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In this article, we study a group (N=98) of students from two campuses of one Finnish university and their responses to a questionnaire that surveyed the respondents' study habits and how they use different kinds of learning materials in university mathematics courses. Our results show that the grades in the national matriculation exams in mathematics and mother tongue do not play a significant role in one's study habits or use of learning materials. The older students are more communicative with their teachers, whereas the younger students ask for help more often from fellow students. The sociomathematical norms that constitute the local study culture have a larger impact on the study habits and on the use of learning materials. For example, the use of videos and studying lecture materials before the lectures were clearly more usual at one campus than at the other. We also found some significant differences between the groups that are based on the study programmes. The students of mathematics without an intention to become a teacher were most traditional in their study habits, whereas the students of applied physics were most active to participate in teaching. The student teachers most often lie in the middle in the issues where the other groups differ from one another. Validerad;2020;Nivå 1;2020-10-08 (johcin)

The ray tracing method is widely used in teaching geometrical optics at the upper secondary and university levels. However, using simple and straightforward examples may lead to a situation in which students use the model of ray tracing too narrowly. Previous studies show that students seem to use the ray tracing method too concretely instead of as a conceptual model. This suggests that introductory physics students need to understand the nature of the ray model more profoundly. In this paper, we show how a virtual ray tracing model can be used as a tool for image formation in more complex and unconventional cases. We believe that this tool has potential in helping students to better appreciate the nature of the ray model.

Fluidized-bed drying of wet granules is a common process in many industrial sectors, and monitoring of the moisture and the hydrodynamics of the granules during drying is important in order to ensure the quality of the final product. Electrical capacitance tomography (ECT) is an electrical imaging modality in which the permittivity distribution inside an object is computed using capacitive measurements from the boundary of the object and mathematical algorithms. Here, three-dimensional moisture distributions were computed through an experimentally determined relationship between granule moisture and relative permittivity. Estimates for the average moisture content were calculated and evaluated. The hydrodynamics of the wet granules were characterized with respect to air velocity and granule moisture.

Flotation is a separation process in which a hydrophobic material is separated from a hydrophilic one. It is commonly used in several branches of processing. The purpose of this paper was to use three dimensional Electrical Resistance Tomography (ERT) for the measurement of the spatial distribution of gas holdup in a mechanical flotation cell. Using an array of metal electrodes installed on the wall of the cell, a set of electric currents was injected to the cell and the resulting voltages were measured. The electrical conductivity within the cell was estimated based on the known currents and measured voltages. The gas holdup distribution was computed based on Maxwell model. The method was tested in a 50 dm 3 mechanical flotation cell filled with water. The gas holdup distribution is presented at various values of the rotor speed and gas superficial velocity in non-frothed and frothed cases. Moreover, the performance of two rotor–stator mechanisms was compared. The results indicate that the 3D gas holdup distribution in a mechanical flotation cell under different conditions can be estimated using ERT. Moreover, differences in the gas holdup distribution can be detected depending on which rotor–stator mechanism is in use.

Electrical capacitance tomography (ECT) was utilized for monitoring of high-shear mixing and high-shear granulation processes. A finite element method (FEM)-based reconstruction algorithm was utilized to take into account the specific geometrical characteristics of the experimental set-up. Two-dimensional ECT tomograms, mixing index curves and permittivity fractions were computed based on the measurements, and their suitability in the analysis of the processes was assessed. It was found that the different mixing processes and the different granulation processes could be analyzed based on these quantities.

The objective of this study was to determine whether electrical impedance tomography (EIT) can be used to image the mixing of two miscible liquids in a turbulent flow, using the Trumpjet mixing system of papermaking chemicals and additives. Trumpjet is a jet injection mixer developed and produced by Wetend Technologies Ltd. An EIT system and a linear impedance sensor, developed at the Department of Physics, University of Kuopio, were used as a monitoring system. In EIT, a difference reconstruction approach was used to produce two- and three-dimensional images of the conductivity distribution with a frame rate about of 40 frames/s. In the linear impedance sensor, 8 impedance values can be measured with a speed of 4 s −1 . In order to make changes in the conductivity distribution a saline solution was used as chemical. In order to evaluate the quality of the mixing, a mixing index was calculated based on the reconstructed images. Laboratory tests were carried out in a pilot plant. The efficiency of the mixing of the chemical into the main flow using Trumpjet technology and using the conventional chemical injection system was compared. It was found that the Trumpjet technology is more efficient, and that EIT and a linear sensor can be used in mixing studies.

In this paper we perform an experimental evaluation of a state estimation approach in process tomography. In particular, we concentrate on the case where a system with rapidly moving target is imaged with electrical impedance tomography. We show experimental results which confirm that non-stationary estimation with proper fluid dynamical models works well even in cases where stationary estimates are completely useless.

