Backward Difference Scheme Research Articles

Nonlinear Vibration of PZT4/PZT-5H Monomorph and Bimorph Beams with Graded Microstructures

Monomorph and bimorph actuators made of piezoelectric materials are widely used in smart materials and structural control. Recent studies show that their performances can be remarkably improved by the use of functionally graded piezoelectric materials (FGPMs) whose compositional profile and effective material properties are graded along the thickness direction. This paper investigates the linear and nonlinear vibration behaviors of monomorph and bimorph made from a mixture of PZT4 and PZT-5H with material composition following a power law distribution. Theoretical formulations are based on von Kármán kinematic relationships and Timoshenko beam theory to account for the effect of transverse shear deformation. The differential quadrature method (DQM) and the second-order backward difference scheme are employed to obtain the linear and nonlinear vibration frequencies. A parametric study is conducted to investigate the influences of volume fraction index, slenderness ratio, nonlinear deformation and different loading conditions.

Numerical bifurcation of a delayed diffusive food‐limited model with Dirichlet boundary condition

This paper studies the Neimark–Sacker bifurcation of a diffusive food‐limited model with a finite delay and Dirichlet boundary condition by the backward Euler difference scheme, Crank‐Nicolson difference scheme, and nonstandard finite‐difference scheme. The existence of Neimark‐Sacker bifurcation at the equilibrium is obtained. Our results show that Crank‐Nicolson and nonstandard finite‐difference schemes are superior to the backward Euler difference scheme under the means of describing approximately the dynamics of the original system. Finally, numerical examples are provided to illustrate the analytical results. Copyright © 2015 John Wiley & Sons, Ltd.

A second-order BDF compact difference scheme for fractional-order Volterra equation

In this paper, a second-order backward differentiation formula compact difference scheme with the truncation error of order for time and 4 for space to fractional-order Volterra equation is considered. The integral term is treated by means of the second-order convolution quadrature suggested by Lubich and fourth-order accuracy compact approximation is applied for the second-order space derivative. The stability and convergence of the compact difference scheme in a new norm are proved by the energy method. Numerical experiments that are in total agreement with our analysis are reported.

A Mathematical Model for Hyperbolic Space-fractional Bioheat Transfer during Thermal Therapy

In this paper, we have developed a hyperbolic space-fractional bioheat transfer model based on the single-phase-lagging constitutive relation. The numerical solution of the present problem has been done using fractional backward finite difference scheme and Legendre wavelet Galerkin approach. The effect of fractional parabolic and hyperbolic bioheat transfer model on temperature profile within living biological tissues has been studied and compared with their respective standard cases. Numerical results are presented graphically in both standard and anomalous case for different values of parameters.

A boundary element formulation for the heat equation with dissipative and heat generation terms

This article presents a formulation of the Boundary Element Method (BEM) for the study of heat diffusion in isotropic and homogeneous media. The proposed formulation has a time independent fundamental solution obtained from the two-dimensional Laplace equation. Consequently, the formulation is called D-BEM since it has domain integrals in the basic integral equation. The first order time derivative that appears in the integral equations is approximated by a backward finite difference scheme. Internal dissipative and heat generation terms are considered in the analyses. The results from the numerical model are compared with the available analytical solutions. The correlation estimator R2 is employed to validate the numerical model and to demonstrate the accuracy of the proposed formulation.

Finite difference method for analyzing band structure in semiconductor heterostructures without spurious solutions

To stably employ multiband k·p model for analyzing the band structure in semiconductor heterostructures without spurious solutions (SSs), the Hermitian forward and backward difference (HFBD) scheme for finite difference method (FDM) is presented. The HFBD is the discretization scheme that eliminates the difference instability and employs the Burt-Foreman Hermitian operator ordering without geometric asymmetry. The difference instability arises from employing Foreman's strategy (FS). FS removes SSs caused by unphysical bowing in bulk dispersion curve meanwhile the HFBD is the only difference scheme that can accurately adapt for it. In comparison with other recent strategies, the proposed method in this paper is as accurate and reliable as FS, along with preserving the rapidness and simplicity of FDM. This difference scheme shows stable convergence without any SSs under variable grid size. Therefore, a wide range of experiment-determined band parameters can be applied to large-scale stable simulation with this method regardless of the SSs they originally generate.

