Second-order Backward Difference Research Articles

A two-grid finite element approximation for a nonlinear time-fractional Cable equation

In this article, a nonlinear fractional Cable equation is solved by a two-grid algorithm combined with finite element (FE) method. A temporal second-order fully discrete two-grid FE scheme, in which the spatial direction is approximated by two-grid FE method and the integer and fractional derivatives in time are discretized by second-order two-step backward difference method and second-order weighted and shifted Gr\"unwald difference (WSGD) scheme, is presented to solve nonlinear fractional Cable equation. The studied algorithm in this paper mainly covers two steps: First, the numerical solution of nonlinear FE scheme on the coarse grid is solved, Second, based on the solution of initial iteration on the coarse grid, the linearized FE system on the fine grid is solved by using Newton iteration. Here, the stability based on fully discrete two-grid method is derived. Moreover, the a priori estimates with second-order convergence rate in time is proved in detail, which is higher than the L1-approximation result with $O(\tau^{2-\alpha}+\tau^{2-\beta})$. Finally, the numerical results by using the two-grid method and FE method are calculated, respectively, and the CPU-time is compared to verify our theoretical results.

An Extrapolated Second Order Backward Difference Time-Stepping Scheme for the Magnetohydrodynamics System

ABSTRACTIn this article, we introduce a fully implicit, linearly extrapolated second-order backward difference time-stepping scheme for solving a time dependent non-homogeneous magnetohydrodynamic system for electrically conducting fluids. The extrapolated time-stepping scheme is used for time discretization and the mixed finite element method is used for spatial discretization. We first prove unconditional energetic stability without introducing an undesirable exponential Gronwall constant. Complete error analysis is provided without assuming any stability condition or restrictions on the time-step size. Numerical experiments are presented to confirm the theoretical convergence results and efficiency of the scheme.

A-Stable Time Discretizations Preserve Maximal Parabolic Regularity

It is shown that for a parabolic problem with maximal $L^p$-regularity (for $1<p<\infty$), the time discretization by a linear multistep method or Runge--Kutta method has maximal $\ell^p$-regularity uniformly in the stepsize if the method is A-stable (and satisfies minor additional conditions). In particular, the implicit Euler method, the Crank--Nicolson method, the second-order backward difference formula (BDF), and the Radau IIA and Gauss Runge--Kutta methods of all orders preserve maximal regularity. The proof uses Weis' characterization of maximal $L^p$-regularity in terms of $R$-boundedness of the resolvent, a discrete operator-valued Fourier multiplier theorem by Blunck, and generating function techniques that have been familiar in the stability analysis of time discretization methods since the work of Dahlquist. The A($\alpha$)-stable higher-order BDF methods have maximal $\ell^p$-regularity under an $R$-boundedness condition in a larger sector. As an illustration of the use of maximal regularity ...

Mixed stabilized finite element methods based on backward difference/Adams–Bashforth scheme for the time-dependent variable density incompressible flows

In this paper, we consider a second-order mixed stabilized finite element method based on pressure projection method for variable density incompressible flows. The originality of the proposed approach is to use a stabilized method based on the difference between a consistent and an under-integrated mass matrix of the pressure for variable density incompressible flows approximated by the lowest equal-order finite element pairs. A second-order backward difference (BDF) for the temporal term and a second-order Adams–Bashforth (AB) for the explicit treatment of the nonlinear term lead to the presented second-order BDF/AB scheme. The stability of the method was proved and the performance of the method is numerically illustrated. Finally, comparison with some established methods, a series of numerical experiments are given to show that this method has better stability and accuracy.

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.

Convergence of a linearized second-order BDF-FEM for nonlinear parabolic interface problems

We present an almost optimal error estimate for the finite element solution of a nonlinear parabolic interface problem, where the coefficient depends on the unknown variable and is discontinuous along an interface inside the computational domain. A linearized second-order backward difference formula is used for the time discretization, and piecewise linear interpolation is used to approximate the interface. We do not assume Lipschitz continuity of the nonlinear coefficient.

