Finite Volume Methods

Nous présentons dans ce travail un algorithme de génération de maillage hexaédrique à partir d'un squelette obtenu depuis une surface qui détermine le domaine à mailler. En tirant parti de cette représentation on peut générer un maillage dont les mailles sont alignées avec la géométrie du squelette et dont le nombre d'irrégularités est réduite au maximum. La principale difficulté réside dans la gestion des embranchements pour laquelle nous proposons une nouvelle solution. Des surfaces de contact, partition de la sphère en faces quadrangulaires, sont construites au niveau de chaque sommet du squelette, encodant la manière de coudre les hexaèdres construits le long des branches. Au sein d'une chaîne de traitement allant de l'extraction de squelette et son rééchantillonnage jusqu'au raffinement et à l'optimisation de la géométrie des mailles, cette méthode permet également des traitements locaux des différentes étapes permettant de faciliter leur exécution parallèle.

Many geophysical flow or wave propagation problems can be modeled with two-dimensional depth-averaged equations, of which the shallow water equations are the simplest example. We describe the GeoClaw software that has been designed to solve problems of this nature, consisting of open source Fortran programs together with Python tools for the user interface and flow visualization. This software uses high-resolution shockcapturing finite volume methods on logically rectangular grids, including latitude-longitude grids on the sphere. Dry states are handled automatically to model inundation. The code incorporates adaptive mesh refinement to allow the efficient solution of large-scale geophysical problems. Examples are given illustrating its use for modeling tsunamis and dam-break flooding problems. Documentation and download information is available at www.clawpack.org/geoclaw.

The Rankine-Hugoniot-Riemann (RHR) solver has been designed to solve steady multidimensional conservation laws with source terms. The solver uses a novel way of incorporating cross fluxes as source terms. The combined source term from the cross fluxes and normal source terms is imposed in the middle of a cell, causing a jump in the solution according to the Rankine-Hugoniot condition. The resulting Riemann problems at the cell faces are then solved by a conventional Riemann solver. We prove that the solver is of second order accuracy for rectangular grids and confirm this by its application to the 2D scalar advection equation, the 2D isothermal Euler equations and the 2D shallow water equations. For these cases, the error of the RHR solver is comparable to or smaller than that of a standard Riemann solver with a MUSCL scheme. The RHR solver is also applied to the 2D full Euler equations for a channel flow with injection, and shown to be comparable to a MUSCL solver.

The Rankine-Hugoniot-Riemann (RHR) solver has been designed to solve steady multidimensional conservation laws with source terms. The solver uses a novel way of incorporating cross fluxes as source terms. The combined source term from the cross fluxes and normal source terms is imposed in the middle of a cell, causing a jump in the solution according to the Rankine-Hugoniot condition. The resulting Riemann problems at the cell faces are then solved by a conventional Riemann solver. We prove that the solver is of second order accuracy for rectangular grids and confirm this by its application to the 2D scalar advection equation, the 2D isothermal Euler equations and the 2D shallow water equations. For these cases, the error of the RHR solver is comparable to or smaller than that of a standard Riemann solver with a MUSCL scheme. The RHR solver is also applied to the 2D full Euler equations for a channel flow with injection, and shown to be comparable to a MUSCL solver.

In this paper we obtain quantitative Bernstein-von Mises type bounds on the normal approximation of the posterior distribution in exponential family models when centering either around the posterior mode or around the maximum likelihood estimator. Our bounds, obtained through a version of Stein's method, are non-asymptotic, and data dependent; they are of the correct order both in the total variation and Wasserstein distances, as well as for approximations for expectations of smooth functions of the posterior. All our results are valid for univariate and multivariate posteriors alike, and do not require a conjugate prior setting. We illustrate our findings on a variety of exponential family distributions, including Poisson, multinomial and normal distribution with unknown mean and variance. The resulting bounds have an explicit dependence on the prior distribution and on sufficient statistics of the data from the sample, and thus provide insight into how these factors may affect the quality of the normal approximation.

