A finite volume method with a modified ENO scheme using a Hermite interpolation to solve advection diffusion equations

In this paper we have developed a finite volume ENO scheme, third‐order accurate, based on cell averages and a TVD Runge–Kutta time discretization to solve advection–diffusion equations in a two‐dimensional spatial domain. We have designed a special interpolating polynomial based on a modified ENO scheme and a Hermite procedure which avoids the excessive smearing in regions with sharpconcentration fronts and the overcompression effects produced by the modified ENO technique. Thesemodifications do not affect the non‐oscillatory philosophy since we compare divided differences inthe modified ENO scheme and in the evaluation of the Hermite polynomial derivatives. Numericalresults compare favourably with their respective analytical solutions. Copyright © 2001 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.     

An essentially non‐oscillatory semi‐Lagrangian method for tidal flow simulations

We develop an essentially non‐oscillatory semi‐Lagrangian method for solving two‐dimensional tidal flows. The governing equations are derived from the incompressible Navier–Stokes equations with assumptions of shallow water flows including bed frictions, eddy viscosity, wind shear stresses and Coriolis forces. The method employs the modified method of characteristics to discretize the convective term in a finite element framework. Limiters are incorporated in the method to reconstruct an essentially non‐oscillatory algorithm at minor additional cost. The central idea consists in combining linear and quadratic interpolation procedures using nodes of the finite element where departure points are localized. The resulting semi‐discretized system is then solved by an explicit Runge–Kutta Chebyshev scheme with extended stages. This scheme adds in a natural way a stabilizing stage to the conventional Runge–Kutta method using the Chebyshev polynomials. The proposed method is verified for the recirculation tidal flow in a channel with forward‐facing step. We also apply the method for simulation of tidal flows in the Strait of Gibraltar. In both test problems, the proposed method demonstrates its ability to handle the interaction between water free‐surface and bed frictions. Copyright © 2009 John Wiley & Sons, Ltd.     

Local RBF‐FD solutions for steady convection–diffusion problems

This paper describes the application of radial basis function (RBF) based finite difference type scheme (RBF‐FD) for solving steady convection–diffusion equations. Numerical studies are made using multiquadric (MQ) RBF. By varying the shape parameter in MQ, the accuracy of the solution is seen to be highly improved for large values of Reynolds' numbers. The developed scheme has been compared with the corresponding finite difference scheme and found that the solutions obtained using the former are non‐oscillatory. Copyright © 2007 John Wiley & Sons, Ltd.     

Comparison of method of lines and finite difference solutions of 2‐D Navier–Stokes equations for transient laminar pipe flow

Performances of method of lines (MOL) and finite difference method (FDM) were tested from the viewpoints of solution accuracy and central processing unit (CPU) time by applying them to the solution of time‐dependent 2‐D Navier–Stokes equations for transient laminar flow without/with sudden expansion and comparing their results with steady‐state numerical predictions and measurements previously reported in the literature. Predictions of both methods were obtained on the same computer by using the same order of spatial discretization and the same uniform grid distribution. Axial velocity and pressure distribution in pipe flow and steady‐state reattachment lengths in sudden expansion flow on uniform grid distribution predicted by both methods were found to be in excellent agreement. Transient solutions of both methods for pipe flow problem show favourable comparison and are in accordance with expected trends. However, non‐physical oscillations were produced by both methods in the transient solution of sudden expansion pipe flow. MOL was demonstrated to yield non‐oscillatory solutions for recirculating flows when intelligent higher‐order discretization scheme is utilized for convective terms. MOL was found to be superior to FDM with respect to CPU and set‐up times and its flexibility for incorporation of other conservation equations. Copyright © 2001 John Wiley & Sons, Ltd.     

Efficient finite element formulation for geothermal heating systems. Part II: transient

This paper presents an extension to the work presented in Part I of this series of two articles to the transient case. Emphasis is placed on the development of a new model for heat flow in a double U‐shape vertical borehole heat exchanger and its thermodynamic interactions with surrounding soil mass. The discretization of the spatial‐temporal domain of the heat pipe model is done by the use of the space–time finite element technique in conjunction with the Petrov–Galerkin method and the finite difference method. The paper shows that the proposed model and the choice of the discretization technique, in addition to the utilization of a sequential numerical algorithm for solving the resulting system of non‐linear equations, have contributed in reducing significantly the required number of finite elements necessary for describing geothermal heating systems. Details of the mathematical derivations and comparison to experimental data are presented. Copyright © 2006 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.     

