achieving minimum length scale in topology optimization using nodal design variables and projection functions a methodology for imposing a minimum length scale on structural members in discretized topology optimization problems is described nodal variables are implemented as the design variables and are projected onto element space to determine the element volume fractions that traditionally define topology the projection is made via mesh independent functions that are based upon the minimum length scale a simple linear projection scheme and a non linear scheme using a regularized heaviside step function to achieve nearly 0 1 solutions are examined the new approach is demonstrated on the minimum compliance problem and the popular simp method is used to penalize the stiffness of intermediate volume fraction elements solutions are shown to meet user defined length scale criterion without additional constraints penalty functions or sensitivity filters no instances of mesh dependence or checkerboard patterns have been observed copyright 2004 john wiley sons ltd

a general mass conservative numerical solution for the unsaturated flow equation numerical approximations based on different forms of the governing partial differential equation can lead to significantly different results for unsaturated flow problems numerical solution based on the standard h based form of richards equation generally yields poor results characterized by large mass balance errors and erroneous estimates of infiltration depth conversely numerical solutions based on the mixed form of richards equation can be shown to possess the conservative property so that mass is perfectly conserved this leads to significant improvement in numerical solution performance while requiring no additional computational effort however use of the mass conservative method does not guarantee good solutions accurate solution of the unsaturated flow equation also requires use of a diagonal time or mass matrix only when diagonal time matrices are used can the solution be shown to obey a maximum principle which guarantees smooth nonoscillatory infiltration profiles this highlights the fact that proper treatment of the time derivative is critical in the numerical solution of unsaturated flow

structural optimization under uncertain loads and nodal locations this paper presents algorithms for solving structural topology optimization problems with uncertainty in the magnitude and location of the applied loads and with small uncertainty in the location of the structural nodes the second type of uncertainty would typically arise from fabrication errors where the tolerances for the node locations are small in relation to the length scale of the structural elements we first review the discrete form of the uncertain loads problem which has been previously solved using a weighted average of multiple load patterns with minor modifications we extend this solution to include loads described by continuous joint probability density functions we then proceed to the main contribution of this paper structural optimization under uncertainty in the nodal locations this optimization problem is computationally difficult because it involves variations of the inverse of the structural stiffness matrix it is shown however that for small uncertainties the problem can be recast into a simpler but equivalent structural optimization problem with equivalent uncertain loads by expressing these equivalent loads in terms of continuous random variables we are able to make use of the extended form of the uncertain loads problem presented in the first part of this paper the optimization algorithms are developed in the context of minimum compliance maximum stiffness design simple examples are presented the results demonstrate that load and nodal uncertainties can have dramatic impact on optimal design for structures containing thin substructures under axial loads it is shown that these uncertainties a are of first order significance influencing the linear elastic response quantities and b can affect designs by avoiding unrealistically optimistic and potentially unstable structures the additional computational cost associated with the uncertainties scales linearly with the number of uncertainties and is insignificant compared to the cost associated with solving the deterministic structural optimization problem

noniterative time stepping schemes with adaptive truncation error control for the solution of richards equation 1 noniterative implicit time stepping schemes with adaptive temporal truncation error control are developed for the solution of the pressure form of richards equation first and second order linearizations of an adaptive backward euler thomas gladwell formulation are introduced and are shown to constrain the temporal truncation errors near a user prescribed tolerance and maintain adequate mass balance numerical experiments demonstrate that accurate noniterative linearizations achieve cost effective solutions of problems where soils are described by highly nonlinear and nonsmooth constitutive functions for these problems many conventional iterative solvers fail to converge the noniterative formulations are considerably more efficient than analogous time stepping schemes with iterative solvers the second order noniterative scheme is found to be more efficient than the first order noniterative scheme the proposed adaptive noniterative algorithms can be easily incorporated into existing backward euler software which is widely used for richards equation and other nonlinear pdes

adaptive backward euler time stepping with truncation error control for numerical modelling of unsaturated fluid flow an automatic time stepping scheme with embedded error control is developed and applied to the moisture based richards equation the algorithm is based on the first order backward euler scheme and uses a numerical estimate of the local truncation error and an efficient time step selector to control the temporal accuracy of the integration local extrapolation equivalent to the use of an unconditionally stable thomas gladwell algorithm achieves second order temporal accuracy at minimal additional costs the time stepping algorithm also provides accurate initial estimates for the iterative non linear solver numerical tests confirm the ability of the scheme to automatically optimize the time step size to match a user prescribed temporal error tolerance an important merit of the proposed method is its conceptual and computational simplicity it can be directly incorporated into existing or new software based on the backward euler scheme currently prevalent in subsurface hydrologic modelling and markedly improves their performance compared with simple fixed or heuristic time step selection the generality of the approach also makes possible its use for solving pdes in other engineering applications where strong non linearity stability or implementation considerations favour a simple and robust low order method or where there is a legacy of backward euler codes in current use copyright 2001 john wiley sons ltd

