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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.

The purpose of this paper is to compare different reduced-order models for models of control of induction brazing process. In the presented application, the problem is to reconstruct temperature at the points of interests (hot spots) from information measured at accessible places.

The paper describes the process of induction brazing. It presents the full field model and evaluates the possibilities for obtaining reduced models for temperature estimation. The primary attention is paid to the model based on proper orthogonal decomposition (POD).

The paper shows that for the given application, it is possible to find low-order estimator. In the examined linear case, the best estimator was created using POD reduced model together with the linear Kalman filter.

The authors are aware of two main limitations of the presented study: material properties are considered linear, which is not a completely realistic assumption. However, if strong coupling and nonlinear material parameters are considered, the model becomes unsolvable. The process and measurement uncertainties are not considered.

The paper deals with POD of multi-physics 3 D application of induction brazing. The paper compares 11 different methods for temperature estimator design.

The purpose of this paper is to present the calibration of a laser welding model suitable for solving problems with input data that are either unknown or known only approximately.

The calibration starts from the measured temperature profile of the weld, and the aim is to get a similar profile by the solution of the model. The corresponding procedure is based on replacing the material characteristics that are known only approximately by polynomial or rational functions whose coefficients are determined using a suitable optimization process. The algorithm is supplemented with a simplified model of the keyhole shape.

The big advantage of the proposed approach is the velocity of solution of the problem and low consumption of the sources (hardware and software). In comparison with solving the full model of laser welding, the methodology provides results of a still acceptable accuracy by several orders faster. On the other hand, the results also depend on the strategy of selecting the points at which the temperature is verified and on “manual” setting of the deformation parameters.

Application of the methodology is conditioned by several experiments with the used material (without experiment it is impossible to carry out the calibration and set the shape of the keyhole), while the full model allows it. On the other hand, the full model is not able to predict the errors in the case when some input data is unknown or known only approximately and the results have to be also confirmed experimentally.

The presented methodology may be used for determining unknown material characteristics and faster modelling of laser welding.

This paper proposes a novel methodology for evaluation of quality of laser welds in cases of unknown or partially unknown material parameters and substantial acceleration (by 2-3 orders) of the numerical solution of the model of laser welding.

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 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.

As far as the authors know, no sufficiently complete model of continual induction hardening was developed and solved so far. The paper presents both mathematical model of the process and algorithm of its solution in the quasi‐coupled formulation.

Computation of electromagnetic and temperature fields is based on the finite element method, while time variable boundary conditions are determined by means of an original theoretically‐empirical procedure.

Substantial are backgrounds for design of the inductor and parameters of the field current as well as parameters of the cooling medium.

The model reached a good level of accuracy validated by suitable experiments. Nevertheless, next work in the field will also have to respect history of the heating before cooling itself (the austenitizing temperature is a function of the velocity of heating). Very important is also appropriate meshing of the investigated region to suppress numerical instabilities appearing during the computation process. Finally, acceleration of numerical schemes is a must, because modelling of one common task (on very fast PCs) takes about 4 h.

The results and conclusions may be used for designing devices for continual induction hardening of axisymmetric bodies.

Complete mathematical and computer model of the process, original methodology for finding the coefficient of convective heat transfer from the cooled part of the heated workpiece to ambient water spray.

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 investigate the parameters and operation characteristics of an actuator working on the principle of thermoelasticity whose structure was designed by the authors.

The mathematical model of the system describes the effects of three physical fields (electromagnetic field, temperature field, and field of mechanical strains and stresses due to thermoelasticity). While the electromagnetic field was solved independently, the thermomechanical task in common with the contact problem was solved in the hard‐coupled formulation. The computations were mostly carried out by own codes.

This type of actuator is characterized by extremely high forces acting in its dilatation element.

The parameters of the system may still be improved using a longer field coil and dilatation element. Attention has to be paid, however to the mechanical stability of the system. Another improvement could be achieved by suitably designed cooling of the coil that would allow increasing parameters of the field current (its frequency or amplitude).

The device is promising for various fixing tasks in the industrial environment.

Although the methods of numerical processing of particular fields are known, the paper provides an algorithm for their simultaneous solution while respecting the temperature dependencies of the material properties and continuous change of the contact surfaces.