generalized multipoint inverse airfoil design in a rather general sense inverse airfoil design can be taken to mean the problem of specifying a desired set of airfoil characteristics such as the airfoil maximum thickness ratio pitching moment part of the velocity distribution or boundary layer development from thie information the corresponding airfoil shape is determined we present a method that approaches the design problem from this perspective in particular the airfoil is divided into segments along which together with the design conditions either the velocity distribution or boundary layer development may be prescribed

convergent series solution of nonlinear equations abstract the author s decomposition method 1 provides a new efficient computational procedure for solving large classes of nonlinear and or stochastic equations these include differential equations containing polynomial exponential and trigonometric terms negative or irrational powers and product nonlinearities 2 also included are partial differential equations 3 delay differential equations 4 algebraic equations 5 and matrix equations 6 which describe physical systems essentially the method provides a systematic computational procedure for equations containing any nonlinear terms of physical significance the procedure depends on calculation of the author s a n a finite set of polynomials 1 13 in terms of which the nonlinearities can be expressed this paper shows important properties of the a n which ensure an accurate and computable convergent solution by the author s decomposition method 1 since the nonlinearities and or stochasticity which can be handled are quite general the results are potentially extremely useful for applications and make a number of common approximations such as linearization unnecessary

an algorithm for solving boundary value problems in this paper a numerical algorithm based on the decomposition technique is presented for solving a class of nonlinear boundary value problems the method is implemented for well known examples including troesch s and bratu s problems which have been extensively studied the scheme is shown to be highly accurate and only a few terms are required to obtain accurate computable solutions

comparison of optimization algorithms for aerodynamic shape design three different optimization algorithms are applied to airfoil design a simplified test problem is considered here so that the distribution of the objective function can be plotted the visualization of the objective function will reveal which optimization strategy is preferable the results of three optimization algorithms will be easy to compare and to evaluate as to performance

multipoint inverse airfoil design method based on conformal mapping a method of multipoint inverse airfoil design for incompressible potential flow is presented multipoint design is handled by dividing the airfoil into a number of desired segments for each segment the velocity distribution is prescribed together with an angle of attack at which the prescribed velocity distribution is to be achieved in this manner multipoint design objectives can be taken into account in the initial specification of the velocity distribution in order for the multipoint inverse airfoil design problem to be well posed three integral constraints and several conditions arise that must be satisfied

on the solutions of stochastic initial value problems in continuum mechanics abstract this work deals with the analysis of a class of initial value problems for nonlinear hyperbolic equations with stochastic process coefficients and four time space independent variables a perturbation procedure is developed leading to a sequence of stochastic partial differential equations which are solved by application of the adomian decomposition method a pertinent application in mechanics is also considered and quantitative results are presented and discussed

on the solution of algebraic equations by the decomposition method abstract the decomposition method g adomian stochastic systems academic press new york 1983 developed to solve nonlinear stochastic differential equations has recently been generalized to nonlinear and or stochastic partial differential equations systems of equations and delay equations and applied to diverse applications as pointed out previously see reference above the methodology is an operator method which can be used for nondifferential operators as well extension has also been made to algebraic equations involving real or complex coefficients this paper deals specifically with quadratic cubic and general higher order polynomial equations and negative or nonintegral powers and random algebraic equations further work on this general subject appears elsewhere g adomian stochastic systems ii academic press new york in press

a review of the decomposition method and some recent results for nonlinear equations abstract the decomposition method can be an effective procedure for solution of nonlinear and or stochastic continous time dynamical systems without usual restrictive assumptions this paper is intended as a convenient tutorial review of the method 1

two point iterative methods for solving nonlinear equations in this study a new root finding technique for solving nonlinear equations is proposed then two new more algorithms are derived from this new technique by employing the adomian decomposition method adm these three algorithms require two starting values that do not necessarily bracketing the root of a given nonlinear equation however when the starting values are closed enough or bracketed the root then the proposed methods converge to the root faster than the secant method another advantage over all iterative methods is that the proposed methods usually converge to two distinct roots when the handled nonlinear equation has more than one root that is the odd iterations of the new techniques converge to a root and the even iterations converge to another root some numerical examples including a sine polynomial equation are solved by using the proposed methods and compared with the secant method perfect agreements were found

adomian s method of decomposition critical review and examples of failure abstract adomian s method of decomposition is considered in application to initial boundary value problems for the one space dimensional spatially homogeneous heat conduction equation it is shown that the fundamental equation of the method is well defined only for certain restricted types of boundary conditions within the class of such boundary conditions examples are given such that the fundamental equation fails to have a unique solution and such that the sequence produced by iteration of this equation is divergent the latter is a counterexample to a published assertion of convergence

geometric filtration using proper orthogonal decomposition for aerodynamic design optimization when carrying out design searches traditional variable screening techniques can find it extremely difficult to distinguish between important and unimportant variables this is particularly true when only a small number of simulations is combined with a parameterization which results in a large number of variables of seemingly equal importance here the authors present a variable reduction technique which employs proper orthogonal decomposition to filter out undesirable or badly performing geometries from an optimization process unlike traditional screening techniques the presented method operates at the geometric level instead of the variable level the filtering process uses the designs which result from a geometry parameterization instead of the variables which control the parameterization the method is shown to perform well in the optimization of a two dimensional airfoil for the minimization of drag to lift ratio producing designs better than those resulting from traditional kriging based surrogate model optimization and with a significant reduction in surrogate tuning cost