experimental design for estimating unknown groundwater pumping using genetic algorithm and reduced order model 1 an optimal experimental design algorithm is developed to select locations for a network of observation wells that provide maximum information about unknown groundwater pumping in a confined anisotropic aquifer the design uses a maximal information criterion that chooses among competing designs the design that maximizes the sum of squared sensitivities while conforming to specified design constraints the formulated optimization problem is non convex and contains integer variables necessitating a combinatorial search given a realistic large scale model the size of the combinatorial search required can make the problem difficult if not impossible to solve using traditional mathematical programming techniques genetic algorithms gas can be used to perform the global search however because a ga requires a large number of calls to a groundwater model the formulated optimization problem still may be infeasible to solve as a result proper orthogonal decomposition pod is applied to the groundwater model to reduce its dimensionality then the information matrix in the full model space can be searched without solving the full model results from a small scale test case show identical optimal solutions among the ga integer programming and exhaustive search methods this demonstrates the ga s ability to determine the optimal solution in addition the results show that a ga with pod model reduction is several orders of magnitude faster in finding the optimal solution than a ga using the full model the proposed experimental design algorithm is applied to a realistic two dimensional large scale groundwater problem the ga converged to a solution for this large scale problem

topology optimization with implicit functions and regularization summary topology optimization is formulated in terms of the nodal variables that control an implicit function description of the shape the implicit function is constrained by upper and lower bounds so that only a band of nodal variables needs to be considered in each step of the optimization the weak form of the equilibrium equation is expressed as a heaviside function of the implicit function the heaviside function is regularized to permit the evaluation of sensitivities we show that the method is a dual of the bendshe kikuchi method the method is applied both to problems of optimizing single material and

an efficient finite element procedure for analyzing three phase porous media based on the relaxed picard method summary effective simulation of the solid liquid gas coupling effect in unsaturated porous media is of great significance in many diverse areas because of the strongly nonlinear characteristics of the fully coupled formulations for the three phase porous media an effective numerical solution scheme such as the finite element method with an efficient iterative algorithm has to be employed in this paper an efficient finite element procedure based on the adaptive relaxed picard method is developed for analyzing the coupled solid liquid gas interactions in porous media the coupled model and the finite element analysis procedure are implemented into a computer code poroush2m and the proposed procedure is validated through comparing the numerical simulations with the experimental benchmarks it is shown that the adaptive relaxed picard method has salient advantage over the traditional one with respect to both the efficiency and the robustness especially for the case of relatively large time step sizes compared with the newton raphson scheme the picard method successfully avoids the unphysical spurious unloading phenomenon under the plastic deformation condition although the latter shows a better convergence rate the proposed procedure provides an important reference for analyzing the fully coupled problems related to the multi phase multi field coupling in porous media copyright 2014 john wiley sons ltd

three dimensional unsaturated flow modeling using cellular automata 1 the parabolic partial differential equation describing fluid flow in partially saturated porous media richards equation is highly nonlinear due to pressure head dependencies in the specific soil moisture capacity and relative hydraulic conductivity terms in order to solve richards equation several numerical techniques have been developed which starting from the discretization of the partial differential equation produced even more accurate models leading to complex and computationally expensive simulations for large scale systems a three dimensional unsaturated flow modeling developed through a simulation environment based on cellular automata ca is described in this paper the proposed model represents an extension of the original computational paradigm of cellular automata because it uses a macroscopic ca approach where local laws with a clear physical meaning govern interactions among automata this ca structure aimed at simulating a large scale system is based on functionalities capable of increasing its computational capacity both in terms of working environment and in terms of the optimal number of processors available for parallel computing the model has been validated with reference multidimensional solutions taken from benchmarks in literature showing a good agreement even in the cases where nonlinearity is very marked furthermore some analyses have been carried out considering quantization techniques aimed at transforming the ca model into an asynchronous structure the use of these techniques in a three dimensional benchmark allowed a considerable reduction in the number of local interactions among adjacent automata without changing the efficiency of the model especially when simulations are characterized by scarce mass exchanges finally from a computational point of view the higher efficiency values were achieved running the model on a parallel architecture obtaining a high speedup very close to the optimal with the maximum number of processors available

