a solution to the stability issues with block norm summation by parts operators finite difference operators approximating first and second derivatives and satisfying a summation by parts sbp rule have been derived for the 4 t h 6 t h 8 t h and 10 t h order case by using the symbolic mathematics software maple the operators are based on block norms to avoid the curse of losing accuracy at the boundaries that are present for corresponding operators based on diagonal norms the reason why block norm operators have not been used in realistic applications before is related to the well known stability issues on curvilinear grids to avoid this problem an additional boundary stabilization is introduced that removes unstable eigenvalues without interfering with accuracy and stiffness the superior accuracy properties of the newly derived block norm sbp operators will be demonstrated for the second order wave equation in 1 d and for the compressible euler equations in complex 2 d geometries

nonlinear analysis of unsteady flows in multistage turbomachines using harmonic balance a harmonic balance technique for the analysis of two dimensional linear small disturbance and nonlinear unsteady flows in multistage turbomachines is presented the present method uses a mixed time domain frequency domain approach that allows one to compute the unsteady aerodynamic response of multistage machines to both blade vibration the flutter problem and wake interaction the forced response problem in general the flowfield may have multiple excitation frequencies that are not integer multiples of each other so that the unsteady flow is sometimes aperiodic in time using our approach we model each blade row using a computational grid spanning a single blade passage in each blade row we store several subtime level solutions for flows that are periodic in time these subtime levels span a single time period for aperiodic flows the temporal period spanned by these subtime level solutions is sufficiently long to sample the relevant discrete frequencies contained in the aperiodic flow in both cases these subtime level solutions are related to each other through the time derivative terms in the euler or navier stokes equations and boundary conditions complex periodicity conditions connect the subtime levels within a blade passage and interrow boundary conditions connect the solutions among blade rows the resulting discretized equations which are mathematically steady because time derivatives have been replaced by a pseudospectral operator in which the excitation frequencies appear as parameters can be solved very efficiently using multigrid acceleration techniques in this paper we apply the technique to both flutter and wake interaction problems and illustrate the influence of neighboring blade rows on the unsteady aerodynamic response of a blade row

discrete adjoint approach for nonlinear unsteady aeroelastic design optimization presented is a discrete adjoint approach for computing nonlinear unsteady aeroelastic geometric design sensitivities which can subsequently be used for nonlinear unsteady aeroelastic geometric des

the computational efficiency of non linear frequency domain methods this paper demonstrates efficient solver technologies applied to the non linear frequency domain nlfd method the basis of the nlfd method is to assume the time period of the solution s oscillation and to transform both the solution and residual using a discrete fourier transform an unsteady residual is formed in the frequency domain and iteratively driven to a negligible value this method is amenable to many of the convergence acceleration techniques used for steady state flows including pseudo local time stepping implicit residual averaging coarse grid viscosity and multigrid this paper will address the implementation of these techniques such that convergence rates of the modified unsteady solver are equivalent to those of the original steady state techniques

natural laminar flow airfoil optimization design using a discrete adjoint approach natural laminar flow wings are one of the most promising technologies for reducing fuel burn and emissions for commercial aviation however there is a lack of tools for performing shape optimizati

a simplified and flexible variant of gcrot for solving nonsymmetric linear systems there is a need for flexible iterative solvers that can solve large scale 10 6 unknowns nonsymmetric sparse linear systems to a small tolerance among flexible solvers flexible gmres fgmres is attractive because it minimizes the residual norm over a particular subspace in practice fgmres is often restarted periodically to keep memory and work requirements reasonable however like restarted gmres restarted fgmres can suffer from stagnation this has led us to develop a flexible variant of the krylov linear solver gcrot generalized conjugate residual with inner orthogonalization and outer truncation unlike the original gcrot algorithm the proposed gcrot variant uses a simplified truncation strategy similar to loose gmres lgmres this modification is motivated by numerical experiments that suggest the specific subspace retained in the outer iteration of gcrot is less important than its size the flexible gcrot variant appears to be well suited for advection dominated problems in particular when applied to an adjoint problem from computational aerodynamics the proposed gcrot variant is robust and efficient compared with several popular truncated krylov subspace methods finally a flexible version of lgmres is easily constructed by recognizing algorithmic similarities to gcrot

cross validation of aerodynamic shape optimization methodologies for aircraft wing body optimization this paper presents the cross validation of two aerodynamic shape optimization methodologies in order to validate characterize and compare the two methodologies both methodologies use gradient b

