Use Of Quadrature Rules Research Articles

The Multi-PN angular discretization method of the neutral-particle transport equation for radiation shielding calculations

The well-known Discrete Ordinates (SN) approximation is little effective in multidimensional radiation shielding calculations. In the presence of geometrically large and low-scattering media, discrete-ordinates solutions are generally affected by the ray effect and an adequate refinement of the angular discretization is sometimes prohibitive in terms of calculation cost. The Multi-PN method (MPN) is an angular discretization technique leading to SN-like transport equations but less affected by ray effect. It relies on a solid angle expansion of the interface fluxes, allowing for a local coupling between different angular degrees of freedom within the same solid angle. This coupling enables to solve the equation by means of a standard sweeping algorithm. Furthermore, flux angular moments are directly integrated without any use of quadrature rules. The MPN measured error convergence rate with respect to the number of angular degrees of freedom is significantly larger than the one measured with SN.

An enriched Galerkin-characteristics finite element method for convection-dominated and transport problems

We propose an enriched Galerkin-characteristics finite element method for numerical solution of convection-dominated problems. The method uses the modified method of characteristics for the integration of the total derivative in time, combined with the finite element method for the spatial discretization on unstructured grids. The L2-projection method is implemented for the evaluation of numerical solutions by tracking the departure points from the integration points in each element. In the present study, a family of quadrature rules are used to enrich the approximation of integrals in the L2-projection method. The use of quadrature rules as an enrichment procedure allows for spatial discretizations on coarse fixed meshes and no need to introduce time-dependent enrichments. This procedure offers a very great advantage over the conventional Galerkin-characteristics finite element method since the same governing matrix representation can be used during the entire time steeping process. We also propose a multilevel adaptive procedure in the enriched Galerkin-characteristics finite element method by monitoring the gradient of the solution in the computational domain during its advection. In comparison with the traditional finite element analysis with h-, p- and hp-version refinements, the present approach is much simpler, more robust and efficient, and it yields more accurate solutions for a fixed number of degrees of freedom without refining the mesh. To examine the performance of the proposed method we solve several test examples for convection-diffusion problems. Comparison to the conventional Galerkin-characteristics finite element method is also carried out in the present work. The aim of such enriched method compared to the classical finite element method is to solve the time-dependent convection-dominated problems efficiently and with an appropriate level of accuracy.

A refined use of the residue theorem for the evaluation of band-averaged input power into linear second-order dynamic systems

Abstract In a recent work, a strategy was proposed which exploits the residue theorem as an efficient tool for evaluating the frequency averaged input power into vibrating systems. In this paper, such a technique is generalised and further insight and improvements are presented. Evaluating band-averages requires the solution of a weighted integral over a real frequency variable. In this paper, the integration path is moved to the complex plane, where the input mobility shows a smoother behaviour. Consequently, a smaller number of quadrature points are needed to perform an accurate integration, which leads to a significant reduction of computational time. In addition, the connection between the use of quadrature rules in the complex plane and weighting functions on real frequencies is considered. Two application examples prove the accuracy of the present strategy for different quadrature rules. Adaptive integration schemes are also investigated.

Piecewise-quasilinearization techniques for singularly perturbed Volterra integro-differential equations

Three families of iterative quasilinear methods for the numerical solution of singularly perturbed Volterra integro-differential equations are presented. Two of these methods are based on the piecewise integration of the differential equations and the use of quadrature rules, while the third one employs a locally frozen approximation for the coefficients and right-hand side of the differential equation. It is shown that these quasilinear methods are first-order accurate for step sizes equal to or larger than the initial layer thickness, and are less accurate than exponentially fitted techniques whose coefficients are selected so that the finite difference equations have truncation errors of second- and third-order, even though they exhibit only first-order accuracy for initial layer thicknesses equal to or smaller than the step size. It is also shown that the quasilinear methods presented in this paper are of exponential type and provide more accurate results than the backward Euler and trapezoidal methods for step sizes larger than the initial layer thickness, and the quasilinear frozen method is less accurate than quasilinear techniques that make use of an integrating factor and numerical quadratures.

Penalty-finite element methods for the analysis of stokesian flows

A study of a class of finite element methods for the analysis of Stokes' problem based on the use of exterior penalty formulations is described. The effects of selective reduced integration (i.e., the use of quadrature rules for integrating the penalty terms which are of lower order than that required to integrate polynomial approximations of these terms exactly) are investigated. Error estimates are derived and the numerical stability of these methods, as depicted by a special Babuska-Brezzi condition, is explored in some detail. The results of several numerical experiments with these methods are also given.