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PurposeThe purpose of this paper is to demonstrate an inverse problem approach for the determination of stress zones in steel plates of electrical machines. Steel plates of electrical machines suffer large mechanical stress by processes like cutting or punching during the fabrication. The mechanical stress has effects on the electrical properties of the steel, and thus on the losses of the machine.Design/methodology/approachIn this paper, the authors present a sensor arrangement and an appropriate algorithm for determining the spatial permeability distribution in steel plates. The forward problem for stress zone imaging is explained and an appropriate numerical solution technique is proposed. Then an inverse problem formulation is introduced and the nature of the problem is analyzed.FindingsBased on sensitivity analysis, different measurement procedures are compared and a measurement setup is suggested. Further the ill‐posed nature of the inverse problem is analyzed by the Picard condition.Practical implicationsBecause of the increased losses due to stress zones, the quantification of stress effects is of interest to adjust the production process. Stress zone imaging is a first approach for the application of an imaging system to quantify these material defects.Originality/valueThis paper presents a simulation study about the applicability of an inverse problem for stress zone imaging and presents first reconstruction results. Further, the paper discusses several issues about stress zone imaging for the ongoing research.

PurposeThe purpose of this paper is to demonstrate improvement of the accuracy of electrical tomography reconstruction by incorporation of a priori knowledge into the inverse problem solution.Design/methodology/approachThe fusion of two different inversion algorithms capable of real‐time operation is discussed, namely a non‐iterative monotonicity‐based approach, determining the a priori knowledge and an iterative Gauss‐Newton (GN)‐based reconstruction algorithm. Furthermore, the method is compared with the unmodified algorithms themselves by means of reconstructions from simulated inclusions at different noise levels.FindingsThe accuracy of the inverse problem reconstructions, especially at the boundary regions of the unknown inclusions, benefit from the investigations of incorporating a priori knowledge about material values and can be considerable improved. The monotonicity method itself, which has low complexity, provides remarkable reconstruction results in electrical tomography.Research limitations/implicationsThe paper is applied to simulated discrete two‐phase scenarios, e.g. gas/oil mixtures. In a further step the method would be tested with measured data. Moreover, investigations have to be carried out in order to make the monotonicity‐based reconstruction principle more robust against disturbing artifacts.Originality/valueThe fusion of the non‐iterative monotonicity‐based method with the GN‐based algorithm demonstrates a novel approach of improving the reconstruction accuracy in electrical tomography.

Whenever one is confronted with the necessity to measure some quantities, which are not accessible directly, however, are linked via a mathematical model to some measurement data, one has to solve an inverse problem. In this context we speak of a direct problem, when expected measurement data are calculated from a mathematical model, when the not directly accessible quantities are given and, on the other hand, of an inverse problem, when these quantities are calculated from measured data via the mathematical model. In this paper the principles of inverse problems are explained on the basis of a one-dimensional image restoration problem, which is a linear problem and hence is easy to understand. Furthermore, the term ill-posedness will be explained and some possibilities to attain reasonable solutions to ill-posed problems are discussed.

PurposeThe investigation of the influence of the modeling error on the solution of the inverse problem given uncertain measured data in electrical capacitance tomography (ECT).Design/methodology/approachThe solution of the nonlinear inverse problem in ECT and hence, the obtainable accuracy of the reconstruction result, highly depends on the numerical modeling of the forward map and on the required regularization. The inherent discretization error propagates through the forward map, the solution of the inverse problem, the subsequent calculation of process parameters and properties and may lead to a substantial estimation error. Within this work different finite element meshes are compared in terms of obtainable reconstruction accuracy. In order to characterize the reconstruction results, two error measures are introduced, a relative integral error and the relative error in material fraction. In addition, the influence of the measurement noise given different meshes is investigated from the statistical point of view using repeated measurements.FindingsThe modeling error, the degree of regularization, and measurement uncertainties are the determining and limiting factors for the obtainable reconstruction accuracy of electrical tomography systems. The impact of these key influence factors on the calculation of process properties given both synthetic as well as measured data is quantified. Practical implications – The obtained results show that especially for measured data, the variability in calculated parameters strongly depends on the efforts put on the forward modeling, i.e. on an appropriate finite element mesh size. Hence, an investigation of the modeling error is highly recommended when real‐world tomography problems have to be solved.Originality/valueThe results presented in this work clearly show how the modeling error as well as inherent measurement uncertainties influence the solution of the inverse problem and the posterior calculation of certain parameters like void fraction in process tomography.

A model-based, nonlinear reconstruction for electrical capacitance tomography (ECT) is considered as an inverse problem to find the spatial distributed permittivities in a pipe. The corresponding forward problem is defined by the Laplace equation and depending on the measurement principle two different sets of boundary data can be obtained for the inverse problem. Dirichlet boundary data correspond to voltage measurements and Neumann boundary data to the measurement of charges on the electrodes. For each type of data different cost functionals are defined. In this paper, the influence of the functionals on the inverse problem is examined. Therefore, two different, nonlinear reconstruction methods validate the functionals. One is based on a fixed grid and the forward problem is solved by the finite-element method. Second, a moving contour described by a level set formulation is used to reconstruct the boundaries of different phases. In this case, the field problem is solved by a boundary element formulation. The reconstructions are based on a Gauss–Newton scheme and the gradient is calculated analytically by the material derivative method. Reconstruction results for measurements with an ECT prototype sensor are presented and good results are reported independent of the implemented cost functional.

