Roger Van Keer

Implementation of Homotopy Perturbation Method to Solve a Population Balance Model in Fluidized Bed

Abstract: A particle population balance formulation for a circulating fluidized bed, involving aggregation and breakage of particles, is solved using the homotopy perturbation method (HPM). The homotopy method deforms a difficult problem into a simple problem, which then can be easily solved. The HPM solution is compared with the solution obtained using a standard finite difference method. Using homotopy, a good approximation of the finite difference solution is obtained within a few iteration steps. The results reveal that the homotopy method is an effective and simple tool to solve nonlinear partial integro-differential equations and has a wide scope and applicability to solve complex engineering problems. The reliability of the algorithm is tested using three different feed inlet particle size (diameter) distributions to indicate the robustness of this method.

Determination of a Robin coefficient in semilinear parabolic problems by means of boundary measurements

We consider a semilinear parabolic initial boundary value problem of second order in a bounded domain Ω ⊂ N with a Lipschitz continuous boundary. We study the problem of determination of the Robin coefficient at the boundary part Γnon from a nonlocal boundary condition. We prove the well posedness of the problem. We design a numerical scheme for its approximation and perform the error analysis.

A numerical approach for the determination of a missing boundary data in elliptic problems

We consider the problem of the recovery of an unknown transfer coefficient in the Robin boundary condition on an inaccessible part of the boundary of an elliptic boundary value problem. The physical process inside the domain is governed by a linear elliptic equation of second order. The unknown boundary data are determined from superfluous information on the accessible part of the boundary. We show that the inverse problem is equivalent to a system of direct problems. We design a numerical approach for thin domains. The method is tested by numerical experiments.

Identification of the relation between the material parameters in the Preisach model and in the Jiles–Atherton hysteresis model

In this article the relation between the material parameters appearing in the Preisach model and those entering the Jiles-Atherton model is discussed. Emphasis is on the variation of the anhysteretic curve and on the magnetization dependency of the pinning parameter, when changing the Preisach distribution function. The discussion rests upon the consideration of the shape of the MH loop on the one hand and upon the identification of the instantaneous loss dissipation on the other hand. The techniques outlined are applied to silicon iron alloys and are compared with results from measurements.

Level set method for the inverse elliptic problem in nonlinear electromagnetism

An inverse problem of inhomogeneity identification inside a nonlinear magnetic material from the local measurements of the magnetic induction is investigated. The representation of the shape of the inhomogeneity and its evolution during an iterative reconstruction process is achieved by the level set method. The reconstruction is realized by the minimization of a cost function using the steepest descent method. The gradient directions are evaluated using the sensitivity equation and the adjoint variable method. Simulations has been performed showing the robustness of the algorithm and its ability to reconstruct single inhomogeneities, convex and non-convex, as well as multiple inhomogeneities.

Adjoint variable method for the study of combined active and passive magnetic shielding

For shielding applications that cannot sufficiently be shielded by only a passive shield, it is useful to combine a passive and an active shield. Indeed, the latter does the “finetuning” of the field reduction that is mainly caused by the passive shield. The design requires the optimization of the geometry of the passive shield, the position of all coils of the active shield, and the real and imaginary components of the currents (when working in the frequency domain). As there are many variables, the computational effort for the optimization becomes huge. An optimization using genetic algorithms is compared with a classical gradient optimization and with a design sensitivity approach that uses an adjoint system. Several types of active and/or passive shields with constraints are designed. For each type, the optimization was carried out by all three techniques in order to compare them concerning CPU time and accuracy.

Space mapping methodology for defect recognition in eddy current testing – type NDT

Purpose – The purpose of this paper is to solve an inverse problem of structure recognition arising in eddy current testing (ECT) – type NDT. For this purpose, the space mapping (SM) technique with an extraction based on the Gauss‐Newton algorithm with Tikhonov regularization is applied.Design/methodology/approach – The aim is to have a computationally fast recognition procedure of defects since the monitoring results in a large amount of data points that need to be analyzed by 3D eddy current model. According to the SM optimization, the finite element method (FEM) is used as a fine model, while the model based on an integral method such as the volume integral method (VIM) serves as a coarse model. This approach, being an example of a two‐level optimization method, allows shifting the optimization load from a time consuming and accurate model to the less precise but faster coarse surrogate.Findings – The application of this method enables shortening of the evaluation time that is required to provide the p...

EEG inverse problem solution using a selection procedure on a high number of electrodes with minimal influence of conductivity

The uncertain conductivity value of skull and brain tissue influences the accuracy of the electroencephalogram (EEG) inverse problem solution. Indeed, when the assumed conductivity in the numerical procedure is different from the actual conductivity then a source localization error is introduced. When using traditional least-squares minimization methods, the number of electrodes in the EEG cap does not influence the spatial resolution. A recently developed reduced conductivity dependence (RCD) methodology, based on the selection of electrodes, is able to increase the spatial resolution of the EEG inverse problem. This paper presents the implications of the RCD method when using a large number of electrodes in the EEG cap on the spatial resolution of the EEG inverse solutions. We show by means of numerical experiments that in contrast to traditional methods, the RCD method enables to increase the spatial resolution. The computations show that the EEG hardware should be modified with as large as possible electrodes.

A 2D finite element procedure for magnetic field analysis taking into account a vector Preisach model.

The main purpose of this paper is to incorporate a refined hysteresis model, viz. a vector Preisach model, in 2D magnetic field computations. To this end the governing Maxwell equations are rewritten in a suitable way, which allows to take into account the proper magnetic material parameters and, moreover, to pass to a variational formulation. The variational problem is solved numerically by a FE approximation, using a quadratic mesh, followed by the time discretisation based upon a modified Cranck Nicholson algorithm. The latter includes a suitable iteration procedure to deal with the nonlinear hysteresis behaviour. Finally, the effectiveness of the presented mathematical tool has been confirmed by several numerical experiments.

Numerical Techniques for the Recovery of an Unknown Dirichlet Data Function in Semilinear Parabolic Problems with Nonstandard Boundary Conditions

We study a semilinear parabolic partial differential equation of second order in a bounded domain ? ? RN , with nonstandard boundary conditions (BCs) on a part ?non of the boundary ??. Here, neither the solution nor the flux are prescribed pointwise. Instead, the total flux through ?non is given and the solution along ?non has to follow a prescribed shape function, apart from an additive (unknown) space-constant ?(t).Using the semidiscretization in time (so called Rothes method) we provide videa numerical scheme for the recovery of the unknown boundary data.