extension operators on tensor product structures in two and three dimensions in this paper a uniformly elliptic second order boundary value problem in two dimensions is discretized by the p version of the finite element method an inexact dirichlet dirichlet domain decomposition preconditioner for the system of linear algebraic equations is investigated the ingredients of such a preconditioner are a preconditioner for the schur complement a preconditioner for the subdomains and an extension operator operating from the edges of the elements into their interior using methods of multiresolution analysis we propose a new method to compute the extension efficiently we prove that this type of extension is optimal i e the h 1 omega norm of the extended function is bounded by the h 0 5 partial omega norm of the given function numerical experiments show the optimal performance of the described extension

dynamic response optimization using an active set rqp algorithm a recently developed recursive quadratic programming rqp algorithm is applied to dynamic response design problems the algorithm incorporates updated hessians of the lagrange function and uses an active set strategy in which only a subset of the constraints is included in the direction finding qp subproblem hessian updating with dynamic response constraints presents some difficulties the primary difficulty is that the total number of constraints and the location of time points where they have to be imposed can change from iteration to iteration this can cause inconsistencies in hessian updating if proper numerical procedures are not used a numerical procedure to handle the situation is developed implemented and evaluated automatic restarting procedures are necessary for proper convergence of the algorithm the new algorithm is robust as well as more efficient than the purely linear algorithm the active set strategy plays an important role for application to dynamic response problems the rqp algorithms that do not use such a strategy are not applicable to this class of problems

an approach to optimal design of structures with non linear response a commonly adopted procedure to solve a structural optimum design problem is to improve the design iteratively by solving a sequence of simpler problems where each of these is obtained as an approximation of the original problem considering structures with linear response this procedure works very well and can be implemented quite straightforwardly however considering structures with non linear response the situation is more sophisticated since structural stability properties have to be taken into account the paper presents an approach to optimal design of statically loaded structures with non linear response it is proposed to replace the original problem with a sequence of similar problems from which corresponding approximate problems can easily be obtained these similar problems additionally contain an estimation of the critical load constraint while other constraints are imposed at some properly chosen load level which is lower or at most equal to the critical load level by this approach the structural stability requirements are taken into account in an effective manner the efficiency is illustrated with several examples involving kinematic as well as material non linear effects

further experiments on supersonic turbulent flow development in a square duct the mean flow structure of supersonic turbulent adiabatic wall flow in a square duct is investigated experimentally over a development length x d 0 50 for a uniform flow mach 3 9 condition at the duct inlet the results show that a secondary flow cell structure develops which is similar to that for the incompressible case development of the primary flow is influenced by the combined effects of the secondary flow and the streamwise adverse pressure gradient total pressure axial mean velocity and mach number profiles are presented which show that the outer flow is sensitive primarily to the streamwise pressure gradient while flow in the near wall region is dominated by the secondary flow axial mean velocity profiles plotted in terms of van driest scaled variables show that a well defined log law region exists in the near wall layer this region exists in the presence of a secondary flow which continuously modifies spanwise wall shear stress behavior along the length of the duct

multi objective optimization of fuzzy structural systems it is recognized that there exists a vast amount of fuzzy information in both the objective and constraint functions of the optimum design of structures since most practical structural design problems involve several often conflicting objectives to be considered a multi objective fuzzy programming method is outlined in this work the fuzzy constraints define a fuzzy feasible domain in the design space and each of the fuzzy objective functions defines the optimum solution by a fuzzy set of points a method of solving a fuzzy multi objective structural optimization problem using ordinary single objective programming techniques is presented the computational approach is illustrated with two numerical examples

experimental and numerical investigation of supersonic turbulent flow through a square duct les resultats experimentaux montrent que deux cellules d ecoulement secondaire se developpent autour des coins de maniere semblable a celle observee dans le cas incompressible les resultats numeriques sont bases sur le modele de baldwin lomay

localization of multi constrained optima and avoidance of local optima in structural optimization problems abstract this paper describes two new methods of structural optimization the first method is seen primarily in conjunction with optimality criteria algorithms it proposes a new procedure for the solution of multi constrained optimization problems based on the calculation of partial optima the partial optima refer to solutions of the optimization problem only for part of the constraints in the simplest case only for a single constraint and can be easily determined by already known procedures the second method is a new mathematical programming method designed to combine a high convergence speed together with a high reliability with respect to the avoidance of local minima

design sensitivity analysis and optimization of dynamic response abstract the paper presents methods of design sensitivity analysis and optimization of dynamic response of mechanical and structural systems a key feature of the paper is the development of procedures to handle point wise state variable constraints difficulties with a previous treatment where such constraints were transformed to equivalent integral constraints are noted and explained from theoretical as well as engineering standpoints an alternate treatment of such constraints is proposed developed and evaluated in this treatment each point wise state variable constraint is replaced by several constraints that are imposed at all the local max points for the original constraint function the differential equations of motion are formulated in the first order form so as to handle more general problems the direct differentiation and adjoint variable methods of design sensitivity analysis to deal with the point wise constraints are presented with the adjoint variable methods there are two ways of calculating design sensitivity coefficients the first approach uses an impulse load and the second approach uses a step load for the corresponding adjoint equation since the adjoint variable methods are better for a large class of problems an efficient computational algorithm with these methods is presented in detail optimum results for several problems are obtained and compared with those available in the literature the new formulation works extremely well as precise optimum designs are obtained

