Euler Equations

In the present work, we propose a reformulation of the fluxes and interpolation calculations in the PISO method, a well-known pressure-correction solver. This new reformulation introduces the AUSM C up flux definition as a replacement for the standard Rhie and Chow method of obtaining fluxes and central interpolation of pressure at the control volume faces. This algorithm tries to compatibilize the good efficiency of a pressure based method for low Mach number applications with the advantages of AUSM C up at high Mach number flows. The algorithm is carefully validated using exact solutions. Results for subsonic, transonic and supersonic axisymmetric flows in a nozzle are presented and compared with exact analytical solutions. Further, we also present and discuss subsonic, transonic and supersonic results for the well known bump test-case. The code is also benchmarked against a very tough test-case for the supersonic and hypersonic flow over a cylinder.

When solving the problems based on more than two variables most of the time relative changes on the variables need to be calculated. The degree of dependency among variables and number of variable quantities make solution finding more complex. It is very obvious to get the help of differential equations. When we deal with the functions frequently changing by nature, so it is highly likely to find local maxima, local minima, slope of tangent and value of the function at given point. Finding solution is not a single process but a set of processes where some binary operations, repetitions of the operations, checking the consistency of function within the interval etc take place. Here it becomes very important that the time taken to execute the problem of differential equation must be least. Thus the needs of pipelining, parallel processing, speculative computation etc become crucial factor. So requirement of solution of differential equations become very necessary. The degree of equation, complexity of equation, no of variables etc are the parameters those play very important roles during the solution process. When this task is assigned to serial processors the time taken to solve a problem is greater than the time taken by the parallel processors to do the same task. Here in this research we are calculating the time taken by serial processors and by parallel processors too. Although entry of vector processors and array processors improved the performance very much but since performance is the subject to frequently improvement. So putting that context in mind our study has tried to minimize the time taken for solving the differential equation by dividing a big problem in small ones in such a way so that dependency for data among processors i.e. communication will be very less. Means research has tried to improve the ratio of computation over communication ratio.

We present a novel discretization method for nonlinear convection-diffusion equations and, in particular, for the compressible Navier-Stokes equations. The method is based on a Discontinuous Galerkin (DG) discretization for convection terms, and a Mixed method using H(div) spaces for the diffusive terms. Furthermore, hybridization is used to reduce the number of globally coupled degrees of freedom. For the scalar case, a local postprocessing procedure is used to enhance the quality of the approximate solution w h. The method reduces to a DG scheme for pure convection, and to a Mixed method for pure diffusion, while for the intermediate case the combined variational formulation requires no additional parameters. We formulate and validate our scheme for nonlinear model problems, as well as compressible flow problems.

This paper focuses on the demonstration of an integrated aerostructural method for the design of aerospace vehicles. Both aerodynamics and structures are represented using high-fidelity models such as the Euler equations for the aerodynamics and a detailed finite element model for the primary structure. The aerodynamic outer-mold line and a structure of fixed topology are parameterized using a large number of design variables. The aerostructural sensitivities of aerodynamic and structural cost functions with respect to both outer-mold line shape and structural variables are computed using an accurate and efficient coupled-adjoint procedure. Kreisselmeier-Steinhauser functions are used to reduce the number of structural constraints in the problem. Results of the aerodynamic shape and structural optimization of a natural laminar-flow supersonic business jet are presented together with an assessment of the accuracy of the sensitivity information obtained using the coupled-adjoint procedure.

Public reporting burden for the collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or any other aspect of this collection of information, including suggestions for reducing this burden, to Washington Headquarters Services, Directorate for Information Operations and Reports, 1215 Jefferson Davis Highway, Suite 1204, Arlington VA 22202-4302. Respondents should be aware that notwithstanding any other provision of law, no person shall be subject to a penalty for failing to comply with a collection of information if it does not display a currently valid OMB control number.

