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Neumann Boundary Conditions (neumann + boundary_condition)
Selected AbstractsA fictitious domain decomposition method for the solution of partially axisymmetric acoustic scattering problems.INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 1 2003Part 2: Neumann boundary conditions Abstract We present a fictitious domain decomposition method for the fast solution of acoustic scattering problems characterized by a partially axisymmetric sound-hard scatterer. We apply this method to the solution of a mock-up submarine problem, and highlight its computational advantages and intrinsic parallelism. A key component of our method is an original idea for addressing a Neumann boundary condition in the general framework of a fictitious domain method. This idea is applicable to many other linear partial differential equations besides the Helmholtz equation. Copyright © 2003 John Wiley & Sons, Ltd. [source] A class of higher order compact schemes for the unsteady two-dimensional convection,diffusion equation with variable convection coefficientsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 12 2002Jiten C. Kalita Abstract A class of higher order compact (HOC) schemes has been developed with weighted time discretization for the two-dimensional unsteady convection,diffusion equation with variable convection coefficients. The schemes are second or lower order accurate in time depending on the choice of the weighted average parameter , and fourth order accurate in space. For 0.5,,,1, the schemes are unconditionally stable. Unlike usual HOC schemes, these schemes are capable of using a grid aspect ratio other than unity. They efficiently capture both transient and steady solutions of linear and nonlinear convection,diffusion equations with Dirichlet as well as Neumann boundary condition. They are applied to one linear convection,diffusion problem and three flows of varying complexities governed by the two-dimensional incompressible Navier,Stokes equations. Results obtained are in excellent agreement with analytical and established numerical results. Overall the schemes are found to be robust, efficient and accurate. Copyright © 2002 John Wiley & Sons, Ltd. [source] A Petrov,Galerkin finite element model for one-dimensional fully non-linear and weakly dispersive wave propagationINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 5 2001Seung-Buhm Woo Abstract A new finite element method is presented to solve one-dimensional depth-integrated equations for fully non-linear and weakly dispersive waves. For spatial integration, the Petrov,Galerkin weighted residual method is used. The weak forms of the governing equations are arranged in such a way that the shape functions can be piecewise linear, while the weighting functions are piecewise cubic with C2 -continuity. For the time integration an implicit predictor,corrector iterative scheme is employed. Within the framework of linear theory, the accuracy of the scheme is discussed by considering the truncation error at a node. The leading truncation error is fourth-order in terms of element size. Numerical stability of the scheme is also investigated. If the Courant number is less than 0.5, the scheme is unconditionally stable. By increasing the number of iterations and/or decreasing the element size, the stability characteristics are improved significantly. Both Dirichlet boundary condition (for incident waves) and Neumann boundary condition (for a reflecting wall) are implemented. Several examples are presented to demonstrate the range of applicabilities and the accuracy of the model. Copyright © 2001 John Wiley & Sons, Ltd. [source] Asymptotics for steady-state voltage potentials in a bidimensional highly contrasted medium with thin layerMATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 4 2008Clair Poignard Abstract We study the behaviour of steady-state voltage potentials in two kinds of bidimensional media composed of material of complex permittivity equal to 1 (respectively, ,) surrounded by a thin membrane of thickness h and of complex permittivity , (respectively, 1). We provide in both cases a rigorous derivation of the asymptotic expansion of steady-state voltage potentials at any order as h tends to zero, when Neumann boundary condition is imposed on the exterior boundary of the thin layer. Our complex parameter , is bounded but may be very small compared to 1, hence our results describe the asymptotics of steady-state voltage potentials in all heterogeneous and highly heterogeneous media with thin layer. The asymptotic terms of the potential in the membrane are given explicitly in local coordinates in terms of the boundary data and of the curvature of the domain, while these of the inner potential are the solutions to the so-called dielectric formulation with appropriate boundary conditions. The error estimates are given explicitly in terms of h and , with appropriate Sobolev norm of the boundary data. We show that the two situations described above lead to completely different asymptotic behaviours of the potentials. Copyright © 2007 John Wiley & Sons, Ltd. [source] Compressible Navier,Stokes system in 1-DMATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 9 2001Piotr Bogus, aw Mucha Abstract The compressible barotropic Navier,Stokes system in monodimensional case with a Neumann boundary condition given on a free boundary is considered. The global existence with uniformly boundedness for large initial data and a positive force is proved. The result concerning an asymptotic behavior shows that the solutions tends to the stationary solution. Copyright © 2001 John Wiley & Sons, Ltd. [source] An integral-collocation-based fictitious-domain technique for solving elliptic problemsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 11 2008N. Mai-Duy Abstract This paper presents a new fictitious-domain technique for numerically solving elliptic second-order partial differential equations (PDEs) in complex geometries. The proposed technique is based on the use of integral-collocation