pseudo spectral least squares method for linear elasticity abstract the first order system least squares legendre and chebyshev spectral method for two dimensional space linear elasticity is investigated the drilling rotation is defined as a new variable and the linear elasticity equation is supplemented with an auxiliary equation the weighted l 2 norm least squares principle is applied to a stress displacement rotation it is shown that the homogeneous least squares functional is equivalent to weighted h 1 norm like for stress and weighted h 1 norm for displacement and rotation this weighted h 1 norm equivalence is uniform spectral convergence for both legendre and chebyshev approaches are given along with some numerical experiments the generalization for three dimensional spaces is also provided

analysis of a modified first order system least squares method for linear elasticity with improved momentum balance a modified first order system least squares formulation for linear elasticity obtained by adding the antisymmetric displacement gradient in the test space is analyzed this approach leads to surprisingly small momentum balance error compared to standard least squares approaches it is shown that the modified least squares formulation is well posed and its performance is illustrated by adaptive finite element computation based on using a closely related least squares functional as a posteriori error estimator the results of our numerical computations show that for the modified least squares approach the momentum balance error converges at a much faster rate than the overall error we prove that this is due to a strong connection of the stress approximation to that obtained from a mixed formulation based on the hellinger reissner principle with exact local momentum balance the practical significance is that our proposed approach is almost momentum conservative at a smaller number of degrees of freedom than mixed approximations with exact local momentum balance

variance reduction analysis this paper presents an algorithm for optimal data collection in random fields the so called variance reduction analysis which is an extension of kriging the basis of variance reduction analysis is an information response function i e the amount of information gain at an arbitrary point due to a measurement at another site the ranking of potential sites is conducted using an information ranking function the optimal number of new points is then identified by an economic gain function the selected sequence of sites for further sampling shows a high degree of stability with respect to noisy inputs

an investigation of the small slope approximation for scattering from rough surfaces part i theory the small slope approximation ssa of voronovich sov phys jetp 62 65 70 1985 is a promising method for application to scattering from many natural surfaces the theory gives a systematic expansion that can be interpreted as a series in generalized surface slope the ssa series for the t matrix satisfies the appropriate reciprocity condition at each order and reduces to the standard perturbation series for small surface roughness in this paper we examine in detail the derivation of the ssa for surfaces subject to the dirichlet zero field boundary condition a number of points are discussed including the requirements for determining the series terms in addition questions have been raised recently about the ssa it has been argued that 1 an assumption in the derivation contradicts the exact formulation of the problem and 2 there is an arbitrariness in determining the series terms these two points are refuted and the assumptions needed to determine the series terms unambiguously are clarified the meaning of slope orders in the ssa series expansion are examined and the concept of generalized slope is discussed a future companion paper part ii numerical studies will present an investigation of the accuracy of the ssa through comparison with exact results

an extended identifiability in aquifer parameter identification and optimal pumping test design an extended identifiability called identifiability is developed for groundwater modeling and management some fundamental concepts are developed for establishing a criterion in connection with the problem of optimal experimental design such as the design of an optimum pumping test to assist aquifer parameter identification in practice the quantity and quality of data prohibits a unique solution of the inverse problem of parameter identification the proposed identifiability is based upon the concept of weak uniqueness the determination of the admissibility of a given design is formulated as a nonlinear programming problem the original constrained problem is transformed into solving a sequence of unconstrained problems by a penalty function method numerical experiments are conducted to illustrate the proposed concept and algorithm

computational electro elasticity and magneto elasticity for quasi incompressible media immersed in free space summary in this work a mixed variational formulation to simulate quasi incompressible electro active or magneto active polymers immersed in the surrounding free space is presented a novel domain decomposition is used to disconnect the primary coupled problem and the arbitrary free space mesh update problem exploiting this decomposition we describe a block iterative approach to solving the linearised multiphysics problem and a physically and geometrically based three parameter method to update the free space mesh several application driven example problems are implemented to demonstrate the robustness of the mixed formulation for both electro elastic and magneto elastic problems involving both finite deformations and quasi incompressible media copyright 2016 john wiley sons ltd