In 2017 the author published the book (1) "Basics of Componential Numbers Theory" where new methodology of study of the natural numbers and its powers, and components of such numbers and its powers were introduced. At present moment the author is working on second edition of the English version and third edition of the Russian version (second version was published in 2019). Euler`s conjecture (1769) as a Pierre de Fermat`s one (1637) are examples of negative statements. They are stating situations when something cannot take place. That is why we call them the negative Conjectures. Most probably, people like Euler can see something, others unavailable to see. That is why we call them "genius"… Years after years scholars tried to find a proof for those statements, but unsuccessful. Then some of them, like Andrew Wiles finally found the proof and confirm the genius negative statement of Pierre de Fermat. But, some of them did not till now.

People these days know the Universe as a Whole, because not knowing the edge between This and That. Its secret is the secret of a forgotten body, the seventh in the series of multifaceted as Life, Seven. We know six of it now. But the Ancestors knew all seven. // Люди ныне не знают Вселенной как Целого, не зная грани меж Этим и Тем. Тайна  ее есть тайна забытого тела, седьмого в строю многогранных как Жизнь, Семь. Мы знаем шесть ныне. А Пращуры знали все семь.

Consideraciones culturales y estéticas de la conocida "Fórmula de Euler" o "Identidad de Euler / Lindeman", incluyendo una demostración simple y elegante de la misma, así como la propuesta de una identidad equivalente y alternativa fácil de valorar y otra fórmula que incluye la Sección Áurea o Proporción Divina.

Abstract.

The present edition of the "Basics of the Componential Numbers Theory" developed by the author is based on the suggestions and modifications offered by the readers related to the previous edition published in 2017 and third edition (in Russian) published in 2021. Particularly, the section devoted to the numerical system of calculation was extended and includes now an information on the Greeк, Slavic and Hebrew systems.
The proof of the extended Euler`s Hypotheses related to the arbitrary exponents is presented in § 17.2. In § 6 (Part 2) the analytical methodology of finding of prime and composite integers was presented. He Part 3 is devoted to the application of the Componential Numbers Theory for the proof of Phermat`s Theorem.
The paper is to show that there are still a lot of options to use the methodology of the initial numbers theory. The presented methods will give the answers to several questions and allow to solve some difficult tasks. In addition, the real solutions to the some complicated tasks are introduced in the paper. Author's work aims to show that the application of the methods of the simple primary numbers theory can still be developed and used. The methods, proposed in the book provide answers to many questions, allow to solve complex tasks by using of the elementary methods.
Apart from the purely theoretical segment, the book provides examples of solutions of problems by use of introduced results.

The first edition was translated and adapted from Russian by Eduard Aleksanyan.
This edition was translated and adapted from modification of the 3rd edition (Russian version)
published in 2021.


В электронной версии третьего  издания данной книги излагаются основы компонентной теории чисел разработанной автором с дополнениями и с учётом замечаний и пожеланий читателей высказанных относительно предыдущего издания опубликованного в 2021 г.
В частности, добавлены разделы посвящённые “алфавитным” системам счисления (греческой, кирилице, ивриту), доказательству обобщённой гипотезы (теоремы) Эйлера для произвольных чётных показателей степени (§17.2, Часть 1).
Разработана и представлена (§ 6, Часть 2) методика определения простых и составных чисел. В 3 Части предлагается методика доказательства Теоремы Ферма на основе Компонентной Теории чисел.
Работа автора преследует цель показать, что использование методов элементарной теории чисел далеко ещё не исчерпало себя.
Предлагаемые в книге методы дают ответы на многие вопросы, позволяют решать сложные задачи элементарными методами.
Помимо чисто теоретической части, в книге приводятся конкретные примеры решения задач с использованием полученных результатов.

As scientists probe deeper into the nature of reality, they are discovering that consciousness is at the root of everything. What we believed were the laws of physics and matter are really the archetypal laws of mind, and the condensation of consciousness that created our universe originated from a dimensionless point in the void known in physics as the “Singularity” and in mathematics and philosophy as the “Monad.

