An assessment of using the predictor–corrector technique to solve reactive transport equations

We examined, through comparison among the full‐coupling (FC), operator‐splitting (OS), and predictor–corrector (PC) techniques, the effectiveness of using the PC technique to solve depth‐averaged reactive transport equations in the shallow water domain. Our investigation has led to three major conclusions. Firstly, both the OS and PC techniques can efficiently solve reactive transport equations because the advection–diffusion transport equations are solved outside the non‐linear iteration loop and the reaction equations are solved node by node. However, these two techniques may risk sacrificing computational accuracy. Secondly, the OS or PC technique incorporated with the Lagrangian–Eulerian (LE) approach can handle boundary sources more precisely than alternatively with the conventional Eulerian (CE) approach. Thirdly, with the LE approach incorporated, the numerical results from the three techniques agreed highly with one another except when diffusion became significant. In this case, the PC technique's result still matched well with the FC technique's result, but differences between the OS and FC techniques' results arose as diffusion increased. Based on this study, we recommend to apply as a first step the PC technique to solving reactive transport equations with respect to both computational efficiency and accuracy. Copyright © 2002 John Wiley & Sons, Ltd.     

An improved numerical scheme for the sine‐Gordon equation in 2+1 dimensions

A rational approximant of order 4, which is applied to a three‐time‐level recurrence relation, is used to transform the initial/boundary‐value problem associated with the two‐dimensional sine‐Gordon (SG) equation arising in the Josephson junctions problem. The resulting non‐linear system, which is analyzed for stability, is solved using an appropriate predictor–corrector (P–C) scheme, in which an explicit scheme of order 2 is used as predictor. For the implementation of the corrector, in order to avoid extended matrix evaluations, an auxiliary vector was successfully introduced. In this P–C scheme, a modification in the corrector has been proposed according to which the already evaluated corrected values are considered. The behavior of this P–C scheme is tested numerically on line and ring solitons known from the bibliography regarding the SG equation and conclusions for both undamped and damped problems are derived. Copyright © 2008 John Wiley & Sons, Ltd.     

A rate‐consistent and intrinsically path‐dependent incremental method for plastic buckling problems

This paper presents an incremental predictor–corrector method which is able to handle the continuous spreading of elastic unloading and, therefore, is particularly well suited to solve plastic bucking problems. The method, which deals explicitly with rate variables and equations, is (i) rate consistent, because it leads to the ‘true’ tangent matrix, and (ii) intrinsically path‐dependent, because it enables an adequate identification and characterization of the onset of elastic unloading within each incremental step. Copyright © 2000 John Wiley & Sons, Ltd.     

Formulation of equations of motion of finite element form for vehicle–track–bridge interaction system with two types of vehicle model

Vehicle, track and bridge are considered as an entire system in this paper. Two types of vertical vehicle model are described. One is a one foot mass–spring–damper system having two‐degree‐of‐freedom, and the other is four‐wheelset mass–spring–damper system with two‐layer suspension systems possessing 10‐degree‐of‐freedom. For the latter vehicle model, the eccentric load of car body is taken into account. The rails and the bridge deck are modelled as an elastic Bernoulli–Euler upper beam with finite length and a simply supported Bernoulli–Euler lower beam, respectively, while the elasticity and damping properties of the rail bed are represented by continuous springs and dampers. The dynamic contact forces between the moving vehicle and rails are considered as internal forces, so it is not necessary to calculate the internal forces for setting up the equations of motion of the vehicle–track–bridge interaction system. The two types of equations of motion of finite element form for the entire system are derived by means of the principle of a stationary value of total potential energy of dynamic system. The proposed method can set up directly the equations of motion for sophisticated system, and these equations can be solved by step‐by‐step integration method, to obtain simultaneously the dynamic responses of vehicle, of track and of bridge. Illustration examples are given. Copyright 2004 © John Wiley & Sons, Ltd.     