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

semilinear stochastic systems analysis with the method of the stochastic green s function and application in mechanics abstract this work deals with the analysis of a class of semilinear vector stochastic operator equations the method of the stochastic green s function due to adomian is extended to the above mentioned class of equations some perturbation techniques are also developed and a pertinent application in nonlinear mechanics is considered

symbolic implementation of the algorithm for calculating adomian polynomials in this paper a symbolic implementation code is developed of a technique proposed by wazwaz appl math comput 111 2000 53 for calculating adomian polynomials for nonlinear operators the algorithm proposed by him appl math comput 111 2000 53 offers a promising approach for calculating adomian polynomials for all forms of nonlinearity but it is not easy to implement due to its huge size of algebraic calculations complicated trigonometric terms and unique summation rules it is well known that the algebraic manipulation language such as mathematica is useful to facilitate such a hard computational work pattern matching capabilities peculiar feature of mathematica are used in index regrouping which is a key role in constructing adomian polynomials the computer algebra software mathematica is used to collect terms to their order and to simplify the terms the symbolic implementation code author developed appearing at appendix has the flexibility that may easily cover any length of adomian polynomial for many forms of nonlinear cases a nonlinear evolution equation is investigated in order to justify the availability of symbolic implementation code

coupled nonlinear stochastic differential equations abstract systems of n coupled linear or nonlinear differential equations which may be deterministic or stochastic are solved by methods of the first author and his co workers examples include two coupled riccati equations coupled linear equations stochastic coupled equations with product terms and n coupled stochastic differential equations

inversion of nonlinear stochastic operators abstract the operator theoretic method adomian and malakian j math anal appl 76 1 1980 183 201 recently extended adomian s solutions of nonlinear stochastic differential equations g adomian stochastic systems analysis in applied stochastic processes nonlinear stochastic differential equations j math anal appl 55 1 1976 441 452 on the modeling and analysis of nonlinear stochastic systems in proceeding international conf on mathematical modeling vol 1 pp 29 40 to provide an efficient computational procedure for differential equations containing polynomial exponential and trigonometric nonlinear terms n y the procedure depends on the calculation of certain quantities an and bn this paper generalizes the calculation of the an and bn to much wider classes of nonlinearities of the form n y y essentially the method provides a systematic computational procedure for differential equations containing any nonlinear terms of physical significance this procedure depends on a recurrence rule from which explicit general formulae are obtained for the quantities an and bn for any order n in a convenient form this paper also demonstrates the significance of the iterative series decomposition proposed by adomian for linear stochastic operators in 1964 and developed since 1976 for nonlinear stochastic operators since both the nonlinear and stochastic behavior is quite general the results are extremely significant for applications processes need not for example be limited to either gaussian processes white noise or small fluctuations

an optimization based approach for solving a time harmonic multiphysical wave problem with higher order schemes this study considers developing numerical solution techniques for the computer simulations of time harmonic fluid structure interaction between acoustic and elastic waves the focus is on the efficiency of an iterative solution method based on a controllability approach and spectral elements we concentrate on the model in which the acoustic waves in the fluid domain are modeled by using the velocity potential and the elastic waves in the structure domain are modeled by using displacement traditionally the complex valued time harmonic equations are used for solving the time harmonic problems instead of that we focus on finding periodic solutions without solving the time harmonic problems directly the time dependent equations can be simulated with respect to time until a time harmonic solution is reached but the approach suffers from poor convergence to overcome this challenge we follow the approach first suggested and developed for the acoustic wave equations by bristeau glowinski and periaux thus we accelerate the convergence rate by employing a controllability method the problem is formulated as a least squares optimization problem which is solved with the conjugate gradient cg algorithm computation of the gradient of the functional is done directly for the discretized problem a graph based multigrid method is used for preconditioning the cg algorithm

concise airfoil representation via case based knowledge capture the cost of exploring a k dimensional design space at a sampling density n is of the order nk in view of this curse of dimensionality the paper explores the possibility of reducing the dimensionality of parametric airfoil definitions to as low as one to minimize their cost impact on full airframe conceptual design studies a formulation based on nonuniform rational b splines is introduced in which the parametric airfoil is constructed in a reduceddimensionality space defined in terms of a small number of basis geometries these bases are airfoils themselves selected to represent key features of their class we use them as learning cases or case based representations of our knowledge of what makes a nonuniform rational b splines curve an airfoil moreover an airfoil belonging to a certain class this study focuses on supercritical sections but the template presented here can also be applied to other types of airfoils and indeed to other classes of curves and surfaces

modification of adomian s decomposition method to solve equations containing radicals abstract adomian s decomposition method has been used in obtaining approximate solutions to a wide class of nonlinear deterministic stochastic operator equations adomian rach and sarafyan have investigated the use of the decomposition method for equations involving radicals but in such equations when the linear operator is of higher degree the direct use of adomian polynomials for the radicals produces computational difficulties while inverting the operator in particular if the initial approximation is not a constant in this paper we propose a method in which the nonlinear term in the equation involving radicals is first expanded formally in a power series the power series is then approximated by pade approximant thereby replacing the radical term by a ratio of polynomials the decomposition method of adomian is now applied to the modified equation without any difficulty as the radical type nonlinearity is now converted into a polynomial type nonlinearity

