numerical solution of volterra integral and integro differential equations of convolution type by using operational matrices of piecewise constant orthogonal functions in this paper we use operational matrices of piecewise constant orthogonal functions on the interval 0 1 to solve volterra integral and integro differential equations of convolution type without solving any system we first obtain laplace transform of the problem and then we find numerical inversion of laplace transform by operational matrices numerical examples show that the approximate solutions have a good degree of accuracy

aeroelastic dynamic analysis of a full f 16 configuration for various flight conditions an overview is given of recent advances in a three field methodology for modeling and solving nonlinear fluid structure interaction problems and its application to the prediction of the aeroelastic frequencies and damping coefficients of a full f 16 configuration in various subsonic transonic and supersonic airstreams is reported in this three field methodology the flow is described by the arbitrary lagrangian eulerian form of the euler equations the structure is represented by a detailed finite element model and the fluid mesh is unstructured dynamic and updated by a robust torsional spring analogy method simulation results are presented for stabilized accelerated low g and high g flight conditions and correlated with flight test data consequently the practical feasibility and potential of the described computational fluid dynamics based computational method for the flutter analysis of high performance aircraft particularly in the transonic regime are discussed

scalar curvature and warped products of riemann manifolds we establish the relationship between the scalar curvature of a warped product m xf n of riemann manifolds and those ones of m and n then we search for weights f to obtain constant scalar curvature on m xf n when m is compact

metric based mathematical derivation of efficient airfoil design variables within an aerodynamic shape optimization framework an efficient shape parameterization and deformation scheme is critical to allow flexible deformation of the surface with the maximum possible design space coverage numerous approaches have been developed for the geometric representation of airfoils a fundamental approach is considered here from the geometric perspective and a method is presented to allow the derivation of efficient generic and orthogonal airfoil geometric design variables this is achieved by the mathematical decomposition of a training library the resulting geometric modes are independent of a parameterization scheme surface and volume mesh and flow solver thus they are generally applicable however these modes are dependent on the training library and so a benchmark performance measure called the airfoil technology factor has also been incorporated into the scheme to allow intelligent metric based filtering or design space reduction of the training library to ensure eff

extrapolation effects on coupled computational fluid dynamics computational structural dynamics simulations between them passing loads from the e ow code to the structural code and displacements back theother way forunsteady problems the sequencing is done within the pseudotime loop used for the e ow solver and removes sequencing errors the mesh movement for the aerodynamic solver is done using transe nite interpolation of displacements 7 we restrict ourselves in this note to the case of a wing modeled structurally by a plate this case is interesting as a building block to more complex structural descriptions transformation of data between the two grids is broken into two components in and out of the plane of the structural model the outplane treatment follows the considerations of ref 5 the in plane treatment turns out to have a much stronger ine uence on the test case considered next the two approaches used are the ine nite plate spline ips 8 and linear interpolation for ips a solution of the thin plate equation is forced to pass through the known structural points and is then evaluated at the projected aerodynamic points for the required displacements for linear interpolation each projected e uid point is associated with a triangle on the structural mesh and the deformation is calculated using a linear approximation within the triangle in both cases the relationship between displacements on the e uid and structural grids is linear and the principle of virtual work is then applied for transferring forces iii test cases

efficient and exact mesh deformation using multiscale rbf interpolation abstract radial basis function rbf interpolation is popular for mesh deformation due to robustness and generality but the cost scales with the number of surface points sourcing the deformation as o n s 3 hence there have been numerous works investigating efficient methods using reduced datasets however although reduced data methods are efficient they require a secondary method to treat an error vector field to ensure surface points not included in the primary deformation are moved to the correct location and the volume mesh moved accordingly a new method is presented which captures global and local motions at multiple scales using all the surface points and so no correction stage is required all surface points are used and a single interpolation built but the cost and conditioning issues associated with rbf methods are eliminated moreover the sparsity introduced is exploited using a wall distance function to further reduce the cost the method is compared to an efficient greedy method and it is shown mesh quality is always comparable with or better than with the greedy method and cost is comparable or cheaper at all stages surface mesh preprocessing is the dominant cost for reduced data methods and this cost is reduced significantly here greedy methods select points to minimise interpolation error requiring repeated system solution and cost o n r e d 4 to select n r e d points the multiscale method has no error and the problem is transferred to a geometric search with cost o n s log n s resulting in an eight orders of magnitude cost reduction for three dimensional meshes furthermore since the method is dependent on geometry not deformation it only needs to be applied once prior to simulation as the mesh deformation is decoupled from the point selection process