applications of the new scattering formalism the dirichlet boundary condition this is the third in a series of articles concerning the theory of scattering from a rough surface consideration of the scattering problem for a scalar wave obeying dirichlet boundary conditions enables one to investigate the formalism developed in the previous work the advantages of this new representation of the scattering amplitude is discussed this result is supplemented with an improved integral equation for the desired solutions on the surface this new equation not only allows one to reproduce the approximations for the scattering amplitude obtained previously but it can also be used to extend those results to higher orders this is illustrated with a derivation of the second born approximation the correction to the scattering amplitude due to a distribution of point scatters is also obtained to the order under consideration this phenomenon is shown to be equivalent to a perturbation of the surface a similar result is found when the index of refraction is allowed to vary near the scattering surface

a direct method of parameter estimation for steady state flow in heterogeneous aquifers with unknown boundary conditions 1 we propose a novel direct method for estimating steady state hydrogeological model parameters and model state variables in an aquifer where boundary conditions are unknown the method is adapted from a recently developed potential theory technique for solving general inverse reconstruction problems unlike many inverse techniques used for groundwater model calibration the new method is not based on fitting and optimizing an objective function which usually requires forward simulation and iterative parameter updates instead it directly incorporates noisy observed data hydraulic heads and flow rates at the measurement points in a single step without solving a boundary value problem the new method is computationally efficient and is robust to the presence of observation errors it has been tested on two dimensional groundwater flow problems with regular and irregular geometries different heterogeneity patterns variances of heterogeneity and error magnitudes in all cases parameters hydraulic conductivities converge to the correct or expected values and are thus unique based on which heads and flow fields are constructed directly via a set of analytical expressions accurate boundary conditions are then inferred from these fields the accuracy of the direct method also improves with increasing amount of observed data lower measurement errors and grid refinement under natural flow i e no pumping the direct method yields an equivalent conductivity of the aquifer suggesting that the method can be used as an inexpensive characterization tool with which both aquifer parameters and aquifer boundary conditions can be inferred

parameter identification of groundwater aquifer models a generalized least squares approach this paper concerns the methods of estimating aquifer transmissivities on the basis of unsteady state hydraulic head data traditionally the criterion of minimizing the sum of the squares of errors has been used to match the observed data with the model response the data used for optimization usually contain noise that is not necessarily uncorrelated it is well understood that the results of identification methods are very sensitive to measurement errors in data in this study the ordinary least squares ols technique is carried out along with a generalized least squares gls technique specifically designed to reduce the effect of correlated errors the trace of the covariance matrix is used as a measure of overall accuracy and reliability of the estimated parameters the effectiveness of the ols and gls techniques in dealing with noisy data is demonstrated by using a hypothetical example the results of numerical experiments suggest that gls offers a promising approach in efficiently improving the reliability of the estimated parameters

a proposed stepwise regression method for model structure identification this paper proposes a new methodology for constructing groundwater models the proposed methodology which determines simultaneously both model structure and model parameters is based on the following ideas 1 when solving the inverse problem different model structures always produce different model parameters 2 since the number of possible model structures of an aquifer is infinite the number of possible representative parameters is also infinite 3 to obtain a set of appropriate representative model parameters we must have an appropriate model structure and 4 an appropriate model structure should be determined not only by observation data and prior information but also by the accuracy requirements of model applications in this proposed methodology we start with a homogeneous model structure and step by step gradually increase the complexity of the model structure at each level of complexity we calculate not only the fitting residual of parameter identification but also the error of model structure to determine if a more complex model structure is needed the model structure error of using one model structure to replace another model structure is defined by a maximum minimum max min problem that is based on the distance between the two models and is measured in parameter observation and prediction or decision spaces this proposed methodology is used to solve a hypothetical remediation design problem in which the true hydraulic conductivity is a random field with a certain trend we have found that for the example problem virtually identical pumping policy is obtained when a five zone model with an optimized zonation pattern is used to represent the nonstationary random field we have also found that observation errors have minimum impact on management solution in comparison with structure errors to calculate the model structure error for this example the inverse solution is coupled with a management problem we have also developed an effective iteration method to handle nonlinear water quality constraints