optimum shape design for unsteady flows with time accurate continuous and discrete adjoint methods this paper presents an adjoint method for the optimal control of unsteady flows the goal is to develop the continuous and discrete unsteady adjoint equations and their corresponding boundary conditions for the time accurate method first this paper presents the complete formulation of the time dependent optimal design problem second we present the time accurate unsteady continuous and discrete adjoint equations third we present results that demonstrate the application of the theory to a two dimensional oscillating airfoil the results are compared with a multipoint approach to illustrate the added benefit of performing full unsteady optimization

review and unification of methods for computing derivatives of multidisciplinary computational models this paper presents a review of all existing discrete methods for computing the derivatives of computational models within a unified mathematical framework this framework hinges on a new equation the unifying chain rule from which all the methods can be derived the computation of derivatives is described as a two step process the evaluation of the partial derivatives and the computation of the total derivatives which are dependent on the partial derivatives finite differences the complex step method and symbolic differentiation are discussed as options for computing the partial derivatives it is shown that these are building blocks with which the total derivatives can be assembled using algorithmic differentiation the direct and adjoint methods and coupled analytic methods for multidisciplinary systems each of these methods is presented and applied to a common numerical example to demonstrate and compare the various approaches the paper ends with a discussion of current challenges and possib

induced drag minimization of nonplanar geometries based on the euler equations the induced drag of several nonplanar configurations is minimized using an aerodynamic shape optimization algorithm based on the euler equations the algorithm is first validated using twist optimization to recover an elliptical lift distribution planform optimization reveals that an elliptical planform is not optimal when side edge separation is present optimized winglet and box wing geometries are found to have span efficiencies that agree well with lifting line analysis provided the bound constraints on the entire geometry are accounted for in the linear analyses for the same spanwise and vertical bound constraints a nonplanar split tip geometry outperforms both the winglet and box wing geometries because it can more easily maximize the vertical extent at the tip the performance of all the optimized geometries is verified using refined grids consisting of 88 152 million nodes

aerodynamic optimization algorithm with integrated geometry parameterization and mesh movement an efficient gradient based algorithm for aerodynamic shape optimization is presented the algorithm consists of several components including a novel integrated geometry parameterization and mesh movement a parallel newton krylov flow solver and an adjoint based gradient evaluation to integrate geometry parameterization and mesh movement generalized b spline volumes are used to parameterize both the surface and volume mesh the volume mesh of b spline control points mimics a coarse mesh a linear elasticity mesh movement algorithm is applied directly to this coarse mesh and the fine mesh is regenerated algebraically using this approach mesh movement time is reduced by two to three orders of magnitude relative to a node based movement the mesh adjoint system also becomes smaller and is thus amenable to complex step derivative approximations when solving the flow adjoint equations using restarted krylov subspace methods a nested subspace strategy is shown to be more robust than truncating the entire subspace optimization is accomplished using a sequential quadratic programming algorithm the effectiveness of the complete algorithm is demonstrated using a lift constrained induced drag minimization that involves large changes in geometry

horizontal and vertical formulas for exponential riordan matrices and their applications abstract in this paper we show that an infinite lower triangular matrix a a i j i j n 0 is an exponential riordan matrix a e g f given by i j a i j z i i g f j j if and only if there exist both a horizontal pair h n h n n 0 and a vertical pair v n v n n 0 of sequences that represent all the elements in the matrix as a consequence we obtain that if the horizontal and vertical pairs of an exponential riordan matrix are identical then the matrix is an involution in addition this concept can be applied to obtain the determinants of the production matrix and some conditions for the d orthogonality of the sheffer polynomial sequences

riordan group involutions we study involutions in the riordan group especially those with combinatorial meaning we give a new determinantal criterion for a matrix to be a riordan involution and examine several classes of examples a complete characterization of involutions in the appell subgroup is developed in another direction we find several examples that generalize the rna matrix but are of independent interest

riordan arrays and d orthogonality abstract using the theory of the riordan group and d orthogonal polynomials we shall show that sequences of d orthogonal polynomials can be also generated by the riordan group we interpret some families of d orthogonal polynomials within the framework of the riordan group

