4. Strong and weak formulations - one dimensional heat equation

Direct Approach

Repitition

Finite Element Method

Differential Equation

Weak Formulation

Approximating Functions

Weighted Residuals

FEM - Formulation

TodayOne-dim. Heat equation

Vector

Classification of Problems

Scalar

1-D

2-D

3-D

T(x)

u(x)

T(x,y)

T(x,y,z)

u(x)

u(x) = v(x)

(x)

u(x) =

u(x,y,z)

v(x,y,z)

w(x,y,z)

u(x,y)

v(x,y)u(x) =

Heat flow

Spring elements

2D Flow

3D Flow

Beam

2D - solid

3D - solid

4.1 One-dimensional heat equation- strong form (stationary flow)

• T(x) – temperature distribution [K]

• A(x) – cross section area [m]

• Q – internal heat source [J/s m] or [W/m]

Strong Form : • Differential equation• Boundary conditions• Region

Balance equation

• Study an infinitely small part of the 1-D body

H - Heat flow [J/s] or [W]

Heat balance:(Input positive)

or

Balance equation, contdbut

The balance equation becomes:

H - Heat flow [J/s] or [W]q – Heat flux [W/m2]

Compare with stresses:A

N

Constitutive relation(Material law)

• Fourier’s law (1822):

(Heat flows from hotter to cooler)

T

x

20

10 Direction of heat flow

k = thermal conductivity [W/mK]

Differential Equation

• Inserting constitutive relation into balance eq.:

Natural, (Neumann)

Essential, (Dirichlet)

Known heat flux:

Known temperature:

Boundary conditions

Boundary conditions must always be known at all boundaries

Fundamental Equations- One dimensional heat flow

Flux vector qn

Gradient T

Material point Body

Constitutive law

dx

dTkq

Balance

QAqdx

d)(

Heat source Q

Temperature T

Differential eq.

0

Q

dx

dTAk

dx

d

Strong Form

1. differential equation

2. region

3. boundary conditions

Integration by Parts(mathematical reminder)

• By definition we have: (Fundamental Theorem of Calculus)

• Assume that

• Eq. (2) in (1) implies

• or

(1)

(2), differentiate

4.4 Weak form of one-dimensional heat flow

• Start with differential eq.

• Multiply with arbitrary weight function, v(x)

• Integrate over region

• Integrate by parts

=

=choose:

(3)

(4)

• Eq. (4) in (3) results in

• Inserting x=0 and x=L in the boundary term and using that

• the boundary term may be written as

4.4 Weak form of one-dimensional heat flow

4.4 Weak form of one-dimensional heat flow

• the weak form of one dimensional heat flow is obtained

1. integral equation

2. essential boundary condition (Temperature boundary cond.)

4.4 Advantages of the weak formulation

• only first derivatives of the temperature

• simpler approximating functions can be used

• discontinuities are allowed

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