real congruences of complex subspaces of 2 2 symmetric complex matrices abstract consider complex subspaces of symmetric 2 2 complex matrices under real congruences sending m to ama t with a real and invertible the orbits for dimension one were classified in different language two decades ago those for dimension two were recently classified by coffman it turns out that these two problems are in fact dual to each other and coffman s classification yields a simpler expression for the orbits in dimension one

asymptotic behavior of solutions of second order differential equations with integrable coefficients the differential equation x a t f x 0 t 0 is considered under the condition that lim x fl a s ds exists and is finite and necessary and or sufficient conditions are given for this equation to have solutions which behave asymptotically like nontrivial linear functions cl c2t

multigrid solution of the two dimensional euler equations on unstructured triangular meshes a multigrid algorithm has been developed to accelerate the convergence of the euler equations to a steady state on unstructured triangular meshes the method operates on a sequence of coarse and fine unstructured meshes and assumes no relation exists between the various meshes a tree search algorithm is used to efficiently identify regions of overlap between coarse and fine grid cells the multigrid method is applied to two different discretizations of the euler equations and results about multi element airfoils are presented for both discretization schemes convergence is accelerated by an order of magnitude resulting in convergence rates comparable to those obtained with structured multigrid algorithms

diagonal implicit multigrid algorithm for the euler equations a multigrid implementation of the alternating direction implicit algorithm has been developed to solve the euler equations of inviscid compressible flow the equations are approximated using a finite volume spatial approximation with added dissipation provided by an adaptive blend of second and fourth differences for computational efficiency the equations are diagonalized by a local similariity transformation so that only a decoupled system of scalar pentadiagonal systems need be solved along each line results are computed for transonic flows past airfoils and include pressure distributions to verify the accuracy of the basic scheme and convergence histories to demonstrate the efficiency of the method

compact difference scheme applied to simulation of low sweep delta wing flow computational simulations are presented of the flow over a 50 deg sweep delta wing for angles of attack a 5 10 and 15 deg the full navier stokes equations are solved using a sixth order compact differencing scheme coupled with an eighth order low pass spatial filter to provide numerical stability

cartesian cut cell method for axisymmetric separating body flows a method for the calculation of unsteady axisymmetric compressible flows involving both fixed and separating bodies is presented the method uses a cartesian cut cell mesh approach and high resolution upwind finite volume scheme a stationary background cartesian mesh is generated on the computational domain with complex solid geometries represented by different types of cut cell solid bodies are allowed to move across the mesh using a finite volume scheme modified to deal with moving boundary problems the flow solver employed is a muscl hancock godunov type scheme in conjunction with an approximate riemann solver of the harten lax and van leer type for flow interfaces and an exact riemann solver for a moving piston for fixed or moving solid faces a cell merging technique is used to maintain numerical stability in the presence of arbitrarily small cut cells and to retain strict conservation at moving boundaries the method is applied to a muzzle blast flow and muzzle break problems involving both fixed and separating bodies

homoclinic orbits and subharmonics for nonlinear second order difference equations abstract we show the existence of a nontrivial homoclinic orbit and subharmonic solutions for a class of second order difference equations by applying the mountain pass theorem relying on ekeland s variational principle and the diagonal method and the homoclinic orbit as the limit of the subharmonics a completely new way is provided for dealing with the existence of solutions for difference equations

integration of navier stokes equations using dual time stepping and a multigrid method efficient acceleration techniques typical of explicit steady state solvers are extended to tune accurate calcu lations stability restrictions are greatly reduced by means of a fully implicit time discretization a four stage runge kutta scheme with local tune stepping residual smoothing and multigridding is used instead of traditional computationally expensive factorizations two applications to natural unsteady viscous flows are presented to check for the capability of the procedure ecent progress in computational fluid dynamics along with the evolution of computer performance encourages scientists to look in to details of flow physics there are practical applications where the unsteadiness of the problem can not be neglected i e vortex shedding natural unsteadiness forced unsteadiness aeroe lasticity turbomachinery rotor stator interaction up to now most of the analysis and designing tools are based on a steady or quasi steady assumption even if the flow is known to be unsteady today due to the improvement in computer resources there is a strong interest in developing methodologies for efficient and reliable sim ulation of unsteady flow features it is a common experience while using time accurate explicit schemes to be forced to choose the time step on the basis of stability restrictions as a consequence unless the problem is a very high frequency one the number of time steps to be performed is much higher than the one required for time accuracy by means of some implicit factorization stability restrictions can be relaxed but the work required at each time step grows rapidly with grid dimension and complexity of the flow equations in addition application of boundary conditions in a fully implicit manner is difficult in viscous flow calculations the grid is clustered close to the shear layer and the characteristic time step varies several orders of mag nitude inside the computational domain even if in several practical applications the characteristic time step of the core flow region is comparable with the one required by time accuracy close to the walls the time step restrictions become extremely severe there fore highly vectorizable schemes with less stability restrictions on the allowable time step would be an interesting combination explicit schemes combined with acceleration techniques have proven to be very effective for solving steady problems 1 3 unfortu nately the computational efficiency of those time marching solvers is achieved by sacrificing the accuracy in time in this paper it is shown that the conventional steady state acceleration techniques specifically the multigrid techniques can still be applied to unsteady viscous problems the basic idea is to introduce a dual time stepping and to reformulate the governing equations so that they can be han dled by an explicit accelerated scheme 4 if the time discretization is made implicit stability restrictions are removed and accelera tion techniques can be used instead of traditional time consuming factorizations i e alternate directional implicit and lower upper

