embedded tori with prescribed mean curvature abstract we construct a sequence of compact oriented embedded two dimensional surfaces of genus one into euclidean 3 space with prescribed almost constant mean curvature of the form h x 1 a x for x large when a 0 and 0 2 such surfaces are close to sections of unduloids with small neck size folded along circumferences centered at the origin and with larger and larger radii the construction involves a deep study of the corresponding jacobi operators an application of the lyapunov schmidt reduction method and some variational argument

parallel preconditioners for monolithic solution of shear bands shear bands are one of the most fascinating instabilities in metals that occur under high strain rates they describe narrow regions on the order of micron scales where plastic deformations and significant heating are localized which eventually leads to fracture nucleation and failure of the component in this work shear bands are described by a set of four strongly coupled thermo mechanical equations discretized by a mixed finite element formulation a thermo viscoplastic flow rule is used to model the inelastic constitutive law and finite thermal conductivity is prescribed which gives rise to an inherent physical length scale governed by competition of shear heating and thermal diffusion the residual equations are solved monolithically by a newton type method at every time step and have been shown to yield mesh insensitive result the jacobian of the system is sparse and has a nonsymmetric block structure that varies with the different stages of shear bands formation the aim of the current work is to develop robust parallel preconditioners to gmres in order to solve the resulting jacobian systems efficiently the main idea is to design schur complements tailored to the specific block structure of the system and that account for the varying stages of shear bands we develop multipurpose preconditioners that apply to standard irreducible discretizations as well as our recent work on isogeometric discretizations of shear bands the proposed preconditioners are tested on benchmark examples and compared to standard state of practice solvers such as gmres ilu and lu direct solvers nonlinear and linear iterations counts as well as cpu times and computational speedups are reported and it is shown that the proposed preconditioners are robust efficient and outperform traditional state of the art solvers

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

efficient numerical differentiation of implicitly defined curves for sparse systems a numerical technique is developed for the efficient numerical differentiation of regular implicitly defined curves existing in high dimensional real space such as those representing homotopies where the system of equations which defines the curve implicitly is assumed to be sparse the calculation is verified numerically through its application to the curve defined implicitly by a homotopy constructed based on a discretization of the equations governing compressible aerodynamic fluid flow consideration is given to computational cost data storage and accuracy this method is applicable to any implicitly defined curves or trajectories which can occur for example in dynamical systems analysis or control applications also exist in the area of homotopy continuation where implicitly defined curves are approximately traced numerically such applications include the analysis of curve traceability and the construction of higher order predictors the latter is investigated numerically and it is found that increasing the order of accuracy of the predictor can significantly improve the curve tracing accuracy within a limited radius

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

bubbles with prescribed mean curvature the variational approach abstract let h r 3 r be a c 1 mapping such that h p h 0 as p we show that when h satisfies some global conditions then there exists an h bubble namely a sphere s in r 3 such that the mean curvature of s at any regular point p s equals h p

application of lattice boltzmann method to sensitivity analysis via complex differentiation sensitivity analysis of flows computed by lattice boltzmann method via complex differentiation is proposed the theoretical work is illustrated and the proposed method assessed considering the d2q9 scheme along with the differentiation of the lattice boltzmann equation boundary condition implementation is also detailed some examples illustrate the capability of the proposed method

embedded constant mean curvature surfaces in euclidean three space in this paper we refine the construction and related estimates for complete constant mean curvature surfaces in euclidean three space developed in kapouleas ann math 131 239 330 1990 by adopting the more precise and powerful version of the methodology which was developed in kapouleas invent math 119 3 443 518 1995 as a consequence we remove the severe restrictions in establishing embeddedness for complete constant mean curvature surfaces in kapouleas ann math 131 239 330 1990 and we produce a very large class of new embedded examples of finite topology