In many reservoirs, the transport and deposition of fine sediments are associated with turbidity cur-rents events. These currents are generated when the sediment-laden river inflow enters a reservoir and plunges below the clear quiescent water and continues as a dense ...

Les panneaux en bois lamellé croisé (CLT-Cross Laminated Timber) sont des éléments multicouches dont les plis sont alternativement croisés à 90° et collés sur leurs faces supérieures et inférieures (cf Fig 1). Ils sont de plus en plus utilisés pour construire des bâtiments de grande hauteur du fait de leur légèreté et de leur facilité d'assemblage. Dans les couches transversales, le module de cisaillement roulant G RT (radial-tangentiel) est mis en jeu. Ce module étant très faible, il pourrait conduire à un assouplissement dans la structure et donc à un voilement prématuré en cas de chargement dans le plan.

The present work focuses on the numerical simulation of an unsteady two-dimensional incompressible flow behind a triangular wedge at an angle of attack 0 . The Reynolds-Averaged Navier-Stokes (RANS) Baseline (BSL) Reynolds-Stress models can accurately predict an unsteady flow, though these models tend to be expensive and numerically strong. Numerical simulations for both steady and unsteady flow around a two
dimensional triangle wedge at a Reynolds number of 4.5 x 104 were produced. The numerical simulation for an unsteady flow with automatic 2-D Delaunay mesh scheme was carried out to calculate the Strouhal number and mean stream-wise velocity profiles. The result so obtained agrees with the experimental data available. Turbulent vortex shedding behind the wedge was perfectly captured which results as a good choice of this turbulence model. Also a longer recirculation zone which was obtained for a steady flow will be presented.
Keywords: Wedge, Turbulence model, Vortex Shedding.

Based on the book "Convection Heat Transfer" by Adrian Bejan, this study examines the transition time of flow from laminar to turbulent in natural convection near turbulent walls. The presented analysis investigates the hydrodynamic stability of an inviscid jet near a wall. The analysis shows that if the mixing length exceeds a certain value, the transition from laminar to turbulent flow will occur. Using the scaling method, it is demonstrated that the oscillation period of the unstable jet is of the same order as the viscous penetration time perpendicular to the jet, and the transition occurs when the Reynolds number of the jet is around 100.

-Une nouvelle formulation de l'élément solide-coque SHB6 est décrite. Il s'agit d'un élément isoparamétrique prismatique à 6 noeuds, interpolation linéaire et intégration réduite dans le plan moyen. Les déplacements sont les seuls d.d.l. et les points d'intégration sont distribués à travers l'épaisseur. L'analyse de hourglass a révélé qu'il n'y a pas de modes à énergie nulle à stabiliser ; néanmoins, la méthode "assumed strain" est adoptée pour améliorer sa convergence. Les performances du nouvel élément, ainsi obtenu, sont évaluées à travers des cas tests standard. Mots clés-Solide-coque SHB6, verrouillage, hourglass, méthode « assumed strain », cas tests.

-Dans cet article, la formulation de deux éléments finis de type coque volumique sera présentée. Il s'agit d'un hexaèdre à vingt noeuds et d'un prisme à quinze noeuds obtenus à partir d'une approche purement tridimensionnelle. Ce concept solide−coque procure de nombreux avantages par rapport aux éléments finis traditionnels solides ou coques, puisqu'il consiste à combiner en une seule formulation certaines bonnes caractéristiques des coques avec quelques bonnes propriétés des solides. A noter que les éléments coques volumiques ainsi obtenus possèdent une direction privilégiée, le long de laquelle les points d'intégration sont disposés, et une intégration réduite dans le plan de l'élément est adoptée pour réduire un certain nombre de verrouillages et pour augmenter l'efficacité de calcul de ces éléments. Au travers de cas tests représentatifs, les performances de ces éléments sont montrées. Mots clés-élément fini solide−coque, principe variationnel mixte, phénomènes de verrouillage, intégration réduite, modes à énergie nulle, cas tests standards

