a eulerian lagrangian numerical scheme for the dispersion convection equation using conjugate space time grids abstract a new numerical scheme is proposed for the dispersion convection equation which combines the utility of a fixed grid in eulerian coordinates with the computational power of the lagrangian method convection is formally decoupled from dispersion in a manner which does not leave room for ambiguity the resulting convection problem is solved by the method of characteristics on a grid fixed in space dispersion is handled by finite elements on a separate grid which may but need not coincide wit the former at selected points in spacetime information is projected from one grid to another by local interpolation the conjugate grid method is implemented by linear finite elements in conjunction with piecewise linear interpolation functions and applied to five problems ranging from predominant dispersion to pure convection the results are entirely free of oscillations numerical dispersion exists but can be brought under control either by reducing the spatial increment or by increasing the time step size of the grid used to solve the convection problem contrary to many other methods best results are often obtained when the courant number exceeds 1

temporal moments revisited why there is no better way for physically based model reduction in time 1 many hydro geo logical problems are highly complex in space and time coupled with scale issues variability and uncertainty especially time dependent models often consume enormous computational resources but model reduction techniques can alleviate this problem temporal moments tm offer an approach to reduce the time demands of transient hydro geo logical simulations tm reduce transient governing equations to steady state and directly simulate the temporal characteristics of the system if the equations are linear and coefficients are time independent this is achieved by an integral transform projecting the dynamic system response onto monomials in time in comparison to classical approaches of model reduction that involve orthogonal base functions however the monomials for tm are nonorthogonal which might impair the quality and efficiency of model reduction thus we raise the question of whether there are more suitable temporal base functions than the monomials that lead to tm in this work we will derive theoretically that there is only a limited class of temporal base functions that can reduce hydro geo logical models by comparing those to tm we conclude that in terms of gained efficiency versus maintained accuracy tm are the best possible choice while our theoretical results hold for all systems of linear partial or ordinary differential equations pdes odes with any order of space and time derivatives we illustrate our study with an example of pumping tests in a confined aquifer for that case we demonstrate that two four tm are sufficient to represent more than 80 90 of the dynamic behavior and that the information content strictly increases with increasing tm order

the laplace transform finite difference method for simulation of flow through porous media a new numerical method the laplace transform finite difference ltfd method was developed to solve the partial differential equation pde of transient flow through porous media ltfd provides a solution which is semianalytical in time and numerical in space by solving the discretized pde in the laplace space and numerically inverting the transformed solution vector the effects of the traditional treatment of the time derivative on accuracy and stability are rendered irrelevant because time is no longer a consideration for a single time step ltfd requires no more than eight matrix solutions and an execution time eight times longer than the analogous finite difference fd requirement without an increase in storage this disadvantage is outweighed by an unlimited time step size without any loss of accuracy a superior accuracy and a stable nonincreasing round off error thus a problem in standard fd format may require several hundred time steps and matrix inversions between the initial condition and the desired solution time but ltfd requires only one time step and no more than eight matrix inversions to achieve a more accurate result

boundary integral equation method for linear porous elasticity with applications to soil consolidation for physical phenomena governed by the biot model of porous elasticity a reciprocal relation similar to the betti s recoprocal theorem in elasticity is constructed in laplace transformed space integrating the reciprocal relation enables one to formulate boundary integral equations the fundamental kernels for the integral equations are solved in closed forms for the case of isotropic material numerical implementation of two dimensional problems includes finite element ideas of discretization and polynomial interpolation and numerical inversion of a laplace transform practical applications of the method are found in consolidation problems in soils which contain compressible as well as incompressible pore fluids also as a numerical experiment consolidation of partially saturated soil is simulated and interesting phenomena are observed the currently developed boundary integral equation method biem for porous elasticity may be viewed as an efficient and accurate alternative of existing finite element and finite difference methods for linear consolidation problems application of biem is always preferred to the other numerical methods whenever possible

an algebraically partitioned feti method for parallel structural analysis algorithm description an algebraically partitioned feti method for the solution of structural engineering problems on parallel computers is presented the present algorithm consists of three attributes an explicit generation of the orthogonal null space matrix associated with the interface nodal forces the floating subdomain rigid body modes computed from the subdomain static equilibrium equation of the classical force method and the identification of redundant interface force constraint operator that emanates when the interface force computations are localized comparisons of the present method with the previously developed differentially partitioned feti method are offered in terms of the saddle point formulations at the end of the paper a companion paper reports implementation details and numerical performance of the proposed algorithm 1997 john wiley sons ltd