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

the exponentially convergent trapezoidal rule it is well known that the trapezoidal rule converges geometrically when applied to analytic functions on periodic intervals or the real line the mathematics and history of this phenomenon are reviewed and it is shown that far from being a curiosity it is linked with computational methods all across scientific computing including algorithms related to inverse laplace transforms special functions complex analysis rational approximation integral equations and the computation of functions and eigenvalues of matrices and operators

minimal curvature trajectories riemannian geometry concepts for slow manifold computation in chemical kinetics in dissipative ordinary differential equation systems different time scales cause anisotropic phase volume contraction along solution trajectories model reduction methods exploit this for simplifying chemical kinetics via a time scale separation into fast and slow modes the aim is to approximate the system dynamics with a dimension reduced model after eliminating the fast modes by enslaving them to the slow ones via computation of a slow attracting manifold we present a novel method for computing approximations of such manifolds using trajectory based optimization we discuss riemannian geometry concepts as a basis for suitable optimization criteria characterizing trajectories near slow attracting manifolds and thus provide insight into fundamental geometric properties of multiple time scale chemical kinetics the optimization criteria correspond to a suitable mathematical formulation of minimal relaxation of chemical forces along reaction trajectories under given constraints we present various geometrically motivated criteria and the results of their application to four test case reaction mechanisms serving as examples we demonstrate that accurate numerical approximations of slow invariant manifolds can be obtained

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

multicomplex newmark beta time integration method for sensitivity analysis in structural dynamics sensitivity analysis of structural systems is important for identifying important parameters that influence the dynamic response of a model this paper describes an extension of the standard newmark beta algorithm to the multicomplex mathematical domain such that time dependent high order high accuracy derivatives of dynamic systems can be obtained along with the traditional response in particular complex and bicomplex analyses were used to compute accurate first and second order derivatives of several example problems these derivatives were shown to be highly accurate when compared to analytical and finite difference estimates the method is general and any order derivative including mixed derivatives can be computed

discrete shape sensitivity equations for aerodynamic problems intensive research and development in computational fluid dynamics cfd has recently produced many powerful cfd codes to simulate complex aerodynamic phenomena however in order to enhance the usefulness of these cfd codes for design practice development of design sensitivity equations compatible to these codes becomes very important this paper represents a part of such an effort to develop a sensitivity analysts methodology that enables the sensitivity equations to be implemented into existing cfd codes with minimal code modification the methodology is based upon a pre elimination procedure which accounts for consistently linearized boundary conditions formulations of both the direct differentiation and the adjoint variable methods will be presented in the paper

implicit multigrid algorithms for incompressible turbulent flows on unstructured grids an implicit code for computing inviscid and viscous incompressible flows on unstructured grids is described the foundation of the code is a backward euler time discretization for which the linear system is approximately solved at each time step with either a point implicit method or a preconditioned generalized minimal residual gmres technique for the gmres calculations several techniques are investigated for forming the matrix vector product convergence acceleration is achieved through a multigrid scheme that uses nonnested coarse grids that are generated using a technique described in the present paper convergence characteristics are investigated and results are compared with an exact solution for the inviscid flow over a four element airfoil viscous results which are compared with experimental data include the turbulent flow over a naca 4412 airfoil a three element airfoil for which mach number effects are investigated and three dimensional flow over a wing with a partial span flap

an adaptive discontinuous galerkin method for very stiff systems of ordinary differential equations abstract we present a discontinuous galerkin dg method for the numerical solution of stiff systems of ordinary differential equations odes we use a standard dg variational formulation with polynomials of degree k in each time interval we show that the method is a stable for every k we then introduce a hierarchical legendre finite element basis and we show that a whole family of approximations can be obtained simply by truncating the last p degrees of freedom from the computed solution we show that these approximations converge to order k 1 p in l2 norm and to order k 1 2 p in supremum norm we then show how this can be used to control the error and the time step length we present numerical examples of solutions on very stiff problems and on stiff problems with very long time integration where the time step length can vary on many orders of magnitude

the singular value expansion of the volterra integral equation associated to a numerical differentiation problem abstract we consider the volterra integral equation of the first kind for the derivative of a given function with one side boundary conditions we give a method to obtain the singular value expansion for the corresponding integral kernel this singular value expansion can be used to give algorithms for the solution of the numerical differentiation problem a numerical experiment shows the results obtained by a simple version of such algorithms

