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Weak Formulation (weak + formulation)

Distribution by Scientific Domains

Engineering	90%

Selected Abstracts

Weak formulation of boundary conditions for scalar conservation laws: an application to highway traffic modelling

INTERNATIONAL JOURNAL OF ROBUST AND NONLINEAR CONTROL, Issue 16 2006
Issam S. Strub
Abstract This article proves the existence and uniqueness of a weak solution to a scalar conservation law on a bounded domain. A weak formulation of the boundary conditions is needed for the problem to be well posed. The existence of the solution results from the convergence of the Godunov scheme. This weak formulation is written explicitly in the context of a strictly concave flux function (relevant for highway traffic). The numerical scheme is then applied to a highway scenario with data from highway Interstate-80 obtained from the Berkeley Highway Laboratory. Finally, the existence of a minimiser of travel time is obtained, with the corresponding optimal boundary control. Copyright © 2006 John Wiley & Sons, Ltd. [source]

Analysis of adiabatic shear bands in heat-conducting elastothermoviscoplastic materials by the meshless local Bubnov,Galerkin method

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 10 2009
R. C. Batra
Abstract Transient finite coupled thermomechanical simple shearing deformations of a block made of an elastothermoviscoplastic material that exhibits strain and strain-rate hardening, and thermal softening are studied by using the meshless local Bubnov,Galerkin method. A local nonlinear weak formulation and a semidiscrete formulation of the problem are derived. The prescribed velocity at the top and the bottom surfaces of the block is enforced by using a set of Lagrange multipliers. A homogeneous solution of the problem is perturbed by superimposing on it a temperature bump at the center of the block, and the resulting nonlinear initial-boundary-value problem is solved numerically. We have developed an integration scheme to numerically integrate the set of coupled nonlinear ordinary differential equations. The inhomogeneous deformations of the block are found to concentrate in a narrow region of intense plastic deformation usually called a shear band. For a material exhibiting enhanced thermal softening, it is shown that as the shear stress within the region of localization collapses, an unloading elastic shear wave emanates outward from the edges of the shear band. In the absence of an analytical solution, the computed results have been compared with those obtained by the finite element and the modified smoothed particle hydrodynamics methods. Copyright © 2008 John Wiley & Sons, Ltd. [source]

Analysis of thick functionally graded plates by local integral equation method

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 8 2007
J. Sladek
Abstract Analysis of functionally graded plates under static and dynamic loads is presented by the meshless local Petrov,Galerkin (MLPG) method. Plate bending problem is described by Reissner,Mindlin theory. Both isotropic and orthotropic material properties are considered in the analysis. A weak formulation for the set of governing equations in the Reissner,Mindlin theory with a unit test function is transformed into local integral equations considered on local subdomains in the mean surface of the plate. Nodal points are randomly spread on this surface and each node is surrounded by a circular subdomain, rendering integrals which can be simply evaluated. The meshless approximation based on the moving least-squares (MLS) method is employed in the numerical implementation. Numerical results for simply supported and clamped plates are presented. Copyright © 2006 John Wiley & Sons, Ltd. [source]

A G space theory and a weakened weak (W2) form for a unified formulation of compatible and incompatible methods: Part I theory

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 9 2010
G. R. Liu
Abstract This paper introduces a G space theory and a weakened weak form (W2) using the generalized gradient smoothing technique for a unified formulation of a wide class of compatible and incompatible methods. The W2 formulation works for both finite element method settings and mesh-free settings, and W2 models can have special properties including softened behavior, upper bounds and ultra accuracy. Part I of this paper focuses on the theory and fundamentals for W2 formulations. A normed G space is first defined to include both continuous and discontinuous functions allowing the use of much more types of methods/techniques to create shape functions for numerical models. Important properties and a set of useful inequalities for G spaces are then proven in the theory and analyzed in detail. These properties ensure that a numerical method developed based on the W2 formulation will be spatially stable and convergent to the exact solutions, as long as the physical problem is well posed. The theory is applicable to any problems to which the standard weak formulation is applicable, and can offer numerical solutions with special properties including ,close-to-exact' stiffness, upper bounds and ultra accuracy. Copyright © 2009 John Wiley & Sons, Ltd. [source]

An energy approach to space,time Galerkin BEM for wave propagation problems

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 9 2009
A. Aimi
Abstract In this paper we consider Dirichlet or Neumann wave propagation problems reformulated in terms of boundary integral equations with retarded potential. Starting from a natural energy identity, a space,time weak formulation for 1D integral problems is briefly introduced, and continuity and coerciveness properties of the related bilinear form are proved. Then, a theoretical analysis of an extension of the introduced formulation for 2D problems is proposed, pointing out the novelty with respect to existing literature results. At last, various numerical simulations will be presented and discussed, showing unconditional stability of the space,time Galerkin boundary element method applied to the energetic weak problem. Copyright © 2009 John Wiley & Sons, Ltd. [source]