continuum sensitivity analysis for structural shape design variables using finite volume method two of the most important requirements when using gradient based optimization for fluid structure interaction problems are efficiency and accuracy in calculating the sensitivities as a result analytical continuum sensitivity formulations are finding their place due to their lower cost and accuracy however the necessary computation of mesh sensitivities in shape optimization is becoming a bottleneck especially when handling complex geometries in this research an algorithm based on continuum sensitivity analysis for structural shape design variables is developed in the proposed method regularized heaviside functions are used to modify the properties of mesh cells therefore mesh dependency is removed from the sensitivity equations this method does not require moving or modifying the mesh to handle the shape changes this method is applied to two different validation cases where continuum sensitivity equations are formulated and solved validation cases are selected as a structural and thermal pro

truncation strategies for optimal krylov subspace methods optimal krylov subspace methods like gmres and gcr have to compute an orthogonal basis for the entire krylov subspace to compute the minimal residual approximation to the solution therefore when the number of iterations becomes large the amount of work and the storage requirements become excessive in practice one has to limit the resources the most obvious ways to do this are to restart gmres after some number of iterations and to keep only some number of the most recent vectors in gcr this may lead to very poor convergence and even stagnation therefore we will describe a method that reveals which subspaces of the krylov space were important for convergence thus far and exactly how important they are this information is then used to select which subspace to keep for orthogonalizing future search directions numerical results indicate this to be a very effective strategy

torsional springs for two dimensional dynamic unstructured fluid meshes abstract dynamic fluid grids are commonly used for the solution of flow problems with moving boundaries they are often represented by a network of fictitious lineal springs that can become unreliable when the fluid mesh undergoes large displacements and or deformations in this paper we propose to control the arbitrary motion of two dimensional dynamic unstructured fluid grids with additional torsional springs we show that such springs can be designed to prohibit the interpenetration of neighboring triangles and therefore to provide the method of spring analogy with the robustness needed for enlarging its range of applications we illustrate our new dynamic mesh motion algorithm with several examples that highlight its advantages in terms of robustness quality and performance

three dimensional aerodynamic shape optimization using discrete sensitivity analysis an aerodynamic shape optimization procedure based on discrete sensitivity analysis is extended to treat three dimensional geometries the function of sensitivity analysis is to directly couple computational fluid dynamics cfd with numerical optimization techniques which facilitates the construction of efficient direct design methods the development of a practical three dimensional design procedures entails many challenges such as 1 the demand for significant efficiency improvements over current design methods 2 a general and flexible three dimensional surface representation and 3 the efficient solution of very large systems of linear algebraic equations it is demonstrated that each of these challenges is overcome by 1 employing fully implicit newton methods for the cfd analyses 2 adopting a bezier bernstein polynomial parameterization of two and three dimensional surfaces and 3 using preconditioned conjugate gradient like linear system solvers whereas each of these extensions independently yields an improvement in computational efficiency the combined effect of implementing all the extensions simultaneously results in a significant factor of 50 decrease in computational time and a factor of eight reduction in memory over the most efficient design strategies in current use the new aerodynamic shape optimization procedure is demonstrated in the design of both two and three dimensional inviscid aerodynamic problems including a two dimensional supersonic internal external nozzle two dimensional transonic airfoils resulting in supercritical shapes three dimensional transport wings and three dimensional supersonic delta wings each design application results in realistic and useful optimized shapes

dual consistency and functional accuracy consider the discretization of a partial differential equation pde and an integral functional that depends on the pde solution the discretization is dual consistent if it leads to a discrete dual problem that is a consistent approximation of the corresponding continuous dual problem consequently a dual consistent discretization is a synthesis of the so called discrete adjoint and continuous adjoint approaches we highlight the impact of dual consistency on summation by parts sbp finite difference discretizations of steady state pdes specifically superconvergent functionals and accurate functional error estimates in the case of functional superconvergence the discrete adjoint variables do not need to be computed since dual consistency on its own is sufficient numerical examples demonstrate that dual consistent schemes significantly outperform dual inconsistent schemes in terms of functional accuracy and error estimate effectiveness the dual consistent and dual inconsistent discretizations have similar computational costs so dual consistency leads to improved efficiency to illustrate the dual consistency analysis of sbp schemes we thoroughly examine a discretization of the euler equations of gas dynamics including the treatment of the boundary conditions numerical dissipation interface penalties and quadrature by sbp norms