the modelling of aerodynamic flows by solution of the euler equations on mixed polyhedral grids in this paper an algorithm for obtaining solutions to the compressible euler equations on mixed polyhedral grids is described these grids which are in general composed of hexahedral pentahedral and tetrahedral elements are used in the modelling of aerodynamic flows over complex three dimensional geometries the hexahedra are grouped into regular blocks while the other elements are part of irregular grid regions giving rise to the term hybrid grid approach the interconnection of the grid elements is defined by an efficient data structure which supplies the required information to drive the euler algorithm in fact cell vertex and cell centre spatial discretization variants of this algorithm are detailed along with an explicit time marching scheme for obtaining steady state solutions results on two geometrically simple configurations provide an evaluation of these solution techniques whilst the focus on a third complex configuration demonstrates the potential of the method in addressing the flow over general aircraft geometries

interfacing of fluid and structural models via innovative structural boundary element method an innovative structural boundary element method bem solver is developed for interfacing the computational fluid dynamics cfd and computational structural dynamics csd grids formulated as a solid mechanics problem with a minimum strain energy requirement the bem solver generates a truly three dimensional universal spline matrix the universal spline matrix is a vector operator that includes the coupling of displacements and forces along all axes based on a similar formulation an exterior bem solver is also developed to account for the flowfield grid deformation thus the bem solver allows a unified treatment of the displacement and force transformation for cfd csd interfacing as well as the computation of the flowfield grid deformation the solution procedure is fully automated and no additional model generation is required therefore is ideally suitable for computational aeroelasticity and multidisciplinary optimization applications

real congruence of complex matrix pencils and complex projections of real veronese varieties quadratically parametrized maps from a real projective space to a complex projective space are constructed as projections of the veronese embedding a classification theorem relates equivalence classes of projections to real congruence classes of complex symmetric matrix pencils the images of some low dimensional cases include certain quartic curves in the riemann sphere models of the real projective plane in complex projective 4 space and some normal form varieties for real submanifolds of complex space with cr singularities

a parallel multiselection greedy method for the radial basis function based mesh deformation summary greedy algorithm has been widely adopted for the point selection procedure of radial basis function based mesh deformation however in large deformation simulations with thousands of points selected the greedy point selection will be too expensive and thus become a performance bottleneck to improve the efficiency of the point selection procedure a parallel multi selection greedy method has been developed in this paper multiple points are selected at each step to accelerate the convergence speed of the greedy algorithm in addition two strategies are presented to determine the specific selecting number the parallelization of the greedy point selection is realized based on a master slave model and a hybrid decomposition algorithm is proposed to address the load imbalance problem numerical benchmarks show that both our multi selection method and the parallelization could obviously improve the point selection efficiency specifically total speedups of 20 and 55 are separately obtained for the 3d undulating fish with 106 cell mesh and the 3d rotating hydrofoil with 11 million cell mesh this article is protected by copyright all rights reserved

explicit navier stokes computation of cascade flows using the k epsilon turbulence model a fully explicit two dimensional flow solver based on a four stage runge kutta scheme has been developed and used to predict two dimensional viscous flow through turbomachinery cascades for which experimental data are available the formulation is applied to the density weighted time averaged navier stokes equations several features of the technique improve the ability of the code to predict high reynolds number flows on highly stretched grids these include a low reynolds number compressible form of the a e turbulence model anisotropic scaling of artificial dissipation terms and locally varying timestep evaluation based on hyperbolic and parabolic stability considerations comparisons between computation and experiment are presented for both a supersonic and a low subsonic compressor cascade these results indicate that the code is capable of predicting steady two dimensional viscous cascade flows over a wide range of mach numbers in reasonable computation times

high quality mesh deformation using quaternions for orthogonality preservation many mesh deformation techniques developed in the past have been widely used but for large deformations the mesh properties can be severely altered in particular the mesh lines orthogonality near the boundaries can be lost in this paper a method initially proposed by samareh application of quaternions for mesh deformation nasa tm 2002 211646 april 2002 is developed to overcome this limitation and to provide a high quality mesh orthogonality preservation technique each node displacement is composed of a translation vector and a quaternion which takes into account local mesh rotations algebraic unitary quaternion properties lie group algebra are used to obtain a robust interpolation method two quaternion interpolation methods are compared respectively the spherical linear interpolation and the lie algebra linear interpolation furthermore thanks to a new process transformation division the method becomes independent of the rotation center used the accuracy and generality of the met

linearized euler predictions of unsteady aerodynamic loads in cascades a linearized euler solver for calculating unsteady flows in turbomachinery blade rows due to both incident gusts and vibratory blade motion is presented using the linearized euler technique one decomposes the flow into a mean or steady flow plus an unsteady harmonically varying small disturbance flow linear variable coefficient equations describe the small disturbance behavior of the flow and are solved using a pseudotime time marching lax wendroff scheme for the blade motion problem a harmonically deforming computational grid that conforms to the motion of vibrating blades eliminates large error producing mean flow gradient terms that would otherwise appear in the unsteady flow tangency boundary condition

diagonally inverted lower upper factored implicit multigrid scheme for the three dimensional navier stokes equations an approximately factored implicit multigrid algorithm is developed to calculate three dimensional compressible viscous flows this scheme solves the full reynolds averaged navier stokes equations with a k e model of turbulence the flow equations are solved within the framework of the multigrid method using a four grid level w cycle while the k e equations are solved only on the finest grid and uncoupled from the flow calculations

asymptotic to polynomials solutions for nonlinear differential equations abstract this article is concerned with solutions that behave asymptotically like polynomials for nth order n 1 nonlinear ordinary differential equations for each given integer m with 1 m n 1 sufficient conditions are presented in order that for any real polynomial of degree at most m there exists a solution which is asymptotic at to this polynomial conditions are also given which are sufficient for every solution to be asymptotic at to a real polynomial of degree at most n 1 the application of the results obtained to the special case of second order nonlinear differential equations leads to improved versions of the ones contained in the recent paper by lipovan glasg math j 45 2003 179 and of other related results existing in the literature