a method of finite element tearing and interconnecting and its parallel solution algorithm a novel domain decomposition approach for the parallel finite element solution of equilibrium equations is presented the spatial domain is partitioned into a set of totally disconnected subdomains each assigned to an individual processor lagrange multipliers are introduced to enforce compatibility at the interface nodes in the static case each floating subdomain induces a local singularity that is resolved in two phases first the rigid body modes are eliminated in parallel from each local problem and a direct scheme is applied concurrently to all subdomains in order to recover each partial local solution next the contributions of these modes are related to the lagrange multipliers through an orthogonality condition a parallel conjugate projected gradient algorithm is developed for the solution of the coupled system of local rigid modes components and lagrange multipliers which completes the solution of the problem when implemented on local memory multiprocessors this proposed method of tearing and interconnecting requires less interprocessor communications than the classical method of substructuring it is also suitable for parallel vector computers with shared memory moreover unlike parallel direct solvers it exhibits a degree of parallelism that is not limited by the bandwidth of the finite element system of equations

a variational principle for the formulation of partitioned structural systems a continuum based variational principle is presented for the formulation of the discrete governing equations of partitioned structural systems this application includes coupled substructures as well as subdomains obtained by mesh decomposition the present variational principle is derived by a series of modifications of a hybrid functional originally proposed by atluri for finite element development the interface is treated by a displacement frame and a localized version of the method of lagrange multipliers interior displacements are decomposed into rigid body and deformational components to handle floating subdomains both static and dynamic versions are considered an important application of the present principle is the treatment of nonmatching meshes that arise from various sources such as separate discretization of substructures independent mesh refinement and global local analysis the present principle is compared with that of a globalized version of the multiplier method copyright 2000 john wiley sons ltd

finite analytic numerical method for unsteady two dimensional navier stokes equations abstract the main purpose of this paper is to develop a finite analytic fa numerical solution for unsteady two dimensional navier stokes equations the fa method utilizes the analytic solution in a small local element to formulate the algebraic representation of partial differential equations in this study the combination of linear and exponential functions that satisfy the governing equation is adopted as the boundary function thereby improving the accuracy of the finite analytic solution two flows one a starting cavity flow and the other a vortex shedding flow behind a rectangular block are solved by the fa method the starting square cavity flow is solved for reynolds numbers of 400 1000 and 2000 to show the accuracy and stability of the fa solution the fa solution for flow over a rectangular block h h 4 predicts the strouhal number for reynolds numbers of 100 and 500 to be 0 156 and 0 125 details of the flow patterns are given in addition to streamlines and vorticity distribution rest streamlines are given to illustrate the vortex motion downstream of the block

essential self adjointness generalized eigenforms and spectra for the neumann problem on g manifolds let m be a complex manifold with boundary satisfying a subelliptic estimate which is also the total space of a principal g bundle with g a lie group and compact orbit space m g here we investigate the neumann laplacian on m we show that it is essentially self adjoint on its restriction to compactly supported smooth forms moreover we relate its spectrum to the existence of generalized eigenforms an energy belongs to if there is a subexponentially bounded generalized eigenform for this energy vice versa there is an expansion in terms of these well behaved eigenforms so that spectrally almost every energy comes with such a generalized eigenform

sensitivity methods for time continuous spatially discrete groundwater contaminant transport models existing sensitivity methods for spatially discrete groundwater contaminant transport models have been developed for time stepping numerical algorithms and cannot be readily used with time continuous approaches to transport simulation such as the laplace transform galerkin technique we develop direct and adjoint sensitivity methods in which sensitivity coefficients are computed in the laplace domain and inverted numerically to the time domain the methods are computationally efficient when used in conjunction with time continuous transport equations the relative efficiency of the two methods depends on the number of model parameters number of performance measures and number of spatial discretization nodes the adjoint method is favored when the number of performance measures is much smaller than the number of model parameters the adjoint method is limited in that performance measures are restricted to being linear functions of state variables a two dimensional transport example is developed in detail and sensitivities with respect to nodal hydraulic conductivities are computed in the problems analyzed the direct and adjoint methods are 9 to 156 times faster than the perturbation method with the computational savings increasing as the size of the problem is increased

