HeatConduction Process

Introduction

This process simulates transient heat conduction to model temperature distribution and heat flow.

It supports various thermal material models and can handle both steady-state and transient heat conduction problems with temperature-dependent properties.

Key Features

The major features of this process are:

• Temperature-dependent material properties
• Mass lumping option for improved numerical stability
• Heat flux computation and output

The process handles both Dirichlet and Neumann boundary conditions and can account for various heat sources and material interfaces.

Theoretical background

The heat conduction process solves the transient heat equation in strong form:

where:

• $\rho$ is the density [M·L⁻³]
• $c_p$ is the specific heat capacity [L²·T⁻²·Θ⁻¹]
• $T$ is the temperature [Θ]
• $\boldsymbol{\lambda}$ is the thermal conductivity tensor [M·L·T⁻³·Θ⁻¹]
• $Q$ is the heat source term [M·L⁻¹·T⁻³]; in SI-units [W·m⁻³].

The heat flux vector is defined by Fourier’s law:

where $\mathbf{q}$ is the heat flux vector [M·T⁻³].

Finite Element Discretization

The heat equation is discretized using the finite element method, resulting in:

where the element matrices are defined as follows.

Element Storage Matrix

The element storage matrix $\mathbf{M}_e$ [M·L⁻¹·Θ⁻¹] represents the heat capacity:

where $\mathbf{N}$ is the shape function matrix.

Element Conductivity Matrix

The element conductivity matrix $\mathbf{K}_e$ [M·L·T⁻³·Θ⁻¹] represents the thermal diffusion:

where $\nabla \mathbf{N}$ is the gradient of shape functions.

Element Load Vector

The load vector $\mathbf{f}_e$ [M·L·T⁻³] includes heat sources and boundary fluxes:

where:

• $Q$ is the volumetric heat source [M·L⁻¹·T⁻³]
• $q_n = \mathbf{q} \cdot \mathbf{n}$ is the surface heat flux [M·T⁻³] with $\mathbf{n}$ the outward normal to the surface.

Definition in the project file

The heat conduction process has to be declared in the project file in the processes block. For example in following way:

<process>
    <name>HeatConduction</name>
    <type>HEAT_CONDUCTION</type>
    <integration_order>2</integration_order>
    <linear>true</linear>
    <mass_lumping>false</mass_lumping>
    <process_variables>
        <process_variable>temperature</process_variable>
    </process_variables>
    <secondary_variables>
        <secondary_variable name="heat_flux"/>
    </secondary_variables>
</process>

For more detailed description of tags used in this snippet, please see Processes.

Process variables

The heat conduction process requires a single scalar temperature process variable. For more details, see Process variables.

Media

The heat conduction process requires properties for the medium only.

Medium properties

Required medium property

See medium properties for more details on defining them.

Features

Mass lumping

Mass lumping can be enabled to improve numerical stability for transient problems:

<mass_lumping>true</mass_lumping>

Linear solver optimization

The process supports linear solver optimizations for improved performance when solving linear problems:

<linear>true</linear>
<linear_solver_compute_only_upon_timestep_change>false</linear_solver_compute_only_upon_timestep_change>

Source terms

The heat conduction process supports various source term types including:

• Volumetric heat sources: Applied to entire domain regions
• Line source terms: For 2D and 3D problems.

Available benchmarks

To gain more insight into this process, you can investigate heat conduction benchmarks.