Showing papers on "Temporal discretization published in 2015"

A novel high order space-time spectral method for the time fractional fokker-planck equation ∗

Abstract: The fractional Fokker--Planck equation is an important physical model for simulating anomalous diffusions with external forces. Because of the nonlocal property of the fractional derivative an interesting problem is to explore high accuracy numerical methods for fractional differential equations. In this paper, a space-time spectral method is presented for the numerical solution of the time fractional Fokker--Planck initial-boundary value problem. The proposed method employs the Jacobi polynomials for the temporal discretization and Fourier-like basis functions for the spatial discretization. Due to the diagonalizable trait of the Fourier-like basis functions, this leads to a reduced representation of the inner product in the Galerkin analysis. We prove that the time fractional Fokker--Planck equation attains the same approximation order as the time fractional diffusion equation developed in [X. Li and C. Xu, SIAM J. Numer. Anal., 47 (2009), pp. 2108--2131] by using the present method. That indicates an e...

Classification-driven temporal discretization of multivariate time series

Abstract: Biomedical data, in particular electronic medical records data, include a large number of variables sampled in irregular fashion, often including both time point and time intervals, thus providing several challenges for analysis and data mining. Classification of multivariate time series data is a challenging task, but is often necessary for medical care or research. Increasingly, temporal abstraction, in which a series of raw-data time points is abstracted into a set of symbolic time intervals, is being used for classification of multivariate time series. In this paper, we introduce a novel supervised discretization method, geared towards enhancement of classification accuracy, which determines the cutoffs that will best discriminate among classes through the distribution of their states. We present a framework for classification of multivariate time series analysis, which implements three phases: (1) application of a temporal-abstraction process that transforms a series of raw time-stamped data points into a series of symbolic time intervals (based on either unsupervised or supervised temporal abstraction); (2) mining these time intervals to discover frequent temporal-interval relation patterns (TIRPs), using versions of Allen's 13 temporal relations; (3) using the patterns as features to induce a classifier. We evaluated the framework, focusing on the comparison of three versions of the new, supervised, temporal discretization for classification (TD4C) method, each relying on a different symbolic-state distribution-distance measure among outcome classes, to several commonly used unsupervised methods, on real datasets in the domains of diabetes, intensive care, and infectious hepatitis. Using only three abstract temporal relations resulted in a better classification performance than using Allen's seven relations, especially when using three symbolic states per variable. Similarly when using the horizontal support and mean duration as the TIRPs feature representation, rather than a binary (existence) representation. The classification performance when using the three versions of TD4C was superior to the performance when using the unsupervised (EWD, SAX, and KB) discretization methods.

A Two-Stage Fourth Order Time-Accurate Discretization for Lax-Wendroff Type Flow Solvers. I. Hyperbolic Conservation Laws

Abstract: In this paper we develop a novel two-stage fourth order time-accurate discretization for time-dependent flow problems, particularly for hyperbolic conservation laws. Different from the classical Runge-Kutta (R-K) temporal discretization for first order Riemann solvers as building blocks, the current approach is solely associated with Lax-Wendroff (L-W) type schemes as the building blocks. As a result, a two-stage procedure can be constructed to achieve a fourth order temporal accuracy, rather than using well-developed four stages for R-K methods. The generalized Riemann problem (GRP) solver is taken as a representative of L-W type schemes for the construction of a two-stage fourth order scheme.

Rolling contact of a rigid sphere/sliding of a spherical indenter upon a viscoelastic half-space containing an ellipsoidal inhomogeneity

Abstract: In this paper, the frictionless rolling contact problem between a rigid sphere and a viscoelastic half-space containing one elastic inhomogeneity is solved. The problem is equivalent to the frictionless sliding of a spherical tip over a viscoelastic body. The inhomogeneity may be of spherical or ellipsoidal shape, the later being of any orientation relatively to the contact surface. The model presented here is three dimensional and based on semi-analytical methods. In order to take into account the viscoelastic aspect of the problem, contact equations are discretized in the spatial and temporal dimensions. The frictionless rolling of the sphere, assumed rigid here for the sake of simplicity, is taken into account by translating the subsurface viscoelastic fields related to the contact problem. Eshelby's formalism is applied at each step of the temporal discretization to account for the effect of the inhomogeneity on the contact pressure distribution, subsurface stresses, rolling friction and the resulting torque. A Conjugate Gradient Method and the Fast Fourier Transforms are used to reduce the computation cost. The model is validated by a finite element model of a rigid sphere rolling upon a homogeneous vciscoelastic half-space, as well as through comparison with reference solutions from the literature. A parametric analysis of the effect of elastic properties and geometrical features of the inhomogeneity is performed. Transient and steady-state solutions are obtained. Numerical results about the contact pressure distribution, the deformed surface geometry, the apparent friction coefficient as well as subsurface stresses are presented, with or without heterogeneous inclusion.