the least squares meshfree method for elasto plasticity and its application to metal forming analysis a new meshfree method for the analysis of elasto plastic deformation is presented the method is based on the proposed first order least squares formulation for elasto plasticity and the moving least squares approximation the least squares formulation for classical elasto plasticity and its extension to an incrementally objective formulation for finite deformation are proposed in the formulation equilibrium equation and flow rule are enforced in least squares sense i e their squared residuals are minimized and hardening law and loading unloading condition are enforced pointwise at each integration point the closest point projection method for the integration of rate form constitutive equation is inherently involved in the formulation and thus the radial return mapping algorithm is not performed explicitly the proposed formulation is a mixed type method since the residuals are represented in a form of first order differential system using displacement and stress components as nodal unknowns also the penalty schemes for the enforcement of boundary and frictional contact conditions are devised and the reshaping of nodal supports is introduced to avoid the difficulties due to the severe local deformation near contact interface the proposed method does not employ structure of extrinsic cells for any purpose through some numerical examples of metal forming processes the validity and effectiveness of the method are discussed

coupled inverse problems in groundwater modeling 1 sensitivity analysis and parameter identification some basic concepts and methods for solving the coupled inverse problem in groundwater modeling are presented in a series of two papers because there are crossover effects between state variables and parmeters coupled inverse problems are different from single inverse problems in several aspects especially in the selection of objective functions identifiability and experimental design this paper discusses the general definition of coupled inverse problems posed in the framework of multiobjective optimization in order to obtain gradient vectors of objective functions as well as all sensitivity coefficients a general procedure for deriving the adjoint state equations and their associated conditions is described using the rules of adjoint operation presented in this paper the derivation of adjoint problems becomes very easy even for nonlinear and transient coupled problems as examples the adjoint problem of flow mass transport or flow heat transport the adjoint problem of saltwater intrusion and the adjoint problem of two phase flow are derived numerical experiments on a two dimensional flow mass transport problem are given to explain the procedure of sensitivity analysis and parameter identification for coupled problems the hydraulic conductivity can be identified by using observations of head and or solute concentration

the inverse problem for confined aquifer flow identification and estimation with extensions the contributions of this work are twofold first a methodology for estimating the elements of parameter matrices in the governing equation of flow in a confined aquifer is developed the estimation techniques for the distributed parameter inverse problem pertain to linear least squares and generalized least squares methods the linear relationship among the known heads and unknown parameters of the flow equation provides the background for developing criteria for determining the identifiability status of unknown parameters under conditions of exact or overidentification it is possible to develop statistically consistent parameter estimators and their asymptotic distributions the estimation techniques namely two stage least squares and three stage least squares are applied to a specific groundwater inverse problem and compared between themselves and with an ordinary least squares estimator the three stage estimator provides the closer approximation to the actual parameter values but it also shows relatively large standard errors as compared to the ordinary and two stage estimators the estimation techniques provide the parameter matrices required to simulate the unsteady groundwater flow equation second a nonlinear maximum likelihood estimation approach to the inverse problem is presented the statistical properties of maximum likelihood estimators are derived and a procedure to construct confidence intervals and do hypothesis testing is given the relative merits of the linear and maximum likelihood estimators are analyzed other topics relevant to the identification and estimation methodologies i e a continuous time solution to the flow equation coping with noise corrupted head measurements and extension of the developed theory to nonlinear cases are also discussed a simulation study is used to evaluate the methods developed in this study

nodal auxiliary space preconditioning in h curl and h div spaces in this paper we develop and analyze a general approach to preconditioning linear systems of equations arising from conforming finite element discretizations of h curl and h div elliptic variational problems the preconditioners exclusively rely on solvers for discrete poisson problems we prove mesh independent effectivity of the preconditioners by using the abstract theory of auxiliary space preconditioning the main tools are discrete analogues of so called regular decomposition results in the function spaces h curl and h div our preconditioner for h curl is similar to an algorithm proposed in r beck algebraic multigrid by component splitting for edge elements on simplicial triangulations tech rep sc 99 40 zib berlin germany 1999

