Finite Difference Formulation Research Articles

A comparison of higher-order finite-difference shock capturing schemes

The efficiency of computational fluid dynamics simulations can be greatly enhanced by employing higher-order accurate numerical schemes which provide superior accuracy for a given cost. For unsteady turbulent flow simulations involving shocks, contacts, and/or material discontinuities, various higher-order shock capturing schemes are available in the literature. The desired numerical scheme should be free of spurious numerical oscillations across discontinuities and it should obtain higher-order accuracy in smooth flow regions in an efficient manner. Sufficient robustness is necessary for effectively utilizing these numerical methods in engineering and science applications. Three classes of higher-order shock capturing schemes are compared in this paper: (1) central finite-difference schemes with explicit artificial dissipation, (2) a compact centered finite-difference scheme with localized artificial diffusivity and (3) weighted essentially non-oscillatory schemes in both explicit and compact finite difference forms. Multiple variations of these methods were implemented and tested using a block-structured Cartesian mesh solver. The current paper assesses shock capturing capabilities as well as effects on the accuracy in smooth flow regions using a variety of test cases that range from canonical shock problems to homogeneous isotropic turbulence at a turbulent Mach number of 0.5 where shocklets are formed. Finally, a computational cost breakdown for each scheme is provided and the overall computational efficiency of the different schemes are compared to each other.

A New Singularity Treatment Approach for Journal-Bearing Mixed Lubrication Modeled by the Finite Difference Method With a Herringbone Mesh

Steady-state mixed hydrodynamic lubrication of rigid journal bearing is investigated by using a finite difference form of the Patir–Cheng average Reynolds equation under the Reynolds boundary condition. Two sets of discretization meshes, i.e., the rectangular and nonorthogonal herringbone meshes, are considered. A virtual-mesh approach is suggested to resolve the problem due to the singularities of pressure derivatives at the turning point of the herringbone mesh. The effectiveness of the new approach is examined by comparing the predicted load with that found in the literature for a smooth-surface case solved in the conventional rectangular mesh. The effects of the skewness angles of symmetric and asymmetric herringbone meshes on the predicted parameters, such as load, friction coefficient, attitude angle, and maximum pressure, are investigated for smooth, rough, and herringbone-grooved bearing surfaces. It is found that the new approach helps to improve the computational accuracy significantly, as demonstrated by comparing the results with and without the treatment of the pressure derivative discontinuity although the latter costs slightly less computational time.

On the failure of a discrete axial chain using a continualized nonlocal Continuum Damage Mechanics approach

SummaryThe failure of a discrete elastic‐damage axial system is investigated using both a discrete and an equivalent continuum approach. The Discrete Damage Mechanics approach is based on a microstructured model composed of a series of periodic elastic‐damage springs (axial Discrete Damage Mechanics lattice system). Such a discrete damage system can be associated with the finite difference formulation of a Continuum Damage Mechanics evolution problem. Several analytical and numerical results are presented for the tensile failure of this axial damage chain under its own weight.The nonlocal Continuum Damage Mechanics models examined in this paper are mainly built from a continualization procedure applied to centered or uncentered finite difference schemes. The asymptotic expansion of the first‐order upward difference equations leads to a first‐order nonlocal model, whereas the asymptotic expansion of the centered finite difference equations leads to a second‐order nonlocal Eringen's approach. To complete this study, a phenomenological nonlocal gradient approach is also examined and compared with the first continualization methods.A comparison of the discrete and the continuous problems for the chains shows the effectiveness of the new micromechanics‐based nonlocal Continuum Damage modeling, especially for capturing scale effects. For both continualized approaches, the length scale of the nonlocal models depends only on the cell size, while for the so‐called phenomenological approach, the length scale may depend on the loading parameter. This apparent load‐dependent length scale, already discussed in the literature with numerical arguments, is found to be sensitive to the postulated structure of the nonlocal model calibrated according to a lattice approach. Copyright © 2015 John Wiley & Sons, Ltd.