Microchannels have been suggested to be a very effective heat transfer device. However, the electrical double layer (EDL) effect in a microchannel is anticipated to be significant. In this paper, two EDL models coupled with Navier–Stokes equations were used to compute a 3‐D developing microchannel flow. The Poisson–Boltzmann model (PBM) has been proven to be a promising tool in studying the EDL effect for developed microchannel flow, with respect to its accuracy and efficiency. However, it was commended that the assumption of Boltzmann distribution in PBM, for electric ion concentration distribution in particular was questionable in the developing flow. Nernst–Planck model (NPM), with its two extra partial differential equations (PDEs) to predict the ion concentration distribution, was reported to be a more appropriate model for developing microchannel flow though increased RAM and CPU were needed as compared to the PBM. The governing equations for both models were discretized for d...

Bu makale, birinci yazar tarafından İTÜ Uçak ve Uzay Bilimleri Fakültesi'nde tamamlanmış "Parallel solution of unsteady, incompressible three-dimensional Navier-Stokes equations with a new implicit method" adlı doktora tezinden hazırlanmıştır. Makale metni 07.01.2003 tarihinde dergiye ulaşmış, 24.01.2003 tarihinde basım kararı alınmıştır. Makale ile ilgili tartışmalar 31.07.2003 tarihine kadar dergiye gönderilmelidir. Özet Üç boyutlu ve zamana bağlı Navier-Stokes denklemlerinin sayısal hassas çözümü, analiz edilecek alanın büyüklüğü ve karmaşıklığı söz konusu olduğunda paralel hesap gereksinimini ortaya çıkartmaktadır. Bu amaca uygun olarak, zamanda ve uzayda ikinci mertebeden hassas kapalı sayısal bir çözüm tekniği bu çalışma çerçevesinde geliştirilmiş ve uygulan yeni bir paralel çözüm tekniği ile sonuçlar elde edilmiştir. Momentum denkleminin kapalı sayısal çözümü, zamana göre yapılan ayrıklaştırmada atılacak zaman adımı büyüklüğüne bir kısıtlama getirmeden koşulsuz kararlı bir şekilde yapılabilmekte ve basınç alanı Poisson denkleminden elde edilmektedir. Çözüm alanı birbirleriyle bitişik çözüm bölgelerine ayrılmakta ve hem momentum hem de basınç denklemi, ayrı ayrı, bu bölgelerde iteratif yöntemlerle çözülmektedir.

-Les éléments finis "rotation free" constituent une alternative séduisante aux éléments plaques-coques classiques. Cependant la principale limitation de ces éléments à notre sens, est leur sensibilité à la distorsion du maillage, seuls des maillages réguliers peuvent conduire à des résultats très précis. Dans ce contexte, l'objectif de ce papier est la présentation d'un nouvel élément (appelé SFE3) qui, tout en étant précis, présente une très faible sensibilité à la distorsion du maillage. Mots clés-plate and shell elements, rotation free, Large strains.

This mini-symposium is dedicated to the discussion of recent developments and applications in the field of Numerical Simulation of Petroleum R eservoirs and related disciplines, including new gridding and numerical formulations. The goal is to bring together researchers, students and professionals in the field of Petroleu m Reservoir Simulation and related areas. The scope of the mini-symposium ranges from the mathematical and computational methods to the modeling and simulation of challenging applications in petroleum reservoir simulation.

