a full bayesian approach to the groundwater inverse problem for steady state flow a full bayesian approach to the estimation of transmissivity from hydraulic head and transmissivity measurements is developed for two dimensional steady state groundwater flow the approach combines both bayesian and maximum entropy viewpoints of probability in the first phase log transmissivity measurements are incorporated into bayes theorem and the prior probability density function is updated yielding posterior estimates of the mean value of the log transmissivity field and covariance the two central moments are generated assuming that the prior mean variance and integral scales are hyperparameters that is they are treated as random variables in themselves which is contrary to classical statistical approaches the probability density functions pdfs of these hyperparameters are in turn determined from maximum entropy considerations in other words pdfs are chosen for each of the hyperparameters that are maximally uncommitted with respect to unknown information this methodology is quite general and provides an alternative to kriging for spatial interpolation the final step consists of updating the conditioned natural logarithm transmissivity ln t field with hydraulic head measurements utilizing a linearized aquifer equation it is assumed that the statistical properties of the noise in the hydraulic head measurements are also uncertain at each step uncertainties in all pertinent hyperparameters are removed by marginalization finally what is produced is a ln t field conditioned on measurements of both hydraulic heads and log transmissivity and covariances of the ln t field in addition we can also produce resolution matrices confidence credibility limits and the like for the ln t field it is shown that the application of the methodology yields good estimates of transmissivities even when hydraulic head measurements are noisy and little or no information is specified on mean values of ln t variance of ln t and integral scales

sensitivity of post critical states to changes in design parameters abstract sensitivity of post buckling paths is studied in the context of the general theory of elastic stability of discrete structural systems it is assumed that the sensitivity of the critical state itself has been computed and a formulation is developed to account for sensitivity of the curvature of the post critical states when there are changes in design parameters a linear fundamental path is considered explicit expressions are obtained for the sensitivities and they take the form of perturbation expansions only first order sensitivity of post critical paths has been developed a simple example of an angle section column with deformable cross section illustrates that although a critical state may be insensitive to changes in certain design parameters the post critical response may be highly sensitive in the first example presented an axially loaded angle section column the postbuckling response even changes from stable to unstable depending on the values of the design parameter considered

elastoplastic truss design using a displacement based optimization abstract a displacement based optimization dbo method is applied to truss design problems with material nonlinearities to explore feasibility and verify efficiency of the method as compared to traditional structural optimization minimum weight truss sizing problems with various path independent elastoplastic laws including elastic perfectly plastic linear strain hardening and ramberg osgood models are investigated this type of material nonlinearity allows us to naturally extend the linear elastic truss sizing in the dbo setting to nonlinear problems to implement the methodology a computer program that uses the commercially available optimizer dot by vr d and imsl linear programming solver by visual numerics is developed several test problems are successfully solved using the dbo approach and solutions are compared to those available in the literature demonstrating significant reduction of computational time in comparison to the traditional structural optimization method in particular the dbo approach is found to be suitable for truss topology design since the method allows member areas to have cross sectional areas equal to zero exactly

estimation of aquifer parameters under transient and steady state conditions 2 uniqueness stability and solution algorithms paper 2 of this three part series starts with a discussion of the question under what conditions is the aquifer inverse problem well posed after defining the terms uniqueness identifiability and stability theoretical considerations and synthetic examples are used to demonstrate that ill posedness can be mitigated by including prior information about the parameters in the estimation criterion to be minimized at the same time the inclusion of such information is shown to be insufficient to guarantee uniqueness and stability in all cases several test problems in the recent literature which have resulted in pessimistic conclusions about the solvability of the aquifer inverse problem are shown to be ill posed a question is thus raised about the validity of these conclusions in the general case various conjugate gradient algorithms coupled with the adjoint state finite element method for computing the gradient of the estimation criterion and with newton s method for determining the optimum step size downgradient are compared a marked improvement in the rate at which these algorithms converge is shown to be achieved by switching from one method to another when the former slows down or fails to converge

review of options for structural design sensitivity analysis part 1 linear systems the paper reviews options for structural design sensitivity analysis including global finite differences continuum derivatives discrete derivatives and computational or automated differentiation the objective is to put these different approaches to design sensitivity analysis in the context of accuracy and consistency computational cost and implementation options and effort linear static analysis and transient dynamic analysis are reviewed in a separate appendix special attention is paid to the semi analytical method a future paper will address design sensitivity analysis in nonlinear structural problems

on model selection criteria in multimodel analysis 1 hydrologic systems are open and complex rendering them prone to multiple conceptualizations and mathematical descriptions there has been a growing tendency to postulate several alternative hydrologic models for a site and use model selection criteria to 1 rank these models 2 eliminate some of them and or 3 weigh and average predictions and statistics generated by multiple models this has led to some debate among hydrogeologists about the merits and demerits of common model selection also known as model discrimination or information criteria such as aic aicc bic and kic and some lack of clarity about the proper interpretation and mathematical representation of each criterion we examine the model selection literature to find that 1 all published rigorous derivations of aic and aicc require that the true model having generated the observational data be in the set of candidate models 2 though bic and kic were originally derived by assuming that such a model is in the set bic has been rederived by cavanaugh and neath 1999 without the need for such an assumption and 3 kic reduces to bic as the number of observations becomes large relative to the number of adjustable model parameters implying that it likewise does not require the existence of a true model in the set of alternatives we explain why kic is the only criterion accounting validly for the likelihood of prior parameter estimates elucidate the unique role that the fisher information matrix plays in kic and demonstrate through an example that it imbues kic with desirable model selection properties not shared by aic aicc or bic our example appears to provide the first comprehensive test of how aic aicc bic and kic weigh and rank alternative models in light of the models predictive performance under cross validation with real hydrologic data

three dimensional steady state flow to a well in a randomly heterogeneous bounded aquifer 1 we consider flow in a confined aquifer of uniform thickness due to a well of zero radius that fully penetrates the aquifer and discharges at a constant rate if the lateral extent of the aquifer is infinite a steady state flow regime never develops it is however well known that if the aquifer is additionally uniform a quasi steady state region extends from the well out to a cylindrical surface whose radius expands as the square root of time on the expanding surface head is uniform and time invariant inside this surface head at any time is described by a steady state solution a rigorous analysis of the analogous situation in a randomly heterogeneous aquifer would require the solution of a three dimensional transient stochastic flow problem in an aquifer of infinite lateral extent here we take a different approach by developing a three dimensional steady solution for mean flow to a well in a randomly heterogeneous aquifer with a cylindrical prescribed head boundary in analogy to the uniform case we expect our solution to approximate a quasi steady state region whose radius is initially small in comparison to the horizontal correlation scale of log conductivity but grows with time to become eventually much larger we treat log conductivity as a statistically homogeneous random field characterized by a gaussian spatial covariance function that may have different horizontal and vertical correlation scales our solution consists of analytical expressions for the ensemble mean and variance of head in the aquifer to second order in the standard deviation of log conductivity it is based on recursive approximations of exact nonlocal moment equations that are free of distributional assumptions and so apply to both gaussian and non gaussian log conductivity fields the analytical solution is supported by numerical monte carlo simulations it clarifies the manner in which relationships between the horizontal and vertical scales of the quasi steady state region and those of statistical anisotropy impact the statistical moments of drawdown and the equivalent and apparent hydraulic conductivities of the aquifer both conductivities are shown to exhibit a scale effect by growing with distance from the well within a radius of one to two horizontal integral scales from it