a review of the decomposition method in applied mathematics abstract the decomposition method can be an effective procedure for analytical solution of a wide class of dynamical systems without linearization or weak nonlinearity assumptions closure approximations perturbation theory or restrictive assumptions on stochasticitiy

a new iterative method to compute nonlinear equations the aim of this paper is to construct a new efficient iterative method to solve nonlinear equations the new method is based on the proposals of abbasbandy on improving the order of accuracy of newton raphson method s abbasbandy improving newton raphson method for nonlinear equations by modified adomian decomposition method applied mathematics and computation 145 2003 887 893 and on the proposals of babolian and biazar on improving the order of accuracy of adomian s decomposition method e babolian j biazar on the order of convergence of adomian method applied mathematics and computation 130 2002 383 387 the convergence of the new scheme is proved and at least the cubic order of convergence is established several examples are presented and compared to other methods showing the accuracy and fast convergence of this new method also it is shown in this paper that the modified adomian s method developed by babolian and biazar to solve nonlinear equations e babolian j biazar solution of nonlinear equations by modified adomian decomposition method applied mathematics and computation 132 2002 167 172 should be slightly modified due to the fact that convergence of adomian s method does not ensure convergence of the modified method an example illustrates this fact which unlike what is claimed by the authors does not converge with their method but with a simple different choice of the zero component becomes convergent

drag minimization based on the navier stokes equations using a newton krylov approach a methodology is presented for performing numerical aerodynamic shape optimization based on the three dimensional reynolds averaged navier stokes rans equations an initial multiblock structured mesh is first fit with b spline volumes that form the basis for a hybrid mesh movement scheme that is tightly integrated with the geometry parameterization based on b spline surfaces the rans equations and the one equation spalart allmaras turbulence model are solved in a fully coupled manner using an efficient parallel newton krylov algorithm with approximate schur preconditioning gradient evaluations are performed using the discrete adjoint approach with analytical differentiation of the discrete flow and mesh movement equations the overall methodology remains robust even in the presence of large shape changes several examples of lift constrained drag minimization are provided including a study of the common research model wing geometry a wing body tail geometry with a prescribed spanwise load distributi

solving elastodynamics in a fluid solid heterogeneous sphere a parallel spectral element approximation on non conforming grids we present a spectral element approach for modeling elastic wave propagation in a solid fluid sphere where the local effects of gravity are taken into account the equations are discretized in terms of the displacement in the solid and the velocity potential in a neutrally stratified fluid the spatial approximation is based upon a spherical mesh of hexahedra in which local refinement allows for adapting the discretization to the variation of elastic parameters in both the solid and the fluid regions continuity constraints across the non conforming interfaces are introduced through lagrange multipliers which are further discretized by the mortar element method due to the spherical nature of the non conforming interfaces the mortar method turns out to be functionally conforming and allows for an equal order interpolation of the primal variables and the lagrange multipliers the method is shown to provide an accurate solution when compared to analytical calculations obtained for radial models of elastic parameters its parallel implementation is based upon a simple domain decomposition strategy which makes it efficient to solve large problems as those imposed by planetary scales

an analytical approach to the fornberg whitham type equations by using the variational iteration method in this paper he s variational iteration method vim is applied to solve the fornberg whitham type equations the vim provides fast converged approximate solutions to nonlinear equations without linearization discretization perturbation or the adomian polynomials this makes the method attractive and reliable for solving the fornberg whitham type equations numerical examples related to two initial value problems are presented to show the efficiency of the vim

controllability method for acoustic scattering with spectral elements we formulate the helmholtz equation as an exact controllability problem for the time dependent wave equation the problem is then discretized in time domain with central finite difference scheme and in space domain with spectral elements this approach leads to high accuracy in spatial discretization moreover the spectral element method results in diagonal mass matrices which makes the time integration of the wave equation highly efficient after discretization the exact controllability problem is reformulated as a least squares problem which is solved by the conjugate gradient method we illustrate the method with some numerical experiments which demonstrate the significant improvements in efficiency due to the higher order spectral elements for a given accuracy the controllability technique with spectral element method requires fewer computational operations than with conventional finite element method in addition by using higher order polynomial basis the influence of the pollution effect is reduced

generalized galerkin approximations of elastic waves with absorbing boundary conditions abstract for the propagation of elastic waves in unbounded domains absorbing boundary conditions abcs at the fictitious numerical boundaries have been proposed in this paper we focus on both first and second order abcs in the framework of variational weak approximations like those stemming from galerkin method or its variants for finite element or spectral approximations 1 in particular we recover first order conditions as natural or neumann conditions whereas we propose a penalty residual method for the treatment of second order abcs the time discretization is based on implicit backward finite differences whereas we use spectral legendre collocation methods set in a variational form for the spatial discretization treatment of finite element or spectral element approximations is completely similar numerical experiments exhibit that the present formulation of second order abcs improves the one based on first order abcs with regard to both the reduction of the total energy in the computational domain and the fourier spectrum of the displacement field at selected points of the elastic medium a stability analysis is developed for the variational problem in the continuous case both for first and second order abcs a suitable treatment of abcs at corners is also proposed

aerodynamic shape optimization using preconditioned conjugate gradient methods in an effort to further improve upon the latest advancements made in aerodynamic shape optimization procedures a systematic study is performed to examine several current solution methodologies as applied to various aspects of the optimization procedure it is demonstrated that preconditioned conjugate gradient like methodologies dramatically decrease the computational efforts required for such procedures the design problem investigated is the shape optimization of the upper and lower surfaces of an initially symmetric naca 0012 airfoil in inviscid transonic flow and at zero degrees angle of attack the complete surface shape is represented using a bezier bernstein polynomial