Suivant la tradition, il est désormais temps d'achever ce manuscrit et ce cycle de vie par le début. Je souhaite dédier cet espace aux remerciements de toutes les personnes qui ont contribué, de près ou de loin, volontairement ou involontairement, à cette aventure (aventure de bureau certes, mais aventure quand même). Mes trois années de thèse sur Toulouse furent à la fois belles et mouvementées, et n'auraient eu que peu de saveurs sans les personnes qui les ont partagé. Je tiens d'abord à remercier chaleureusement mon Directeur de thèse, Sébastien Pernet. Ces quelques lignes ne sauraient exprimer pleinement ma profonde reconnaissance pour son aide inconditionnelle tout au long de la thèse, sa profonde gentillesse, ainsi que la confiance qu'il m'a accordé. Ce travail de thèse lui doit énormément. Qu'il soit ici assuré de ma gratitude et de mon amitié. De manière plus prosaïque, un grand merci pour toutes les dosettes de café piquées qui m'ont sauvé quand les cafés du distributeur apparaissaient trop infects ! Merci aussi pour la bonne humeur et toutes nos discussions (professionnelles ou pas !) pendant tout ce temps de cohabitation au bureau 209. J'exprime ma profonde gratitude à Marion Darbas et Xavier Antoine d'avoir accepté de rapporter ma thèse, et de l'avoir fait avec une extrême minutie, m'ayant permis une amélioration notable du manuscrit final. J'adresse également mes remerciements aux autres membres du jury, Samuel Nosal, Sébastien Tordeux, Nicolas Zerbib, ainsi que Patrick Ciarlet pour avoir accepté d'en être le président. Je tiens également à remercier très chaleureusement Francois Rogier pour son extrême gentillesse, pour son aide ainsi que ses conseils et les discussions enrichissantes que nous avons pu avoir durant ces quelques années. A Daniela qui, malgré tout, m'a soutenu, accompagné et aidé en ces moments si particuliers, une phrase ne saurait suffire à exprimer la reconnaissance et la gratitude qui m'habitent. Enfin, il me faut absolument remercier l'ensemble des stagiaires et doctorants sans qui la vie au laboratoire aurait été bien fade ! Je commence par "l'ancienne génération", qui a su si bien m'accueillir au sein du DTIS. Je les remercie pour tous les moments (et ils sont nombreux !) passés ensemble, les débats interminables et souvent absurdes, les pauses de 4h, tous ces repas partagés et enfin-comment les oublier !-toutes ces soirées mémorables (petite pensée pour les cantalous du Danu et nos raclettes !

A voxel based methodology to create high quality finite element meshes of large city models for urban computational physics is proposed. Initial data are limited to 3D points cloud or STL files. A voxel grid which provides a rough representation of the outer (visible) surfaces of geometry is first created. The voxel model is then smoothed by a subdivision technique which provides a better representation of the normal vector to the surface than the voxel model. The points smoothed models are thereafter projected on the initial data. The ground, building blocks and the air are identified as volumes and their corresponding closed conforming surface mesh is created. Examples of complex reconstructions are presented.

Aims. This work presents a new hydrodynamical algorithm to study astrophysical detonations. A prime motivation of this development is the description of a carbon detonation in conditions relevant to superbursts, which are thought to result from the propagation of a detonation front around the surface of a neutron star in the carbon layer underlying the atmosphere. Methods. The algorithm we have developed is a finite-volume method inspired by the original MUSCL scheme of van Leer (1979). The algorithm is of second-order in the smooth part of the flow and avoids dimensional splitting. It is applied to some test cases, and the time-dependent results are compared to the corresponding steady state solution. Results. Our algorithm proves to be robust to test cases, and is considered to be reliably applicable to astrophysical detonations. The preliminary one-dimensional calculations we have performed demonstrate that the carbon detonation at the surface of a neutron star is a multiscale phenomenon. The length scale of liberation of energy is 10 6 times smaller than the total reaction length. We show that a multi-resolution approach can be used to solve all the reaction lengths. This result will be very useful in future multi-dimensional simulations. We present also thermodynamical and composition profiles after the passage of a detonation in a pure carbon or mixed carbon-iron layer, in thermodynamical conditions relevant to superbursts in pure helium accretor systems.

