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Finite-element Methods (finite-element + methods)

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Engineering	66%

Selected Abstracts

Parsimonious finite-volume frequency-domain method for 2-D P,SV -wave modelling

GEOPHYSICAL JOURNAL INTERNATIONAL, Issue 2 2008
R. Brossier
SUMMARY A new numerical technique for solving 2-D elastodynamic equations based on a finite-volume frequency-domain approach is proposed. This method has been developed as a tool to perform 2-D elastic frequency-domain full-waveform inversion. In this context, the system of linear equations that results from the discretization of the elastodynamic equations is solved with a direct solver, allowing efficient multiple-source simulations at the partial expense of the memory requirement. The discretization of the finite-volume approach is through triangles. Only fluxes with the required quantities are shared between the cells, relaxing the meshing conditions, as compared to finite-element methods. The free surface is described along the edges of the triangles, which can have different slopes. By applying a parsimonious strategy, the stress components are eliminated from the discrete equations and only the velocities are left as unknowns in the triangles. Together with the local support of the P0 finite-volume stencil, the parsimonious approach allows the minimizing of core memory requirements for the simulation. Efficient perfectly matched layer absorbing conditions have been designed for damping the waves around the grid. The numerical dispersion of this FV formulation is similar to that of O(,x2) staggered-grid finite-difference (FD) formulations when considering structured triangular meshes. The validation has been performed with analytical solutions of several canonical problems and with numerical solutions computed with a well-established FD time-domain method in heterogeneous media. In the presence of a free surface, the finite-volume method requires 10 triangles per wavelength for a flat topography, and fifteen triangles per wavelength for more complex shapes, well below the criteria required by the staircase approximation of O(,x2) FD methods. Comparisons between the frequency-domain finite-volume and the O(,x2) rotated FD methods also show that the former is faster and less memory demanding for a given accuracy level, an attractive feature for frequency-domain seismic inversion. We have thus developed an efficient method for 2-D P,SV -wave modelling on structured triangular meshes as a tool for frequency-domain full-waveform inversion. Further work is required to improve the accuracy of the method on unstructured meshes. [source]

The maximum principle violations of the mixed-hybrid finite-element method applied to diffusion equations

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 12 2002
H. Hoteit
Abstract The abundant literature of finite-element methods applied to linear parabolic problems, generally, produces numerical procedures with satisfactory properties. However, some initial,boundary value problems may cause large gradients at some points and consequently jumps in the solution that usually needs a certain period of time to become more and more smooth. This intuitive fact of the diffusion process necessitates, when applying numerical methods, varying the mesh size (in time and space) according to the smoothness of the solution. In this work, the numerical behaviour of the time-dependent solutions for such problems during small time duration obtained by using a non-conforming mixed-hybrid finite-element method (MHFEM) is investigated. Numerical comparisons with the standard Galerkin finite element (FE) as well as the finite-difference (FD) methods are checked. Owing to the fact that the mixed methods violate the discrete maximum principle, some numerical experiments showed that the MHFEM leads sometimes to non-physical peaks in the solution. A diffusivity criterion relating the mesh steps for an artificial initial,boundary value problem will be presented. One of the propositions given to avoid any non-physical oscillations is to use the mass-lumping techniques. Copyright © 2002 John Wiley & Sons, Ltd. [source]

Developing implicit pressure-weighted upwinding scheme to calculate steady and unsteady flows on unstructured grids

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 2 2008
M. Darbandi
Abstract The finite-volume methods normally utilize either simple or complicated mathematical expressions to interpolate the fluxes at the cell faces of their unstructured volumes. Alternatively, we benefit from the advantages of both finite-volume and finite-element methods and estimate the advection terms on the cell faces using an inclusive pressure-weighted upwinding scheme extended on unstructured grids. The present pressure-based method treats the steady and unsteady flows on a collocated grid arrangement. However, to avoid a non-physical spurious pressure field pattern, two mass flux per volume expressions are derived at the cell interfaces. The dual advantages of using an unstructured-based discretization and a pressure-weighted upwinding scheme result in obtaining high accurate solutions with noticeable progress in the performance of the primitive method extended on the structured grids. The accuracy and performance of the extended formulations are demonstrated by solving different standard and benchmark problems. The results show that there are excellent agreements with both benchmark and analytical solutions as well as experimental data. Copyright © 2007 John Wiley & Sons, Ltd. [source]

Positivity-preserving, flux-limited finite-difference and finite-element methods for reactive transport

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 2 2003
Robert J. MacKinnon
Abstract A new class of positivity-preserving, flux-limited finite-difference and Petrov,Galerkin (PG) finite-element methods are devised for reactive transport problems. The methods are similar to classical TVD flux-limited schemes with the main difference being that the flux-limiter constraint is designed to preserve positivity for problems involving diffusion and reaction. In the finite-element formulation, we also consider the effect of numerical quadrature in the lumped and consistent mass matrix forms on the positivity-preserving property. Analysis of the latter scheme shows that positivity-preserving solutions of the resulting difference equations can only be guaranteed if the flux-limited scheme is both implicit and satisfies an additional lower-bound condition on time-step size. We show that this condition also applies to standard Galerkin linear finite-element approximations to the linear diffusion equation. Numerical experiments are provided to demonstrate the behavior of the methods and confirm the theoretical conditions on time-step size, mesh spacing, and flux limiting for transport problems with and without nonlinear reaction. Copyright © 2003 John Wiley & Sons, Ltd. [source]

A posteriori error estimator for expanded mixed hybrid methods,

NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 2 2007
Dongho Kim
Abstract In this article, we construct an a posteriori error estimator for expanded mixed hybrid finite-element methods for second-order elliptic problems. An a posteriori error analysis yields reliable and efficient estimate based on residuals. Several numerical examples are presented to show the effectivity of our error indicators. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 23: 330,349, 2007 [source]