a new theory for scattering from a surface a new formal approach to the problem of scattering from a rough surface is derived the first ingredient of this approach is a formal statement of the composite model this consists of an approximate expression for the change in the scattering amplitude corresponding to a perturbation of a reference surface it is good to first order in the perturbation and exact with respect to the reference surface the derivation of this result makes use of the general asymptotic form of the scattering solution the expression is simplified using the boundary conditions when the formal composite model is applied to an unfinitesimal translation a new representation of the scattering amplitude is obtained unlike the traditional representation this expression depends on two full solutions to the wave equation and it manifestly exhibits reciprocity this article is the first in a series application of the results will be considered in future papers

an adaptive least squares mixed finite element method for elasto plasticity a least squares mixed finite element method for the incremental formulation of elasto plasticity using a plastic flow rule of von mises type with isotropic hardening is presented this approach is based on the use of the stress tensor in addition to the displacement field as independent process variables the nonlinear least squares functional is shown to constitute an a posteriori error estimator on which an adaptive refinement strategy may be based for the finite element implementation under plane strain conditions quadratic i e next to lowest order raviart thomas elements are used for the stress approximation while the displacement is represented by standard quadratic conforming elements computational results for a benchmark problem of elasto plasticity under plane strain conditions are presented in order to illustrate the effectiveness of the least squares approach

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

least squares mixed finite elements for small strain elasto viscoplasticity the main aim of this contribution is to provide a mixed finite element for small strain elasto viscoplastic material behavior based on the least squares method the l2 norm minimization of the residuals of the given first order system of differential equations leads to a two field functional with displacements and stresses as process variables for the continuous approximation of the stresses lowest order raviart thomas elements are used whereas for the displacements standard conforming elements are employed it is shown that the non linear least squares functional provides an a posteriori error estimator which establishes ellipticity of the proposed variational approach further on details about the implementation of the least squares mixed finite elements are given and some numerical examples are presented copyright 2008 john wiley sons ltd

an examination of the composite roughness scattering model a perturbation technique is applied to derive a composite roughness theory for acoustic scattering from the sea surface the leading term in the expansion obtained is the well known result obtainable using ad hoc arguments higher order terms are evaluated to assess their contribution to the high frequency monostatic backscattering strength of the sea it is concluded that the leading term in the perturbation expansion provides an excellent approximation

simultaneous inversion of hydrogeologic and thermal data 1 theory and application using hydraulic head data one goal in solving the inverse problem in groundwater hydrology is to obtain parameter estimates for use in a simulation model a philosophy about the inverse is adopted here that takes the form of constructing a variety of models under different norms large differences in parameter estimates obtained using different norms suggests the possibility of an incorrect interpretational model an optimization procedure is used that is based on the constrained simplex method this algorithm easily handles constraints does not require derivatives of the functional with respect to the parameters and can be employed to minimize complicated functionals the algorithm produces a cluster of search points in parameter space about the global minimum of the objective function each of these points generates data that matches observed values within a prescribed tolerance the centroid of the search points is taken as the best estimate of the parameters the covariances of the estimated parameters and estimated data can also be computed this inverse technique is applied to a sparse inaccurate hydraulic head data set observed at downie slide british columbia canada for this cross sectional problem with noisy data and a strong drift component interpolation of hydraulic head values using kriging techniques was not possible the domain is represented as a homogeneous isotropic medium four parameters are estimated the free surface position of the water table and three boundary conditions further simulations using a synthetic data set with assumed properties similar to that of downie slide show that with noise free data and an adequate spacing between measurement points it is possible to interpolate an unbiased estimate of hydraulic head data at all nodes in the equivalent discretized domain when the inverse technique is applied the hydraulic conductivity structure is correctly identified the implications for downie slide are that in order to resolve more than a simple hydraulic conductivity structure accurate field measurements of hydraulic head a better spacing between measurements and more detailed knowledge of the boundary conditions are required

a composite l1 parameter estimator for model fitting in groundwater flow and solute transport simulation this paper proposes a composite l1 parameter estimator to solve the inverse problems in groundwater flow and solute transport modeling the estimator is formulated using a weighted l1 norm as the error measure between the vector of the observed hydraulic heads and solute concentrations and the corresponding vector of the heads and concentrations computed from a finite element model for steady state groundwater flow and solute transport in a two dimensional areal confined aquifer system the parameters currently considered are the hydraulic conductivities the dispersivities the porosities of the aquifer system and the solute source concentration s the gradients of the state variables with respect to the parameters are computed analytically using a formulation stemming from the finite element model for the state variables the solutions to two hypothetical groundwater parameter estimation problems are presented to illustrate that the proposed estimator does have the property of being fairly accurate and more significantly being robust in handling observation data containing certain outliers