implementation of the jacobian free newton krylov method for solving the first order ice sheet momentum balance we have implemented the jacobian free newton krylov jfnk method for solving the first order ice sheet momentum equation in order to improve the numerical performance of the glimmer community ice sheet model glimmer cism the land ice component of the community earth system model cesm our jfnk implementation is based on significant re use of existing code for example our physics based preconditioner uses the original picard linear solver in glimmer cism for several test cases spanning a range of geometries and boundary conditions our jfnk implementation is 1 8 3 6 times more efficient than the standard picard solver in glimmer cism importantly this computational gain of jfnk over the picard solver increases when refining the grid global convergence of the jfnk solver has been significantly improved by rescaling the equation for the basal boundary condition and through the use of an inexact newton method while a diverse set of test cases show that our jfnk implementation is usually robust for some problems it may fail to converge with increasing resolution as does the picard solver globalization through parameter continuation did not remedy this problem and future work to improve robustness will explore a combination of picard and jfnk and the use of homotopy methods

techniques for making finite elements competitve in modeling flow in variably saturated porous media a galerkin finite element formulation is developed for the numerical simulation of water flow in variably saturated soil systems included in this formulation is a solution strategy based on picard and newton raphson algorithms both algorithms are designed especially to cope with severely nonlinear field problems the two algorithms are formulated for both rectangular and triangular elements the element matrices are evaluated in a simple and efficient manner using a technique referred to as the influence coefficient technique this technique avoids numerical integration and leads to a substantial saving of computational cost four examples are presented to demonstrate the effectiveness of the present finite element approach these examples show that the nonlinear solution schemes are capable of accomodating cases involving large variations in the saturated hydraulic conductivity as well as highly nonlinear soil moisture characteristics a comparative study of the picard and the newton raphson algorithms is also provided the study indicates that despite the higher cost per iteration of the newton raphson scheme it usually requires a substantially smaller number of iterations than the picard scheme in some instances where convergence difficulties are experienced with the latter scheme it is desirable to use the newton raphson scheme in order to obtain a cost effective solution to the problem

topology design of structures subjected to periodic loading although a lot of attention in the topology optimization literature has focused on the optimization of eigenfrequencies in free vibration problems relatively little work has been done on the optimization of structures subjected to periodic loading in this paper we propose two measures one global and the other local for the minimization of vibrations of structures subjected to periodic loading the global measure which we term as the dynamic compliance reduces the vibrations in an overall sense and thus has important implications from the viewpoint of reducing the noise radiated from a structure while the local measure reduces the vibrations at a user defined point both measures bring about a reduction in the vibration level by moving the natural frequencies which contribute most significantly to the measures away from the driving frequencies although as expected in different ways quite surprisingly the structure of the dynamic compliance optimization problem turns out to be very similar to the structure of the static compliance optimization problem the availability of analytical sensitivities results in an efficient algorithm for both measures we show the effectiveness of the measures by presenting some numerical examples

parameter estimation in flow through partially saturated porous materials a class of numerical simulators were developed and critically evaluated to be incorporated as the solver of a forward problem in the framework of an inverse modeling strategy the strategy couples a mass lumped galerkin linear finite element solution of the mixed form richards equation with an experimental time space series and the osborne more revised form of the levenberg marquardt algorithm to retrieve hydraulic parameters of a partially saturated porous medium the numerical simulator shows excellent agreement with a reference solution obtained on a dense grid and infinitesimal time step in terms of fluid pressure head fluid content and fluid volumetric flux density and perfectly conserves the global mass an adaptive algorithm was implemented to estimate sensitivity matrix in the inverse algorithm a multi criterion stopping rule was developed and successfully implemented to end the inverse code at the solution the result of the optimization was compared with a large scale in situ soil moisture space time series measured during the course of a drainage experiment and excellent agreements were found analysis of the parameter response surfaces and hyper space plots closeness of the gradient of the penalty function at minimum to zero and positive definiteness of the approximation for the hessian at the solution eigs h 0 indicate that the obtained solution is a strong local minimum a state of the art sensitivity analysis carried out to quantify sensitivity of the state variable with respect to uncertainty and changes in different model parameters

finite element formulation of unilateral boundary conditions for unsaturated flow in porous continua this paper presents the numerical resolution of unilateral boundary conditions able to effectively model several problems of unsaturated flow as those involving rainfall infiltration and seepage faces besides the penalty technique we also consider the novel regularization of these conditions by means of the more effective augmented lagrangian method the performance of the so obtained finite element method is carefully investigated in terms of accuracy and ill conditioning effects including comparisons with analytical solutions and a complete identification of the analogies with the problem of frictionless contact in this way we provide a priori estimates of optimal and admissible ranges for the penalty coefficient as functions of permeability and spatial discretization the proposed method and the estimated coefficient ranges are validated in further numerical examples involving the propagation of a wetting front due to rainfall and the partial saturation of an aged concrete dam these applications show that the proposed regularizations do not induce any detrimental effect on solution accuracy and on convergence rate of the employed newton raphson method hence the present approach should be preferred to the commonly used iterative switching between the imposed flow and the imposed pressure conditions which often leads to spurious oscillations and convergence failures