schemes and Chebyshev polynomials. The boundary conditions on the actual boundary are implemented by means of integration constants. The method works for both Dirichlet and Neumann boundary conditions. Several test problems are considered to verify the technique. Numerical results show that the present method yields spectral accuracy for smooth (analytic) problems. Copyright © 2007 John Wiley & Sons, Ltd. [source] Numerical investigation of the reliability of a posteriori error estimation for advection,diffusion equationsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 9 2008A. H. ElSheikh Abstract A numerical investigation of the reliability of a posteriori error estimation for advection,diffusion equations is presented. The estimator used is based on the solution of local problems subjected to Neumann boundary conditions. The estimated errors are calculated in a weighted energy norm, a stability norm and an approximate fractional order norm in order to study the effect of the error norm on both the effectivity index of the estimated errors and the mesh adaptivity process. The reported numerical results are in general better than what is available in the literature. The results reveal that the reliability of the estimated errors depends on the relation between the mesh size and the size of local features in the solution. The stability norm is found to have some advantages over the weighted energy norm in terms of producing effectivity indices closer to the optimal unit value, especially for problems with internal sharp layers. Meshes adapted by the element residual method measured in the stability norm conform to the sharp layers and are shown to be less dependent on the wind direction. Copyright © 2007 John Wiley & Sons, Ltd. [source] The boundary element method for solving the Laplace equation in two-dimensions with oblique derivative boundary conditionsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 12 2007D. Lesnic Abstract In this communication, we extend the Neumann boundary conditions by adding a component containing the tangential derivative, hence producing oblique derivative boundary conditions. A variant of Green's formula is employed to translate the tangential derivative to the fundamental solution in the boundary element method (BEM). The two-dimensional steady-state heat conduction with the imposed oblique boundary condition has been tested in smooth, piecewise smooth and multiply connected domains in which the Laplace equation is the governing equation, producing results at the boundary in excellent agreement with the available analytical solutions. Convergence of the normal and tangential derivatives at the boundary is also achieved. The numerical boundary data are then used to successfully calculate the values of the solution at interior points again. The outlined test cases have been repeated with various boundary element meshes, indicating that the accuracy of the numerical results increases with increasing boundary discretization. Copyright © 2006 John Wiley & Sons, Ltd. [source] Solution of two-dimensional Poisson problems in quadrilateral domains using transfinite Coons interpolationINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 7 2004Christopher G. Provatidis Abstract This paper proposes a global approximation method to solve elliptic boundary value Poisson problems in arbitrary shaped 2-D domains. Using transfinite interpolation, a symmetric finite element formulation is derived for degrees of freedom arranged mostly along the boundary of the domain. In cases where both Dirichlet and Neumann boundary conditions occur, the numerical solution is based on bivariate Coons interpolation using the boundary only. Furthermore, in case of only Dirichlet boundary conditions and no existing axes of symmetry, it is proposed to use at least one internal point and apply transfinite interpolation. The theory is sustained by five numerical examples applied to domains of square, circular and elliptic shape. Copyright © 2004 John Wiley & Sons, Ltd. [source] An element-wise, locally conservative Galerkin (LCG) method for solving diffusion and convection,diffusion problemsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 5 2008C. G. Thomas Abstract An element-wise locally conservative Galerkin (LCG) method is employed to solve the conservation equations of diffusion and convection,diffusion. This approach allows the system of simultaneous equations to be solved over each element. Thus, the traditional assembly of elemental contributions into a global matrix system is avoided. This simplifies the calculation procedure over the standard global (continuous) Galerkin method, in addition to explicitly establishing element-wise flux conservation. In the LCG method, elements are treated as sub-domains with weakly imposed Neumann boundary conditions. The LCG method obtains a continuous and unique nodal solution from the surrounding element contributions via averaging. It is also shown in this paper that the proposed LCG method is identical to the standard global Galerkin (GG) method, at both steady and unsteady states, for an inside node. Thus, the method, has all the advantages of the standard GG method while explicitly conserving fluxes over each element. Several problems of diffusion and convection,diffusion are solved on both structured and unstructured grids to demonstrate the accuracy and robustness of the LCG method. Both linear and quadratic elements are used in the calculations. For convection-dominated problems, Petrov,Galerkin weighting and high-order characteristic-based temporal schemes have been implemented into the LCG formulation. Copyright © 2007 John Wiley & Sons, Ltd. [source] Imposing Dirichlet boundary conditions in the extended finite element methodINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 12 2006Nicolas Moës Abstract This paper is devoted to the imposition of Dirichlet-type conditions within the extended finite element method (X-FEM). This method allows one to easily model surfaces of discontinuity or domain boundaries on