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

optimization of high order diagonally implicit runge kutta methods abstract this article presents constrained numerical optimization of high order linearly and algebraically stable diagonally implicit runge kutta methods after satisfying the desired order conditions undetermined coefficients are optimized with respect to objective functions which consider accuracy stability and computational cost constraints are applied during the optimization to enforce stability properties to ensure a well conditioned method and to limit the domain of the abscissa two promising third order methods are derived using this approach labelled sdirk 3 1 2 2 3 l 14 and sdirk 3 1 4 l sa 5 both optimized schemes have a good balance of properties the relative error norm of the latter the l 2 norm scaled by a function of the number of implicit stages is a factor of two smaller than comparable methods found in the literature variations on these methods are discussed relative to trade offs in their accuracy and stability properties a novel fifth order scheme sdirk 5 1 5 l 02 is derived with a significantly lower relative error norm than the comparable fifth order a stable reference method in addition the optimized scheme is l stable the accuracy and relative efficiency of the runge kutta methods are verified through numerical simulation of van der pol s equation as well as numerical simulation of vortex shedding in the laminar wake of a circular cylinder and in the turbulent wake of a naca 0012 airfoil these results demonstrate the value of numerical optimization for selecting undetermined coefficients in the construction of high order runge kutta methods with a balance between competing objectives

an improved removal method for estimating animal abundance summary the problem of estimating animal populations from the numbers in successive catches has been addressed by many authors most have assumed asymptotic normality of maximum likelihood estimates to construct confidence intervals this assumption can be shown to fail in many common practical situations we construct intervals based on the profile likelihood ratio and show by simulation that their true confidence is much closer to the nominal value than that of previous methods the removal method whereby the total number of animals in an enclosed area is estimated from the numbers in successive removals i e captures without replacement has been addressed by a number of authors since it was introduced by moran 1951 and zippin 1956 all authors have attempted using a series of more or less justifiable assumptions to obtain both a point estimate and a confidence interval for the number of animals in the area in this paper we aim to bring together the most recent theory and present a method of obtaining confidence intervals which a simulation study shows gives intervals of true confidence closer to the nominal value than previous methods

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

verified predictions of shape sensitivities in wall bounded turbulent flows by an adaptive finite element method a continuous sensitivity equation cse method is presented for shape parameters in turbulent wall bounded flows modeled with the standard email protected turbulence model with wall functions differentiation of boundary conditions and their complex dependencies on shape parameters including the two velocity scale wall functions is presented in details along with the appropriate methodology required for the cse method to ensure accuracy grid convergence and to reduce computational time an adaptive finite element method driven by asymptotically exact error estimations is used the adaptive process is controlled by error estimates on both flow and sensitivity solutions firstly the proposed approach is applied on a problem with a closed form solution derived using the method of the manufactured solution to perform code verification results from adaptive grid refinement studies show verification of flow and sensitivity solvers error estimators and the adaptive strategy secondly we consider turbulent flows around a square cross section cylinder in proximity of a solid wall we examine the quality of the numerical solutions by performing solution verification and validation then sensitivity analysis of these turbulent flows is performed to investigate the ability of the method to deal with non trivial geometrical changes sensitivity information is used to estimate uncertainties in the flow solution caused by uncertainties in the shape parameter and to perform fast evaluation of flows on nearby configurations

optimal shape design of a two dimensional asymmetric diffuser in turbulent flow an optimal shape of two dimensional asymmetric diffuser with maximum pressure recovery at the exit is numerically obtained using a mathematical theory based on the variational calculus and gradient algorithm the initial diffuser is taken to be a two dimensional asymmetric diffuser for which much experimental and numerical data are available the reynolds number based on the bulk mean velocity and the channel height at the diffuser entrance is 1 8 10 4 f rom this initial shape optimal diffuser shapes are designed for six different geometric constraints such as the streamwise length and height of the diffuser the optimality condition for maximum pressure recovery is obtained to be zero skin friction along the diffuser wall the turbulent flow inside the diffuser is predicted using the k v2 f model and optimal shapes are obtained through iterative procedures to satisfy the optimality condition with the shape design flow separation that appeared in the initial diffuser is completely removed or significantly reduced for one of the optimal diffuser shapes obtained large eddy simulation is carried out to validate the result of the shape design the wall shear stress wall pressure mean velocity and turbulence quantities obtained from large eddy simulation are in good agreement with those from the simulation using the turbulence model

multiobjective aerodynamic optimization by variable fidelity models and response surface surrogates a computationally efficient procedure for multiobjective design optimization with variable fidelity models and response surface surrogates is presented the proposed approach uses the multiobjective evolutionary algorithm that works with a fast surrogate model obtained with kriging interpolation of the low fidelity model data enhanced by space mapping correction exploiting a few high fidelity training points the initial pareto front generated by multiobjective optimization of the surrogate using the multiobjective evolutionary algorithm can be iteratively refined by local enhancements of the surrogate model the latter are realized with a space mapping response correction based on a limited number of high fidelity training points allocated along the initial pareto front the proposed method allows us to obtain at a low computational cost a set of designs representing tradeoffs between the conflicting objectives the current approach is illustrated using examples of airfoil design one in transonic flo

