaw • Poland

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Notation remarks Local differential formulation Global integral formulations Matrix formulations

Outline

1

Notation remarks

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Notation remarks Local differential formulation Global integral formulations Matrix formulations

Outline

1

Notation remarks

2

Local differential formulation Partial Differential Equation Initial and boundary conditions Initial-Boundary-Value Problem

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Notation remarks Local differential formulation Global integral formulations Matrix formulations

Outline

1

Notation remarks

2

Local differential formulation Partial Differential Equation Initial and boundary conditions Initial-Boundary-Value Problem

3

Global integral formulations Test functions Weighted formulation and weak variational form

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Notation remarks Local differential formulation Global integral formulations Matrix formulations

Outline

1

Notation remarks

2

Local differential formulation Partial Differential Equation Initial and boundary conditions Initial-Boundary-Value Problem

3

Global integral formulations Test functions Weighted formulation and weak variational form

4

Matrix formulations Approximation Transient heat transfer (ordinary differential equations) Stationary heat transfer (algebraic equations)

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Notation remarks Local differential formulation Global integral formulations Matrix formulations

Outline

1

Notation remarks

2

Local differential formulation Partial Differential Equation Initial and boundary conditions Initial-Boundary-Value Problem

3

Global integral formulations Test functions Weighted formulation and weak variational form

4

Matrix formulations Approximation Transient heat transfer (ordinary differential equations) Stationary heat transfer (algebraic equations)

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Notation remarks Local differential formulation Global integral formulations Matrix formulations

Notation remarks

The index notation is used with summation over the index i. Consequently, the summation rule is also applied for the approximation expressions, that is, over the indices r, s = 1, . . . N (where N is the number of degrees of freedom). The symbol (. . .)|i means a (generalized) invariant partial differentiation over the i-th coordinate: (. . .)|i = ∂(. . .) ∂xi . The invariance involves the so-called Christoffel symbols (in the case of curvilinear systems of reference). Symbols dV and dS are completely omitted in all the integrals presented below since it is obvious that one integrates over the specified domain or boundary. Therefore, one should understand that:

  • B

(. . .) =

  • B

(. . .) dV(x) ,

  • ∂B

(. . .) =

  • ∂B

(. . .) dS(x) .

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Notation remarks Local differential formulation Global integral formulations Matrix formulations

Outline

1

Notation remarks

2

Local differential formulation Partial Differential Equation Initial and boundary conditions Initial-Boundary-Value Problem

3

Global integral formulations Test functions Weighted formulation and weak variational form

4

Matrix formulations Approximation Transient heat transfer (ordinary differential equations) Stationary heat transfer (algebraic equations)

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Notation remarks Local differential formulation Global integral formulations Matrix formulations

PDE for Heat Transfer Problem

B (̺, c, k) ∂B n

heat source

f > 0

heat sink

f < 0

∂T ∂x · n < 0

(cold)

∂T ∂x · n > 0

(warm)

Material data: ̺ = ̺(x) – the density kg

m3

  • c = c(x) – the thermal capacity
  • J

kg·K

  • k = k(x) – the thermal conductivity

W

m·K

  • Known fields:

f = f(x, t) – the heat production rate W

m3

  • ui = ui(x, t) – the convective velocity

m

s

  • The unknown field:

T = T(x, t) =? – the temperature

  • K
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Notation remarks Local differential formulation Global integral formulations Matrix formulations

PDE for Heat Transfer Problem

B (̺, c, k) ∂B n

heat source

f > 0

heat sink

f < 0

∂T ∂x · n < 0

(cold)

∂T ∂x · n > 0

(warm)

Material data: ̺ = ̺(x) – the density kg

m3

  • c = c(x) – the thermal capacity
  • J

kg·K

  • k = k(x) – the thermal conductivity

W

m·K

  • Known fields:

f = f(x, t) – the heat production rate W

m3

  • ui = ui(x, t) – the convective velocity

m

s

  • The unknown field:

T = T(x, t) =? – the temperature

  • K
  • Heat transfer equation

̺ c

  • T + qi|i − f = 0

where the heat flux vector W

K

  • :

qi = qi(T) =

  • −k T|i

– for conduction (only), −k T|i + ̺ c ui T – for conduction and convection, and

  • T = ∂T

∂t is the time rate of change of temperature

K

s

  • .
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Notation remarks Local differential formulation Global integral formulations Matrix formulations

Initial and boundary conditions

The initial condition (at t = t0) T(x, t0) = T0(x) in B Prescribed field: T0 = T0(x) – the initial temperature

  • K
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Notation remarks Local differential formulation Global integral formulations Matrix formulations

Initial and boundary conditions

The initial condition (at t = t0) T(x, t0) = T0(x) in B Prescribed field: T0 = T0(x) – the initial temperature