pde constrained optimization with error estimation and control the paper describes an algorithm for pde constrained optimization that controls numerical errors using error estimates and grid adaptation during the optimization process a key aspect of the algorithm is the use of adjoint variables to estimate errors in the first order optimality conditions multilevel optimization is used to drive the optimality conditions and their estimated errors below a specified tolerance the error estimate requires two additional adjoint solutions but only at the beginning and end of each optimization cycle moreover the adjoint systems can be formed and solved with limited additional infrastructure beyond that found in typical pde constrained optimization algorithms the approach is general and can accommodate both reduced space and full space formulations of the optimization problem the algorithm is illustrated using the inverse design of a nozzle constrained by the quasi one dimensional euler equations

two dimensional legendre wavelet method for travelling wave solutions of time fractional generalized seventh order kdv equation in this paper a new method based on the legendre wavelets expansion together with operational matrices of fractional integration and derivative of these basis functions is proposed to solve time fractional seventh order kdv skdv equation two dimensional legendre wavelet method is applied to compute the numerical solution of nonlinear time fractional skdv equation the approximate solutions of nonlinear time fractional skdv equation thus obtained by legendre wavelet method are compared with the exact solutions as well as homotopy analysis method ham the present scheme is very simple effective and convenient for obtaining numerical solution of the skdv equation

shape optimal design using b splines abstract shape optimal design of an elastic structure is formulated using a design element technique it is shown that bezier and b spline curves typical of the cad philosophy are well suited to the definition of design elements complex geometries can be described in a very compact way by a small set of design variables and a few design elements because of the b splines flexibility it is no longer necessary to piece design elements together in order to agree with the shape complexity nor to restrict the shape variations moreover the additional optimization constraints that are most often needed to avoid unrealistic designs when the shape variables are the nodal coordinates of a finite element mesh are automatically taken into account in the new formulation an analytical derivation of the sensitivity analysis will be established giving rise to numerical efficiency it will be seen that the resulting optimization problem does not involve highly nonlinear functions with respect to the shape variables so that simple mathematical programming algorithms can be applied to solve it some numerical examples are offered to demonstrate the power and generality of the new approach presented in this paper

legendre wavelets method for solving differential equations of lane emden type a numerical method for solving the lane emden equations as singular initial value problems is presented using integral operator and convert lane emden equations to integral equations and then applying legendre wavelet approximations the properties of legendre wavelet are first presented these properties together with the gaussian integration method are then utilized to reduce the integral equations to the solution of algebraic equations illustrative examples are included to demonstrate the validity and applicability of the technique

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

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

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

on the parabolic kernel of the schrodinger operator etude des equations paraboliques du type q x t t u x t 0 sur une variete riemannienne generale introduction estimations de gradients inegalites de harnack majorations et minorations des solutions fondamentales equation de la chaleur et noyau de green operateur de schrodinger

an implicit difference method for the numerical solution of coupled systems of partial differential equations an explicit and computable solution for coupled partial difference systems appearing when one considers difference approximations of coupled systems of partial differential equations is given our approach is based on the consideration of a discrete matrix separation of variables method by using discrete fourier series we avoid the problem of solving algebraic linear systems

special standard static space times abstract essentially some conditions for the riemannian factor and the warping function of a standard static space time are obtained in order to guarantee that no nontrivial warping function on the riemannian factor can make the standard static space time einstein