estimation of spatial covariance structures by adjoint state maximum likelihood cross validation 1 theory this series of three papers describes a cross validation method to estimate the spatial covariance structure of intrinsic or nonintrinsic random functions from point or spatially averaged data that may be corrupted by noise any number of relevant parameters including nugget effect can be estimated the theory described in this paper is based on a maximum likelihood approach which treats the cross validation errors as gaussian various a posteriori statistical tests are used to verify this hypothesis and to show that in many cases correlation between these errors is weak the log likelihood criterion is optimized through a combination of conjugate gradient algorithms an adjoint state theory is used to efficiently calculate the gradient of the estimation criterion optimize the step size downgradient and compute a lower bound for the covariance matrix of the estimation errors issues related to the identifiability stability and uniqueness of the resulting adjoint state maximum likelihood cross validation asmlcv method are discussed this paper also describes the manner in which asmlcv allows one to use model structure identification criteria to select the best covariance model among a given set of alternatives practical aspects of asmlcv and its application to synthetic data are presented in paper 2 samper and neuman this issue a applications to real hydrogeological data transmissivities and groundwater levels have been presented elsewhere while hydrochemical and isotopic data are analyzed by asmlcv in paper 3 samper and neuman this issue b

exponentially fitted runge kutta methods abstract exponentially fitted runge kutta methods with s stages are constructed which exactly integrate differential initial value problems whose solutions are linear combinations of functions of the form x j exp x x j exp x r or i r j 0 1 j max where 0 j max s 2 1 the lower bound being related to explicit methods the upper bound applicable for collocation methods explicit methods with s 2 3 4 belonging to that class are constructed for these methods a study of the local truncation error is made out of which follows a simple heuristic to estimate the value error and step length control is introduced based on richardson extrapolation ideas some numerical experiments show the efficiency of the introduced methods it is shown that the same techniques can be applied to construct implicit exponentially fitted runge kutta methods

design sensitivity of post buckling states including material constraints this paper reports theoretical studies on the design sensitivity of problems with geometrically non linear behavior leading to buckling and post buckling of thin walled structural members for the class of problems considered buckling occurs in the form of a stable bifurcation and it is assumed that changes in the design parameters do not break the bifurcation behavior the specific focus of the research is the first yield or first failure of the material as part of the sensitivity study of equilibrium states along the post critical path the investigation employs a discrete model of a structure in terms of generalized coordinates suitable for finite element analysis and a single load parameter and perturbation techniques to classify the critical state and to approximate the post critical path the problem of material behavior is modeled by means of constraints on the post critical path based on a yield criterion for simplicity the presentation uses the von mises yield criterion but other more complex criteria such as those employed in composite materials first ply failure can also be represented two forms of the constraints are formulated and the problem of sensitivity with respect to changes in a design parameter is discussed a simple example of a circular plate is presented to illustrate the use of the formulation for the sensitivity with respect to a single design parameter

optimization of imperfection sensitive symmetric systems for specified maximum load factor a method is presented for optimum design of a symmetric structure which reaches an unstable bifurcation point as the load factor is increased reduction of the maximum load level due to the antisymmetric imperfection is considered and a straightforward algorithm is proposed for calculating the magnitude of reduction of the load factor corresponding to the most critical mode of antisymmetric imperfection the sensitivity coefficients of the bifurcation load factor are calculated by using the interpolation method developed by the authors and an approximate formulation is presented for sensitivity analysis of the maximum load factor it is shown in the examples that the errors in the sensitivity coefficients do not lead to any significant difference in the optimum design

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

conditions for symmetric antisymmetric and zero stiffness bifurcation in view of imperfection sensitivity and insensitivity abstract in general it is not clear how to design structures such that they are ab initio imperfection insensitive i e without modifications of the original design after the diagnosis of imperfection sensitivity symmetry and antisymmetry of bifurcation paths representing qualitative properties of a structure not only simplify the postbuckling analysis but also have an influence on the behavior of real i e imperfect structures the special case of a zero stiffness postbuckling path incorporates both symmetry and antisymmetry mathematical definitions of the three categories symmetric antisymmetric and zero stiffness equilibrium paths are given it is shown that symmetry as well as antisymmetry causes the vanishing of specific coefficients in an asymptotic series expansion following from koiter s initial postbuckling analysis thereupon methods of checking for the three categories of bifurcation behavior are discussed finally the three categories are investigated in terms of necessary and or sufficient conditions for imperfection insensitivity for instance a horizontal tangent of the postbuckling path at the bifurcation point is required for imperfection insensitivity four examples illustrate non symmetric symmetric and antisymmetric bifurcation as well as zero stiffness postbuckling behavior for the first two examples the approach of increasing the stiffness results in the conversion of the system from imperfection sensitivity into insensitivity remarkably this happens without changing the prebuckling behavior and the buckling load of the structure the third example demonstrates that imperfection sensitivity is an inevitable implication of antisymmetric bifurcation

inverse groundwater modeling with emphasis on model parameterization 1 this study develops an inverse method aiming to circumvent the subjective decision regarding model parameterization and complexity in inverse groundwater modeling the number of parameters is included as a decision variable along with parameter values a parameterization based on b spline surfaces bss is selected to approximate transmissivity and genetic algorithms were selected to perform error minimization a transform based on linear least squares lls is developed so that different parameterizations may be combined by standard genetic algorithm operators first three applications with isotropic anisotropic and zoned aquifer parameters are examined in a single objective optimization problem and the estimated transmissivity is found to be near the true one interestingly in the anisotropic case the algorithm converged to a solution with an anisotropic distribution of control points next a single objective optimization with regularization penalizing complex models is considered and last the problem is expressed in a multiobjective optimization framework moo where the goals are simultaneous minimization of calibration error and model complexity the result of moo is a pareto set of potential solutions where the user can examine the tradeoffs between calibration error and model complexity and select the most suitable model by comparing calibration with prediction errors it appears that the most promising models are the ones near a region where the rate of decrease of calibration error as model complexity increases drops bend of error curve this is a useful result of practical interest in real inverse modeling applications

an explicit numerov type method for second order differential equations with oscillating solutions in this paper a new explicit numerov type method is introduced the construction is based on a modification of a sixth order explicit numerov type method recently developed by tsitouras ch tsitouras explicit numerov type methods with reduced number of stages comput math appl 45 2003 37 42 two free parameters are added in order to nullify the phase lag and the amplification the method is useful only when a good estimate of the frequency of the problem is known in advance the parameters depend on the product of the estimated frequency and the stepsize numerical results obtained for well known test problems show the efficiency of the new method

mixed collocation methods for y f x y inline equatio abstract the collocation methods introduced here are based on linear combinations of trigonometric functions and powers the motivation is to provide better approximations for oscillatory solutions of initial value problems for differential equations of the special form y f x y the resulting methods for two or more collocation points are implicit runge kutta nystrom methods with coefficients which depend on both the fitted angular frequency and the steplength algebraic and trigonometric order conditions are considered and the stability properties of some methods are examined particular mixed collocation methods and other methods for the same class of problems are compared by applying them to a variety of test problems

stochastic modeling of groundwater flow by unconditional and conditional probabilities the inverse problem the inverse problem is defined here as follows determine the transmissivity at varius points given the shape and boundary of the aquifer and recharge intensity and given a set of measured log transmissivity y and head h values at a few points the log transmissivity distribution is regarded as a realization of a random function of normal and stationary unconditional probability density function pdf the solution of the inverse problem is the conditional normal pdf of y conditioned on measured h and y which is expressed in terms of the unconditional joint pdf of y and h the problem is reduced to determining the unconditional head log transmissivity covariance and head variogram for a selected y covariance which depends on a few unknown parameters this is achieved by solving a first order approximation of the flow equations the method is illustrated for an exponential y covariance and the effect of head and transmissivity measurements upon the reduction of uncertainty of y is investigated systematically it is shown that measurement of h has a lesser impact than those of y but a judicious combination may lead to significant reduction of the predicted variance of y possible applications to real aquifers are outlined