accuracy of galerkin finite elements for groundwater flow simulations in two and three dimensional triangulations summary in standard nite element simulations of groundwater 4ow the correspondence between hydraulic head gradients and groundwater 4uxes is represented by the sti ness matrix in two dimensional problems the use of linear triangular elements on delaunay triangulations guarantees a sti ness matrix of type m this implies that the local numerical 4uxes are physically consistent with darcy s law this condition is fundame ntal to avoid theoccurre nceof local maxima or minima and is of crucial importancewhe n the calculated 4ow eld is used in contaminant transport simulations or pathline evaluation in three spatial dimensions the linear galerkin approach on tetrahedra does not lead to m matrices even on delaunay meshes by interpretation of the galerkin approach as a subdomain collocation scheme we develop a new approach osc orthogonal subdomain collocation that is shown to produce m matrices in three dimensional delaunay triangulations in case of heterogeneous and anisotropic coeacients extra mesh properties required for m sti ness matrices will also be discussed copyright 2001 john wiley sons ltd

complex hill s equation and the complex periodic korteweg de vries equations on etudie la classe spectrale des operateurs differentiels l d 2 dx 2 q avec un potentiel q de periode un on traite des potentiels complexes et des operateurs auto adjoints

topology design with optimized self adaptive materials significant performance improvements can be obtained if the topology of an elastic structure is allowed to vary in shape optimization problems we study the optimal shape design of a two dimensional elastic continuum for minimum compliance subject to a constraint on the total volume of material the macroscopic version of this problem is not well posed if no restrictions are placed on the structure topology relaxation of the optimization problem via quasiconvexification or homogenization methods is required the effect of relaxation is to introduce a perforated microstructure that must be optimized simultaneously with the macroscopic distribution of material

an interblock conductivity scheme for finite difference models of steady unsaturated flow in heterogeneous media with the increased ability to model subsurface heterogeneity provided by stochastic simulation methods there is a corresponding increase in the need for numerical flow models to accomodate the spatial variability of hydrodynamic parameters these trends make it necessary to reexamine methods for calculating finite difference interblock conductivities originally developed for use in homogeneous media this study describes a new method for calculating interblock conductivities better suited to the numerical modeling of unsaturated flow in heterogeneous media an expression for interblock conductivity is derived by equating the finite difference flux between adjacent blocks with the exact analytical flux for one dimensional steady unsaturated flow assuming an exponential model for relative conductivity this expression appears as a function of the saturated conductivities alpha parameters and matric potentials of the adjacent blocks and the potential at the interblock boundary since this last quantity is not directly featured in the discretized flow equation it must be evaluated iteratively the interblock conductivity calculation scheme is implemented within a block centered seven point finite difference model of steady unsaturated flow where nonlinearity is resolved using a newton raphson iterative approach in a series of numerical experiments this scheme is compared with the more familiar geometric averaging method the proposed interblock conductivity scheme is shown to provide perfect agreement between numerical and analytical solutions for the one dimensional case of steady infiltration into a perfectly layered medium in problems of two dimensional unsaturated flow in heterogeneous media comparisons with the geometric averaging method show that the proposed scheme allows more rapid convergence of newton raphson iterations and consideration of greater parameter spatial variability before the onset of divergence in these iterations

a mass conservative switching algorithm for modeling fluid flow in variably saturated porous media numerical solutions of flow equation in fluid content based form or in fluid pressure head based form are often tradeoffs between speed accuracy and convenience the fluid content based form can be solved quite rapidly with low cpu time and perfect mass balance however it cannot be used in saturated regions as diffusivity function becomes infinite and strictly becomes invalid in composite layered and real heterogeneous porous materials due to singularity and discontinuity in fluid content profile this formulation also gives misleading impression that gradient in fluid content causes the flow of fluid in porous materials where in reality gravity and fluid pressure potential gradient produce the motion the pressure head based form on the other hand is more flexible but due to its highly nonlinear nature is much more time consuming and produces poor global mass balance for dry initial conditions very fine spatial and temporal discretizations are needed to maintain mass balance property for these scenarios the mixed form of the flow equation partially solves these issues as it maintains acceptable mass balance and is applicable to layered heterogeneous and composite fractured foundations however it is only applicable in unsaturated zones in this study a switching algorithm was proposed and implemented in which the mass conservative mixed form and the pressure head based form were respectively used in the unsaturated and saturated zones of an initial boundary value flow problem involving a variably saturated porous medium the algorithm showed excellent agreement with a reference solution obtained on a very fine spatiotemporal mesh the simulator was then calibrated with several real world large scale experimental datasets in all cases the proposed algorithm exhibited close agreements with the experimental time space series the algorithm poses excellent mass balance property and can easily be used in both saturated and unsaturated regions without special treatment of fluid content discontinuities in heterogeneous and layered porous media the proposed algorithm can also be extended to simulate multiphase and multidimensional flow problems

