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This paper aims to propose a fast and accurate simulation of large-scale induction heating problems by using nonlinear reduced-order models.

A projection space for model order reduction (MOR) is quickly generated from the first kernels of Volterra’s series to the problem solution. The nonlinear reduced model can be solved with time-harmonic phasor approximation, as the nonlinear quadratic structure of the full problem is preserved by the projection.

The solution of induction heating problems is still computationally expensive, even with a time-harmonic eddy current approximation. Numerical results show that the construction of the nonlinear reduced model has a computational cost which is orders of magnitude smaller than that required for the solution of the full problem.

Only linear magnetic materials are considered in the present formulation.

The proposed MOR approach is suitable for the solution of industrial problems with a computing time which is orders of magnitude smaller than that required for the full unreduced problem, solved by traditional discretization methods such as finite element method.

The most common technique for MOR is the proper orthogonal decomposition. It requires solving the full nonlinear problem several times. The present MOR approach can be built directly at a negligible computational cost instead. From the reduced model, magnetic and temperature fields can be accurately reconstructed in whole time and space domains.

The purpose of this paper is to introduce a perturbation method for the A‐χ geometric formulation to solve eddy‐current problems and apply it to the feasibility design of a non‐destructive evaluation device suitable to detect long‐longitudinal volumetric flaws in hot steel bars.

The effect of the flaw is accurately and efficiently computed by solving an eddy‐current problem over an hexahedral grid which gives directly the perturbation due to the flaw with respect to the unperturbed configuration.

The perturbation method, reducing the cancelation error, produces accurate results also for small variations between the solutions obtained in the perturbed and unperturbed configurations. This is especially required when the tool is used as a forward solver for an inverse problem. The method yields also to a considerable speedup: the mesh used in the perturbed problem can in fact be reduced at a small fraction of the initial mesh, considering only a limited region surrounding the flaw in which the mesh can be refined. Moreover, the full three‐dimensional unperturbed problem does not need to be solved, since the source term for computing the perturbation is evaluated by solving a two‐dimensional flawless configuration having revolution symmetry.

A perturbation method for the A‐χ geometric formulation to solve eddy‐current problems has been introduced. The advantages of the perturbation method for non‐destructive testing applications have been described.