improved point selection method for hybrid unstructured mesh deformation using radial basis functions in this paper an efficient mesh deformation technique for hybrid unstructured grids based on radial basis functions has been developed the principle procedure adopted for this scheme can be divided into two steps first a series of radial basis functions are constructed by an interpolation method according to the displacements of moving boundary points later the displacements of all points in the computation domain are determined by the aforementioned radial basis function series to improve the efficiency data reduction should be introduced into the interpolation process consequently a multilevel subspace radial basis function interpolation method based on a double edged greedy algorithm is presented to create an approximate interpolation for all moving boundary points this method is computationally efficient preserves orthogonality and has no dependency on the type of flow solver typical deformation problems of a lann wing dlr f6 wing body nacelle pylon configuration and dlr f11 high li

efficient mesh motion using radial basis functions with data reduction algorithms mesh motion using radial basis functions has been demonstrated previously by the authors to produce high quality meshes suitable for use within unsteady and aeroelastic cfd codes in the aeroelastic case the structural mesh may be used as the set of control points governing the deformation which is efficient since the structural mesh is usually small however as a stand alone mesh motion tool where the surface mesh points control the motion radial basis functions may be restricted by the size of the surface mesh as an update of a single volume point depends on all surface points in this paper a method is presented that allows an arbitrary deformation to be represented to within a desired tolerance by using a significantly reduced set of surface points intelligently identified in a fashion that minimises the error in the interpolated surface this method may be used on much larger cases and is successfully demonstrated here for a 10 6 cell mesh where the initial solve phase cost reduces by a factor of eight with the new scheme and the mesh update by a factor of 55 it has also been shown that the number of surface points required to represent the surface is only geometry dependent i e grid size independent and so this reduction factor actually increases for larger meshes

unsteady transonic two dimensional euler solutions using finite elements a finite element solution of the unsteady euler equations is presented and demonstrated for two dimensional airfoil configurations oscillating in transonic flows computations are performed by spatially discretizing the conservation equations using the galerkin weighted residual method and then employing a multistage rungekutta scheme to march forward in time triangular finite elements are employed in an unstructured o mesh computational grid surrounding the airfoil grid points are fixed in space at the far field boundary and are constrained to move with the airfoil surface to form the near field boundary a mesh deformation scheme has been developed to efficiently move interior points in a smooth fashion as the airfoil undergoes rigid body pitch and plunge motion both steady and unsteady results are presented and a comparison is made with solutions obtained using finite volume techniques the effects of using either a lumped or consistent mass matrix were studied and are presented results show the finite element method provides an accurate solution for unsteady transonic flows about isolated airfoils he accurate determination of the unsteady flowfield surrounding an oscillating airfoil or wing in transonic flow is of paramount importance in the prediction of the flutter characteristics of the body transonic flutter remains an active research topic because of the continued interest by both the military and commercial sectors to operate in these flight regimes with vehicles that are ever lighter and as a result more flexible typically analysis of fluid structure interaction in flight vehicles involves describing the vehicle structure using finite elements whereas finite difference methods are used to model the surrounding fluid coupling of the two dissimilar models then presents problems because of the disparity between the two solution techniques additional complexities arise because of phase mismatching between the fluid pressures and the displacements of the structure it seems a natural progression to use finite element methods to describe the behavior of both the fluid and structure thus bringing more commonality into the analysis of the two media early research efforts concentrated on using the transonic small disturbance equation or the nonlinear full potential equation as the model for the gas dynamic behavior in transonic flows unfortunately this idealization leads to errors when strong shocks are present because of their inability to account for the production of entropy and vorticity to properly account for these effects within the framework of the conservation laws the euler equations must be used these equations although they neglect fluid viscosity have become the standard when unsteady transonic solutions are desired the absence of a viscosity representation is an important limitation of the present formulation and must be considered

euler solution using cartesian grid with a gridless least squares boundary treatment an approach that uses gridless or meshless methods to address the problem of boundary implementation associated with the use of cartesian grid is discussed this method applies the gridless concept only at the interface whereas a standard structured grid method is used everywhere else the cartesian grid is used to specify and distribute the computational points on the boundary surface but not to define the geometrical properties euler fluxes for the neighbors of cut cells are computed using the gridless method involving a local least squares curve fit of a cloud of grid points the boundary conditions implemented on the surface points are automatically satisfied in the process of evaluating the surface values in a similar least squares fashion the present method does not require the use of halo points subsonic transonic and supersonic flows are computed for the naca 0012 and rae 2822 airfoils and the results compare well with solutions obtained by a standard euler solver on body fitted grids the method is also used to calculate the flow over a three element airfoil configuration and the result is compared with the exact solution for this configuration obtained by conformal mapping

time efficient aeroelastic simulations based on radial basis functions aeroelasticity studies the interaction between aerodynamic forces and structural responses and is one of the fundamental problems to be considered in the design of modern aircraft the fluid structure interpolation fsi and mesh deformation are two key issues in the cfd csd coupling approach the partitioned approach which is the mainstream numerical strategy in aeroelastic simulations in this paper a time efficient coupling scheme is developed based on the radial basis function interpolations during the fsi process the positive definite system of linear equations is constructed with the introduction of pseudo structural forces the acting forces on the structural nodes can be calculated more efficiently via the solution of the linear system avoiding the costly computations of the aerodynamic structural coupling matrix the multi layer sequential mesh motion algorithm msm is proposed to improve the efficiency of the volume mesh deformations which is adequate for large scale time dependent applications with frequent mesh updates two dimensional mesh motion cases show that the msm algorithm can reduce the computing cost significantly compared to the standard rbf based method the computations of the agard 445 6 wing flutter and the static deflections of the three dimensional high aspect ratio aircraft demonstrate that the developed coupling scheme is applicable to both dynamic and static aeroelastic problems