  • K
  • The boundary conditions (on ∂B)

the Dirichlet type: T(x, t) = ˆ T(x, t) on ∂BT the Neumann type: −qi

  • T(x, t)
  • ni = ˆ

q(x, t) on ∂Bq Prescribed fields: ˆ T = ˆ T(x, t) – the temperature

  • K
  • ˆ

q = ˆ q(x, t) – the inward heat flux

W

m2

  • B

(̺, c, k) ∂B n

heat source

f > 0

heat sink

f < 0

∂T ∂x · n < 0

(cold)

∂T ∂x · n > 0

(warm)

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Notation remarks Local differential formulation Global integral formulations Matrix formulations

Initial-Boundary-Value Problem

IBVP of the heat transfer Find T = T(x, t) for x ∈ B and t ∈ [t0, t1] satisfying the equation of heat transfer by conduction (a), or by conduction and convection (b): ̺ c

  • T + qi|i − f = 0

where qi = qi(T) =

  • −k T|i

← (a) −k T|i + ̺ c ui T ← (b) with the initial condition (at t = t0): T(x, t0) = T0(x) in B , and subject to the boundary conditions: T(x, t) = ˆ T(x, t) on ∂BT , −qi

  • T(x, t)
  • ni = ˆ

q(x, t) on ∂Bq , where ∂BT ∪ ∂Bq = ∂B and ∂BT ∩ ∂Bq = ∅.

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Notation remarks Local differential formulation Global integral formulations Matrix formulations

Outline

1

Notation remarks

2

Local differential formulation Partial Differential Equation Initial and boundary conditions Initial-Boundary-Value Problem

3

Global integral formulations Test functions Weighted formulation and weak variational form

4

Matrix formulations Approximation Transient heat transfer (ordinary differential equations) Stationary heat transfer (algebraic equations)

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Notation remarks Local differential formulation Global integral formulations Matrix formulations

Test functions

T(x, t), δT(x) x ∂BT Dirichlet b.c. T = ˆ T, δT = 0 ∂Bq Neumann b.c. ˆ T solution and trial functions, T test functions, δT

Test function δT(x) is an arbitrary (but sufficiently regular) function defined in B, which meets the admissibility condition: δT = 0

  • n ∂BT .

Notice that test functions are always time-independent.

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Notation remarks Local differential formulation Global integral formulations Matrix formulations

Weighted formulation and weak variational form

Weighted integral formulation

✎ ✍ ☞ ✌

  • B
  • ̺ c
  • T + qi|i − f
  • δT = 0

(for every δT)

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Notation remarks Local differential formulation Global integral formulations Matrix formulations

Weighted formulation and weak variational form

Weighted integral formulation

✎ ✍ ☞ ✌

  • B

̺ c

  • T δT +
  • B

qi|i δT −

  • B

f δT = 0 (for every δT) The term qi|i introduces the second derivative of T: qi|i = −k T|ii + . . . . However, the heat PDE needs to be satisfied in the integral sense. Therefore, the requirements for T can be weaken as follows.

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Notation remarks Local differential formulation Global integral formulations Matrix formulations

Weighted formulation and weak variational form

Weighted integral formulation

✎ ✍ ☞ ✌

  • B

̺ c

  • T δT +
  • B

qi|i δT −

  • B

f δT = 0 (for every δT) The term qi|i introduces the second derivative of T: qi|i = −k T|ii + . . . . However, the heat PDE needs to be satisfied in the integral sense. Therefore, the requirements for T can be weaken as follows. Integrating by parts (using the divergence theorem)

  • B

qi|i δT =

  • B

(qi δT)|i −

  • B

qi δT|i =

  • ∂B

qi δT ni −

  • B

qi δT|i

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Notation remarks Local differential formulation Global integral formulations Matrix formulations

Weighted formulation and weak variational form

Weighted integral formulation

✎ ✍ ☞ ✌

  • B

̺ c

  • T δT +
  • B

qi|i δT −

  • B

f δT = 0 (for every δT) The term qi|i introduces the second derivative of T: qi|i = −k T|ii + . . . . However, the heat PDE needs to be satisfied in the integral sense. Therefore, the requirements for T can be weaken as follows. Integrating by parts (using the divergence theorem)

  • B

qi|i δT =

  • B

(qi δT)|i −

  • B

qi δT|i =

  • ∂B

qi δT ni −

  • B

qi δT|i Using the Neumann b.c and the property of test function

  • ∂B

qi ni δT =

  • ∂Bq

qi ni

  • −ˆ

q

δT +

  • ∂BT

qi ni δT

  • = −
  • ∂Bq

ˆ q δT

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Notation remarks Local differential formulation Global integral formulations Matrix formulations