A finite element formulation for thermoelastic damping analysis

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 6 2009
Enrico Serra
Abstract We present a finite element formulation based on a weak form of the boundary value problem for fully coupled thermoelasticity. The thermoelastic damping is calculated from the irreversible flow of entropy due to the thermal fluxes that have originated from the volumetric strain variations. Within our weak formulation we define a dissipation function that can be integrated over an oscillation period to evaluate the thermoelastic damping. We show the physical meaning of this dissipation function in the framework of the well-known Biot's variational principle of thermoelasticity. The coupled finite element equations are derived by considering harmonic small variations of displacement and temperature with respect to the thermodynamic equilibrium state. In the finite element formulation two elements are considered: the first is a new 8-node thermoelastic element based on the Reissner,Mindlin plate theory, which can be used for modeling thin or moderately thick structures, while the second is a standard three-dimensional 20-node iso-parametric thermoelastic element, which is suitable to model massive structures. For the 8-node element the dissipation along the plate thickness has been taken into account by introducing a through-the-thickness dependence of the temperature shape function. With this assumption the unknowns and the computational effort are minimized. Comparisons with analytical results for thin beams are shown to illustrate the performances of those coupled-field elements. Copyright © 2008 John Wiley & Sons, Ltd. [source]

Approximation of Cahn,Hilliard diffuse interface models using parallel adaptive mesh refinement and coarsening with C1 elements

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 5 2008
Roy H. Stogner
Abstract A variational formulation and C1 finite element scheme with adaptive mesh refinement and coarsening are developed for phase-separation processes described by the Cahn,Hilliard diffuse interface model of transport in a mixture or alloy. The adaptive scheme is guided by a Laplacian jump indicator based on the corresponding term arising from the weak formulation of the fourth-order non-linear problem, and is implemented in a parallel solution framework. It is then applied to resolve complex evolving interfacial solution behavior for 2D and 3D simulations of the classic spinodal decomposition problem from a random initial mixture and to other phase-transformation applications of interest. Simulation results and adaptive performance are discussed. The scheme permits efficient, robust multiscale resolution and interface characterization. Copyright © 2008 John Wiley & Sons, Ltd. [source]

Practical evaluation of five partly discontinuous finite element pairs for the non-conservative shallow water equations

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 6 2010
Richard Comblen
Abstract This paper provides a comparison of five finite element pairs for the shallow water equations. We consider continuous, discontinuous and partially discontinuous finite element formulations that are supposed to provide second-order spatial accuracy. All of them rely on the same weak formulation, using Riemann solver to evaluate interface integrals. We define several asymptotic limit cases of the shallow water equations within their space of parameters. The idea is to develop a comparison of these numerical schemes in several relevant regimes of the subcritical shallow water flow. Finally, a new pair, using non-conforming linear elements for both velocities and elevation (P,P), is presented, giving optimal rates of convergence in all test cases. P,P1 and P,P1 mixed formulations lack convergence for inviscid flows. P,P2 pair is more expensive but provides accurate results for all benchmarks. P,P provides an efficient option, except for inviscid Coriolis-dominated flows, where a small lack of convergence is observed. Copyright © 2009 John Wiley & Sons, Ltd. [source]

A new high-order finite volume element method with spectral-like resolution

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 3-4 2002
F. Sarghini
Abstract In this work, a new high-order finite volume element method with good spatial resolution characteristics is presented. The method is based on a functional representation of the unknowns based on the finite element method, a balance of physical quantities in weak formulation obtained by using the finite volume method, and an implicit reduction of some of the unknowns obtained by enforcing functional relations between some of them. Applications to hyperbolic and elliptic operators in 1D and 2D, as well as to Navier,Stokes equations for incompressible flows are presented. Copyright © 2002 John Wiley & Sons, Ltd. [source]

Weak formulation of boundary conditions for scalar conservation laws: an application to highway traffic modelling

On non-stationary viscous incompressible flow through a cascade of profiles

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 16 2006
Miloslav Feistauer
Abstract The paper deals with theoretical analysis of non-stationary incompressible flow through a cascade of profiles. The initial-boundary value problem for the Navier,Stokes system is formulated in a domain representing the exterior to an infinite row of profiles, periodically spaced in one direction. Then the problem is reformulated in a bounded domain of the form of one space period and completed by the Dirichlet boundary condition on the inlet and the profile, a suitable natural boundary condition on the outlet and periodic boundary conditions on artificial cuts. We present a weak formulation and prove the existence of a weak solution. Copyright © 2006 John Wiley & Sons, Ltd. [source]