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.     

Analysis of transient flow in pipelines with fluid–structure interaction using method of lines

A mathematical model is presented for transient flow in a pipeline with fluid–structure interaction. Water hammer theory and equations of axial motion for the pipeline are employed and the Poisson, junction and transient shear stress couplings are taken into account, which give rise to four coupled non‐linear, first‐order hyperbolic partial differential equations governing the fluid flow and pipe motion. These equations are discretized in space using the Keller box scheme and the method of lines is employed to reduce the partial differential equations to a system of ordinary differential equations. The resulting system is solved using a backward differentiation formulation method. The effect of transient shear stress on transient flow is investigated and the mechanisms underlying this effect are explored. The results revealed that the influence of transient shear stress can be significant and varies considerably, depending on the boundary conditions, viz, valve closure time. Copyright © 2005 John Wiley & Sons, Ltd.     

Numerical effects in the simulation of Ginzburg–Landau models for superconductivity

Inadequate spatial mesh resolution for simulation of the time‐dependent Ginzburg–Landau equations is shown to give rise to spurious solutions. Phenomenological studies to examine the effect of the physical parameters and boundary conditions in 2D and 3D are presented to illustrate the solution structure and to highlight non‐linear effects related to evolving vortex patterns. We illustrate and explore this issue further by considering a related simplified model in which the number of vortices is equal to the ‘winding number’ that is associated with the applied boundary conditions. Using this model we demonstrate that the solution structure is non‐unique for several values of winding number. Copyright © 2004 John Wiley & Sons, Ltd.     

Energy–momentum consistent finite element discretization of dynamic finite viscoelasticity

This paper is concerned with energy–momentum consistent time discretizations of dynamic finite viscoelasticity. Energy consistency means that the total energy is conserved or dissipated by the fully discretized system in agreement with the laws of thermodynamics. The discretization is energy–momentum consistent if also momentum maps are conserved when group motions are superimposed to deformations. The performed approximation is based on a three‐field formulation, in which the deformation field, the velocity field and a strain‐like viscous internal variable field are treated as independent quantities. The new non‐linear viscous evolution equation satisfies a non‐negative viscous dissipation not only in the continuous case, but also in the fully discretized system. The initial boundary value problem is discretized by using finite elements in space and time. Thereby, the temporal approximation is performed prior to the spatial approximation in order to preserve the stress objectivity for finite rotation increments (incremental objectivity). Although the present approach makes possible to design schemes of arbitrary order, the focus is on finite elements relying on linear Lagrange polynomials for the sake of clearness. The discrete energy–momentum consistency is based on the collocation property and an enhanced second Piola–Kirchhoff stress tensor. The obtained coupled non‐linear algebraic equations are consistently linearized. The corresponding iterative solution procedure is associated with newly proposed convergence criteria, which take the discrete energy consistency into account. The iterative solution procedure is therefore not complicated by different scalings in the independent variables, since the motion of the element is taken into account for solving the viscous evolution equation. Representative numerical simulations with various boundary conditions show the superior stability of the new time‐integration algorithm in comparison with the ordinary midpoint rule. Both the quasi‐rigid deformations during a free flight, and large deformations arising in a dynamic tensile test are considered. Copyright © 2009 John Wiley & Sons, Ltd.     

Sensitivity analysis and design optimization of three‐dimensional non‐linear aeroelastic systems by the adjoint method

We consider the problem of optimizing a non‐linear aeroelastic system in steady‐state conditions, where the structure is represented by a detailed finite element model, and the aerodynamic loads are predicted by the discretization of the non‐linear Euler equations. We present a solution method for this problem that is based on the three‐field formulation of fluid–structure interaction problems, and the adjoint approach for coupled sensitivity analysis. We discuss the computational complexity of the proposed solution method, describe its implementation on parallel processors, and illustrate its computational efficiency with the aeroelastic optimization of various wings. Copyright © 2002 John Wiley & Sons, Ltd.     