Weighted formulation and weak variational form

Weighted integral formulation

✎ ✍ ☞ ✌

  • B

̺ c

  • T δT +
  • B

qi|i δT −

  • B

f δT = 0 (for every δT) The term qi|i introduces the second derivative of T: qi|i = −k T|ii + . . . . However, the heat PDE needs to be satisfied in the integral sense. Therefore, the requirements for T can be weaken as follows. After integrating by parts and using the Neumann boundary condition

  • B

qi|i δT = −

  • ∂Bq

ˆ q δT −

  • B

qi δT|i Weak variational form

✎ ✍ ☞ ✌

  • B

̺ c

  • T δT−
  • B

qi δT|i −

  • ∂Bq

ˆ q δT −

  • B

f δT = 0 (for every δT)

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Notation remarks Local differential formulation Global integral formulations Matrix formulations

Weighted formulation and weak variational form

Weighted integral formulation

✎ ✍ ☞ ✌

  • B

̺ c

  • T δT +
  • B

qi|i δT −

  • B

f δT = 0 (for every δT) The term qi|i introduces the second derivative of T: qi|i = −k T|ii + . . . . However, the heat PDE needs to be satisfied in the integral sense. Therefore, the requirements for T can be weaken as follows. Weak variational form

✎ ✍ ☞ ✌

  • B

̺ c

  • T δT−
  • B

qi δT|i −

  • ∂Bq

ˆ q δT −

  • B

f δT = 0 (for every δT) Now, only the first order spatial-differentiability of T is required. In this formulation the Neumann boundary condition is already met (it has been used in a natural way). Therefore, the only additional requirements are the Dirichlet boundary condition and the initial condition.

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Notation remarks Local differential formulation Global integral formulations Matrix formulations

Outline

1

Notation remarks

2

Local differential formulation Partial Differential Equation Initial and boundary conditions Initial-Boundary-Value Problem

3

Global integral formulations Test functions Weighted formulation and weak variational form

4

Matrix formulations Approximation Transient heat transfer (ordinary differential equations) Stationary heat transfer (algebraic equations)

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Notation remarks Local differential formulation Global integral formulations Matrix formulations

Approximation functions and space

The spatial approximation of solution in the domain B is accomplished by a linear combination of (global) shape functions, φs = φs(x), T(x, t) = θs(t) φs(x) (s = 1, . . . N; summation over s) where θs(t)

  • K

are (time-dependent) coefficients – the degrees of

freedom (N is the total number of degrees of freedom). Consistent result is obtained now for the time rate of temperature

  • T(x, t) =
  • θs(t) φs(x)

where

  • θs(t) = dθs(t)

dt

K

s

.

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Notation remarks Local differential formulation Global integral formulations Matrix formulations

Approximation functions and space

The spatial approximation of solution in the domain B is accomplished by a linear combination of (global) shape functions, φs = φs(x), T(x, t) = θs(t) φs(x) (s = 1, . . . N; summation over s) where θs(t)

  • K

are (time-dependent) coefficients – the degrees of

freedom (N is the total number of degrees of freedom). Consistent result is obtained now for the time rate of temperature

  • T(x, t) =
  • θs(t) φs(x)

where

  • θs(t) = dθs(t)

dt

K

s

.

Distinctive feature of the Galerkin method: The same shape functions are used to approximate the solution as well as the test function, namely δT(x) = δθr φr(x) (r = 1, . . . N; summation over r) .

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Notation remarks Local differential formulation Global integral formulations Matrix formulations

Transient heat transfer (system of ODEs)

To reduce the “regularity” requirements for solution the approximations T = θs φs

T =

  • θs φs
  • ,

δT = δθr φr are used for the weak variational form of the heat transfer problem

  • B

̺ c

  • T δT −
  • B

qi δT|i −

  • ∂Bq

ˆ q δT −

  • B

f δT = 0 .

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Notation remarks Local differential formulation Global integral formulations Matrix formulations

Transient heat transfer (system of ODEs)

To reduce the “regularity” requirements for solution the approximations T = θs φs

T =

  • θs φs
  • ,

δT = δθr φr are used for the weak variational form of the heat transfer problem

  • B

̺ c

  • T δT −
  • B

qi δT|i −

  • ∂Bq

ˆ q δT −

  • B

f δT = 0 .

1

  • B

̺ c

  • T δT =
  • θs δθr
  • B

̺ c φs φr =

  • θs δθr Mrs
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Notation remarks Local differential formulation Global integral formulations Matrix formulations

Transient heat transfer (system of ODEs)

To reduce the “regularity” requirements for solution the approximations T = θs φs

T =

  • θs φs
  • ,

δT = δθr φr are used for the weak variational form of the heat transfer problem

  • B

̺ c

  • T δT −
  • B

qi δT|i −

  • ∂Bq

ˆ q δT −

  • B

f δT = 0 .