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

an improved subspace selection algorithm for meshless collocation methods choosing data points is a common problem for researchers who employ various meshless methods for solving partial differential equations on the one hand high accuracy is always desired on the other ill conditioning problems of the resultant matrices which may lead to unstable algorithms prevent some researchers from using meshless methods for example the optimal placements of source points in the method of fundamental solutions or of the centers in the radial basis functions method are always unclear intuitively such optimal locations will depend on many factors the partial differential equations the domain the trial basis used i e the employed method itself 1pt the computational precisions some user defined parameters and so on such complexity makes the hope of having an optimal centers placement unpromising in this paper we provide a data dependent algorithm that adaptively selects centers based on all the other variables copyright 2009 john wiley sons ltd

application of wavelet collocation method for hyperbolic partial differential equations via matrices abstract in this work we developed an efficient computational method based on legendre and chebyshev wavelets to find an approximate solution of one dimensional hyperbolic partial differential equations hpdes with the given initial conditions the operational matrices of integration for legendre and chebyshev wavelets are derived and utilized to transform the given pde into the linear system of equations by combining collocation method convergence analysis and error estimation associated to the presented idea are also investigated under several mild conditions numerical experiments confirm that the proposed method has good accuracy and efficiency moreover the use of legendre and chebyshev wavelets are found to be accurate simple and fast

convergence analysis of bernoulli matrix approach for one dimensional matrix hyperbolic equations of the first order abstract in this paper an approximate approach based on bernoulli operational matrices has been presented to obtain the numerical solution of first order matrix hyperbolic partial differential equations under the given initial conditions after using the operational matrices we use the completeness of bernoulli basis in which the main problem reduces to a system of algebraic equations the solutions of this algebraic system are the coefficients of the truncated double bernoulli series which are defined in the interval 0 1 also convergence analysis associated to the presented idea is provided under several mild conditions several numerical examples are considered to show the efficiency of the technique

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

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

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

gradient limiting shape control for efficient aerodynamic optimization local shape control methods such as b spline surfaces are well conditioned such that they allow high fidelity design optimization however this comes at the cost of degraded optimization converg

wavelet methods for fast resolution of elliptic problems this paper shows that the use of wavelets to discretize an elliptic problem with dirichlet or neumann boundary conditions has two advantages an explicit diagonal preconditioning makes the condition number of the corresponding matrix become bounded by a constant and the order of approximation is locally of spectral type in contrast with classical methods using a conjugate gradient method one thus obtains fast numerical algorithms of resolution a comparison is also drawn between wavelet and classical methods

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

the operational matrices of bernstein polynomials for solving the parabolic equation subject to specification of the mass abstract some physical problems in science and engineering are modelled by the parabolic partial differential equations with nonlocal boundary specifications in this paper a numerical method which employs the bernstein polynomials basis is implemented to give the approximate solution of a parabolic partial differential equation with boundary integral conditions the properties of bernstein polynomials and the operational matrices for integration differentiation and the product are introduced and are utilized to reduce the solution of the given parabolic partial differential equation to the solution of algebraic equations illustrative examples are included to demonstrate the validity and applicability of the new technique

wavelet approximation scheme for distributed order fractional differential equations abstract this paper is concerned with the study of wavelet approximation scheme based on legendre and chebyshev wavelets for finding the approximate solutions of distributed order linear differential equations the operational matrix for distributed order fractional differential operator is derived for legendre and chebyshev wavelets basis furthermore the obtained operational matrix along with gauss legendre quadrature formula and standard tau method are utilized to reduce the distributed order linear differential equations into the system of linear algebraic equations for the better understanding of the method numerical algorithms are also provided for the considered problems in order to verify the desired accuracy of the proposed method five test examples are included and numerical experiments confirm the theoretical results and illustrate applicability and efficiency of the proposed method also the convergence analysis error bounds error estimate and numerical stability of presented method via legendre and chebyshev wavelets are investigated moreover comparison of the numerical results obtained by the proposed approach is provided with the results of existing method

