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Solution Techniques (solution + techniques)
Selected AbstractsSolution of the unsaturated soil moisture equation using repeated transformsINTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICS, Issue 15 2001S. G. Fityus Abstract An alternative method of solution for the linearized ,theta-based' form of the Richards equation of unsaturated flow is developed in two spatial dimensions. The Laplace and Fourier transformations are employed to reduce the Richards equation to an ordinary differential equation in terms of a transformed moisture content and the transform variables, s and ,. Separate analytic solutions to the transformed equation are developed for initial states which are either in equilibrium or dis-equilibrium. The solutions are assembled into a finite layer formulation satisfying continuity of soil suction, thereby facilitating the analysis of horizontally stratified soil profiles. Solution techniques are outlined for various boundary conditions including prescribed constant moisture content, prescribed constant flux and flux as a function of moisture change. Example solutions are compared with linearized finite element solutions. The agreement is found to be good. An adaptation of the method for treating the quasilinearized Richards equation with variable diffusivity is also described. Comparisons of quasilinear solutions with some earlier semi-analytical, finite element and finite difference results are also favourable. Copyright © 2001 John Wiley & Sons, Ltd. [source] Review of the Integrated Groundwater and Surface-Water Model (IGSM)GROUND WATER, Issue 2 2003Eric M. LaBolle Development of the finite-element-based Integrated Groundwater and Surface-Water Model (IGSM) began in the 1970s. Its popularity grew in the early 1990s with its application to California's Central Valley Groundwater Surface-Water Model in support of the Central Valley Project Improvement Act. Since that time, IGSM has been applied by federal, state, and local agencies to model a number of major basins in California. Our review of the recently released version 5.0 of IGSM reveals a solution methodology that deviates from established solution techniques, potentially compromising its reliability under many circumstances. One difficulty occurs because of the semi-explicit time discretization used. Combined with the fixed monthly time step of IGSM, this approach can prevent applications from accurately converging when using parameter values typically found in nature. Additionally, IGSM fails to properly couple and simultaneously solve ground water and surface water models with appropriate mass balance and head convergence under the reasonable conditions considered herein. As a result, IGSM-predicted streamflow is error prone, and errors could exceed 100%. IGSM does not inform the user that there may be a convergence problem with the solution, but instead generally reports good mass balance. Although our review touches on only a few aspects of the code, which exceeds 17,000 lines, our experience is that similar problems arise in other parts of IGSM. Review and examples demonstrate the potential consequences of using the solution methods in IGSM for the prediction, planning, and management of water resources, and provide perspective on the roles of standards and code validation in ground water modeling. [source] A priori pivoting in solving the Navier,Stokes equationsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 10 2002S. Ø. Wille Abstract Mixed finite element formulations of incompressible Navier,Stokes Equations leads to non-positive definite algebraic systems inappropriate for iterative solution techniques. However, introducing a suitable preconditioner, the mixed finite element equation system becomes positive definite and solvable by iterative techniques. The present work suggests a priori pivoting sequences for parallel and serial implementations of incomplete Gaussian factorization. Tests are performed for the driven cavity problem in two and three dimensions. Copyright © 2002 John Wiley & Sons, Ltd. [source] A 3D mortar method for solid mechanics,INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 3 2004Michael A. Puso Abstract A version of the mortar method is developed for tying arbitrary dissimilar 3D meshes with a focus on issues related to large deformation solid mechanics. Issues regarding momentum conservation, large deformations, computational efficiency and bending are considered. In particular, a mortar method formulation that is invariant to rigid body rotations is introduced. A scheme is presented for the numerical integration of the mortar surface projection integrals applicable to arbitrary 3D curved dissimilar interfaces. Here, integration need only be performed at problem initialization such that coefficients can be stored and used throughout a quasi-static time stepping process even for large deformation problems. A degree of freedom reduction scheme exploiting the dual space interpolation method such that direct linear solution techniques can be applied without Lagrange multipliers is proposed. This provided a significant reduction in factorization times. Example problems which touch on the aforementioned solid mechanics related issues are presented. Published in 2003 by John Wiley & Sons, Ltd. [source] Efficient solution techniques for implicit finite element schemes with flux limitersINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 7 2007M. MöllerArticle first published online: 20 MAR 200 Abstract The algebraic flux correction (AFC) paradigm is equipped with efficient solution strategies for implicit time-stepping schemes. It is shown that Newton-like techniques can be applied to the nonlinear systems of equations resulting from the application of