artificial dissipation models for the euler equations various artificial dissipation models that are used with central difference algorithms for the euler equations are analyzed for their effect on accuracy stability and convergence rates in particular linear and nonlinear models are investigated using an implicit approximate factorization code arc2d for transonic airfoils fully implicit application of the dissipation models is shown to improve robustness and convergence rates the treatment of dissipation models at boundaries will be examined it will be shown that accurate error free solutions with sharp shocks can be obtained using a central difference algorithm coupled with an appropriate nonlinear artificial dissipation model i introduction t he solution of the euler equations using numerical techniques requires the use of either a differencing method with inherent dissipation or the addition of dissipation terms to a nondissipative scheme this is because the euler equations do not provide any natural dissipation mechanism such as viscosity in the navier stokes equations that would eliminate high frequencies which are caused by nonlinearitie s and especially shocks a variety of numerical algorithms and computer codes for the euler equations have been developed methods such as maccormack s1 explicit

oscillation of solutions of second order nonlinear differential equations of euler type we consider the nonlinear euler differential equation t 2 x g x 0 here g x satisfies xg x 0 for x 0 but is not assumed to be sublinear or superlinear we present implicit necessary and sufficient condition for all nontrivial solutions of this system to be oscillatory or nonoscillatory also we prove that solutions of this system are all oscillatory or all nonoscillatory and cannot be both we derive explicit conditions and improve the results presented in the previous literature we extend our results to the extended

pencils of complex and real symmetric and skew matrices abstract this expository paper establishes the canonical forms under congruence for pairs of complex or real symmetric or skew matrices the treatment is in the spirit of the well known book of gantmacher on matrix theory and may be regarded as a supplement to gantmacher s chapters on pencils of matrices

arbitrary lagrangian eulerian unstructured finite volume lattice boltzmann method for computing two dimensional compressible inviscid flows over moving bodies the objective of this study is to develop and apply an arbitrary lagrangian eulerian unstructured finite volume lattice boltzmann method ale fvlbm for solving two dimensional compressible inviscid flows around moving bodies the two dimensional compressible form of the lb equation is considered and the resulting lb equation is formulated in the ale framework on an unstructured body fitted mesh to correctly model the body shape and properly incorporate the mesh movement due to the body motion the spatial discretization of the resulting system of equations is performed by a second order cell centered finite volume method on arbitrary quadrilateral meshes and an implicit dual time stepping method is utilized for the time integration to stabilize the numerical solution appropriate numerical dissipation terms are added to the formulation at first the shock tube problem is computed to examine the accuracy of the solution obtained by applying the proposed fvlbm for this unsteady test case which includes shock expansion wave and contact discontinuity in the flow domain then the stationary isentropic vortex is simulated on both the stationary and moving meshes to assess the implementation of the geometric conservation law in enhancing the solution accuracy of the ale fvlbm the compressible inviscid flow in the transonic regime is then computed around the stationary naca0012 airfoil in order to further study the sensitivity of the solution method to the user defined parameters now the transonic inviscid flow is simulated over the pitching or plunging naca0012 airfoil to investigate the accuracy and capability of the proposed solution method ale fvlbm for the computation of the compressible flows over moving bodies finally the pitching or plunging naca0012 airfoil near the ground in the transonic inviscid flow is simulated as a practical and challenging problem to study the ground effect on the aerodynamic characteristics of the airfoil it is indicated that the solution methodology proposed based on the finite volume lbm formulated in the arbitrary lagrangian eulerian framework ale fvlbm is capable of accurately computing the compressible inviscid flows around the moving bodies with and without the ground effect

parallel efficient mesh motion using radial basis functions with application to multi bladed rotors radial basis functions are used to provide a solution to the problem of mesh motion for unsteady aerodynamic simulation the method is independent of connectivity and produces high quality meshes but is expensive for large meshes in its full form hence the efficiency of the technique has been greatly improved here by reducing the number of surface points used to define deformations of the surface and the minor error in position that this implies at other surface points is corrected with a simple decaying perturbation thus splitting the method into a primary basis function method and a secondary local correction method this means that the exact surface is retained but the mesh motion is significantly faster while splitting the motion into two stages allows both the methods to work on appropriate problems given their relative strengths an example deformation for a 5 x10 6 cell helicopter rotor mesh with an exaggerated cyclic pitch motion shows excellent mesh quality thus validating a scheme that is also simple robust and readily parallelized

an application of integral inequality to second order nonlinear oscillation abstract a simple result concerning integral inequalities enables us to give an alternative proof of waltman s theorem lim t t 0 a s ds implies oscillation of the second order nonlinear equation y t a t f y t 0 to prove an analog of wintner s theorem that relates the nonoscillation of the second order nonlinear equations to the existence of solutions of some integral equations assuming that lim t t 0 a s ds exists and to give an alternative proof and to extend a result of butler an often used condition on the coefficient a t is given a more familiar equivalent form and an oscillation criterion involving this condition is established

adaptive radial basis function mesh deformation using data reduction radial basis function rbf mesh deformation is one of the most robust mesh deformation methods available using the greedy data reduction method in combination with an explicit boundary correction results in an efficient method as shown in literature however to ensure the method remains robust two issues are addressed 1 how to ensure that the set of control points remains an accurate representation of the geometry in time and 2 how to use automate the explicit boundary correction while ensuring a high mesh quality in this paper we propose an adaptive rbf mesh deformation method which ensures the set of control points always represents the geometry displacement up to a certain user specified criteria by keeping track of the boundary error throughout the simulation and re selecting when needed opposed to the unit displacement and prescribed displacement selection methods the adaptive method is more robust user independent and efficient for the cases considered secondly the analysis of a single high aspect ratio cell is used to formulate an equation for the correction radius needed depending on the characteristics of the correction function used maximum aspect ratio minimum first cell height and boundary error based on the analysis two new radial basis correction functions are derived and proposed this proposed automated procedure is verified while varying the correction function reynolds number and thus first cell height and aspect ratio and boundary error finally the parallel efficiency is studied for the two adaptive methods unit displacement and prescribed displacement for both the cpu as well as the memory formulation with a 2d oscillating and translating airfoil with oscillating flap a 3d flexible locally deforming tube and deforming wind turbine blade generally the memory formulation requires less work due to the large amount of work required for evaluating rbf s but the parallel efficiency reduces due to the limited bandwidth available between cpu and memory in terms of parallel efficiency scaling the different studied methods perform similarly with the greedy algorithm being the bottleneck in terms of absolute computational work the adaptive methods are better for the cases studied due to their more efficient selection of the control points by automating most of the rbf mesh deformation a robust efficient and almost user independent mesh deformation method is presented