Aims. This work presents a new hydrodynamical algorithm to study astrophysical detonations. A prime motivation of this development is the description of a carbon detonation in conditions relevant to superbursts, which are thought to result from the propagation of a detonation front around the surface of a neutron star in the carbon layer underlying the atmosphere. Methods. The algorithm we have developed is a finite-volume method inspired by the original MUSCL scheme of van Leer (1979). The algorithm is of second-order in the smooth part of the flow and avoids dimensional splitting. It is applied to some test cases, and the time-dependent results are compared to the corresponding steady state solution. Results. Our algorithm proves to be robust to test cases, and is considered to be reliably applicable to astrophysical detonations. The preliminary one-dimensional calculations we have performed demonstrate that the carbon detonation at the surface of a neutron star is a multiscale phenomenon. The length scale of liberation of energy is 10 6 times smaller than the total reaction length. We show that a multi-resolution approach can be used to solve all the reaction lengths. This result will be very useful in future multi-dimensional simulations. We present also thermodynamical and composition profiles after the passage of a detonation in a pure carbon or mixed carbon-iron layer, in thermodynamical conditions relevant to superbursts in pure helium accretor systems.

In generale, l'individuazione di aree sottoposte al pericolo di inondazione richiede la conoscenza dettagliata dell'area di studio (corso d'acqua ed aree di espansione delle piene), l'analisi idrologica e la modellazione idraulica. La stima del pericolo di inondazione è oggi agevolata da sistemi di rilievo che consentono di creare modelli digitali del terreno di estremo dettaglio (COBBY et alii, 2001), da modelli matematici sofi sticati e da uno sviluppo hardware e software che garantisce sempre più elabora-zioni complesse in tempi accettabili. Tuttavia, la costruzione del model-lo di simulazione rappresenta sempre un compromesso tra potenzialità degli strumenti utilizzati e grado di dettaglio che si vuole raggiungere, pertanto può essere utile verifi care l'entità delle variazioni dei parame-tri idraulici di output rispetto alla scelta operata sui parametri in input. Questo è ancora più vero quando non si possiedono suffi cienti dati os-servati di eventi di pien...

Present paper investigates the numerical solution of two-dimensional unsteady compressible Navier-Stokes equations by a new scheme based on the finite volume method. Kinetic Energy Preserving (KEP) scheme is introduced for solving the supersonic and Transonic external compressible flow field on very fine grids (with a number of cells of the order of the Reynolds number) without artificial dissipation terms even in place of shock waves. It should be noted that the solution of flow field with this scheme in this range of speed, is presented for the first time. By discretization of the governing equations based on KEP scheme and elimination of dissipative effects, the Direct Numerical Simulation (DNS) of the flow is possible. The results of this solution for supersonic flow over flat plate and Transonic flow over the airfoil at low Reynolds numbers show that the KEP method can be presented stable and non-oscillatory solution by no artificial dissipation even in areas with shock waves. Therefore, the KEP method can be used for DNS of turbulent flows (without a modeling the turbulence phenomena itself).

The finite volume (FV) method is the dominating discretization technique for computational fluid dynamics (CFD), particularly in the case of compressible fluids. The discontinuous Galerkin (DG) method has emerged as a promising highaccuracy alternative. The standard DG method reduces to a cell-centered FV method at lowest order. However, many of today's CFD codes use a vertex-centered FV method in which the data structures are edge based. We develop a new DG method that reduces to the vertex-centered FV method at lowest order, and examine here the new scheme for scalar hyperbolic problems. Numerically, the method shows optimal-order accuracy for a smooth linear problem. By applying a basic hp-adaption strategy, the method successfully handles shocks. We also discuss how to extend the FV edge-based data structure to support the new scheme. In this way, it will in principle be possible to extend an existing code employing the vertex-centered and edge-based FV discretization to encompass higher accuracy through the new DG method.