a flexible inner outer preconditioned gmres algorithm a variant of the gmres algorithm is presented that allows changes in the preconditioning at every step there are many possible applications of the new algorithm some of which are briefly discussed in particular a result of the flexibility of the new variant is that any iterative method can be used as a preconditioner for example the standard gmres algorithm itself can be used as a preconditioner as can cgnr or cgne the conjugate gradient method applied to the normal equations however the more appealing utilization of the method is in conjunction with relaxation techniques possibly multilevel techniques the possibility of changing preconditioners may be exploited to develop efficient iterative methods and to enhance robustness a few numerical experiments are reported to illustrate this fact

a comparison of optimization based approaches for a model computational aerodynamics design problem abstract the objective of this paper is to compare three optimization based methods for solving aerodynamic design problems we use the euler equations for one dimensional duct flow as a model problem the optimization methods are i the black box method with finite difference gradients ii a modification where gradients are found by an algorithm based on the implicit function theorem and iii an all atonce method where the flow and design variables are simultaneously altered the three methods are applied to the model problem and compared for efficiency robustness and implementation difficulty we also show that the black box implicit gradient method is equivalent to applying the variational or optimal control approach to design optimization directly to the discretized analysis problem rather than to the continuous problem as is usually done the black box method with implicit gradients seems to provide a good compromise of features and can be retrofitted to most existing analysis codes to turn them into design codes although the all at once method was found to be less robust than the black box methods when it succeeded it was considerably more efficient

recurrence relations for polynomial sequences via riordan matrices we give recurrence relations for any family of generalized appell polynomials unifying so some known recurrences for many classical sequences of polynomials our main tool to get our goal is the riordan group we use the product of riordan matrices to interpret some relationships between different polynomial families moreover using the hadamard product of series we get a general recurrence relation for the polynomial sequences associated to the so called generalized umbral calculus

algorithm developments for discrete adjoint methods this paper presents a number of algorithm developments for adjoint methods using the discrete approach in which the discretisation of the non linear equations is linearised and the resulting matrix is then transposed with a new iterative procedure for solving the adjoint equations exact numerical equivalence is maintained between the linear and adjoint discretisations the incorporation of strong boundary conditions within the discrete approach is discussed as well as a new application of adjoint methods to linear unsteady flow in turbomachinery

multipoint and multi objective aerodynamic shape optimization a newton krylov algorithm is presented for the aerodynamic optimization of singleand multi element airfoil configurations the flow is governed by the compressible navier stokes equations in conjunction with a one equation turbulence model the preconditioned generalized minimum residual method is applied to solve the discreteadjoint equation leading to a fast computation of accurate objective function gradients optimization constraints are enforced through a penalty formulation and the resulting unconstrained problem is solved via a quasi newton method design examples include lift enhancement and multi point lift constrained drag minimization problems furthermore the new algorithm is used to compute a pareto front for a multi objective problem and the results are validated using a genetic algorithm overall the new algorithm provides an ecient and robust approach for addressing the issues of complex aerodynamic

demonstration of nonlinear frequency domain methods method in which users deliberately choose not to resolve temporal modes in the solution a variable time period method has been proposed such that the nonlinear frequency domain method can be applied to problems where the time period of the unsteadiness is either known or unknown a priori to validate the latter case results from this method have been compared with experimental results of vortex shedding in low reynolds number flows past cylinders validation of the first case utilizes experimental data of a pitching airfoil in transonic flow these comparisons demonstrate the efficiency of the nonlinear frequency domain method in representing complex nonlinear flow field physics with a limited number of temporal modes

mesh movement for a discrete adjoint newton krylov algorithm for aerodynamic optimization a grid movement algorithm based on the linear elasticity method with multiple increments is presented the methodisrelativelycomputationallyexpensivebutisexceptionallyrobust producinghigh qualityelementsevenfor large shape changes it is integrated with an aerodynamic shape optimization algorithm that uses an augmented adjoint approach for gradient calculation the discrete adjoint equations are augmented to explicitly include the sensitivities of the mesh movement resulting in an increase in efficiency and numerical accuracy this gradient computation method requires less computational time than a function evaluation and leads to significant computational savings as dimensionality is increased the results of the application of these techniques to several large deformation and optimization cases are presented

convergence error and higher order sensitivity estimations differential equation with respect to the design variables the accuracies of the convergence error and higher order sensitivity estimation methods are verified using laplace euler and navier stokes equations the developed methods are used to improve the accuracy of the finite difference sensitivity calculations in iteratively solved problems aboundonthenormvalueofthe finite differencesensitivityerrorinthestatevariablesisminimizedwith respect to the finite difference step size the optimum finite difference step size is formulated as a function of the norm values of both convergenceerror andhigher order sensitivities thesensitivities calculated with the analytical and the finite difference methods are compared the performance of the proposed methods on the convergence of inverse design optimization is evaluated