Numerical solution of non‐linear Klein–Gordon equations by variational iteration method

In this paper, numerical solution of non‐linear Klein–Gordon equations with power law non‐linearities are obtained by the new application of He's variational iteration method. Numerical illustrations that include non‐linear Klein–Gordon equations and non‐linear partial differential equations are investigated to show the pertinent features of the technique. Copyright © 2006 John Wiley & Sons, Ltd.     

A new predictor–corrector approach for the numerical integration of coupled electromechanical equations

In this paper, a new approach for the numerical solution of coupled electromechanical problems is presented. The structure of the considered problem consists of the low‐frequency integral formulation of the Maxwell equations coupled with Newton–Euler rigid‐body dynamic equations. Two different integration schemes based on the predictor–corrector approach are presented and discussed. In the first method, the electrical equation is integrated with an implicit single‐step time marching algorithm, while the mechanical dynamics is studied by a predictor–corrector scheme. The predictor uses the forward Euler method, while the corrector is based on the trapezoidal rule. The second method is based on the use of two interleaved predictor–corrector schemes: one for the electrical equations and the other for the mechanical ones. Both the presented methods have been validated by comparison with experimental data (when available) and with results obtained by other numerical formulations; in problems characterized by low speeds, both schemes produce accurate results, with similar computation times. When high speeds are involved, the first scheme needs shorter time steps (i.e., longer computation times) in order to achieve the same accuracy of the second one. A brief discussion on extending the algorithm for simulating deformable bodies is also presented. An example of application to a two‐degree‐of‐freedom levitating device based on permanent magnets is finally reported. Copyright © 2015 John Wiley & Sons, Ltd.     

Solving linear and non‐linear space–time fractional reaction–diffusion equations by the Adomian decomposition method

In this paper, we consider linear and non‐linear space–time fractional reaction–diffusion equations (STFRDE) on a finite domain. The equations are obtained from standard reaction–diffusion equations by replacing a second‐order space derivative by a fractional derivative of order β∈(1, 2], and a first‐order time derivative by a fractional derivative of order α∈(0, 1]. We use the Adomian decomposition method to construct explicit solutions of the linear and non‐linear STFRDE. Finally, some examples are given. Copyright © 2007 John Wiley & Sons, Ltd.     

A 3D beam‐to‐beam contact finite element for coupled electric–mechanical fields

In this paper the formulation of an electric–mechanical beam‐to‐beam contact element is presented. Beams with circular cross‐sections are assumed to get in contact in a point‐wise manner and with clean metallic surfaces. The voltage distribution is influenced by the contact mechanics, since the current flow is constricted to small contacting spots. Therefore, the solution is governed by the contacting areas and hence by the contact forces. As a consequence the problem is semi‐coupled with the mechanical field influencing the electric one. The electric–mechanical contact constraints are enforced with the penalty method within the finite element technique. The virtual work equations for the mechanical and electric fields are written and consistently linearized to achieve a good level of computational efficiency with the finite element method. The set of equations is solved with a monolithic approach. Copyright © 2005 John Wiley & Sons, Ltd.     

A rapid simulation of nano‐particle transport in a two‐dimensional human airway using POD/Galerkin reduced‐order models

An approach is proposed for the rapid prediction of nano‐particle transport and deposition in the human airway, which requires the solution of both the Navier–Stokes and advection–diffusion equations and for which computational efficiency is a challenge. The proposed method builds low‐order models that are representative of the fully coupled equations by means of the Galerkin projection and proper orthogonal decomposition technique. The obtained reduced‐order models (ROMs) are a set of ordinary differential equations for the temporal coefficients of the basis functions. The numerical results indicate that the ROMs are highly efficient for the computation (the speedup factor is approximately 3 × 10³) and have reasonable accuracy compared with the full model (relative error of ≈7 × 10^?3). Using ROMs, the deposition of particles is studied for 1≤d_n≤100 nm, where d_n is the diameter of a nano‐particle. The effectiveness of this approach is promising for applications of health risk assessment. Copyright © 2015 John Wiley & Sons, Ltd.     