The finite volume (FV) method is the dominating discretization technique for computational fluid dynamics (CFD), particularly in the case of compressible fluids. The discontinuous Galerkin (DG) method has emerged as a promising highaccuracy alternative. The standard DG method reduces to a cell-centered FV method at lowest order. However, many of today's CFD codes use a vertex-centered FV method in which the data structures are edge based. We develop a new DG method that reduces to the vertex-centered FV method at lowest order, and examine here the new scheme for scalar hyperbolic problems. Numerically, the method shows optimal-order accuracy for a smooth linear problem. By applying a basic hp-adaption strategy, the method successfully handles shocks. We also discuss how to extend the FV edge-based data structure to support the new scheme. In this way, it will in principle be possible to extend an existing code employing the vertex-centered and edge-based FV discretization to encompass higher accuracy through the new DG method.

A finite volume solver is developed for the simulation of the hydrodynamic and water recirculation in the Nador lagoon. The physical model is based on the 2D shallow water equations that include slope variations and friction losses. The convective fluxes are approximated with a Non Homogeneous Riemann Solver (SRNH) on unstructured meshes, whereas an operator splitting procedure is used for the friction terms. Second-order accuracy in space is achieved through variable reconstruction. The algorithm can be applied to model flows on complicated geometries and preserves the balance between the pressure gradient and source term due to the slope variation. Numerical tests are presented here and discussed for the study of the water circulations in the Nador lagoon considering the new “pass” with the Mediterranean Sea. Keywords Shallow water equations, Finite volume, SRNH solver, Unstructured meshes, Water recirculation, Nador lagoon

In this paper, the problem of perforation of the human cranial bone surface in the shape of a hemisphere using a water jet is numerically simulated in three dimensions. The simulation involves both the flow and heat transfer of the impinging water jet, and surface perforation process. The volume of fluid method was used to model the jet two-phase flow, and Johnson-Cook equation in finite element method was applied for perforating the bone surface. The obtained results show that the pressure and friction coefficients on the surface depend on the nozzle distance from the surface, and the pressure at the stagnation point increases as the nozzle distance decreases. Also, by increasing the nozzle distance, local Nusselt number along the hemisphere radius as well as maximum Nusselt decreases at the stagnation point. The effect of nozzle diameter on pressure and friction coefficients was also investigated and found that the pressure at stagnation point is increased by the nozzle diameter. The change in the jet velocity also showed that a 20% change in velocity has no significant effect on the pressure and friction coefficients, while increases the Nusselt number. In modeling the perforation process, actual properties and coefficients of bone material, such as Poisson's coefficient and Young's modulus were used. Comparison of on-site stress distribution based on Tresca and von Mises criteria showed that according to the proposed parameters, the perforation is performed properly.

Dans cet article, la méthode des éléments finis étendue (X-FEM) est utilisée pour décrire le comportement vibratoire des plaques présentant une discontinuité à différentes positions. La théorie utilisée est celle de Mindlin, où les effets de l'inertie de rotation et des déformations en ...

Reactive transport plays an important role in various subsurface applications, including carbon dioxide sequestration, nuclear waste storage, biogeochemistry and the simulation of hydro–thermal reservoirs. The model couples a set of partial differential equations, describing the transport of chemical species, to nonlinear algebraic or differential equations, describing the chemical reactions. Solution methods for the resulting large nonlinear system can be either fully coupled or can iterate between transport and chemistry. This paper extends previous work by the authors where an approach based on the Newton–Krylov method applied to a reduced system has been developed. The main feature of the approach is to solve the nonlinear system in a fully coupled manner while keeping transport and chemistry modules separate. Here we extend the method in two directions. First, we take into account mineral precipitation and dissolution reactions by using an interior point Newton method, so as to...

Je remercie Dieu le tout puissant qui m'a permis d'achever ce modeste travail. Je tiens à remercier infiniment l'encadreur Mr :A.badereddine qui m'ont accompagné et aidé dans ma recherche, grâce à ses précieux conseils et ses orientations importantes. Je tiens aussi à remercier tous les membres du jury qui vont certainement enrichir cette recherche et la rendre plus performante. Je remercie également tous ceux qui ont participé de près ou de loin à la réalisation de ce travail, notamment mes collègues. Je n'oublie pas de remercier toute ma famille et tous mes amis pour leur support.