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 finite difference solution of non‐linear systems of radiative–conductive heat transfer equations

A finite difference solution for a system of non‐linear integro–differential equations modelling the steady‐state combined radiative–conductive heat transfer is proposed. A new backward–forward finite difference scheme is formulated for the Radiative Transfer Equation. The non‐linear heat conduction equation is solved using the Kirchhoff transformation associated with a centred finite difference scheme. The coupled system of equations is solved using a fixed‐point method, which relates to the temperature field. An application on a real insulator composed of silica fibres is illustrated. The results show that the method is very efficient. Copyright © 2002 John Wiley & Sons, Ltd.     

A simultaneous solution procedure for strong interactions of generalized Newtonian fluids and viscoelastic solids at large strains

The paper presents a methodology for numerical analyses of coupled systems exhibiting strong interactions of viscoelastic solids and generalized Newtonian fluids. In the monolithic approach, velocity variables are used for both solid and fluid, and the entire set of model equations is discretized with stabilized space–time finite elements. A viscoelastic material model for finite deformations, which is based on the concept of internal variables, describes the stress‐deformation behaviour of the solid. In the generalized Newtonian approach for the fluid, the viscosity depends on the shear strain rate, leading to common non‐Newtonian fluid models like the power‐law. The consideration of non‐linear constitutive equations for solid and fluid documents the capability of the monolithic space–time finite element formulation to deal with complex material models. The methodology is applied to fluid‐conveying cantilevered pipes in order to determine the influence of material non‐linearities on stability characteristics of coupled systems. Copyright © 2005 John Wiley & Sons, Ltd.     

A combined fluid–structure interaction and multi–field scalar transport model for simulating mass transport in biomechanics

Mass transport processes are known to play an important role in many fields of biomechanics such as respiratory, cardiovascular, and biofilm mechanics. In this paper, we present a novel computational model considering the effect of local solid deformation and fluid flow on mass transport. As the transport processes are assumed to influence neither structure deformation nor fluid flow, a sequential one‐way coupling of a fluid–structure interaction (FSI) and a multi‐field scalar transport model is realized. In each time step, first the non‐linear monolithic FSI problem is solved to determine current local deformations and velocities. Using this information, the mass transport equations can then be formulated on the deformed fluid and solid domains. At the interface, concentrations are related depending on the interfacial permeability. First numerical examples demonstrate that the proposed approach is suitable for simulating convective and diffusive scalar transport on coupled, deformable fluid and solid domains. Copyright © 2014 John Wiley & Sons, Ltd.     

Numerical modelling of fluids mixing,heat transfer and non‐equilibrium redox chemical reactions in fluid‐saturated porous rocks

Non‐equilibrium redox chemical reactions of high orders are ubiquitous in fluid‐saturated porous rocks within the crust of the Earth. The numerical modelling of such high‐order chemical reactions becomes a challenging problem because these chemical reactions are not only produced strong non‐linear source/sink terms for reactive transport equations, but also often coupled with the fluids mixing, heat transfer and reactive mass transport processes. In order to solve this problem effectively and efficiently, it is desirable to reduce the total number of reactive transport equations with strong non‐linear source/sink terms to a minimum in a computational model. For this purpose, the concept of the chemical reaction rate invariant is used to develop a numerical procedure for dealing with fluids mixing, heat transfer and non‐equilibrium redox chemical reactions in fluid‐saturated porous rocks. Using the proposed concept and numerical procedure, only one reactive transport equation, which is used to describe the distribution of the chemical product and has a strong non‐linear source/sink term, needs to be solved for each of the non‐equilibrium redox chemical reactions. The original reactive transport equations of the chemical reactants with strong non‐linear source/sink terms are turned into the conventional mass transport equations of the chemical reaction rate invariants without any non‐linear source/sink terms. A testing example, for some aspects of which the analytical solutions are available, is used to validate the proposed numerical procedure. The related numerical solutions have demonstrated that (1) the proposed numerical procedure is useful and applicable for dealing with the coupled problem between fluids mixing, heat transfer and non‐equilibrium redox chemical reactions of high orders in fluid‐saturated porous rocks; (2) the interaction between the solute diffusion, solute advection and chemical kinetics is an important mechanism to control distribution patterns of chemical products in an ore‐forming process; and (3) if the pore‐fluid pressure gradient is lithostatic, it is difficult for the chemical equilibrium to be attained within permeable cracks and geological faults within the crust of the Earth. Copyright © 2005 John Wiley & Sons, Ltd.     