It is well-known that the shortest curve on surface of sphere is great circle. However, finding of the shortest curve on other surface is not much studied. Currently, many researchers are more interested in studying about this field. A curve which is called geodesic is a curve that accelerates to surface. The definition of geodesic is 0 α α ⋅ = & && and orthogonal to the tangent space. In this paper, we offered the shortest path on some surfaces of revolution. Then, we analyzed the parametric equation to determine the shortest curve by using the application of Euler equation. Finally, the given examples can be used to apply in clinical application and other.

La belleza y la elegancia se suelen vincular vulgarmente con la subjetividad, pero bien puede relacionarse a principios naturales, instintivos y formulados matemáticamente, en otras palabras enlazados a la ciencia. Se explora aquí un acercamiento al concepto de la belleza y la elegancia en la ciencia en general, la matemática en particular y extensible a lo cotidiano.

Public reporting burden for the collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing the collection of ...

Les méthodes de Runge-Kutta (ou RK), l'ordre 2 ou 4, sont très couramment utilisées pour la résolution d'équations différentielles ordinaires (EDO). Ce sont des méthodes à pas unique, directement dérivées de la méthode d'Euler, qui est une méthode RK1.

The aim of this work is to accelerate an existing two-dimensional explicit Euler solver implemented on a single CPU. The solver based on Discontinuous-Galerkin method for the spatial discretization is ported to run on a graphical processing units (GPU) using the CUDA programming language.
The input has been rearranged in one-dimensional buffers to maximize the profit from the GPU SIMD architecture. A GPU implementation of the explicit solver has been performed. The GPU implementation runs on a Tesla C1060 GPU and a Tesla K20
GPU. The CPU implementation has been also extended to run on multi-core CPU over OpenMP. The multi-CPU program runs on a hyper-threaded E5540 Intel Xeon processor.
Peak speedup of 66x has been achieved for the simulation of an acoustic pulse between GPU and single CPU implementation and peak speedup of 6x between the GPU and multiple CPUs implementation.

G.H. Hardy once said “[b]eauty is the first test: there is no place in the world for ugly mathematics.” If one is privileged to be literate enough to see it, one would, nevertheless, see it everywhere. Mathematics’ true beauty lies in its generosity. This text goes about that. Bidding against the incomprehensibility of the quantum domain, the simple grasp at uncovering the mathematical splendour finds itself needed to expand its horizon towards the legendary. Hermes and Hestia are figures improbable to be coupled other than antagonistically, yet here they are lovers. Their union saves us from the misfortune of an ugly mathematics.

Euler’s formula gives an exponential and a constant growth rate in the form of a  trigonometric function and the exponential function. It is essentially two waves that have interfered or combined. One wave is the real number e and the other is imaginary (i), together these two waves propagate on both sides of the complex plane due to this unique relationship.
I present that the imaginary wave is real. It is the other side of the infinite harmonic series. Thus, by finding this other side we arrive at a wave and frequency, a real number that replaces the imaginary wave or number used by Euler and mathematicians. I am suggesting that the imaginary number (i) is a real number that has not been discovered until now.

The public reporting burden for this collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or any other aspect of this collection of information, including suggestions for reducing this burden, to Department of Defense, Washington Headquarters Services, Directorate for Information Operations and Reports (0704-0188), 1215 Jefferson Davis Highway, Suite 1204, Arlington, VA 22202-4302. Respondents should be aware that notwithstanding any other provision of law, no person shall be subject to any penalty for failing to comply with a collection of information if it does not display a currently valid OMB control number. PLEASE DO NOT RETURN YOUR FORM TO THE ABOVE ADDRESS.