on product nonlinearities in stochastic differential equations the author s methods for solution of stochastic differential equations shown elsewhere to be applicable for nonlinear cases including polynomial exponential and trigonometric terms is shown to be applicable to product nonlinearities as well

an algorithm for adomian decomposition method in this paper a reliable technique for calculating adomian polynomials for nonlinear operators is developed the algorithm is fulfilled with symbolic computation so that the adomian decomposition methods for solving differential equations can be mechanized in maple ten examples are presented to illustrate the implementation of the program

time harmonic solution for acousto elastic interaction with controllability and spectral elements the classical way of solving the time harmonic linear acousto elastic wave problem is to discretize the equations with finite elements or finite differences this approach leads to large scale indefinite complex valued linear systems for these kinds of systems it is difficult to construct efficient iterative solution methods that is why we use an alternative approach and solve the time harmonic problem by controlling the solution of the corresponding time dependent wave equation in this paper we use an unsymmetric formulation where fluid structure interaction is modeled as a coupling between pressure and displacement the coupled problem is discretized in space domain with spectral elements and in time domain with central finite differences after discretization exact controllability problem is reformulated as a least squares problem which is solved by the conjugate gradient method

numerical method for the solution of special nonlinear fourth order boundary value problems the sinc galerkin method is used to approximate solution of nonlinear problems this work deals with the sinc galerkin method for solving nonlinear fourth order differential equations with homogeneous and nonhomogeneous boundary conditions the scheme is tested on three nonlinear problems and a comparison with finite difference methods and fourth order multiderivative method is made it is shown that the sinc galerkin method yields better results

a convenient computational form for the adomian polynomials abstract recent important generalizations by g adomian stochastic systems academic press 1983 have extended the scope of his decomposition method for nonlinear stochastic operator equations see also iterative method inverse operator method symmetrized method or stochastic green s function method very considerably so that they are now applicable to differential partial differential delay and coupled equations which may be strongly nonlinear and or strongly stochastic or linear or deterministic as subcases thus for equations modeling physical problems solutions are obtained rapidly easily and accurately the methodology involves an analytic parametrization in which certain polynomials an dependent on the nonlinearity are derived this paper establishes simple symmetry rules which yield adomian s polynomials quickly to high orders

airfoil shape optimization using a nonuniform rational b splines parameterization under thickness constraint results for 2 d airfoil shape optimization in transonic regime are presented airfoil shapes are represented by nonuniform rational b splines with appropriate regularity properties a navier stokes flow solver is used to compute the flow field and to obtain aerodynamic coefficients a design of experiment is conducted to select the most sensitive design variables among the nonuniform rational b splines parameters to reduce their number in the final optimization process single point and multipoint formulations of the optimization problem are proposed and compared the nonuniform rational b splines parameterization guarantees smooth optimized airfoils the multipoint optimization formulation combined with the nonuniform rational b splines parameterization leads to airfoils with good performance over a specified mach range

time harmonic elasticity with controllability and higher order discretization methods the time harmonic solution of the linear elastic wave equation is needed for a variety of applications the typical procedure for solving the time harmonic elastic wave equation leads to difficulties solving large scale indefinite linear systems to avoid these difficulties we consider the original time dependent equation with a method based on an exact controllability formulation the main idea of this approach is to find initial conditions such that after one time period the solution and its time derivative coincide with the initial conditions the wave equation is discretized in the space domain with spectral elements the degrees of freedom associated with the basis functions are situated at the gauss lobatto quadrature points of the elements and the gauss lobatto quadrature rule is used so that the mass matrix becomes diagonal this method is combined with the second order central finite difference or the fourth order runge kutta time discretization as a consequence of these choices only matrix vector products are needed in time dependent simulation this makes the controllability method computationally efficient

improving the efficiency of aerodynamic shape optimization the computational efficiency of an aerodynamic shape optimization procedure that is based on discrete sensitivity analysis is increased through the implementation of two improvements the first improvement involves replacing a grid point based approach for surface representation with a bezier bernstein polynomial parameterization of the surface explicit analytical expressions for the grid sensitivity terms are developed for both approaches the second improvement proposes the use of newton s method in lieu of an alternating direction implicit methodology to calculate the highly converged flow solutions that are required to compute the sensitivity coefficients

a new approach to the wave equation an application of the decomposition method abstract in the present paper the inhomogeneous wave equation is solved quickly easily and elegantly by adomian s decomposition method

swarm algorithm for single and multiobjective airfoil design optimization shape optimization of airfoils involves highly expensive nonlinear objective s and constraint functions often with functional and slope discontinuity that limits the efficient use of gradient based methods for its solution gradient based methods are not capable of generating a set of pareto solutions as required in multiobjective problems as they work with a single solution and improve it through successive iterations population based zero order stochastic optimization methods are therefore an attractive choice for shape optimization problems as they are easy to implement and effective for highly nonlinear problems we present a swarm algorithm that is applicable for optimization problems in general but is here explored for airfoil design optimization studies

