There is great need to understand the mechanical properties of lead-free alloys—an alternative of lead-based alloys—to address the environmental problems associated with the use of lead-based materials in microelectronics. In this work, the microstructures of Bi57In26Sn17 lead-free alloy were examined using Optical Microscopy and Energy Dispersive X-ray Spectroscopy analysis. The micro-indentation technique was used to study the mechanical properties of Bi57In26Sn17 lead-free alloy. The experimental results of the hardness and contact modulus were presented and discussed. Local creep during the indentation was observed from the load-displacement curves. The Vickers hardness (HV) increases with the decrease of the indentation depth, suggesting that the alloy exhibits indentation size effect.

Thesis (MScEng) --University of Stellenbosch, 2010. / ENGLISH ABSTRACT: This thesis deals with the development of a new process for the production of Direct Reduced Iron (DRI), intended for use specifically by small scale Electric Arc Furnace (EAF) based steel mills, who require small volumes of DRI. The term development as used here is taken to include such aspects as conceptual design, theoretical verification and initial practical testing. The rise of EAF steelmaking brought about the metamorphosis of steel scrap from a waste product into a valuable raw material. Scrap prices rose steeply during the period 1995 to 2009 compelling EAF steelmakers, wishing to have more control over the cost of their input material, to seek for scrap supplements or alternatives. DRI has become an accepted and sought after supplement, or even complete alternative, to steel scrap. Adding DRI to an EAF charge has a range of advantages, including the dilution of tramp elements and possible cost benefits, but it does have negative effects. These include the lowering of the scrap to liquid metal yield and an increase in power consumed. The effect of charging DRI to a small EAF is quantified. The maximum DRI that may be added to the burden whilst still maintaining the present steelmaking volume, is shown to be as high as 50% if charged continuously, and the maximum price payable for DRI, is shown to be approximately 80% of base grade scrap price. Finally other requirements unique to small scale EAF operators are considered in order to prepare a schedule of requirements for a DRI plant specifically for small scale EAF steel mills. A review of published information on existing DRI production technology, processes and plants is undertaken is establish the fit of existing processes to the requirements set. Initially the thermodynamics and kinetics of iron ore reduction and coal gasification, specifically downdraft gasification are reviewed. Thereafter existing processes are reviewed. Shaft based processes and rotary kiln based processes are identified as possible suitors to the requirements. Limitations of these processes, specifically heat transfer in rotary kilns and the pressure drop over a reduction shafts are investigated. Finally a typical process in each of the main process classes is adjudicated against the set requirements. None is found to match the set requirements. A new process is proposed that is claimed to better suit to small scale operation. The uniqueness of the process is embodied in the combination of existing technologies of downdraft gasification and iron ore reduction in a shaft, in a single reactor. The process consists of two shafts, one placed above the other. Iron ore is charged into the top shaft, called the pre-heat shaft, where it is pre-heated and lightly reduced to wustite with gas from the bottom shaft, called the reduction shaft. The pre-heated ore is then charged together with coal into the reduction shaft. Gasification air is drawn into the top of the reduction shaft where the coal is gasified in a downdraft gasifier, generating reduction gas which reduces the ore as the gas moves concurrently with the iron ore. The exit gas is cleaned and pumped to the pre-heat shaft where it combusted with air to pre-heat the iron ore in the pre-heat shaft. The concept is analysed thermodynamically using amongst others, FactSage, and is shown to be thermodynamically viable. To test the concept process concept practically, an extremely small pilot plant with a production rate of 2kg DRI/h, consisting of only a gasifier/reduction shaft, was designed and constructed using reduction rate data obtained from literature supplemented with data obtained from thermogravimetric analysis of CO reduction of lump Sishen hematite. Pilot Plant trials were performed using various reductant sources. The degree of metallizaion was analysed using visual inspection of cut and polished samples compared to calibrated standards. Analysis of the results indicate that coal rate and production rate influence the degree of reduction positively and negatively. The conclusions arrived at include the fact that the process is thermodynamically viable, that it was possible to reduce iron ore in a simplified pilot plant, and