Discrete variational Lie group formulation of geometrically exact beam dynamics

Abstract: The goal of this paper is to derive a structure preserving integrator for geometrically exact beam dynamics, by using a Lie group variational integrator. Both spatial and temporal discretization are implemented in a geometry preserving manner. The resulting scheme preserves both the discrete momentum maps and symplectic structures, and exhibits almost-perfect energy conservation. Comparisons with existing numerical schemes are provided and the convergence behavior is analyzed numerically.

On Time Discretizations of Fluid-Structure Interactions

Abstract: In this contribution, time discretizations of fluid-structure interactions are considered. We explore two specific complexities: first, the stiffness of the coupled system including different scales of the Navier-Stokes equations of parabolic type and the structure equation of hyperbolic type and second, the problem of moving domains that is inherent to fluid-structure interactions.

A Jacobian-free Newton–Krylov method for thermalhydraulics simulations

Abstract: Summary The current paper is focused on investigating a Jacobian-free Newton–Krylov (JFNK) method to obtain a fully implicit solution for two-phase flows. In the JFNK formulation, the Jacobian matrix is not directly evaluated, potentially leading to major computational savings compared with a simple Newton's solver. The objectives of the present paper are as follows: (i) application of the JFNK method to two-fluid models; (ii) investigation of the advantages and disadvantages of the fully implicit JFNK method compared with commonly used explicit formulations and implicit Newton–Krylov calculations using the determination of the Jacobian matrix; and (iii) comparison of the numerical predictions with those obtained by the Canadian Algorithm for Thermaulhydraulics Network Analysis 4. Two well-known benchmarks are considered, the water faucet and the oscillating manometer. An isentropic two-fluid model is selected. Time discretization is performed using a backward Euler scheme. A Crank–Nicolson scheme is also implemented to check the effect of temporal discretization on the predictions. Advection Upstream Splitting Method+ is applied to the convective fluxes. The source terms are discretized using a central differencing scheme. One explicit and two implicit formulations, one with Newton's solver with the Jacobian matrix and one with JFNK, are implemented. A detailed grid and model parameter sensitivity analysis is performed. For both cases, the JFNK predictions are in good agreement with the analytical solutions and explicit profiles. Further, stable results can be achieved using high CFL numbers up to 200 with a suitable choice of JFNK parameters. The computational time is significantly reduced by JFNK compared with the calculations requiring the determination of the Jacobian matrix. Copyright © 2015 John Wiley & Sons, Ltd.

Transient dynamic interface crack analysis in magnetoelectroelastic bi-materials by a time-domain BEM

Abstract: This paper aims to gain a deeper insight into the effects of the poling directions, the volume fractions and the combined loading parameters on the dynamic field intensity factors at the interfacial crack-tips in two dimensional, piecewise homogeneous and magnetoelectroelastic composite bi-materials under the combined magnetoelectromechanical impact loadings. For this purpose, a time-domain boundary element method (BEM) together with the sub-domain technique is applied. The present time-domain BEM uses a quadrature formula for the temporal discretization to approximate the convolution integrals and a collocation method for the spatial discretization. Quadratic quarter-point elements are implemented at the interfacial crack-tips. An explicit displacement extrapolating formula is proposed to determine the dynamic field intensity factors of stresses, electrical displacements and magnetic inductions at the interfacial crack-tips. Numerical examples for different poling directions, different volume fractions and different combined loading parameters are presented and discussed. Based on these numerical results, some useful conclusions are drawn on the effects of various material and loading combinations on the dynamic field intensity factors.

Lane formation by side-stepping

Abstract: In this paper we study a system of nonlinear partial differential equations, which describes the evolution of two pedestrian groups moving in opposite direction. The pedestrian dynamics are driven by aversion and cohesion, i.e. the tendency to follow individuals from the own group and step aside in the case of contraflow. We start with a 2D lattice based approach, in which the transition rates reflect the described dynamics, and derive the corresponding PDE system by formally passing to the limit in the spatial and temporal discretization. We discuss the existence of special stationary solutions, which correspond to the formation of directional lanes and prove existence of global in time bounded weak solutions. The proof is based on an approximation argument and entropy inequalities. Furthermore we illustrate the behavior of the system with numerical simulations.

Reduced Numerical Model for Methane Hydrate Formation under Conditions of Variable Salinity. Time-Stepping Variants and Sensitivity

Abstract: In this paper, we consider a reduced computational model of methane hydrate formation in variable salinity conditions, and give details on the discretization and phase equilibria implementation. We describe three time-stepping variants: Implicit, Semi-implicit, and Sequential, and we compare the accuracy and efficiency of these variants depending on the spatial and temporal discretization parameters. We also study the sensitivity of the model to the simulation parameters and in particular to the reduced phase equilibria model.