nonlinear plasma response abstract the decomposition method g adomian stochastic systems academic press new york london 1983 g adomian stochastic systems ii in press and earlier work of adomian referenced therein allows solution without linearization of problems normally requiring the self consistent field technique the latter technique can provide a sufficiently accurate approximation in weakly coupled plasma but the decomposition method of adomian provides a superior and physically preferable methodology for the general case

the decomposition method applied to chandrasekhar h equation in this paper adomian s decomposition method is applied to the nonlinear chandrasekhar h equation the solution is expressed as an infinite series it is shown that two or three iterations are sufficient to obtain good approximation of the exact solution adomian s method has been illustrated by considering some particular cases of this nonlinear integral equation that are of physical interest

revisit on partial solutions in the adomian decomposition method solving heat and wave equations the adomian decomposition method is considered in application to heat and wave equations the so called partial solution technique is used it is shown that the fundamental equation of the method is well defined only for certain types of boundary conditions in cases involving inhomogeneous boundary conditions improper results may be obtained by former method this paper presents a further insight into partial solutions in the decomposition method and the resolution of such cases

adomian decomposition method for burger s huxley and burger s fisher equations the approximate solutions for the burger s huxley and burger s fisher equations are obtained by using the adomian decomposition method solving frontier problems of physics the decomposition method kluwer boston 1994 the algorithm is illustrated by studying an initial value problem the obtained results are presented and only few terms of the expansion are required to obtain the approximate solution which is found to be accurate and efficient

new exact traveling wave solutions of the 4 1 dimensional fokas equation abstract in this paper two distinct methods are applied to look for exact traveling wave solutions of the 4 1 dimensional nonlinear fokas equation namely the modified simple equation method msem and the extended simplest equation method esem some new exact traveling wave solutions involving some parameters are obtained the solitary wave solutions can be extracted by assigning special values of these parameters the obtained solutions show the simplicity and efficiency of the used approaches that can be applied for nonlinear equations as well as linear ones

algebraic equations with exponential terms abstract the decomposition method is applied to algebraic equations containing exponential terms the n term approximation n is rapidly damped as n increases yielding an oscillating convergence of superior accuracy n 5 and n 9 for k 2

optimal aerodynamic design of airfoils in unsteady viscous flows abstract a continuous adjoint formulation is used to determine optimal airfoil shapes in unsteady viscous flows at re 1 10 4 the reynolds number is based on the free stream speed and the chord length of the airfoil a finite element method based on streamline upwind petrov galerkin supg and pressure stabilized petrov galerkin pspg stabilizations is used to solve both the flow and adjoint equations the airfoil is parametrized via a non uniform rational b splines nurbs curve three different objective functions are used to obtain optimal shapes maximize lift minimize drag and minimize ratio of drag to lift the objective functions are formulated on the basis of time averaged aerodynamic coefficients the three objective functions result in diverse airfoil geometries the resulting airfoils are thin with the largest thickness to chord ratio being only 5 4 the shapes obtained are further investigated for their aerodynamic performance maximization of time averaged lift leads to an airfoil that produces more than six times more lift compared to the naca 0012 airfoil the excess lift is a consequence of the large peak and extended region of high suction on the upper surface and high pressure on the lower surface minimization of drag results in an airfoil with a sharp leading edge the flow remains attached for close to 70 of the chord length minimization of the ratio of drag to lift results in an airfoil with a shallow dimple on the upper surface it leads to a fairly large value of the time averaged ratio of lift to drag 17 8 the high value is mostly achieved by a 447 increase in lift and 16 reduction in drag compared to a naca 0012 airfoil imposition of volume constraint for the cases studied is found to result in airfoils that have lower aerodynamic performance

numerical solution of linear elastic problems by spectral collocation methods abstract numerical approximations to the linear elastic system are traditionally based on the finite element method here we propose a new formulation based on the spectral collocation ethod a rigorous theoretical analysis is developed in order to prove the stability and convergance properties of the collocation scheme we also consider domain decomposition methods in order to handle complex geometries and non smooth data finally we present several numerical results for some examples concerning benchmark problems in geomechanics

dimension reduction for aerodynamic design optimization the search for an optimal design in a high dimensional design space of a multivariate problem requires a sample size proportional or even exponential to the number of variables of the problem this curse of dimensionality places a computational burden on the cost of optimization especially when the problem uses expensive high fidelity simulations and may force one to try to reduce the dimensions of a problem traditional variable screening techniques reduce the dimensionality of the problem by removing variables that seem irrelevant to the design problem this practice fails when all the variables are equally relevant in the problem or when some variables are relevant only in some parts of the design space the present work describes a dimension reduction method called generative topographic mapping based on non linear latent models which transform a high dimensional data set into a low dimensional latent space without removing any variables it is first illustrated on a two dimensional branin function and then applied to a thirty dimensional airfoil problem the method is then compared with a global optimizer a genetic algorithm other dimension reduction methods principle component analysis and gaussian process latent variable models and with kriging surrogate models the method improves when the initial sample used for dimension reduction is filtered to contain only good designs

on a class of nonlinear stochastic dynamical systems analysis of the transient behaviour abstract a class of nonlinear dynamical systems with random parameters and initial conditions is considered the transient time evolution of the mean value of the dependent variable is analyzed both by means of the stochastic averaging method and by adomian s decomposition method a proof of the equivalence of the decomposition method to the time series approximation of the dynamical response is provided an application is made to the van der pol equation and finally a comparison between the two previously discussed methods is given