an adjoint view on flux consistency and strong wall boundary conditions to the navier stokes equations inconsistent discrete expressions in the boundary treatment of navier stokes solvers and in the definition of force objective functionals can lead to discrete adjoint boundary treatments that are not a valid representation of the boundary conditions to the corresponding adjoint partial differential equations the underlying problem is studied for an elementary 1d advection diffusion problem first using a node centred finite volume discretisation the defect of the boundary operators in the inconsistently defined discrete adjoint problem leads to oscillations and becomes evident with the additional insight of the continuous adjoint approach a homogenisation of the discretisations for the primal boundary treatment and the force objective functional yields second order functional accuracy and eliminates the defect in the discrete adjoint boundary treatment subsequently the issue is studied for aerodynamic reynolds averaged navier stokes problems in conjunction with a standard finite volume discretisation on median dual grids and a strong implementation of noslip walls found in many unstructured general purpose flow solvers going out from a base line discretisation of force objective functionals which is independent of the boundary treatment in the flow solver two improved flux consistent schemes are presented based on either body wall defined or farfield defined control volumes they resolve the dual inconsistency the behaviour of the schemes is investigated on a sequence of grids in 2d and 3d

eigenmode analysis of unsteady flows about airfoils we present a reduced order modelling technique for analyzing the unsteady subsonic aerodynamic flow about isolated airfoils to start we model the flow using the time linearized full potential equation the linearized potential equation is discretized on a computational mesh composed of quadrilateral elements using a variational finite element technique the resulting discretized equations are linear in the unknown potential but quadratic in the reduced frequency of vibration we compute the dominant low frequency eigenfrequencies and mode shapes of the unsteady fluid motion using a nonsymmetric lanczos algorithm and then we use these eigenmodes to construct a low degree of freedom reduced order model of the unsteady flow field a static correction technique is used to account for the high frequency eigenmodes not retained in the model we show that the unsteady flow can be modelled accurately using a relatively small number of eigenmodes

implicit method for the computation of unsteady flows on unstructured grids an implicit method for the computation of unsteady flows on unstructured grids is presented following a finite difference approximation for the time derivative the resulting nonlinear system of equations is solved at each time step by using an agglomeration multigrid procedure the method allows for arbitrarily large time steps and is efficient in terms of computational effort and storage inviscid and viscous unsteady flows are computed to validate the procedure the issue of the mass matrix which arises with vertex centered finite volume schemes is addressed the present formulation allows the mass matrix to be inverted indirectly a mesh point movement and reconnection procedure is described that allows the grids to evolve with the motion of bodies as an example of flow over bodies in relative motion flow over a multi element airfoil system undergoing deployment is computed

efficient mesh deformation based on cartesian background mesh abstract moving mesh is widely used in the simulation of aerodynamic shape optimization multibody relative motion aircraft icing and aeroelasticity the efficient and high quality mesh deformation is the key technology of moving mesh this paper presented a new mesh deformation method based on cartesian background mesh mdcbm first the cartesian background mesh is deformed with radial basis functions rbf second the displacement of cartesian background mesh is algebraically interpolated onto all meshes in the computing domain since the background mesh is coarse the background mesh deformation can be finished fast because the background mesh of mdcbm is regular the mapping relationship between background mesh and the computing mesh is simple so the time spent on mapping search is substantially reduced the examples including naca0012 airfoil multi element airfoil with structured unstructured mesh and dlr f4 wing body show the good performance of mdcbm we highlight the advantages of mdcbm with respect to its computational efficiency and high quality of deformed mesh

existence and nonoscillation theorems for an emden fowler equation with deviating argument on donne des conditions suffisantes pour qu une equation d emden fowler generalisee a argument devie possede des solutions non oscillatoires a comportement asymptotique prescrit pour t

isolated singularities of the solutions of the schrodinger equation with a radial potential we study the behavior near an isolated singularity say x 0 of solutions of the equation 1 u x v x u x defined in a neighborhood of 0 in terms of the potential v which we assume to be radially symmetric for a class of potentials p1 that we characterize completely there is a unique singularity with constant sign behaving like the fundamental solution of the laplace operator for a larger class p2 we obtain a unique type of nonnegative singularity that is radially symmetric if v r cr with c 0 e r the class p1 is characterized by 2 and the class p2 by 2 the results fail for 2

influence of mesh sensitivities on computational fluid dynamics based derivative calculation to clarify the influence of mesh sensitivities on the computational fluid dynamics based derivatives a theoretical deduction and treatment of the mesh sensitivities are carried out yielding an explicit mathematical formulation for issues of concern numerical tests focusing on aerodynamic shape derivatives within the context of a transonic viscous flow passing through an rae2822 airfoil are conducted to validate the theoretical analysis the automatic differentiation the adjoint and tangent modes as well as the finite difference are applied to resolve the gradient information various mesh deformation strategies including the finite element analog the spring analog the original radial basis function interpolation and a new modified radial basis function method are applied to provide different mesh sensitivity solutions the numerical results confirm that mesh sensitivities are essential for the geometry related numerical derivatives and the reliance is independent of the perturbation magnitu

oscillation of solutions of second order nonlinear self adjoint differential equations abstract we are concerned with the oscillation problem for the nonlinear self adjoint differential equation a t x b t g x 0 here g x satisfied the signum condition xg x 0 if x 0 but is not imposed such monotonicity as superlinear or sublinear we show that certain growth conditions on g x play an essential role in a decision whether all nontrivial solutions are oscillatory or not our main theorems extend recent results in a serious of papers and are best possible for the oscillation of solutions in a sense to accomplish our results we use sturm s comparison method and phase plane analysis of systems of lienard type we also explain an analogy between our results and an oscillation criterion of kneser hille type for linear differential equations