Numerical realization of mathematical models always induces errors to the computational models, thus affecting both predictive simulations and related inversion results. Especially, inverse problems are typically sensitive to modeling and measurement errors, and therefore the accuracy of the numerical model is a crucial issue in inverse computations. For instance, in problems related to partial differential equation models, the implementation of a numerical model with high accuracy necessitates the use of fine discretization and realistic boundary conditions. However, in some cases realistic boundary conditions can be posed only for very large or even unbounded computational domains. Fine discretization and large domains lead to very high-dimensional models that may be of prohibitive computational cost. Therefore, it is often necessary in practice to use coarser discretization and smaller computational domains with more or less incorrect boundary conditions in order to decrease the dimensionality of the model. In this paper we apply the recently proposed approximation error approach to the problem of incorrectly posed boundary conditions. As a specific computational example we consider the imaging of conductivity distribution of soil using electrical resistance tomography. We show that the approximation error approach can also be applied to domain truncation problems and that it allows one to use significantly smaller scale forward models in the inversion.

In many industrial applications the aim is to obtain information on three-dimensional (3D) material distribution within the process vessels. With standard two-dimensional (2D) techniques only vague cross-sectional information can be obtained. It could be possible to carry out several 2D reconstructions on different layers and in this way to obtain 3D information. However, in this approach errors are induced since no real 3D information is utilized in the image reconstruction. In this paper we describe an approach to measure, reconstruct and visualize three-dimensional electrical impedance tomography images in real time. The reconstruction is based on a difference imaging scheme. An efficient current injection and voltage measurement protocol is used in order to increase the sensitivity and reduce the data collection time. The proposed approach can produce and visualize up to 15 3D EIT images per second when 80 measurement electrodes are used. Imaging results from a stirred vessel and a flow loop will be shown. The reconstructions show, for example, that 3D air/liquid distribution in the stirred vessel can reliably be visualized in real time and material flow can be monitored in a 3D section of the flow loop. Reconstructions can be visualized and analysed in many different ways in order to produce essential information on the behaviour of the processes.

For many industrial processes the bulk volume fraction of one phase such as gas or solid can be of considerable importance. For this reason it is often controlled within certain limits and therefore is known at least vaguely. Tomographic imaging of volume fraction can provide very valuable information not only on the bulk volume fraction but also on the spatial distribution of the phase of interest. Prior information on the bulk volume fraction can assist in tomographic inversion, even if this information is not precise. In this paper we propose an electrical impedance tomography approach for imaging volume fraction distribution. A vague bulk volume fraction prior is utilized in the estimation. Two-dimensional numerical examples are used to demonstrate the proposed method. Copyright © 2005 John Wiley & Sons, Ltd.

Electrical impedance tomography (EIT) is a diffuse imaging modality in which the resistivity distribution inside the object is estimated based on electrical measurements made on the boundary. Several applications can be found in geophysics, medicine and industry. Image reconstruction is an iterative procedure in which the norm between the computed and measured voltages is minimized. Also, an additional regularization term is included in the minimized functional due to the ill-posedness of the problem. In the reconstruction process, the geometry of the object is assumed to be known. Geometry is known in many cases but there are various situations in which the shape of the domain is unknown. For example, in medical applications the shape of the domain, a part of the human body on which the measurement electrodes are attached, is unknown unless some other imaging modality is used for receiving the shape. In industrial applications, such as in the imaging of a stirrer vessel for detecting air distribution or detecting large air bubbles in pipelines, the free surface between the liquid and air is unknown and should be estimated. Within the domain we may also have 'voids' having zero conductivity and we might be interested in detecting the shapes and locations of the voids. An example could be the detection of corrosion faults in metallic plates. In this paper, two approaches for shape and free surface estimation are proposed. The approaches taken here are based on the idea used in shape optimization problems. In the first approach, the unknown shape is parametrized using the mesh nodes as parameters. In the second approach, we define new 'design variables' which are used as parameters. These design variables are the coefficients of a Bezier curve that defines the shape of the surface. In this paper, we show results of the free surface estimation from both computer simulations and tank measurements. Also, results of simultaneous reconstruction of the resistivity distribution and the free surface are shown. Comparison of the results between these two approaches will be given. The comparison shows better performance in the Bezier curve approach.

In this paper, we examine how the estimation method of the electrode contact impedances proposed earlier works in two different laboratory experiments. In the first experiment, the performance of the method was studied for a tank filled with a homogeneous liquid. In this case a single parameter for the contact impedances was used. In the second experiment the method was tested with a single target within a tank. In this case the contact impedances on all the electrodes were estimated. It was found that in the two-electrode measurement protocol the estimation of the electrode contact impedances improves reconstructions substantially. If only a single electrode contact impedance value for all the electrodes was estimated it was found that in certain cases it was not adequate.

