Effects Of Discretization Errors Research Articles

Molecular dynamics with rigid bodies: Alternative formulation and assessment of its limitations when employed to simulate liquid water.

Sets of atoms collectively behaving as rigid bodies are often used in molecular dynamics to model entire molecules or parts thereof. This is a coarse-graining strategy that eliminates degrees of freedom and supposedly admits larger time steps without abandoning the atomistic character of a model. In this paper, we rely on a particular factorization of the rotation matrix to simplify the mechanical formulation of systems containing rigid bodies. We then propose a new derivation for the exact solution of torque-free rotations, which are employed as part of a symplectic numerical integration scheme for rigid-body dynamics. We also review methods for calculating pressure in systems of rigid bodies with pairwise-additive potentials and periodic boundary conditions. Finally, simulations of liquid phases, with special focus on water, are employed to analyze the numerical aspects of the proposed methodology. Our results show that energy drift is avoided for time step sizes up to 5 fs, but only if a proper smoothing is applied to the interatomic potentials. Despite this, the effects of discretization errors are relevant, even for smaller time steps. These errors induce, for instance, a systematic failure of the expected equipartition of kinetic energy between translational and rotational degrees of freedom.

Spectrum of the Wilson Dirac operator at finite lattice spacings

We consider the effect of discretization errors on the microscopic spectrum of the Wilson Dirac operator using both chiral Perturbation Theory and chiral Random Matrix Theory. A graded chiral Lagrangian is used to evaluate the microscopic spectral density of the Hermitian Wilson Dirac operator as well as the distribution of the chirality over the real eigenvalues of the Wilson Dirac operator. It is shown that a chiral Random Matrix Theory for the Wilson Dirac operator reproduces the leading zero-momentum terms of Wilson chiral Perturbation Theory. All results are obtained for fixed index of the Wilson Dirac operator. The low-energy constants of Wilson chiral Perturbation theory are shown to be constrained by the Hermiticity properties of the Wilson Dirac operator.

Dispersion Analysis of a Three-Layered Waveguide with Finite Element and Matrix Methods

The study of sound wave propagation in multilayered plate structures represents an important part of acoustics. In this paper, we compare two different modeling techniques based on the finite element method (FEM) and the matrix methods using an example of a three-layered waveguide with a middle liquid layer. First, the theory of the matrix methods is summarized with a focus on multilayered waveguides containing liquid layers. Furthermore, we present a numerical procedure for obtaining modal solutions, and we discuss sample solutions for the three-layered waveguide. Second, FEM is applied to simulate the waveguide setup including real-world transducers mounted on wedges. We introduce a 2D model of the setup and a procedure for obtaining group velocity dispersion diagrams. The model is validated using theoretical and experimental results. In addition, the effect of discretization error is demonstrated. We present sample dispersion diagrams and compare them to those obtained with the matrix methods. Finally, we utilize the FEM simulation results to depict the displacement profiles across the waveguide. We show that the results of both modeling techniques are in good agreement and that they provide interesting insights into the wave mechanics of the three-layered waveguide.

A computational error-assessment of central finite-volume discretizations in large-eddy simulation using a Smagorinsky model