Ce travail de these concerne en premier lieu la formulation et l'evaluation de deux elements finis brique bases sur l'approche en deformation: Le premier est un element brique a neuf nœuds nomme SBB (Strain Based Brik), pour l'analyse lineaire des plaques minces et epaisses. Cet element possede seulement trois degres de liberte externes essentiels (trois translations) a chacun des huit nœuds du sommet ainsi qu'au nœud central. La technique de condensation statique est utilisee pour cet element. Le deuxieme est un element brique a huit nœuds nomme SBBE (Strain Based Brik Equilibrium) qui a trois degres de liberte externes essentiels (u, v et w) a chacun des huit nœuds de sommet. Le champ de deplacement de chacun de ces elements a ete developpe en utilisant l'approche en deformation. Ils sont bases sur des fonctions assumees pour les differentes composantes de deformation qui satisfont les equations de compatibilites, ainsi que les equations d'equilibre pour l&...

-Un méta-modèle est proposé basé sur une combinaison de technique de décomposition aux valeurs propres (POD) et fonction à base radiale (RBF) pour représenter la dépendance des paramètres sur les procédées de mise en forme. L'objectif est de présenter la POD pour la représentation des champs de déplacement après retour élastique. Après avoir trouvé une base réduit et une interpolation de POD coefficient par réseau à RBF permettant l'évaluation d'influence des paramètres sur la qualité de forme de retours élastique et optimisation des paramètres procédées. Mots clés-décomposition aux valeurs propres (POD), fonction à base radiale (RBF), mise en forme, emboutissage.

AUTHORS' CONTRIBUTIONS This work was carried out in collaboration between both authors. Author AKO gathered the initial data and help to interpret the results. Author MPM designed the study, wrote the algorithm, methodology programming and interpreted the results. Both authors read and approved the final manuscript.

II. Linear finite elements ______________________________________________________________ C. Linear elastic beams _____________________________________________________________ C.1. Hierarchic levels and reference systems _________________________________________ C.2. Basic reference system ________________________________________________________ C.3. The remaining squares of the "monopoly" ________________________________________ C.4. Governing beam differential equations : consistent derivation of 3D beam, theory from 3D equilibrium and compatibility equations _____________________________________________ C.5. Governing beam differential equations : three-dimensional Timoshenko beam theory ___ C.6. Governing beam differential equations : extended 3D timoshenko beam theory with distortion and warping ____________________________________________________________ C.7. Governing Beam differential equations : three-dimensional Euler-Bernoulli beam theory _ C.8. A 2D Timoshenko beam FE : Development, characteristics and mesh refinement________ C.9. A 2D Timoshenko beam FE : shear locking ________________________________________ C.10. A 2D Timoshenko beam FE : reduced integration _________________________________ C.11. 2D Euler-Bernoulli beam formulations: boundary conditions expressed at the basic reference system _________________________________________________________________ C.12. 2D Euler-Bernoulli displacement-based beam formulation __________________________ C.13. 2D Euler-Bernoulli force-based beam formulations ________________________________ C.14. DB VS FB Eléments-Summary _________________________________________________ D. Finite Element Mehtod for linear elastic shells and solids _______________________________ D.1.Linear elastic solid ____________________________________________________________ D.2. Linear elastic shell/plates _____________________________________________________ III. Nonlinear Finite Elements __________________________________________________________ E.

Karachi is the largest city of Pakistan. The temperature change in Karachi is studied in this research by analyzing the time series data of mean maximum temperature (MMxT), mean minimum temperature (MMiT) and mean annual temperature (MAT) from 1947 to 2005 (59 years). Data is analyzed in three parts by running linear regression and by taking anomalies of all time periods: (a) whole