Fast BEM–FEM mortar coupling for acoustic–structure interaction

A coupling algorithm based on Lagrange multipliers is proposed for the simulation of structure–acoustic field interaction. Finite plate elements are coupled to a Galerkin boundary element formulation of the acoustic domain. The interface pressure is interpolated as a Lagrange multiplier, thus, allowing the coupling of non‐matching grids. The resulting saddle‐point problem is solved by an approximate Uzawa‐type scheme in which the matrix–vector products of the boundary element operators are evaluated efficiently by the fast multipole boundary element method. The algorithm is demonstrated on the example of a cavity‐backed elastic panel. Copyright © 2005 John Wiley & Sons, Ltd.     

Truncation error and stability analysis of iterative and non‐iterative Thomas–Gladwell methods for first‐order non‐linear differential equations

The consistency and stability of a Thomas–Gladwell family of multistage time‐stepping schemes for the solution of first‐order non‐linear differential equations are examined. It is shown that the consistency and stability conditions are less stringent than those derived for second‐order governing equations. Second‐order accuracy is achieved by approximating the solution and its derivative at the same location within the time step. Useful flexibility is available in the evaluation of the non‐linear coefficients and is exploited to develop a new non‐iterative modification of the Thomas–Gladwell method that is second‐order accurate and unconditionally stable. A case study from applied hydrogeology using the non‐linear Richards equation confirms the analytic convergence assessment and demonstrates the efficiency of the non‐iterative formulation. Copyright © 2004 John Wiley & Sons, Ltd.     

Error bounds on block Gauss–Seidel solutions of coupled multiphysics problems

Mathematical models in many fields often consist of coupled sub‐models, each of which describes a different physical process. For many applications, the quantity of interest from these models may be written as a linear functional of the solution to the governing equations. Mature numerical solution techniques for the individual sub‐models often exist. Rather than derive a numerical solution technique for the full coupled model, it is therefore natural to investigate whether these techniques may be used by coupling in a block Gauss–Seidel fashion. In this study, we derive two a posteriori bounds for such linear functionals. These bounds may be used on each Gauss–Seidel iteration to estimate the error in the linear functional computed using the single physics solvers, without actually solving the full, coupled problem. We demonstrate the use of the bound first by using a model problem from linear algebra, and then a linear ordinary differential equation example. We then investigate the effectiveness of the bound using a non‐linear coupled fluid‐temperature problem. One of the bounds derived is very sharp for most linear functionals considered, allowing us to predict very accurately when to terminate our block Gauss–Seidel iteration. Copyright © 2011 John Wiley & Sons, Ltd.     

A coupled two‐scale shell model with applications to layered structures

In this paper a coupled two‐scale shell model is presented. A variational formulation and associated linearization for the coupled global–local boundary value problem is derived. For small strain problems, various numerical solutions are computed within the so‐called FE ² method. The discretization of the shell is performed with quadrilaterals, whereas the local boundary value problems at the integration points of the shell are discretized using 8‐noded or 27‐noded brick elements or so‐called solid shell elements. At the bottom and top surface of the representative volume element stress boundary conditions are applied, whereas at the lateral surfaces the in‐plane displacements are prescribed. For the out‐of‐plane displacements link conditions are applied. The coupled nonlinear boundary value problems are simultaneously solved within a Newton iteration scheme. With an important test, the correct material matrix for the stress resultants assuming linear elasticity and a homogeneous continuum is verified.Copyright © 2013 John Wiley & Sons, Ltd.     

