Academic literature on the topic 'Nedelec Finite Elements'

AbstractThe problem of approximating the band structure of electromagnetic Bloch modes in a three-dimensional periodic medium is studied. We analyze a mixed finite element approximation technique based on a variation of Nedelec edge elements. The usual conditions for convergence of the static problem are first verified. Subsequently, convergence of approximate eigenvalues to those of the continuous system is proved.

Microwave ablation (MWA) is a cancer thermal ablation treatment that uses electromagnetic waves to generate heat within the tissue. The goal of this treatment is to eliminate tumor cells while leaving healthy cells unharmed. During MWA, excess heat generation can kill healthy cells. Hence, mathematical models and numerical techniques are required to analyze the heat distribution in the tissue before the treatment. The aim of this research is to explain the implementation of the 3D vector finite element method in a wave propagation model that simulates the specific absorption rate in the liver. The 3D Nedelec elements from H(curl; Ω) space are used to discretize the wave propagation model, and this implementation is helpful in solving many real-world problems that involve electromagnetic propagation with perfect conducting and absorbing boundary conditions. One of the difficulties in ablation treatment is creating a large ablation zone for a large tumor (diameter greater than 3 cm) in a short period of time with minimum damage to the surrounding tissue. This article addresses the aforementioned issue by introducing four antennas into the different places of the tumor sequentially and producing heat uniformly over the tumor. The results demonstrated that 95.5% of the tumor cells were killed with minimal damage to the healthy cells when the heating time was increased to 4 minutes at each position. Subsequently, we studied the temperature distribution and localised tissue contraction in the tissue using the three-dimensional bio-heat equation and temperature-time dependent model, respectively. The local tissue contraction is measured at arbitrary points in the domain and is more noticeable at temperatures higher than 102°C. The thermal damage in the liver during MWA treatment is investigated using the three-state cell death model. The system of partial differential equations is solved numerically due to the complex geometry of the domain, and the results are compared with experimental data to validate the models and parameters.

We propose continuous and discrete-time finite-element (FE) methods to solve an initial boundary-value problem (IBVP) for the thermo-poroelasticity wave equation based on the combined Biot/Lord-Shulman (LS) theories to describe the porous and thermal effects, respectively. In particular, the LS model, that includes a Maxwell-Vernotte-Cattaneo (MVC) relaxation term, leads to a hyperbolic heat equation, thus avoiding infinite signal velocities. The FE methods are formulated on a bounded domain with absorbing boundary conditions at the artifical boundaries. The dynamical equations predict four propagation modes, a fast P (P1) wave, a Biot slow (P2) wave, a thermal (T) wave, and a shear (S) wave. The spatial discretization uses globally continuous bilinear polynomials to represent the solid displacements and the temperature, while the vector part of the Raviart-Thomas Nedelec of zero order is used to represent the fluid displacements.First, priori optimal error estimates are derived for the continuous-time FE method, and then an explicit conditionally stable discrete-time FE method is defined and analyzed. The explicit FE algorithm is implemented in 1D to analyze the behavior of the P1, P2 and T waves. The algorithms can be useful for a better understanding of seismic waves in hydrocarbon reservoirs and crustal rocks, whose description is mainly based on the assumption of isothermal wave propagation.

AISI 420 martensitik paslanmaz çelik, yüksek korozyon direnci nedeniyle makine, petrol ve petro kimya endüstrilerinde, gıda ve gıda üretim tesislerinde, otomotiv sanayinde, buhar türbin kanatlarında ve tıbbi aletlerin üretiminde sıklıkla kullanılmaktadır. Bu çelik türünün tornalanması, özellikle yüksek yüzey kalitesine sahip parçaların üretiminde önemli bir prosestir. Son ürünün yüzey kalitesi, ürünün genel kalitesini ve işleme sürecinin verimliliğini belirleyen kritik bir faktördür. Bu nedenle bu çalışmada sonlu elemanlar yöntemi ve Taguchi deney tasarımı kullanılarak AISI 420 martensitik paslanmaz çeliğin tornalanmasında kesme parametrelerinin (kesme hızı, ilerleme miktarı ve kesme derinliği) mutlak yüzey pürüzlülük değerine (Rz) etkileri araştırılmıştır. İşleme deneyleri sonlu elemanlar analizi yazılımı olan ThirdWave AdvantEdge programında yapılmıştır. Çalışmanın sonlu elemanlar analizi sonucunda kesme hızının artırılması ile Rz değerinin azaldığı, ilerleme miktarının ve kesme derinliğinin artırılması ile Rz değerlerinin arttığı tespit edilmiştir. Yapılan Taguchi deney tasarımı sonucu elde edilen istatiksel analizler sonucu optimum kesme parametreleri 0.1 mm/rev ilerleme miktarı, 230 m/min kesme hızı ve 0.9 mm kesme derinliği olarak belirlenmiştir. Ayrıca, Rz’ye etki eden en önemli kesme parametresinin ilerleme miktarı olduğu tespit edilmiştir.