exponentially fitted explicit runge kutta nystrom methods exponentially fitted runge kutta nystrom efrkn methods for the numerical integration of second order ivps with oscillatory solutions are derived these methods integrate exactly differential systems whose solutions can be expressed as linear combinations of the set of functions exp t exp t c or equivalently sin t cos t when i r explicit efrkn methods with two and three stages and algebraic orders 3 and 4 are constructed in addition a 4 3 embedded pair of explicit efrkn methods based on the fsal technique is obtained which permits to introduce an error and step length control with a small cost added some numerical experiments show the efficiency of our explicit efrkn methods when they are compared with other exponential fitting type codes proposed in the scientific literature

nonlocal and localized analyses of conditional mean steady state flow in bounded randomly nonuniform domains 2 computational examples in paper 1 guadagnini and neuman this issue of this two part series we presented exact nonlocal equations for first and second conditional ensemble moments of hydraulic head and flux under steady state groundwater flow in bounded randomly nonuniform domains in the presence of uncertain source and boundary terms we derived recursive approximations for these equations based on expansion in powers of a small parameter y which represents the standard estimation error of log hydraulic conductivity developed a two dimensional finite element scheme for their solution under the action of deterministic forcing terms to first order in y2 showed how to localize the conditional mean flow equations so as to yield familiar looking darcian differential equations in which however all quantities are conditional on data and therefore nonunique and proposed that traditional deterministic flow models be reinterpreted in light of these localized equations we stressed that all quantities which enter into our nonlocal and localized conditional moment equations are defined on a consistent support scale thereby obviating the need for upscaling from support to grid scale or the a priori definition of a computational grid that such a grid can therefore be defined a posteriori based on the degree of smoothness one expects the conditional moments to exhibit and that conditioning points are often spaced widely enough to render these moments relatively smooth and hence resolvable on a coarser grid than is necessary for monte carlo simulations this notwithstanding we use in this paper a high resolution grid to explore superimposed mean uniform and convergent flows in unconditional and conditional log hydraulic conductivity domains and to compare the relative accuracies of corresponding nonlocal localized and monte carlo finite element solutions whereas our nonlocal solution is nominally restricted to mildly nonuniform media with y2 1 we find that it actually yields remarkably accurate results for strongly nonuniform media with y2 at least as large as 4 localized solutions deteriorate relative to nonlocal results as one approaches some points of conditioning and singularity or as y2 increases though they require much less computer time than do nonlocal solutions localized solutions have the added disadvantage that they yield no information about predictive uncertainty

a comparison of seven geostatistically based inverse approaches to estimate transmissivities for modeling advective transport by groundwater flow this paper describes the first major attempt to compare seven different inverse approaches for identifying aquifer transmissivity the ultimate objective was to determine which of several geostatistical inverse techniques is better suited for making probabilistic forecasts of the potential transport of solutes in an aquifer where spatial variability and uncertainty in hydrogeologic properties are significant seven geostatistical methods fast fourier transform ff fractal simulation fs linearized cokriging lc linearized semianalytical ls maximum likelihood ml pilot point pp and sequential self calibration ss were compared on four synthetic data sets each data set had specific features meeting or not classical assumptions about stationarity amenability to a geostatistical description etc the comparison of the outcome of the methods is based on the prediction of travel times and travel paths taken by conservative solutes migrating in the aquifer for a distance of 5 km four of the methods ls ml pp and ss were identified as being approximately equivalent for the specific problems considered the magnitude of the variance of the transmissivity fields which went as high as 10 times the generally accepted range for linearized approaches was not a problem for the linearized methods when applied to stationary fields that is their inverse solutions and travel time predictions were as accurate as those of the nonlinear methods nonstationarity of the true transmissivity field or the presence of anomalies such as high permeability fracture zones was however more of a problem for the linearized methods the importance of the proper selection of the semivariogram of the log10 t field or the ability of the method to optimize this variogram iteratively was found to have a significant impact on the accuracy and precision of the travel time predictions use of additional transient information from pumping tests did not result in major changes in the outcome while the methods differ in their underlying theory and the codes developed to implement the theories were limited to varying degrees the most important factor for achieving a successful solution was the time and experience devoted by the user of the method

ode solvers using band limited approximations we use generalized gaussian quadratures for exponentials to develop a new ode solver nodes and weights of these quadratures are computed for a given bandlimit c and user selected accuracy e so that they integrate functions e i b x for all b

a new theoretical approach to runge kutta methods runge kutta rk methods are treated here as linear multistage methods within the author s concept of a methods as a consequence the traditional difference in treatment of rk methods and linear multistep methods vanishes in particular it is not necessary to consider elementary differentials it is shown how the order of a rk method depends on the error constants of its stages and the conditions nonlinear equations for order p are generated from a very simple recursion

an application of the geostatistical approach to the inverse problem in two dimensional groundwater modeling the geostatistical approach to the estimation of transmissivity from head and transmissivity measurements is developed for two dimensional steady flow the field of the logarithm of transmissivity log transmissivity is represented as a zero order intrinsic random field its spatial structure is described in this application through a two term covariance function that is linear in the parameters 1 and 2 linearization of the discretized flow equations allows the construction of the joint covariance matrix of the head and log transmissivity measurements as a linear function of 1 and 2 in this particular application the coefficient matrices are calculated numerically in a noniterative fashion maximum likelihood estimation is employed to estimate 1 and 2 as well as additional parameters from measurements linear estimation theory cokriging then yields point or block averaged estimates of transmissivity the approach is first applied to a test case with favorable results it is shown that the application of the methodology gives good estimates of transmissivities it is also shown that when the transmissivities are used in a numerical model they reproduce the head measurements quite well results from the application of the methodology to the jordan aquifer in iowa are also presented

estimation of aquifer parameters under transient and steady state conditions 1 maximum likelihood method incorporating prior information in this series of three papers a method is presented to estimate the parameters of groundwater flow models under steady and nonsteady state conditions the parameters include values and directions of principal hydraulic conductivities or transmissivities in anisotropic media specific storage or storativity interior and boundary recharge or leakage rates coefficients of head dependent interior and boundary sources and boundary heads in transient situations the initial head distribution can also be estimated if the system is originally at a steady state paper 1 of the series discusses some of the advantage in treating the inverse problem statistically and in regularizing its solution by means of penalty criteria based on prior estimates of the parameters the inverse problem is posed in the framework of maximum likelihood theory cast in a manner that accounts for prior information about the parameters since not all the factors which contribute to the prior errors can be quantified statistically at the outset the covariance matrices of these errors are expressed in terms of several parameters which if unknown can be estimated jointly with the hydraulic parameters by a stagewise optimization process when transient head data are separated by a fixed time interval the temporal structure of these data is approximated by a lag one autoregressive model with a correlation coefficient that can be treated as another unknown parameter estimation errors are analyzed by examining the lower bound of their covariance matrix in the eigenspace paper 1 concludes by suggesting that certain model identification criteria developed in the time series literature all of which are based on the maximum likelihood concept might be useful for selecting the best groundwater model or the best method of parameterizing a particular model among a number of given alternatives

nonlinear structural design sensivitity analysis for path dependent problems part 1 general theory abstract two procedures direct differentiation and adjoint structure for design sensitivity analysis of nonlinear structural systems with path dependent effects are described in this paper continuum formulation is used and both geometric and material nonlinearities are included in the derivations incremental nonlinear analysis based on the total lagrangian concept is adopted the reference volume concept is used to unify the shape and nonshape design problems to show the generality of the theory the derived expressions are reduced to some special cases that have been treated previously in the literature in part 2 of the paper a few analytical example problems will be solved to demonstrate the use of the design sensitivity analysis theory the theory has also beendiscretized elsewhere using finite element approximations for implementation into structural analysis programs these developments offer a firm foundation for design optimization of complex structural systems

