New Math Tool Helps Engineers Predict Hidden Temperature Changes in Materials
Researchers have developed a mathematical method to determine internal temperature patterns in materials undergoing phase transitions—like metals melting or water freezing—using only surface measurements. The technique could accelerate design of better heat management systems in manufacturing, aerospace, and battery technology.
Originaltitel: Solving an inverse heat conduction problem with phase transitions using Tikhonov regularization
<p>In our previous work, we introduced an iterative algorithm for the nonlinear inverse heat conduction problem in which material parameters may depend on temperature, and demonstrated its effectiveness in practical applications (Ngendahayo et al., Citation2021). A formal convergence analysis was provided in (Ngendahayo et al., Citation2025), thereby establishing a solid theoretical foundation for the method. In practical applications, such as heat treatment of steel, phase transitions often occur within the material. In these cases, the temperature dependence of the material parameters cannot be expressed as a simple functional relationship of the form <img src="//liu.diva-portal.org:0" data-formula-source="{"type":"image","src":"/cms/asset/f55166aa-98a8-480e-958c-abb1d10cb5e7/oama_a_2670816_ilm0001.gif"}" />𝜅=𝜅(𝑢). Instead, the full temperature history <img src="//liu.diva-portal.org:0" data-formula-source="{"type":"image","src":"/cms/asset/291d7171-036d-4dbe-a775-01410bf217fe/oama_a_2670816_ilm0002.gif"}" />𝑢(𝑥,𝑡) determines which phase transitions actually occur within the material. Consequently, the material properties at a given point <img src="//liu.diva-portal.org:0" data-formula-source="{"type":"image","src":"/cms/asset/a8b887c0-15e1-45bb-9c2f-ec6486f189c1/oama_a_2670816_ilm0003.gif"}" />(𝑥,𝑡) depend not only on the temperature <img src="//liu.diva-portal.org:0" data-formula-source="{"type":"image","src":"/cms/asset/30798662-3824-4c31-8cab-0b8dd4c884af/oama_a_2670816_ilm0004.gif"}" />𝑢(𝑥,𝑡), but also on the initial state of the material at <img src="//liu.diva-portal.org:0" data-formula-source="{"type":"image","src":"/cms/asset/debc6467-5872-42af-bceb-54d767a780fe/oama_a_2670816_ilm0005.gif"}" />𝑡=0 and the entire temperature history <img src="//liu.diva-portal.org:0" data-formula-source="{"type":"image","src":"/cms/asset/af88617c-cb6e-4edf-bf2e-4f26bdf91a96/oama_a_2670816_ilm0006.gif"}" />𝑢(𝑥,𝜏) for <img src="//liu.diva-portal.org:0" data-formula-source="{"type":"image","src":"/cms/asset/a6d25b16-e72a-4fdd-ab8a-e023a51d8a2b/oama_a_2670816_ilm0007.gif"}" />0≤𝜏≤𝑡. In this work, we aim to introduce a more precise mathematical description of the nonlinearity caused by the presence of phase transitions in the material. We also extend the convergence analysis in (Ngendahayo et al., Citation2025) to account for phase transitions. In addition, we present numerical experiments demonstrating that the iterative algorithm performs well under these conditions.</p>