bayesian inversion for facies detection an extensible level set framework 1 in many cases it has been assumed that the variability in hydrologic parameters can be adequately described through a simple geostatistical model with a given variogram in other cases variability may be best described as a series of jumps in parameter behavior e g those that occur at geologic facies contacts when using indirect measurements such as pump tests to try to map such heterogeneity during inverse modeling the resulting images of the subsurface are always affected by the assumptions invoked in this paper we discuss inversion for parameter fields where prior information has suggested that major variability can be described by boundaries between geologic units or facies in order to identify such parameter fields we propose a bayesian level set inversion protocol framework which allows for flexible zones of any shape size and number we review formulas for defining facies locations using the level set method and for moving the boundaries between zones using a gradient based technique that improves fit through iterative deformation of the boundaries we describe the optimization algorithm employed when multiple level set functions are used to represent a field with more than two facies we extend these formulas to the investigation of the inverse problem in a bayesian context in which prior information is taken into account and through which measures of uncertainty can be derived we also demonstrate that the level set method is well suited for joint inversion problems and present a strategy for integrating different data types such as hydrologic and geophysical without assuming strict petrophysical relations our framework for joint inversion also contrasts with many previous methods in that all data sources e g both hydrologic and geophysical contribute to boundary delineation at once

aquifer parameter identification with kriging and optimum parameterization a new parameter identification method is developed for a two dimensional unsteady state groundwater flow model the method is a direct approach as it requires no iterative searching of parameters the unknown functional coefficients parameters are optimally determined by a generalized matrix inversion scheme kriging an unbiased minimum variance estimator is used as a presampling filter to reconstruct the head distribution for the entire flow domain the ill posedness of the inverse problem as characterized by instability and nonuniqueness is controlled by reparameterization utilizing the technique of finite elements a set of statistics are developed to quantify explicitly the tradeoff between the system modeling error and the error associated with the parameter uncertainty these two types of errors are shown to be a function of the parameter dimension residual mean square and trace of the covariance matrix of the estimated parameters serve as good indicators for the determination of the optimum parameter dimension a set of numerical experiments are conducted to illustrate the methodology

stabilization of bi linear mixed finite elements for tetrahedra with enhanced interpolation using volume and area bubble functions in order to overcome the oscillatory effects of the mixed bi linear galerkin formulation for tetrahedral elements a stabilization approach is presented to this end the mixed method of incompatible modes and the mixed method of enhanced strains are reformulated thus giving both the interpretation of a mixed finite element method with stabilization terms for non linear problems these are non linearly dependent on the current deformation state and therefore are replaced by linearly dependent stabilization terms the approach becomes most attractive for the numerical implementation since the use of quantities related to the previous newton iteration step typically arising for mixed enhanced elements is completely avoided the stabilization matrices for the mixed method of incompatible modes and the mixed method of enhanced strains are obtained with volume and area bubble functions various numerical examples are presented which illustrate successfully the stabilization effect for bi linear tetrahedral elements copyright 2007 john wiley sons ltd

least squares methods for linear elasticity this paper develops least squares methods for the solution of linear elastic problems in both two and three dimensions our main approach is defined by simply applying the l2 norm least squares principle to a stress displacement system the constitutive and the equilibrium equations it is shown that the homogeneous least squares functional is elliptic and continuous in the h rm div omega d times h 1 omega d norm this immediately implies optimal error estimates for finite element subspaces of h rm div omega d times h 1 omega d it admits optimal multigrid solution methods as well if raviart thomas finite element spaces are used to approximate the stress tensor our method does not degrade when the material properties approach the incompressible limit least squares methods that impose boundary conditions weakly and use an inverse norm are also considered numerical results for a benchmark test problem of planar elasticity are included in order to illustrate the robustness of our