on combining feasibility descent and superlinear convergence in inequality constrained optimization extension of quasi newton techniques from unconstrained to constrained optimization via sequential quadratic programming sqp presents several difficulties among these are the possible inconsistency away from the solution of first order approximations to the constraints resulting in infeasibility of the quadratic programs and the task of selecting a suitable merit function to induce global convergence in ths case of inequality constrained optimization both of these difficulties disappear if the algorithm is forced to generate iterates that all satisfy the constraints and that yield monotonically decreasing objective function values feasibility of the successive iterates is in fact required in many contexts such as in real time applications or when the objective function is not well defined outside the feasible set it has been recently shown that this can be achieved while preserving local two step superlinear convergence in this note the essential ingredients for an sqp based method exhibiting the desired properties are highlighted correspondingly a class of such algorithms is described and analyzed tests performed with an efficient implementation are discussed

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

a new mesh deformation method based on disk relaxation algorithm with pre displacement and post smoothing graphical abstractdisplay omitted highlights the new dra method is composed of pre displacement disk relaxation and post smoothing the method is valid for large and irregular deformation with complex boundary the mesh quality decreases slowly as the boundary deformation increases the method is more efficient and robust than the semi torsional spring method the method results in high quality mesh especially near the deformed boundary mesh deformation is crucial to the accuracy and efficiency of the numerical simulations involving moving boundaries in this paper the disk relaxation algorithm and background mesh are applied to develop a new mesh deformation method the disk relaxation algorithm rearranges the nodes in the mesh by repulsive and attractive interactions to preserve high mesh quality with the help of background mesh the computational region is divided into layers to transfer the boundary deformation uniformly and effectively in the pre displacement procedure we further introduce the post smoothing procedure to avoid the negative volume of elements numerical applications including the rotation of airfoil rotation and translation of rectangle relative motion of multi body and complex boundary with irregular deformation are carried out the results demonstrate that the new method generates the deformed mesh with high quality even in large deformations especially concerning the boundary meshes comparisons with previous methods show that the deformation capability and efficiency of the new method are better than the widely used semi torsional spring analogy and the mesh quality of the deformed mesh obtained is superior to that of the spring method as well as some new approaches presented recently when they undergo the same deformation in the cases

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

improved radial basis function fluid structure coupling via efficient localized implementation previous work by the authors has developed a universal interpolation scheme using radial basis functions rbfs which results in a unified formulation for robust fluid structure interpolation and high quality mesh motion the method has several significant advantages primarily all volume mesh structural mesh and flow solver type dependence is removed entirely as all operations are performed on totally arbitrary point clouds of any form hence all connectivity requirements are removed from both the coupling and mesh motion problems furthermore only matrix vector multiplications are required during unsteady simulation because dependence relations are computed once prior to any simulation and then remain constant this property means that the method is both perfectly parallel and totally independent from the flow solver however the full method is expensive since the dependence matrix between two sets of points is n points1 n points2 the fluid structure coupling behaviour can also be influenced by parameters used in the interpolation to alleviate these difficulties a more efficient form of the rbf fluid structure coupling is presented which also greatly reduces the interpolation parameter influence a pointwise form of the partition of unity approach is developed that localizes the interpolation with results presented for static aeroelastic simulations of the brite euram multi disciplinary optimization wing using a very fine mesh containing 58000 surface points it is shown that a 58 reduction in data size is achieved and equally importantly the interpolation has a much smaller influence on final aeroelastic results

an arbitrary lagrangian eulerian formulation for solving moving boundary problems with large displacements and rotations an arbitrary lagrangian eulerian ale formulation based on the unstructured finite volume method is proposed for solving moving boundary problems with large displacements and rotations the numerical method is based on the side centered arrangement of the primitive variables that does not require any ad hoc modifications in order to enhance pressure coupling the continuity equation is satisfied within each element at machine precision and the summation of the continuity equations can be exactly reduced to the domain boundary which is important for the global mass conservation a special attention is given to construct an ale algorithm obeying the discrete geometric conservation law dgcl the mesh deformation algorithm is based on the indirect radial basis function rbf algorithm at each time level while avoiding remeshing in order to enhance numerical robustness for the parallel solution of resulting large scale algebraic equations in a fully coupled form a matrix factorization is introduced similar to that of the projection method for the whole system and the parallel algebraic multigrid solver boomeramg is used for the scaled discrete laplacian provided by the hypre library which we access through the petsc library the present numerical algorithm is initially validated for the decaying taylor green vortex flow the flow past an oscillating circular cylinder in a channel and the flow induced by an oscillating sphere in a cubic cavity then the numerical algorithm is applied to the numerical simulation of flow field around a pair of flapping drosophila wings in hover flight the time variation of the eulerian coherent structures in the near wake is shown along with the aerodynamic loads