minimum weight design of non linear elastic structures with multimodal buckling constraints it well known that multimodal instability is an event particularly relevant in structural optimization here in the context of non linear stability theory an exact method is developed for minimum weight design of elastic structures with multimodal buckling constraints given an initial design the method generates a sequence of improved designs by determining a sequence of critical equilibrium points related to decreasing values of the structural weight multimodal buckling constraints are imposed without repeatedly solving an eigenvalue problem and the difficulties related to the non differentiability in the common sense of state variables in multimodal critical states are overcome by means of the lagrange multiplier method further constraints impose that only the first critical equilibrium states local maxima or bifurcation points on the initial equilibrium path of the actual designs are taken into account copyright 2002 john wiley sons ltd

design sensitivity analysis and optimization for nonlinear buckling of finite dimensional elastic conservative structures the purpose of this review paper is to summarize the existing methods of design sensitivity analysis and optimization of elastic conservative finite dimensional systems with respect to nonlinear buckling behavior difficulties related to geometrical nonlinear singular behaviors are discussed in detail characteristics of optimized structures are demonstrated in reference to snapthrough behavior hill top branching and degenerate critical points a new optimization result of a flexible truss that fully utilizes the snapthrough behavior is also presented

nonlocal and localized analyses of conditional mean steady state flow in bounded randomly nonuniform domains 1 theory and computational approach we consider the effect of measuring randomly varying hydraulic conductivitiesk x on one s ability to predict numerically without resorting to either monte carlo simulation or upscaling steady state flow in bounded domains driven by random source and boundary terms our aim is to allow optimum unbiased prediction of hydraulic heads h x and fluxes q x by means of their ensemble moments h x c and q x c respectively conditioned on measurements of k x these predictors have been shown by neuman and orr 1993a to satisfy exactly an integrodifferential conditional mean flow equation in which q x c is nonlocal and non darcian here we develop complementary integrodifferential equations for second conditional moments of head and flux which serve as measures of predictive uncertainty obtain recursive closure approximations for both the first and second conditional moment equations through expansion in powers of a small parameter y which represents the standard estimation error of ln k x and show how to solve these equations to first order in y2 by finite elements on a rectangular grid in two dimensions in the special case where one treats k x as if it was locally homogeneous and mean flow as if it was locally uniform one obtains a localized darcian approximation q x c kc x h x c in which kc x is a space dependent conditional hydraulic conductivity tensor this leads to the traditional deterministic darcian steady state flow equation which however acquires a nontraditional meaning in that its parameters and state variables are data dependent and therefore inherently nonunique it further explains why parameter estimates obtained by traditional inverse methods tend to vary as one modifies the database localized equations yield no information about predictive uncertainty our stochastic derivation of these otherwise standard deterministic flow equations makes clear that uncertainty measures associated with estimates of head and flux obtained by traditional inverse methods are generally smaller often considerably so than measures of corresponding predictive uncertainty which can be assessed only by means of stochastic models such as ours we present a detailed comparison between finite element solutions of nonlocal and localized moment equations and monte carlo simulations under superimposed mean uniform and convergent flow regimes in two dimensions paper 1 presents the theory and computational approach and paper 2 guadagnini and neuman this issue describes unconditional and conditional computational results

optimization of shallow trusses against limit point instability the problem of the maximization of the critical load limit point instability of shallow space trusses of constant volume or weight is considered exact solutions being possible in simple cases a detailed investigation of the question of the uniqueness of the optimum designs for two and four bar shallow truss problems is carried out the optimized trusses satisfy the constant strain energy density criteria and the load maximization problem is verified to be the dual of the minimization of the total potential energy

minimum weight design of truss structures with geometric nonlinear behavior the paper presents an optimization method based on optimality criterion for minimum weight design of structures with geometric nonlinear behavior the nonlinear critical load is determined by finding the load level at which the hessian of the potential energy ceases to be positive definite a recurrence relation based on the criterion that at optimum the nonlinear strain energy density be equal in all the members is used to develop an algorithm sample problems are given to illustrate the application of the method to truss type structures with a large number of design variables number of algorithms based on the optimality criterion approach have been developed to design a minimum weight structure with specified constraints on nodal displacements element stresses system stability etc 1 2 the optimization algorithm consisted of analyzing the structure by the finite element method and using a recurrence relation derived from the appropriate optimality criterion to modify the design variables the optimality criterion was derived by differentiating the lagrangian with respect to the design variables the displacements were assumed to be small and in the finite element analysis linear equilibrium equations were solved to determine the response of the structure to the ap plied loads in the case of system stability the constraints were defined see refs 3 6 by the associated linear eigenvalue problem this definition of system stability for some structures may not be valid because of the nonlinear behavior of the structure which may be due to the geometry of the structure or the presence of geometric imperfections the correct procedure for these structures would be to analyze the structure by using nonlinear equilibrium equations this will be particularly true for large space structures lss in this paper an optimization method based on the optimality criterion approach is presented for structures with geometric nonlinear behavior in the case of a structure optimized with constraints on linear stability the optimum structure can have more than one critical buckling mode this design tends to become im perfection sensitive and a small deviation in the geometry of the structure can reduce its load carrying capacity sub stantially the imperfection sensitivity of the optimized structure can be reduced by designing the structure so that buckling loads associated with the critical buckling modes are not equal see ref 7 a real structure has geometric im perfections and these structures tend to have nonlinear

regression and time series model selection in small samples summary a bias correction to the akaike information criterion aic is derived for regression and autoregressive time series models the correction is of particular use when the sample size is small or when the number of fitted parameters is a moderate to large fraction of the sample size the corrected method called aicc is asymptotically efficient if the true model is infinite dimensional furthermore when the true model is of finite dimension aicc is found to provide better model order choices than any other asymptotically efficient method applications to nonstationary autoregressive and mixed autoregressive moving average time series models are also discussed

an analysis of the pilot point methodology for automated calibration of an ensemble of conditionally simulated transmissivity fields an analysis of the pilot point method for automated calibration of an ensemble of conditionally simulated transmissivity fields was conducted on the basis of the simplifying assumption that the flow model is a linear function of log transmissivity the analysis shows that the pilot point and conditional simulation method of model calibration and uncertainty analysis can produce accurate uncertainty measures if it can be assumed that errors of unknown origin in the differences between observed and model computed water pressures are small when this assumption is not met the method could yield significant errors from overparameterization and the neglect of potential sources of model inaccuracy the conditional simulation part of the method is also shown to be a variant of the percentile bootstrap method so that when applied to a nonlinear model the method is subject to bootstrap errors these sources of error must be considered when using the method

weights of the exponential fitting multistep algorithms for first order odes abstract we describe a numerical method for the calculation of the weights of the linear multistep algorithms for solving first order differential equations the main novelties are that i we admit nonequidistant mesh points in the partition and ii the weights are determined on the basis of the exponential functions exp i x i 1 2 3 rather than on the power function set as it is done for the classical weights in this way the method allows computing not only the weights of the well established algorithms but also those of new ones another novelty consists in the construction of a general scheme for the error analysis of this kind of algorithms some relevant numerical illustrations are given

refined semi analytical design sensitivities efficient structural optimization routines require availability of gradient information semi analytical sa design sensitivities are rather popular as they combine ease of implementation with computational efficiency their main drawback however is their well known inaccuracy problem for shape design sensitivities it was found that the inaccuracies are especially unacceptable for slender structures and become more pronounced when relatively large rigid body motions can be identified for individual finite elements based on these observations the authors recently developed a refined sa method taking full advantage of analytical differentiation of rigid body modes the present article presents a sound and unified formulation of refined semi analytical rsa design sensitivities for linear linearized buckling geometrically nonlinear and limit point analyses numerical results are presented in order to demonstrate the efficiency of the proposed method it is concluded that the refined sa method possesses the advantages of the traditional sa method whereas it does not exhibit its unacceptable inaccuracies