delaunay graph and radial basis function for fast quality mesh deformation a novel mesh deformation technique is developed based on the delaunay graph mapping dgm method and the radial basis function rbf method the algorithm combines the advantages of the efficiency of dgm mesh deformation and the better control of the near surface mesh quality from the rbf method the delaunay graph is used to divide the mesh domain into a number of sub domains on each of the sub domains the radial basis function is applied to build a much smaller sized translation matrix between the original mesh and the deformed mesh resulting in a similar efficiency for the mesh deformation as compared to the fast delaunay graph mapping method furthermore by separating the translation and rotation motion the mesh quality near the wall can be substantially improved for extremely large rotational deformation the paper will show how the near wall mesh quality is controlled and improved by the new method while the computational time is maintained to be comparable to the original delaunay graph mapping method

upwind schemes and boundary conditions with applications to euler equations in general geometries abstract a unified treatment of several upwind shock capturing algorithms is presented each algorithm has a riemann initial value problem as its basis the treatment of boundaries involves solving the associated riemann initial boundary value problem the first author s algorithm applied to multidimensional euler equations in general geometries is then presented its worth is verified by various calculations which include mach 8 supersonic flow past a circular cylinder

nonconsistent mesh movement and sensitivity calculation on adjoint aerodynamic optimization this paper presents an investigation of the influence of a nonconsistent approach in terms of mesh movement and mesh sensitivity calculation in a discrete adjoint based optimization some mesh movement methods are more robust or of higher quality whereas others can be more efficient for calculating mesh sensitivity it is found that a nonconsistent approach gives comparable results when compared to a consistent approach therefore an appropriate combination of nonconsistent approaches can be achieved for efficient adjoint optimization this paper investigates and compares various consistent and nonconsistent combinations by using linear elasticity delaunay graph mapping and radial basis function mesh movement methods an investigation is presented using a lift constrained drag minimization to assess which step of the chain introduces a deviation if any and to which degree this affects the final result

coupling dynamic mesh technique and heuristic algorithms in 3 d tapered wing design in this study onera m6 wing has been optimized by two parameters the wing section and the taper ratio by combining the recent mostly preferable popular approaches i e parallel computing and evolutionary techniques for the 3 d models developed during the optimization stages the mesh required has been generated by using dynamic mesh technique the code developed for this is robust and faster than the codes that produce mesh only by classical techniques an euler flow solver acer3d is used to obtain the flow parameters for each member from the results it is observed that the optimization process is working as expected during the optimization process the lift coefficient and the thickness ratio are tried to be maintained close to the design values determined at the beginning the taper ratio becomes smaller and converges to a certain value while the code tries to minimize the drag force additionally this study can be used as a reference for 2 d or 3 d aerodynamic body optimization by using heuristic type algorithms since all details are outlined referenced and interpreted with their advantages and disadvantages copyright 2007 john wiley sons ltd

mesh deformation using the biharmonic operator the use of the biharmonic operator for deforming a mesh in an arbitrary lagrangian eulerian simulation is investigated the biharmonic operator has the advantage that two conditions can be specified on each boundary of the mesh this allows both the position and the normal mesh spacing along a boundary to be controlled which is important for two fluid interfaces and periodic boundaries at these boundaries we can simultaneously fix the position of the boundary and ensure that the normal mesh spacing is continuous across the boundary in addition results for deforming surfaces show that greater surface deformation can be tolerated when using biharmonic equations compared to approaches using second order partial differential equations a final advantage is that with the biharmonic operator the integrity of a grid in a moving boundary layer can be preserved as the boundary moves the main disadvantage of the approach is its increased computational expense

asymptotic behavior of solutions of second order quasilinear differential equations with delay depending on the unknown function the asymptotic behavior of the nonoscillatory solutions of quasilinear differential equations of second order with delay depending on the unknown function is considered the main results given by bainov et al j comput appl math 91 1998 87 96 and wong funkcial ekvac 11 1968 207 234 are improved and generalized

efficient mesh motion using radial basis functions with volume grid points reduction algorithm abstract as one of the most robust mesh deformation technique available the radial basis function rbf mesh deformation has been accepted widely however for volume mesh deformation driven by surface motion the rbf system may become impractical for large meshes due to the large number of both surface control points and volume points surface points selection procedure based on the greedy algorithm results in an efficient implementation of the rbf based mesh deformation procedure the greedy algorithm could reduce the number of surface points involved in the rbf interpolation while acquire an acceptable accuracy as shown in literature to improve the efficiency of the rbf method furthermore an issue that how to reduce the number of the volume points needed to be moved is addressed in this paper we propose an algorithm for volume points reduction based on a wall distance based restricting function which is added to the formulation of the rbf based interpolation this restricting function is firstly introduced by the current article to support large deformation a multi level subspace interpolation is essentially needed although this technique was used to improve the efficiency of the surface points selection procedure in the existed literature the key point of this technique is setting the error of previous interpolation step as the object of current step and restricting interpolation region gradually because the tolerance of the error is decreased hierarchically the number of the surface points is increased but the number of the volume points needed to be moved is reduced gradually therefore the cpu cost of updating the mesh motion could be reduced eventually since it scales with the product of these two numbers the computational requirement of the proposed procedure is reduced evidently compared with the standard procedure as proved by some examples

fast dynamic grid deformation based on delaunay graph mapping a simple and efficient dynamic grid deformation technique is proposed for computing unsteady flow problems with geometrical deformation relative body movement or shape changes due to aerodynamic optimisation and fluid structure interaction a delaunay graph of the solution domain is first generated which can be moved easily during the geometric dynamic deformation even for very large distortion a one to one mapping between the delaunay graph and the computational grid is maintained during the movement therefore the new computational grid after the dynamic movement can be generated efficiently through the mapping while maintaining the primary qualities of the grid while most dynamic grid deformation techniques are iterative based on the spring analogy the present method is non iterative and much more efficient on the other hand in comparison with dynamic grid techniques based on transfinite interpolation for structured grids it offers both geometric and cell topology flexibility which is crucial for many unsteady flow problems involving geometric deformation and relative motions the method is demonstrated through some typical unsteady flow test cases including a pitching aerofoil in a fixed domain boundary relative movement between multi element aerofoils due to flap deployment for high lift a deformable sphere in a fixed cube and a three dimensional flexible wing with large deformation