comment on a space time accurate method for solving solute transport problems by s g li f ruan and d mclaughlin solve the advection diffusion equation with spatially variable coefficients the laplace transform is used to evaluate the temporal derivative in the advection diffusion equation ana lytically thereby eliminating the effects of the time deriva tive on accuracy and stability because the temporal deriv ative is evaluated using the laplace transform there is no need for the time stepping that is associated with more traditional techniques for the solution of the advection diffusion equation therefore it is a potentially more efficient technique the authors do not provide numerical simulations involv ing the spatially variable coefficient advection diffusion equation a reason for this might be that there are very few analytical solutions to the advection diffusion equation with spatially variable coefficients the purpose of this discussion is to present what we believe to be a new analytical solution to a particular form of the advection diffusion equation with spatially variable co efficients and to highlight a potential mass conservation

unitary representations of unimodular lie groups in bergman spaces for an arbitrary unimodular lie group g we construct strongly continuous unitary representations in the bergman space of a strongly pseudoconvex neighborhood of g in the complexification of its underlying manifold these representation spaces are infinite dimensional and have compact kernels in particular the bergman spaces of these natural manifolds are infinite dimensional

efficient time stepping methods for miscible displacement problems in porous media efficient procedures for time stepping galerkin methods for approximating the solution of a coupled system for c c x t and p p x t with nonlinear neumann boundary conditions of the form begin gathered nabla cdot a x c nabla p y x c nabla g equiv nabla cdot u f 1 x t quad x in omega quad t in 0 t hfill nabla cdot b x c nabla p nabla c u cdot nabla c phi x frac partial c partial t f 2 x t c quad x in omega quad t in 0 t hfill u cdot v q 1 x t quad x in partial omega quad t in 0 t hfill b frac partial c partial v q x t c quad x in partial omega quad t in 0 t hfill c x 0 c 0 x quad x in omega hfill end gathered where omega subset mathbb r d 2 leqq d leqq 3 are presented and analyzed this system is a possible model system for describing the miscible displacement of one incompressible fluid by another in a porous medium when flow conditions a

numerical methods for convection dominated diffusion problems based on combining the method of characteristics with finite element or finite difference procedures finite element and finite difference methods are combined with the method of characteristics to treat a parabolic problem of the form cu t bu x au x x f optimal order error estimates in l 2 and w 1 2 are derived for the finite element procedure various error estimates are presented for a variety of finite difference methods the estimates show that for convection dominated problems b gg a these schemes have much smaller time truncation errors than those of standard methods extensions to n space variables and time dependent or nonlinear coefficients are indicated along with applications of the concepts to certain problems described by systems of differential equations

the two level feti method part ii extension to shell problems parallel implementation and performance results the two level feti method presented in part i of this work is a mathematically optimal and scalable substructuring algorithm for solving iteratively large scale systems of equations arising from the finite element discretization of static and dynamic plate bending problems in this paper we extend the two level feti method to shell problems describe a variant approach revisit the preconditioning problem address the efficient implementation of the corresponding iterative solvers on massively parallel processors highlight the computational price of mathematical optimality and discuss its consequences and report some impressive performance results on two paragon xp s and ibm sp2 massively parallel processors for several realistic plate and shell problems

a component mode selection method based on a consistent perturbation expansion of interface displacement abstract a mode selection method is presented for the reduced order modeling rom of structural systems in conjunction with the craig bampton component mode synthesis technique the proposed method is derived by using a consistent expansion of the interface displacement in terms of a frequency dependent small parameter as applied to a craig bampton like rom formulation it is found that this procedure yields a coupling mechanism of the modes of the full model to those of substructures the present mode selection method employs this coupling mechanism as an indicator labeled as the cms method for the substructural modal contributions to the full model the performance of the proposed method is demonstrated by various numerical examples and compared favorably with existing method such as the cms method