supersonic corner flow predictions using the quadratic constitutive relation a series of reynolds averaged navier stokes simulations is performed for a supersonic wall bounded turbulent corner flow these simulations are compared to a high order implicit large eddy simulation for the same geometry and flow conditions inclusion of the quadratic constitutive relation in the reynolds averaged navier stokes simulation results in significant improvement in qualitative agreement with the large eddy simulation specifically the presence of secondary flow a counter rotating vortex pair the range of valid values for the constant in the quadratic constitutive relation formulation is explored additionally the effects on this range from different turbulence models and momentum thickness reynolds numbers for the corner are also examined these effects of the quadratic constitutive relation are explored using both the finite difference fluid solver overflow and the finite volume fluid solver us3d the results indicate that the quadratic constitutive relation term directly affects the str

construction of minimum energy high order helmholtz bases for structured elements we present a construction procedure for high order expansion bases for structured finite elements specific for the operator under consideration the procedure aims to obtain bases in such way that the condition numbers for the element matrices are almost constant or have a moderate increase in terms of the polynomial order the internal modes of the mass and stiffness matrices are made simultaneously diagonal and the minimum energy concept is used to make the boundary modes orthogonal to the internal modes the performance of the proposed bases is compared to the standard basis using jacobi polynomials this is performed through numerical examples for helmholtz problem and transient linear elasticity employing explicit and implicit time integration algorithms and the conjugate gradient method with diagonal ssor and gauss seidel pre conditioners the sparsity patterns conditioning and solution costs are investigated a significant speedup and reduction in the number of iterations are obtained when compared to the standard basis

preconditioning on element interfaces for the p version finite element method and spectral element method in the preconditioned iterative method based on nonoverlapping domain decompositions the preconditioner is usually of block diagonal type to calculate its action on a residual vector it is required to solve a linear system of algebraic equations with the coefficient matrix being the schur complement block corresponding to the interface to reduce the computational cost the schur complement blocks in the preconditioners can be replaced with other simpler matrices which are called the interface preconditioners in this paper we discuss various interface preconditioners for the p version finite element method and spectral element method in r2 and r3 emphases are placed on 1 the lower and upper bounds of the condition numbers for the preconditioned schur complement blocks using the interface preconditioners which characterize the enlargement of the overall condition numbers and 2 the algorithms to efficiently implement the interface preconditioners numerical results are presented for the condition numbers using various interface preconditioners

experiments on supersonic turbulent flow development in a square duct les experiences sont concues de telle sorte qu on minimise les effets d onde et de distorsions induites par la tuyere dans l ecoulement determination des contours de pression totale et des distributions du coefficient de frottement superficiel local comparaison avec le cas incompressible

preconditioned spectral multi domain discretization of the incompressible navier stokes equations in this paper we present a preconditioned multi domain algorithm applied to the elliptic kernels arising from the spectral collocation of the incompressible navier stokes equations in three space dimensions with one homogeneous direction the technique based on the iterative solution of the schur complement matrix allows for efficient numerical solution of the operators in complex geometries consisting of a collection of non overlapping rectangular subdomains the method is shown to be nearly optimal in terms of condition number behavior in a double path of refinement strategy i e whenever both the number of legendre modes and the number of subdomains are significantly increased it is thus well suited for engineering applications in the fields of direct numerical simulation and large eddy simulation of turbulence

weight minimisation of displacement constrained truss structures using a strain energy criterion this paper discusses a new method for the solution of the general truss weight minimisation problem under simultaneous stress and displacement constraints the method introduces a novel strain energy density criterion that refers to the ratio of the virtual strain energy per unit volume in each structural member to its average value on the whole structure the virtual strain energy comes from the unit load theorem and it is proportional to the product of the axial member forces due to both the actual loads and a virtual unit load that is applied at the node with the maximum displacement a simple recursive formula for updating the cross sectional areas based on displacement constraints is presented a general subsequent algorithm applicable to both single and multiple load cases follows this formula the results are encouraging since in all test cases the method was found to be robust and generally led to the same weight level as the literature in both small and large structures

on condition numbers in hp fem with gauss lobatto based shape functions sharp bounds on the condition number of stiffness matrices arising in hp spectral discretizations for two dimensional problems elliptic problems are given two types of shape functions that are based on lagrange interpolation polynomials in the gauss lobatto points are considered these shape functions result in condition numbers o p and o p ln p for the condensed stiffness matrices where p is the polynomial degree employed locally refined meshes are analyzed for the discretization of dirichlet problems on meshes that are refined geometrically toward singularities the conditioning of the stiffness matrix is shown to be independent of the number of layers of geometric refinement