that the process was found to be stable and controllable. It is recommended that a larger scale pilot plant, embodying the full proposed flow sheet be erected to test the process more completely. / AFRIKAANSE OPSOMMING: Die tesis handel oor die the ontwikkeling van ‘n nuwe proses vir die vervaardiging van sponsyster. Die proses is beoog spesifiek vir gebruik deur kleinskaalse Elektriese Boogoond (EBO) gebaseerde staal aanlegte, wat kleiner hoeveelhede sponsyster benodig. Die term ontwikkeling soos hier gebruik word aanvaar om aspekte soos konseptuele ontwerp, teoretiese verifikasie en aanvanklike toetsing te behels. Die vinnige groei van EBO staalvervaardiging het skroot getransformeer van weggooiproduk tot waardevolle grondstof. Die prys van skroot het skerp gestyg gedurende die periode 1995 to 2009. EBO gebaseerde staal produsente, in ‘n poging om meer beheer te hê oor die koste van hul insetmateriaal, het hul in ‘n toenemende mate tot skrootalternatiewe gewend. Sponsyster het ‘n aanvaarde en gewaardeerde byvoeging, en selfs alternatief tot staalskroot geword. Die byvoeging van sponsyster by die lading van ‘n tipiese EBO het besliste voordele, maar het dit ook nadelige effekte. Die voordele sluit die verdunning van reselemente en moontlike kostevoordele in, terwyl van die nadele die verlaging van die skroot tot vloeistaal opbrengs, en ‘n verhoging in kragverbruik, is. Die effek van die byvoeging van sponsyster tot ‘n EBO lading word gekwantifiseer. Daar word getoon dat die maksimum hoeveelheid sponsyster wat by ‘n EBO lading gevoeg kan word terwyl die hoeveelheid staal geproduseer konstant gehou word, ongeveer 50% is indien die sponsyster kontinue gelaai word, en die maksimum prys wat vir die sponsyster betaal kan word, word bereken op ongeveer 80% van die prys van basisgraad skroot. Ander vereistes uniek aan kleinskaal EBO bedrywers word oorweeg ten einde ‘n lys van vereistes vir ‘n sponsysteraanleg, uniek aan kleinskaal EBO bedrywers, te kan bepaal. ‘n Oorsig van gepubliseerde inligting oor sponsysterproduksietegnologie word onderneem ten einde die passing van bestaande prosesse met die gestelde vereistes te kan bepaal. Nadat die termodinamika en kinetika van ysterertsreduksie en steenkoolvergassing be-oordeel is, word bestaande sponsysterprosesse beskou. Skag- en Roterende oond gebaseerde prosesse word as moontlik gepaste prosesse identifiseer. Hitte-oordrag en die drukval oor gepakte beddens, synde tipiese beperkings eie aan die twee prosesse, woord beskou. Tipiese prosesse in elk van die hoofklasse van prosesse word ten laaste be-oordeel aan die gestelde kriteria. Daar word bevind dat geeneen van die bestaande prosesse aan die vereistes voldoen nie. ‘n Nuwe proses, wat skynbaar die behoefte van kleinskaalse EBO gebaseerde staalprodusente beter bevredig, word voorgestel. Bestaande tegnolgie word in ‘n unieke opstelling geïntegreer. Reduksie word in ‘n reduksiekag gedoen as gevolg van die ooglopende massa- en hitte-oordragvoordele van ‘n skag. Reduksiegas word verkry van steenkoolvergassing in ‘n afstroomvergasser ten einde teerverwydering in ‘n naverwerkingsstap oorbodig te maak. Die uniekheid van die proses is beliggaam in die kombinasie van ‘n steenkoolvergasser en reduksieskag in ‘n enkele reaktor. Die proses bestaan uit twee skagte, een bo die ander. Ystererts word in die boonste skag, wat die voorverhitskag genoem word, gelaai. Hier word die erts voorverhit en moontlik lig gereduseer tot wustiet met gas van die onderste skag, wat die reduksieskag genoem word. Die voorverhitte erts word saam met steenkool in die reduksieskag gelaai. Vergassingslug, word in die reduksieskag gesuig waar die steenkool in ‘n afstroomvergasser vergas word. Hierdeur word reduksiegas gegenereer wat die erts verder reduseer soos dit saamstromend met die erts af beweeg. Die uitlaatgas word gesuiwer en na die voorverhitskag gepomp waar dit verbrand word om die erts te voorverhit. Die konsep is termodinamies analiseer met gebruikmaking van onder andere FactSage, en werkbaar bevind. ‘n Baie klein, vereenvoudigde proefaanleg, met ‘n produksievermoë van 2kg DRY/uur, bestaande uit slegs ‘n reduksiekag, is ontwerp en gebou met gebruikmaking van kinetika inligting uit die literatuur aangevul met inligting uit termogravimetriese analise van die CO reduksie van Sishen hematiet. Proefaanleglopies is uitgevoer met ‘n reeks reduktantbronne. Die metallisasiegraad is bepaal deur visuele inspeksie van gesnyde, gepoleerde monsters wat vergelyk is met gekalibreerde standaarde. Analise van die resultate toon dat die steenkoolkoers ‘n positiewe verband, en die produksiekoers ‘n negatiewe verband met die metallisasiegraad het. Die slotsom waartoe gekom is, is dat die proses termodinamies werkbaar is, dat reduksie van ystererts in ‘n vereenvoudigde proefaanleg bewerk kon word, en dat die prose stabiel en beheerbaar voorgekom het. Die aanbeveling word gemaak dat ‘n groter proefaanleg wat die volledige voorgestelde vloeiskema verteenwoordig, opgerig behoort te word, ten einde die proses meer volledig te kan toets.