h bubbles in a perturbative setting the finite dimensional reduction s method given a regular function h colon mathbb r 3 to mathbb r we look for h bubbles that is regular surfaces in mathbb r 3 parametrized on the sphere mathbb s 2 with mean curvature h at every point here we study the case of h u h 0 varepsilon h 1 u h varepsilon u where h 0 is a nonzero constant varepsilon is the smallness parameter and h 1 is any c 2 function we prove that if bar p in mathbb r 3 is a good stationary point for the melnikov type function gamma p int q p h 0 1 h 1 q dq then for varepsilon small there exists an h varepsilon bubble omega varepsilon that converges to a sphere of radius h 0 1 centered at bar p as varepsilon to 0

nonintrusive continuum sensitivity analysis for fluid applications abstract continuum sensitivity analysis csa provides an analytic method of computing derivatives for structures fluids and fluid structure interaction problems with respect to shape or value parameters csa does not require mesh sensitivity to calculate local shape sensitivity thereby contributing to its computational efficiency csa involves solving linear sensitivity equations with the corresponding sensitivity boundary conditions spatial gradients of flow variables are required to construct the sensitivity boundary conditions the accuracy of which significantly affects the flow shape derivatives a method called spatial gradient reconstruction sgr is used to accurately compute the flow spatial gradients that are further used to set the sensitivity boundary conditions here we present a nonintrusive csa formulation for calculating the material derivatives of flow variables with respect to shape design parameters nonintrusive csa means csa without any modification to the black box program used for the cfd flow analysis moreover nonintrusive csa is flow solver agnostic in that knowledge of the solver s discretization is not needed an example of two dimensional flow over a naca0012 airfoil demonstrates the application of nonintrusive csa to steady flow problems involving euler compressible inviscid equations over unstructured grids with finite volume spatial discretization accuracy was studied by investigating convergence for a family of six high quality grids ranging from four thousand to four million cells factors such as the accuracy of the flow spatial gradients flow sensitivity boundary conditions weak implementation of the boundary conditions in the finite volume framework and the use of an approximate flux jacobian matrix all of which affect the accuracy of the material derivatives are discussed a novelty of this work is that the sensitivity analysis is done nonintrusively using fluent and su2 software to establish the use of black box codes for obtaining flow derivatives using the csa approach

quaternion and octonion based finite element analysis methods for computing multiple first order derivatives abstract the complex taylor series expansion method for computing accurate first order derivatives is extended in this work to quaternion octonion and any order cayley dickson algebra the advantage of this new approach is that highly accurate multiple first order derivatives can be obtained in a single analysis quaternion and octonion based finite element analysis methods were developed in order to compute up to three quaternion and up to seven octonion first order derivatives of shape material properties and or loading conditions in a single analysis the traditional finite element formulation was modified such that each degree of freedom was augmented with three or seven additional imaginary nodes the quaternion and octonion based methods were integrated within the abaqus commercial finite element code through a user element subroutine numerical examples are presented for thermal conductivity and linear elasticity however the methodology is general the results indicate that the quaternion and octonion based methods provide derivatives of the same high accuracy as the complex finite element method but are significantly more efficient a fortran code to solve a simple seven variable quaternion example is given in the appendix

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

geometry parameterization method for multidisciplinary applications a technique is described for parameterizing geometries in a manner suitable for numerical simulations used in design optimization the new methodology referred to as control grids is flexible easy to use extendable and applicable to multidisciplinary design optimization although significantly different than free form deformation the present method provides many similar capabilities comparisons between the current technique and free form deformation are provided to highlight the advantages and disadvantages of each method the new parameterization technology is described and demonstrated in both two and three dimensions for single and multidisciplinary applications