some algebraic structure of the riordan group abstract in this paper we highlight some of the algebraic properties of the riordan group we present some new properties and provide proofs for some known properties for instance proofs of properties of the power bell and derivative subgroups are presented we also give a new property for generating stochastic riordan matrices which are elements of the stochastic riordan subgroup centralizers stabilizers and certain riordan group isomorphisms are also given and proved we conclude by proving some properties connecting similar riordan matrices and pseudo involutions in the riordan group

accurate high order representation of complex three dimensional surfaces via fourier continuation analysis we present a new method for construction of high order parametrizations of surfaces starting from point clouds the method we propose can be used to produce full surface parametrizations by sets of local charts each one representing a large surface patch which typically contains thousands of the points in the original point cloud for complex surfaces of scientific and engineering relevance the proposed approach accurately renders both smooth and non smooth portions of a surface it yields super algebraically convergent fourier series approximations to a given surface up to and including all points of geometric singularity such as corners edges conical points etc in view of their c smoothness except at true geometric singularities and their properties of high order approximation the surfaces produced by this method are suitable for use in conjunction with high order numerical methods for boundary value problems in domains with complex boundaries including pde solvers integral equation solvers etc our approach is based on a very simple concept use of fourier analysis to continue smooth portions of a piecewise smooth function into new functions which defined on larger domains are both smooth and periodic the continuation functions arising from a function f converge super algebraically to f in its domain of definition as discretizations are refined we demonstrate the capabilities of the proposed approach for a number of surfaces of engineering relevance

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

comparison of b spline surface and free form deformation geometry control for aerodynamic optimization two aerodynamic shape optimization geometry control methods b spline surface control and free form deformation are applied to three optimization problems and compared on the bases of optimal shap

an algebraic structure for faber polynomials abstract let be the set of functions convergent for all z 1 with a laurent series of the form f z z n 0 a n z n in this paper we prove that the set of faber polynomial sequences over and the set of their normalized kth derivative sequences form groups which are isomorphic to the hitting time subgroup and the bell k subgroup of the riordan group respectively further a relationship between such faber polynomial sequences and lucas and sheffer polynomial sequences is derived

on approximate solutions of the linear functional equation of higher order abstract we show that under some assumptions every approximate solution of the linear functional equation of higher order in single variable generates a solution of the equation that is close to it we also give a description of a procedure that yields such a solution estimate the distance between those approximate and exact solutions to the equation and discuss the problem of uniqueness moreover as a consequence we obtain some results concerning the hyers ulam stability of the equation

airfoil and wing design through hybrid optimization strategies real world design problems need robust and effective system level optimization tools inasmuch as they are ruled by several criteria most often in multidisciplinary environments in this work a hybrid optimization algorithm has been obtained by adding a gradient based technique to the set of operators of a multiobjective genetic algorithm this makes it possible to increase the computational efficiency of the genetic algorithm while preserving its favorable features of robustness problem independence and multiobjective optimization capabilities aerodynamic shape design problems including both airfoil and wing designs are considered

two level free form and axial deformation for exploratory aerodynamic shape optimization an intuitive shape parameterization and control technique suitable for high fidelity aerodynamic shape optimization is presented it relies on the principles of free form and axial deformation enabling thorough exploration of the design space while keeping the number of design variables manageable surface sensitivities to the design variables are readily available their inclusion in a highly efficient and robust adjoint based optimization methodology involving linearly elastic volume mesh deformation and a newton krylov solver for the euler equations is described the flexibility of the proposed approach is demonstrated through the exploratory shape optimization of a three pronged feathered winglet leading to a span efficiency of 1 19 under a height to span ratio constraint of 0 1 and an optimization of a regional jet wing at transonic speed where a winglet is allowed to develop starting from a planar wingtip extension leading to an 18 8 reduction in drag