mixed finite elements and newton type linearizations for the solution of richards equation we present the development of a two dimensional mixed hybrid finite element mhfe model for the solution of the non linear equation of variably saturated flow in groundwater on unstructured triangular meshes by this approach the darcy velocity is approximated using lowest order raviart thomas rt0 elements and is exactly mass conserving hybridization is used to overcome the ill conditioning of the mixed system the scheme is globally first order in space nevertheless numerical results employing non uniform meshes show second order accuracy of the pressure head and normal fluxes on specific grid points the non linear systems of algebraic equations resulting from the mhfe discretization are solved using picard or newton iterations realistic sample tests show that the mhfe newton approach achieves fast convergence in many situations in particular when a good initial guess is provided by either the picard scheme or relaxation techniques copyright 1999 john wiley sons ltd

an efficient unconditionally energy stable local discontinuous galerkin scheme for the cahn hilliard brinkman system in this paper we present an efficient and unconditionally energy stable fully discrete local discontinuous galerkin ldg method for approximating the cahn hilliard brinkman chb system which is comprised of a cahn hilliard type equation and a generalized brinkman equation modeling fluid flow the semi discrete energy stability of the ldg method is proved firstly due to the strict time step restriction t o x 4 of explicit time discretization methods for stability we introduce a semi implicit scheme which consists of the implicit euler method combined with a convex splitting of the discrete cahn hilliard energy strategy for the temporal discretization the unconditional energy stability of this fully discrete convex splitting scheme is also proved obviously the fully discrete equations at the implicit time level are nonlinear and to enhance the efficiency of the proposed approach the nonlinear full approximation scheme fas multigrid method has been employed to solve this system of algebraic equations we also show the nearly optimal complexity numerically numerical experiments based on the overall solution method of combining the proposed ldg method convex splitting scheme and the nonlinear multigrid solver are given to validate the theoretical results and to show the effectiveness of the proposed approach for the chb system

innovative modelling of 3d unsaturated flow in porous media by coupling independent models for vertical and lateral flows unsaturated groundwater flows are mathematically represented by the richards equation hitherto in hydrology solutions of this equation mainly serve as an alimentation of the source term for the surface runoff modelling therefore the complete resolution of the 3d model looks surplus to requirements and the infiltration is dealt either thanks to 1d vertical modelling of the richards equation or through derivate models like e g the green ampt infiltration model or the horton law thus ignoring eventual horizontal transfers nowadays the request for more detailed information is real and the physics of groundwater unsaturated flow needs to be represented more reliably this information could be furnished by the resolution of the complete 3d model but although numerically mastered and well documented it is very costly for large scale both in time and space real applications climate change adaptation of watersheds the authors propose an original solution decoupling the 3d equations into 1d vertical equations and a 2d depth integrated horizontal equation the aim is to consider independent vertical columns of infiltration coupled with lateral transfer of mass through the boundary conditions on this basis they postulate that the mass transfers in the three dimensions are correctly represented this way problematic like the supply of the aquifers the re emergence of groundwater to surface water or especially the capability of memorization of past rainy events could be reliably depicted the two coupled models are solved on a unique numerical frame a cell centred finite volume method is used to solve the parabolic partial differential equations the spatial derivatives are approximate by a second order central difference scheme while the time splitting follows an implicit backward euler scheme coupled with picard iteration the method has been tested and its reliability assessed on different theoretical two dimensional cross sectional test cases representing infiltration phenomena

implementation of the jacobian free newton krylov method for solving the for solving the first order ice sheet momentum balance we have implemented the jacobian free newton krylov jfnk method for solving the rst order ice sheet momentum equation in order to improve the numerical performance of the community ice sheet model cism the land ice component of the community earth system model cesm our jfnk implementation is based on signicant re use of existing code for example our physics based preconditioner uses the original picard linear solver in cism for several test cases spanning a range of geometries and boundary conditions our jfnk implementation is 1 84 3 62 times more efficient than the standard picard solver in cism importantly this computational gain of jfnk over the picard solver increases when rening the grid global convergence of the jfnk solver has been signicantly improved by rescaling the equation for the basal boundary condition and through the use of an inexact newton method while a diverse set of test cases show that our jfnk implementation is usually robust for some problems it may fail to converge with increasing resolution as does the picard solver globalization through parameter continuation did not remedy this problem and future work to improve robustness will explore a combination of picard and jfnk and the use of homotopy methods