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

fast wavelet transforms and numerical algorithms i a class of algorithms is introduced for the rapid numerical application of a class of linear operators to arbitrary vectors previously published schemes of this type utilize detailed analytical information about the operators being applied and are specific to extremely narrow classes of matrices in contrast the methods presented here are based on the recently developed theory of wavelets and are applicable to all calderon zygmund and pseudo differential operators the algorithms of this paper require order o n or o n log n operations to apply an n n matrix to a vector depending on the particular operator and the version of the algorithm being used and our numerical experiments indicate that many previously intractable problems become manageable with the techniques presented here

efficient mesh deformation based on radial basis function interpolation by means of the inverse fast multipole method abstract radial basis function interpolation is often employed in mesh deformation algorithms for unstructured meshes for example in fluid structure interaction or design optimization problems this is known to be a robust methodology that results in high quality deformed meshes the applicability of this method to large problems is currently hampered by its prohibitive computational cost however which is caused by the need to solve a dense system of equations the computation time grows as o n b 3 if a direct solver is employed for the solution where n b denotes the number of boundary nodes in the mesh while alternative iterative solvers often suffer from an unfavorable convergence behavior in this paper we present the inverse fast multipole method as a novel fast approximate direct solver with a computational cost scaling as o n b the linear complexity is achieved by transforming the dense system into an extended sparse system along with the compression of certain matrix blocks into low rank factorizations the solver is inexact although the error can be controlled and made as small as needed a low accuracy solver can hence be used as an efficient preconditioner in an iterative scheme numerical benchmarks are presented demonstrating that the proposed approach enhances the computational efficiency of mesh deformation algorithms based on radial basis function interpolation significantly without jeopardizing their robustness and quality

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

a geometric comparison of aerofoil shape parameterisation methods a comprehensive review of aerofoil shape parameterization methods that can be used for aerodynamic shape optimization is presented seven parameterization methods are considered for a range of desi

numerical solution of fredholm integral equations of the first kind by using legendre wavelets abstract an efficient method based on the legendre wavelet approach is proposed for approximate the solution of fredholm integral equations of the first kind the continues legendre wavelets constructed on 0 1 are utilized as a basis in galerkin method to reduce the solution of linear integral equations to a system of algebraic equations for solving this system we use the cg method furthermore we suggest a convergence analysis and error estimation for this method for showing efficiency of method some test problems for which the exact solution is known are considered

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

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

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

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

a fast mesh deformation method using explicit interpolation a novel mesh deformation algorithm for unstructured polyhedral meshes is developed utilizing a tree code optimization of a simple direct interpolation method the algorithm is shown to provide mesh quality that is competitive with radial basis function based methods with markedly better performance in preserving boundary layer orthogonality in viscous meshes the parallelization of the algorithm is described and the algorithm cost is demonstrated to be o nlogn the parallel implementation was used to deform meshes of 100 million nodes on nearly 200 processors demonstrating that the method scales to large mesh sizes results are provided for a simulation of a high reynolds number fluid structure interaction case using this technique