eigenanalysis of unsteady flow about airfoils cascades and wings a general technique for constructing reduced order models of unsteady aerodynamic flows about two dimensional isolated airfoils cascades of airfoils and three dimensional wings is developed the starting point is a the domain computational model of the unsteady small disturbance flow for illustration purposes we apply the technique to an unsteady incompressible vortex lattice model the eigenmodes of the system which may be thought of as aerodynamic states are computed and subsequently used to construct computationally efficient reduced order models of the unsteady flowfield only a handful of the most dominant eigenmodes are retained in the reduced order model the effect of the remaining eigenmodes is included approximately using a static correction technique an important advantage of the present method is that once the eigenmode information has been computed reduced order models can be constructed for any number of arbitrary modes of airfold motion very inexpensively numerical examples are presented that demonstrate the accuracy and computational efficiency of the present method finally we show how the reduced order model may be incorporated into an aeroelastic flutter model

mild integrability conditions for global solutions of an elliptic equation abstract it is shown that the equation u p x u 0 has positive radially symmetric solutions on a given exterior domain e a xe r n x a 0 which behave asymptotically as x like constant multiples of the radial solutions v1 1 and v2 x n 2 of v 0 provided that p p t satisfies certain integrability conditions on a the integrability conditions are weaker than those usually imposed

finite element euler computations in three dimensions an explicit finite element solution procedure for the three dimensional euler equations is presented the solution domain is automatically meshed using a tetrahedral mesh generator which is an extension of our previous two dimensional work several examples are included to illustrate the performance of the generator and solver an adaptive mesh regeneration procedure is used for the first time in three dimensions

real classification of complex quadrics abstract canonical forms are given for complex quadric surfaces and conics under real changes of variable the basic idea is to use known results on pairs of real quadratic forms

finite volume solution of the two dimensional euler equations on a regular triangular mesh the two dimensional euler equations have been solved on a triangular grid by a multigrid scheme using the finite volume approach by careful construction of the dissipative terms the scheme is designed to be secondorder accurate in space provided the grid is smooth except in the vicinity of shocks where it behaves as firstorder accurate in its present form the accuracy and convergence rate of the triangle code are comparable to that of the quadrilateral mesh code of jameson

aeroelastic design optimization for laminar and turbulent flows abstract the design of flexible structures undergoing aeroelastic deformations is a challenging task in particular if the prediction of the fluid loads and the evaluation of the aerodynamic performance criteria require accounting for laminar and turbulent effects this paper presents a general optimization methodology for fluid structure interaction problems based on turbulent flow models the structure is modeled via a geometrically nonlinear finite element method the flow field is computed by solving a finite volume approximation of the navier stokes equations on moving grids augmented by turbulence models the global design sensitivity equations are presented and solution methods for computing the numerically consistent and approximated design sensitivities are introduced the accuracy of the sensitivity analysis methods are studied by numerical examples for three dimensional problems the potential of the overall optimization methodology is illustrated by the shape and thickness optimization of a realistic wing the numerical examples suggest that the derivatives of the turbulence variables can be neglected in the sensitivity analysis without noticeable errors comparing the optimization results for euler and navier stokes flow models shows the importance of accounting for viscous laminar and turbulent effects for fine tuning the design and obtaining reliable optimization results

unified fluid structure interpolation and mesh motion using radial basis functions a multivariate interpolation scheme using radial basis functions is presented which results in a completely unified formulation for the fluid structure interpolation and mesh motion problems the method has several significant advantages primarily all volume mesh structural mesh and flow solver type dependence is removed and all operations are performed on totally arbitrary point clouds of any form hence all connectivity and user input requirements are removed from the computational fluid dynamics computational structural dynamics cfd csd coupling problem as only point clouds are required to determine the coupling also it may equally well be applied to structured and unstructured grids or structural and aerodynamic grids that intersect again because no connectivity information is required furthermore no expensive computations are required during an unsteady simulation just matrix vector multiplications since the required dependence relations are computed only once prior to any simulation and then remain constant this property means that the method is both perfectly parallel since only the data relevant to each structured block or unstructured partition are required to move those points and totally independent from the flow solver hence a completely generic black box tool can be developed which is ideal for use in an optimization approach aeroelastic behaviour of the brite euram mdo wing is analysed in terms of both static deflection and dynamic responses and it is demonstrated that responses are strongly dependent on the exact cfd csd interpolation used mesh quality is also examined during the motion resulting from a large surface deformation global grid quality is shown to be preserved well with local grid orthogonality also being maintained well particularly at and near the moving surface where the original orthogonality is retained copyright 2007 john wiley sons ltd

existence and multiplicity of solutions for nonlinear second order difference boundary value problems in the present paper the following nonlinear second order difference boundary value problems d p n 1 du n 1 q n u n f n u n 0 n z 1 n u 0 u n p 0 du 0 p n du n are studied by using the critical theory and three new multiple results of solutions are obtained where z 1 n 1 2 n f z 1 n xr r is a continuous function in the second variable p q z 0 n r and p n 0