<p>In aeromechanical design one of the major rules is to operate under High Cyclic Fatigue (HCF) margins and away from flutter. The level of dynamic excitations and risk of HCF can be estimated by performing forced response analyses from blade row interaction forces or Low Engine Order (LEO) excitation mechanisms. On the other hand, flutter stability prediction can be assessed by calculation of aerodynamic damping forces due to blade motion. In order to include these analyses as regular practices in an industrial aeromechanical design process, interaction between the fields of fluid and structural dynamics must be established in a rather simple yet accurate manner. Effects such as aerodynamic and structural mistuning should also be taken into account where parametric and probabilistic studies take an important role.</p><p>The present work presents the development and validation of a numerical tool for aeromechanical design. The tool aims to integrate in a standard and simple manner regular aeromechanical analysis such as forced response analysis and aerodynamic damping analysis of bladed disks.</p><p>Mistuning influence on forced response and aerodynamic damping is assessed by implementing existing model order reduction techniques in order to decrease the computational effort and assess results in an industrially applicable time frame.  The synthesis program solves the interaction of structure and fluid from existing Finite Element Modeling (FEM) and Computational Fluid Dynamics (CFD) solvers inputs by including a mapping program which establishes the fluid and structure mesh compatibility. Blade row interaction harmonic forces and/or blade motion aerodynamic damping forces are inputs from unsteady fluid dynamic solvers whereas the geometry, mass and stiffness matrices of a blade alone or bladed disk sector are inputs from finite element solvers. Structural and aerodynamic damping is also considered.</p><p>Structural mistuning is assessed by importing different sectors and any combinations of the full disk model can be achieved by using Reduced Order Model (ROM) techniques. Aerodynamic mistuning data can also be imported and its effects on the forced response and stability assessed. The tool is developed in such a way to allow iterative analysis in a simple manner, being possible to realize aerodynamically and structurally coupled analyses of industrial bladed disks. A new method for performing aerodynamic coupled forced response and stability analyses considering the interaction of different mode families has also been implemented. The method is based on the determination of the aerodynamic matrices by means of least square approximations and is here referred as the Multimode Least Square (MLS) method.</p><p>The present work includes the program description and its applicability is assessed on a high pressure ratio transonic compressor blade and on a simple blisk.</p> / Turbopower / AROMA

房屋抵押貸款的評價，因為需考慮到貸款人的提前清償及違約風險造成現金流量之不確定性，決定房屋抵押貸款的價格比決定一般違約證券的價格更具困難度。因此，如何合理評估抵押貸款證券的價值實為一值得深入探討之課題，本文即針對此議題進行研究。傳統文獻在進行房屋抵押貸款的評價方法，主要可區分為兩種：結構模型(structural-form approach)及縮減式模型(reduced-form approach)。目前的文獻上，其評價的封閉解只存在於結構式模型，但在此模型下的評價，存在著違約與提前清償條件的設定問題，這將對評價的準確性造成很大的影響，在實務的應用上有一定的限制。再者，結構式模型在處理多變數且變數間具相關性的情況，存在一定的複雜性與困難度，而縮減式模型在此情況的處理上是較容易的。本研究將從縮減式模型的角度，引入 Jarrow (2001)的概念，在包含多重變數並考慮變數間相關係數之縮減模型下，進行房屋抵押貸款封閉解的推導。透過此方法可協助資產管理者從事投資組合配置最適化與避險策略的分析，亦期望能提供實務界一個更具可行性與效率性之房屋抵押貸款評價模型。 / Valuing mortgage-related securities is more complicated than valuing regular defaultable claims due to the borrower’s prepayment behavior as well as the possibility of default. In general, the methods that are applied to investigate mortgage value and termination risk can be divided into two categories: a structural-form approach and a reduced-form approach. Some researchers use a structural-form model to obtain the closed-form formulae for the mortgage value. With this method, however, it is difficult to identify the critical region of early exercise and deal with the situation including multivariable and their correlations correlation among variables. As an alternative, the reduced-form model developed in this study is able to value the mortgage without setting boundary conditions, and can thereby accurately handle the multi-dimensional space of correlated state variables. This study extends Jarrow’s (2001) model to examine mortgage valuations. The purpose of this article is to derive a closed-form solution of the mortgage valuation equation under a general reduced-form model that embeds relevant economic variables. This new approach enables portfolio managers to undertake sophisticated portfolio optimization and hedging analyses, and makes it possible to more accurately and efficiently value the complicated mortgage.