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

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

dynamic unstructured method for flows past multiple objects in relative motion a new methodology is developed to simulate unsteady flows about objects in relative motion and compute the trajectory of their motion as determined by this flowfield the method couples the fluid dynamics and rigid body dynamics equations to capture the time dependent interference between stationary and moving boundaries the unsteady compressible inviscid euler equations are solved on dynamic unstructured grids by an explicit finite volume upwind method for efficiency the grid adaptation is perfomed within a window around the moving object the eulerian equations of the rigid body dynamics are solved by a runge kutta method in a noninertial frame of reference this dynamic unstructured flow solver is validated by computing the flow past a sinusoidally pitching airfoil and comparing these results with the experimental data the trajectory code is tested by computing the six degree of freedom trajectory of a store separating from a wing using the experimentally determined force and moment fields then comparing with the experimental trajectory finally the overall methodology is used for two two dimensional examples the flow past a pitching and plunging airfoil and the free call of a store after separation from a wing section

incremental approach for radial basis functions mesh deformation with greedy algorithm mesh deformation is an important element of any fluidstructure interaction simulation in this article a new methodology is presented for the deformation of volume meshes using incremental radial basis function rbf based interpolation a greedy algorithm is used to select a small subset of the surface nodes iteratively two incremental approaches are introduced to solve the rbf system of equations 1 block matrix inversion based approach and 2 modified lu decomposition approach the use of incremental approach decreased the computational complexity of solving the system of equations within each greedy algorithm s iteration from o n3 to o n2 results are presented from an accuracy study using specified deformations on a 2d surface mesh deformations for bending and twisting of a 3d rectangular supercritical wing have been demonstrated outcomes showed the incremental approaches reduce the cpu time up to 67 as compared to a traditional rbf matrix solver finally the proposed mesh deformation approach was integrated within a fluidstructure interaction solver for investigating a flow induced cantilever beam vibration

aerodynamic shape optimization using geometry surrogates and adjoint method nowadays classical configurations of civil transportation airplanes are well optimized with further improvements expected to be of the order of a single drag count to achieve it the entire airplane has to be taken into account during the aerodynamic shape optimization which requires handling complex geometries with computer aided design software so far the integration of this software and aerodynamic shape optimization has been difficult due to the framework complexity and problematic gradient evaluation in this paper the authors propose to substitute the computer aided design engine with a surrogate model of the surface deformation the resulting framework does not require calls to the computer aided design software during the optimization in addition the surrogate model has been analytically differentiated and coupled with a mesh adjoint method based on the radial basis functions to enable an efficient gradient based optimization the methodology is validated using two cases a three dimensi

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

a monolithic approach to fluid structure interaction using space time finite elements abstract the paper presents a simultaneous solution procedure for fluid structure interaction problems the structural motion is described by geometrically nonlinear elastodynamics the fluid is modeled by the incompressible navier stokes equations the space time finite element method is applied to both continua leading to an almost uniform discretization in which velocity variables are used for fluid and structure in order to enforce momentum conservation and geometric continuity at the interface a weighted residual formulation of coupling conditions is introduced the discretized model equations for fluid structure and coupling conditions are formulated in a single equation system the nonlinear system and the fluid mesh movement are solved in a single iteration loop two dimensional examples of membrane flutter and vortex excited plate vibrations demonstrate the efficiency of the methodology

shifted jacobi spectral galerkin method for solving hyperbolic partial differential equations abstract herein we propose a numerical scheme to solve spectrally hyperbolic partial differential equations hpdes using galerkin method and approximate the solutions using double shifted jacobi polynomials the main characteristic behind this approach is that it reduces such problems to those of solving systems of algebraic equations which greatly simplifies the problem the validity and efficiency of the proposed method are investigated and verified through several examples

a multi dimensional high order dg ale method based on gas kinetic theory with application to oscillating bodies this paper presents a multi dimensional high order discontinuous galerkin dg method in an arbitrary lagrangian eulerian ale formulation to simulate flows over variable domains with moving and deforming meshes it is an extension of the gas kinetic dg method proposed by the authors for static domains x ren et al 2015 22 a moving mesh gas kinetic dg method is proposed for both inviscid and viscous flow computations a flux integration method across a translating and deforming cell interface has been constructed differently from the previous ale type gas kinetic method with piecewise constant mesh velocity at each cell interface within each time step the mesh velocity variation inside a cell and the mesh moving and rotating at a cell interface have been accounted for in the finite element framework as a result the current scheme is applicable for any kind of mesh movement such as translation rotation and deformation the accuracy and robustness of the scheme have been improved significantly in the oscillating airfoil calculations all computations are conducted in a physical domain rather than in a reference domain and the basis functions move with the grid movement therefore the numerical scheme can preserve the uniform flow automatically and satisfy the geometric conservation law gcl the numerical accuracy can be maintained even for a largely moving and deforming mesh several test cases are presented to demonstrate the performance of the gas kinetic dg ale method