displacement pressure based mixed finite element formulations for acoustic fluid structure interaction problems summary we present reliable finite element discretizations based on displacement pressure interpolations for the analysis of acoustic fluid structure interaction problems the finite element interpolations are selected using the inf sup condition and emphasis is given to the fact that the boundary conditions must satisfy the mass and momentum conservation we show that with our analysis procedure no spurious non zero frequencies are encountered as heretofore calculated with other displacement based discretizations 1997 by john wiley sons ltd

random heat equation solutions by the stochastic adaptive interpolation method abstract the one dimensional nonlinear heat equation is considered with stochastic coefficients and initial boundary values in a finite strip and in the half space approximated solutions are obtained by the stochastic adaptive interpolation method which corresponds first to transforming the original partial differential equation into a system of ordinary differential equations via a generalization to the stochastic case of the bellman s differential quadrature method the system of ordinary differential equations is then solved by a continuous approximation following by adomian s decomposition method and the solutions are compared with those obtained by more standard numerical techniques

an algorithm for solving bond pricing problem the nonlinear bond pricing problem has been extensively studied in the literature since an analytical solution is not readily available we will seek to find an approximate solution we will present a decomposition method due to adomian and show how to obtain a reasonable numerical solution to the bond pricing problem

controllability methods for the computation of time periodic solutions application to scattering we discuss in this article the application of controllability techniques to the computation of the time periodic solutions of evolution equations the basic principles of the computational methods are presented in a fairly general context where the time discretization aspect is also discussed then this general methodology is applied to the solution of scattering problems for harmonic planar waves by two and three dimensional purely reflecting nonconvex obstacles numerical results obtained by the above method and comparisons with the results obtained by more classical methods show the superiority of the former ones

a new algorithm for calculating adomian polynomials for nonlinear operators in this paper a reliable technique for calculating adomian polynomials for nonlinear operators will be developed the new algorithm offers a promising approach for calculating adomian polynomials for all forms of nonlinearity the algorithm will be illustrated by studying suitable forms of nonlinearity a nonlinear evolution model will be investigated

spectral domain decomposition methods for the solution of acoustic and elastic wave equations a new spectral domain decomposition method is presented for acoustic and elastodynamic wave propagation in 2 d heterogeneous media starting from a variational formulation of the problem two different approaches are proposed for the spatial discretization a mixed fourier legendre and a full legendre collocation the matching conditions at subdomain interfaces are carefully analyzed and the stability and efficiency of time advancing schemes are investigated the numerical validation with some significant test cases illustrates the accuracy flexibility and robustness of our methods these allow the treatment of complex geometries and heterogeneous media while retaining spectral accuracy

genetic optimization of target pressure distributions for inverse design methods a genetic algorithm has been applied to optimize target pressure distributions for inverse design methods pressure distributions around airfoils are parameterized by b spline polygons and the airfoil drag is minimized under constraints on lift airfoil thickness and other design principles once target pressure distributions are obtained corresponding airfoil wing geometries can be computed by an inverse design code coupled with a navier stokes solver successful design results were obtained for transonic cases with and without a shock wave

spectral methods for hyperbolic problems we review the current state of fourier and chebyshev collocation methods for the solution of hyperbolic problems with an eye to basic questions of accuracy and stability of the numerical approximations throughout the discussion we emphasize recent developments in the area such as spectral penalty methods the use of filters the resolution of the gibbs phenomenon and issues related to the solution of nonlinear conservations laws such as conservation and convergence we also include a brief discussion on the formulation of multi domain methods for hyperbolic problems and conclude with a few examples of the application of pseudospectral collocation methods for solving nontrivial systems of conservation laws

analytical solution of the linear fractional differential equation by adomian decomposition method in this paper we consider the n term linear fractional order differential equation with constant coefficients and obtain the solution of this kind of fractional differential equations by adomian decomposition method with the equivalent transmutation we show that the solution by adomian decomposition method is the same as the solution by the green s function finally we illustrate our result with some examples

a new symbolic computational approach to singular initial value problems in the second order ordinary differential equations this paper presents a new symbolic algorithm to compute the singular initial value problem of second order ordinary differential equations using adomian decomposition method the algorithm has been implemented in the computer algebra system maple to facilitate the application of this method a number of examples are presented to demonstrate the computation efficiency of the program

on the adomian decomposition method and comparisons with picard s method abstract this paper shows that the adomian decomposition method stochastic systems academic press new york 1983 is not to be identified with any of the earlier methods since it solves a wide class of nonlinear and stochastic equations without the often unphysical assumptions which have become customary similarities between the picard method and the decomposition method are purely superficial

inverse and direct airfoil design using a multiobjective genetic algorithm some of the advantages and drawbacks of genetic algorithms applications to aerodynamic design are demonstrated a numerical procedure for the aerodynamic design of transonic airfoils by means of genetic algorithms with single point multipoint and multiobjective optimization capabilities is presented in the rst part an investigation on the relative ef ciency of different genetic operators combinations is carried out on an aerodynamic inverse design problem it is shown how an appropriate tuning of the algorithm can provide improved performances better adaption to design space size and topology and variables cross correlation in the second part the multiobjective approach to design is introduced the problem of the optimization of the drag rise characteristics of a transonic airfoil is addressed and dealt with using a single point a multipoint and a multiobjective approach a comparison between the results obtained using the three different strategies is nally established showing the advantages of multiobjective optimization