stochastic finite element method for probabilistic analysis of flow and transport in a three dimensional heterogeneous porous formation 1 a probabilistic study is attempted to analyze the flow and transport in a three dimensional 3 d porous formation where the governing parameters are varying randomly in space it is assumed that the soil parameters namely hydraulic conductivity dispersivity molecular diffusion porosity sorption coefficient and decay rate are random fields a stochastic finite element method sfem which is based on perturbation technique is developed the method developed here uses an alternate approach for obtaining improved computational efficiency the derivatives of the concentration with respect to random parameters are obtained by using the derivatives of local matrices instead of global matrices this approach increases the computational efficiency of the present method by several orders with respect to standard sfem both accuracy and computational efficiency of this method are compared with that of commonly used monte carlo simulation method mcsm it is observed that for moderate values of coefficient of variations of the random parameters the mean and standard deviation match reasonably well with mcsm results using this method the excessive computational effort required by mcsm can be avoided in the present study both 1 d as well as 3 d problems are solved to show the advantages of sfem over mcsm the correlation scale of the random field is found to be an important parameter for the range of this parameter studied here it is found that as correlation scale increases the standard deviation increases the results obtained for two particular problems in this study show that the coefficient of variation of concentration is higher for the 1 d problem than the 3 d problem

finite analytic numerical solution for two dimensional groundwater solute transport a new numerical method called the finite analytic fa method is used to solve a groundwater solute transport problem the basic idea of the finite analytic method is the incorporation of local analytic solution in the numerical solution of the partial differential equation when the local analytic solution is evaluated at a given nodal point it gives an algebraic relationship between a nodal value in an element and its neighboring nodal points the assemble of the linear system equations results in a tridiagonal matrix like most finite difference method the advantages of using efficient iterative techniques for solving tridiagonal matrices are equally applicable to fa method the automatic localized upstream shift and the analytic property of the fa method eliminates the difficulty of numerical dispersion locally and suppresses the overall numerical dispersion for large peclet number for small peclet number fa method yields excellent results in comparison with the analytic solution for large peclet number fa solutions are oscillation free with some degree of numerical dispersion the results are comparable with those obtained using upstream weighted finite element method

partitioned component mode synthesis via a flexibility approach a new flexibility based component mode synthesis method is presented which is derived by approximating the partitioned equations of motion that employ a localized method of lagrange multipliers the use of the localized lagrange multipliers leads to unlike the classical lagrange multipliers a linearly independent set of interface forces without any redundancies at multiply connected interface nodes hence the resulting interface flexibility matrices are uniquely determined and well suited for the present method development an attractive feature of the present method is its substructural mode selection criterion that is independent of loading conditions numerical experiments show that the present modal selection criterion is reliable and that the proposed flexibility based component mode synthesis method performs well relative to the classical craig bampton method in terms of accuracy and of its ability to include dominant substructural modes for simple plates and a relatively complex solid ring

eulerian lagrangian solution of the convection dispersion equation in natural coordinates the vast majority of numerical investigations of transport phenomena use an eulerian formulation for the convenience that the computational grids are fixed in space an eulerian lagrangian method elm of solution for the convection dispersion equation is discussed and analyzed the elm uses the lagrangian concept in an eulerian computational grid system the values of the dependent variable off the grid are calculated by interpolation when a linear interpolation is used the method is a slight improvement over the upwind difference method at this level of approximation both the elm and the upwind difference method suffer from large numerical dispersion however if second order lagrangian polynomials are used in the interpolation the elm is proven to be free of artificial numerical dispersion for the convection dispersion equation the concept of the elm is extended for treatment of anisotropic dispersion in natural coordinates in this approach the anisotropic properties of dispersion can be conveniently related to the properties of the flow field several numerical examples are given to further substantiate the results of the present analysis

a modified method of characteristics technique and mixed finite elements method for simulation of groundwater solute transport a comprehensive groundwater solute transport simulator is developed based on the modified method of characteristics mmoc combined with the galerkin finite element method for the transport equation and the mixed finite element mfe method for the groundwater flow equation the preconditioned conjugate gradient algorithm is used to solve the two large sparse algebraic system of equations arising from the mmoc and mfe discretizations the mmoc takes time steps in the direction of flow along the characteristics of the velocity field of the total fluid the physical diffusion and dispersion terms are treated by a standard finite element scheme the crucial aspect of the mmoc technique is that it looks backward in time along an approximate flow path instead of forward in time as in many method of characteristics or moving mesh techniques the mfe procedure involves solving for both the hydraulic head and the specific discharge simultaneously one order of convergence is gained by the mfe method as compared with other standard finite element methods and therefore more accurate velocity fields are simulated the overall advantages of the mmoc mfe method include minimum numerical oscillation or grid orientation problems under steep concentration gradient simulations and material balance errors are greatly reduced due to a very accurate velocity simulation by the mfe method in addition much larger time steps with courant number well in excess of 1 as compared with the standard galerkin finite element method can be taken on a fixed spatial grid system without significant loss of accuracy