reduced surface point selection options for efficient mesh deformation using radial basis functions previous work by the authors has developed an efficient method for using radial basis functions rbfs to achieve high quality mesh deformation for large meshes for volume mesh deformation driven by surface motion the rbf system can become impractical for large meshes due to the large number of surface control points and so a particularly effective data reduction scheme has been developed to vastly reduce the number of surface points used the method uses a chosen error function on the surface mesh to select a reduced subset of the surface points this subset contains a sufficiently small number of points so as to make the volume deformation fast and a correction function is used to correct those surface points not included hence the scheme is split such that both parts are working on appropriate problems rbfs are an excellent way of finding smooth orthogonality preserving global deformations but are less suitable for enforcing an exact geometry for a large number of points while a simpler approach is ideal for diffusing small changes evenly but has quality and possibly expense drawbacks if used for the entire volume however alternatives exist for the error function used to select the reduced data set so here a comparison is made between three different options the surface error function the unit function and the power function tests run on structured and unstructured meshes show that the surface error function gives the lowest errors but this also requires a deformed surface shape to be known in advance of the simulation the unit and power functions both avoid the need for a deformed surface and the unit function is shown to be superior

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

conservative load projection and tracking for fluid structure problems the loose coupling of computational fluid dynamics and computational structural dynamics solvers introduces some problems related to the information transfer between the codes some techniques developed to solve the problems of the load transfer and interface surface tracking are presented the main criterion is to achieve conservation of total loads and total energy the load projection scheme is based on gaussian integration and fast interpolation algorithms for unstructured grids the surface tracking algorithm also based on interpolation is important for many applications including aeroelastic deformation of wings due to aerodynamic loads the methodologies not only improve present fluid structure interaction simulations but also increase the range of their applicability these techniques are of general character and can be used in other multidisciplinary applications as well

a variational flexible multibody formulation for partitioned fluid structure interaction application to bat inspired drones and unmanned air vehicles abstract we present a three dimensional 3d partitioned aeroelastic formulation for a flexible multibody system interacting with incompressible turbulent fluid flow while the incompressible navier stokes system is discretized using a stabilized petrov galerkin procedure the multibody structural system consists of a generic interaction of multiple components such as rigid body beams and flexible thin shells along with various types of joints and connections among them a co rotational framework is utilized for the category of small strain problems where the displacement of the body is decomposed into a rigid body rotation and a small strain component this assumption simplifies the structural equations and allows for the incorporation of multiple bodies rigid as well as flexible in the system the displacement and rotation constraints at the joints are imposed by a lagrange multiplier method the equilibrium conditions at the fluid structure interface are satisfied by the transfer of tractions and structural displacements via the radial basis function approach a scattered data interpolation technique which is globally conservative for the coupled stability in low structure to fluid mass ratio regimes a nonlinear iterative force correction scheme is employed in the partitioned staggered predictor corrector scheme the convergence and generality of the radial basis function mapping are analyzed by carrying out systematic error analysis of the transfer of fluid traction across the non matching fluid structure interface where a third order of convergence is observed the proposed aeroelastic framework is then validated by considering a flow across a flexible pitching plate configuration with serration at the trailing edge finally we demonstrate the flow across a flexible flapping wing of a bat modeling the bone fingers as beams and the flexible membrane as thin shells in the multibody system along with the joints