A new projection method is developed for the Euler equations to determine the thermodynamic state in computational cells. It consists in the resolution of a mechanical relaxation problem between the various sub-volumes present in a computational cell. These sub-volumes correspond to the ones traveled by the various waves that produce states with different pressures, velocities, densities and temperatures. Contrarily to Godunov type schemes the relaxed state corresponds to mechanical equilibrium only and remains out of thermal equilibrium. The pressure computation with this relaxation process replaces the use of the conventional equation of state (EOS). A simplified relaxation method is also derived and provides a specific EOS (named the Numerical EOS). The use of the Numerical EOS gives a cure to spurious pressure oscillations that appear at contact discontinuities for fluids governed by real gas EOS. It is then extended to the computation of interface problems separating fluids with different EOS (liquid-gas interface for example) with the Euler equations. The resulting method is very robust, accurate, oscillation free and conservative. For the sake of simplicity and efficiency the method is developed in a Lagrange-projection context and is validated over exact solutions. In a companion paper [F. Petitpas, E. Franquet, R. Saurel, A relaxation-projection method for compressible flows. Part II: computation of interfaces and multiphase mixtures with stiff mechanical relaxation. J. Comput. Phys. (submitted for publication)], the method is extended to the numerical approximation of a non-conservative hyperbolic multiphase flow model for interface computation and shock propagation into mixtures.

Neste trabalho foi apresentado o Método de Euler, que é uma resolução numérica para EDO's, através de uma aproximação da reta tangente à curva original. Este método é um truncamento da expansão da série de Taylor, que gera resultados mais aproximados, porém sua implementação é muito mais complexa. Para compreensão deste método, foram estudados dois casos que necessitam de uma resolução numérica. O primeiro caso foi a obtenção do Número de Euler, onde se teve apenas o Problema de Valor Inicial, e uma função que contém uma derivada desconhecida. Para o segundo caso, foi analisado um circuito RL em série. Neste caso, o maior interesse foi encontrar o valor da corrente elétrica, e em que momento ela se estabiliza. Neste estudo houve também a implementação de outros métodos para se comparar com o Método de Euler. Houve a implementação de uma melhoria para o método de Euler, descrevendo se tal método é efetivo ou não. Assim como a passagem do código para a linguagem Portugol. Ao final foram apresentados os resultados de todos os métodos, mostrando qual foi o mais eficiente, levando em consideração os dados obtidos e visualizados por tabelas e gráficos.

In several papers of Bouchut, Bourdarias, Perthame and Coquel, Le Floch ((13), (7)...), a general methodology has been developed to construct second order flnite volume schemes for hyperbolic systems of conservation laws satisfying the entire family of entropy inequalities. This approach is mainly based on the construction of an entropy diminishing projection. Unfor- tunately, the explicit computation of this projection

An optimization methodology which uses the conservative ®eld variables, is developed to solve a design optimization problem in¯uid dynamical distributed parameter systems. This approach which is completely based on the variational method, is employed to derive the costate partial dierential equations (pdes) and their transversality (boundary) conditions from the continuous pdes of the¯uid¯ow. The costate equations coupled with the¯ow ®eld equations are solved iteratively to get the functional derivative coecients. Then, these derivative coecients combined with the¯ow ®eld variables are used to ®nd the boundary shape which extremizes the performance index (functional).

The paper describes a mathematical model on the physical principles of the Pitot device for acquiring flows speed. The relation between speed and static pressures is derived from simple integration of the fluid dynamics equations for an inviscid flow. In this way, the treatment is general and is not based on the Bernoulli theorem on energy conservation, as usually done in scientific and technical literature. The application to Formula 1 cars is discussed.