semi analytical static nonlinear structural sensitivity analysis the paper discusses two uses of the semi analytical method for nonlinear sensitivity analysis first the application to noncritical response is analyzed it is shown that the semi analytical method is equivalent to a particular form of the overall finite difference approach it is concluded that the difference between overall finite differences and the semi analytical approach is blurred for the case of nonlinear static response next the application of the semi analytical method to calculating sensitivities of limit loads is discussed a formulation that is easy to implement in general purpose finite element programs is derived three examples are used to demonstrate the application of the formulation and to explore its accuracy

parameter sensitivity of inelastic buckling and post buckling response this paper is concerned with recent sensitivity results for buckling and post buckling problems the article starts with an overview of the fem based approach to inelastic structural mechanics systems with singular points along the equilibrium path the existing sensitivity techniques are reviewed their inherent drawbacks are indicated and a new methodology is suggested the discretized formalism is mainly employed throughout but a continuum formulation appropriate for this class of problems is also given numerical studies illustrate the presentation

structural optimization of guyed trusses here is a strong trend in the design of skeletal structures to wherever practical make increasing use of tensioned cables one of the best examples of the convergence of structural efficiency to a best design can probably be found when studying the evolution of the configurations of power transmission towers from the early days of cantilever truss structures these have now evolved to the much lighter cross rope type in which two separate vertical cable stayed lattice struts support the power conductors through suspension from a cross rope apart from the savings in material significant savings in transportation cost can also be obtained especially since erection often has to occur in remote areas although the cable stayed structures are easy to analyze the determination of a best or least weight design can be complex due to the variety of design variables available these are typically configuration geometry member sizing and prestraining the optimization of a configuration is complex and is best accomplished manually perhaps with the aid of layout optimization methods on the contrary geometric and prestrain optimization and sizing can be treated by direct sensitivity analysis of a chosen topology this paper describes the sizing and prestrain optimization of guyed trusses for the nonlinear analysis the total lagrangian method is used in combination with the pure newton method and adaptive arc length procedures because of the lack of stiffness of a cable structure in its undeformed state soft spring elements are used to stabilize the structure temporarily during the load stepping phase the choice of prestrain instead of prestress variables is justified by both the simplicity of inclusion of prestrain in the sensitivity analysis and by the independence of the prestrain quantity with respect to the sequence of pretensioning of the structure the prestraining of a member can also simply be seen as the lack of fit of a member in a fixed geometry and can in practice be accomplished by specifying member lengths to create a predetermined lack of fit or by specifying turns of a turnbuckle included in the member for the optimization process two optimization methods are used namely the sequential quadratic programming sqp method idesign plba authored by arora1 and the successive approximation method sam algorithm of snyman and stander2 and stander et al 3 using two methods gives optima in which greater confidence can be placed additionally some idea of the degree of difficulty of the newly formulated optimization problems is gained five examples are given to demonstrate the analysis and optimization methodology the optimization of example structures subjected to second order effects show that unexpected results can be obtained a number of papers 4 8 related to the general topic of the sensitivity analysis and optimization of structures with nonlinear geometric

design sensitivity analysis of non linear structures in regular and critical states abstract a general design sensitivity formulation is introduced for non linear elastic structures reaching critical equilibrium states a discretized formulation is considered first and the sensitivity analysis is discussed for regular and critical states including analysis of post critical behavior next a variational approach is presented for beam and surface structures for which generalized stress and strain operator formulation is applied the formulae for sensitivity of bifurcation loads are derived in terms of adjoint fields which are shown to be equivalent to post buckling fields in the case of symmetric bifurcation

a class of stabilized extended one step methods for the numerical solution of odes abstract to overcome the order barrier imposed by a stability on linear multistep methods lmms usmani and agarwal 1 had constructed a third order a stabilized extended one step method by coupling two lmms in the present paper we introduce a class of extended one step methods generalizing the method of usmani and agarwal methods of this class of orders three and four which are a and or l stable have been given previously in 2 3 it is natural to ask the maximum attainable order for methods of this class which are also a stable or l stable the purpose of this paper is to show that the maximum attainable order of a method in this class is five we derive fifth order extended one step methods and show the existence of sub families of these methods which are a stable or l stable these methods are illustrated by considering two numerical examples

piecewise linearized methods for initial value problems with oscillating solutions two piecewise linearized methods for the numerical solution of nonlinear second order differential equations which contain the first order derivative are presented these methods provide piecewise analytical solutions in open intervals result in explicit finite difference equations for both the displacement and the velocity or only the displacement at nodal points and continuous solutions everywhere one of the piecewise linearized methods is a two level method that provides smooth solutions and is self starting the other is a non self starting three level technique the paper also presents several iterative schemes a time linearized three point technique and a time linearized numerov method an extensive assessment of the piecewise linearized methods is carried out in order to assess their accuracy in nine examples and it is shown that in some cases these techniques provide as accurate results as exponentially or trigonometrically fitting methods that do require an estimate of the dominant frequency of the solution the piecewise linearized methods presented here do not require such an estimate at all but may exhibit spurious fixed points attractors in nonlinear problems and their accuracy is a strong function of both the nonlinearities and the time step however they are applicable to nonlinear second order differential equations which contain the first order derivative

inverse stochastic moment analysis of steady state flow in randomly heterogeneous media 1 nonlocal stochastic moment equations have been used successfully to analyze steady state and transient flow in randomly heterogeneous media conditional on measured values of medium properties we present a nonlinear geostatistical inverse algorithm for steady state flow that makes it possible to further condition such analyses on measured values of state variables our approach accounts for all scales of spatial variability resolvable by the computational grid it is based on recursive finite element approximations of exact nonlocal first and second conditional moment equations hydraulic conductivity is parameterized geostatistically on the basis of measured values at discrete locations and unknown values at discrete pilot points prior estimates of pilot point values are obtained optionally by generalized kriging posterior estimates at pilot points and optionally at measurement points are obtained by calibrating mean flow against measured values of head the parameters are projected onto a computational grid via kriging maximum likelihood calibration allows estimating not only hydraulic but also optionally unknown variogram parameters with or without prior information about the former the approach yields covariance matrices for parameter estimation as well as head and flux prediction errors the latter being obtained a posteriori we implement our inverse approach on highly and mildly heterogeneous media under superimposed mean uniform and convergent flows in a bounded two dimensional domain our examples illustrate how conductivity and head data act separately and jointly to reduce parameter estimation errors and to model predictive uncertainty we also evaluate the functional form of the log conductivity variogram and its parameters using likelihood based model discrimination criteria

scheme dependence and equivalence of sensitivity for nonlinear problems in static nonlinear structural analysis there are several schemes widely employed to calculate the responses of structures examples are the newton raphson method displacement control method arc length method and work control method it is known that the responses obtained are independent of the schemes chosen for the analysis problem it is shown in this paper that if sensitivities of response variables are computed different schemes do give rise in general to different results however these sensitivities are all equivalent a systematic approach is derived in order to show this equivalence this approach is derived in incremental form using the analytical differentiation approach numerical examples for validating the present approach include a nonlinear elastic spring model and geometric nonlinear analysis of trusses the finite element method is used in the truss examples interestingly it is also found that the finite difference method performs poorly in obtaining the sensitivities corresponding to the arc length and work control methods especially for structures with limit point response

numerical solutions of moment equations for flow in heterogeneous composite aquifers 1 we analyze flow in heterogeneous media composed of multiple materials whose hydraulic properties and geometries are uncertain our analysis relies on the composite media theory of winter and tartakovksy 2000 2002 which allows one to derive and solve moment equations even when the medium is highly heterogeneous we use numerical solutions of darcy flows through a representative composite medium to investigate the robustness of perturbation approximations in porous medium with total log conductivity variances as high as 20 we also investigate the relative importance of the two sources of uncertainty in composite media material properties and geometry in our examples the uncertain geometry by itself captures the main features of the mean head estimated by the full composite model even when the within material conductivities are deterministic however neglecting randomness within materials leads to head variance estimates that are qualitatively and quantitatively wrong we compare the composite media approach to approximations that replace statistically inhomogeneous conductivity fields with pseudohomogeneous random fields with deterministic trends we demonstrate that models with a deterministic trend can be expected to give a poor estimate of the statistics of head

