An improved unsteady,unstructured, artificial compressibility,finite volume scheme for viscous incompressible flows: Part I. Theory and implementation

A robust, artificial compressibility scheme has been developed for modelling laminar steady state and transient, incompressible flows over a wide range of Reynolds and Rayleigh numbers. Artificial compressibility is applied in a consistant manner resulting in a system of preconditioned governing equations. A locally generalized preconditioner is introduced, designed to be robust and offer good convergence rates. Free artificial compressibility parameters in the equations are automated to allow ease of use while facilitating improved or comparable convergence rates as compared with the standard artificial compressibility scheme. Memory efficiency is achieved through a multistage, pseudo‐time‐explicit time‐marching solution procedure. A node‐centred dual‐cell edge‐based finite volume discretization technique, suitable for unstructured grids, is used due to its computational efficiency and high‐resolution spatial accuracy. In the interest of computational efficiency and ease of implementation, stabilization is achieved via a scalar‐valued artificial dissipation scheme. Temporal accuracy is facilitated by employing a second‐order accurate, dual‐time‐stepping method. In this part of the paper the theory and implementation details are discussed. In Part II, the scheme will be applied to a number of example problems to solve flows over a wide range of Reynolds and Rayleigh numbers. Copyright © 2002 John Wiley & Sons, Ltd.     

An implicit–explicit solution method for electro‐osmotic flow through three‐dimensional micro‐channels

This paper presents a comprehensive finite‐element modelling approach to electro‐osmotic flows on unstructured meshes. The non‐linear equation governing the electric potential is solved using an iterative algorithm. The employed algorithm is based on a preconditioned GMRES scheme. The linear Laplace equation governing the external electric potential is solved using a standard pre‐conditioned conjugate gradient solver. The coupled fluid dynamics equations are solved using a fractional step‐based, fully explicit, artificial compressibility scheme. This combination of an implicit approach to the electric potential equations and an explicit discretization to the Navier–Stokes equations is one of the best ways of solving the coupled equations in a memory‐efficient manner. The local time‐stepping approach used in the solution of the fluid flow equations accelerates the solution to a steady state faster than by using a global time‐stepping approach. The fully explicit form and the fractional stages of the fluid dynamics equations make the system memory efficient and free of pressure instability. In addition to these advantages, the proposed method is suitable for use on both structured and unstructured meshes with a highly non‐uniform distribution of element sizes. The accuracy of the proposed procedure is demonstrated by solving a basic micro‐channel flow problem and comparing the results against an analytical solution. The comparisons show excellent agreement between the numerical and analytical data. In addition to the benchmark solution, we have also presented results for flow through a fully three‐dimensional rectangular channel to further demonstrate the application of the presented method. Copyright © 2007 John Wiley & Sons, Ltd.     

An adaptive hybrid time‐stepping scheme for highly non‐linear strongly coupled problems

This paper deals with the design and implementation of an adaptive hybrid scheme for the solution of highly non‐linear, strongly coupled problems. The term ‘hybrid’ refers to a composite time stepping scheme where a controller decides whether a monolithic scheme or a fractional step (splitting) scheme is appropriate for a given time step. The criteria are based on accuracy and efficiency. The key contribution of this paper is the development of a framework for incorporating error criteria for stepsize selection and a mechanism for choosing from splitting or monolithic possibilities. The resulting framework is applied to silylation, a highly non‐linear, strongly coupled problem of solvent diffusion and reaction in deforming polymers. Numerical examples show the efficacy of our new hybrid scheme on both two‐ and three‐dimensional silylation simulations in the context of microlithography. Copyright © 2005 John Wiley & Sons, Ltd.     