A case study is presented where electrical impedance tomography is used to image paste during extrusion. This indicates regions of high moisture at the wall of the barrel permitting slip and so making extrusion more efficient. Multi-step Gauss–Newton image reconstruction methods are used. Techniques include (i) axi-symmetric smoothing that is coupled with ultrasound measurement to provide discretization information, (ii) full impedance imaging and (iii) full three-dimensional field modelling. These provide clear images of the paste that can be used to gauge extrudability of a paste formulation.

In this paper we propose a method for detecting faults in resistive thin films and coatings with an electrical-impedance-tomography-related approach. The motivation of our study is the need to develop methods for the evaluation of coating properties and quality of implants. The method is based on the complete electrode model. We model the coating as one electrode with spatially varying contact impedance. Using a set of current and voltage measurements in a specially developed measurement cell including the coating and additional electrodes, we estimate the spatially varying contact impedance on the coating. The estimated contact impedance distribution can be regarded as a measure of the condition of the coating; that is, low contact impedance indicates a fault in the coating. The necessary modifications of the complete electrode model are given. The suggested method is tested with simulations and real measurements and it is shown that, with feasible regularization, faults in the coating can be detected with good precision.

One possible technique to monitor sedimentation and froth flotation processes is electrical impedance tomography (EIT). When EIT is used in these applications, the measurements are usually conducted using a probe sensor that is positioned inside the process vessel. Full three-dimensional reconstruction is possible, however, it leads to time-consuming reconstructions. For simplifying the reconstruction procedure and reducing the computation time, two different reduced forward models are proposed. In the first model, the conductivity is considered to be rotationally symmetric, that is, the conductivity and the potential do not change in the angular direction. This model is accurate as far as the given assumptions are fulfilled. In the second model, the conductivity is considered to be invariant also in the radial direction. This one-dimensional (1D) forward model is highly approximative and therefore the reconstructed conductivity profile is useless with conventional reconstruction approach. The approximati...

Classical electrical impedance tomography is an imaging modality in which the internal impedivity distribution is reconstructed from the known injected currents and voltages measured on the surface of the object. However, in many industrial cases there are a priori known internal structures inside the vessels which could be used as internal electrodes in tomographical imaging. In this paper we consider modelling of certain types of internal structures and using them as internal electrodes in two-dimensional electrical process tomography. We also propose an inversion approach in which directional smoothness of the resistivity distribution can be taken into account. Simulations and laboratory experiments show that, by utilizing internal structural information and using these internal structures as additional electrodes, we can achieve significant improvements in image reconstruction. Improvements are shown for the cases in which the electrodes are placed at the centre of the object and are surrounded by a resistive layer.

Tomographic imaging of fluidized bed reactor is of great interest in pharmaceutical, chemical and food industries. A special application is the detection of granule moisture distribution inside the reactor. In this paper we consider electrical capacitance tomography to be used in in-line imaging of conical shaped fluidized bed reactor. Results of determining the granule moisture distribution are shown.

Electrical impedance tomography (EIT) was adapted to monitor drug release three-dimensionally as a function of time. EIT is an electrical imaging modality in which the three-dimensional conductivity distribution inside an object is computed based on electrical measurements from the boundaries. Here, the three-dimensional concentration distribution was monitored with the help of the experimentally determined relationship between drug concentration and conductivity. The EIT monitoring was carried out with propranolol hydrochloride tablets in an apparatus similar to USP dissolution apparatus 2. The release profiles estimated using EIT matched well with the UV/VIS spectrophotometric analyses that were performed as a reference. There are several benefits conferred by three-dimensional monitoring, i.e., comprehensive information about the release process; no need to take samples during experiments; and not essential to assume homogenous concentration distribution in the drug release analysis. EIT is an in-line technique, and moreover, it is non-intrusive and non-invasive. The possibilities and the characteristics of the EIT monitoring are described in detail, and some potential drug release applications are proposed. EIT is especially encouraged to be exploited for research and development purposes.

An approach to monitor in vitro dissolution process from pharmaceutical tablets utilizing electrical impedance tomography (EIT) is introduced. In the demonstration, a tablet containing sodium chloride (NaCl) was dissolution tested using tap water as a dissolution medium within an apparatus similar to the United States Pharmacopoeia dissolution apparatus II. During the process, the three-dimensional sodium chloride concentration distribution was monitored with EIT measurements as a function of time. For EIT measurements, an array of electrodes was attached on the boundary of the dissolution vessel, a set of alternating electric currents was injected through the electrodes, and the resulting voltages were measured. With these data and by applying mathematical algorithms, an approximation for the spatial/temporal concentration distribution inside the vessel was computed. It was found that the computed distributions were relatively homogeneous. A NaCl release curve was computed by integrating the concentration distribution over the vessel volume, and the final value of the curve matched well with the reference point based on the weight loss of the tablet. Finally, EIT monitoring is suggested to be used for research and product development purposes.

