Use Of Nonuniform Grids Research Articles

L1‐type finite element method for time‐fractional diffusion‐wave equations on nonuniform grids

AbstractAs is well known, one of the main challenges in dealing with the time‐fractional diffusion‐wave equation is that the solutions have weak regularity at the initial values. This necessitates the use of nonuniform grids for error analysis. However, the theoretical analysis of nonuniform grids is relatively complex and lacks efficient and stable numerical methods. The main research problem addressed in this article is the time‐fractional diffusion‐wave equation on nonuniform grids. We use the order reduction method and discrete complementary convolution kernel to discretize the Caputo derivative of the equation and obtain the L1‐type finite element method on nonuniform grids. The properties related to discrete complementary convolutional kernels are often used for algorithm convergence analysis, which simplifies the process of finite element theory analysis and is efficient and innovative. This article demonstrates the solvability, stability, and convergence of the L1‐type finite element schemes on nonuniform grids. Finally, the consistency between the proposed finite element scheme and the convergence order results obtained from theoretical analysis was verified through experiments.

Analysis and reduction of spurious noise generated at grid refinement interfaces with the lattice Boltzmann method

The present study focuses on the unphysical effects induced by the use of non-uniform grids in the lattice Boltzmann method. In particular, the convection of vortical structures across a grid refinement interface is likely to generate spurious noise that may impact the whole computation domain. This issue becomes critical in the case of aeroacoustic simulations, where accurate pressure estimations are of paramount importance. The purpose of this article is to identify the issues occurring at the interface and to propose possible solutions yielding significant improvements for aeroacoustic simulations. More specifically, this study highlights the critical involvement of non-physical modes in the generation of spurious vorticity and acoustics. The identification of these modes is made possible thanks to linear stability analyses performed in the fluid core, and non-hydrodynamic sensors specifically developed to systematically emphasize them during a simulation. Investigations seeking pure acoustic waves and sheared flows allow for isolating the contribution of each mode. An important result is that spurious wave generation is intrinsically due to the change in the grid resolution (i.e. aliasing) independently of the details of the grid transition algorithm. Finally, the solution proposed to minimize spurious wave amplitude consists of choosing an appropriate collision model in the fluid core so as to cancel the non-hydrodynamic mode contribution regardless the grid coupling algorithm. Results are validated on a convected vortex and on a turbulent flow around a cylinder where a huge reduction of both spurious noise and vorticity are obtained.

On Green’s function-based time stepping for semilinear initial–boundary value problems

The Fourier spectral approach is central to the analysis of semilinear initial–boundary value problems when considered with periodic boundary conditions. It aids theoretical arguments, such as renormalisation group analysis, as well as numerical simulation through standard pseudo-spectral methods. Fixed boundary conditions, in contrast, require the use of non-uniform grids, usually generated by bases of orthogonal polynomials. On such bases, numerical differentiation is ill-conditioned and can potentially lead to a catastrophic blow-up of round-off error. In this paper, we apply ideas explored by Viswanath (2013) in the context of Navier–Stokes solvers to completely eliminate numerical differentiation and linear solving from the time-stepping algorithm in favour of numerical quadrature. We propose a concrete quadrature rule and test its stability, scalability and spectral accuracy on a Kuramoto–Sivashinsky template problem over a range of five orders of magnitude of the viscosity.

A meshfree unification: reproducing kernel peridynamics

This paper is the first investigation establishing the link between the meshfree state-based peridynamics method and other meshfree methods, in particular with the moving least squares reproducing kernel particle method (RKPM). It is concluded that the discretization of state-based peridynamics leads directly to an approximation of the derivatives that can be obtained from RKPM. However, state-based peridynamics obtains the same result at a significantly lower computational cost which motivates its use in large-scale computations. In light of the findings of this study, an update to the method is proposed such that the limitations regarding application of boundary conditions and the use of non-uniform grids are corrected by using the reproducing kernel approximation.

Lattice Boltzmann method for population balance equations with simultaneous growth, nucleation, aggregation and breakage

Lattice Boltzmann method (LBM) is developed for solution of one-dimensional population balance equations (PBEs) with simultaneous growth, nucleation, aggregation and breakage. Aggregation and breakage, which act as source terms in PBEs, are included as force terms in LBM formulation. The force terms representing aggregation and breakage are evaluated by fixed pivot (FP) method. Multiscale analysis is used to derive the kinetic equations associated with LBM, whose long-time large-scale solution provides the solution of the PBE. A coordinate transformation is proposed, which allows the use of non-uniform grid for LBM to obtain accurate solution of PBE with moderate number of grid points. The performance of the proposed LBM-FP method is compared with finite volume (FV) and method of characteristics (MOC) combined with FP (MOC-FP) methods. Using benchmark examples, the proposed LBM-FP method is shown to be useful for solving PBEs due to its computational efficiency and ability to handle a wide range of problems.

