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Space Discretization (space + discretization)
Selected AbstractsA robust algorithm for configurational-force-driven brittle crack propagation with R-adaptive mesh alignmentINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 2 2007C. Miehe Abstract The paper considers a variational formulation of brittle fracture in elastic solids and proposes a numerical implementation by a finite element method. On the theoretical side, we outline a consistent thermodynamic framework for crack propagation in an elastic solid. It is shown that both the elastic equilibrium response as well as the local crack evolution follow in a natural format by exploitation of a global Clausius,Planck inequality in the sense of Coleman's method. Here, the canonical direction of the crack propagation associated with the classical Griffith criterion is the direction of the material configurational force which maximizes the local dissipation at the crack tip and minimizes the incremental energy release. On the numerical side, we exploit this variational structure in terms of crack-driving configurational forces. First, a standard finite element discretization in space yields a discrete formulation of the global dissipation in terms configurational nodal forces. As a consequence, the constitutive setting of crack propagation in the space-discretized finite element context is naturally related to discrete nodes of a typical finite element mesh. Next, consistent with the node-based setting, the discretization of the evolving crack discontinuity is performed by the doubling of critical nodes and interface segments of the mesh. Critical for the success of this procedure is its embedding into an r-adaptive crack-segment reorientation procedure with configurational-force-based directional indicator. Here, successive crack releases appear in discrete steps associated with the given space discretization. These are performed by a staggered loading,release algorithm of energy minimization at frozen crack state followed by the successive crack releases at frozen deformation. This constitutes a sequence of positive-definite discrete subproblems with successively decreasing overall stiffness, providing an extremely robust algorithmic setting in the postcritical range. We demonstrate the performance of the formulation by means of representative numerical simulations. Copyright © 2007 John Wiley & Sons, Ltd. [source] Semi-Lagrangian advection on a spherical geodesic gridINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 2 2007Maria Francesca Carfora Abstract A simple and efficient numerical method for solving the advection equation on the spherical surface is presented. To overcome the well-known ,pole problem' related to the polar singularity of spherical coordinates, the space discretization is performed on a geodesic grid derived by a uniform triangulation of the sphere; the time discretization uses a semi-Lagrangian approach. These two choices, efficiently combined in a substepping procedure, allow us to easily determine the departure points of the characteristic lines, avoiding any computationally expensive tree-search. Moreover, suitable interpolation procedures on such geodesic grid are presented and compared. The performance of the method in terms of accuracy and efficiency is assessed on two standard test cases: solid-body rotation and a deformation flow. Copyright © 2007 John Wiley & Sons, Ltd. [source] A fully implicit method for steady and unsteady viscous flow simulationsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 2 2003Jie Li Abstract In this paper a time-accurate, fully implicit method has been applied to solve a variety of steady and unsteady viscous flow problems. It uses a finite volume cell-centred formulation on structured grids and employs central space discretization with artificial dissipation for the residual computation. In order to obtain a second-order time-accurate implicit scheme, a Newton-like subiteration is performed in the original LU-SGS method to converge the calculations at each physical time step by means of a dual-time approach proposed by Jameson. The numerical experiments show that the present method is very efficient, reliable, and robust for steady and unsteady viscous flow simulations, especially for some low speed flow problems. Copyright © 2003 John Wiley & Sons, Ltd. [source] Numerical simulation of three-dimensional free surface flowsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 7 2003V. Maronnier Abstract A numerical model is presented for the simulation of complex fluid flows with free surfaces in three space dimensions. The model described in Maronnier et al. (J. Comput. Phys. 1999; 155(2) : 439) is extended to three dimensional situations. The mathematical formulation of the model is similar to that of the volume of fluid (VOF) method, but the numerical procedures are different. A splitting method is used for the time discretization. At each time step, two advection problems,one for the predicted velocity field and the other for the volume fraction of liquid,are to be solved. Then, a generalized Stokes problem is solved and the velocity field is corrected. Two different grids are used for the space discretization. The two advection problems are solved on a fixed, structured grid made out of small cubic cells, using a forward characteristic method. The generalized Stokes problem is solved using continuous, piecewise linear stabilized finite elements on a fixed, unstructured mesh of tetrahedrons. The three-dimensional implementation is discussed. Efficient postprocessing algorithms enhance the quality of the numerical solution. A hierarchical data structure reduces memory requirements. Numerical results are presented for complex geometries arising in mold filling. Copyright © 2003 John Wiley & Sons, Ltd. [source] Numerical evaluation of two discontinuous Galerkin methods for the compressible Navier,Stokes equationsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 1-2 2002F. Bassi This paper presents a critical comparison between two recently proposed discontinuous Galerkin methods for the space discretization of the viscous terms of the compressible Navier,Stokes equations. The robustness and accuracy of the two methods has been numerically evaluated by considering simple but well documented classical two-dimensional test cases, including the flow around the NACA0012 airfoil, the flow along a flat plate and the flow through a turbine nozzle. Copyright © 2002 John Wiley & Sons, Ltd. [source] Simulation of Rayleigh waves in cracked platesMATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 1 2007M. T. Cao Abstract The aim of this paper is to develop new numerical procedures to detect micro cracks, or superficial imperfections, in thin plates using excitation by Rayleigh waves. We shall consider a unilateral contact problem between the two sides of the crack in an elastic plate subjected to suitable boundary conditions in order to reproduce a single Rayleigh wave cycle. An approximate solution of this problem will be calculated by using one of the Newmark methods for time discretization and a finite element method for space discretization. To deal with the nonlinearity due to the contact condition, an iterative algorithm involving one multiplier will be used; this multiplier will be approximated by using Newton's techniques. Finally, we will show numerical simulations for both cracked and non-cracked plates. Copyright © 2006 John Wiley & Sons, Ltd. [source] A finite element modified method of characteristics for convective heat transportNUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 3 2008Mofdi El-Amrani Abstract We propose a finite element modified method of characteristics for numerical solution of convective heat transport. The flow equations are the incompressible Navier-Stokes equations including density variation through the Boussinesq approximation. The solution procedure consists of combining an essentially non-oscillatory modified method of characteristics for time discretization with finite element method for space discretization. These numerical techniques associate the geometrical flexibility of the finite elements with the ability offered by modified method of characteristics to solve convection-dominated flows using time steps larger than its Eulerian counterparts. Numerical results are shown for natural convection in a squared cavity and heat transport in the strait of Gibraltar. Performance and accuracy of the method are compared to other published data. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008 [source] | |||||||||||||