first order system least squares fosls for convection diffusion problems numerical results the focus of this paper is on planar linear convection diffusion problems to which we apply a special form of first order system least squares fosls cai et al siam j numer anal 31 1994 pp 1785 1799 cai manteuffel and mccormick siam j numer anal 34 1997 pp 425 454 this we do by introducing the gradient of the primary variable scaled by certain exponential functions the convection diffusion equation is then recast as a minimization principle for a functional corresponding to a sum of weighted l2 norms of the resulting first order system discretization is accomplished by a rayleigh ritz method based on standard finite element subspaces and the resulting linear systems are solved by basic multigrid algorithms the main goal here is to obtain optimal discretization accuracy and solver speed that is essentially uniform in the size of convection our results show that the fosls approach achieves this goal in general when the performance is measured either by the functional or by an equivalent weighted h1 norm included in our study is a multilevel adaptive refinement method based on locally uniform composite grids and local error estimates based on the functional itself

a modified least squares mixed finite element with improved momentum balance the main goal of this contribution is to provide an improved mixed finite element for quasi incompressible linear elasticity based on a classical least squares formulation a modified weak form with displacements and stresses as process variables is derived this weak form is the basis for a finite element with an advanced fulfillment of the momentum balance and therefore with a better performance for the continuous approximation of stresses and displacements on the triangular and tetrahedral elements lowest order raviart thomas and linear standard lagrange interpolations can be used it is shown that coercivity and continuity of the resulting asymmetric bilinear form could be established with respect to appropriate norms further on details about the implementation of the least squares mixed finite elements are given and some numerical examples are presented in order to demonstrate the performance of the proposed formulation copyright 2009 john wiley sons ltd

least squares finite element methods and algebraic multigrid solvers for linear hyperbolic pdes least squares finite element methods lsfems for scalar linear partial differential equations pdes of hyperbolic type are studied the space of admissible boundary data is identified precisely and a trace theorem and a poincare inequality are formulated the pde is restated as the minimization of a least squares functional and the well posedness of the associated weak formulation is proved finite element convergence is proved for conforming and nonconforming discontinuous lsfems that are similar to previously proposed methods but for which no rigorous convergence proofs have been given in the literature convergence properties and solution quality for discontinuous solutions are investigated in detail for finite elements of increasing polynomial degree on triangular and quadrilateral meshes and for the general case that the discontinuity is not aligned with the computational mesh our numerical studies found that higher order elements yield slightly better convergence properties when measured in terms of the number of degrees of freedom standard algebraic multigrid methods that are known to be optimal for large classes of elliptic pdes are applied without modifications to the linear systems that result from the hyperbolic lsfem formulations they are found to yield complexity that grows only slowly relative to the size of the linear systems

aquifer parameter identification with optimum dimension in parameterization this paper presents a systematic procedure whereby the functional coefficients imbedded in a two dimensional partial differential equation which governs unsteady groundwater flow are optimally identified the coefficients to be identified are transmissivities which vary spatially finite elements are used to represent the unknown transmissivity function parametrically in terms of nodal values over a suitable discretization of a flow region a modified gauss newton algorithm is used for parameter optimization covariance analysis is used to estimate the reliability of the estimated parameters as the dimension of the unknown parameter increases the modeling error represented by a least squares criterion will generally decrease but errors in data would be propagated to a greater degree into the estimated parameters thus reducing the reliability of estimation the reliability of the estimated parameters is characterized by a norm of the covariance matrix this information is used for the determination of the optimum dimension in parameterization

global local optimization for parameter structure identification in three dimensional groundwater modeling 1 this paper develops two global local optimization methods for three dimensional parameter structure identification in groundwater modeling parameter structure identification is formulated in terms of solving a generalized inverse problem which allows for a determination of an appropriate level of parameter structure complexity and the identification of its pattern and the associated parameter values the structure complexity is determined by calculating the parameter structure error measured in the prediction space or management space while the structure pattern and the associated parameter values are identified simultaneously by minimizing the fitting residual measured in the observation space the former requires solving a continuous maximum minimum max min optimization problem and the latter requires solving a combinatorial optimization problem in this paper voronoi zonation is used to parameterize the unknown distributed parameter for each given level of structure complexity a sequential global local optimization method which consists of a genetic algorithm a quasi newton method and local search is developed to solve the combinatorial problem in addition the continuous max min problem is transformed to a hierarchical optimization problem and solved by a hybrid global local method sensitivities of state variables to the unknown parameters are calculated by the sensitivity equations the validity and applicability of the proposed methodology are demonstrated by numerical experiments we have shown that the choice of an objective function in model application impacts the determination of the parameter structure complexity