adjoint based airfoil optimization with adaptive isogeometric discontinuous galerkin method abstract in this work an adjoint based airfoil shape optimization algorithm is developed based on the adaptive isogeometric discontinuous galerkin method for compressible euler equations to investigate the significance of each design variable of airfoil b spline parameterization we first parameterize the airfoil by b spline curve approximation with some control points viewed as design variables and build the b spline representation of the flow field with the curve to apply the goal oriented h adaptive isogeometric dg method for flow solution then we compute and employ the discrete adjoint solutions for both multi target error estimation in adaptive mesh refinement with the isogeometric nature not only all the geometrical cells but also the numerical basis functions can be analytically expressed by the design variables indicating that the numerical solutions and objective could be differentiable with respect to those variables consequently the gradient is totally computed in an accurate approach and the sensitivity analysis is thus improved by reducing the spatial discretization error and introducing the analytical expression of derivative to reveal the key parameters for optimization in an intuitive and efficient manner although the sqp optimization algorithm is adopted in the paper the given accurate gradient can be applied to any gradient based optimization algorithm the proposed algorithm is demonstrated on rae2822 airfoil with inviscid transonic flow where the shape is optimized to minimize the drag coefficient at a constrained lift and airfoil area the numerical results show that the drag is much more sensitive to the design variables near the tailing edge at the beginning but sensitivity is reduced when optimal

a new approximate lu factorization scheme for the reynolds averaged navier stokes equations a new approximate lu factorization scheme is developed to solve the steady state reynolds averaged navierstokes ns equations central differencing is used for both implicit and explicit operators and special care is taken to obtain well conditioned factors on the implicit side the scheme is then analyzed and optimized according to a simple linear analysis it is unconditionally stable for the model hyperbolic equation in both two and three dimensions however the requirement for well conditioned factors has essentially limited the effective time step that the scheme can achieve supersonic and transonic 3 d flows past a hemisphere cylinder are computed to demonstrate the convergence characterstics of the scheme a good convergence rate is achieved in the inviscid case finally an explicit eigenvector annihilation procedure is employed successfully to remove the stiffness caused by the fine grid spacing for viscous flows

unsteady euler algorithm with unstructured dynamic mesh for complex aircraft aerodynamic analysis an euler solution algorithm is presented for unsteady aerodynamic analysis of complex aircr aft configurations the flow solver involves a multistage runge kutta time stepping scheme that uses a finite volume spatial discretization on an unstructured grid made up of tetrahedra a significant contribution of the research is the development and implementation of a moving mesh algorithm that is employed for problems involving static or dynamic deformation of the aircraft the mesh algorithm is a general procedure that can treat realistic motions and deformations of complex aircraft configurations steady and unsteady results are presented for a supersonic fighter configuration to demonstrate applications of the euler solver and dynamic mesh algorithm the unsteady flow results were obtained for the aircraft oscillating harmonically in a complete vehicle bending mode effects of angle of attack and reduced frequency on instantaneous pressures and force responses were investigated the paper presents descriptions of the euler solver and dynamic mesh algorithm along with results that assess the capability

on the stability of the positive radial steady states for a semilinear cauchy problem abstract this paper is a contribution to the analysis of the cauchy problem 0 1 u t u i 1 k c i x l i u p i in r n 0 t u x 0 x in r n with initial function 0 the stability and asymptotic stability of the positive radial steady states which are positive solutions of 0 2 u i 1 k c i x l i u p i 0 have been discussed with in weighted banach spaces

diagonal inversion of lower upper implicit schemes a new diagonally inverted lower upper lu implicit scheme is developed for the three dimensional euler equations the matrix systems that are to be inverted in the scheme are treated by local diagonalizing transformations that decouple them into systems of scalar equations unlike the diagonalized alternating direction implicit method the time accuracy of the lu scheme is not reduced since the diagonalizing procedure does not destroy time conservation furthermore this diagonalization reduces the computational effort required to solve the implicit approximation and therefore transforms it into a more efficient method of numerically solving the three dimensional euler equations i ncreasingly attention is being directed towards implicit schemes as the need to develop more efficient numerical methods becomes apparent an implicit approximation is an attractive way to increase the efficiency of the time marching technique because implicit schemes permit much larger time steps than most explicit methods normally allow the drawback of most implicit schemes is that they are computationally more expensive per time step than explicit methods as a result interest is being focused upon decreasing the computational effort required to solve the implicit approximation examples of schemes that address this issue are chaussee and pulliam s diagonalized alternating direction implicit adi scheme1 and obayashi and fujii s lower upper lu alternating direction implicit scheme 2 the euler equations can be discretized into an implicit approximation that when written in delta form produces a large block banded matrix system that is impractical to solve without first being approximately factored in the present work the lu implicit multigrid algorithm developed by yokota and caughey3 for the three dimensional euler equations has been made more efficient through a local diagonalizing procedure that reduces the computational effort required to solve the lu approximation the lu factorization produces two block triangular operators one upper and one lower which through back substitution can be solved by effectively explicit sweeps that require matrix inversions at every mesh cell in the domain the efficiency of this lu scheme can be increased by reducing the computational costs associated with these matrix inversions this reduction can be achieved by a local diagonalizing transformation that decouples the matrix systems into scalar equations time conservation and stability remain unaltered by the decoupling since the lu scheme s differencing operators are unaffected by the diagonalizing transformations this result is unlike the diagonalized adi method which is produced at the expense of time accuracy the euler equations for the generalized coordinate system 7 f can be written

efficient method to eliminate mesh sensitivity in adjoint based optimization a method based on the delaunay graph mapping mesh movement is proposed to eliminate mesh sensitivity which is required in the discrete adjoint optimization framework the method makes use of a one to one explicit algebraic mapping between the volume and surface mesh nodes for which the relative volume coefficients based on the delaunay graph are computed only once this procedure also results in a straightforward computation of the mesh sensitivity without the need to invert the large sparse matrix generally associated with implicit mesh movements such as the spring analogy torsional spring and linear elastic analogy the advantage of the method comes from the fact that the solution of the second linear mesh adjoint system of equations is no longer required the method has been verified for the two dimensional rae 5243 airfoil and for the three dimensional onera m6 wing for two cases 1 sensitivities of the objective function with respect to each surface mesh point as a design variable compared with

entire solutions of a class of even order quasilinear elliptic equations on demontre l existence de solutions entieres radialement symetriques et on decrit leur comportement asymptotique pour x