A general conjugate‐gradient‐based predictor–corrector solver for explicit finite‐element contact

A Lagrange‐multiplier based approach is presented for the general solution of multi‐body contact within an explicit finite element framework. The technique employs an explicit predictor step to permit the detection of interpenetration and then utilizes a corrector step, whose solution is obtained with a pre‐conditioned matrix‐free conjugate gradient projection method, to determine the Lagrange multipliers necessary to eliminate the predicted penetration. The predictor–corrector algorithm is developed for deformable bodies based upon the central difference method, and for rigid bodies from momentum and energy conserving approaches. Both frictionless and Coulomb‐based frictional contact idealizations are addressed. The technique imposes no time‐step constraints and quickly mitigates velocity discontinuities across closed interfaces. Special attention is directed toward contact between rigid bodies. Algorithmic moment arms conserve the translational and angular momentums of the system in the absence of external loads. Elastic collisions are captured with a two‐phase predictor–corrector approach and a geometrically approximate velocity jump criterion. The first step solves the inelastic contact problem and identifies inactive constraints between rigid bodies, while the second step generates the necessary velocity jump condition on the active constraints. The velocity criterion is shown to algorithmically preserve the system kinetic energy for two unconstrained rigid bodies. Copyright © 1999 John Wiley & Sons, Ltd. This paper was produced under the auspices of the U.S. Government and it is therefore not subject to copyright in the U.S.     

The enriched space–time finite element method (EST) for simultaneous solution of fluid–structure interaction

The paper introduces a weighted residual‐based approach for the numerical investigation of the interaction of fluid flow and thin flexible structures. The presented method enables one to treat strongly coupled systems involving large structural motion and deformation of multiple‐flow‐immersed solid objects. The fluid flow is described by the incompressible Navier–Stokes equations. The current configuration of the thin structure of linear elastic material with non‐linear kinematics is mapped to the flow using the zero iso‐contour of an updated level set function. The formulation of fluid, structure and coupling conditions uniformly uses velocities as unknowns. The integration of the weak form is performed on a space–time finite element discretization of the domain. Interfacial constraints of the multi‐field problem are ensured by distributed Lagrange multipliers. The proposed formulation and discretization techniques lead to a monolithic algebraic system, well suited for strongly coupled fluid–structure systems. Embedding a thin structure into a flow results in non‐smooth fields for the fluid. Based on the concept of the extended finite element method, the space–time approximations of fluid pressure and velocity are properly enriched to capture weakly and strongly discontinuous solutions. This leads to the present enriched space–time (EST) method. Numerical examples of fluid–structure interaction show the eligibility of the developed numerical approach in order to describe the behavior of such coupled systems. The test cases demonstrate the application of the proposed technique to problems where mesh moving strategies often fail. Copyright © 2007 John Wiley & Sons, Ltd.     

A novel numerical method of high‐order accuracy for flow in unsaturated porous media

We construct finite volume schemes of very high order of accuracy in space and time for solving the nonlinear Richards equation (RE). The general scheme is based on a three‐stage predictor–corrector procedure. First, a high‐order weighted essentially non‐oscillatory (WENO) reconstruction procedure is applied to the cell averages at the current time level to guarantee monotonicity in the presence of steep gradients. Second, the temporal evolution of the WENO reconstruction polynomials is computed in a predictor stage by using a global weak form of the governing equations. A global space–time DG FEM is used to obtain a scheme without the parabolic time‐step restriction caused by the presence of the diffusion term in the RE. The resulting nonlinear algebraic system is solved by a Newton–Krylov method, where the generalized minimal residual method algorithm of Saad and Schulz is used to solve the linear subsystems. Finally, as a third step, the cell averages of the finite volume method are updated using a one‐step scheme, on the basis of the solution calculated previously in the space–time predictor stage. Our scheme is validated against analytical, experimental, and other numerical reference solutions in four test cases. A numerical convergence study performed allows us to show that the proposed novel scheme is high order accurate in space and time. Copyright © 2011 John Wiley & Sons, Ltd.     