high-resolution flux limiting schemes. To this end, the Jacobian matrix is approximated by means of first- or second-order finite differences. The edge-based formulation of AFC schemes can be exploited to devise an efficient assembly procedure for the Jacobian. Each matrix entry is constructed from a differential and an average contribution edge by edge. The perturbation of solution values affects the nodal correction factors at neighbouring vertices so that the stencil for each individual node needs to be extended. Two alternative strategies for constructing the corresponding sparsity pattern of the resulting Jacobian are proposed. For nonlinear governing equations, the contribution to the Newton matrix which is associated with the discrete transport operator is approximated by means of divided differences and assembled edge by edge. Numerical examples for both linear and nonlinear benchmark problems are presented to illustrate the superiority of Newton methods as compared to the standard defect correction approach. Copyright © 2007 John Wiley & Sons, Ltd. [source] Iterative solution techniques for unsteady flow computations using higher order time integration schemesINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 8-9 2005H. Bijl Abstract In this paper iterative techniques for unsteady flow computations with implicit higher order time integration methods at large time steps are investigated. It is shown that with a minimal coding effort the standard non-linear multigrid method can be combined with a Newton,Krylov method leading to speed-ups in the order of 30%. Copyright © 2005 John Wiley & Sons, Ltd. [source] A novel search framework for multi-stage process scheduling with tight due datesAICHE JOURNAL, Issue 8 2010Yaohua He Abstract This article improves the original genetic algorithm developed by He and Hui (Chem Eng Sci. 2007; 62:1504,1527) and proposes a novel global search framework (GSF) for the large-size multi-stage process scheduling problems. This work first constructs a comprehensive set of position selection rules according to the impact factors analysis presented by He and Hui (in this publication in 2007), and then selects suitable rules for schedule synthesis. In coping with infeasibility emerging during the search, a penalty function is adopted to force the algorithm to approach the feasible solutions. The large-size problems with tight due dates are challenging to the current solution techniques. Inspired by the gradient used in numerical analysis, we treat the deviation existing among the computational tests of the algorithm as evolutionary gradient. Based on this concept, a GSF is laid out to fully utilize the search ability of the current algorithm. Numerical experiments indicate that the proposed search framework solves such problems with satisfactory solutions. © 2010 American Institute of Chemical Engineers AIChE J, 2010 [source] Stochastic modeling of particle motion along a sliding conveyorAICHE JOURNAL, Issue 1 2010Kevin Cronin Abstract The sliding conveyor consists of a plane surface, known as the track, along which particles are induced to move by vibrating the bed sinusoidal with respect to time. The forces on the particle include gravity, bed reaction force and friction. Because friction coefficients are inherently variable, particle motion along the bed is erratic and unpredictable. A deterministic model of particle motion (where friction is considered to be known and invariant) is selected and its output validated by experiment. Two probabilistic solution techniques are developed and applied to the deterministic model, in order to account for the randomness that is present. The two methods consider particle displacement to be represented by discrete time and continuous time random processes, respectively, and permits analytical solutions for mean and variance in displacement versus time to be found. These are compared with experimental measurements of particle motion. Ultimately this analysis can be employed to calculate residence-time distributions for such items of process equipment. © 2009 American Institute of Chemical Engineers AIChE J, 2010 [source] Recursive estimation in constrained nonlinear dynamical systemsAICHE JOURNAL, Issue 3 2005Pramod Vachhani In any modern chemical plant or refinery, process operation and the quality of product depend on the reliability of data used for process monitoring and control. The task of improving the quality of data to be consistent with material and energy balances is called reconciliation. Because chemical processes often operate dynamically in nonlinear regimes, techniques such as extended-Kalman filter (EKF) and nonlinear dynamic data reconciliation (NDDR) have been developed for reconciliation. There are various issues that arise with the use of either of these techniques. EKF cannot handle inequality or equality constraints, whereas the NDDR has high computational cost. Therefore, a more efficient and robust method is required for reconciling process measurements and estimating parameters involved in nonlinear dynamic processes. Two solution techniques are presented: recursive nonlinear dynamic data reconciliation (RNDDR) and a combined predictor,corrector optimization (CPCO) method for efficient state and parameter estimation in nonlinear systems. The proposed approaches combine the efficiency of EKF and the ability of NDDR to handle algebraic inequality and equality constraints. Moreover, the CPCO technique allows deterministic parameter variation, thus relaxing another restriction of EKF where the parameter changes are modeled through a discrete stochastic equation. The proposed techniques are compared against the EKF and the NDDR formulations through simulation studies on a continuous stirred tank reactor and a polymerization reactor. In