the cell analytical numerical method for solution of the advection dispersion equation two dimensional problems in this paper we present the development and evaluation of a new numerical scheme for efficient solution of groundwater solute transport problems the scheme which we have named the cell analytical numerical can method is an extension of the so called nodal methods which have been developed in the nuclear engineering area for solving neutron diffusion problems the can method is based upon decomposition of the solution domain into a number of rectangular cells volume subdomains each cell is homogeneous so that a local analytic solution to the solute transport equation can be obtained a first order accurate finite difference approximation of the time derivative followed by a transverse averaging procedure is used to transform the governing partial differential equation into a set of coupled one dimensional ordinary differential equations for the transverse moments of the concentration the local analytical solution is therefore found in terms of these moments solute mass flux continuity across cell surfaces along with the local analytical solution is used to construct an algebraic relationship between concentration moment values at adjacent cell surfaces assembling all the cells together results in a set of coupled tridiagonal matrix equations one set for each spatial direction which can be solved very efficiently the can method is applied to several simple test problems involving uniform flow in homogeneous aquifers while the method lacks geometrical flexibility it offers a spatial approximation that is demonstrated to have high accuracy and minimal grid orientation error even when applied to coarse meshes

on the substructure synthesis method substructure synthesis is a model reduction method whereby a complex structure is regarded as an assemblage of substructures the motion of each of the substructures is represented by a series of admissible functions vectors and the substructures are made to act as a single structure by imposing certain geometric compatibility conditions at the boundary between any two adjacent substructures because the series of admissible functions vectors represents the motion of the substructures only approximately and because the geometric compatibility conditions are only approximations of the true conditions the computed eigensolution is only an approximation of the actual one convergence of computed eigenvalues to actual eigenvalues requires two limiting processes one in which the number of admissible functions vectors is increased and the other in which the number of geometric compatibility conditions is increased

the laplace transform galerkin technique a time continuous finite element theory and application to mass transport in groundwater a time continuous numerical technique referred to as the laplace transform galerkin ltg method is developed and applied to the problem of solute transport in porous media after application of galerkin s procedure and subdivision of the domain into finite elements the method involves a simple application of the laplace transformation to eliminate the temporal derivatives appearing in the space discretized set of ordinary differential equations then by solving the resulting transformed system of algebraic equations in laplace p space numerical inversion of the laplace transformed nodal concentration is performed using the robust and accurate crump 1976 algorithm the crump algorithm permits the concentration to be evaluated from a range of time values from a single set of laplace p space solutions because each of the needed p space solutions are independent the algorithm is well suited for execution on multiprocessor parallel computers it is demonstrated by means of a series of examples that the ltg scheme is capable of providing highly accurate solutions essentially devoid of numerical dispersion for grid peclet numbers in excess of 30 examination of the complex valued laplace domain concentration profiles reveal that they are generally smooth well behaved oscillatory functions compared to the profiles in the time domain thus permitting the use of a coarse finite element grid because of the nature of the laplace transformation the ltg method is particularly well suited to the problem of transient groundwater flow and solute transport in fractured porous media or multiple aquifer aquitard systems based on the dual porosity integrodifferential equation approach

a space time accurate method for solving solute transport problems solute transport studies frequently rely on numerical solutions of the classical advection diffusion equation unfortunately solutions obtained with traditional finite difference and finite element techniques typically exhibit spurious damping or oscillation when advection dominates recently developed variants of these techniques such as the finite analytic method chen and li 1979 chen and chen 1984 and the optimal test function method celia et al 1989a b c perform well for steady state problems extensions of these methods to the transient case have however not been successful primarily because of inadequate approximations of the temporal derivative the new numerical method proposed in this paper avoids this difficulty by taking the laplace transform of the transient equation the transformed expression behaves like a steady state advection diffusion equation with a first order decay term this expression can be solved with either the finite analytic or optimal test function method and the time dependence recovered with an efficient inverse laplace transform algorithm the result is an accurate and robust transient solution which performs well over a very wide range of peclet numbers we demonstrate this approach by applying the finite analytic method to a laplace transformed one dimensional model problem a comparison with other competing techniques shows that good approximations are required in both space and time in order to obtain accurate solutions to advection dominated problems a good space approximation combined with a poor temporal approximation or vice versa does not give satisfactory results the method we propose provides a balanced space time approximation which works very well for one dimensional problems extensions to multiple dimensions are conceptually straightforward and briefly discussed