Investigation of fluid–structure interactions using a velocity‐linked P2/P1 finite element method and the generalized‐ α method

A velocity‐linked algorithm for solving unsteady fluid–structure interaction (FSI) problems in a fully coupled manner is developed using the arbitrary Lagrangian–Eulerian method. The P2/P1 finite element is used to spatially discretize the incompressible Navier–Stokes equations and structural equations, and the generalized‐ α method is adopted for temporal discretization. Common velocity variables are employed at the fluid–structure interface for the strong coupling of both equations. Because of the velocity‐linked formulation, kinematic compatibility is automatically satisfied and forcing terms do not need to be calculated explicitly. Both the numerical stability and the convergence characteristics of an iterative solver for the coupled algorithm are investigated by solving the FSI problem of flexible tube flows. It is noteworthy that the generalized‐ α method with small damping is free from unstable velocity fields. However, the convergence characteristics of the coupled system deteriorate greatly for certain Poisson's ratios so that direct solvers are essential for these cases. Furthermore, the proposed method is shown to clearly display the advantage of considering FSI in the simulation of flexible tube flows, while enabling much larger time‐steps than those adopted in some previous studies. This is possible through the strong coupling of the fluid and structural equations by employing common primitive variables. Copyright © 2012 John Wiley & Sons, Ltd.     

An alternative approach for numerical solutions of the Navier–Stokes equations

Conventional approaches for solving the Navier–Stokes equations of incompressible fluid dynamics are the primitive‐variable approach and the vorticity–velocity approach. In this paper, an alternative approach is presented. In this approach, pressure and one of the velocity components are eliminated from the governing equations. The result is one higher‐order partial differential equation with one unknown for two‐dimensional problems or two higher‐order partial differential equations with two unknowns for three‐dimensional problems. A meshless collocation method based on radial basis functions for solving the Navier–Stokes equations using this approach is presented. The proposed method is used to solve a two‐ and a three‐dimensional test problem of which exact solutions are known. It is found that, with appropriate values of the method parameters, solutions of satisfactory accuracy can be obtained. Copyright © 2006 John Wiley & Sons, Ltd.     

A dual‐primal FETI method for solving a class of fluid–structure interaction problems in the frequency domain

The dual‐primal finite element tearing and interconnecting method (FETI‐DP) is extended to systems of linear equations arising from a finite element discretization for a class of fluid–structure interaction problems in the frequency domain. A preconditioned generalized minimal residual method is used to solve the linear equations for the Lagrange multipliers introduced on the subdomain boundaries to enforce continuity of the solution. The coupling between the fluid and the structure on the fluid–structure interface requires an appropriate choice of coarse level degrees of freedom in the FETI‐DP algorithm to achieve fast convergence. Several choices are proposed and tested by numerical experiments on three‐dimensional fluid–structure interaction problems in the mid‐frequency regime that demonstrate the greatly improved performance of the proposed algorithm over the standard FETI‐DP method. Copyright © 2011 John Wiley & Sons, Ltd.     

Unified fractional step method for Lagrangian analysis of quasi‐incompressible fluid and nonlinear structure interaction using the PFEM

The fractional step method (FSM) is an efficient solution technique for the particle finite element method, a Lagrangian‐based approach to simulate fluid–structure interaction (FSI). Despite various refinements, the applicability of the FSM has been limited to low viscosity flow and FSI simulations with a small number of equations along the fluid–structure interface. To overcome these limitations, while incorporating nonlinear response in the structural domain, an FSM that unifies structural and fluid response in the discrete governing equations is developed using the quasi‐incompressible formulation. With this approach, fluid and structural particles do not need to be treated separately, and both domains are unified in the same system of equations. Thus, the equations along the fluid–structure interface do not need to be segregated from the fluid and structural domains. Numerical examples compare the unified FSM with the non‐unified FSM and show that the computational cost of the proposed method overcomes the slow convergence of the non‐unified FSM for high values of viscosity. As opposed to the non‐unified FSM, the number of iterations required for convergence with the unified FSM becomes independent of viscosity and time step, and the simulation run time does not depend on the size of the FSI interface. Copyright © 2016 John Wiley & Sons, Ltd.     