Use of Non-uniform Grids for Reduced Spatial Sampling in FDTD Simulation

Use of Non-uniform Grids for Reduced Spatial Sampling in FDTD Simulation

Level set method for numerical simulation of a cavitation bubble, its growth, collapse and rebound near a rigid wall

A level set method of non-uniform grids is used to simulate the whole evolution of a cavitation bubble, including its growth, collapse and rebound near a rigid wall. Single-phase Navier–Stokes equation in the liquid region is solved by MAC projection algorithm combined with second-order ENO scheme for the advection terms. The moving interface is captured by the level set function, and the interface velocity is resolved by “one-side” velocity extension from the liquid region to the bubble region, complementing the second-order weighted least squares method across the interface and projection inside bubble. The use of non-uniform grid overcomes the difficulty caused by the large computational domain and very small bubble size. The computation is very stable without suffering from large flow-field gradients, and the results are in good agreements with other studies. The bubble interface kinematics, dynamics and its effect on the wall are highlighted, which shows that the code can effectively capture the “shock wave”-like pressure and velocity at jet impact, toroidal bubble, and complicated pressure structure with peak, plateau and valley in the later stage of bubble oscillating.

Improved Velocity Projection for the Material Point Method

The standard velocity projection scheme for the Material Point Method (MPM) and a typical form of the GIMP Method are exam­ ined. It is demonstrated that the fidelity of in­ formation transfer from a particle representation to the computational grid is strongly dependent on particle density and location. In addition, use of non-uniform grids and even non-uniform par­ ticle sizes are shown to introduce error. An en­ hancement to the projection operation is devel­ oped which makes use of already available veloc­ ity gradient information. This enhancement facil­ itates exact projection of linear functions and re­ duces the dependence of projection accuracy on particle location and density for non-linear func­ tions. The efficacy of this formulation for reduc­ ing error is demonstrated in solid mechanics sim­ ulations in one and two dimensions.

Numerical analysis of the big bounce in loop quantum cosmology

Loop quantum cosmology (LQC) homogeneous models with a massless scalar field show that the big-bang singularity can be replaced by a big quantum bounce. To gain further insight on the nature of this bounce, we study the semidiscrete loop quantum gravity Hamiltonian constraint equation from the point of view of numerical analysis. For illustration purposes, we establish a numerical analogy between the quantum bounces and reflections in finite difference discretizations of wave equations triggered by the use of nonuniform grids or, equivalently, reflections found when solving numerically wave equations with varying coefficients. We show that the bounce is closely related to the method for the temporal update of the system and demonstrate that explicit time-updates in general yield bounces. Finally, we present an example of an implicit time-update devoid of bounces and show back-in-time, deterministic evolutions that reach and partially jump over the big-bang singularity.

A compact 2-D FDFD method for modeling microstrip structures with nonuniform grids and perfectly matched layer

Full-wave analysis of the microstrip structures is performed by using the compact two-dimensional (2-D) finite-difference frequency-domain (FDFD) method with nonuniform grids and perfectly matched layer (PML). The use of nonuniform grids can significantly reduce the computational matrix size. Less memory and CPU time are required as comparing with the original compact 2-D FDFD method. For the analysis of the microstrip structures with an absorbing boundary condition, the compact 2-D FDFD method with PML is presented. The performances of different PML thickness are studied. Numerical examples are presented to demonstrate the accuracy and efficiency of this method

Computer simulations of tsunamis due to sector collapse at Stromboli, Italy

Stromboli is an island volcano of the Aeolian Volcanic Arc, characterised by persistent activity. The cone rises about 2500–3000 m from its submarine base with very steep slopes; its summit at 924 m above the sea level. The subaerial growth of Stromboli, occurred in the last 100 ka, has been marked by repeated episodes of large gravitational collapses especially affecting the NW flank of the island in the last 13 ka. The last one occurred less than 5000 years ago forming the deep depression on the NW seaward flank, named Sciara del Fuoco (SdF), and it produced very likely large water waves. This paper envisages a scenario where a huge mass of volcanic material collapses into the sea in the same sector in which the SdF collapse took place in Holocenic times and it computes possible tsunami evolutions assuming the present-day bathymetry. Numerical simulations are performed by means of two distinct models; one for the mass collapse and one for the tsunami. Slope failure dynamics are calculated with the aid of a Lagrangian model: the landslide is subdivided into blocks, and the motion of each constituent block is calculated by applying the basic principle of mechanical momentum conservation, with block–block and block–ambient interactions being taken into account. Water waves are computed by solving a system of shallow-water equations including a forcing term dependent on the sliding mass motion. The finite-element (FE) technique is employed since it permits the use of non-uniform grids, which are adequate to account for marine basins with irregular coastlines. In addition to the sensitivity analysis concerning the main parameters governing the slide motion, two main cases are explored, differing in the slide path followed by the mass. The resulting tsunami is very large, with giant waves as high as several meters (tens of meters in the worst cases) impinging the coast. Due to the strong wave refraction induced by bathymetry, waves travel around the island, affecting even the island coast opposite the source.