An artificial compressibility method for viscous incompressible and low Mach number flows

Within the framework of the pressure‐based algorithm, an artificial compressibility method is developed on a non‐orthogonal grid for incompressible and low Mach number fluid flow problems, using cell‐centered finite‐volume approximation. Resorting to the traditional pseudo‐compressibility concept, the continuity constraint is perturbed by the time derivative of pressure, the physical relevance of which is to invoke matrix preconditionings. The approach provokes density perturbations, assisting the transformation between primitive and conservative variables. A dual‐dissipation scheme for the pressure–velocity coupling is contrived, which has the expediences of greater flexibility and increased accuracy in a way similar to the monotone upstream‐centered schemes for conservation laws approach. To account for the flow directionality in the upwinding, a rotational matrix is introduced to evaluate the convective flux. Numerical experiments in reference to a few well‐documented laminar flows demonstrate that the entire contrivance expedites enhanced robustness and improved overall damping properties of the factored pseudo‐time integration procedure. Copyright © 2008 John Wiley & Sons, Ltd.     

Adaptive backward Euler time stepping with truncation error control for numerical modelling of unsaturated fluid flow

An automatic time stepping scheme with embedded error control is developed and applied to the moisture‐based Richards equation. The algorithm is based on the first‐order backward Euler scheme, and uses a numerical estimate of the local truncation error and an efficient time step selector to control the temporal accuracy of the integration. Local extrapolation, equivalent to the use of an unconditionally stable Thomas–Gladwell algorithm, achieves second‐order temporal accuracy at minimal additional costs. The time stepping algorithm also provides accurate initial estimates for the iterative non‐linear solver. Numerical tests confirm the ability of the scheme to automatically optimize the time step size to match a user prescribed temporal error tolerance. An important merit of the proposed method is its conceptual and computational simplicity. It can be directly incorporated into existing or new software based on the backward Euler scheme (currently prevalent in subsurface hydrologic modelling), and markedly improves their performance compared with simple fixed or heuristic time step selection. The generality of the approach also makes possible its use for solving PDEs in other engineering applications, where strong non‐linearity, stability or implementation considerations favour a simple and robust low‐order method, or where there is a legacy of backward Euler codes in current use. Copyright © 2001 John Wiley & Sons, Ltd.     

Impact of Artificial Compressibility on the Numerical Solution of Incompressible Nanofluid Flow

The numerical solution of compressible flows has become more prevalent than that of incompressible flows. With the help of the artificial compressibility approach, incompressible flows can be solved numerically using the same methods as compressible ones. The artificial compressibility scheme is thus widely used to numerically solve incompressible Navier-Stokes equations. Any numerical method highly depends on its accuracy and speed of convergence. Although the artificial compressibility approach is utilized in several numerical simulations, the effect of the compressibility factor on the accuracy of results and convergence speed has not been investigated for nanofluid flows in previous studies. Therefore, this paper assesses the effect of this factor on the convergence speed and accuracy of results for various types of thermo-flow. To improve the stability and convergence speed of time discretizations, the fifth-order Runge-Kutta method is applied. A computer program has been written in FORTRAN to solve the discretized equations in different Reynolds and Grashof numbers for various grids. The results demonstrate that the artificial compressibility factor has a noticeable effect on the accuracy and convergence rate of the simulation. The optimum artificial compressibility is found to be between 1 and 5. These findings can be utilized to enhance the performance of commercial numerical simulation tools, including ANSYS and COMSOL.     

Spatial stability for the monolithic and sequential methods with various space discretizations in poroelasticity

We investigate spatial stability with various numerical discretizations in displacement and pressure fields for poroelasticity. We study 2 sources of the early time instability: discontinuity of pressure and violation of the inf‐sup condition. We consider both compressible and incompressible fluids by employing the monolithic, stabilized monolithic, and fixed‐stress sequential methods. Four different spatial discretization schemes are used: Q1Q1, Q2Q1, Q1P0, and Q2P0. From mathematic analysis and numerical tests, the piecewise constant finite volume method for flow provides stability at the early time for the case of the pressure discontinuity. On the other hand, a piecewise continuous (or higher‐order) interpolation of pressure shows spatial oscillation, having lower limits of time step size, although lower approximations of pressure than displacement can alleviate the oscillation. For an incompressible fluid, Q2Q1 can be better than Q1P0, because Q1P0 might not satisfy the inf‐sup condition. However, regardless of fluid compressibility and the pressure discontinuity, the fixed‐stress method can effectively stabilize the oscillation without an artificial stabilizer. Even when Q1P0 and Q1Q1 with the monolithic method cannot satisfy the inf‐sup condition, the fixed‐stress method can yield the full‐rank linear system, providing stability. Thus, the fixed‐stress method with Q1P0 can effectively circumvent the aforementioned 2 types of instability.     