1

  • B

̺ c

  • T δT =
  • θs δθr
  • B

̺ c φs φr =

  • θs δθr Mrs

2

  • B

qi δT|i =

  • B
  • −k T|i + ̺ c ui T
  • δT|i = θs δθr
  • B
  • −k φs|i + ̺ c ui φs
  • φr|i

= θs δθr Krs

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Notation remarks Local differential formulation Global integral formulations Matrix formulations

Transient heat transfer (system of ODEs)

To reduce the “regularity” requirements for solution the approximations T = θs φs

T =

  • θs φs
  • ,

δT = δθr φr are used for the weak variational form of the heat transfer problem

  • B

̺ c

  • T δT −
  • B

qi δT|i −

  • ∂Bq

ˆ q δT −

  • B

f δT = 0 .

1

  • B

̺ c

  • T δT =
  • θs δθr
  • B

̺ c φs φr =

  • θs δθr Mrs

2

  • B

qi δT|i =

  • B
  • −k T|i + ̺ c ui T
  • δT|i = θs δθr
  • B
  • −k φs|i + ̺ c ui φs
  • φr|i

= θs δθr Krs

3

  • ∂Bq

ˆ q δT = δθr

  • ∂Bq

ˆ q φr = δθr Qr

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Notation remarks Local differential formulation Global integral formulations Matrix formulations

Transient heat transfer (system of ODEs)

To reduce the “regularity” requirements for solution the approximations T = θs φs

T =

  • θs φs
  • ,

δT = δθr φr are used for the weak variational form of the heat transfer problem

  • B

̺ c

  • T δT −
  • B

qi δT|i −

  • ∂Bq

ˆ q δT −

  • B

f δT = 0 .

1

  • B

̺ c

  • T δT =
  • θs δθr
  • B

̺ c φs φr =

  • θs δθr Mrs

2

  • B

qi δT|i =

  • B
  • −k T|i + ̺ c ui T
  • δT|i = θs δθr
  • B
  • −k φs|i + ̺ c ui φs
  • φr|i

= θs δθr Krs

3

  • ∂Bq

ˆ q δT = δθr

  • ∂Bq

ˆ q φr = δθr Qr

4

  • B

f δT = δθr

  • B

f φr = δθr Fr

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Notation remarks Local differential formulation Global integral formulations Matrix formulations

Transient heat transfer (system of ODEs)

Matrix formulation of the heat transfer problem

  • Mrs
  • θs − Krs θs − (Qr + Fr)
  • δθr = 0

for every δθr. This produces the following system of first-order ordinary differential equations (for θs = θs(t) =?): ✞ ✝ ☎ ✆ Mrs

  • θs − Krs θs = (Qr + Fr)

(r, s = 1, . . . N).

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Notation remarks Local differential formulation Global integral formulations Matrix formulations

Transient heat transfer (system of ODEs)

Matrix formulation of the heat transfer problem

  • Mrs
  • θs − Krs θs − (Qr + Fr)
  • δθr = 0

for every δθr. This produces the following system of first-order ordinary differential equations (for θs = θs(t) =?): ✞ ✝ ☎ ✆ Mrs

  • θs − Krs θs = (Qr + Fr)

(r, s = 1, . . . N). Mrs =

  • B

̺ c φs φr – the thermal capacity matrix

J

K

,

Krs =

  • B
  • −k φs|i + ̺ c ui φs
  • φr|i

– the heat transfer matrix

W

K

,

Qr =

  • ∂Bq

ˆ q φr – the inward heat flow vector

  • W

,

Fr =

  • B

f φr – the heat production vector

  • W

.

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Notation remarks Local differential formulation Global integral formulations Matrix formulations

Stationary heat transfer (algebraic equations)

T = T(x) , f = f(x) , ui = ui(x) (for x ∈ B) . BVP of stationary heat flow: Find T = T(x) satisfying (in B) qi|i − f = 0 where: qi = qi(T) =

  • −k T|i

(no convection), −k T|i + ̺ c ui T (with convection), with boundary conditions: T = ˆ T

  • n ∂BT (Dirichlet),

− qi(T) ni = ˆ q

  • n ∂Bq (Neumann).

The weak variational form lacks the rate integrand −

  • B

qi δT|i −

  • ∂Bq

ˆ q δT −

  • B

f δT = 0 . The approximations T(x) = θs φs(x), δT(x) = δθr φr(x) lead to the following system of linear algebraic equations (for θs =?):

✞ ✝ ☎ ✆

−Krs θs = (Qr + Fr) (r, s = 1, . . . N).