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Inverse problems and optimization in low frequency electromagnetism

Research project P6/21 (Research action P6)

Persons :

Description :

In this project, efficient numerical modelling of electromagnetic forward problems, investigated in previous IAP phases, is approached as a constraint for inverse problems and extended to optimization schemes. Even if the inverse problem treatment is indeed fundamentally different, our common experience and the results achieved till now give the consortium the capabilities to tackle this kind of problems. Furthermore, new numerical tools, optimization algorithms, regularization techniques and experimental procedures are developed, paying special attention to the questions of uniqueness and stability of the solution. Already existing methods in the field of inverse problems and optimization are adapted to low frequency electromagnetic systems. Finally, the developed methods are validated by means of real-life applications. Among these applications are considered for validation, are: non-destructive testing (eddy current, hysteresis), optimal electromagnetic microsystems (microsensors), design of active and passive shielding of electromagnetic systems taking into account the safety levels, optimal induction heaters (surface thermal treatment, thixoforming, …), power electronics and non-linear power distribution systems, high-efficiency motors, non-classical machines (axial flux, flywheels), design of magnetic and piezoelectric actuators, control, generators for renewable energy (wind generators, biogas turbines), etc.

The present project concerns fundamental research. Three main parts can be distinguished:

• Inverse problems and optimization: theoretical study, in particular with respect to CPU-time and accuracy
• Stochastic uncertainty and external models to be developed with the aim of optimization and inverse problems (electrical circuits and control, materials, direct methods, thermal modelling, stochastic uncertainty (geometric and material parameters))
• Validation environment (rotation and motion, electromagnetic processing of materials, low-frequency screening, power electronic circuits, non-destructive testing, non-linear power distribution systems)

1. Inverse problems and optimization
Many methodologies for solving inverse problems use optimization. The large variety of optimization algorithms available makes the choice for solving a specific problem non-evident. The determination (development if necessary) of the most suitable optimization scheme for each problem is considered.
The sensitivity analysis plays an important role in optimization processes. The solution methods of the forward problems (finite element method, integral methods…), partially still to be developed, have to be flexible enough to cope with as many applications as possible. Moreover, they must be sufficiently fast to allow for a practical solution of the inverse problem or optimization problem at hand, both regarding to computation time and precision. A reformulation of the forward problems might be necessary.
Special attention is given to development and implementation of approximation methods for the parameter identification in the following situations:

• Reconstruction of an unknown data function in one or more of the differential equations or boundary conditions, starting from over-specified measured data;
• Determination of a source term of a magnetic field vector in the physical problem;
• Non-standard boundary value problems.

The data searched are approximated in terms of a finite array of parameters. Optimization techniques and recent techniques of automatic differentiation can be used to build the gradient and the Hessian of the cost-functional behind it. To obtain this, it is necessary to use very efficient algorithms for the corresponding direct boundary value problems. Particular focus is on the topic of accuracy versus CPU-time for inverse and optimization problems mentioned above. In particular, two families of algorithms will be investigated and extended: space mapping methods and regularization techniques.

2. External models and stochastic uncertainty
Physical models are developed with respect to optimization and inverse problems. As an example, the magneto-mechanical properties of magnetic materials are investigated. Improved or new Preisach type hysteresis models are constructed in order to include in a proper way magneto-mechanical coupling under arbitrary unidirectional or rotational magnetic excitation. Magnetostriction is a material characteristic, which has to be determined experimentally, and which may display (frequency-dependent) hysteresis. Magnetostriction models, suitable for describing the complex magneto-elastic behaviour in specific applications, are developed for determining the contribution to the observed deformations or vibrations. The models are applied for the optimization of the (geometrical) parameters and the choice of materials with respect to their magneto-elastic behaviour in order to minimize vibrations and acoustic noise.