An efficient method for seismic analysis of structures

Purpose – The purpose of this paper is to present an efficient method for dynamic analysis of structures utilizing a modal analysis with the main purpose of decreasing the computational complexity of the problem. In traditional methods, the solution of initial-value problems (IVPs) using numerical methods like finite difference method leads to step by step and time-consuming recursive solutions. Design/methodology/approach – The present method is based on converting the IVP into boundary-value problems (BVPs) and utilizing the features of the latter problems in efficient solution of the former ones. Finite difference formulation of BVPs leads to matrices with repetitive tri-diagonal and block tri-diagonal patterns wherein the eigensolution and matrix inversion are obtained using graph products rules. To get advantage of these efficient solutions for IVPs like the dynamic analysis of single DOF systems, IVPs are converted to boundary-value ones using mathematical manipulations. The obtained formulation is then generalized to the multi DOF systems by utilizing modal analysis. Findings – Applying the method to the modal analysis leads to a simple and efficient formulation. The laborious matrix inversion and eigensolution operations, of computational complexities of O(n2.373) and O(n3), respectively, are converted to a closed-form formulation with summation operations. Research limitations/implications – No limitation. Practical implications – Swift analysis has become possible. Originality/value – Suitability of solving IVPs and modal analysis using conversion and graph product rules is presented and applied to efficient seismic optimal analysis and preliminary design.

Nonlocal Continuum Damage Mechanics Approach of a Discrete Axial Chain under Non-Uniform Axial Load

The failure of a discrete elastic-damage axial system is investigated using both a discrete and anequivalent continuum approach. The discrete damage mechanics (DDM) approach is based on amicrostructured model composed of a series of periodic elastic-damage springs (axial DDM latticesystem). Such a damage discrete system can be associated with the finite difference formulation of aContinuum Damage Mechanics (CDM) evolution problem.The nonlocal CDM models considered in this paper are mainly built from a continualizationprocedure applied to centered finite difference schemes. A comparison of the discrete and thecontinuous problems for the chains shows the effectiveness of the new micromechanics-basednonlocal Continuum Damage modeling, especially for capturing scale effects.

A nonlinear pyrolysis layer model for analyzing thermal behavior of charring ablator

Understanding the pyrolysis phenomena experienced by charring ablators used in thermal protection systems for manned reentry vehicles is crucial for their design. A one-dimension nonlinear pyrolysis layer model without surface recession has been developed to explore the thermal behavior of charring ablator when subjected to an aerodynamic hyper-thermal environment. The charring ablator in this model consists of three distinct zones: char, pyrolysis and virgin material. The heat and mass transfer, the two moving interfaces and the temperature-dependent thermal properties in charring materials undergoing pyrolysis are considered in the formulation of the model. The governing differential equations are derived, and its implicit finite difference formulations are programmed in MATLAB. Examples are given to demonstrate the effectiveness and accuracy of this model. The thermal response of charring material with antioxidants is also predicted under actual service conditions.

Discrete and non-local elastica

In this paper, the buckling and post-buckling behavior of an elastic lattice system referred to as the discrete elastica problem is investigated using an equivalent non-local continuum approach. The geometrically exact post-buckling analysis of the elastic chain, also called Hencky system, is first numerically solved using the shooting method. This discrete physical model is also mathematically equivalent to a finite difference formulation of the continuum elastica. Starting from the exact difference equations of the discrete problem, a continualization method is applied for approximating the difference operators by differential ones, in order to better characterize the discrete system by an enriched continuous one. It is shown that the new continuum associated with the discrete system exactly fits the discrete elastica post-buckling problem, where the non-locality is of Eringen׳s type (also called stress gradient non-local model). An asymptotic expansion is performed for both the discrete and the non-local continuum models, in order to approximate the post-buckling branches of the discrete system. Some numerical investigations show the efficiency of the non-local approach, especially for capturing the scale effects inherent to the cell size of the lattice model.