Backward Difference Scheme for Simulating Unsteady Compressible Flow on Deforming Mesh in an Implicit Adaptive Solver

The paper describes the derivation of the numerical formulation of a second order time accurate and Geometrically Conservative Backward Difference Scheme (BDF) for transient flow simulation of Arbitrary Lagrangian Eulerian (ALE) problems using the control volume approach. The required modification to implement the scheme in an implicit adaptive flow solver is explained. The accuracy and robustness of the current formulation is demonstrated by simulating unsteady flow field over a sinusoidally pitching NACA0012 airfoil with larger allowable timestep in comparison to an existing Mid-point scheme.

A FEM approach to biomagnetic fluid flow in multiple stenosed channels

The unsteady and laminar biomagnetic fluid flow of a viscous, incompressible, Newtonian and electrically conducting fluid is numerically investigated. Specifically, the two-dimensional flow driven in an infinite channel containing multiple stenosis and subject to a spatially varying external magnetic field is considered. The numerical method is based on the use of finite element method (FEM) in spatial discretization and unconditionally stable backward finite difference scheme for the time integration, for solving coupled nonlinear differential equations in terms of stream function, vorticity and temperature. Due to nonlinearity, an iterative process is employed between the equations. The study focuses on the alteration of the flow and its temperature behaviors due to the irregular constrictions along the channel. Also, the effects of the external magnetic field, its location and its intensity on the biomagnetic fluid flow are analyzed. The numerical results are presented in terms of streamlines, equivorticity and temperature contours for several values of magnetic field intensity in irregularly and multiply stenosed channels. It is shown that the lengths of the vortices appeared due to the stenosis and magnetic source, increase with an increase in the intensity of the magnetic field and the severity of the constriction.

Shape memory polymers: three-dimensional isotropic modeling

This paper presents a comprehensive three-dimensional isotropic numerical simulation for a thermo-mechanical constitutive model of shape memory polymers (SMPs). In order to predict the thermo-mechanical behavior of SMPs, a one-dimensional rheological thermo-mechanical constitutive model is adopted, translated into a three-dimensional form and a time discrete form of the three-dimensional model is then presented. Numerical simulation of this model was developed using the UMAT subroutine capabilities of the finite element software ABAQUS. Evolution of the analysis was conducted by making use of the backward difference scheme, which was applied to all quantities within the model, including the material properties. A comparison of the numerical simulation results was carried out with the available experimental data. Numerical simulation results clearly exhibit the thermo-mechanical properties of the material which include shape fixity, shape recovery, and recovery stress. Finally, a prediction for the transverse and shear directions of the material is presented.

Internal and subspace correction approximations of implicit variational inequalities

The aim of this paper is to study the existence of solutions and some approximations for a class of implicit evolution variational inequalities that represents a generalization of several quasistatic contact problems in elasticity. Using appropriate estimates for the incremental solutions, the existence of a continuous solution and convergence results are proved for some corresponding internal approximation and backward difference scheme. To solve the fully discrete problems, general additive subspace correction algorithms are considered, for which global convergence is proved and some error estimates are established.

Microstructural Alterations in Bearing Steels under Rolling Contact Fatigue: Part 2—Diffusion-Based Modeling Approach

The microstructure of rolling element bearings can experience significant transformation when subjected to repetitive contact cycling. In Part 1 of this article, a detailed overview of the known microstructure alteration phenomena was presented and the different mechanisms for describing them proposed by various investigators were critically reviewed. It is agreed generally that the primary path of these structural changes is the decay of martensite due to carbon diffusion, leading to the formation of ferrite and lenticular carbides. Furthermore, these altered regions in the material microstructure can be stress concentration regions leading to crack initiation and propagation. In this article, a J2-based elastic–plastic Voronoi finite element model (previously developed by the authors) is coupled with a carbon diffusion–based model. Using Fick's law for stress-assisted diffusion, the dispersion of carbon in the bearing microstructure is evaluated. A backward Euler finite difference scheme is employed to solve the partial differential equations. The model can accurately predict the onset of martensitic decay and formation of the white etching bands along with their distinctive orientation. A comparison of the numerical results shows good corroboration with experimental observations.