As social network information keeps growing, there is always a need for better information visualization techniques for carrying out multifarious analysis, which is commonly called as social network analysis. Social networks have become a big platform for advertisements, where a company targets highly connected than to more isolated users. To target highly connected users and their activities, visualizations such as Euler diagrams, Treemap diagrams, graphs and their combinations were tested by various researchers. Though Euler diagrams combined with graph and Treemap diagrams with graph have provided some interesting results, the one question yet to be answered is the scalability. In this paper, we propose a novel visual technique, " Spherule diagrams with graph " , which addresses the scalability issue. The novel technique is then compared with the traditional Euler diagrams with graph using a twitter case study in an empirical form. Twenty-eight participants were exposed to eighteen diagrams (nine diagrams of each type of visualization) in a software which recorded accuracy and response time (i.e., performance measure). The results of 504 observations indicate that (a) there is no significant difference between the visualizations in terms of accuracy and (b) there is a significant difference between the visualizations in terms of response time. Also, users were asked to aggrandize between the two visualizations (i.e., preference measure), where 75% preferred Spherule diagrams with graph for its simplicity, comprehensiveness, navigation, alignment (layout), set ordering and data items ordering characteristics. We conclude that, Spherule diagrams with graph will be beneficial for researchers and practitioners in the information visualization community who are exploring social networks for business.

The report documents the modelling of a symmetrical, 4-pole, 1.5kW, 1725rpm motor with rated current at 6.1A and 230 line voltage. A dynamic transient analysis is conducted in this investigation by developing the machine equations such that a steady space model is formulated for a computer simulation The results of the model is compared to the steady state analysis model. The report concludes that the synchronous reference frame is the prime option to select for the control of an induction motor.

The relaxation-projection method developed in Saurel et al. [R. Saurel, E. Franquet, E. Daniel, O. Le Metayer, A relaxation-projection method for compressible flows. Part I: The numerical equation of state for the Euler equations, J. Comput. Phys. (2007) 822-845] is extended to the non-conservative hyperbolic multiphase flow model of Kapila et al. [A.K. Kapila, Menikoff, J.B. Bdzil, S.F. Son, D.S. Stewart, Two-phase modeling of deflagration to detonation transition in granular materials: reduced equations, Physics of Fluids 13(10) (2001) 3002-3024]. This model has the ability to treat multitemperatures mixtures evolving with a single pressure and velocity and is particularly interesting for the computation of interface problems with compressible materials as well as wave propagation in heterogeneous mixtures. The non-conservative character of this model poses however computational challenges in the presence of shocks. The first issue is related to the Riemann problem resolution that necessitates shock jump conditions. Thanks to the Rankine-Hugoniot relations proposed and validated in Saurel et al. [R. Saurel, O. Le Metayer, J. Massoni, S. Gavrilyuk, Shock jump conditions for multiphase mixtures with stiff mechanical relaxation, Shock Waves 16 -232] exact and approximate 2-shocks Riemann solvers are derived. However, the Riemann solver is only a part of a numerical scheme and non-conservative variables pose extra difficulties for the projection or cell average of the solution. It is shown that conventional Godunov schemes are unable to converge to the exact solution for strong multiphase shocks. This is due to the incorrect partition of the energies or entropies in the cell averaged mixture. To circumvent this difficulty a specific Lagrangian scheme is developed. The correct partition of the energies is achieved by using an artificial heat exchange in the shock layer. With the help of an asymptotic analysis this heat exchange takes a similar form as the 'pseudoviscosity' introduced by Von Neumann and Richtmyer [J. Von Neumann, R.D. Richtmyer, A method for the numerical calculation of hydrodynamic shocks, J. Appl. Phys. 21 (1950) 232-237]. The present Lagrangian numerical scheme thus combines Riemann solvers and artificial heat exchanges. An Eulerian variant is then obtained by using the relaxation-projection method developed earlier by the authors for the Euler equations. The method is validated against exact solutions based on the multiphase shock relations as well as exact solutions of the Euler equations in the context of interface problems. The method is able to solve interfaces separating pure fluids or heterogeneous mixtures with very large density ratio and with very strong shocks.

Let e^v = (1 + v/n)^n (p. 181) e^iv = (1 + iv/n)^n e^iv = cos v + isin v e^itheta = cos theta + isin theta e^ipi = cos pi + isin pi e^ipi = 1 + 0 (p. 77) e^ipi – 1 = 0 which implies that e^ipi = 1. How so, take the ipi th root of 1 : e = 1 [ unit circle ] (p. 127) III Conclusion This is written on Euler's Formula.