In a first part, we study numerical schemes using the finite-element method to discretize the Oldroyd-B system of equations, modelling a viscoelastic fluid under no flow boundary condition in a 2- or 3- dimensional bounded domain. The goal is to get schemes which are stable in the sense that they dissipate a free-energy, mimicking that way thermodynamical properties of dissipation similar to those actually identified for smooth solutions of the continuous model. This study adds to numerous previous ones about the instabilities observed in the numerical simulations of viscoelastic fluids (in particular those known as High Weissenberg Number Problems). To our knowledge, this is the first study that rigorously considers the numerical stability in the sense of an energy dissipation for Galerkin discretizations. In a second part, we adapt and use ideas of a numerical method initially developped in the works of Y. Maday, A.T. Patera et al., the reduced-basis method, in order to efficiently simulate some multiscale models. The principle is to numerically approximate each element of a parametrized family of complicate objects in a Hilbert space through the closest linear combination within the best linear subspace spanned by a few elementswell chosen inside the same parametrized family. We apply this principle to numerical problems linked : to the numerical homogenization of second-order elliptic equations, with two-scale oscillating diffusion coefficients, then ; to the propagation of uncertainty (computations of the mean and the variance) in an elliptic problem with stochastic coefficients (a bounded stochastic field in a boundary condition of third type), last ; to the Monte-Carlo computation of the expectations of numerous parametrized random variables, in particular functionals of parametrized Itô stochastic processes close to what is encountered in micro-macro models of polymeric fluids, with a control variate to reduce its variance. In each application, the goal of the reduced-basis approach is to speed up the computations without any loss of precision

For repeated transient solutions of geometrically nonlinear structures the numerical effort often poses a major obstacle. Thus, the introduction of a reduced order model, which takes the nonlinear effects into account and accelerates the calculations considerably, is often necessary.This work yields a method that allows for rapid, accurate and parameterisable solutions by means of a reduced model of the original structure. The structure is discretised and its dynamic equilibrium described by a matrix equation. The projection on a reduced basis is introduced to obtain the reduced model. A comprehensive numerical study on several common reduced bases shows that the simple introduction of a constant basis is not sufficient to account for the nonlinear behaviour. Three requirements for an rapid, accurate and parameterisable solution are derived. The solution algorithm has to take into account the nonlinear evolution of the solution, the solution has to be independent of the nonlinear finite element terms and the basis has to be adapted to external parameters.Three approaches are provided, each responding to one requirement. These approaches are assembled to the integrated method. The approaches are the update and augmentation of the basis, the polynomial formulation of the nonlinear terms and the interpolation of the basis. A Newmark-type time-marching algorithm provides the frame of the integrated method. The application of the integrated method on test-cases with geometrically nonlinear finite elements confirms that this method leads to the initial aim of a rapid, accurate and parameterisable transient solution.

En 1940, Paul Erdős énonça une conjecture sur la distribution des classes inversibles modulo un entier. La présente thèse étudie la distribution des k-uplets de classes inversibles propose une preuve de la conjecture d'Erdős étendue au cas des k-uplets. / In 1940 Paul Erdos made a conjecture about the distribution of reduced residues. Here, in this thesis we studied the distribution of k-tuples of reduced residues, and proved extension of Erdos's conjecture for them.