An artificial compressibility algorithm for modelling natural convection in saturated packed pebble beds: A heterogeneous approach

This work is concerned with the modelling of heat and fluid flow through saturated packed pebble beds. A volume‐averaged set of local thermal disequilibrium governing equations is employed to describe the latter as a heterogeneous porous medium with porosity varying from 0.39 to 0.99. The thermal disequilibrium approach, together with stated porosity upper limit, allows for the modelling of wall effects such as wall channeling and wall–bed radiative heat transfer. The resulting set of coupled non‐linear partial differential equations is solved via a locally preconditioned artificial compressibility method, where spatial discretization is effected with a compact finite volume edge‐based discretization scheme. The latter was done in the interest of accuracy. Stabilization is effected via JST scalar‐valued artificial dissipation. This is the first instance in which an artificial compressibility algorithm is applied to modelling heat and fluid flow through heterogeneous porous materials. For this reason, special attention was given to the calculation of acoustic velocities, stabilization scaling factors, as well as allowable time‐step sizes. The developed technology is validated by application to the modelling of a number of benchmark test cases. Copyright © 2008 John Wiley & Sons, Ltd.     

Fractional step methods for thermo‐mechanical damage analyses at transient elevated temperatures

In the interest of computational efficiency this paper describes the implementation of a coupled thermo‐damage constitutive model into a coupled time‐stepping analysis using fractional step methods. To begin it is demonstrated that a thermo‐damage model can be presented in a thermodynamic framework with the evolution equations satisfying the first and second laws of thermodynamics. The equations of evolution are partitioned in two ways, thus defining two fractional step methods: an isothermal method and an isentropic method. When implemented into a time‐stepping algorithm the isentropic method maintains a precise energy balance for the entire analysis where as the isothermal method can only provide an energy balance at the end of each thermal time step. In addition, a stability analysis shows that the isentropic analysis is unconditionally stable while a isothermal analysis is at best conditionally stable. Simulations of thermal fracture in a restrained specimen under heat show stable growth of damage to failure. Copyright © 2000 John Wiley & Sons, Ltd.     

A dual reciprocity boundary element formulation using the fractional step method for the incompressible Navier–Stokes equations

This paper presents a dual reciprocity boundary element solution method for the unsteady Navier–Stokes equations in two-dimensional incompressible flow, where a fractional step algorithm is utilized for the time advancement. A fully explicit, second-order, Adams–Bashforth scheme is used for the nonlinear convective terms. We performed numerical tests for two examples: the Taylor–Green vortex and the lid-driven square cavity flow for Reynolds numbers up to 400. The results in the former case are compared to the analytical solution, and in the latter to numerical results available in the literature. Overall the agreement is excellent demonstrating the applicability and accuracy of the fractional step, dual reciprocity boundary element solution formulations to the Navier–Stokes equations for incompressible flows.     

Solution of the second-order one-dimensional hyperbolic telegraph equation by using the dual reciprocity boundary integral equation (DRBIE) method

In this paper, we use a numerical method based on the boundary integral equation (BIE) and an application of the dual reciprocity method (DRM) to solve the second-order one space-dimensional hyperbolic telegraph equation. Also the time stepping scheme is employed to deal with the time derivative. In this study, we have used three different types of radial basis functions (cubic, thin plate spline and linear RBFs), to approximate functions in the dual reciprocity method (DRM). To confirm the accuracy of the new approach and to show the performance of each of the RBFs, several examples are presented. The convergence of the DRBIE method is studied numerically by comparison with the exact solutions of the problems.     