a mesh not necessarily conforming to these surfaces. Imposing Neumann boundary conditions on boundaries running through the elements is straightforward and does preserve the optimal rate of convergence of the background mesh (observed numerically in earlier papers). On the contrary, much less work has been devoted to Dirichlet boundary conditions for the X-FEM (or the limiting case of stiff boundary conditions). In this paper, we introduce a strategy to impose Dirichlet boundary conditions while preserving the optimal rate of convergence. The key aspect is the construction of the correct Lagrange multiplier space on the boundary. As an application, we suggest to use this new approach to impose precisely zero pressure on the moving resin front in resin transfer moulding (RTM) process while avoiding remeshing. The case of inner conditions is also discussed as well as two important practical cases: material interfaces and phase-transformation front capturing. Copyright © 2006 John Wiley & Sons, Ltd. [source] Simple modifications for stabilization of the finite point methodINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 3 2005B. Boroomand Abstract A stabilized version of the finite point method (FPM) is presented. A source of instability due to the evaluation of the base function using a least square procedure is discussed. A suitable mapping is proposed and employed to eliminate the ill-conditioning effect due to directional arrangement of the points. A step by step algorithm is given for finding the local rotated axes and the dimensions of the cloud using local average spacing and inertia moments of the points distribution. It is shown that the conventional version of FPM may lead to wrong results when the proposed mapping algorithm is not used. It is shown that another source for instability and non-monotonic convergence rate in collocation methods lies in the treatment of Neumann boundary conditions. Unlike the conventional FPM, in this work the Neumann boundary conditions and the equilibrium equations appear simultaneously in a weight equation similar to that of weighted residual methods. The stabilization procedure may be considered as an interpretation of the finite calculus (FIC) method. The main difference between the two stabilization procedures lies in choosing the characteristic length in FIC and the weight of the boundary residual in the proposed method. The new approach also provides a unique definition for the sign of the stabilization terms. The reasons for using stabilization terms only at the boundaries is discussed and the two methods are compared. Several numerical examples are presented to demonstrate the performance and convergence of the proposed methods. Copyright © 2005 John Wiley & Sons, Ltd. [source] Exponential finite elements for diffusion,advection problemsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 15 2005Abbas El-Zein Abstract A new finite element method for the solution of the diffusion,advection equation is proposed. The method uses non-isoparametric exponentially-varying interpolation functions, based on exact, one- and two-dimensional solutions of the Laplace-transformed differential equation. Two eight-noded elements are developed and tested for convergence, stability, Peclet number limit, anisotropy, material heterogeneity, Dirichlet and Neumann boundary conditions and tolerance for mesh distortions. Their performance is compared to that of conventional, eight- and 12-noded polynomial elements. The exponential element based on two-dimensional analytical solutions fails basic tests of convergence. The one based on one-dimensional solutions performs particularly well. It reduces by about 75% the number of elements and degrees of freedom required for convergence, yielding an error that is one order of magnitude smaller than that of the eight-noded polynomial element. The exponential element is stable and robust under relatively high degrees of heterogeneity, anisotropy and mesh distortions. Copyright © 2005 John Wiley & Sons, Ltd. [source] Using vorticity to define conditions at multiple open boundaries for simulating flow in a simplified vortex settling basinINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 1 2007A. N. Ziaei Abstract In this paper a method is developed to define multiple open boundary (OB) conditions in a simplified vortex settling basin (VSB). In this method, the normal component of the momentum equation is solved at the OBs, and tangential components of vorticity are calculated by solving vorticity transport equations only at the OBs. Then the tangential vorticity components are used to construct Neumann boundary conditions for tangential velocity components. Pressure is set to its ambient value, and the divergence-free condition is satisfied at these boundaries by employing the divergence as the Neumann condition for the normal-direction momentum equation. The 3-D incompressible Navier,Stokes equations in a primitive-variable form are solved using the SIMPLE algorithm. Grid-function convergence tests are utilized to verify the numerical results. The complicated laminar flow structure in the VSB is investigated, and preliminary assessment of two popular turbulence models, k,, and k,,, is conducted. Copyright © 2006 John Wiley & Sons, Ltd. [source] Does a ,volume-filling effect' always prevent chemotactic collapse?MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 1 2010Michael Winkler Abstract The parabolic,parabolic Keller,Segel system for chemotaxis phenomena, is considered under homogeneous Neumann boundary conditions in a smooth bounded domain ,,,n with n,2. It is proved that if ,(u)/,(u) grows faster than u2/n as u,, and some further technical conditions are fulfilled, then there exist solutions that blow up in either finite or infinite time. Here, the total mass ,,u(x, t)dx may attain arbitrarily small positive values. In particular, in the framework of chemotaxis models incorporating a volume-filling effect in the sense of Painter and Hillen (Can. Appl. Math. Q. 2002; 10(4):501,543), the results indicate