Les bâtiments résidentiels et industriels munis d'un réseau de ventilation constituent des installations complexes, susceptibles d'être le siège de transferts de masse et d'énergie variés, selon les situations de fonctionnement. Afin d'étudier ces transferts de masse, une méthodologie permettant d'établir des expérimentations à échelle réduite pour l'étude des écoulements isothermes, en régime permanent ou transitoire, a été développée. Cette méthodologie a été validée numériquement et expérimentalement sur des configurations simples, puis appliquée à deux configurations de référence, représentatives de celles rencontrées dans le domaine nucléaire.L'influence du vent sur les transferts de masse au sein de ces configurations, en situation de fonctionnement normale, dégradée (arrêt de la ventilation) ou accidentelle (surpression interne), a été étudiée dans la soufflerie climatique Jules Verne du CSTB. Les effets du vent, couplés ou non à une surpression interne, peuvent alors entraîner une perte partielle ou globale du confinement des polluants au sein des installations. De plus, la turbulence du vent peut induire des inversions instantanées des débits de fuite, qui ne sont pas identifiées en régime permanent. Par ailleurs, l'analyse de sollicitations transitoires montre la faible influence de l’inertie des branches sur les écoulements transitoires, pour des grandeurs caractéristiques d'une installation réelle. Enfin, des essais de traçage gazeux ont été réalisés afin d’étudier la dispersion d’un polluant au sein d’une configuration de référence soumise aux effets couplés du vent, de la ventilation mécanique et d'une surpression interne.La robustesse du code à zones SYLVIA, utilisé notamment pour appuyer les évaluations de sûreté des installations nucléaires, a été analysée à partir de ces résultats expérimentaux. La prise en compte des phénomènes physiques observés expérimentalement a été validée, en régimes permanent et transitoire. Toutefois, quelques limitations ont été identifiées pour l'étude de la dispersion d'un scalaire passif, du fait des hypothèses utilisées dans le code SYLVIA, comme dans tout code à zones (concentration homogène dans les locaux, propagation instantanée dans les branches et dans les locaux). / Residential and industrial buildings equipped with a ventilation system are complex facilities, where various heat and mass transfers could occur according to the operating conditions. In order to study these mass transfers, a methodology has been developed so as to carry out reduced-scale experiments for the study of isothermal flows, in steady or transient state. This methodology has been numerically and experimentally validated on simple configurations, and then applied to two standard configurations, representing nuclear facilities.The wind influence on mass transfers inside these configurations, in normal, damaged (stopping ventilation) or accidental (internal overpressure) situations, has been studied in the Jules Verne climatic wind tunnel of the CSTB. The wind effects, coupled or not with an internal overpressure, can lead to a partial or a total loss of the pollutant's containment inside buildings. Moreover, the wind turbulence can bring about instantaneous reversal leakage flowrates, which cannot be identified in steady state. In addition, the study of transient phenomena has highlighted the low influence of the branch inertia on transient flows, for typical values of real facilities. Finally, tracer tests have been carried out in order to study the pollutant dispersion inside a standard configuration subjected to wind, mechanical ventilation and internal overpressure effects.The reliability of the zonal code SYLVIA, used notably to support safety assessment in nuclear buildings, has been analyzed from these experimental results. The modelling of the physical phenomena experimentally observed has been validated, in steady and transient states. However, limitations have been identified for the study of pollutant dispersion, due to hypothesis used in SYLVIA code, as in all zonal codes (homogenous concentration inside rooms, instantaneous propagation inside branches and rooms).