Global instabilities and transient growth in Blasius boundary-layer flow over a compliant panel

We develop a hybrid of computational and theoretical approaches suited to study the fluid–structure interaction (FSI) of a compliant panel, flush between rigid upstream and downstream wall sections, with a Blasius boundary-layer flow. The ensuing linear-stability analysis is focused upon global instability and transient growth of disturbances. The flow solution is developed using a combination of vortex and source boundary-element sheets on a computational grid while the dynamics of a plate-spring compliant wall are couched in finite-difference form. The fully coupled FSI system is then written as an eigenvalue problem and the eigenvalues of the various flow- and wall-based instabilities are analysed. It is shown that coalescence or resonance of a structural eigenmode with either a flow-based Tollmien–Schlichting Wave (TSW) or wall-based travelling-wave flutter (TWF) modes can occur. This can render the nature of these well-known convective instabilities to become global for a finite compliant wall giving temporal growth of system disturbances. Finally, a non-modal analysis based on the linear superposition of the extracted temporal modes is presented. This reveals a high level of transient growth when the flow interacts with a compliant panel that has structural properties which render the FSI system prone to global instability. Thus, to design stable finite compliant panels for applications such as boundary-layer transition postponement, both global instabilities and transient growth must be taken into account.

Coupled finite element-finite difference formulation for long-term analysis of timber–concrete composite structures

This paper focuses on development of a simple frame finite element model for nonlinear analysis of timber–concrete composite (TCC) beams under long-term loads. The element formulation within the context of the force interpolation and modified fibre element is derived in which the partial interaction between the layers can be captured by discrete nonlinear springs at ends of the element. Further, a finite difference scheme is also developed to analyse variation of diffusion within the timber section with respect to relative humidity changes. Different long-term effects of timber, concrete and connection are taken into account and their formulations within the total secant approach and efficient step-by-step analysis are derived. The implementation of the proposed model in a 1D FE model is discussed and the long-term behaviour of TCC components and beams is verified with available test results from literature.

Multiple time scales and pressure forcing in discontinuous Galerkin approximations to layered ocean models

This paper addresses some issues involving the application of discontinuous Galerkin (DG) methods to ocean circulation models having a generalized vertical coordinate. These issues include the following. (1) Determine the pressure forcing at cell edges, where the dependent variables can be discontinuous. In principle, this could be accomplished by solving a Riemann problem for the full system, but some ideas related to barotropic–baroclinic time splitting can be used to reduce the Riemann problem to a much simpler system of lower dimension. Such splittings were originally developed in order to address the multiple time scales that are present in the system. (2) Adapt the general idea of barotropic–baroclinic splitting to a DG implementation. A significant step is enforcing consistency between the numerical solution of the layer equations and the numerical solution of the vertically-integrated barotropic equations. The method used here has the effect of introducing a type of time filtering into the forcing for the layer equations, which are solved with a long time step. (3) Test these ideas in a model problem involving geostrophic adjustment in a multi-layer fluid. In certain situations, the DG formulation can give significantly better results than those obtained with a standard finite difference formulation.

Numerical study of laminar flow in a sudden expansion obstacled channel

In the present work, a numerical study has been conducted to investigate the flow heat transfer through an obstacled sudden expansion channel. Rectangular adiabatic obstacles mounted behind the expansion region on the upper and lower wall of the channel used. The effects of obstacles length, obstacles thickness and number of obstacles on flow and thermal fields for different Reynolds number and expansion ratio examined. Three values of expansion ratio (ER) equal to 1.5, 1.75 and 2 were used. The choice of values of Reynolds number takes in consideration the symmetry state. The governing equations of continuity, momentum and energy discretized by using the finite difference formulation and the resulting algebraic equations solved by using Gauss-Seidle iteration method. The obtained results show that the obstacles have a considerable effect on dynamics of the flow and enhancement of heat transfer. In addition, it is found that the heat transfer is enhanced more as the obstacles thickness increases and this trend is decreased as the obstacles length increases.

From discrete to nonlocal continuum damage mechanics: Analysis of a lattice system in bending using a continualized approach