exponential fitting bdf algorithms and their properties we present two families of explicit and implicit bdf formulas capable of the exact integration with only round off errors of differential equations whose solutions belong to the space generated by the linear combinations of exponential of matrices products of the exponentials by polynomials and products of those matrices by ordinary polynomials those methods are suitable for stiff and highly oscillatory problems then we will study their properties of local truncation error and stability plots of their 0 stability regions in terms of ah are provided plots of their regions of absolute stability that include all the negative real axis in the implicit case are provided exponential fitting algorithms depend on a parameter ah how can we find this parameter is a big question here we show different ways to find a good parameter finally numerical examples show the efficiency of the proposed codes specially when we are integrating stiff problems where the jacobian of the function has complex eigenvalues or problems where the jacobian has positive eigenvalues but the solutions of the problems have not positive exponentials

a direct application of higher order parametric programming techniques to structural optimization computational methods based on a sequence of parametric programming problems are presented for solving constrained optimization problems cop without any parameter an auxiliary parametric programming problem appp is formulated in order to solve cop the procedure is started with an arbitrary initial solution which is the trivial solution of appp corresponding to the initial value of the parameter then the optimal solution for the final value of the parameter which is the optimal solution of cop is estimated by taylor s expansion with respect to the parameter where higher order terms are incorporated it is shown that the incorporation of the higher order terms indeed leads to a faster convergence of the solution as an extension of the method a general algorithm is presented for optimum design problems with state variable constraints which are implicit functions of the design variables logarithmic penalty functions are incorporated and the weight coefficients for the penalty terms are updated continuously the derivatives of the state variables with respect to the parameter and their sensitivity coefficients are expressed explicitly in terms of those of the design variables finally a method of simultaneous analysis and optimization is developed for trusses with geometrical non linearity

groundwater flow in heterogeneous composite aquifers 1 we introduce a stochastic model of flow through highly heterogeneous composite porous media that greatly improves estimates of pressure head statistics composite porous media consist of disjoint blocks of permeable materials each block comprising a single material type within a composite medium hydraulic conductivity can be represented through a pair of random processes 1 a boundary process that determines block arrangement and extent and 2 a stationary process that defines conductivity within a given block we obtain second order statistics for hydraulic conductivity in the composite model and then contrast them with statistics obtained from a standard univariate model that ignores the boundary process and treats a composite medium as if it were statistically homogeneous next we develop perturbation expansions for the first two moments of head and contrast them with expansions based on the homogeneous approximation in most cases the bivariate model leads to much sharper perturbation approximations than does the usual model of flow through an undifferentiated material when both are applied to highly heterogeneous media we make this statement precise we illustrate the composite model with examples of one dimensional flows which are interesting in their own right and which allow us to compare the accuracy of perturbation approximations of head statistics to exact analytical solutions we also show the boundary process of our bivariate model is equivalent to the indicator functions often used to represent composite media in monte carlo simulations

multimodel bayesian analysis of groundwater data worth we explore the way in which uncertain descriptions of aquifer heterogeneity and groundwater flow impact one s ability to assess the worth of collecting additional data we do so on the basis of maximum likelihood bayesian model averaging mlbma by accounting jointly for uncertainties in geostatistical and flow model structures and parameter hydraulic conductivity as well as system state hydraulic head estimates given uncertain measurements of one or both variables previous description of our approach was limited to geostatistical models based solely on hydraulic conductivity data here we implement the approach on a synthetic example of steady state flow in a two dimensional random log hydraulic conductivity field with and without recharge by embedding an inverse stochastic moment solution of groundwater flow in mlbma a moment equations based geostatistical inversion method is utilized to circumvent the need for computationally expensive numerical monte carlo simulations the approach is compatible with either deterministic or stochastic flow models and consistent with modern statistical methods of parameter estimation admitting but not requiring prior information about the parameters it allows but does not require approximating lead predictive statistical moments of system states by linearization while updating model posterior probabilities and parameter estimates on the basis of potential new data both before and after such data are actually collected

exponential fitted runge kutta methods of collocation type fixed or variable knot points two different classes of exponential fitted runge kutta collocation methods are considered methods with fixed points and methods with frequency dependent points for both cases we have obtained extensions of the classical two stage gauss radauiia and lobattoiiia methods numerical examples reveal important differences between both approaches

controlling the error growth in long term numerical integration of perturbed oscillations in one or several frequencies in this paper we present a simple technique that allows us to limit the error growth in the long term numerical integration of perturbed multi dimensional oscillators while using highly efficient and accurate special multi step codes after laying down the theoretical basis of their behaviour we then illustrate them with some numerical examples including a case with non resonant frequencies seldom found in the related literature

conversion from imperfection sensitive into imperfection insensitive elastic structures i theory abstract a qualitative improvement of the initial postbuckling behavior of imperfection sensitive elastic structures is their conversion into imperfection insensitive structures by means of modifications of the original design such a conversion is restricted to symmetric bifurcation koiter s initial postbuckling analysis is applied in the framework of the fem to deduce mathematical relations for the transition from imperfection sensitivity to insensitivity which may be achieved by additional supports of the structure this conclusion as well as several other conclusions from the theoretical investigation reported in part i of this paper are corroborated by the results from a comprehensive numerical investigation documented in part ii of this work

frequency determination and step length control for exponentially fitted runge kutta methods an exponentially tted runge kutta efrk fth order method with six stages is constructed which exactly integrates rst order di8erential initial value problems whose solutions are linear combinations of functions of the form exp x exp x r or ir by combining this efrk method with an equivalent classical embedded 4 5 runge kutta method a technique is developed for the estimation of the occurring values error and step length control is carried out by using the richardson extrapolation procedure some numerical experiments show the e ciency of the introduced methods c 2001 elsevier science b v all rights reserved

new explicit adapted numerov methods for second order oscillatory differential equations this paper is concerned with a novel class of explicit adapted numerov an methods for solving initial value problems of second order oscillatory differential equations of the form y email protected 2y f x y the scheme of the methods incorporate the special structure of the problem carried by the term w 2y so that they can integrate the linear equation y email protected 2y 0 without truncation error a new type of simplifying assumption is proposed on the linear operators related to the quadratures that lead to the an scheme a practical an method of order five and three an methods of order six are derived the phase and stability properties of the new methods are analyzed the results of numerical experiments show the robustness and competence of the new methods compared with some highly efficient codes in the recent literature

nonlinear analysis of optimized structure with constraints on systemstability an optimization algorithm based on an optimality criterion was used to design a minimum weight space truss with different constraint requirements on system stability the constraints were specified so that the eigenvalues associated with all the critical buckling modes are either equal or separated by a specified factor for the second case the critical buckling mode was preselected from all the possible critical modes the designs obtained for the various constraint conditions were analyzed with and without specified geometric imperfections using a nonlinear finite element program which accounts for geometric nonlinearity the results obtained for various designs were compared for their imperfection sensitivity

exact sensitivity matrix and influence of the number of pilot points in the geostatistical inversion of moment equations of groundwater flow received 3 august 2009 revised 12 july 2010 accepted 21 july 2010 published 4 november 2010 1 we present novel equations for the exact sensitivity matrix of the ensemble mean hydraulic head under steady state groundwater flow conditions these equations are embedded in a geostatistical inverse procedure to condition approximations of stochastic moment equations of flow on measured hydraulic conductivities and heads our formulation allows considerable improvement of the methodology proposed by hernandez et al 2003 2006 and renders the inversion of moment equations feasible for a large number of unknown hydraulic parameters the spatial distribution of the natural logarithm y of conductivity is parameterized within the pilot points framework whereas prior values of y at pilot points are obtained by a variant of kriging posterior estimates at pilot points are obtained through a maximum likelihood fit of computed to measured heads the maximum likelihood function also includes a regularization term by means of a synthetic example and upon adopting formal model information criteria we explore the influence of 1 the number of pilot points and 2 the order of approximation of the governing mean flow equation on our ability to properly estimate the log conductivity and head fields and identify the relative weight of the regularization term and the parameters of the underlying y variogram we find that none of the adopted information criteria can identify the optimum number of pilot points and the plausibility weight and variogram parameters values can be determined by the kashyap s bayesian measure