bounds for decoupled design and analysis discretizations in topology optimization summary topology optimization formulations using multiple design variables per finite element have been proposed to improve the design resolution this paper discusses the relation between the number of design variables per element and the order of the elements used for analysis we derive that beyond a maximum number of design variables certain sets of material distributions cannot be discriminated based on the corresponding analysis results this makes the design description inefficient and the solution of the optimization problem non unique to prevent this we establish bounds for the maximum number of design variables that can be used to describe the material distribution for any given finite element scheme without introducing non uniqueness copyright 2016 the authors international journal for numerical methods in engineering published by john wiley sons ltd

optimal design and relaxation of variational problems iii abstract this paper has attempted a synthesis of three subjects that have developed separately optimal design homogenization and relaxation in the calculus of variations the underlying theme is that of oscillations they occur with certain definite patterns in minimizing sequences for the variational problem the same patterns recur in nearly optimal designs and in the coefficients of the equation being homogenized each problem is resolved by determining the structure of those oscillations and above all their macroscopic averages it is those averages that solve the relaxed problem they also determine the optimal design and the solution of the homogenized equation we emphasize also the role of optimization the homogenized equations and composite materials of interest here come not from arbitrary mixtures but from those that are extremal since they correspond to minimizing sequences for the variational problem the effective properties of these composites attain the optimal bounds for given materials in fixed proportions

locally conservative stabilized finite element methods for variably saturated flow standard galerkin finite element methods for variably saturated groundwater flow have several deficiencies for instance local oscillations can appear around sharp infiltration fronts without the use of mass lumping and velocity fields obtained from differentiation of pressure fields are discontinuous at element boundaries here we consider conforming finite element discretizations based on a multiscale formulation along with recently developed local postprocessing schemes the resulting approach maintains the basic flexibility and appeal of traditional finite element methods while controlling nonphysical oscillations and producing element wise mass conservative velocity fields accuracy and efficiency of the proposed schemes are evaluated through a series of steady state and transient variably saturated groundwater flow problems in homogeneous as well as heterogeneous domains the schemes are formulated for a generic nonlinear advection diffusion equation and are thus applicable to many other flow models

stability of finite element models for distributed parameter optimization and topology design abstract we address a problem of numerical instability that is often encountered in finite element solutions of distributed parameter optimization and variable topology shape design problems we show that the cause of this problem is numerical rather than physical in nature we consider a two field distributed parameter optimization problem involving a design field and a response field and show that the optimization problem corresponds to a mixed variational problem an improper selection of the discrete function spaces for these two fields leads to grid scale anomalies in the numerical solutions to optimization problems similar to those that are sometimes encountered in mixed formulations of the stokes problem we present a theoretical framework to explain the cause of these anomalies and present stability conditions for discrete models the general theoretical framework is specialized to analyze the stability of specific optimization problems and stability results for various mixed finite element models are presented we propose patch tests that are useful in identifying unstable elements

the quasilinear approximation for unsaturated porous media flow this paper gives a review of the quasilinear approximation an exponential relationship between hydraulic conductivity and moisture potential and the uses this has been put to since the time of its creation one of the major attractions of this approximation is that under steady regimes the highly nonlinear richards equation governing unsaturated flow can be linearized this simplification has enabled analytic and semianalytic solutions to be found for many unsaturated flow problems these are reviewed and the current research areas in which the quasilinear approximation is being used are discussed