We present a framework for the computational assessment and comparison of large-eddy simulation methods. We apply this to large-eddy simulation of homogeneous isotropic decaying turbulence using a Smagorinsky subgrid model and investigate the combined effect of discretization and model errors at coarse subgrid resolutions. We compare four different central finite-volume methods. These discretization methods arise from the four possible combinations that can be made with a second-order and a fourth-order central scheme for either the convective and the viscous fluxes. By systematically varying the simulation resolution and the Smagorinsky coefficient, we determine parameter regions for which a desired number of flow properties is simultaneously predicted with approximately minimal error. We include both physics-based and mathematics-based error definitions, leading to different error-measures designed to emphasize either errors in large- or in small-scale flow properties. It is shown that the evaluation of simulations based on a single physics-based error may lead to inaccurate perceptions on quality. We demonstrate however that evaluations based on a range of errors yields robust conclusions on accuracy, both for physics-based and mathematics-based errors. Parameter regions where all considered errors are simultaneously near-optimal are referred to as ‘multi-objective optimal’ parameter regions. The effects of discretization errors are particularly important at marginal spatial resolution. Such resolutions reflect local simulation conditions that may also be found in parts of more complex flow simulations. Under these circumstances, the asymptotic error-behavior as expressed by the order of the spatial discretization is no longer characteristic for the total dynamic consequences of discretization errors. We find that the level of overall simulation errors for a second-order central discretization of both the convective and viscous fluxes (the ‘2–2’ method), and the fully fourth-order (‘4–4’) method, is equivalent in their respective ‘multi-objective optimal’ regions. Mixed order methods, i.e. the ‘2–4’ and ‘4–2’ combinations, yield errors which are considerably higher.

Effect of discretization error and adaptive mesh generation in diffuse optical absorption imaging: I

In diffuse optical tomography (DOT), the discretization error in the numerical solutions of the forward and inverse problems results in error in the reconstructed optical images. In this first part of our work, we analyse the error in the reconstructed optical absorption images, resulting from the discretization of the forward and inverse problems. Our analysis identifies several factors which influence the extent to which the discretization impacts on the accuracy of the reconstructed images. For example, the mutual dependence of the forward and inverse problems, the number of sources and detectors, their configuration and their orientation with respect to optical absorptive heterogeneities, and the formulation of the inverse problem. As a result, our error analysis shows that the discretization of one problem cannot be considered independent of the other problem. While our analysis focuses specifically on the discretization error in DOT, the approach can be extended to quantify other error sources in DOT and other inverse parameter estimation problems.

Effect of discretization error and adaptive mesh generation in diffuse optical absorption imaging: II

In part I (Guven et al 2007 Inverse Problems 23 1115–33), we analysed the error in the reconstructed optical absorption images resulting from the discretization of the forward and inverse problems. Our analysis led to two new error estimates, which present the relationship between the optical absorption imaging accuracy and the discretization error in the solutions of the forward and inverse problems. In this work, based on the analysis presented in part I, we develop new adaptive discretization schemes for the forward and inverse problems in order to reduce the error in the reconstructed images resulting from discretization. The proposed discretization schemes lead to adaptively refined composite meshes that yield the desired level of imaging accuracy while reducing the size of the discretized forward and inverse problems. We present numerical experiments to validate the error estimates developed in part I and show the improvement in the accuracy of the reconstructed optical images with the new adaptive mesh generation algorithms.

The effect of discretization error on bandwidth selection for kernel density estimation

SUMMARY It is well known that the cross-validation score function of discretized data often tends to -oo as the bandwidth tends to zero. This disturbing property causes some difficulty in applying cross-validation to discretized data. Since all data are rounded to some degree, the phenomenon limits the application of cross-validation in practice. Based on characteristic functions, the cause of the difficulty is explained, and a simple modification is suggested. Under some conditions, it is shown that the bandwidth selected by the proposed procedure is asymptotically equivalent to the bandwidth selected by cross-validation of the original unrounded data. The advantages of the proposed procedure are demonstrated in two examples.

Sensitivity analysis with plate and shell finite elements

AbstractThis paper addresses the problem of calculating sensitivity data by direct methods for isoparametric plate or shell elements. Sensitivity parameters of interest include intrinsic properties such as material modulus and plate thickness, as well as geometry variables which influence the size and shape of a structure. The sensitivity calculation therefore must consider the parametric mapping within an element, as well as the influence of geometric variables on the orientation of an element in space. The methods presented specialize directly to continuum elements, in which the co‐ordinate transformation is omitted, or to simple structural members situated arbitrarily in space. Numerical examples are presented which illustrate the accuracy of the proposed techniques, and the effect of discretization error on computed sensitivities.