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An inverse problem for the one-phase Stefan problem with varying melting temperature

Marc Dambrine, Helmut Harbrecht

TL;DR

The article addresses forward and inverse problems for a transient one-phase Stefan problem with a time-varying melting temperature. It develops a moving-mesh finite element framework on a space-time tube and uses a front-tracking approach to evolve the domain, together with a v=u-u_m transformation to handle the time dependence of the interface temperature. An inverse reconstruction strategy is proposed to recover the melting temperature from boundary observations, including identifiability discussion and a least-squares update mechanism. Numerical experiments in two dimensions demonstrate the feasibility of reconstructing u_m from evolving boundaries, albeit with sensitivity to noise and inherent identifiability challenges. The work provides a proof-of-concept pathway for inferring interface physics from free-boundary data, with potential relevance to experiments such as methane hydrate systems.

Abstract

The present article is dedicated to the forward and backward solution of a transient one-phase Stefan problem. In the forward problem, we compute the evolution of the initial domain for a Stefan problem where the melting temperature varies over time. This occurs in practice, for example, when the pressure in the external space changes in time. In the corresponding backward problem, we then reconstruct the time-dependent melting temperature from the knowledge of the evolving geometry. We develop respective numerical algorithms using a moving mesh finite element method and provide numerical simulations.

An inverse problem for the one-phase Stefan problem with varying melting temperature

TL;DR

The article addresses forward and inverse problems for a transient one-phase Stefan problem with a time-varying melting temperature. It develops a moving-mesh finite element framework on a space-time tube and uses a front-tracking approach to evolve the domain, together with a v=u-u_m transformation to handle the time dependence of the interface temperature. An inverse reconstruction strategy is proposed to recover the melting temperature from boundary observations, including identifiability discussion and a least-squares update mechanism. Numerical experiments in two dimensions demonstrate the feasibility of reconstructing u_m from evolving boundaries, albeit with sensitivity to noise and inherent identifiability challenges. The work provides a proof-of-concept pathway for inferring interface physics from free-boundary data, with potential relevance to experiments such as methane hydrate systems.

Abstract

The present article is dedicated to the forward and backward solution of a transient one-phase Stefan problem. In the forward problem, we compute the evolution of the initial domain for a Stefan problem where the melting temperature varies over time. This occurs in practice, for example, when the pressure in the external space changes in time. In the corresponding backward problem, we then reconstruct the time-dependent melting temperature from the knowledge of the evolving geometry. We develop respective numerical algorithms using a moving mesh finite element method and provide numerical simulations.

Paper Structure

This paper contains 22 sections, 1 theorem, 35 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Stefan problem
  3. Motivation and background
  4. Our contribution
  5. Content
  6. A nonstationary Stefan problem
  7. Classical one-phase Stefan problem
  8. Varying the melting temperature
  9. Generation of the space-time tube
  10. Solution algorithm for the transient Stefan problem
  11. Numerical simulation of the forward solution
  12. Discretization of the space-time tube
  13. Discretization of the heat equation
  14. Updating the domain
  15. Numerical results of the forward simulation
  16. ...and 7 more sections

Key Result

Lemma 4.2

For any $R_T>0$ and for any final time $T$, there exists a temperature trajectory $u_m(t)$ that transforms, through the Stefan dynamics, the initial disk into a disk with radius $R_T$ at time $T$.

Figures (9)

  • Figure 1: Geometric setup of the Stefan problem: The space-time tube $Q_T$ is given by the time-evolution of the initial domain $\Omega_0$. The cut area at time $t$ is the domain $\Omega_t$ with boundary $\Gamma_t = \partial\Omega_t$.
  • Figure 2: Generation of the space-time tube by the mapping ${\bf T}(t, \cdot)$ induced by the velocity field ${\bf V}$. In particular, the relation $\Omega_t = {\bf T}(t,\Omega_{\text{ref}})$ always applies.
  • Figure 3: The finite element mesh on the circle and its mapped counterpart on the deformed domain $\Omega_k$.
  • Figure 4: The space-time tube in case of the melting temperature $u_m(t) = 1/20(t-5/2)^2$.
  • Figure 5: The space-time tube in case of the melting temperature $u_m(t) = 1/20(\cos(2t)-1)$ on the right.
  • ...and 4 more figures

Theorems & Definitions (2)

  • Example 4.1
  • Lemma 4.2