时间自适应在海洋平台桩靴上拔模拟中的应用

该文通过模型试验和时间自适应有限元分析的方法模拟了海洋平台桩靴的上拔过程。在有限元数值分析中引入了启发式和基于误差评估的两种时间自适应方法,有效解决了因时间步长的选取而引起的不收敛和计算效率低等问题。启发式算法通过控制收敛速度调整时间步长,有效预防了不收敛或收敛过慢,但时步调整较为粗糙。基于误差评估的时间自适应有效控制了计算误差,能够平滑地调整时间步长,相比于启发式算法更具有精确性和稳定性。通过试验和数值方法得出海洋平台在上拔桩靴时需克服海床土体吸附力,采用时间自适应方法可以高效模拟桩靴位移时程的非线性问题。     

An efficient artificial compressibility (AC) scheme based on the characteristic based split (CBS) method for incompressible flows

In this paper, an artificial compressibility scheme using the finite element method is introduced. 2002 Zienkiewicz Silver Medal and Prize winning paper. The multi‐purpose CBS scheme is implemented in its fully explicit form to solve incompressible fluid dynamics problems. It is important to note that the scheme developed here includes split and velocity correction. The proposed method takes advantage of good features from both velocity correction and standard artificial compressibility schemes. Unlike many other artificial compressibility schemes, the proposed one works on a variety of grids and gives results for a wide range of Reynold's numbers. The paper presents some bench mark two‐ and three‐dimensional steady and unsteady incompressible flow solutions obtained from the proposed scheme. Copyright © 2003 John Wiley & Sons, Ltd.     

A variationally consistent fractional time‐step integration method for incompressible and nearly incompressible Lagrangian dynamics

In this paper we present a fractional time‐step method for Lagrangian formulations of solid dynamics problems. The method can be interpreted as belonging to the class of variational integrators which are designed to conserve linear and angular momentum of the entire mechanical system exactly. Energy fluctuations are found to be minimal and stay bounded for long durations. In order to handle incompressibility, a mixed formulation in which the pressure appears explicitly is adopted. The velocity update over a time step is split into deviatoric and volumetric components. The deviatoric component is advanced using explicit time marching, whereas the pressure correction for each time step is computed implicitly by solving a Poisson‐like equation. Once the pressure is known, the volumetric component of the velocity update is calculated. In contrast with standard explicit schemes, where the time‐step size is determined by the speed of the pressure waves, the allowable time step for the proposed scheme is found to depend only on the shear wave speed. This leads to a significant advantage in the case of nearly incompressible materials and permits the solution of truly incompressible problems. Copyright © 2005 John Wiley & Sons, Ltd.     

Finite element/finite volume approaches with adaptive time stepping strategies for transient thermal problems

Transient thermal analysis of engineering materials and structures by space discretization techniques such as the finite element method (FEM) or finite volume method (FVM) lead to a system of parabolic ordinary differential equations in time. These semidiscrete equations are traditionally solved using the generalized trapezoidal family of time integration algorithms which uses a constant single time step. This single time step is normally selected based on the stability and accuracy criteria of the time integration method employed. For long duration transient analysis and/or when severe time step restrictions as in nonlinear problems prohibit the use of taking a larger time step, a single time stepping strategy for the thermal analysis may not be optimal during the entire temporal analysis. As a consequence, an adaptive time stepping strategy which computes the time step based on the local truncation error with a good global error control may be used to obtain optimal time steps for use during the entire analysis. Such an adaptive time stepping approach is described here. Also proposed is an approach for employing combinedFEM/FVM mesh partitionings to achieve numerically improved physical representations. Adaptive time stepping is employed thoughout to practical linear/nonlinear transient engineering problems for studying their effectiveness in finite element and finite volume thermal analysis simulations. Additional support and computing times were furnished by Minnesota Supercomputer Institute at the University of Minnesota.     