chebyshev wavelets approach for nonlinear systems of volterra integral equations in this paper a new approach for solving nonlinear systems of volterra integral equations has been proposed the method is based on chebyshev wavelets approximations the method is described and after that the error is analyzed at the end some examples are presented to illustrate the ability and simplify of the method and the results reveal the effectiveness of the technique

adjoint based optimal variable stiffness mesh deformation strategy based on bi elliptic equations there are many recent advances in mesh deformation methods for computational fluid dynamics simulation in deforming geometries we present a method of constructing dynamic mesh around deforming objects by solving the bi elliptic equation an extension of the biharmonic equation we show that introducing a stiffness coefficient fielda x in the bi elliptic equation can enable mesh deformation for very large boundary movements an indicator of the mesh quality is constructed as an objective function of a numerical optimization procedure to find the best stiffness coefficient field a x the optimization is efficiently solved using steepest descent along adjoint based integrated sobolev gradients a multi scenario optimization procedure is performed to calculate the optimal stiffness coefficient fielda x for a priori unpredictable boundary movements simulation of fluid structure interaction problems requires solving the governing equation of the unsteady flow inside a deforming physical domain whose geometry is determined by the deformation of the structure in numerical methods for solving such problems arbitrary lagrangian eulerian ale 1 method is one of the most popular types of approaches it formulates the governing equation in a coordinate system that moves with the boundary and discretizes the deforming domain with a deformable body conforming mesh ale methods are capable of using anisotropic mesh to efficiently resolve boundary layers however these methods requires an algorithm to deform the mesh so that it conforms to the boundary movement while maintaining good mesh quality there has been many recent advances in algorithms for computing mesh deformation in the context of computational fluid dynamics simulations in moving domains two of the most popular categories of such algorithms are interpolation based methods and structure analogy methods interpolation based methods determines the position or displacement of mesh points in the interior of the domain by interpolating the position or displacement of the mesh points on the boundary of the domain an example of interpolation based methods is the radial basis function interpolation method 23 structural analogy methods include methods based on structural models of the discrete computational mesh and methods based on modeling the computational domain as a continuous elastic media recent developments in discrete mesh structural model includes the torsional spring method 45 recent developments in the continuous elasticity model include the biharmonic equation method 6 and the quasi elliptic transformation method 7 in this paper we present a mesh deformation method based on continuous elasticity model our contribution is introducing a variable stiffness coefficient in t he biharmonic elastic model and developing a method to efficiently obtain the optimal stiffness coefficient d istribution we show that the variable stiffness coefficient algorithm can produce a good deformed mesh whilethe original scheme with constant coefficient produces unacceptable mesh to obtain a suitable stiffness coefficient we define a simple global mesh

higher order time integration through smooth mesh deformation for 3d fluid structure interaction simulations in this paper we present a higher order accurate in time partitioned integration scheme imex for fluid structure interaction the scheme is based on a combination of an implicit l stable multi stage runge kutta scheme and an explicit runge kutta scheme fluid and structure dynamics are integrated using the implicit scheme and only the pressure loads acting on the structure are integrated explicitly for an academic problem we show that mesh optimization functions which are often necessary in standard mesh deformation algorithms can have a detrimental effect on the temporal order and accuracy we use a radial basis function rbf interpolation with a thin plate spline to create a smooth displacement field for the whole fluid domain which does not affect the order of the imex time integration scheme for reasonable accuracies the imex schemes outperform a second order staggered scheme by a factor of 2 3 as an example for a three dimensional real world problem a simulation of a transonic wing flutter case the agard 445 6 wing is performed for this test case a clear third order time accuracy is observed for imex3

survey of shape parameterization techniques for high fidelity multidisciplinary shape optimization a survey is provided of shape parameterization techniques formultidisciplinary optimization and some emerging ideas are highlighted the survey focuses on the suitability of available techniques for multidisciplinary applications of complex cone gurations using high e delity analysis tools such as computational e uid dynamics and computational structural mechanics the suitability criteria are based on the efe ciency effectiveness ease of implementation and availability of analytical sensitivities for geometry and grids a section on sensitivity analysis grid regeneration and grid deformation techniques is also provided