an algorithm for solving a nonlinear integro differential equation in this paper an algorithm based on adomian s decomposition method is developed to approximate the solution of the nonlinear integro differential equationu t x t email protected t 0a email protected email protected email protected u x x t email protected f x t 0

a space time finite element method for structural acoustics in infinite domains part 1 formulation stability and convergence abstract a space time finite element method for solution of the exterior structural acoustics problem involving the interaction of vibrating elastic structures submerged in an infinite acoustic fluid is formulated in particular time discontinuous galerkin and galerkin least squares gls variational formulations for coupled structural acoustics in unbounded domains are developed and analyzed for stability and convergence the formulation employs a finite computational fluid domain surrounding the structure and incorporates time dependent non reflecting boundary conditions on the fluid truncation boundary energy estimates are obtained which allow us to prove the unconditional stability of the method for the coupled fluid structure problem with absorbing boundaries the methods developed are especially useful for the application of adaptive solution strategies for transient acoustics in which unstructured space time meshes are used to track waves propagating along space time characteristics an important feature of the space time formulation is the incorporation of temporal jump operators which allow for finite element interpolations that are discontinaous in time for additional stability least squares operators based on local residuals of the structural acoustics equations including the non reflecting boundary conditions are incorporated the energy decay estimates and high order accuracy predicted by our a priori error estimates are demonstrated numerically in a simple canonical example

a new approach to nonlinear partial differential equations abstract the author s decomposition method for the solution of operator equations which may be nonlinear and or stochastic is generalized to multidimensional cases

airfoil shape design and optimization using multifidelity analysis and embedded inverse design design optimization using high fidelity computational fluid dynamics simulations is becoming increasingly popular sustaining the desire to make these methods more computationally efficient a reduction in problem dimensions as a result of improved parameterization techniques is a common contributor to this efficiency the focus of this paper is on the high fidelity aerodynamic design of airfoil shapes a multifidelity design search method is presented which uses a parameterization of the airfoil pressure distribution followed by inverse design giving a reduction in the number of design variables used in optimization although an expensive analysis code is used in evaluating airfoil performance computational cost is reduced by using a low fidelity code in the inverse design process this method is run side by side with a method which is considered to be a current benchmark in design optimization the two methods are described in detail and their relative performance is compared and discussed the newly presented method is found to converge towards the optimum design significantly more quickly than the benchmark method providing designs with greater performance for a given computational expense

variational iteration method for solving a nonlinear system of second order boundary value problems the variational iteration method is introduced to solve a nonlinear system of second order boundary value problems numerical results demonstrate that this method is promising and readily implemented

an adjoint method for shape optimization in unsteady viscous flows a new method for shape optimization for unsteady viscous flows is presented it is based on the continuous adjoint approach using a time accurate method and is capable of handling both inverse and direct objective functions the objective function is minimized or maximized subject to the satisfaction of flow equations the shape of the body is parametrized via a non uniform rational b splines nurbs curve and is updated by using the gradients obtained from solving the flow and adjoint equations a finite element method based on streamline upwind petrov galerkin supg and pressure stabilized petrov galerkin pspg stabilization techniques is used to solve both the flow and adjoint equations the method has been implemented and tested for the design of airfoils based on enhancing its time averaged aerodynamic coefficients interesting shapes are obtained especially when the objective is to produce high performance airfoils the effect of the extent of the window of time integration of flow and adjoint equations on the design process is studied it is found that when the window of time integration is insufficient the gradients are most likely to be erroneous

sinc galerkin method for solving nonlinear boundary value problems the sinc galerkin method is used to approximate solutions of nonlinear problems involving nonlinear second fourth and sixth order differential equations with homogeneous and nonhomogeneous boundary conditions the scheme is tested on four nonlinear problems the results demonstrate the reliability and efficiency of the algorithm developed

numerical study of modified adomian s method applied to burgers equation the application of adomian s decomposition method to partial differential equations when the exact solution is not reached demands the use of truncated series but the solution s series may have small convergence radius and the truncated series may be inaccurate in many regions in order to enlarge the convergence domain of the truncated series pade approximants pas to the adomian s series solution have been tested and applied to partial and ordinary differential equations with good results in this paper pas both in x and t directions applied to the truncated series solution given by adomian s decomposition technique for burgers equation are tested numerical and graphical illustrations show that this technique can improve the accuracy and enlarge the domain of convergence of the solution it is also shown in this paper that the application of adomian s method to the ordinary differential equations set arising from the discretization of the spatial derivatives by finite differences the so called method of lines may reduce the convergence domain of the solution s series

an algebraic multigrid based shifted laplacian preconditioner for the helmholtz equation a preconditioner defined by an algebraic multigrid cycle for a damped helmholtz operator is proposed for the helmholtz equation this approach is well suited for acoustic scattering problems in complicated computational domains and with varying material properties the spectral properties of the preconditioned systems and the convergence of the gmres method are studied with linear quadratic and cubic finite element discretizations numerical experiments are performed with two dimensional problems describing acoustic scattering in a cross section of a car cabin and in a layered medium asymptotically the number of iterations grows linearly with respect to the frequency while for lower frequencies the growth is milder the proposed preconditioner is particularly effective for low frequency and mid frequency problems