NONUNIFORM GRIDS FOR REDUCED BASIS DESIGN OF LOW ORDER FEEDBACK CONTROLLERS FOR NONLINEAR CONTINUOUS SYSTEMS

In this paper, we discuss the use of nonuniform grids in reduced basis design for developing low order nonlinear feedback controllers for hybrid distributed parameter systems. The reduced basis approach was presented in an earlier paper by Burns and King; therein, all approximations were based upon uniform grids. In this paper, we explore the effect on control design of using nonuniform grids in the fundamental step of approximating the functional controller gains. We illustrate the process using a weakly nonlinear distributed parameter system.

Analyses of Archival [Fe/H] Results for Cool Evolved Stars. I. Basic Rules and [Fe/H] Corrections

ABSTRACTThis paper is the first in a series based on published values of[Fe/H] for late‐type evolved stars. Only results from high‐dispersion spectroscopy or related techniques are used. From these data, an “analysis catalog” is to be produced containing mean values of [Fe/H] and accidental errors. Such a catalog was published by Taylor in 1991 and is now to be updated. Basic protocols for selecting and analyzing input values of [Fe/H] are stated here. A procedure is described for calculating preferred temperatures and adjusting input values of [Fe/H] to correspond to those temperatures. Ten other possible [Fe/H] corrections are also mentioned, with three corrections that can be applied being described in some detail. These are (1) corrections for use of nonuniform grids of model atmospheres, (2) corrections for use of solar equivalent widths that are not on the Liège system, and (3) corrections for non‐LTE effects.

Efficient FDTD analysis of discontinuities in a square coaxial waveguide

The finite-difference-time-domain (FDTD) method is applied to model some discontinuities that are present in C-band beam-forming networks. In particular, a branch line coupler, a phase shifter, and a slope equalizer are modeled. A FDTD algorithm, which makes use of nonuniform grids and a synthetic excitation, gives good results and saves computer resources. We achieve an efficient algorithm for analyzing BFN structures on a personal computer. Numerical results are given for the scattering parameters of three devices in 3.4(SINGLEBOND)4.2(SINGLEBOND)GHz band. © 1996 John Wiley & Sons, Inc.

The lattice Boltzmann equation on irregular lattices

A general framework to extend the lattice Boltzmann equation to arbitrary lattice geometries is presented and numerically demonstrated for the case of a two-dimensional Poiseuille flow. The new scheme considerably extends the range of applicability of the Boltzmann method to problems requiring the use of nonuniform grids.

Application of Continuous Dynamic Grid Adaption Techniques to Meteorological Modeling. Part I: Basic Formulation and Accuracy

Abstract The continuous dynamic grid adaption (CDGA) technique developed in astrophysics and aeronautics is applied, to our knowledge, for the first time to meteorological modeling. The aim of CDGA is to improve the accuracy of numerical solutions of partial differential equations (typically those governing fluid flow) by the use of nonuniform grids that have higher local resolution in regions where the numerical error is presumed to be large. Conceptually, CDGA has some relationship to the well-known technique of grid stretching, but its power lies in its ability to determine an appropriate spatial distribution of grid points automatically and to update this distribution in response to changes in the evolving numerical solution. Application of the technique is facilitated by transforming the governing equations from physical space in which the grid is nonuniform, nonorthogonal and for which the individual grid points are in continuous motion to computational space, which by definition has both a regular ...

Optimized numerical models for semiconductor devices: Part I

This article presents some results from a study concerned with the trade-off between efficiency of numerical computation and accuracy of numerical models for semiconductor devices. The approach taken in this study was to use nonuniform grids in the finite difference equations approximating the semiconductor partial differential equations. A constrained minimization problem was formulated which analytically optimizes the numerical models. Several performance functions were found useful in optimizing the numerical models; the one presented here matches computed and "exact" terminal current responses. An example is presented to illustrate the results obtained with this performance function. The example model optimization shows that the optimized grids should have their greatest density in the space-charge region near the junction. An interesting side result is that when one performance function is optimized, apparently other desired features are optimized for numerical models for semiconductor devices. The Fletcher-Powell optimization algorithm was used with an adjoint gradient algorithm specifically developed for this study, without which the use of nonuniform grids would not have been practical.

On the use of nonuniform grids in finite-difference equations

A study of the truncation errors introduced by the use of nonuniform grids is presented. It is shown that the use of certain stretched coordinates has several advantages for the numerical study of flows with boundary layers.