nurbs based least squares finite element methods for fluid and solid mechanics summary this contribution investigates the performance of a least squares finite element method based on non uniform rational b splines nurbs basis functions the least squares functional is formulated directly in terms of the strong form of the governing equations and boundary conditions thus the introduction of auxiliary variables is avoided but the order of the basis functions must be higher or equal to the order of the highest spatial derivatives the methodology is applied to the incompressible navier stokes equations and to linear as well as nonlinear elastic solid mechanics the numerical examples presented feature convective effects and incompressible or nearly incompressible material the numerical results which are obtained with equal order interpolation and without any stabilisation techniques are smooth and accurate it is shown that for p and h refinement the theoretical rates of convergence are achieved copyright 2014 john wiley sons ltd

first order system least squares for the stokes equations with application to linear elasticity following our earlier work on general second order scalar equations here we develop a least squares functional for the two and three dimensional stokes equations generalized slightly by allowing a pressure term in the continuity equation by introducing a velocity flux variable and associated curl and trace equations we are able to establish ellipticity in an h1 product norm appropriately weighted by the reynolds number this immediately yields optimal discretization error estimates for finite element spaces in this norm and optimal algebraic convergence estimates for multiplicative and additive multigrid methods applied to the resulting discrete systems both estimates are naturally uniform in the reynolds number moreover our pressure perturbed form of the generalized stokes equations allows us to develop an analogous result for the dirichlet problem for linear elasticity where we obtain the more substantive result that the estimates are uniform in the poisson ratio

review of parameter identification procedures in groundwater hydrology the inverse problem the purpose of this survey is to review parameter identification procedures in groundwater hydrology and to examine computational techniques which have been developed to solve the inverse problem parameter identification methods are classified under the error criterion used in the formulation of the inverse problem the problem of ill posedness in connection with the inverse problem is addressed typical inverse solution techniques are highlighted the review also includes the evaluation of methods used for computing the sensitivity matrix statistics which can be used to estimate the parameter uncertainty are outlined attempts have been made to compare and contrast representative inverse procedures and direction for future research is suggested

kirchhoff scattering from non penetrable targets modeled as an assembly of triangular facets frequency and time domain solutions for the scattering of acoustic waves from an arbitrarily shaped target using the kirchhoff approximation are developed in this method the scattering amplitude is analytically evaluated on a single triangle and scattering from a triangularly facetted target is computed by coherently summing the contributions from all the triangles that make up its surface in the frequency domain the solution is expressed in terms of regular non singular functions which only require the knowledge of the directions of the incident and scattered fields the edge vectors for the triangles and position vectors to one of their vertices to derive representations using regular functions in the time domain the scattered signal is expressed by different expressions for various limiting cases the frequency domain solution is validated by comparing its results to the solutions of problems for which the kirchhoff approximation has analytic solutions in order of increasing complexity they include the square plate the circular plate the finite cylinder and the sphere the time domain solution is validated by comparing it to the time domain solution of the kirchhoff approximation for a rigid sphere

multigrid in h div and h curl summary we consider the solution of systems of linear algebraic equations which arise from the finite element discretization of variational problems posed in the hilbert spaces bf h div and bf h curl in three dimensions we show that if appropriate finite element spaces and appropriate additive or multiplicative schwarz smoothers are used then the multigrid v cycle is an efficient solver and preconditioner for the discrete operator all results are uniform with respect to the mesh size the number of mesh levels and weights on the two terms in the inner products