bergman spaces of natural g manifolds let g be a unimodular lie group x a compact manifold with boundary and m the total space of a principal bundle g m x so that m is also a strongly pseudoconvex complex manifold in this work we show that if there exists a point p bm such that tpg is contained in the complex tangent space tpcbm of bm at p then the bergman space of m is large natural examples include the gauged g complexifications of heinzner huckleberry and kutzschebauch

the construction of free free flexibility matrices for multilevel structural analysis abstract this article considers a generalization of the classical structural flexibility matrix it expands on previous papers by taking a deeper look at computational considerations at the substructure level direct or indirect computation of flexibilities as influence coefficients has traditionally required pre removal of rigid body modes by imposing appropriate support conditions mimicking experimental arrangements with the method presented here the flexibility of an individual element or substructure is directly obtained as a particular generalized inverse of the free free stiffness matrix this generalized inverse preserves the stiffness spectrum the definition is element independent and only involves access to the stiffness generated by a standard finite element program and the separate construction of an orthonormal rigid body mode basis the free free flexibility has proven useful in special application areas of finite element structural analysis notably massively parallel processing model reduction and damage localization it can be computed by solving sets of linear equations and does not require processing an eigenproblem or performing a singular value decomposition if substructures contain thousands of d o f exploitation of the stiffness sparseness is important for that case this paper presents a computation procedure based on an exact penalty method and a projected rank regularized inverse stiffness with diagonal entries inserted by the sparse factorization process these entries can be physically interpreted as penalty springs this procedure takes advantage of the stiffness sparseness while forming the full free free flexibility or a boundary subset and is backed by an in depth null space analysis for robustness

finite element methods for first order hyperbolic systems with particular emphasis on the compressible euler equations a petrov galerkin finite element formulation is presented for first order hyperbolic systems of conservation laws with particular emphasis on the compressible euler equations applications of the methodology are made to one and two dimensional steady and unsteady flows with shocks results obtained suggest the potential of the type of methods developed

a characteristic based finite analytic method for solving the two dimensional steady state advection diffusion equation 1 this paper develops an improved finite analytic fa solution method to the advection diffusion equation ade for solving advection dominated steady state transport problems the fa method solves the ade analytically in localized discrete elements with each element linked through local boundary conditions previous fa methods have suffered from complex solution formulations as well as from numerical dispersion stemming from inaccuracies in the local boundary estimations here we use finite difference approximations of the dispersion terms to reduce solution complexity implement an improved particle tracking scheme to account for velocity variations within each element and use a hermite interpolation scheme to estimate the local boundary conditions the new fa method previous fa methods and a finite difference upwinding method are compared in both homogeneous and heterogeneous velocity fields at different peclet numbers results show the new fa method exhibits little numerical dispersion without undue complexity or computational effort across all tested flow conditions

a lagrangian eulerian method with zoomable hidden fine mesh approach to solving advection dispersion equations a lagrangian eulerian method with zoomable hidden fine mesh approach lezoom that can be adapted with either finite element or finite difference methods is used to solve the advection dispersion equation the approach is based on automatic adaptation of zooming a hidden fine mesh in regions where the sharp front is located application of lezoom to four bench mark problems indicates that it can handle the advection dispersion diffusion problems with mesh peclet numbers ranged from 0 to infinity and with mesh courant numbers well in excess of 1 difficulties that can be resolved with lezoom include numerical dispersion oscillations the clipping of peaks and the effect of grid orientation nonuniform grid as well as spatial temporally variable flow pose no problems with lezoom both initial and boundary value problems can be solved accurately with lezoom it is shown that although the mixed lagrangian eulerian le approach lezoom without zooming also produces excessive numerical dispersion as the upstream finite element ufe method the le approach is superior to the ufe method