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 complementary study of analytical and computational fluid-structure interaction

This paper introduces new initial value problems for the design and analysis of fluid-structure interaction algorithms. The problems are constructed from a finite number of sinusoidal perturbations of thin structures superposed on incompressible fluids. Explicit analytical solutions are provided for rigorous quantitative comparisons including viscous effects. The equation governing a single inviscid mode is split into fluid and structure subproblems and numerical parameters are derived for several partitioned algorithms based on a Dirichlet-Neumann decomposition and a second-order backward difference time discretization. Numerical experiments are performed using a stabilized fractional-step finite element method for the fluid and membrane finite elements for the structure. The convergence behaviour of the finite element solutions to the analytical solutions is demonstrated. Finally, the stability and performance of the numerical parameters derived from a single inviscid mode are presented for more general problems including multiple modes and viscous effects.

An analysis of the Rayleigh-Stokes problem for a generalized second-grade fluid.

We study the Rayleigh–Stokes problem for a generalized second-grade fluid which involves a Riemann–Liouville fractional derivative in time, and present an analysis of the problem in the continuous, space semidiscrete and fully discrete formulations. We establish the Sobolev regularity of the homogeneous problem for both smooth and nonsmooth initial data v, including vin L^2(Omega ). A space semidiscrete Galerkin scheme using continuous piecewise linear finite elements is developed, and optimal with respect to initial data regularity error estimates for the finite element approximations are derived. Further, two fully discrete schemes based on the backward Euler method and second-order backward difference method and the related convolution quadrature are developed, and optimal error estimates are derived for the fully discrete approximations for both smooth and nonsmooth initial data. Numerical results for one- and two-dimensional examples with smooth and nonsmooth initial data are presented to illustrate the efficiency of the method, and to verify the convergence theory.

A second-order accurate in time IMplicit–EXplicit (IMEX) integration scheme for sea ice dynamics

Current sea ice models use numerical schemes based on a splitting in time between the momentum and continuity equations. Because the ice strength is explicit when solving the momentum equation, this can create unrealistic ice stress gradients when using a large time step. As a consequence, noise develops in the numerical solution and these models can even become numerically unstable at high resolution. To resolve this issue, we have implemented an iterated IMplicit–EXplicit (IMEX) time integration method. This IMEX method was developed in the framework of an already implemented Jacobian-free Newton–Krylov solver. The basic idea of this IMEX approach is to move the explicit calculation of the sea ice thickness and concentration inside the Newton loop such that these tracers evolve during the implicit integration. To obtain second-order accuracy in time, we have also modified the explicit time integration to a second-order Runge–Kutta approach and by introducing a second-order backward difference method for the implicit integration of the momentum equation. These modifications to the code are minor and straightforward. By comparing results with a reference solution obtained with a very small time step, it is shown that the approximate solution is second-order accurate in time. The new method permits to obtain the same accuracy as the splitting in time but by using a time step that is 10 times larger. Results show that the second-order scheme is more than five times more computationally efficient than the splitting in time approach for an equivalent level of error.

Full Eulerian finite element method of a phase field model for fluid–structure interaction problem

In this paper we present a full Eulerian model for a dynamic fluid–structure interaction (FSI) problem in terms of phase field approach, and design its full Eulerian finite element discretization and effective iterative method. The present full Eulerian FSI model effectively demonstrates the interaction between fluid flow and solid structure in terms of a uniform system of governing equations defined in a single domain, thus the computational grid is fixed, and the re-meshing and interpolation techniques which are always required by other FSI modeling approaches are no longer needed here. We develop a new stable scheme to discretize the Euler equation of an incompressible hyperelastic structure in Eulerian description, and employ Galerkin/least-square (GLS) stabilization scheme, streamline-upwind/Petrov–Galerkin (SUPG) method, and the second-order backward difference formula (BDF) to solve the derived transient nonlinear system of Navier–Stokes equations and transport equations. Numerical experiment is carried out for a cross spinning around its rotation of axis due to the passing flow field, and the numerical results dramatically show the spinning motion of the cross due to the interaction with the fluid, showing that our model and numerical methods are effective to simulate the dynamic fluid–structure interaction phenomena.