a comparison of picard and newton iteration in the numerical solution of multidimensional variably saturated flow problems picard iteration is a widely used procedure for solving the nonlinear equation governing flow in variably saturated porous media the method is simple to code and computationally cheap but has been known to fail or converge slowly the newton method is more complex and expensive on a per iteration basis than picard and as such has not received very much attention its robustness and higher rate of convergence however make it an attractive alternative to the picard method particularly for strongly nonlinear problems in this paper the picard and newton schemes are implemented and compared in one two and three dimensional finite element simulations involving both steady state and transient flow the eight test cases presented highlight different aspects of the performance of the two iterative methods and the different factors that can affect their convergence and efficiency including problem size spatial and temporal discretization initial solution estimates convergence error norm mass lumping time weighting conductivity and moisture content characteristics boundary conditions seepage faces and the extent of fully saturated zones in the soil previous strategies for enhancing the performance of the picard and newton schemes are revisited and new ones are suggested the strategies include chord slope approximations for the derivatives of the characteristic equations relaxing convergence requirements along seepage faces dynamic time step control nonlinear relaxation and a mixed picard newton approach the tests show that the picard or relaxed picard schemes are often adequate for solving richards equation but that in cases where these fail to converge or converge slowly the newton method should be used the mixed picard newton approach can effectively overcome the newton scheme s sensitivity to initial solution estimates while comparatively poor performance is reported for the various chord slope approximations finally given the reliability and efficiency of current conjugate gradient like methods for solving linear nonsymmetric systems the only real drawback of using newton rather than picard iteration is the algebraic complexity and computational cost of assembling the derivative terms of the jacobian matrix and it is suggested that both methods can be effectively implemented and used in numerical models of richards equation

a robust dual algorithm for topology design of structures in discrete variables dual optimization algorithms for the topology optimization of continuum structures in discrete variables are gaining popularity in recent times since in topology design problems the number of constraints is small in comparison to the number of design variables good topologies can be obtained for the minimum compliance design problem when the perimeter constraint is imposed in addition to the volume constraint however when the perimeter constraint is relaxed the dual algorithm tends to give bad results even with the use of higher order finite element models as we demonstrate in this work since a priori one does not know what a good value of the perimeter to be specified is it is essential to have an algorithm which generates good topologies even in the absence of the perimeter constraint we show how the dual algorithm can be made more robust so that it yields good designs consistently in the absence of the perimeter constraint in particular we show that the problem of checkerboarding which is frequently observed with the use of lower order finite elements is eliminated

efficient numerical methods for infiltration using richards equation two efficient finite difference methods for solving richards equation in one dimension are presented and their use in a range of soils and conditions is investigated large time steps are possible when the mass conserving mixed form of richards equation is combined with an implicit iterative scheme while a hyperbolic sine transform for the matric potential allows large spatial increments even in dry inhomogeneous soil infiltration in a range of soils can be simulated in a few seconds on a personal computer with errors of only a few percent in the amount and distribution of soil water one of the methods adds points to the space grid as an infiltration or redistribution front advances thus gaining considerably in efficiency over the other fixed grid method for infiltration problems in 17 s computing this advancing front method simulated infiltration redistribution and drainage for 50 days in an inhomogeneous soil with nonuniform initial conditions only 16 space and 21 time steps were needed for the simulation which included early ponding with the development and dissipation of a perched water table

a transpose free quasi minimal residual algorithm for non hermitian linear systems the biconjugate gradient method bcg for solving general non hermitian linear systems ax b and its transpose free variant the conjugate gradients squared algorithm cgs both typically exhib

solutions to shape and topology eigenvalue optimization problems using a homogenization method a solution strategy to find the shape and topology of structures that maximize a natural frequency is presented the methodology is based on a homogenization method and the representation of the shape of the structure as a material property the problem is formulated as a reinforcement problem in which a given structure is reinforced using a prescribed amount of material two dimensional plane elasticity problems are considered examples are presented for illustration

finite volume integrated surface subsurface flow modeling on nonorthogonal grids in this paper we present an innovative finite volume surface subsurface integrated flow model on nonorthogonal grids the shallow water equation with diffusion wave approximation is used to formulate the surface flow system while the richards equation is used to formulate the saturated unsaturated subsurface flow system these two flow systems are discretized using a finite volume method and are then coupled by enforcing the continuity of pressure and flux at the surface subsurface interface which does not require unphysical parameters such as the interface permeability and thickness the numerical instability caused by enforcing the continuity of pressure and flux at the interface is resolved using a cell centered finite volume discretization the coupled systems are solved simultaneously by the newton iterative method a battery of benchmark analyses and laboratory experiments verify the proposed model s superior performance relative to existing models two numerical experiments over irregular terrain show that the nonorthogonal grids and diffusive wave approximation used in the proposed model accurately represent the interaction between surface and subsurface flows for irregular topographies in particular they capture the significant topographical effects on runoff discharges especially where gentle slopes are involved