Power electronic subsystems within electrical energy systems containing electromagnetic conversion elements consist of power switches, quasi-linear passive elements and programmable control devices. As such, fundamental topics to consider are adequate models to solve inverse problems associated to power electronic systems. Special attention has to be paid to the very high frequencies that can be expected in the near future. Not only the active components are important, but even more the behaviour of passive components under these new constraints.
In the field of electromagnetic system modelling, the input data are often supposed to be perfectly known or to obey deterministic laws. However, mechanical parts are produced with dimensional tolerances. Some dimensions, such as the air gap are critical as they strongly influence the performances. Uncertainties in material composition, the evolution in time due to environmental factors (humidity, pressure) and thermal and mechanical solicitations are often hardly known. Material features become stochastic. A stochastic approach in the numerical modelling will be envisaged.
The focus is on two important kinds of methods studied in mechanics but that have been fundamentally adapted for electromagnetics:

• Methods embedding deterministic numerical models in an environment of stochastic procedures. The Monte Carlo method is very efficient, but very time consuming requiring much calculation time with finite element models. Other methods have been proposed enabling to reduce the number of calculations, e.g. surface response method and collocation methods.
• Methods can consist in discretizing the variables of the stochastic mathematical model in spatial and stochastic domain simultaneously. Applying the weighted residual method, a system of equations to be solved is built. All models allow to estimate the effects of the uncertainties on the performance and to assess the risks of faults, to set up plans of maintenance and to insure a maximal reliability.

3. Validation environment
Rotation and linear motion — Electrical machines and actuators are used in drives mainly aiming at high efficiency and fast dynamics, both for control and efficiency. Fast dynamics imply: high torque, small inertia and possibly high speed. For the optimal design of machines, material characterisation and modelling geometric uncertainties are key items. Especially for small machines the exact prediction (and modelling) of the magnetic material properties is of uttermost importance as the area affected by machining is an important fraction of the total magnetic material. The air gap and its uncertainty are even more critical for small machines (micro motors and micro generators including sensors).
Electromagnetic processing of materials — Electromagnetic Processing of Materials (EPM) deals with manufacturing of new materials and products by the interactions of electromagnetic fields with materials. EPM is involved in a broad range of material production, e.g. metals and alloys, high added value ceramics and glasses, semiconductors, high purity materials, medical treatment. The different physical systems with different time scales make EPM a challenge for numerical modelling.

Low frequency shielding — The goal of the study of LF magnetic shielding is the development of a numerical knowledge platform for electromagnetic screening in the frequency domain between 50 Hz and 400 kHz. The mitigation of the magnetic stray field can be obtained by using either passive or active shielding.
Typical generic inverse problems are:

• The search for the optimal position, dimensions and material properties of the passive shield is translated into the reconstruction of an unknown data function. Losses in the passive shield are taken into account.
• The optimization of an active shield corresponds to the determination of position and amplitude of a source term leading to a prescribed magnetic field vector in the physical domain.
• Partially unknown boundary data for the physical domain, crucial for the influence of ambient effects on the fields, have to be determined
Power electronics in grids — Due to the introduction of power electronic circuits in the power distribution system, a highly non-linear behaviour is encountered. Due to the highly different time constants involved, mathematically very stiff problems are found. The mixing of continuous and discrete models brings along an extra complication. Also the stochastic nature is clearly present. The tools for the described inverse problems have a similar nature as those found in the other applications discussed here.

Non-destructive evaluation (NDE) — Electromagnetic-based NDE techniques are very attractive due to their handiness what measurements concern and to their large analysis capabilities assisted by advanced numerical methods. Among others, it is worth mentioning: eddy current testing, pulsed eddy current testing, magnetic flux leakage testing, remote field testing. Eddy current inspection is widely used for a variety of applications such as detection of cracks (discontinuities), measurement of metal thickness, detection of metal thinning due to corrosion and/or erosion, measurement of electrical conductivity and magnetic permeability. The enhancement of eddy current testing technology is focused on the optimized design of the probe, the inverse analysis and the modelling of natural cracks and multicracks. These three areas overlap and must be considered in a combined way. Mechanical and magnetic hysteresis properties of steels are directly linked to their microstructure. Therefore, the measurement of magnetic hysteresis loops in a scale of metal grains can serve as a magnetic based NDE method. NDE techniques provide an excellent balance between quality control and cost-effectiveness.

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