It is shown herein that the bending problem of a discrete damage system, also called microstructured damage system or lattice damage system, can be rigorously handled by a nonlocal continuum damage mechanics approach. It has been already shown that Eringen’s nonlocal elasticity was able to capture the scale effects induced by the discreteness of a microstructured system. This paper generalizes such results for inelastic materials and first presents some results for engineering problems modelled within continuum damage mechanics. The microstructured model is composed of rigid periodic elements connected by rotational elastic damage springs (discrete damage mechanics). Such a discrete damage system can be associated with the finite difference formulation of a continuum damage mechanics problem, i.e. the Euler–Bernoulli damage beam problem. Starting from the discrete equations of this structural problem, a continualization method leads to the formulation of an Eringen’s type nonlocal model with full coupling between nonlocal elasticity and nonlocal continuum damage mechanics. Indeed, the nonlocality appears in this continualized approach both in the constitutive law and in the damage loading function. A comparison of the discrete and the continuous problems for the cantilever shows the efficiency of the new micromechanics-based nonlocal continuum damage modelling for capturing scale effects. The length scale of the nonlocal continuum damage mechanics model is rigorously calibrated from the size of the cell of the discrete repetitive damage system. The new micromechanics-based nonlocal damage mechanics model is also analysed with respect to available nonlocal damage mechanics models.

Multilayered Discrete Green's Functions Based on Mixed-Potential Finite-Difference Formulation

An approach to calculating discrete Green’s functions (DGFs) in multilayered media is proposed based on the finite-difference scheme of time-domainmixed-potential equations. The calculated DGFs have very high accuracy in comparison to the direct FDTD solution. Furthermore, the steady-state values of the DGFs are properly estimated in terms of the scalar potential, ensuring stability out of the truncating window. These DGFs are applicable to the analysis of antennas with multilayered planar structures. In such analysis, the proposed method has the advantage of the method of moments in that the computations are performed only on the antenna, regardless of the white space around it. Moreover, the broadband frequency characteristic of the antenna is achieved with a single simulation run, such as in the FDTDmethod. The theoretical results have been verified by the experimental results on a wideband coplanar waveguide (CPW)-fed monopole antenna.

Flat-Plate Solar Collector in Transient Operation: Modeling and Measurements

This paper describes a mathematical model for simulating the transient processes which occur in liquid flat-plate solar collectors. A discrete nodal model that represents the flat-plate solar collector's layers and the storage tank is employed. The model is based on solving a system of coupled differential equations which describe the energy conservation for the glass cover, air gap, absorber, fluid, insulation, and the storage tank. Inputs to the model include the time-varying liquid flow rate, incident solar radiation, and the ambient air temperature, as well as the volume of liquid in the storage tank and initial temperature of the system. The system of differential equations is solved iteratively using an implicit, finite-difference formulation executed with Matlab software. In order to verify the proposed method, an experiment was designed and conducted on different days with variable ambient conditions and flow rates. The comparison between the computed and measured results of the transient fluid temperature at the collector outlet shows good agreement. The proposed method is extremely general and flexible accounting for variable ambient conditions and flow rates and allowing for a geometrical and thermophysical description of all major components of the solar collector system, including the storage tank. The validated, general model is suitable to investigate the effectiveness of various components without the necessity of carrying out experimental work, and the flexible computational scheme is useful for transient simulations of energy systems.

Improved stiffness confinement method within the coarse mesh finite difference framework for efficient spatial kinetics calculation

An improved Stiffness Confinement Method (SCM) is formulated within the framework of the coarse mesh finite difference (CMFD) formulation for efficient multigroup spatial kinetics calculation. The algorithm for searching for the amplitude frequency that makes the dynamic eigenvalue unity is developed in a systematic way along with the methods for determining the shape and precursor frequencies. A nodal calculation scheme is established within the CMFD framework to incorporate the cross section changes due to thermal feedback and dynamic frequency update. The conditional nodal update scheme is employed such that the transient calculation is performed mostly with the CMFD formulation and the CMFD parameters are conditionally updated by intermittent nodal calculations. A quadratic representation of amplitude frequency is introduced as another improvement. The performance of the improved SCM within the CMFD framework is assessed by comparing the solution accuracy and computing times for the NEACRP control rod ejection benchmark problems with those obtained with the Crank–Nicholson method with exponential transform (CNET). It is demonstrated that the improved SCM is beneficial for large time step size calculations with stability and accuracy enhancement.