two mixed finite element formulations with area bubble functions for tetrahedral elements abstract we address the solution of geometrically linear elastic problems for three dimensional solid mechanics using tetrahedral finite elements starting from a five field weak formulation involving fields for compatible displacements incompatible displacements pressure enhanced strains and stresses both the mixed method of incompatible modes and the mixed method of enhanced strains are considered as special cases as a key idea area bubble functions are used for both mixed finite element formulations in order to enrich the displacement field and the enhanced strain field respectively appropriate conditions for satisfaction of the patch test and unique solution of the discrete equation are verified we also observe that contrary to a the standard enhanced assumed strain method applied to triangles and tetrahedra our mixed formulations with pressure as an independent variable are not degenerate in the representative examples firstly a numerical verification of the patch test is obtained two additional examples cook s membrane problem and a plate with a ring hole illustrate the good performance of the presented approaches in comparison to existing finite element formulations

multigrid methods for the pure traction problem of linear elasticity mixed formulation a multigrid method using conforming p 1 finite elements is developed for the two dimensional pure traction boundary value problem of linear elasticity the convergence is uniform even as the material becomes nearly incompressible a heuristic argument for acceleration of the multigrid method is discussed as well numerical results with and without this acceleration as well as performance estimates on a parallel computer are included

a weighted h div least squares method for second order elliptic problems this paper presents analysis of a weighted norm least squares finite element method for elliptic problems with boundary singularities we use h div conforming raviart thomas elements and continuous piecewise polynomial elements with only a rough estimate of the power of the singularity we employ a simple locally weighted l 2 norm to eliminate the pollution effect and recover better rates of convergence theoretical results are carried out in weighted sobolev spaces and include ellipticity bounds of the homogeneous least squares functional new weighted raviart thomas interpolation results and error estimates in both weighted and nonweighted norms numerical tests are given to confirm the theoretical estimates and to illustrate the practicality of the method

approximate representations of the scattering amplitude this is the second in a series of articles about the theory of scattering from a rough surface a symmetric representation of the scattering amplitude and a formal statement of the composite model both derived in the previous article are used to approximate the scattering amplitude in terms of the known results for a reference surface when the reference surface is a plane the small slope approximation follows while the traditional composite model is obtained if the curvature of the reference surface is neglected the medium is assumed to be homogeneous and the calculation is performed for scalar waves obeying dirichlet or neumann boundary conditions electromagnetic waves obeying perfect conductor boundary conditions and the interface between two media for both types of waves finally the requirement that the formal composite model must reproduce the traditional composite model in the appropriate limit is used to obtain a new approximation to the scattering amplitude this is done for the dirichlet neumann and perfect conductor problems the error goes as the square of the curvature

identification of parameter structure in groundwater inverse problem this paper develops an optimization method that can be used to identify parameter structure as well as its values in connection with the inverse problem in groundwater modeling the complexity of the parameter structure is determined by the quantity and quality of observation data output least square error is used as the criterion of identification sensitivity coefficients at finite element nodes are evaluated by the variational method the problem of identifying the parameter structure is formulated as a problem of combinatorial optimization a new parameterization technique is presented for describing the parameter structure it can represent various distributions of parameters from piecewise constant to continuous variability based on this representation the combinatorial optimization problem for determining the parameter structure is then solved by a simple one at a time searching procedure numerical experimentations are used to demonstrate the efficiency and practicality of the proposed method

development of objective oriented groundwater models 1 robust parameter identification 1 when the structure of an aquifer is complex and unknown determining the complexity of model structure and identifying its spatially distributed parameters become very difficult this two paper series describes an approach for objective oriented model construction that can identify a model with simplified structure and assure its reliability for predetermined model applications the theoretical basis of the methodology includes the definition of structure error the conditions of structure reducibility the calculation of the worst case parameter wcp and the solution of a generalized inverse problem in the first paper of the series we extend the basic theory established in our previous works from zonation parameterization to any kind of parameterization the extension makes the methodology more general and complete we present an effective direct search algorithm that can form a nonnested structure series with an optimized pattern for solving the generalized inverse problem we use numerical examples to show how the wcp depends on the structure complexity prior information boundary conditions and the objectives of model application generally speaking with more prior information simpler boundary conditions fewer objectives and lower accuracy requirement the wcp will have a simpler shape as a result a reliable model can be identified with less observation data in part 2 of this series we will give the conditions of structure reducibility and the method for robust experimental design a robust experimental design is independent of the true parameter and can provide sufficient information for identifying an objective oriented model