Aeroelastic Modeling and Analysis of a Horizontal Axis Wind Turbine

Dynamics of the horizontal axis wind turbine is investigated in this paper. The wind turbine rotor modelling is based on the nonlinear Eular beam model, the generalized dynamic wake model and the Beddoes-Leishman dynamic stall model are used to predict the aerodynamic loads. The Newton-Rafson iteration is adopted to get the solution in each time step and the second-order backward difference is used in time marching. A horizontal axis wind turbine is analysed in fixed rotor speed and the structural responses in rated conditions and critical wind conditions are obtained. The critical wind velocity is acquired from simulation. The influences of the rotor speed and yaw angle on the critical wind velocity of the rotor are reserched.

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.

VALIDATION AND APPLICATION OF AN HIGH-ORDER SPECTRAL DIFFERENCE METHOD FOR FLOW INDUCED NOISE SIMULATION

The main goal of this paper is to develop an efficient numerical algorithm to compute the radiated far field noise provided by an unsteady flow field from bodies in arbitrary motion. The method computes a turbulent flow field in the near fields using a high-order spectral difference method coupled with large-eddy simulation approach. The unsteady equations are solved by advancing in time using a second-order backward difference formulae scheme. The nonlinear algebraic system arising from the time discretization is solved with the nonlinear lower–upper symmetric Gauss–Seidel algorithm. In the second step, the method calculates the far field sound pressure based on the acoustic source information provided by the first step simulation. The method is based on the Ffowcs Williams–Hawkings approach, which provides noise contributions for monopole, dipole and quadrupole acoustic sources. This paper will focus on the validation and assessment of this hybrid approach using different test cases. The test cases used are: a laminar flow over a two-dimensional (2D) open cavity at Re = 1.5 × 103 and M = 0.15 and a laminar flow past a 2D square cylinder at Re = 200 and M = 0.5. In order to show the application of the numerical method in industrial cases and to assess its capability for sound field simulation, a three-dimensional turbulent flow in a muffler at Re = 4.665 × 104 and M = 0.05 has been chosen as a third test case. The flow results show good agreement with numerical and experimental reference solutions. Comparison of the computed noise results with those of reference solutions also shows that the numerical approach predicts noise accurately.

Towards a three-dimensional moving body incompressible flow solver with a linear deformable model

In this study, a three-dimensional fluid–structured parallelized solver is extended from the previous work (Niu et al., 2009 [1]) for moving body simulations. Based on the unified Eulerian and Lagrangian coordinate transformations, the unsteady three-dimensional incompressible Navier–Stokes equations with artificial compressibility (Chorin, 1967 [2]) in a dual-time stepping approach are first derived. To implement unsteady flow calculations, the dual-time stepping strategy including the LU decomposition method is used in the pseudo-time iteration and the second-order accurate backward difference is adopted to discretize the unsteady flow terms. Also, a third-order Roe type flux limited splitting is derived to evaluate the spatial difference of the convective fluxes. The original FORTRAN code is converted to the MPI code and tested on a 64-CPU IBM SP2. The parallel strategy here is based on the partitions of all do-loops in the original FORTRAN code and transferring the calculations inside the do-loop into different CPUs. The partition of the do-loop can be applied on the innermost loop, only or the last two inner loops depending on two-dimensional or three-dimensional problems. This kind of the parallel data partition of the loops is independent of what kind of the explicit or implicit type numerical algorithm used. Therefore, the current parallel approach can take advantage of the MPI language fully to transfer data efficiently among CPUs even for solving the governing equation implicitly. The test results show that a significant reduction of computing time in running the model and a near-linear speed up rate is achieved up to 32 CPUs at IBM SP2. The speed up rate is as high as 31 for using 64 IBM SP2 processors The test shows efficient parallel processing to provide prompt simulation of 3D cavity, unsteady dropping airfoil and blood flows in an aortic tube with a linear elastic modeling of wall motion is included here.

Transient dynamic response of fluid-saturated soil under a moving cyclic loading

Based on the Theory of Porous Media (TPM), a mathematical model of a two-dimensional incompressible fluid-saturated elastic soil is established, and the periodic boundary conditions are presented to analyze the transient dynamic response of this soil under a moving cyclic loading. The differential quadrature method (DQM) and the second-order backward difference scheme are applied to discretize the governing equations on the spatial and temporal domains, respectively. As application, a typical two-dimensional wave-induced transient problem with a seabed of finite thickness is analyzed, and the numerical results are compared with the analytical results presented in the present work. In addition, a transient dynamic response of fluid-saturated soil under limit moving vehicle loadings is studied. The effects of the velocity of vehicle and the volume fraction on the settlement and the pore water pressure are studied.