trust region methods for structural optimization using exact second order sensitivity the performance of multiplier algorithms for structural optimization has been significantly improved by using trust regions the trust regions are constructed using analytical second order sensitivity and within this region the augmented lagrangian is minimized subject to bounds evaluation of first and second derivatives of by the adjoint method does not require derivations of individual implicit constraint functions which makes the method economical eight test problems are considered and a vast improvement over previously used multiplier algorithms has been noted also the algorithm is robust with respect to scaling input parameters and starting designs

additive schwarz algorithms for the p version of the galerkin boundary element method we study some additive schwarz algorithms for the p version galerkin boundary element method applied to some weakly singular and hypersingular integral equations of the first kind both non overlapping and overlapping methods are considered we prove that the condition numbers of the additive schwarz operators grow at most as 1 log p 2 independently of h where p is the degree of the polynomials used in the galerkin boundary element schemes and h is the mesh size thus we show that additive schwarz methods which were originally designed for finite element discretisation of differential equations are also efficient preconditioners for some boundary integral operators which are non local operators

a recursive quadratic programming method with active set strategy for optimal design recursive quadratic programming methods have become popular in the field of mathematical programming owing to their excellent convergence characteristics there are two recursive quadratic programming methods that have been published in the literature one is by han and the other is by pshenichny published in 1977 and 1970 respectively the algorithm of pshenichny had been undiscovered until now and is examined here for the first time it is found that the proof of global convergence by han requires computing sensitivity coefficients derivatives of all constraint functions of the problem at every iteration this is prohibitively expensive for large scale applications in optimal design in contrast pshenichny has proved global convergence of his algorithm using only an active set strategy this is clearly preferable for large scale applications the method of pshenichny has been coded into a fortran program applications of this method to four example problems are presented the method is found to be very reliable however the method is found to be very sensitive to local minima i e it converges to a local minimum nearest to the starting design thus for optimal design problems which usually possess multiple local minima it is suggested that pshenichny s method be used as part of a hybrid method

improvements to the quadratic constitutive relation based on nasa juncture flow data modifications to the quadratic constitutive relations qcrs of spalart strategies for turbulence modelling and simulation international journal of heat and fluid flow vol 21 2000 pp 252 2

timely communication diagonal edge preconditioners in p version and spectral element methods domain decomposition preconditioners for high order galerkin methods in two dimensions are often built from modules associated with the decomposition of the discrete space into subspaces of functions related to the interior of elements individual edges and vertices the restriction of the original bilinear form to a particular subspace gives rise to a diagonal block of the preconditioner and the action of its inverse on a vector has to be evaluated in each iteration each block can be replaced by a preconditioner in order to decrease the cost knowledge of the quality of this local preconditioner can be used directly in a study of the convergence rate of the overall iterative process the schur complement of an edge with respect to the variables interior to two adjacent elements is considered the assembly and factorization of this block matrix are potentially expensive especially for polynomials of high degree it is demonstrated that the diagonal preconditioner of one such block has a condition number that increases approximately linearly with the degree of the polynomials numerical results demonstrate that the actual condition numbers are relatively small

efficient approximation concepts using second order information the primary goal of this paper is to show how second derivative information can be used in an effective way in structural optimization problems the basic idea is to generate such an information at the expense of only one more virtual load case in the sensitivity analysis part of the finite element code to achieve this goal a primal dual approach is employed that can also be interpreted as a sequential quadratic programming method another objective is to relate the proposed method to the well known family of approximation concepts techniques where the primary optimization problem is transformed into a sequence of non linear explicit subproblems when restricted to diagonal second derivatives the new approach can be viewed as a recursive convex programming method similar to the convex linearization method conlin and to its recent generalization the method of moving asymptotes mma this new method has been successfully tested on simple problems that can be solved in closed form as well as on sizing optimization of trusses in all cases the method converges faster than conlin mma or other approximation techniques based on reciprocal variables

a basis change strategy for the reduced gradient method and the optimum design of large structures this paper proposes a basis change strategy within the reduced gradient method for optimization under linear constraints it ensures a non singular basis matrix at every iteration the same strategy can reliably be used within the generalized reduced gradient method for optimization under non linear constraints this method is applied to the minimum weight design of large structures under displacement and stress constraints exploiting the sparsity of the constraint jacobian matrix

semi automatic sparse preconditioners for high order finite element methods on non uniform meshes high order finite elements often have a higher accuracy per degree of freedom than the classical low order finite elements however in the context of implicit time stepping methods high order finite elements present challenges to the construction of efficient simulations due to the high cost of inverting the denser finite element matrix there are many cases where simulations are limited by the memory required to store the matrix and or the algorithmic components of the linear solver we are particularly interested in preconditioned krylov methods for linear systems generated by discretization of elliptic partial differential equations with high order finite elements using a preconditioner like algebraic multigrid can be costly in terms of memory due to the need to store matrix information at the various levels we present a novel method for defining a preconditioner for systems generated by high order finite elements that is based on a much sparser system than the original high order finite element system we investigate the performance for non uniform meshes on a cube and a cubed sphere mesh showing that the sparser preconditioner is more efficient and uses significantly less memory finally we explore new methods to construct the sparse preconditioner and examine their effectiveness for non uniform meshes we compare results to a direct use of algebraic multigrid as a preconditioner and to a two level additive schwarz method