Global‐basis two‐level method for indefinite systems. Part 2: computational issues

Algorithmic aspects and computational efficiency of the global‐basis two‐level method are investigated in the context of symmetric indefinite system of equations. The algorithm includes efficient construction of the global‐basis prolongator using Lanczos vectors, predictor–corrector smoothing procedures, and a heuristic two‐level feedback loop aimed at ensuring convergence. Numerical experiments consisting of 3D Helmholtz equations and shear banding problems with strain softening demonstrate the excellent performance of the method. Copyright © 2000 John Wiley & Sons, Ltd.     

Coupled finite element – hierarchical boundary element methods for dynamic soil–structure interaction in the frequency domain

This paper discusses the coupling of finite element and fast boundary element methods for the solution of dynamic soil–structure interaction problems in the frequency domain. The application of hierarchical matrices in the boundary element formulation allows considering much larger problems compared to classical methods. Three coupling methodologies are presented and their computational performance is assessed through numerical examples. It is demonstrated that the use of hierarchical matrices renders a direct coupling approach the least efficient, as it requires the assembly of a dynamic soil stiffness matrix. Iterative solution procedures are presented as well, and it is shown that the application of such schemes to dynamic soil–structure interaction problems in the frequency domain is not trivial, as convergence can hardly be achieved if no relaxation procedure is incorporated. Aitken's Δ²‐method is therefore employed in sequential iterative schemes for the calculation of an optimized interface relaxation parameter, while a novel relaxation technique is proposed for parallel iterative algorithms. It is demonstrated that the efficiency of these algorithms strongly depends on the boundary conditions applied to each subdomain; the fastest convergence is observed if Neumann boundary conditions are imposed on the stiffest subdomain. The use of a dedicated solver for each subdomain hence results in a reduced computational effort. A monolithic coupling strategy, often used for the solution of fluid–structure interaction problems, is also introduced. The governing equations are simultaneously solved in this approach, while the assembly of a dynamic soil stiffness matrix is avoided. Copyright © 2013 John Wiley & Sons, Ltd.     

A new element‐by‐element method for trajectory calculations with tetrahedral finite element meshes

A new method to track massless particles in a three‐dimensional flow field is presented. The method is based on an element‐by‐element approach coupled with a predictor–corrector shooting scheme and does not use any time step. By analogy with time‐dependent schemes, the number of shootings is related to an equivalent number of time steps. The method has been implemented in a finite element framework using unstructured tetrahedral finite element meshes. However, it is general enough so that it can be implemented in finite difference and finite volume frameworks as well. It has been tested on a variety of flow systems namely: the Poiseuille flow in an empty circular pipe, the rotating flow in a stirred tank, the shear flow in a square tank and the flow through a static mixer. Accuracy has been found to depend on the accuracy of the velocity computation, the number of points per element and the level of mesh refinement. Copyright © 2006 John Wiley & Sons, Ltd.     

Arbitrary discontinuities in space–time finite elements by level sets and X‐FEM

An enriched finite element method with arbitrary discontinuities in space–time is presented. The discontinuities are treated by the extended finite element method (X‐FEM), which uses a local partition of unity enrichment to introduce discontinuities along a moving hyper‐surface which is described by level sets. A space–time weak form for conservation laws is developed where the Rankine–Hugoniot jump conditions are natural conditions of the weak form. The method is illustrated in the solution of first order hyperbolic equations and applied to linear first order wave and non‐linear Burgers' equations. By capturing the discontinuity in time as well as space, results are improved over capturing the discontinuity in space alone and the method is remarkably accurate. Implications to standard semi‐discretization X‐FEM formulations are also discussed. Copyright © 2004 John Wiley & Sons, Ltd.