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This paper seeks to describe the solution of a simple electrostatic problem using an adaptive hp‐FEM and to show the benefits of this approach. Numerical experiments are presented to demonstrate its superiority.

Adaptive hp‐FEM is used. In contrast with standard FEM, the automatic adaptivity procedure can choose from a variety of refinement candidates. An element with over estimated error can be refined in space, or its polynomial degree can be increased. Arbitrary level hanging nodes are allowed, so that no unnecessary refinements are performed in order to keep a mesh regular.

Numerical solution of a singular electrostatic problem is presented. From the comparison it can be seen that the hp‐FEM outperforms both the standard linear and quadratic elements significantly. The accuracy of an hp‐FEM solution would be hard to attain by standard means due to the limited capacity of the computer memory.

The paper describes results obtained from an original and innovative implementation of the adaptive hp‐FEM.

Most eddy current problems are solved using numerical schemes based on the differential approach. Nevertheless, there exist several classes of tasks where use of this approach may be complicated (problems characterized by geometrical incommensurability of individual subdomains, motion, etc.). In such cases, application of the integrodifferential approach may be an advantage. The paper seeks to present the theoretical background of the method.

The mathematical model consists of a system of integrodifferential equations for current densities in electrically conductive domains.

The methodology is illustrated on an example. All computations are realized by a code developed and written by the authors.

The presented algorithm based on the integrodifferential approach makes it possible to solve problems that are only hardly solvable by classical differential methods.

The purpose of this paper is to present a methodology of high‐precision finite element modeling of induction heating of rotating nonferromagnetic cylindrical billets in static magnetic field produced by appropriately arranged permanent magnets.

The mathematical model consisting of two partial differential equations describing the distribution of the magnetic and temperature fields are solved by a fully adaptive higher‐order finite element method in the monolithic formulation and selected results are validated experimentally.

The method of solution realized by own code is very fast, robust and exhibits much more powerful features when compared with classical low‐order numerical methods implemented in existing commercial codes.

For sufficiently long arrangements the method provides good results even for 2D model. The principal limitation consists in problems with determining correct boundary conditions for the temperature field (generalized coefficient of convective heat transfer as a function of the temperature and revolutions).

The methodology can successfully be used for design of devices for induction heating of cylindrical nonmagnetic bodies by rotation and determination of their operation parameters.

The paper is a presentation of the fully adaptive higher‐order finite element and its utilization for a monolithic numerical solution of a relatively complicated coupled problem.

The purpose of this paper is to present a model of induction heating of aluminium billets rotating in a static magnetic field generated by permanent magnets. The model is solved by the authors' own software and the results are verified experimentally.

The mathematical model of the problem given by two partial differential equations describing the distribution of the magnetic and temperature fields in the system is solved by a fully adaptive higher‐order finite element method in the hard‐coupled formulation. All material nonlinearities are taken into account.

The method of solution realized by the code is reliable and works faster in comparison with the existing low‐order finite element codes.

The method works for 2D arrangements with an extremely high accuracy. Its limitations consist mainly in problems of determining the coefficients of convection and radiation for temperature field in the system (respecting both temperature and revolutions).

The methodology can successfully be used for design of devices for induction heating of cylindrical nonmagnetic bodies by rotation and anticipation of their operation parameters.

The paper presents a fully adaptive higher‐order finite element and its utilization for a hard‐coupled numerical solution of the problem of induction heating.

The paper presents the principal elements of automatic adaptivity built in our 2D software for monolithic solution of multiphysics problems based on a fully adaptive finite element method of higher order of accuracy. The adaptive techniques are illustrated by appropriate examples.

Presented are algorithms for realization of the h‐adaptivity, p‐adaptivity, hp‐adaptivity, creation of curvilinear elements for modelling general boundaries and interfaces. Indicated also is the possibility of combining triangular and quadrilateral elements (both classical and curved).

The presented higher‐order adaptive processes are reliable, robust and lead to a substantial reduction of the degrees of freedom in comparison with the techniques used in low‐order finite element methods. They allow solving examples that are by classical approaches either unsolvable or solvable at a cost of high memory and time of computation.

The adaptive processes described in the paper are still limited to 2D computations. Their computer implementation is highly nontrivial (every physical field in a multiphysics task is generally solved on a different mesh satisfying its specific features) and in 3D the number of possible adaptive steps is many times higher.

The described adaptive techniques may represent a powerful tool for the monolithic solution of complex multiphysics problems.

The presented higher‐order adaptive approach of solution is shown to provide better results than the schemes implemented in professional codes based on low‐order finite element methods. Obtaining the results, moreover, requires less time and computer memory.

A novel technique for control of complex physical processes based on the solution of their sufficiently accurate models is presented. The technique works with the model order reduction (MOR), which significantly accelerates the solution at a still acceptable uncertainty. Its advantages are illustrated with an example of induction brazing.

The complete mathematical model of the above heat treatment process is presented. Considering all relevant nonlinearities, the numerical model is reduced using the orthogonal decomposition and solved by the finite element method (FEM). It is cheap compared with classical FEM.

The proposed technique is applicable in a wide variety of linear and weakly nonlinear problems and exhibits a good degree of robustness and reliability.

The quality of obtained results strongly depends on the temperature dependencies of material properties and degree of nonlinearities involved. In case of multiphysics problems characterized by low nonlinearities, the results of solved problems differ only negligibly from those solved on the full model, but the computation time is lower by two and more orders. Yet, however, application of the technique in problems with stronger nonlinearities was not fully evaluated.

The presented model and methodology of its solution may represent a basis for design of complex technologies connected with induction-based heat treatment of metal materials.

Proposal of a sophisticated methodology for solution of complex multiphysics problems established the MOR technology that significantly accelerates their solution at still acceptable errors.