[Show abstract][Hide abstract] ABSTRACT: Numerical methods based on unstructured grids, with irregular cells, usually require discrete shape functions to approximate the distribution of quantities across cells. For control-volume mixed finite-element methods, vector shape functions are used to approximate the distribution of velocities across cells and vector test functions are used to minimize the error associated with the numerical approximation scheme. For a logically cubic mesh, the lowest-order shape functions are chosen in a natural way to conserve intercell fluxes that vary linearly in logical space. Vector test functions, while somewhat restricted by the mapping into the logical reference cube, admit a wider class of possibilities. Ideally, an error minimization procedure to select the test function from an acceptable class of candidates would be the best procedure. Lacking such a procedure, we first investigate the effect of possible test functions on the pressure distribution over the control volume; specifically, we look for test functions that allow for the elimination of intermediate pressures on cell faces. From these results, we select three forms for the test function for use in a control-volume mixed method code and subject them to an error analysis for different forms of grid irregularity; errors are reported in terms of the discrete L2 norm of the velocity error. Of these three forms, one appears to produce optimal results for most forms of grid irregularity.
[Show abstract][Hide abstract] ABSTRACT: SUMMARYA two-dimensional control volume mixed finite element method is applied to the elliptic equation. Discretization of the computational domain is based in triangular elements. Shape functions and test functions are formulated on the basis of an equilateral reference triangle with unit edges. A pressure support based on the linear interpolation of elemental edge pressures is used in this formulation. Comparisons are made between results from the standard mixed finite element method and this control volume mixed finite element method. Published 2011. This article is a US Government work and is in the public domain in the USA.
International Journal for Numerical Methods in Engineering 02/2012; 89(7):846-868. · 1.96 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: The mixed finite-element approximation to a second-order elliptic PDE results in a saddle-point problem and leads to an indefinite
linear system of equations. The mixed system of equations can be transformed into coupled symmetric positive-definite matrix
equations, or a Schur complement problem, using block Gauss elimination. A preconditioned conjugate-gradient algorithm is
used for solving the Schur complement problem. The mixed finite-element method is closely related to the cell-centered finite
difference scheme for solving second-order elliptic problems with variable coefficients. For the cell-centered finite difference
scheme, a simple multigrid algorithm can be defined and used as a preconditioner. For distorted grids, an additional iteration
is needed. Nested iteration with a multigrid preconditioned conjugate gradient inner iteration results in an effective numerical
solution technique for the mixed system of linear equations arising from a discretization on distorted grids. Numerical results
show that the preconditioned conjugate-gradient inner iteration is robust with respect to grid size and variability in the
hydraulic conductivity tensor.
[Show abstract][Hide abstract] ABSTRACT: Temporal variations in bedload transport rates that occur at a variety of timescales, even under steady flow conditions, are accepted as an inherent component of the bedload transport process. Rarely, however, has the cause of such variations been explained clearly. We consider three data sets, obtained from laboratory experiments, that refer to measurements of bedload transport made with continuously recording bedload traps. Each data set is characterized by a predominant low-frequency oscillation, on which additional higher-frequency oscillations generally are superimposed. The period of these oscillations, as isolated through the use of spectral analysts, ranged between 0·47 and 168 minutes, and was associated unequivocally with the migration of bedforms such as ripples, dunes, and bars. The extent to which such oscillatory behaviour may be recognized in a data set depends on the duration of sampling and the length of the sampling time, with respect to the period of a given bedform.Several theoretical probability distribution functions have been developed to describe the frequency distributions of (relative) bedload transport rates that are associated with the migration of bedforms (Einstein, 1937b; Hamamori, 1962; Carey and Hubbell, 1986). These distribution functions were derived without reference to a sampling interval. We present a modification of Hamamori's (1962) probability distribution function, generated by Monte Carlo simulation, which permits one to specify the sampling interval, in relation to the length of a bedform. Comparisons between the simulated and observed frequency distributions, that were undertaken on the basis of the data described herein, are good (significant at the 90 per cent confidence level). Finally, the implications that temporal variability, which is associated with the migration of bedforms, have for the accurate determination of bedload transport rates are considered.
[Show abstract][Hide abstract] ABSTRACT: Numerical methods for grids with irregular cells require discrete shape functions to approximate the distribution of quantities across cells. For control-volume mixed finite-element (CVMFE) methods, vector shape functions approximate velocities and vector test functions enforce a discrete form of Darcy''s law. In this paper, a new vector shape function is developed for use with irregular, hexahedral cells (trilinear images of cubes). It interpolates velocities and fluxes quadratically, because as shown here, the usual Piola-transformed shape functions, which interpolate linearly, cannot match uniform flow on general hexahedral cells. Truncation-error estimates for the shape function are demonstrated. CVMFE simulations of uniform and non-uniform flow with irregular meshes show first- and second-order convergence of fluxes in the L
2 norm in the presence and absence of singularities, respectively.
[Show abstract][Hide abstract] ABSTRACT: A control volume mixed finite element scheme for a triangular discretization of a 2-D domain is presented; several control-volume scenarios for use with the scheme are explored.