elasto acoustic and acoustic acoustic coupling on non matching grids flexible discretization techniques for the approximative solution of coupled wave propagation problems are investigated focussing on aero acoustic and elasto acoustic coupling in particular the advantages of using non matching grids are presented when one subregion has to be resolved by a substantially finer grid than the other subregion for the elasto acoustic coupling the problem formulation remains essentially the same as for the matching situation while for the aero acoustic coupling the formulation is enhanced with lagrange multipliers within the framework of mortar finite element methods several numerical examples are presented to demonstrate the flexibility and applicability of the approach copyright 2006 john wiley sons ltd

finite element analysis of pressure formulation of the elastoacoustic problem in this paper we analyze the non symmetric pressure displacement formulation of the elastoacoustic vibration problem and show its equivalence with the symmetric stiffness coupling formulation we introduce discretizations for these problems based on lagrangian finite elements we show that both formulations are also equivalent at discrete level and prove optimal error estimates for eigenfunctions and eigenvalues both formulations are rewritten such as to be solved with a standard matlab eigensolver we report numerical results comparing the efficiency of the methods over some test examples

optimized nonuniform rational b spline geometrical representation for aerodynamic design of wings the geometric representation and parameterization used in an aerodynamic wing design process determines the number of design variables and influences the smoothness of the wing representation in an attempt to reduce the number of design variables while preserving good smoothness properties the present research investigates the performance of an optimized nonuniform rational b spline nurbs geometrical representation for the aerodynamic design of wings as a first step an approach is described whereby optimal spatial positions and weights of a fixed number of nurbs control points is determined using a quasi newton optimization algorithm to approximate a general airfoil section the resulting optimized nurbs representation significantly reduces the number of design variables needed to define accurately a wing section while ensuring good smoothness properties in a second step the nurbs control point positions and weights are used as design variables in an aerodynamic optimization problem this methodology results in a rapid and robust design process as illustrated by examples of aerodynamic optimization for two and three dimensional cases

the modified decomposition method applied to unsteady flow of gas through a porous medium in this paper an analytic solution for the unsteady flow of gas through a porous medium is constructed our approach rests on the recently developed modification of the adomian decomposition method the diagonal pade approximants are effectively used in the analysis to capture the essential behavior of y x and to determine the initial slope y 0 the proposed scheme avoids the complexity provided by using perturbation techniques

the decomposition method for approximate solution of the goursat problem abstract in this paper the adomian s decomposition method is effectively implemented to establish an analytic solution and a reliable numerical approximation to the goursat partial differential equation linear and nonlinear goursat examples are examined and the method supplied quantitatively reliable results for these types of problems the accuracy level of the results obtained indicates the superiority of the decomposition method over existing numerical methods that were applied to the goursat problem

an analytical solution for the nonlinear energy spectrum equation by the decomposition method we discuss the isotropic turbulence decay and solve the energy density spectrum eds equation considering the inertial transfer energy and viscosity terms using the heisenberg parameterization in the present approach buoyant and shear terms are neglected and turbulence is assumed to be homogeneous and isotropic the nonlinear integro differential equation is solved by adomian s generic decomposition method which yields an analytical recursive expression and upon truncation gives an approximate solution we show the resulting eds and the time dependent decay of the intensity of the turbulent kinetic energy our results prove consistent the heisenberg parameterization for the transfer term of the inertial energy the analytical character of the solution permits a validation of the nonlinear details of the physical model

controllability method for the helmholtz equation with higher order discretizations we consider a controllability technique for the numerical solution of the helmholtz equation the original time harmonic equation is represented as an exact controllability problem for the time dependent wave equation this problem is then formulated as a least squares optimization problem which is solved by the conjugate gradient method such an approach was first suggested and developed in the 1990s by french researchers and we introduce some improvements to its practical realization we use higher order spectral elements for spatial discretization which leads to high accuracy and lumped mass matrices higher order approximation reduces the pollution effect associated with finite element approximation of time harmonic wave equations and mass lumping makes explicit time stepping schemes for the wave equation very efficient we also derive a new way to compute the gradient of the least squares functional and use algebraic multigrid method for preconditioning the conjugate gradient algorithm numerical results demonstrate the significant improvements in efficiency due to the higher order spectral elements for a given accuracy spectral element method requires fewer computational operations than conventional finite element method in addition by using higher order polynomial basis the influence of the pollution effect is reduced

a new approach to the nonlinear advection problem an application of the decomposition method abstract in this paper the nonlinear advection problem is solved easily and elegantly by using the adomian s decomposition method the exact solution is quickly obtained by observing the appearance of the self canceling noise terms whose sum vanishes in the limit comparison with other existing techniques shows that the present approach is highly accurate

a learning approach to the two person decentralized team problem with incomplete information a learning approach to the two person decentralized team problem with incomplete information and with a 2x2 payoff matrix is considered it is shown that if the payoff matrix is unimodal there exists a proper choice of parameters of the learning algorithm that will ensure asymptotically an expected payoff as close to maximum payoff as desired it is shown that a team problem with multimodal payoff matrix gives rise to interesting class of open problems

exact solutions of the generalized 2 1 dimensional nonlinear evolution equations via the modified simple equation method in this paper the modified simple equation method msem is applied to construct exact solutions of the generalized 2 1 dimensional nonlinear evolution equations nlees involving parameters via the 2 1 dimensional calogero bogoyavlenskii schiff cbs equation the 2 1 dimensional breaking soliton equation and the 2 1 dimensional bogoyavlenskii s breaking soliton equation the solitary wave solutions are derived from the exact solutions by assigning special values of the parameters