The present edition of the "Basics of the Componential Numbers Theory" developed by the author is based on the suggestions and modifications offered by the readers related to the previouse edition published in 2017 and third edition (in Russian) published in 2021. Particularly, the section devoted to the numerical system of calculation was extended and includes now an information on the Greeк, Slavic and Hebrew systems. The proof of the extended Euler`s Hypotheses related to the arbitrary exponents is presented in § 17.2. In § 6 (Part 2) the analytical methodology of finding of prime and composite integers was presented. He Part 3 is devoted to the application of the Componential Numbers Theory for the proof of Phermat`s Theorem. The paper is to show that there are still a lot of options to use the methodology of the initial numbers theory. The presented methods will give the answers to several questions and allow to solve some difficult tasks. In addition, the real solutions to the some complicated tasks are introduced in the paper. Author's work aims to show that the application of the methods of the simple primary numbers theory can still be developed and used. The methods, proposed in the book provide answers to many questions, allow to solve complex tasks by using of the elementary methods. Apart from the purely theoretical segment, the book provides examples of solutions of problems by use of introduced results.

Piezoelectric transduction has received great attention for vibration-to-electric energy conversion over the last five years. A typical piezoelectric energy harvester is a unimorph or a bimorph cantilever located on a vibrating host structure, to generate electrical energy from base excitations. Several authors have investigated modeling of cantilevered piezoelectric energy harvesters under base excitation. The existing mathematical modeling approaches range from elementary single-degree-of-freedom models to approximate distributed parameter solutions in the sense of Rayleigh-Ritz discretization as well as analytical solution attempts with certain simplifications. Recently, the authors have presented the closed-form analytical solution for a unimorph cantilever under base excitation based on the Euler-Bernoulli beam assumptions. In this paper, the analytical solution is applied to bimorph cantilever configurations with series and parallel connections of piezoceramic layers. The base excitation is assumed to be translation in the transverse direction with a superimposed small rotation. The closed-form steady state response expressions are obtained for harmonic excitations at arbitrary frequencies, which are then reduced to simple but accurate single-mode expressions for modal excitations. The electromechanical frequency response functions (FRFs) that relate the voltage output and vibration response to translational and rotational base accelerations are identified from the multi-mode and single-mode solutions. Experimental validation of the single-mode coupled voltage output and vibration response expressions is presented for a bimorph cantilever with a tip mass. It is observed that the closed-form single-mode FRFs obtained from the analytical solution can successfully predict the coupled system dynamics for a wide range of electrical load resistance. The performance of the bimorph device is analyzed extensively for the short circuit and open circuit resonance frequency excitations and the accuracy of the model is shown in all cases.

We study the weak boundary layer phenomenon of the Navier-Stokes equations with generalized Navier friction boundary conditions, u·n = 0, [S(u)n] tan +Au = 0, in a bounded domain in R 3 when the viscosity, ε > 0, is small. Here, A is a type (1, 1) tensor on the boundary: when A = αI we obtain Navier boundary conditions, and when A is the shape operator we obtain the conditions, u · n = (curl u) × n = 0. By constructing an explicit corrector, we prove the convergence, as ε tends to zero, of the Navier-Stokes solutions to the Euler solution. We do this both in the natural energy norm with a rate of order ε 3/4 as well as uniformly in time and space with a rate of order ε 3/8−δ near the boundary and ε 3/4−δ ′ in the interior, where δ, δ ′ decrease to 0 as the regularity of the initial velocity increases. This work simplifies an earlier work of Iftimie and Sueur, , as we use a simple and explicit corrector (which is more easily implemented in numerical applications). It also improves a result of Masmoudi and Rousset, , who obtain convergence uniformly in time and space via a method that does not yield a convergence rate.