Adaptive methods for thermomechanical coupled contact problems

In this paper an adaptive method for the analysis of thermomechanical coupled multi‐body contact problems is presented. The method is applied to non‐linear elastic solids undergoing finite (thermal) deformations. The contact model considers non‐linear pressure‐dependent heat flux as well as frictional heating in the interface. A time–space‐finite element discretization of the governing equations is formulated including unilateral constraints due to contact. A staggered solution algorithm has been constructed that allows an independent spatial discretization of the coupled subproblems. A posteriori projection‐based error estimators, which enforce implicitly the special boundary conditions due to thermal contact, are used to control the spatial discretization as well as the adaptive time stepping. Numerical examples are presented to corroborate the applicability of the adaptive algorithm to the considered problem type. Copyright © 2004 John Wiley & Sons, Ltd.     

Efficient non‐linear solid–fluid interaction analysis by an iterative BEM/FEM coupling

An iterative coupling of finite element and boundary element methods for the time domain modelling of coupled fluid–solid systems is presented. While finite elements are used to model the solid, the adjacent fluid is represented by boundary elements. In order to perform the coupling of the two numerical methods, a successive renewal of the variables on the interface between the two subdomains is performed through an iterative procedure until the final convergence is achieved. In the case of local non‐linearities within the finite element subdomain, it is straightforward to perform the iterative coupling together with the iterations needed to solve the non‐linear system. In particular a more efficient and a more stable performance of the new coupling procedure is achieved by a special formulation that allows to use different time steps in each subdomain. Copyright © 2005 John Wiley & Sons, Ltd.     

Numerical solution of eddy current problems in ferromagnetic bodies travelling in a transverse magnetic field

Eddy currents are investigated in a ferromagnetic bar travelling in a transverse magnetic field. Such an open boundary field problem is analysed by a hybrid approach based on Galerkin finite element formulation coupled with a separation of variables. A steady state is considered, introducing time‐periodic boundary conditions. The resultant system of non‐linear equations is solved by an iterative procedure based on Brouwer's fixed‐point theorem. Numerical results are presented for a bar of circular cross‐section made of cast steel or cast iron. Selected examples of the field distribution and characteristics of eddy‐current power losses are enclosed in graphic form. Copyright © 2003 John Wiley & Sons, Ltd.     

An efficient solution method for finite element ring‐rolling simulation

Finite element ring‐rolling simulation gives rise to poor conditioned non‐linear equations that require repeated solution. The associated computational costs are extreme making analysis impracticable in industry. This paper is concerned with a solution strategy that addresses this problem and involves the combined use of an arbitrary Lagrangian–Eulerian (ALE) formulation and a successive preconditioned conjugate gradient method (SPCGM). This approach, coupled to a finite element flow formulation, is shown to offer considerable computational savings. Through the combined use of the ALE flow formulation and the SPCGM the stability and condition of the non‐linear systems is enhanced. This purely iterative approach takes advantage of the slowly evolving velocity field and the self‐preconditioning offered by the SPCGM. The performance of the solver is compared against well‐known alternatives for varying problem sizes. The approach is shown to be generic but in particular makes ring‐rolling simulation a more practicable proposition. Copyright © 2000 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.     

Post buckling analysis of shear deformable shallow shells by the boundary element method

This paper presents four boundary element formulations for post buckling analysis of shear deformable shallow shells. The main differences between the formulations rely on the way non‐linear terms are treated and on the number of degrees of freedom in the domain. Boundary integral equations are obtained by coupling boundary element formulation of shear deformable plate and two‐dimensional plane stress elasticity. Four different sets of non‐linear integral equations are presented. Some domain integrals are treated directly with domain discretization whereas others are dealt indirectly with the dual reciprocity method. Each set of non‐linear boundary integral equations are solved using an incremental approach, where loads and prescribed boundary conditions are applied in small but finite increments. The resulting systems of equations are solved using a purely incremental technique and the Newton–Raphson technique with the Arc length method. Finally, the effect of imperfections (obtained from a linear buckling analysis) on the post‐buckling behaviour of axially compressed shallow shells is investigated. Results of several benchmark examples are compared with the published work and good agreement is obtained. Copyright © 2010 John Wiley & Sons, Ltd.