Le développement de nouvelles technologies pour le transport aérien moins polluant est de plus en plus basé sur la simulation numérique, qui nécessite alors une description fiable de la chimie. Pour la plupart des carburants, la description de la combustion nécessite des mécanismes détaillés mais leur utilisation dans une simulation numérique de combustion turbulente est limitée par le coût calcul. Des mécanismes cinétiques réduits et des méthodes de tabulation ont été proposés pour surmonter ce problème. Ces descriptions chimiques simplifiées ayant été développées dans le cadre de configurations laminaires, cette thèse propose de les évaluer dans des configurations turbulentes: une DNS de flamme prémélangée méthane/air de type Bunsen et une LES d’un brûleur expérimental. Les mécanismes sont analysés en termes de structure de flamme, paramètres de flamme globaux, longuer de flamme, prediction des concentrations en espèces majoritaires et des émissions polluantes. Une méthodologie pour évaluer a priori la capacité d’un mécanisme à prédire correctement des phénomènes chimiques tridimensionnels est proposée en se basant sur les résultats de flammes laminaires monodimensionnelles non étirées et étirées. Il ressort que, d’une part, pour construire un mécanisme réduit, il est nécessaire de faire un compromis entre coût calcul, robustesse et qualité des résultats. D’autre part, la qualité des résultats de DNS et LES de configurations tridimensionnelles turbulentes peut être anticipée par une analyse du comportement des schémas réduits dans des configurations simplifiées de flammes monodimensionnelles laminaires non étirées et étirées. / A growing need for numerical simulations based on reliable chemistries has been observed in the last years in order to develop new technologies which could guarantee the reduction of the enviromental impact on air transport. The description of combustion requires the use of detailed kinetic mechanisms for most hydro-carbons. Their use in turbulent combustion simulation is still prohibitive because of their high computational cost. Reduced chemistries and tabulation methods have been proposed to over-come this problem. Since all these reductions have been developed for laminar configurations, this thesis proposes to evaluate their performances in simulations of turbulent configurations such as a DNS of a premixed Bunsen methane/air flame and a LES of an experimental PREC-CINSTA burner. The mechanisms are analysed in terms of flame structure, global burning parameters, flame length, prediction of major species concentrations and pollutant emissions. An a priori methodology based on one-dimensional unstrained and strained laminar flames to evaluate the mechanism capability to predict three-dimensional turbulent flame features is therefore proposed. On the one hand when building a new reduced scheme, its requirements should be fixed compromising the computational cost, the robustness of the chemical description and the desired quality of results. On the other hand, the quality of DNS or LES results in three-dimensional configurations could be anticipated testing the reduced mechanism on laminar one-dimensional premixed unstrained and strained flames.

Les instabilités fluides peuvent dégrader le confinement du plasma au sein des tokamaks. Étant données les échelles spatio-temporelles, on choisit les modèles fluides obtenus à partir de la dérivation des modèles cinétiques. On dérive plusieurs modèles hiérarchiques de la MagnétoHydroDynamique (MHD) et en particulier les modèles de la MHD réduite du Current Hole et de l'équilibre de Grad-Shafranov. Une des difficulté de l'ensemble de ces modèles est de respecter l'équation modélisant l'absence de monopôles magnétiques. Pour assurer cette condition en tout point du domaine, le champ magnétique est réécrit avec un potentiel vecteur. L'utilisation de potentiels fait apparaître des équations faisant intervenir des dérivées d'ordre supérieurs. La stratégie numérique développée est l'utilisation de la méthode des éléments finis avec des éléments C1 de Bell. Sur un maillage non structuré, ces éléments ont l'intérêt de présenter une base réduite définir exclusivement avec des variables aux noeuds du maillage. Les modèles de MHD réduite du Current Hole et de Grad-Shafranov ont été résolus avec ces éléments. La résolution du cas test de Grad-Shafranov avec les conditions de bords exactes a permis d'obtenir l'ordre optimale de 5. La résolution du système du Current Hole avec ces éléments, validée par l'obtention du paramètre η1/3, a permis l'observation de développement d'instabilités en dents de scies. / Fluid instabilities can degrade plasma confinement in tokamaks. Given the spatial and temporal scales, we choose the fluid models obtained from the derivation of kinetic models. We derived several hierarchical models of MagnetoHydroDynamic (MHD) and in particular models of reduced MHD like the Current Hole and the Grad-Shafranov equilibrium. One of the difficulty of all these models is to respect the absence of magnetic monopoles equation. To ensure this condition at any point, the magnetic field is rewritten with a vector potential. The use of vector portential implies that higher order derivatives appear in the equation. The numerical strategy is developed using the finite element method with C1 Bell's elements. On a unstructured mesh, these have the advantage to present a reduced basis with degrees of freedom defined exclusively on the nodes of the mesh. The reduced MHD models of the Current Hole and Grad-Shafranov have thus been resolved with these elements. The resolution of a Grad-Shafranov test case with exact boundary conditions yields the optimal order of 5. The resolution of the Current Hole system with thesse elements has been validated by obtaining physical parameter η1/3 and allowed the observation of the development of sawtooth instabilities.