In this study, an exact Eulerian solution is derived for the bidirectional vortex in a conical chamber. Our model is applicable to idealized representations of cyclone separators and liquid rocket engines with slowly expanding chamber cross-sections. The bulk fluid motion is assumed to be nonreactive, steady, rotational, inviscid, and incompressible. Our approach is founded on the Bragg-Hawthorne equation and seeks to overcome some of the deficiencies encountered by Bloor and Ingham (Bloor, M. I. G., and Ingham, D. B., “The Flow in Industrial Cyclones,” Journal of Fluid Mechanics, Vol. 178, 1987, pp. 507-519). Despite inevitable similarities with Bloor and Ingham’s model, ours is constructed using a judicious framework that connects the solution to the swirl number and the cone divergence angle. In consequence, a self-similar formulation is produced that is independent of the cone’s finite body length. closed-form representations for the principal variables This enables us able to c...

Cantilevered beams with piezoceramic layers have been frequently used as piezoelectric vibration energy harvesters in the past five years. The literature includes several single degree-of-freedom models, a few approximate distributed parameter models and even some incorrect approaches for predicting the electromechanical behavior of these harvesters. In this paper, we present the exact analytical solution of a cantilevered piezoelectric energy harvester with Euler–Bernoulli beam assumptions. The excitation of the harvester is assumed to be due to its base motion in the form of translation in the transverse direction with small rotation, and it is not restricted to be harmonic in time. The resulting expressions for the coupled mechanical response and the electrical outputs are then reduced for the particular case of harmonic behavior in time and closed-form exact expressions are obtained. Simple expressions for the coupled mechanical response, voltage, current, and power outputs are ...

This article presents numerical simulation of planar potential flow around an airfoil with possibility of changing its shape. Two-dimensional unsteady flow model with scalar velocity potential, which allows us to calculate pressure distribution along an airfoil from Cauchy-Lagrange integral, is used. For this purpose, an airfoil contour is approximated by a complex cubic spline with possibility of displacement its vertices. This algorithm has been used in the context of fluid-structure interaction and has been applied successfully to determination of stability of an elastic airfoil segment interacting with a flow stream, so-called panel flutter problem. Calculation of external flow is carried out by vortex panel method with Kutta-Joukowski trailing edge condition, which makes mathematical solution unique. Using this method of approximation of an airfoil in combination with the method of discrete vortices provides a semi-analytical solution for complex potential for whole computational domain of air flow. This solution significantly accelerates process of numerical computation of time-averaged aerodynamic force as well as the dynamic stability problem for aeroelastic wing design and temporal evolution of its natural disturbances.

It is well known that numerical methods designed to solve the compressible Euler equations, when written in terms of conservation variables behave poorly in the incompressible limit, that is, when density variations are negligible. However, a change to pressure based variables seem to, partly, eliminate the problem by making the Jacobian matrices fully invertible whatever the ow regime may be. Despite this apparent beneÿt, the stabilization matrix plus discontinuity capturing operator (for the compressible regime) still need attention, since they tend to be ill behaved for either conservation or pressure variables. In this paper, we introduce a simple way of balancing two stabilizing matrices, one of them suitable for low Mach ows and the other one for supersonic ows, so that a wide range of ow regimes are covered with only one formulation. Comparison between conservation and pressure variables is made and numerical examples are shown to validate the method.

This paper investigates monetary policy transmission at the microeconometric level. The credit rationing literature suggests that monetary policy has a larger effects on firms which are credit constrained. I use a large panel of German industrial firms to investigate whether this is empirically true in the German case. Whereas interest rates have only a weak effect on investment, I find strong evidence that monetary policy affects constrained firms more than others. Furthermore, investment is more sensitive to financial structure in periods of tight monetary policy. This contrasts earlier findings by St�� (1996). The same result applies for periods of monetary tightness. I conclude that the credit channel is dominant in Germany. Given the findings for other countries this result indicates that financial systems in the Euro area are less different than suggested in previous studies.

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