An Iteration Free Backward Semi-Lagrangian Scheme for Guiding Center Problems

Abstract: In this paper, we develop an iteration free backward semi-Lagrangian method for nonlinear guiding center models. We apply the fourth-order central difference scheme for the Poisson equation and employ the local cubic interpolation for the spatial discretization. A key problem in the time discretization is to find the characteristic curve arriving at each grid point which is the solution of a system of highly nonlinear ODEs with a self-consistency imposed by the Poisson equation. The proposed method is based on the error correction method recently developed by the authors. For the error correction method, we introduce a modified Euler's polygon and solve the induced asymptotically linear differential equation with the midpoint quadrature rule to get the error correction term. We prove that the proposed iteration free method has convergence order at least $3$ in space and $2$ in time in the sense of the $L_{2}$-norm. In particular, it is shown that the proposed method has a good performance in computational...

Efficient matrix-free implementation of discontinuous Galerkin methods for compressible flow problems

Abstract: We discuss the matrix-free implementation of Discontinuous Galerkin methods for compressible flow problems, i.e. the compressible Navier-Stokes equations. For the spatial discretization the CDG2 method and for temporal discretization an explicit Runge-Kutta method is used. For the presented matrix-free approach we discuss asynchronous communication, shared memory parallelization, and automated code generation to increase the oating point performance of the code.

Principles of Solution of the Governing Equations

Abstract: We briefly describe the most important methodologies for the solution of the governing equations and provide a large number of bibliographical references for each. The discussion starts with the presentation of structured and unstructured grids employed by the principal spatial discretization approaches, namely finite-differences, finite-elements, and finite-volumes. We also briefly discuss discretization schemes such as spectral elements, lattice Boltzmann, and gridless methods. We discuss central and upwind schemes including their respective pros and cons. We further present the various possibilities for temporal discretization of the governing equations by either explicit or implicit schemes. We proceed with a discussion of the different approaches employed for turbulence modeling. We conclude with a few notes regarding the application of initial and boundary conditions.

Nonlinear model order reduction of Burgers' Equation using proper orthogonal decomposition

Abstract: In this paper, we examine a model order reduction approach for dynamic systems governed by Burgers' equation with Neumann boundary conditions. The proper orthogonal decomposition (POD) method is employed here that provides a reliable and accurate modeling approach, while the temporal discretization of the continuous error function leads to a more accurate estimation of the defined cost function. We will investigate the accuracy of the reduced-order model compared to the finite element (FE) model by choosing an adequate number of basis functions for the approximating subspace. The derived lumped-parameter model for Burgers' equation is then described by a nonlinear state-space model. We finally demonstrate the accuracy of the reduced-order model through a numerical example, where we show that a 7-dimensional POD can accurately estimate the system output.

Real-Time Aero-elasticity Simulation of Open Rotors With Slender Blades for the Multidisciplinary Design of Rotorcraft

Abstract: This paper elaborates on the theoretical development of a mathematical approach, targeting the real-time simulation of aero-elasticity for open rotors with slender blades, as employed in the majority of rotorcraft. A Lagrangian approach is formulated for the rapid estimation of natural vibration characteristics of rotor blades with nonuniform structural properties. Modal characteristics obtained from classical vibration analysis methods are utilized as assumed deformation functions. Closed form integral expressions are incorporated, describing the generalized centrifugal forces and moments acting on the blade. The treatment of three-dimensional elastic blade kinematics in the timedomain is thoroughly discussed. In order to ensure robustness and establish applicability in real time, a novel, second-order accurate, finite-difference scheme is utilized for the temporal discretization of elastic blade motion. The developed mathematical approach is coupled with a finite-state induced flow model, an unsteady blade element aerodynamics model, and a dynamic wake distortion model. The combined formulation is implemented in an existing helicopter flight mechanics code. The aero-elastic behavior of a full-scale hingeless helicopter rotor has been investigated. Results are presented in terms of rotor blade resonant frequencies, rotor trim performance, oscillatory structural blade loads, and transient rotor response to control inputs. Extensive comparisons are carried out with wind tunnel (WT) and flight test (FT) measurements found in the open literature as well as with nonreal-time comprehensive analysis methods. It is shown that the proposed approach exhibits good agreement with measured data regarding trim performance and transient rotor response characteristics. Accurate estimation of structural blade loads is demonstrated, in terms of both amplitude and phase, up to the third harmonic component of oscillatory loading. It is shown that the developed model can be utilized for real-time simulation on a modern personal computer. The proposed methodology essentially constitutes an enabling technology for the multidisciplinary design of rotorcraft, when a compromise between simulation fidelity and computational efficiency has to be sought for in the process of model development. [DOI: 10.1115/1.4028180]