transient-thermal-1756651292

Single Element Transient Thermal with deal.II

The single element transient thermal model has been created as the base for the implementation of kMC-FEA and JMAK.
The model is built with deal.II.

Classes

TimeDependentTemperature

This class is used to take input from a file describing the time and the temperature.

• The constructor reads a file and populates a temperature_data vector with (time, temperature) pairs.
• Temperatures are converted to Kelvin by adding 273.15.
• The private member interpolate_temperature performs linear interpolation between the nearest values.
• Extrapolation is not performed — values outside the range are clamped to the minimum or maximum provided.

TransientThermalProblem

This object houses the main problem definition. The class has the following member functions:

1. setup_system

   • Creates a cube mesh using GridGenerator::subdivided_hyper_rectangle between coordinates $(0,0,0)$ and $(1,1,1)$, subdivided into {2,2,2} blocks (→ 512 hexahedral elements).
   • Uses quadratic (Q2) elements (FE_Q<3>(2)).
   • Initialises DoFs, solution vectors, sparsity pattern, and the mass, stiffness, and system matrices.

2. assemble_system

   • Assembles the consistent mass matrix and conductivity matrix.
   • Material properties (c_p(T) and k(T)) are interpolated from tables and evaluated at the average temperature per element.
   • Forms the system matrix using the θ-method with θ = 1 (implicit Euler).
   • Right-hand side (RHS) is computed as $M\cdot u^n$.

3. solve_time_step

   • Uses the direct solver UMFPACK to solve the linear system for the current time step.

4. load_material_tables

   • Loads c_p(T) and k(T) from external .dat files.
   • Tables are linearly interpolated in temperature.

5. output_results

   • Exports results each step into Solution/solution-<timestep>.vtu for ParaView and .gpl for Gnuplot.
   • Both the initial condition (step 0) and subsequent time steps are written.

6. run

   • Calls setup_system and load_material_tables.
   • Initialises the solution with a uniform temperature of 293.15 K.
   • Outputs the initial condition.
   • Advances in time steps until total_time = 555.0 s, applying Dirichlet boundary conditions from temperature_input.dat via TimeDependentTemperature.
   • Each step assembles the system, solves, updates the solution, and outputs results.

---
title: Schematic representation of run()
---
flowchart TD
    A["Start Simulation"] --> B["Setup System<br>- Mesh cube (0,1)^3<br>- FE_Q<3>(2)<br>- Init matrices"]
    B --> C["Load Material Tables<br>cp(T), k(T) from files"]
    C --> D["Set Initial Condition<br>T = 293.15 K everywhere"]
    D --> E["Output Initial Condition"]
    E --> F["Time Loop: while t < total_time"]
    F --> |yes| M["t += time_step"]
    M --> G["`Assemble System<br>- Mass Matrix<br>- Stiffness Matrix<br>- Apply cp(T), k(T)`"]
    G --> H["Update Boundary Temp<br>from file"]
    H --> I["Apply Boundary Conditions"]
    I --> J["Solve Linear System<br>UMFPACK"]
    J --> K["Update Solution<br>uⁿ -> uⁿ⁺¹"]
    K --> L["Output Results<br>.vtu + .gpl"]
    L --> F
    F -.-> |No| N["End Simulation"]

Main function

Instantiates TransientThermalProblem and calls run() to execute the simulation.

Summary of Model Assumptions

• Mesh: cube domain [0,1]^3, subdivided into 512 quadratic hexahedral elements.
• Time integration: backward Euler (θ = 1 implicit).
• Boundary condition: Dirichlet temperature from file (clamped outside range).
• Initial condition: uniform 293.15 K.
• Material data:
  • ρ = 4620 kg/m³ (constant),
  • cp(T), k(T) read from files and interpolated linearly.
• Outputs: .vtu for ParaView and .gpl for Gnuplot at every step.