Dynamics of numerical discretization in a delayed diffusive Nicholson’s blowflies equation

In this article, we study the dynamics of a diffusive Nicholson’s blowflies equation with a finite delay and Dirichlet boundary condition by the forward and backward Euler difference schemes and Crank–Nicolson scheme. The existence of Hopf bifurcation at the equilibrium is obtained. Through analyzing the distribution of the eigenvalues we find that the unstable equilibrium state without dispersion may become stable with dispersion under certain conditions. Our results show that Crank–Nicolson and the backward Euler difference schemes are superior to the forward Euler difference scheme under the means of describing approximately the dynamics of the original system. Finally, numerical examples are provided to illustrate the analytical results.

Finite volume method solution of pollutant transport in catchment sheet flow

A finite volume numerical method was employed in the solution of two-dimensional pollutant transport in catchment sheet flow. The full dynamic wave constituted the sheet flow while the advection–diffusion equation with sink/source terms was the pollutant transport model. It is assumed that the solute in the surface active layer is uniformly distributed and the exchange rate of the solute between the active layer and overland flow are proportional to the difference between the concentrations in soil and water volume. Decrease of the solute transfer rate in the active surface layer caused by the transfer of solutes from soil to the overlying runoff is assumed to follow an exponential law. The equations governing sheet flow and pollutant transport are discretized using the finite volume method in space and an implicit backward difference scheme in time. The model was subjected to several numerical tests involving varying microtopographic surface, roughness, and infiltration. The results revealed that spatially varying microtopography plays an important role unlike the roughness and infiltration with respect to the total pollutant rate from the outlet of a catchment.

Finite element study of biomagnetic fluid flow in a symmetrically stenosed channel

The two-dimensional unsteady, laminar flow of a viscous, Newtonian, incompressible and electrically conducting biofluid in a channel with a stenosis, under the influence of a spatially varying magnetic field, is considered. The mathematical modeling of the problem results in a coupled nonlinear system of equations and is given in stream function–vorticity–temperature formulation for the numerical treatment. These equations together with their appropriate boundary conditions are solved iteratively using the finite element method for the spatial discretization, and an unconditionally stable backward difference scheme is employed for the time integration. The numerical results obtained are illustrated using streamlines, vorticity and temperature contours. The behavior of the biofluid along the thin channel is investigated for a symmetric stenosis of degrees 40% and 60%, with a magnetic source placed below the lower plate. The results indicate that the flow is appreciably affected by the presence of stenosis and the magnetic source in terms of vortices. The lengths of the vortices and the temperature increase with increase in the intensity of the magnetic field and the degree of the constriction.

Computational modeling of chemo-electro-mechanical coupling: a novel implicit monolithic finite element approach.

Computational modeling of the human heart allows us to predict how chemical, electrical, and mechanical fields interact throughout a cardiac cycle. Pharmacological treatment of cardiac disease has advanced significantly over the past decades, yet it remains unclear how the local biochemistry of an individual heart cell translates into global cardiac function. Here, we propose a novel, unified strategy to simulate excitable biological systems across three biological scales. To discretize the governing chemical, electrical, and mechanical equations in space, we propose a monolithic finite element scheme. We apply a highly efficient and inherently modular global-local split, in which the deformation and the transmembrane potential are introduced globally as nodal degrees of freedom, whereas the chemical state variables are treated locally as internal variables. To ensure unconditional algorithmic stability, we apply an implicit backward Euler finite difference scheme to discretize the resulting system in time. To increase algorithmic robustness and guarantee optimal quadratic convergence, we suggest an incremental iterative Newton-Raphson scheme. The proposed algorithm allows us to simulate the interaction of chemical, electrical, and mechanical fields during a representative cardiac cycle on a patient-specific geometry, robust and stable, with calculation times on the order of 4 days on a standard desktop computer.