Finite difference computational modeling of marine impact pile driving

Computational models based on the finite difference (FD) method can be successfully used to predict underwater pressure waves generated by marine impact pile driving. FD-based models typically discretize the equations of motion for a cylindrical shell to model the vibrations of a submerged pile in the time-domain. However, because the dynamics of a driven pile are complex, realistic models must also incorporate physics of the driving hammer and surrounding acousto-elastic media into the FD formulation. This paper discusses several of the different physical phenomena involved, and shows some approaches to simulating them using the FD method. Topics include dynamics of the hammer and its coupling to the pile head, transmission of axial pile vibrations into the soil, energy dissipation at the pile wall due to friction, acousto-elastic coupling to the surrounding media, and near-field versus far-field propagation modeling. Furthermore, this paper considers the physical parameters required for predictive modeling of pile driving noise in conjunction with some practical considerations about how to determine these parameters for real-world scenarios.

On Some Aspects of the Complex–Envelope Finite–Differences Simulation of Wave Propagation in One–Dimensional Case

Abstract Some aspects of the numerical modeling of the electromagnetic waves propagation using the “complex-envelope” finitedifferences formulation in the one-dimensional case are here reviewed and discussed in comparison with the standard finitedifferences in time-domain (FDTD) approach. The main focus is put on the stability and the numerical dispersion issues of the “complex envelope” explicit and implicit methods

Finite Difference Hermite WENO Schemes for Hyperbolic Conservation Laws

In this paper, a class of weighted essentially non-oscillatory (WENO) schemes based on Hermite polynomials, termed HWENO (Hermite WENO) schemes, for solving one and two dimensional nonlinear hyperbolic conservation law systems is presented. The construction of HWENO schemes is based on a finite difference formulation, Hermite interpolation, and nonlinearly stable Runge---Kutta methods. The idea of the reconstruction in the HWENO schemes comes from the original WENO schemes, however both the function and its first derivative values are evolved in time and used in the reconstruction, while only the function values are evolved and used in the original WENO schemes. Comparing with the original finite difference WENO schemes of Jiang and Shu (J Comput Phys 126:202---228, 1996), one major advantage of HWENO schemes is its compactness in the reconstruction. For example, five points are needed in the stencil for a fifth order WENO (WENO5) reconstruction, while only three points are needed for a fifth order HWENO (HWENO5) reconstruction. Some benchmark numerical experiments are presented to illustrate efficiency of HWENO schemes.

On lateral-torsional buckling of discrete elastic systems: A nonlocal approach

The lateral-torsional buckling of a discrete repetitive elastic beam-like structure, composed of rigid links connected by bending and torsional elastic springs, is investigated herein using discrete and continuum approaches. It is shown that the governing equations of the microstructured model are equivalent to a finite difference formulation of a continuous lateral-torsional buckling problem. The discrete equations are introduced through variational arguments and solved exactly for the hinged-hinged boundary conditions using a finite difference approach. A nonlocal equivalent continuum is sought via a continualization method. It is shown that the equivalent continuum is an Eringen's based elastic nonlocal continuum, which perfectly fits the exact discrete problem. The length scale effect related to the size of the repetitive cell tends to soften the lateral-torsional buckling limit of the asymptotically local continuum. Prandtl's lateral-torsional buckling solution is a particular case, associated with an infinite number of cells. Warping can also be included into the discrete thin-walled structural model, leading to a microstructured-based nonlocal thin-walled model. Love's equations augmented by the so-called Vlasov effect (warping bimoment) are used to compare the nonlocal approach with the one arising from the continualization of the discrete equation. It is shown that, in this last case, the length scale effect may depend on the warping stiffness.

Graph-Theoretic Sensitivity Analysis of Multibody Systems

A graph-theoretic formulation to perform sensitivity analysis on multibody systems is presented in this article. In this formulation, linear graphs are used to capture the system topologies from which a graph-theoretic formulation simultaneously generates the system equations and the sensitivity equations. This ensures the automated, accurate, and efficient generation of sensitivity equations. The basic formulation steps are outlined to illustrate the process of the generation of sensitivity equations. The salient aspects of multibody systems are presented along with a brief description of the software platform that has been used to implement the algorithm. A 3D pendulum and a double-wishbone suspension system are analyzed to demonstrate the application of the algorithm. The results are validated by using a finite difference formulation. Finally, the efficiency of the software implementation is assessed by comparing the computational costs associated with the proposed method and that of existing methods.