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

adjoint sensitivity formulation for discontinuous galerkin discretizations in unsteady inviscid flow problems this paper presents an unsteady discrete adjoint algorithm for high order implicit discontinuous galerkin discretizations in time dependent inviscid flow problems the major function of the adjoint approach is to obtain the sensitivity information in a time dependent functional output which in turn is used to drive an unsteady shape optimization process to deliver a minimum of the objective functional a gradient based optimization strategy is investigated in which the sensitivity derivatives of the objective functional with respect to input variables are formulated in the context of high order discontinuous galerkin discretizations while special emphasis is given to the variations and linearizations of curvilinear boundary elements implicit temporal discretizations consisting of a second order backward euler scheme and a fourth order implicit runge kutta scheme are considered exclusively in this work where the corresponding adjoint problem is required to be solved in a backward time integration manner due to the associated transpose operation two numerical examples for the unsteady shape design techniques are presented to verify the derived sensitivity formulations and to demonstrate the performance of the adjoint approach the first involves an inverse shape optimization case by matching a time dependent target pressure profile for a two dimensional periodic vortical gust impinging on a rae 2822 airfoil and the second considers minimization of the acoustic noise produced by subsonic flow over a naca0012 airfoil with a 0 03c thick blunt trailing edge

sensitivity derivatives for advanced cfd algorithm and viscous modeling parameters via automatic differentiation the computational technique of automatic differentiation ad is applied to a complicated computer program to illustrate the simplicity efficiency and versatility of ad with complex algorithms for use within a sensitivity analysis many algorithmic and physics modeling coefficients appear in computer programs that are routinely set in an ad hoc manner ad can be used to enhance computer programs with derivative information suitable for guiding formal sensitivity analyses which allows these coefficient values to be chosen in a rigorous manner to achieve particular program properties such as an improved convergence rate or improved accuracy in this paper ad is applied to a three dimensional thin layer navier stokes multigrid flow solver to assess the feasibility and computational impact of obtaining exact sensitivity derivatives with respect to algorithmic and physics modeling parameters typical of those needed for sensitivity analyses calculations are performed for an onera m6 wing in transonic flow with both the baldwin lomax and johnson king turbulence models the wing lift drag and pitching moment coefficients are differentiated with respect to two different groups of input parameters the first group consists of the second and fourth order damping coefficients of the computational algorithm whereas the second group consists of two parameters in the viscous turbulent flow physics modeling results obtained via ad are compared for both accuracy and computational efficiency with the results obtained with divided differences dd the ad results are accurate extremely simple to obtain and show significant computational advantage over those obtained by dd for some cases

complex step derivative approximation in noisy environment abstract the complex step derivative approximation is a powerful method for derivative approximations which has been successfully implemented in deterministic numerical algorithms we explore and analyze its implementation in noisy environment through examples error analysis and application to optimization methods numerical results show a promising performance of the complex step gradient approximation in noisy environment

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

an adjoint based lattice boltzmann method for noise control problems in this paper optimal control of acoustic problems is addressed within the lattice boltzmann method framework to this end an adjoint based lattice boltzmann method is proposed to solve the adjoint problem the adjoint state provides an easy access to the optimization gradients the line search step in newton s descent method is performed through a combination of complex differentiation and adjoint problem in the lattice boltzmann method the implementation of an active noise reduction method for two dimensional weakly compressible low mach number flows is discussed and the applicability of the method is assessed

using an adjoint approach to eliminate mesh sensitivities in computational design an adjoint algorithm for efficiently incorporating the effects of mesh sensitivities in a computational design framework is introduced the method eliminates the need for explicit linearizations of the mesh movement scheme with respect to the geometric parameterization variables an expense that has hindered large scale design optimization for practical applications the effects of the mesh sensitivities can be accounted for through the solution of an adjoint problem equivalent in cost to a single mesh movement computation followed by an explicit matrix vector product whose cost scales with the number of design variables and the resolution of the parameterized surface grid the methodology augments the current practice of using adjoints solely for the flowfield and leads to a dramatic computational savings the accuracy of the implementation is established and several sample design optimizations are shown