how strongly the cellular movement must be inhibited at large cell densities in order to rule out chemotactic collapse. Copyright © 2009 John Wiley & Sons, Ltd. [source] Minimal regularity of the solution of some boundary value problems of Signorini's type in polygonal domainsMATHEMATISCHE NACHRICHTEN, Issue 6 2005Denis Mercier Abstract We study the regularity in Sobolev spaces of the solution of transmission problems in a polygonal domain of the plane, with unilateral boundary conditions of Signorini's type in a part of the boundary and Dirichlet or Neumann boundary conditions on the remainder part. We use a penalization method combined with an appropriated lifting argument to get uniform estimates of the approximated solutions in order to obtain some minimal regularity results for the exact solution. The same method allows us to consider problems with thin obstacles. It can be easily extended to 3D problems. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] Analysis of block matrix preconditioners for elliptic optimal control problemsNUMERICAL LINEAR ALGEBRA WITH APPLICATIONS, Issue 4 2007T. P. Mathew Abstract In this paper, we describe and analyse several block matrix iterative algorithms for solving a saddle point linear system arising from the discretization of a linear-quadratic elliptic control problem with Neumann boundary conditions. To ensure that the problem is well posed, a regularization term with a parameter , is included. The first algorithm reduces the saddle point system to a symmetric positive definite Schur complement system for the control variable and employs conjugate gradient (CG) acceleration, however, double iteration is required (except in special cases). A preconditioner yielding a rate of convergence independent of the mesh size h is described for , , R2 or R3, and a preconditioner independent of h and , when , , R2. Next, two algorithms avoiding double iteration are described using an augmented Lagrangian formulation. One of these algorithms solves the augmented saddle point system employing MINRES acceleration, while the other solves a symmetric positive definite reformulation of the augmented saddle point system employing CG acceleration. For both algorithms, a symmetric positive definite preconditioner is described yielding a rate of convergence independent of h. In addition to the above algorithms, two heuristic algorithms are described, one a projected CG algorithm, and the other an indefinite block matrix preconditioner employing GMRES acceleration. Rigorous convergence results, however, are not known for the heuristic algorithms. Copyright © 2007 John Wiley & Sons, Ltd. [source] Compact difference schemes for heat equation with Neumann boundary conditionsNUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 6 2009Zhi-Zhong Sun Abstract In this article, two recent proposed compact schemes for the heat conduction problem with Neumann boundary conditions are analyzed. The first difference scheme was proposed by Zhao, Dai, and Niu (Numer Methods Partial Differential Eq 23, (2007), 949,959). The unconditional stability and convergence are proved by the energy methods. The convergence order is O(,2 + h2.5) in a discrete maximum norm. Numerical examples demonstrate that the convergence order of the scheme can not exceeds O(,2 + h3). An improved compact scheme is presented, by which the approximate values at the boundary points can be obtained directly. The second scheme was given by Liao, Zhu, and Khaliq (Methods Partial Differential Eq 22, (2006), 600,616). The unconditional stability and convergence are also shown. By the way, it is reported how to avoid computing the values at the fictitious points. Some numerical examples are presented to show the theoretical results. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009 [source] A fourth-order compact algorithm for nonlinear reaction-diffusion equations with Neumann boundary conditionsNUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 3 2006Wenyuan Liao Abstract In this article, we discuss a scheme for dealing with Neumann and mixed boundary conditions using a compact stencil. The resulting compact algorithm for solving systems of nonlinear reaction-diffusion equations is fourth-order accurate in both the temporal and spatial dimensions. We also prove that the standard second-order approximation to zero Neumann boundary conditions provides fourth-order accuracy when the nonlinear reaction term is independent of the spatial variables. Numerical examples, including an application of this algorithm to a mathematical model describing frontal polymerization process, are presented in the article to demonstrate the accuracy and efficiency of the scheme. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005 [source] On the mixed finite element method with Lagrange multipliersNUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 2 2003Ivo Babu Abstract In this note we analyze a modified mixed finite element method for second-order elliptic equations in divergence form. As a model we consider the Poisson problem with mixed boundary conditions in a polygonal domain of R2. The Neumann (essential) condition is imposed here in a weak sense, which yields the introduction of a Lagrange multiplier given by the trace of the solution on the corresponding boundary. This approach allows to handle nonhomogeneous Neumann boundary conditions, theoretically and computationally, in an alternative and usually easier way. Then we utilize the classical Babu,ka-Brezzi theory to show that the resulting mixed variational formulation is well posed. In addition, we use Raviart-Thomas spaces to define the associated finite element method and, applying some elliptic regularity results, we prove the stability, unique solvability, and convergence of this discrete scheme, under appropriate assumptions on the mesh sizes. Finally, we provide numerical results illustrating the performance of the algorithm for smooth and singular problems. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 192,210, 2003 [source] | |||||||||||||