InverseHeatTransfer

Important

This package is under active development! Features and API are subject to change. ❗

InverseHeatTransfer.jl is a Julia package designed to solve direct and inverse heat conduction problems. It provides numerical solvers for temperature distribution and robust optimization wrappers for parameter estimation.

Installation

Since this package and its core dependency are not yet in the General registry, install them directly via URL:

using Pkg
# Required dependency
Pkg.add(url="https://github.com/Manarom/ScaledPolynomials.jl.git")
# This package
Pkg.add(url="https://github.com/Manarom/InverseHeatTransfer.jl.git")

Mathematical Model

The package solves the 1D heat equation in the following form:

$$ \frac{C}{\lambda} \frac{\partial T}{\partial t} = \frac{\partial^2 T}{\partial x^2} + \frac{\lambda'}{\lambda} \left( \frac{\partial T}{\partial x} \right)^2 $$

With the initial conditions:

$$ T(x, 0) = T_i(x) $$

Physical Parameters:

• $\lambda$ — Thermal conductivity $[W/(m \cdot K)]$
• $C$ — Volumetric heat capacity $[J/(m^3 \cdot K)]$, where $C = C_p \cdot \rho$
• $C_p$ — Specific heat $[J/(kg \cdot K)]$
• $\rho$ — Density $[kg/m^3]$

Inverse Problem & Regularization

The main interface for a single inverse problem (see SingleInverseProblem type) supports the optimization of the following parameters of the heat transfer equation:

• Initial temperature distribution
• Boundary conditions (Dirichlet and Neuman)
• Physical properties of the material (both $\lambda$ and $C$)

To make the parameter optimizable it should be wrapped into OptimizableVariable type. For example, Bernstein polynomials can be used to approximate the temperature dependence of $\lambda$ and $C$.

For ill-posed inverse problems, the package implements advanced estimation and regularization techniques:

• Tikhonov Regularization: Used to ensure solution stability by penalizing high-frequency oscillations in the parameter space.
• Weighted Regression: Supports various weighting strategies for the discrepancy function to handle measurement noise:
  • Diagonal Weighting: Independent weights for each data point.
  • Proportional Weighting: Weights scaled according to the magnitude of the measured values.
  • AR(1) Weighting Function: Accounts for first-order autoregressive correlations in measurement errors.

Parallel Inverse Solver

The library includes a specialized type (see ParallelInverseProblems) for the parallel solving of inverse problems.

This allows for:

*   Simultaneous estimation of $\lambda$ and $C$ across multiple experimental datasets.
*   A **joint discrepancy function** that aggregates residuals from several measurements with different heating regimes.
*   Improved parameter identifiability by leveraging diverse thermal loading scenarios within a single optimization framework.

Data formats compatibility

*   **Experimental data  :** Import from HDF5,ASCII and binary formats for import and export experimental data.
*   **Inverse problem solution:** Saving to HDF5.

Numerical Solvers (Direct Problem)

The direct problem can be solved using the following Finite Difference Schemes:

1. Fully Explicit
2. Fully Implicit
3. Crank-Nicolson (default)
4. BDF-Implicit

Roadmap (TODO)

• Integration of Spectral Methods via OrdinaryDiffEq.jl.
• Advanced Sensitivity Analysis for inverse problems.