An implicit high-order spectral difference approach for large eddy simulation

The filtered fluid dynamic equations are discretized in space by a high-order spectral difference (SD) method coupled with large eddy simulation (LES) approach. The subgrid-scale stress tensor is modelled by the wall-adapting local eddy-viscosity model (WALE). We solve the unsteady equations by advancing in time using a second-order backward difference formulae (BDF2) scheme. The nonlinear algebraic system arising from the time discretization is solved with the nonlinear lower–upper symmetric Gauss–Seidel (LU-SGS) algorithm. In order to study the sensitivity of the method, first, the implicit solver is used to compute the two-dimensional (2D) laminar flow around a NACA0012 airfoil at Re = 5 × 10 5 with zero angle of attack. Afterwards, the accuracy and the reliability of the solver are tested by solving the 2D “turbulent” flow around a square cylinder at Re = 10 4 and Re = 2.2 × 10 4. The results show a good agreement with the experimental data and the reference solutions.

Tear film dynamics on an eye-shaped domain. Part 2. Flux boundary conditions

We model the dynamics of the human tear film during relaxation (after a blink) using lubrication theory and explore the effects of viscosity, surface tension, gravity and boundary conditions that specify the flux of tear fluid into or out of the domain. The governing nonlinear partial differential equation is solved on an overset grid by a method of lines using finite differences in space and an adaptive second-order backward difference formula solver in time. Our simulations in a two-dimensional domain are computed in the Overture computational framework. The flow around the boundary is sensitive to both our choice of flux boundary condition and the presence of gravity. The simulations recover features seen in one-dimensional simulations and capture some experimental observations of tear film dynamics around the lid margins. In some instances, the influx from the lacrimal gland splits with some fluid going along the upper lid towards the nasal canthus and some travelling around the temporal canthus and then along the lower lid. Tear supply can also push through some parts of the black line near the eyelid margins.

Tear film dynamics on an eye-shaped domain I: pressure boundary conditions

We study the relaxation of a model for the human tear film after a blink on a stationary eye-shaped domain corresponding to a fully open eye using lubrication theory and explore the effects of viscosity, surface tension, gravity and boundary conditions that specify the pressure. The governing non-linear partial differential equation is solved on an overset grid by a method of lines using a finite-difference discretization in space and an adaptive second-order backward-difference formula solver in time. Our 2D simulations are calculated in the Overture computational framework. The computed flows show sensitivity to both our choices between two different pressure boundary conditions and the presence of gravity; this is particularly true around the boundary. The simulations recover features seen in 1D simulations and capture some experimental observations including hydraulic connectivity around the lid margins.

Navier–Stokes spectral solver in a sphere

The paper presents the first implementation of a primitive variable spectral method for calculating viscous flows inside a sphere. A variational formulation of the Navier–Stokes equations is adopted using a fractional-step time discretization with the classical second-order backward difference scheme combined with explicit extrapolation of the nonlinear term. The resulting scalar and vector elliptic equations are solved by means of the direct spectral solvers developed recently by the authors. The spectral matrices for radial operators are characterized by a minimal sparsity – diagonal stiffness and tridiagonal mass matrix. Closed-form expressions of their nonzero elements are provided here for the first time, showing that the condition number of the relevant matrices grows as the second power of the truncation order. A new spectral elliptic solver for the velocity unknown in spherical coordinates is also described that includes implicitly the Coriolis force in a rotating frame, but requires a minimal coupling between the modal velocity components in the Fourier space. The numerical tests confirm that the proposed method achieves spectral accuracy and ensures infinite differentiability to all orders of the numerical solution, by construction. These results indicate that the new primitive variable spectral solver is an effective alternative to the spectral method recently proposed by Kida and Nakayama, where the velocity field is represented in terms of poloidal and toroidal functions.