Molecular dynamics‐smoothed molecular dynamics (MD‐SMD) adaptive coupling method with seamless transition

Smoothed molecular dynamics (SMD) method is a recently proposed efficient molecular simulation method by introducing one set of background mesh and mapping process into molecular dynamics (MD) flow chart. SMD can sharply enlarge MD time step size while maintaining global accuracy. MD‐SMD coupling method was proposed to improve the capability to describe local atom disorders. The coupling method is greatly improved in this paper in two essential aspects. Firstly, a transition scheme is proposed to avoid artificial wave reflection at the interface of MD and SMD regions. The new transition scheme has simple formulation and high efficiency, and the wave reflection can be well suppressed. Secondly, an adaptive scheme is proposed to automatically identify the regions requiring MD simulation. Two adaptive criteria, the centro‐symmetry parameter criterion and the displacement criterion, are also proposed. It is found that both the two criteria can achieve good accuracy but the efficiency of the displacement criterion is much better. The coupling method does not demand reduction in mesh size near the interface, and a multiple time stepping scheme is adopted to ensure high efficiency. Numerical results including wave propagation, nano‐indentation, and crack propagation validate the method and show nice accuracy. Copyright © 2016 John Wiley & Sons, Ltd.     

基于分数步长法的压铸充型过程数值模拟

为了模拟型腔内流体的流动形态并预测气孔缺陷,应用充型过程数值模拟方法,采用分数步长法来求解非稳态的Navier-Stokes方程,可以在每个时间步长内仅求解一次动量和压力方程.动量方程求解中采用Adams-Bashforth方法处理对流项,采用Crank-Nicolson方法处理扩散项.搭建了压铸过程水模拟系统,设计了型腔尺寸沿充填方向逐渐递减的有机玻璃模具.通过彩色高速摄像机以500帧/秒的速度记录充填形貌.将数值模拟结果和水模拟结果相比较,两者吻合较好.同时采用SOLA-VOF法模拟了同一充型过程,结果表明,分数步长法和VOF法相结合要比SOLA-VOF法更能准确地预测压铸过程中的卷气现象.     

Finite element solution of free‐surface ship‐wave problems

An unstructured finite element solver to evaluate the ship‐wave problem is presented. The scheme uses a non‐structured finite element algorithm for the Euler or Navier–Stokes flow as for the free‐surface boundary problem. The incompressible flow equations are solved via a fractional step method whereas the non‐linear free‐surface equation is solved via a reference surface which allows fixed and moving meshes. A new non‐structured stabilized approximation is used to eliminate spurious numerical oscillations of the free surface. Copyright © 1999 John Wiley & Sons, Ltd.     

A dual-time method for two-dimensional unsteady incompressible flow calculations

A method for computing unsteady incompressible viscous flows on moving or deforming meshes is described. It uses a well-established time-marching finite-volume flow solver, developed for steady compressible flows past rigid bodies. Time-marching methods cannot be applied directly to incompressible flows because the governing equations are not hyperbolic. Such methods can be extended to steady incompressible flows using an artificial compressibility scheme. A time-accurate scheme for unsteady incompressible flows is achieved by using an implicit real-time discretization and a dual-time approach, which uses a technique similar to the artificial compressibility scheme. Results are presented for test cases on both fixed and deforming meshes. Experimental, numerical and theoretical data have been included for comparison where available and reasonable agreement has been achieved. © 1998 John Wiley & Sons, Ltd.     

Improving mass conservation in simulation of incompressible flows

We propose a technique for improving mass‐conservation features of fractional step schemes applied to incompressible flows. The method is illustrated by using a Lagrangian fluid formulation, where the mass loss effects are particularly apparent. However, the methodology is general and could be used for fixed grid approaches. The idea consists in reflecting the incompressibility condition already in the intermediate velocity. This is achieved by predicting the end‐of‐step pressure and using this prediction in the fractional momentum equation. The resulting intermediate velocity field is thus much closer to the final incompressible one than that of the standard fractional step scheme. In turn, the predicted pressure can be used as the boundary condition necessary for the solution of the pressure Poisson equation in case a continuous Laplacian matrix is employed. Using this approximation of the end‐of‐step incompressible pressure as the essential boundary condition considerably improves the conservation of mass, specially for the free surface flows of fluids with low viscosity. The pressure prediction does not require the resolution of any additional equations system. The efficiency of the method is shown in two examples. The first one shows the performance of the method with respect to mass conservation. The second one tests the method in a challenging fluid–structure interaction benchmark, which can be naturally resolved by using the presented approach. Copyright © 2012 John Wiley & Sons, Ltd.