domain element method for aerodynamic shape optimization applied to a modern transport wing generic wraparound aerodynamic shape optimization technology is presented and applied to a modern commercial aircraft wing in transonic cruise the wing geometry is parameterized by a novel domain element method which uses efficient global interpolation functions to deform both the surface geometry and corresponding computational fluid dynamics volume mesh the technique also provides a method that allows geometries to be parameterized at various levels ranging from global three dimensional planform alterations to detailed local surface changes combining all levels of parameterization allows for free form design control with very few design variables the method provides an efficient combined shape parameterization and high quality mesh deformation technique that is totally independent of mesh type structured or unstructured optimization independence from the flow solver is achieved by obtaining sensitivity information for an advanced gradient based optimizer by finite differences the entire optimization suite has also been parallelized to allow optimization with highly flexible parameterization in practical times results are presented for highly constrained optimizations of the modern aircraft wing in transonic cruise using three levels of parameterization number of design variables to assess the effect of parameterization level on the optimization the highest level optimization results in a totally shock free geometry with an associated substantial reduction in drag

adjoint an approach for the rapid development of discrete adjoint solvers an automatic differentiation tool is used to develop the adjoint code for a three dimensional computational fluid dynamics solver rather than using automatic differentiation to differentiate the entire source code of the computational fluid dynamics solver we have applied it selectively to produce code that computes the flux jacobian matrix and the other partial derivatives that are necessary to compute total derivatives using an adjoint method the resulting linear discrete adjoint system is then solved using the portable extensible toolkit for scientific computation this selective application of automatic differentiation is the central idea behind the automatic differentiation adjoint adjoint approach this approach has the advantage that it is applicable to arbitrary sets of governing equations and cost functions and that it is exactly consistent with the gradients that would be computed by exact numerical differentiation of the original solver furthermore the approach is largely automatic thus avoiding the lengthy development times usually required to develop adjoint solvers for partial differential equations these significant advantages come at the cost of increased memory requirements for the adjoint solver derivatives of drag and lift coefficients are validated and the low computational cost and ease of implementation of the method are shown

using complex variables to estimate derivatives of real functions a method to approximate derivatives of real functions using complex variables which avoids the subtractive cancellation errors inherent in the classical derivative approximations is described numerical examples illustrating the power of the approximation are presented

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

the stability of d alembert and jensen type functional equations abstract the aim of this paper is to study the stability problem of the d alembert type and jensen type functional equations f x y f x y 2 g x f y f x y f x y 2 f x g y f x y f x y 2 f x

computing stability derivatives and their gradients for aerodynamic shape optimization aerodynamic shape optimization of aircraft configurations often ignores stability considerations to address this a method for the computation of static dynamic and transient aircraft stability derivatives and their sensitivities for use in gradient based optimization is introduced and evaluated computational fluid dynamics in the form of a three dimensional structured grid multiblock flow solver with both euler and reynolds averaged navier stokes equations is used to compute the stability derivatives a time spectral formulation is used to compute an oscillating solution for the configuration of interest from this oscillating solution a series of linear regressions is performed to calculate the various stability derivatives because the solution is time dependent it contains the information required to compute the transient or dot derivatives for the configuration an adjoint method is used to compute the gradients of the stability derivatives of interest enabling gradient based stability co

bijections and the riordan group one of the cornerstone ideas in mathematics is to take a problem and to look at it in a bigger space in this paper we examine combinatorial sequences in the context of the riordan group various subgroups of the riordan group each give us a different view of the original sequence in many cases this leads to both a combinatorial interpretation and to eco rewriting rules in this paper we will concentrate on just four of the subgroups of the riordan group to demonstrate some of the possibilities of this approach

efficient hessian calculations using automatic differentiation and the adjoint method with applications in this paper an efficient general algorithm to calculate the hessian of a steady or unsteady functional of interest in the context of computational fluid dynamics is outlined verified and applied to an aerodynamic optimization and to an extrapolation example the successful extrapolation is then applied to approximate monte carlo simulations for artificial geometric uncertainty analysis the presented optimization and extrapolation examples demonstrate that the combination of automatic differentiation and an adjoint method to calculate the hessian of a steady or unsteady objective function and thereby obtaining second order information can be an efficient tool for optimization and uncertainty quantification

output error estimation for summation by parts finite difference schemes the paper develops a posteriori error estimates of integral output functionals for summation by parts finite difference methods the error estimates are based on the adjoint weighted residual method and take advantage of a variational interpretation of summation by parts discretizations the estimates are computed on a fixed grid and do not require an embedded grid or explicit interpolation operators for smooth boundary value problems containing first and second derivatives the error estimates converge to the exact error as the mesh is refined the theory is verified using linear boundary value problems and the euler equations