a comparison of the jacobian free newton krylov method and the evp model for solving the sea ice momentum equation with a viscous plastic formulation a serial algorithm study numerical convergence properties of a recently developed jacobian free newton krylov jfnk solver are compared to the ones of the widely used evp model when solving the sea ice momentum equation with a viscous plastic vp formulation to do so very accurate reference solutions are produced with an independent picard solver with an advective time step of 10s and a tight nonlinear convergence criterion on 10 20 40 and 80 km grids approximate solutions with the jfnk and evp solvers are obtained for advective time steps of 10 20 and 30min because of an artificial elastic term the evp model permits an explicit time stepping scheme with a relatively large subcycling time step the elastic waves excited during the subcycling are intended to damp out and almost entirely disappear such that the approximate solution should be close to the vp solution results show that residual elastic waves cause the evp approximate solution to have notable differences with the reference solution and that these differences get more important as the grid is refined compared to the reference solution additional shear lines and zones of strong convergence divergence are seen in the evp approximate solution the approximate solution obtained with the jfnk solver is very close to the reference solution for all spatial resolutions tested

some new procedures for numerical solution of variably saturated flow problems persistent difficulties that arise in forming numerical solutions of variably saturated flow problems include controlling the stability of the nonlinear equation solvers and devising a reliable yet efficient method for determining the positions of seepage surfaces new techniques for addressing these problems are applied to a subdomain finite element discretization of the governing flow equations a series of test problems demonstrates that the techniques are reliable and efficient for a wide variety of problems

numerical evaluation of iterative and noniterative methods for the solution of the nonlinear richards equation several noniterative procedures for solving the nonlinear richards equation are introduced and compared to the conventional newton and picard iteration methods noniterative strategies for the numerical solution of transient nonlinear equations arise from explicit or linear time discretizations or they can be obtained by linearizing an implicit differencing scheme we present two first order accurate linearization methods a second order accurate two level implicit factored scheme and a second order accurate three level lees method the accuracy and computational efficiency of these four schemes and of the newton and picard methods are evaluated for a series of test problems simulating one dimensional flow processes in unsaturated porous media the results indicate that first order accurate schemes are inefficient compared to second order accurate methods that second order accurate noniterative schemes can be quite competitive with the iterative newton and picard methods and that the newton scheme is no less efficient than the picard method and for strongly nonlinear problems can outperform the picard scheme the two second order accurate noniterative schemes appear to be attractive alternatives to the iterative methods although there are concerns regarding the stability behavior of the three level scheme which need to be resolved we conclude that of the four noniterative strategies presented the implicit factored scheme is the most promising and we suggest improved formulations of the method

the mast edge centred lumped scheme for the flow simulation in variably saturated heterogeneous porous media a novel methodology is proposed for the solution of the flow equation in a variably saturated heterogeneous porous medium the computational domain is descretized using triangular meshes and the governing pdes are discretized using a lumped in the edge centres numerical technique the dependent unknown variable of the problem is the piezometric head a fractional time step methodology is applied for the solution of the original system solving consecutively a prediction and a correction problem a scalar potential of the flow field exists and in the prediction step a marching in space and time mast formulation is applied for the sequential solution of the ordinary differential equation of the cells ordered according to their potential value computed at the beginning of the time step in the correction step the solution of a large linear system with order equal to the number of edges is required a semi analytical procedure is also proposed for the solution of the prediction step the computational performance the order of convergence and the mass balance error have been estimated in several tests and compared with the results of other literature models

an engineer s approach to optimal material distribution and shape finding abstract the fundamental aim of shape optimization involves the optimal distribution of mass in space observing global and local design constraints the practical realization in applied mechanics can be performed by introducing a foam like material with variable density as discussed by bendsoe and kikuchi this paper uses for the evaluation of material properties a simple energy approach the application is extended to three dimensional domains multiple loading cases are also considered further the approach is implemented in a preoptimizer for optimal shape finding

a perturbative view on the subsurface water pressure response at hillslope scale 1 this work deals with a rearrangement of richards equation to obtain a better understanding of the components of the subsurface water and pressure flows and their interactions in a hillslope the basic idea proposed in this paper is the normalization of the spatial and temporal coordinates according to the physical knowledge of the processes and the geometry of a hillslope the pressure head and all the other quantities contained in richards equation are rescaled to obtain a dimensionless equation the pressure head is also split into an equilibrium component and a transient component and expanded in a perturbation series subsequently richards equation itself is conveniently split into two equations connected by a source sink term results of these manipulations show that for the first order approximation the water and pressure flow is slope normal while slope parallel effects only occur at successive orders integrating the perturbed equation obtained for the long term pressure head component results in the well known boussinesq equation for the water table level further integrations then lead to the hillslope storage boussinesq model and finally to a generalization of the o loughlin formula for the slope parallel long term subsurface flow the introduced source sink term is estimated at the first order of approximation to be proportional to the temporal variation of water pressure head at the bottom of the integration domain the treatment of the equations clarifies some of the mechanisms acting in water pressure redistribution at hillslope scale even without solving the equation