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

reduced order models for parameter dependent geometries based on shape sensitivity analysis the proper orthogonal decomposition pod is widely used to derive low dimensional models of large and complex systems one of the main drawback of this method however is that it is based on reference data when they are obtained for one single set of parameter values the resulting model can reproduce the reference dynamics very accurately but generally lack of robustness away from the reference state it is therefore crucial to enlarge the validity range of these models beyond the parameter values for which they were derived this paper presents two strategies based on shape sensitivity analysis to partially address this limitation of the pod for parameters that define the geometry of the problem at hand design or shape parameters we first detail the methodology to compute both the pod modes and their lagrangian sensitivities with respect to shape parameters from them we derive improved reduced order bases to approximate a class of solutions over a range of parameter values secondly we demonstrate the efficiency and limitations of these approaches on two typical flow problems 1 the one dimensional burgers equation 2 the two dimensional flows past a square cylinder over a range of incidence angles

coupled adjoint based sensitivities in large displacement fluid structure interaction using algorithmic differentiation summary a methodology for the calculation of gradients with respect to design parameters in general fluid structure interaction problems is presented it is based on fixed point iterations on the adjoint variables of the coupled system using algorithmic differentiation this removes the need for the construction of the analytic jacobian for the coupled physical problem which is the usual limitation for the computation of adjoints in most realistic applications the formulation is shown to be amenable to partitioned solution methods for the adjoint equations it also poses no restrictions to the nonlinear physics in either the fluid or structural field other than the existence of a converged solution to the primal problem from which to compute the adjoints we demonstrate the applicability of this procedure and the accuracy of the computed gradients on coupled problems involving viscous flows with geometrical and material non linearities in the structural domain this article is protected by copyright all rights reserved

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

variational approach to first and second order sensitivity analysis of elastic structures for a linear elastic structure the first variations of arbitrary stress strain and displacement functionals were explicitly expressed in terms of variations of material stiffness parameters within specified domain in references 1 and 2 by using the solutions for primary and adjoint systems in this paper explicit expressions for second variations are derived for the case when the material stiffness or compliance variations depend on a set of parameters these expressions arc obtained in terms of variations of stiffness or compliance parameters by using the solutions of two additional sets of adjoint systems furthermore the variation of local stress strain or displacement is considered several illustrative examples are provided for the first and second order sensitivity analysis of beams

adaptive optimization based improvement of tetrahedral meshes a tetrahedral mesh improvement method is presented that uses an optimization based node perturbation technique to minimize a cost function the cost function for each tetrahedron is based on the element condition number this condition number incorporates a weight matrix that transforms the element from physical space to a reference space the condition number is a measure of the conformity of the physical element with respect to the reference element feature based adaptation is incorporated into the element weight matrix to influence the node perturbation scheme to favor nonunit aspect ratio elements the mesh improvement scheme produces meshes that cluster grid nodes in high gradient regions of the selected adaptation function results for several inviscid three dimensional cases are included

on the generalization of the complex step method we generalize the well known complex step method for computing derivatives by introducing a complex step in a strict sense exploring different combinations of terms we derive 52 approximations for computing the first order derivatives and 43 for the second order derivatives for an appropriate combination of terms and appropriate choice of the step size in the real and imaginary directions fourth order accuracy can be achieved in a very simple and efficient scheme on a compact stencil new different ways of computing second order derivatives in one single step are shown many of the first order derivative approximations avoid the problem of subtractive cancellation inherent to the classic finite difference approximations for real valued steps and the superior accuracy and stability of the generalized complex step approximations are demonstrated for an analytic test function

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

on the accuracy of the complex step finite difference method abstract the complex step finite difference method csfdm is a very simple methodology that can be implemented in well known numerical techniques helping to improve for instance in the wave propagation problem time and or space derivative based wavefields we clarify differences between the csfdm and previous implementations of the complex step cs derivative approximation in well known numerical techniques we study dispersion properties for the one way and two way wave equations using the finite difference method fdm the pseudospectral method psm the finite element method fem and the csfdm under the influence of a plane wave and ricker source time functions we show the gain in numerical accuracy offered by the methodology of the csfdm over the fdm psm and fem we finally discuss consequences of the csfdm in future scenarios and propose directions of study in this area

complex variable method for eigensolution sensitivity analysis the application of complex variable method cvm for eigenvalue and eigenvector sensitivity analysis is presented gradi ent based methods usually used in structural optimization require accurate sensitivity information cvm is accurate robust and easy to implement as compared to other approximate sensitivity analysis methods by cvm the first order modal sensitivity can b e calculated using an existing structural analysis package this is achieved by making a small complex perturbation to a design variable the resulting eigensolutions would be complex the sensitivity of eigensolutions with respect to that design variable can then be obtained by taking the complex part of the eigensolutions divided by the perturbation since there is no subtraction involved in the sensitivity calculation the approximation is not sensitive to step size thus unlike finite difference method s cvm does not suffer the effect of subtraction cancellation error when a very small step size is used the computation of 1 st order modal sensitivity is not hampered by repeated eigenvalue problem as is the case with other sensitivity analysis methods for repeated eigenvalues cvm automatically provides correct eigenvalue sensitivity and the associated eigenvector sensitivity for a differentiable mode a 5 dof system with an eigenvalue of multiplicity 3 is used to illustrate the salient features of cvm in eigen solution sensitivity analysis nomenclature k stiffness matrix for spring mass system m mass matrix for spring mass system eigenvalue for spring mass system eigenvector for spring mass system derivative of w r t design variable