DQEM for analyzing dynamic characteristics of layered fluid-saturated porous elastic media

Based on the Porous Media Theory presented by de Boer, the governing differential equations for a layered space-axisymmetrical fluid-saturated porous elastic body are firstly established, in which the suitable interface conditions between layers are presented. Then, a differential quadrature element method (DQEM) is developed, and the DQEM and the second-order backward difference scheme are applied to discretize the governing differential equations of the problem in the spatial and temporal domain, respectively. In order to show the validity of the present analysis, the dynamic response of a fluid-saturated porous medium is analyzed, and the obtained numerical results are directly compared with the existing analytical results. The effects of the numbers of the elements and grid points on the convergence of the numerical results are considered. Finally, the dynamic characteristics of a layered fluid-saturated elastic soil cylinder subjected to a water pressure or a dynamic loading are studied, and the effects of material parameters are considered in detail. From the above numerical results, it can be found that the DQEM has advantages, such as little amount in computation, good stability and convergence as well as high accuracy, so it is a very efficient method for solving the problems in soil mechanics, especially such problems with discontinuities.

Modeling pollutant transport in overland flow over non-planar and non-homogenous infiltrating surfaces

Pollutant transport in overland flow over surfaces with spatially varying microtopography, roughness, and infiltration was investigated using the diffusion wave equation and transport rate-based equation. The finite volume method in space and an implicit backward difference scheme in time were employed in the numerical solution of the 2D governing equations. The developed model was first tested against an analytical solution and an experimental study involving overland flow and the associated pollutant transport, subsequently a series of numerical tests were carried out. Non-point source pollution was investigated under spatially varying microtopography, roughness, and infiltration. The simulation results showed that microtopography and roughness were the dominant factors causing significant spatial variations in solute concentration. When the spatially varying microtopography was replaced by a smooth surface, the result was an overestimation of the solute rate at the outlet of the upland. On the other hand, when the spatially varying roughness was replaced by the average roughness and spatially varying infiltration rate by the average infiltration rate, the pollutant discharge at the outlet of the upland was not significantly affected. The numerical results further showed that one cannot ignore the spatial variations of slope and roughness when investigating the local pollutant concentration distribution.

An Error Estimates for Convection Dominated Diffusion Problems

In classical mixed finite element method, the choice of the finite element approximating spaces is restricted by the imposition of the LBB consistency condition. The method of H1- Galerkin mixed finite element method avoids completely the imposition of such a condition on the approximating spaces. In this article, we discuss and analyze error estimates for Convection- dominated diffusion problems using H1-Galerkin mixed finite element method, along with the method of characteristics. Optimal order of convergence has been achieved for the error estimates of a single-step Euler backward difference scheme. Keywords. H1- Galerkin mixed finite element method, Characteristics method, LBB condition, optimal error estimates, and Euler backward difference scheme.

Optimal Convergence Analysis for Convection Dominated Diffusion Problems

In classical mixed finite element method, the choice of the finite element approximating spaces is restricted by the imposition of the LBB consistency condition. The method of H1-Galerkin mixed finite element method avoids completely the imposition of such a condition on the approximating spaces. In this article, we discuss and analyze error estimates for Convection-dominated diffusion problems using H1-Galerkin mixed finite element method, along with the method of characteristics. Optimal order of convergence has been achieved for the error estimates of a two-step Euler backward difference scheme.

Second order global uniformly convergent numerical method for a coupled system of singularly perturbed initial value problems

We consider a coupled system of singularly perturbed first order ordinary differential equations. The equations have small positive parameters of different magnitudes associated with them. A decomposition of the exact solution into smooth and singular components is constructed that is useful for the analysis of the numerical method. We apply the HODIE (High Order Difference approximation with Identity Expansions) technique to construct a second order finite difference scheme and combine this with standard backward Euler difference scheme in a special way on a piecewise-uniform Shishkin mesh to solve the system numerically. It is proved that the method is second order (up to a logarithmic factor) convergent in the maximum norm uniformly in both perturbation parameters. Numerical experiments are conducted to support the theoretical results.