general, the RNDDR performs as well as the two traditional approaches, whereas the CPCO formulation provides more accurate results than RNDDR at a marginal increase in computational cost. © 2005 American Institute of Chemical Engineers AIChE J, 51: 946,959, 2005 [source] Iterative versus direct parallel substructuring methods in semiconductor device modellingNUMERICAL LINEAR ALGEBRA WITH APPLICATIONS, Issue 1 2005L. Giraud Abstract The numerical simulation of semiconductor devices is extremely demanding in term of computational time because it involves complex embedded numerical schemes. At the kernel of these schemes is the solution of very ill-conditioned large linear systems. In this paper, we present the various ingredients of some hybrid iterative schemes that play a central role in the robustness of these solvers when they are embedded in other numerical procedures. On a set of two-dimensional unstructured mixed finite element problems representative of semiconductor simulation, we perform a fair and detailed comparison between parallel iterative and direct linear solution techniques. We show that iterative solvers can be robust enough to solve the very challenging linear systems that arise in those simulations. Copyright © 2004 John Wiley & Sons, Ltd. [source] Structure determination without Fourier inversion.ACTA CRYSTALLOGRAPHICA SECTION A, Issue 6 2009The parameter-space concept for solving crystal structures from reflection amplitudes (without employing or searching for their phases) is described on a theoretically oriented basis. Emphasis is placed on the principles of the method, on selecting one of three types of parameter spaces discussed in this paper, and in particular on the structure model employed (equal-atom point model, however usually reduced to one-dimensional projections) and on the system of `isosurfaces' representing experimental `geometrical structure amplitudes' in an orthonormal parameter space of as many dimensions as unknown atomic coordinates. The symmetry of the parameter space as well as of the imprinted isosurfaces and its effect on solution methods is discussed. For point atoms scattering with different phases or signs (as is possible in the case of X-ray resonant or of neutron scattering) it is demonstrated that the `landscape' of these isosurfaces remains invariant save certain shifts of origin known beforehand (under the condition that all atomic scattering amplitudes have been reduced to 1 thus meeting the requirement of the structure model above). Partly referring to earlier publications on the subject, measures are briefly described which permit circumventing an analytical solution of the system of structure-amplitude equations and lead to either a unique (unequivocal) approximate structure solution (offering rather high spatial resolution) or to all possible solutions permitted by the experimental data used (thus including also all potential `false minima'). A simple connection to Patterson vectors is given, also a first hint on data errors. References are given for practical details of various solution techniques already tested and for reconstruction of three-dimensional structures from their projections by `point tomography'. We would feel foolish if we tried to aim at any kind of `competition' to existing methods. Having mentioned `pros and cons' of our concept, some ideas about potential applications are nevertheless offered which are mainly based on its inherent resolution power though demanding rather few reflection data (use of optimal intensity contrast included) and possibly providing a result proven to be unique. [source] An inverse estimation of initial temperature profile in a polymer processPOLYMER ENGINEERING & SCIENCE, Issue 1 2008Ali A. Ranjbar Since one of the most important parameter in polymer processing such as injection stretch blow molding is temperature distribution in the thickness direction, an inverse method has been applied to estimate this profile. This process comprises of four steps. In the first step the preform is injection molded, and in the second and third step it is stretched by a rod to its final length and then inflated and in the last step it is discharged from the mold. In such kind of polymer flows viscous dissipation plays a remarkable role in the evolution of temperature profile. Some theoretical temperature profile has been applied to confirm the validation of the inverse algorithm. Different solution techniques are applied in this article to the inverse problem under consideration, namely: the conjugate gradient and Levenberg,Marquardt method. After the preform is injection molded, which is the first step, it is removed from the mold, which corresponds to time t = 0. At this moment an infrared camera is used to record the surface temperature of the preform with a certain time step. With regard to variation of thermal properties with temperature, the inverse problem becomes nonlinear. These experimental data provided by the infrared camera are then used to estimate the temperature profile at the end of injection process before stretching and inflation took place. POLYM. ENG. SCI., 48:133,140, 2008. © 2007 Society of Plastics Engineers [source] The Subtle Influence of Binary versus Homoatomic Zintl Ions: The Phenyl-Ligated Trimetallic Cage [Sn2Sb5(ZnPh)2]3,CHEMISTRY - A EUROPEAN JOURNAL, Issue 47 2009Felicitas Lips Dipl.-Chem. Zintlating stuff! The first binary Sn/Sb Zintl anions produced by solution techniques were obtained in [K([2.2.2]crypt)]2[Sn2Sb2],en (en=ethylenediamine) and [K6(NH3)9][Sn3Sb4] upon extraction of K8SnSb4 by en or liquid ammonia. Addition of ZnPh2 to the en/[2.2.2]crypt extract resulted in the formation of a heterotrimetallic complex in [K([2.2.2]crypt)]3[Sn2Sb5(ZnPh)2],en,0.5tol (see scheme). [source] | |||||||||||||