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

on the extension of the complex step derivative technique to pseudospectral algorithms the complex step derivative csd technique is a convenient and highly accurate strategy to perform a linearized perturbation analysis to determine a directional derivative via a minor modification of an existing nonlinear simulation code the technique has previously been applied to nonlinear simulation codes such as finite element codes which employ real arithmetic only the present note examines the suitability of this technique for extension to efficient pseudospectral simulation codes which nominally use the fast fourier transform fft to convert back and forth between the physical and transformed representations of the system it is found that if used carefully this extension retains the remarkable accuracy of the csd approach however to perform this extension without sacrificing this accuracy particular care must be exercised specifically the state real and perturbation imaginary components of the complexified system must be transformed separately and arranged in such a manner that they are kept distinct during the process of differentiation in the transformed space in order to avoid the linear combination of the large and small quantities in the analysis it is shown that this is relatively straightforward to implement even in complicated nonlinear simulation codes thus positioning the csd approach as an attractive and relatively simple alternative to hand coding a perturbation a k a tangent linear code for determining the directional derivative even when pseudospectral algorithms are employed

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

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

nonlinear robust performance analysis using complex step gradient approximation in this paper the complex step method is applied in the setting of numerical optimisation problems involving dynamical systems modelled as nonlinear differential equations the main advantage of the complex step method for gradient approximation is that it entails no subtractive cancellation error and therefore the truncation error can be made arbitrarily to machine precision small the method is applied to two robust performance analysis problems the accuracy and convergence rate of the solutions computed using the proposed approach are seen to be significantly better than those achieved using standard gradient approximation methods

interval estimation for mark recapture studies of closed populations textbooks continue to recommend the use of an asymptotic normal distribution to provide an interval estimate for the unknown size n of a closed population studied by a mark recapture experiment or multiple record system a likelihood interval approach is proposed and its implementation demonstrated for a range of models for such studies including all main effect and interaction models for incomplete contingency tables

unified maximum likelihood estimates for closed capture recapture models using mixtures summary agresti 1994 biometrics50 494 500 and norris and pollock 1996a biometrics52 639 649 suggested using methods of finite mixtures to partition the animals in a closed capture recapture experiment into two or more groups with relatively homogeneous capture probabilities this enabled them to fit the models mh mbh norris and pollock and mth agresti of otis et al 1978 wildlife monographs62 1 135 in this article finite mixture partitions of animals and or samples are used to give a unified linear logistic framework for fitting all eight models of otis et al by maximum likelihood likelihood ratio tests are available for model comparisons for many data sets a simple dichotomy of animals is enough to substantially correct for heterogeneity induced bias in the estimation of population size although there is the option of fitting more than two groups if the data warrant it

discrete adjoint approach for modeling unsteady aerodynamic design sensitivities a discrete adjoint approach is presented for computing steady and unsteady aerodynamic design sensitivities for compressible viscous flows about airfoil configurations the nominal flow solver method is based on a harmonic balance solution technique which is capable of modeling both steady and nonlinear periodic unsteady flows the computer code for the discrete adjoint solver which is derived from the nominal harmonic balance solver has been generated with the aid of the advanced automatic differentiation software tool known as taf transformation of algorithms in fortran

numerical differentiation for local and global tangent operators in computational plasticity in this paper numerical differentiation is applied to integrate plastic constitutive laws and to compute the corresponding consistent tangent operators the derivatives of the constitutive equations are approximated by means of difference schemes these derivatives are needed to achieve quadratic convergence in the integration at gauss point level and in the solution of the boundary value problem numerical differentiation is shown to be a simple robust and competitive alternative to analytical derivatives quadratic convergence is maintained provided that adequate schemes and stepsizes are chosen this point is illustrated by means of some numerical examples

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