The EIDORS (electrical impedance and diffuse optical reconstruction software) project aims to produce a software system for reconstructing images from electrical or diffuse optical data. MATLAB is a software that is used in the EIDORS project for rapid prototyping, graphical user interface construction and image display. We have written a MATLAB package (http://venda.uku.fi/ vauhkon/) which can be used for two-dimensional mesh generation, solving the forward problem and reconstructing and displaying the reconstructed images (resistivity or admittivity). In this paper we briefly describe the mathematical theory on which the codes are based on and also give some examples of the capabilities of the package.

Inverse problems can be characterized as problems that tolerate measurement and modelling errors poorly. Typical sources of modelling errors include (pure) approximation errors related to numerical discretization, unknown geometry and boundary data, and possibly sensor locations. With electrical impedance tomography (EIT), the unknown contact impedances are an additional error source. Recently, a Bayesian approach to the treatment of approximation and modelling errors for inverse problems has been proposed. This approach has been shown to be applicable to a variety of modelling and approximation errors, at least with simulations. Recently, it was shown that recovery from significant model reduction and moderate mismodelling of geometry in EIT was also possible with laboratory EIT data. In this paper, we show that the errors due to the unknown contact impedances can also be compensated for by employing the approximation error approach. Furthermore, the recovery from simultaneous contact impedance, domain truncation and discretization-related errors is also feasible.

There are many different electrical impedance tomography (EIT) systems which are either non-commercial (in-house products) or commercial products. However, these systems are usually designed for specific applications and therefore the functionality of the systems might be limited. Nowadays there are commercially available many low-cost, efficient and accurate multifunctional components for data acquisition and signal processing. Therefore, it should be possible to construct an EIT system which is mainly built from commercially available components. The main goal of this work was to study the performance of a PXI-based EIT system. In this work, a PXI-based EIT system with 16 independent current injection channels and 80 independent measurement channels was constructed and tested. The results indicate that an EIT system can be constructed using a PXI platform which decreases the construction time of the system. Moreover, the system is efficient, accurate, modular, and it is not limited to any predetermined measurement protocols.

Inverse problems can be characterized as problems that tolerate measurement and modelling errors poorly. While the measurement error issue has been widely considered as a solved problem, the modelling errors have remained largely untreated. The approximation and modelling errors can, however, be argued to dominate the measurement errors in most applications. There are several applications in which the temporal and memory requirements dictate that the computational complexity of the forward solver be radically reduced. For example, in process tomography the reconstructions have to be carried out typically in a few tens of milliseconds. Recently, a Bayesian approach for the treatment of approximation and modelling errors for inverse problems has been proposed. This approach has proven to work well in several classes of problems, but the approach has not been verified in any problem with real data. In this paper, we study two different types of modelling errors in the case of electrical impedance tomography: one related to model reduction and one concerning partially unknown geometry. We show that the approach is also feasible in practice and may facilitate the reduction of the computational complexity of the nonlinear EIT problem at least by an order of magnitude.

The paper considers the Galerkin finite-element approximation of the Boltzmann transport equation for charged particles. The aim is to model the electron transport in biological tissue for the dose calculation and to apply the method to planning of the treatment. The model takes into account multiple Coulomb scattering but it omits brehmsstrahlung and pair production. The differential multiple-scattering cross sections are computed from the screened Rutherford scattering formula by repeated convolutions. In the inverse problem a feasible solution is sought by the Cimmino algorithm. The preliminary simulation results show that refinements for practical situations are worth studying.

In process tomography the aim is to obtain information typically from the interior of the process vessels based on the measurements made on the surface of the vessel. Electrical impedance tomography (EIT) is an imaging modality in which the internal resistivity distribution is reconstructed based on the known injected currents and measured voltages on the surface. Since the reconstructed image represents the resistivity distribution of the interior, certain internal structures such as highly conductive mixing paddles in a stirred vessel may entail difficulties in the image reconstruction. This is because EIT is a diffuse, nonlinear imaging modality which makes it difficult to reconstruct high contrasts in the internal resistivity distribution. In this paper the effects of internal conductive structures on the reconstructed images in two-dimensional cases are considered and possible improvements in the reconstructions by taking into account the locations and resistivities of these structures are studied. I...