Download Akantu User's Guide - Computational Solid Mechanics Laboratory

Chapter 6
Heat Transfer Model
The heat transfer model is a specific implementation of the Model interface dedicated to handle
the dynamic heat equation.
6.1
Theory
The strong form of the dynamic heat equation can be expressed as
ρcv Ṫ + ∇ · κ∇T = b
(6.1)
with T the scalar temperature field, cv the specific heat capacity, ρ the mass density, κ the conductivity tensor, and b the heat generation per unit of volume. The discretized weak form with a finite
number of elements is
!
!
Z
Z
X Z
X Z
∀i
ρcv N j Ni dΩ Ṫ j −
bNi dΩ
(6.2)
κ∇N j ∇Ni dΩ T j = − Ni q · ndΓ +
Ω
j
j
Γ
Ω
Ω
with i and j the node indices, n the normal field to the surface Γ = ∂Ω. To simplify, we can define
the capacity and the conductivity matrices as
Z
Z
Ci j =
ρcv N j Ni dΩ
and
Ki j = −
κ∇N j ∇Ni dΩ
(6.3)
Ω
Ω
and the system to solve can be written
C · Ṫ = Qext − K · T ,
(6.4)
with Qext the consistent heat generated.
6.2
Using the Heat Transfer Model
Currently, the HeatTransferModel object uses dynamic analysis with an explicit time integration
scheme. It can simply be created like this
HeatTransferModel model(mesh, spatial_dimension);
while an existing mesh has been used (see 2.4). Then the model object can be initialized with:
model.initFull()
where a material file name has to be provided. This function will load the material properties, and
allocate / initialize the nodes and element Array s More precisely, the heat transfer model contains
4 Arrays :
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