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Variational Equations (variational + equation)
Selected AbstractsAn extended finite element framework for slow-rate frictional faulting with bulk plasticity and variable frictionINTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICS, Issue 13 2009Fushen Liu Abstract We present an extended finite element (FE) approach for the simulation of slow-rate frictional faulting in geologic media incorporating bulk plasticity and variable friction. The method allows the fault to pass through the interior of FEs without remeshing. The extended FE algorithm for frictional faulting, advocated in two recent articles, emanates from a variational equation formulated in terms of the relative displacement on the fault. In the present paper we consider the combined effects of bulk plasticity and variable friction in a two-dimensional plane strain setting. Bulk plasticity is localized to the fault tip and could potentially be used as a predictor for the initiation and propagation of new faults. We utilize a variable velocity- and state-dependent friction, known as the Dieterich,Ruina or ,slowness' law, formulated in a slip-weakening format. The slip-weakening/variable friction model is then time-integrated according to the generalized trapezoidal rule. We present numerical examples demonstrating the convergence properties of a global Newton-based iterative scheme, as well as illustrate some interesting properties of the variable friction model. Copyright © 2009 John Wiley & Sons, Ltd. [source] Non-linear version of stabilized conforming nodal integration for Galerkin mesh-free methodsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 12 2002Jiun-Shyan Chen Abstract A stabilized conforming (SC) nodal integration, which meets the integration constraint in the Galerkin mesh-free approximation, is generalized for non-linear problems. Using a Lagrangian discretization, the integration constraints for SC nodal integration are imposed in the undeformed configuration. This is accomplished by introducing a Lagrangian strain smoothing to the deformation gradient, and by performing a nodal integration in the undeformed configuration. The proposed method is independent to the path dependency of the materials. An assumed strain method is employed to formulate the discrete equilibrium equations, and the smoothed deformation gradient serves as the stabilization mechanism in the nodally integrated variational equation. Eigenvalue analysis demonstrated that the proposed strain smoothing provides a stabilization to the nodally integrated discrete equations. By employing Lagrangian shape functions, the computation of smoothed gradient matrix for deformation gradient is only necessary in the initial stage, and it can be stored and reused in the subsequent load steps. A significant gain in computational efficiency is achieved, as well as enhanced accuracy, in comparison with the mesh-free solution using Gauss integration. The performance of the proposed method is shown to be quite robust in dealing with non-uniform discretization. Copyright © 2002 John Wiley & Sons, Ltd. [source] A numerical-variational procedure for laminar flow in curved square ductsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 12 2004P. M. Hatzikonstantinou Abstract A new numerical method is presented for the solution of the Navier,Stokes and continuity equations governing the internal incompressible flows. The method denoted as the CVP method consists in the numerical solution of these equations in conjunction with three additional variational equations for the continuity, the vorticity and the pressure field, using a non-staggered grid. The method is used for the study of the characteristics of the laminar fully developed flows in curved square ducts. Numerical results are presented for the effects of the flow parameters like the curvature, the Dean number and the stream pressure gradient on the velocity distributions, the friction factor and the appearance of a pair of vortices in addition to those of the familiar secondary flow. The accuracy of the method is discussed and the results are compared with those obtained by us, using a variation of the velocity,pressure linked equation methods denoted as the PLEM method and the results obtained by other methods. Copyright © 2004 John Wiley & Sons, Ltd. [source] Appropriate SCF basis sets for orbital studies of galaxies and a ,quantum-mechanical' method to compute themMONTHLY NOTICES OF THE ROYAL ASTRONOMICAL SOCIETY, Issue 3 2008Constantinos Kalapotharakos ABSTRACT We address the question of an appropriate choice of basis functions for the self-consistent field (SCF) method of simulation of the N -body problem. Our criterion is based on a comparison of the orbits found in N -body realizations of analytical potential,density models of triaxial galaxies, in which the potential is fitted by the SCF method using a variety of basis sets, with those of the original models. Our tests refer to maximally triaxial Dehnen ,-models for values of , in the range 0 ,,, 1, i.e. from the harmonic core up to the weak cusp limit. When an N -body realization of a model is fitted by the SCF method, the choice of radial basis functions affects significantly the way the potential, forces or derivatives of the forces are reproduced, especially in the central regions of the system. We find that this results in serious discrepancies in the relative amounts of chaotic versus regular orbits, or in the distributions of the Lyapunov characteristic exponents, as found by different basis sets. Numerical tests include the Clutton-Brock and the Hernquist,Ostriker basis sets, as well as a family of numerical basis sets which are ,close' to the Hernquist,Ostriker basis set (according to a given definition of distance in the space of basis functions). The family of numerical basis sets is parametrized in terms of a quantity , which appears in the kernel functions of the Sturm,Liouville equation defining each basis set. The Hernquist,Ostriker basis set is the ,= 0 member of the family. We demonstrate that grid solutions of the Sturm,Liouville equation yielding numerical basis sets introduce large errors in the variational equations of motion. We propose a quantum-mechanical method of solution of the Sturm,Liouville equation which overcomes these errors. We finally give criteria for a choice of optimal value of , and calculate the latter as a function of the value of ,, i.e. of the power-law exponent of the radial density profile at the central regions of the galaxy. [source] Using switching detection and variational equations for the shooting methodOPTIMAL CONTROL APPLICATIONS AND METHODS, Issue 2 2007Pierre Martinon Abstract We study in this paper the resolution by single shooting of an optimal control problem with a bang-bang control involving a large number of commutations. We focus on the handling of these commutations regarding the precise computation of the shooting function and its Jacobian. We first observe the impact of a switching detection algorithm on the shooting method results. Then, we study the computation of the Jacobian of the shooting function, by comparing classical finite differences to a formulation using the variational equations. We consider as an application a low thrust orbital transfer with payload maximization. This kind of problem presents a discontinuous optimal control, and involves up to 1800 commutations for the lowest thrust. Copyright © 2007 John Wiley & Sons, Ltd. [source] hp -Adaptive Finite Element Methods for Variational InequalitiesPROCEEDINGS IN APPLIED MATHEMATICS & MECHANICS, Issue 1 2008Andreas SchröderArticle first published online: 25 FEB 200 In this work, we combine an hp,adaptive strategy with a posteriori error estimates for variational inequalities, which are given by contact problems. The a posteriori error estimates are obtained using a general approach based on the saddle point formulation of contact problems and making use of a yposteriori error estimates for variational equations. Error estimates are presented for obstacle problems and Signorini problems with friction. Numerical experiments confirm the reliability of the error estimates for finite elements of higher order. The use of the hp,adaptive strategy leads to meshes with the same characteristics as geometric meshes and to exponential convergence. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] | ||||||||||