frequency evaluation for exponentially fitted runge kutta methods this paper can be seen as a further investigation of the frequency evaluation techniques which are very recently proposed by ixaru et al for exponentially fitted multistep algorithms for first order ordinary differential equations odes the question answered was how the frequencies should be tuned in order to have a maximal benefit from exponentially fitted methods in a previous paper this frequency evaluation algorithm was successfully applied in a direct way to a second order exponentially fitted runge kutta efrk method of collocation type but such a direct application is impossible for higher order efrk methods to overcome this difficulty we develop an efficient extension of ixaru s frequency evaluation algorithm for the exponentially fitted radauiia method of third order it is an adaption of ixaru s algorithm in the sense that instead being applied globally it is applied stagewise numerical experiments illustrate the properties of the developed algorithm

role of model selection criteria in geostatistical inverse estimation of statistical data and model parameters 1 we analyze theoretically the ability of model quality sometimes termed information or discrimination criteria such as the negative log likelihood nll bayesian criteria bic and kic and information theoretic criteria aic aicc and hic to estimate 1 the parameter vector of the variogram of hydraulic log conductivity y ln k and 2 statistical parameters and proportional to head and log conductivity measurement error variances respectively in the context of geostatistical groundwater flow inversion our analysis extends the work of hernandez et al 2003 2006 and riva et al 2009 who developed nonlinear stochastic inverse algorithms that allow conditioning estimates of steady state and transient hydraulic heads fluxes and their associated uncertainty on information about conductivity and head data collected in a randomly heterogeneous confined aquifer their algorithms are based on recursive numerical approximations of exact nonlocal conditional equations describing the mean and co variance of groundwater flow log conductivity is parameterized geostatistically based on measured values at discrete locations and unknown values at discrete pilot points optionally the maximum likelihood function on which the inverse estimation of y at pilot points is based may include a regularization term reflecting prior information about y the relative weight assigned to this term and its components and as well as are evaluated separately from other model parameters to avoid bias and instability this evaluation is done on the basis of criteria such as nll kic bic hic aic and aicc we demonstrate theoretically that whereas all these six criteria make it possible to estimate kic alone allows one to estimate validly and and thus we illustrate this discriminatory power of kic numerically by using a differential evolution genetic search algorithm to minimize it in the context of a two dimensional steady state groundwater flow problem we find that whereas and the integral scale of y can be estimated on the basis of a zero order mean flow equation the sill of the y variogram is estimated more accurately by a second order approximation of flow this notwithstanding kic prefers the simpler zero order moment over the more complex second order version

prediction of steady state flow in nonuniform geologic media by conditional moments exact nonlocal formalism effective conductivities and weak approximation we consider the effect of measuring randomly varying local hydraulic conductivities k x on one s ability to predict steady state flow within a bounded domain driven by random source and boundary functions more precisely we consider the prediction of local hydraulic head h x and darcy flux q x by means of their unbiased ensemble moments h x and q x conditioned on measurements of k x these predictors satisfy a deterministic flow equation in which q x x h x r x where x is a relatively smooth unbiased estimate of k x and r x is a residual flux we derive a compact integral expression for r x which is rigorously valid for a broad class of k x fields including fractals it demonstrates that q x is nonlocal and non darcian so that an effective hydraulic conductivity does not generally exist we show analytically that under uniform mean flow the effective conductivity may be a scalar a symmetric or a nonsymmetric tensor or a set of directional scalars which do not form a tensor we demonstrate numerically that in two dimensional mean radial flow it may increase from the harmonic mean of k x near interior and boundary sources to the geometric mean far from such sources for cases where r x can neither be expressed nor approximated by a local expression we propose a weak integral approximation closure which appears to work well in media with pronounced heterogeneity and improves as the quantity and quality of k x measurements increase the nonlocal deterministic flow equation can be solved numerically by standard methods our theory shows clearly how the scale of grid discretization should relate to the scale quantity and quality of available data after providing explicit approximations for the second moments of head and flux prediction errors we conclude by discussing practical methods to compute x from noisy measurements of k x and to calculate required second moments of the associated estimation errors when k x is lognormal

phase fitted and amplification fitted two step hybrid methods for y f x y this paper provides a theoretical framework for a new type of phase fitted and amplification fitted two step hybrid ftsh methods which is introduced by the author in h van de vyver a phase fitted and amplification fitted explicit two step hybrid method for second order periodic initial value problems internat j modern phys c 17 2006 663 675 the methods constitute a modification of dissipative two step hybrid methods in the sense that two free parameters are added to eliminate the phase lag and the amplification error the methods are useful only when a good estimate of the frequency of the problem is known in advance the parameters depend on the product of the estimated frequency and the stepsize the algebraic order zero stability stability and phase properties are examined the theory is illustrated with sixth order explicit ftsh methods numerical results carried out on an assortment of test problems show the relevance of the theory

an embedded pair of exponentially fitted explicit runge kutta methods an embedded pair of exponentially fitted explicit runge kutta rk methods for the numerical integration of ivps with oscillatory solutions is derived this pair is based on the exponentially fitted explicit rk method constructed in vanden berghe et al and we confirm that the methods which constitute the pair have algebraic order 4 and 3 some numerical experiments show the efficiency of our pair when it is compared with the variable step code proposed by vanden berghe et al j comput appl math 125 2000 107

exponentially fitted methods and their stability functions we investigate the properties of stability functions of exponentially fitted runge kutta methods and we show that it is possible to some extent to determine the stability function of a method without actually constructing the method itself to focus attention examples are given for the case of one stage methods we also make the connection with so called integrating factor methods and exponential collocation methods various approaches are given to construct these methods

optimization of structures with nonlinear behavior using equivalent loads abstract nonlinear response optimization using the equivalent loads nroel method algorithm is proposed to perform optimization of non linear response structures the conventional method spends most of the design time on nonlinear analysis the nroel algorithm makes the equivalent load cases for each response and repeatedly performs linear response optimization and uses them as multiple loading conditions the equivalent loads are defined as the loads in linear analysis which generate the same response fields as those in nonlinear analysis the algorithm is validated for convergence and optimality the proposed algorithm is applied to a simple mathematical problem to verify the convergence and the optimality the nroel algorithm is applied to several structural problems with geometric and or material nonlinearity conventional optimization with sensitivity analysis using the finite difference method is also applied to the same examples the results of the optimizations are compared the proposed nroel method is found to be very efficient and good solutions are derived

sensitivity analysis for frictional contact problems in the augmented lagrangian formulation direct differentiation method of sensitivity analysis is developed for frictional contact problems as a result of the augmented lagrangian treatment of contact constraints the direct problem is solved simultaneously for the displacements and lagrange multipliers using the newton method the main purpose of the paper is to show that this formulation of the augmented lagrangian method is particularly suitable for sensitivity analysis because the direct differentiation method leads to a non iterative exact sensitivity problem to be solved at each time increment the approach is applied to a general class of three dimensional frictional contact problems and numerical examples are provided involving large deformations multibody contact interactions and contact smoothing techniques