Humanitarian clearance of minefields is most often carried out by hand, conventionally using a a metal detector and a probe. Detection is a very slow process, as every piece of detected metal must treated as if it were a landmine and carefully probed and excavated, while many of them are not. The process can be safely sped up by use of Ground-Penetrating Radar (GPR) to image the subsurface, to verify metal detection results and safely ignore any objects which could not possibly be a landmine. In this thesis, we explore the possibility of using Full Wave Inversion (FWI) to improve GPR imaging for landmine detection. Posing the imaging task as FWI means solving the large-scale, non-linear and ill-posed optimisation problem of determining the physical parameters of the subsurface (such as electrical permittivity) which would best reproduce the data. This thesis begins by giving an overview of all the mathematical and implementational aspects of FWI, so as to provide an informative text for both mathematicians (perhaps already familiar with other inverse problems) wanting to contribute to the mine detection problem, as well as a wider engineering audience (perhaps already working on GPR or mine detection) interested in the mathematical study of inverse problems and FWI.We present the first numerical 3D FWI results for GPR, and consider only surface measurements from small-scale arrays as these are suitable for our application. The FWI problem requires an accurate forward model to simulate GPR data, for which we use a hybrid finite-element boundary-integral solver utilising first order curl-conforming N\'d\'{e}lec (edge) elements. We present a novel `line search' type algorithm which prioritises inversion of some target parameters in a region of interest (ROI), with the update outside of the area defined implicitly as a function of the target parameters. This is particularly applicable to the mine detection problem, in which we wish to know more about some detected metallic objects, but are not interested in the surrounding medium. We may need to resolve the surrounding area though, in order to account for the target being obscured and multiple scattering in a highly cluttered subsurface. We focus particularly on spatial sensitivity of the inverse problem, using both a singular value decomposition to analyse the Jacobian matrix, as well as an asymptotic expansion involving polarization tensors describing the perturbation of electric field due to small objects. The latter allows us to extend the current theory of sensitivity in for acoustic FWI, based on the Born approximation, to better understand how polarization plays a role in the 3D electromagnetic inverse problem. Based on this asymptotic approximation, we derive a novel approximation to the diagonals of the Hessian matrix which can be used to pre-condition the GPR FWI problem.