sensitivity analysis for navier stokes equations on unstructured meshes using complex variables the use of complex variables for determining sensitivity derivatives for turbulent flows is examined although a step size parameter is required the numerical derivatives are not subject to subtractive cancellation errors and therefore exhibit true second order accuracy as the step size is reduced as a result this technique guarantees two additional digits of accuracy each time the step size is reduced one order of magnitude this behavior is in contrast to the use of finite differences which suffer from inaccuracies due to subtractive cancellation errors in addition the complex variable procedure is easily implemented into existing codes

on the stability of the pexiderized trigonometric functional equation abstract the aim of this paper is to investigate the superstability problem of the cosine d alembert the wilson and the sine functional equation from the pexiderized trigonometric functional equation t gh under the condition f x y f x y 2 g x h y x or y and also will investigated the stability of the jensen type functional equation j y under the condition f x y f x y 2 f y x y

derivatives for time spectral computational fluid dynamics using an automatic differentiation adjoint sensitivities for an oscillating onera m6 wing the sensitivities are shown to be accurate to 8 12 digits and the computational cost of the adjoint computations is shown to scale well up to problems of more than 41 million state variables

scalable parallel approach for high fidelity steady state aeroelastic analysis and adjoint derivative computations aeroelastic systems achieve the best performance when the aerodynamic shape and structural sizing are optimized concurrently but such an optimization is challenging when high fidelity aerodynamic and structural models are required this paper addresses this challenge through several significant improvements fully coupled newton krylov methods are presented for the solution of aerostructural systems and for the corresponding adjoint systems the coupled adjoint method presented can compute gradients with respect to thousands of multidisciplinary design variables accurately and efficiently this is enabled by several improvements in the computation of the multidisciplinary terms in the coupled adjoint the parallel scalability of the methods is demonstrated for a full aircraft configuration using an euler computational fluid dynamics model with more than 8 106 state variables and a detailed structural finite element model of the wing with more than 1 106 degrees of freedom the coupled newton krylov metho

recurrence relations for the sheffer sequences abstract in this paper using the production matrix of an exponential riordan array g t f t we give a recurrence relation for the sheffer sequence for the ordered pair g t f t we also develop a new determinant representation for the general term of the sheffer sequence as applications determinant expressions for some classical sheffer polynomial sequences are derived

distributed schur complement techniques for general sparse linear systems this paper presents a few preconditioning techniques for solving general sparse linear systems on distributed memory environments these techniques utilize the schur complement system for deriving the preconditioning matrix in a number of ways two of these preconditioners consist of an approximate solution process for the global system which exploits approximate lu factorizations for diagonal blocks of the schur complement another preconditioner uses a sparse approximate inverse technique to obtain certain local approximations of the schur complement comparisons are reported for systems of varying difficulty

stability of a generalized trigonometric functional equation the stability of the functional equation f x y g x y 2h x k y over the domain of an abelian group g and the range of the complex field is investigated several related results extending a number of previously known ones such as the ones dealing with the sine functional equation the d alembert functional equation and wilson functional equation are derived as direct consequences applying the main result to the setting of banach algebra it is shown that if their operators satisfy a functional inequality and are subject to certain natural requirements then these operators must be solutions of some well known functional equations

parallel newton krylov schur flow solver for the navier stokes equations the objective of the present paper is to demonstrate the effectiveness of a spatial discretization based on summation by parts operators with simultaneous approximation terms in combination with a parallel newton krylov schur algorithm for solving the three dimensional reynolds averaged navier stokes equations coupled with the spalart allmaras one equation turbulence model the algorithm employs second order summation by parts operators on multiblock structured grids with simultaneous approximation terms to enforce block interface coupling and boundary conditions the discrete equations are solved iteratively with an inexact newton method while the linear system at each newton iteration is solved using a flexible krylov subspace iterative method with an approximate schur parallel preconditioner the algorithm is verified and validated through the solution of two dimensional model problems highlighting the correspondence of the current algorithm with several established flow solvers a transonic solution

some discrete d orthogonal polynomial sets in this paper we characterize all polynomial sets which are at the same time d orthogonal and appell the resulting polynomials reduce to charlier polynomials for d 1 1 various properties of the obtained polynomials are singled out generating function recurrence relation of order d 1 and a difference equation of order d 1 we also explicitly express the d dimensional functional for which the d orthogonality holds

a stability of the generalized sine functional equations abstract in this paper we will investigate the stability of the generalized pexiderized sine functional equation g x h y f x y 2 2 f x y 2 2