structural design optimization of shallow frame structures undergoing large deflections an optimization algorithm for structural design against instability is developed for shallow beam structures undergoing large deflections the algorithm is based on the maximization of the limit load under specified volume constraint the analysis for obtaining the limit load involves coupling of axial and bending deformations and is based on the nonlinear finite element analysis using the displacement control technique the optimization is carried out using both the sequential quadratic programming sqp and optimality criterion oc techniques and the results are compared for the sqp technique the sensitivity derivatives of the critical load factor are calculated using the adjoint method based on the information obtained from the nonlinear buckling analysis a shallow plane arch illustrates the structural design optimization methodology and the results are compared with those in the literature it is shown that a design based on the generalized eigenvalue problem linear buckling gives an optimum limit load less than the initial limit load whereas the optimization using the nonlinear buckling analysis obtains a larger value for the optimum limit load compared to the initial limit load it has also been demonstrated that the optimum results obtained using oc technique are in good agreement with those obtained through sqp technique however the computational time for the oc is significantly lower than that of sqp which requires search techniques and sensitivity of the limit load for its successful completion and termination

efficient handling of stability problems in shell optimization by asymmetric worst case shape imperfection the paper presents an approach to shape optimization of proportionally loaded elastic shell structures under stability constraints to reduce the stability related problems a special technique is utilized by which the response analysis is always terminated before the first critical point is reached in this way the optimization is always related to a precritical structural state the necessary load carrying capability of the optimal structure is assured by extending the usual formulation of the optimization problem by a constraint on an estimated critical load factor since limit points are easier to handle the possible presence of bifurcation points is avoided by introducing imperfection parameters they are related to an asymmetric shape perturbation of the structure during the optimization the imperfection parameters are updated to get automatically the worst case pattern and amplitude of the imperfection both the imperfection parameters and the design variables are related to the structural shape via the design element technique a gradient based optimizer is employed to solve the optimization problem three examples illustrate the proposed approach copyright 2007 john wiley sons ltd

elastic postbuckling analysis via finite element and perturbation techniques part 1 formulation the main equations for the equilibrium stability and critical state analysis of discrete elastic systems are presented following the works of thompson but in such a way that the original set of generalized coordinates and loads are preserved in the total potential energy this introduces differences in the resulting equations in bifurcation analysis but does not introduce any new feature regarding the physics of the problem the new formulation is approximated by means of a standard finite element approach based on interpolation of displacements in which the derivatives of the potential energy are approximated the terms retained are those of moderately large rotation theory the energy analysis is finally related to the more conventional finite element notation in terms of stiffness matrices and it is shown how in such a way it can be included in present day codes part 2 of the paper deals with applications to the analysis of shells of revolution using a semi analytical approximation two cases are presented in detail bifurcation in axisymmetric and in asymmetric modes and the results show good correlation with those of other authors the influence of load and geometric imperfections is evaluated

frequency evaluation in exponential fitting multistep algorithms for odes we consider the linear multistep algorithms for first order odes and examine the problem of how the frequencies should be tuned in order to obtain the maximal benefit from the exponential fitting versions of such algorithms we find out that the key of the answer consists in analysing the behaviour of the error on further investigating the simple case of two step bdf algorithms we produce formulae for the optimal s and show that if the optimal s are used the order of the method is increased by one unit the reported numerical illustrations suggest that further investigations along these lines deserve a real attention

stochastic identification of transmissivity and effective recharge in steady groundwater flow 2 case study the study is a continuation and extension of a previous work dagan 1985a whose aim was to identify the values of the log transmissivity y for steady flow the common basic assumptions are that y is a normal and stationary random space function the aquifer is unbounded and a first order approximation of the flow equation is adopted the expected value of the water head h as well as the y unconditional autocovariance are supposed to have analytical expressions which depend on a parameters vector the proposed solution of the inverse problem consists of identifying with the aid of the model and of the measurements of y and h and subsequently computing the statistical moments of y conditioned on the same data the additional features of the present study are 1 incorporation of a constant but random effective recharge and its identification and 2 accounting for the fact that estimation is associated with some uncertainty whereas before was assumed to be identified with certainty analytical expressions are derived for the y and h covariances for an exponential autocovariance of y paper 2 rubin and dagan this issue of the study illustrates the applications of the method to a real life case

remarkable postbuckling paths analyzed by means of the consistently linearized eigenproblem in addition to the determination of load levels at critical points of stability problems frequently the postbuckling behavior is of interest for non linear problems the so called consistently linearized eigenvalue problem is a suitable tangent linearization method which facilitates determination of stability limits the solution process of the eigenproblem is significantly simplified by appropriate coordinate transformations within this process characteristic shapes of eigenvalue curves allow identification of bifurcation buckling modes snap through modes and hilltop buckling modes mathematical properties of the eigenvalue curves are addressed and conclusions regarding the shape of postbuckling paths are drawn considerations also touch upon the conversion from imperfection sensitivity into insensitivity the theoretical findings are corroborated by examples dealing with a von mises truss and a similar discrete system showing a remarkable postbuckling behavior such as a zero stiffness equilibrium path for both systems the same approach of stiffness increase allows conversion from imperfection sensitivity into insensitivity copyright 2008 john wiley sons ltd

estimation of spatial covariance structures by adjoint state maximum likelihood cross validation 2 synthetic experiments paper 2 of this three part series uses synthetic data to investigate the properties of the adjoint state maximum likelihood cross validation asmlcv method presented in paper 1 samper and neuman this issue a more than 40 synthetic experiments are performed to compare various conjugate gradient algorithms investigate the manner in which computer time varies with asmlcv parameters study the effect of sample size and choice of kriging points on asmlcv estimates evaluate the ability of various model structure identification criteria to help select the most appropriate semivariogram model among given alternatives study the conditions required for parameter identifiability uniqueness and stability quantify the statistics of cross validation errors test hypotheses concerning the distribution and autocorrelation of these errors and illustrate the computation of approximate quality indicators for asmlcv parameter estimates

highly parameterized inverse estimation of hydraulic conductivity and porosity in a three dimensional heterogeneous transport experiment 1 assessing the impact of parameter estimation accuracy in models of heterogeneous three dimensional 3 d groundwater systems is critical for predictions of solute transport a unique experimental data set provides concentration breakthrough curves btcs measured at a 0 253 cm3 scale over the 13 8 8 cm3 domain 53 000 measurement locations advective transport is used to match the first temporal moments of btcs or mean arrival times m1 averaged at 0 253 and 1 0 cm3 scales through simultaneous inversion of highly parameterized heterogeneous hydraulic conductivity k and porosity fields pilot points parameterize the fields within eight layers of the 3 d medium and estimations are completed with six different models of the k relationship parameter estimation through advective transport shows accurate estimation of the observed m1 values results across the six different k relationships have statistically similar fits to the observed m1 values and similar spatial estimates of m1 along the main flow direction the resulting fields provide the basis for forward transport modeling of the advection dispersion equation ade using the estimated k and fields demonstrates that advective transport coupled with inversion using dense spatial field parameterization provides an efficient surrogate for the ade these results indicate that there is not a single set of model parameters or a single k relationship that leads to a best representation of the actual experimental sand packing pattern i e nonuniqueness additionally knowledge of the individual sand k and values along with their arrangement in the 3 d experiment does not reproduce the observed transport results at small scales small scale variation in the packing and mixing of the sands causes large deviations from the expected transport results as highlighted in forward ade simulations highly parameterized inverse estimation is able to identify those regions where variations in mixing and packing alter the expected property values and significantly improve results relative to the naive application of the experimentally derived property values impacts of the observation scale the scale over which results are averaged and the number of observations and parameters on the final estimations are also examined results indicate existence of a representative element volume rev at 0 25 cm3 the existence of subgrid scale heterogeneity that impacts transport and the accuracy of highly parameterized models with even relatively small amounts of observations finally this work suggests that local heterogeneity features below the rev scale are difficult to incorporate into parameterized models highlighting the importance of addressing prediction uncertainty for small scale variability i e uncaptured variability in modeling practice