PurposeFor the spatial reconstruction of a two phase flow, as it might occur in a pipe, the main problem has always been the blurring of the resulting images.Design/methodology/approachIn this paper, we present a method where blurring is implicitly avoided by the use of level sets. The level set method describes the iteratively evolving interface between different materials. The underlying field problem is solved with the boundary element method formulated in the region, where the degrees of freedom are present and the finite element method in all other regions.FindingsFinally reconstruction results of an electrical capacitance tomography sensor are presented to show the validity of the method.Originality/valuePresents a method where blurring is avoided by the use of level sets.

In order to reduce the influence of axial displacement and/or to ease the assembly, certain capacitive angular sensors use symmetric designs limiting their absolute measuring range to (360/spl deg//n) where n is the number of blades. This paper presents a combined sensor that utilizes a Wiegand sensor to obtain additional information needed to resolve 360/spl deg/. Since magnetic and electric fields required by the sensing principles do not interfere, one single rotor design can be used for both fields. When the receiver coil on a Wiegand sensor has to drive a current, this current influences the shape and duration of the pulse generated by the sensor. We found that shape and amplitude of the pulse vary depending on the strength of the applied fields and the location of the initial inverse nucleation. Since most of design optimization for capacitive sensors is done by means of simulations, an adequate model of the combined sensor is needed. For the capacitive part, numerical approaches such as the finite element method are available. For the pulse wire part, an extended Ising model has been developed; results of measurements and numerical simulations are reported.

In reconstruction (i.e., determining the states of a model from measurements of model outputs), one is often forced to search for a regularized solution due to poor sensitivity of model outputs with respect to the model states. The amount of regularization is controlled by the regularization parameter, a scalar value multiplied with the so-called regularization term. The choice of the regularization parameter is crucial for the reconstruction process. In this paper, a new method to estimate the regularization parameter in an adaptive way is proposed. A condition-number based estimate of the regularization parameter for the first iteration step is required to choose the weighting factor for adapting the regularization parameter iteratively. By virtue of controlling the regularization term, a kind of edge preservation can be achieved. The validity of this method will be demonstrated for a capacitance tomography problem, which is solved applying a Gauss-Newton scheme.

Electrical capacitance tomography (ECT) is a technique for reconstructing information about the spatial distribution of the contents of closed pipes by measuring variations in the dielectric properties of the material inside the pipe. In this paper, we propose a method that solves the non‐linear reconstruction problem directly leading to less iterations and higher accuracy than linear back projection algorithms currently in use in most ECT systems.

The accurate simulation of battery cells is of growing interest in automotive industry especially in hybrid vehicle technology. Conventional lumped parameter models are not able to predict the battery voltage accurately. Thus models describing the physics of the battery cell are searched. In this paper a model consisting of six partial differential equations is proposed, which predicts the state of charge (SOC) and the battery voltage for given charge and discharge current densities.

Electrical impedance tomography is a noninvasive method, which is particularly useful for patient monitoring. For reconstructing the interior of the human body, a finite-element model is established and the inverse problem is solved by means of a Gauss-Newton method, which requires the Jacobian, describing the mapping between voltage distribution on the body's surface and the conductivity distribution in the interior. The calculation of the Jacobian has to meet an accuracy requirement as well as a minimal processing time requirement. In this paper, the reciprocity principle is applied to meet both requirements leading to good results for the reconstruction.

A major drawback of electrical impedance tomography is the poor quality of the conductivity images, i.e., the low spatial resolution as well as large errors in the reconstructed conductivity values. The main reason is the necessity for regularization of the ill-conditioned inverse problem which results in excessive spatial low-pass filtering. A novel regularization method (SMORR (spectral modelling regularized reconstructor)) is proposed, which is based on the inclusion of spectral a priori information in the form of appropriate tissue models (e.g. Cole models). This approach reduces the ill-posedness of the inverse problem, when multifrequency data are available. An additional advantage is the direct reconstruction of the (physiological) tissue parameters of interest instead of the conductivities. SMORR was compared with posterior fitting of a Cole model to the conductivity spectra obtained with a classical iterative reconstruction scheme at various frequencies. SMORR performed significantly better than the reference method concerning robustness against noise in the data.

In most real world optimization problems, one tries to determine the global among some or even numerous local solutions within the feasible region of parameters. Nevertheless, it could be worthwhile to investigate some of the local solutions as well. A most desirable behavior would be that the optimization strategy behaves globally and yields additional information about local minima detected on the way to the global solution. In this paper, a clustering algorithm has been implemented into an extended higher order evolution strategy in order to achieve these goals. Multimodal two-dimensional test problems, namely, Rastrigin's function and the 4-parameter die mold press benchmark problem (Takahashi, 1996), are solved using this approach.