Cette thèse s'inscrit dans un projet de recherche qui a pour ambition de développer, dans une démarche écologique, une méthodologie permettant de retrouver la densité des matériaux du génie civil. L'objectif est de remplacer une méthode invasive et nucléaire par une approche non destructive et électromagnétique. Les travaux de cette thèse sont issues d'une collaboration Cifre entre le Cerema, Routes de France et le Laboratoire de Mathématiques de l'INSA de Rouen Normandie (LMI). Des premiers travaux ont permis d'établir un lien entre la densité et la permittivité diélectrique d'un matériau, ce qui a conduit l'équipe ENDSUM du Cerema Normandie à réaliser un banc permettant d'émettre et de recevoir des ondes électromagnétiques. Il est équipé de moteurs pas à pas pour les antennes et un moteur pour le support, permettant d'accéder à des mesures de type tomographie. L'objectif de cette thèse est de mettre en place un solveur permettant de réaliser une inversion sur les données générées par ce banc afin de retrouver la permittivité et in fine la compacité. Cela implique la modélisation et la simulation numérique de ce système, basée sur la diffraction des ondes électromagnétiques régie par les équations de Maxwell que nous avons étudiés en ordre 2. La réalisation de ce solveur 3D a nécessité l'implémentation d'une méthode type Élément Finis, basée sur les Éléments Finis de Nédelec. La prise en compte du caractère non borné du domaine a été réalisée grâce à l'implémentation de Perfectly Matched Layers. Afin d'optimiser l'implémentation, nous avons également mis en place une vectorisation de l'assemblage des matrices de discrétisation et implémenté une méthode de décomposition de domaine. Finalement, la résolution du problème de minimisation s'est faite par une approche de type Gauss-Newton utilisant la méthode d'état adjoints pour le calcul de la matrice Hessienne. Cette résolution est combinée avec une régularisation de Tikhonov dite semi-quadratique permettant d'accentuer le contraste dans la permittivité recherchée. La modélisation du banc a également nécessité des travaux sur le calibrage des antennes utilisées. Nous avons réadapté les travaux dans le but de considérer les antennes comme une source ponctuelle associée à une onde sphérique et mis en place un procédé expérimental permettant de corriger les signaux reçus
This thesis is part of a research project aiming to develop, in an ecological approach, a methodology for retrieving the density of civil engineering materials. The objective is to replace an invasive and nuclear method with a non-destructive and electromagnetic approach. The work of this thesis stems from a CIFRE collaboration between Cerema, Routes de France, and the Laboratory of Mathematics at INSA Rouen Normandie (LMI).The initial work has established a relationship between the density and the dielectric permittivity of a material, leading the ENDSUM team at Cerema Normandie to develop a bench capable of emitting and receiving electromagnetic waves. It is equipped with stepper motors for the antennas and a motor for the support, enabling tomography-type measurements. The objective of this thesis is to implement a solver capable of performing inversion on the data generated by this bench to retrieve the permittivity and ultimately the compactness. This involves the numerical modeling and simulation of this system, based on the diffraction of electromagnetic waves governed by the Maxwell equations we studied in second order. The development of this 3D solver required the implementation of a Finite Element type method, based on Nedelec Finite Elements. The consideration of the unbounded nature of the domain was achieved through the implementation of Perfectly Matched Layers. To optimize the implementation, we also introduced vectorization of the discretization matrix assembly and implemented a domain decomposition method. Finally, the resolution of the minimization problem was carried out using a Gauss-Newton approach utilizing the adjoint state method for computing the Hessian matrix. This resolution is combined with a semi-quadratic Tikhonov regularization method to enhance the contrast in the desired permittivity.The modeling of the bench also required work on the calibration of the antennas used. We have readapted previous work to consider the antennas as a point source associated with a spherical wave and implemented an experimental process to correct the received signals

High-frequency wave phenomena is observed in many physical settings, most notably in acoustics, electromagnetics, and elasticity. In all of these fields, numerical simulation and modeling of the forward propagation problem is important to the design and analysis of many systems; a few examples which rely on these computations are the development of metamaterial technologies and geophysical prospecting for natural resources. There are two modes of modeling the forward problem: the frequency domain and the time domain. As the title states, this work is concerned with the former regime. The difficulties of solving the high-frequency wave propagation problem accurately lies in the large number of degrees of freedom required. Conventional wisdom in the computational electromagnetics commmunity suggests that about 10 degrees of freedom per wavelength be used in each coordinate direction to resolve each oscillation. If K is the width of the domain in wavelengths, the number of unknowns N grows at least by O(K^2) for surface discretizations and O(K^3) for volume discretizations in 3D. The memory requirements and asymptotic complexity estimates of direct algorithms such as the multifrontal method are too costly for such problems. Thus, iterative solvers must be used. In this dissertation, I will present fast algorithms which, in conjunction with GMRES, allow the solution of the forward problem in O(N) or O(N log N) time.
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Abstract Numerical analysis of electromagnetic problems is now quite important in wide fields of science and engineering. The application of the finite element method (FEM) to such ends is expected to be very effective especially for 3-D problems since FEM is well suited to deal with complex regions and various boundary conditions. By appropriate modeling of the Maxwell equations of electromagnetics, we have various problems describing electromagnetic phenomena in practice.