Electrical impedance tomography (EIT) is a non‐invasive method to monitor conductivity changes in regions of the human body. Its robust, miniaturizable instrumentation makes EIT particularly suitable for online‐monitoring without too much inconvenience for the patient. A major methodological problem is the poor quality of the conductivity images, which is due to the low spatial resolution and low sensitivity for structures far away from the object’s surface as well as large qualitative errors in the reconstructed conductivity values. This paper outlines the advantages of multi‐frequency EIT for a simple two‐dimensional model. In the first part of the paper we assume that some a priori information from MR images is at hand, providing good starting values for the reconstruction process, while in the second part it is assumed that no a priori information about regions of different material values is available.

Considers, without loss of generality, a simple linear problem, where in a certain domain the magnetic field, generated by infinitely long conductors, whose locations as well as the currents are unknown, has to meet a certain figure. The problem is solved by applying hierarchical simulated annealing, which iteratively reduces the dimension of the search space to save computational cost. A Gauss‐Newton scheme, making use of analytical Jacobians, preceding a sequential quadratic program (SQP), will be applied as a second approach to tackle this severely ill‐posed problem. The results of these two techniques will be analyzed and discussed and some comments on future work will be given.

This paper gives an overview of some stochastic optimization strategies, namely, evolution strategies, genetic algorithms, and simulated annealing, and how these methods can be applied to problems in electrical engineering. Since these methods usually require a careful tuning of the parameters which control the behavior of the strategies (strategy parameters), significant features of the algorithms implemented by the authors are presented. An analytical comparison among them is performed. Finally, results are discussed on three optimization problems.

This paper describes a non-invasive approach for a combined flow profile and velocity measurement of powdery solids and fluids. The measurement is based on Electrical Capacitance Tomography (ECT) techniques, where disturbances in terms of aluminum tracer particles are detected by monitoring capacitance changes caused by the metal tracers. The proposed algorithm for the localization of disturbances forms the basis for a later cross-correlation of layer images and for velocity and profile measurement consequently.

Laser speckle strain and displacement measurement is a well established method for noncontacting sensing in material science applications. This paper deals with a simulation of the forming of laser speckles for free field propagation of the diffracted light-wave-field whose intensity distribution on the observation plane is the objective speckle pattern. A need for simulation of laser speckles is the analysis of accuracy and sensitivity of different measurement set-ups as well as the analysis of decorrelation effects due to structural changes on the specimen's surface, known and unknown object translation and rotation and changes in the illumination conditions.

Capacitive technologies play an increasingly important role in the fields of industrial and automotive sensors. The non-contact working principle, on the one hand, is the main advantage of this technology, on the other hand the sensor may be sensitive to electromagnetic disturbances. This paper gives an overview of capacitive sensors which can reliably be operated in harsh industrial and automotive environments summarizing the author's work on this topic.

Magnetic induction tomography (MIT) is used for reconstructing the changes of the conductivity in a target object using alternating magnetic fields. Applications include, for example, the non-invasive monitoring of oedema in the human brain. A powerful software package has been developed which makes it possible to generate a finite element (FE) model of complex structures and to calculate the eddy currents in the object under investigation. To validate our software a model of a previously published experimental arrangement was generated. The model consists of a coaxial coil system and a conducting sphere which is moved perpendicular to the coil axis (a) in an empty space and (b) in a saline-filled cylindrical tank. The agreement of the measured and simulated data is very good when taking into consideration the systematic measurement errors in case (b). Thus the applicability of the simulation algorithm for two-compartment systems has been demonstrated even in the case of low conductivities and weak contrast. This can be considered an important step towards the solution of the inverse problem of MIT.

In order to reduce the influence of axial displacement, certain capacitive angular sensors use a symmetric design that limits their absolute measuring range to 180/spl deg/. This paper presents a combined sensor that utilizes a Wiegand sensor to obtain additional information needed to resolve 360/spl deg/. Since magnetic and electric fields required by the sensing principles do not interfere, one single rotor design can be used for both fields. When the receiver coil on a Wiegand sensor has to drive a current, this current influences the shape and duration of the pulse generated by the sensor. We found that shape and amplitude of the pulse vary depending on the strength of the applied fields and the location of the initial inverse nucleation. However the stability of the pulse amplitude and the pulse duration are important for the robustness of the sensor. A modified Ising model that simulates the domain wall propagation is presented and results of measurements and numerical simulations are reported.

Global optimization in electrical engineering using stochastic methods requires usually a large amount of CPU time to locate the optimum, if the objective function is calculated either with the finite element method (FEM) or the boundary element method (BEM). One approach to reduce the number of FEM or BEM calls using neural networks and another one using multiquadric functions have been introduced recently. This paper compares the efficiency of both methods, which are applied to a couple of test problems and the results are discussed.