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.     

A conservative high‐order discontinuous Galerkin method for the shallow water equations with arbitrary topography

A conservative high‐order Godunov‐type scheme is presented for solving the balance laws of the 1D shallow water equations (SWE). The scheme adopts a finite element Runge–Kutta (RK) discontinuous Galerkin (DG) framework. Based on an overall third‐order accurate formulation, the model is referred to as RKDG3. Treatment of topographic source term is built in the DG approximation. Simplified formulae for initializing bed data at a discrete level are derived by assuming a local linear bed function to ease practical flow simulations. Owing to the adverse effects caused by using an uncontrolled global limiting process in an RKDG3 scheme (RKDG3‐GL), a new conservative RKDG3 scheme with user‐parameter‐free local limiting method (RKDG3‐LL) is designed to gain better accuracy and conservativeness. The advantages of the new RKDG3‐LL model are demonstrated by applying to several steady and transient benchmark flow tests with irregular (either differentiable or non‐differentiable) topography. Copyright © 2010 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.     

Improving multigrid performance for unstructured mesh drift–diffusion simulations on 147,000 cores

This study considers the scaling of three algebraic multigrid aggregation schemes for a finite element discretization of a drift–diffusion system, specifically the drift–diffusion model for semiconductor devices. The approach is more general and can be applied to other systems of partial differential equations. After discretization on unstructured meshes, a fully coupled multigrid preconditioned Newton–Krylov solution method is employed. The choice of aggregation scheme for generating coarser levels has a significant impact on the performance and scalability of the multigrid preconditioner. For the test cases considered, the uncoupled aggregation scheme, which aggregates/combines the immediate neighbors, followed by repartitioning and data redistribution for the coarser level matrices on a subset of the Message Passing Interface (MPI) processes, outperformed the two other approaches, including the baseline aggressive coarsening scheme. Scaling results are presented up to 147,456 cores on an IBM Blue Gene/P platform. A comparison of the scaling of a multigrid V‐cycle and W‐cycle is provided. Results for 65,536 cores demonstrate that a factor of 3.5 × reduction in time between the uncoupled aggregation and baseline aggressive coarsening scheme can be obtained by significantly reducing the iteration count due to the increased number of multigrid levels and the generation of better quality aggregates. Copyright © 2012 John Wiley & Sons, Ltd.     

Stabilized global–local X‐FEM for 3D non‐planar frictional crack using relevant meshes

A stabilized global–local quasi‐static contact algorithm for 3D non‐planar frictional crack is presented in the X‐FEM/level set framework. A three‐field weak formulation is considered and allows an independent discretization of the bulk and the crack interface. Then, a fine discretization of the interface can be defined according to the possible complex contact state along the crack faces independently from the mesh in the bulk. Furthermore, an efficient stabilized non‐linear LATIN solver dedicated to contact and friction is proposed. It allows solving in a unified framework frictionless and frictional contact at the crack interface with a symmetric formulation, no iterations on the local stage (unilateral contact law with/without friction), no calculation of any global tangent operator, and improved convergence rate. 2D and 3D patch tests are presented to illustrate the relevance of the proposed model and an actual 3D frictional crack problem under cyclic fretting loading is modeled. Copyright © 2011 John Wiley & Sons, Ltd.     

Non‐linear model reduction for uncertainty quantification in large‐scale inverse problems

We present a model reduction approach to the solution of large‐scale statistical inverse problems in a Bayesian inference setting. A key to the model reduction is an efficient representation of the non‐linear terms in the reduced model. To achieve this, we present a formulation that employs masked projection of the discrete equations; that is, we compute an approximation of the non‐linear term using a select subset of interpolation points. Further, through this formulation we show similarities among the existing techniques of gappy proper orthogonal decomposition, missing point estimation, and empirical interpolation via coefficient‐function approximation. The resulting model reduction methodology is applied to a highly non‐linear combustion problem governed by an advection–diffusion‐reaction partial differential equation (PDE). Our reduced model is used as a surrogate for a finite element discretization of the non‐linear PDE within the Markov chain Monte Carlo sampling employed by the Bayesian inference approach. In two spatial dimensions, we show that this approach yields accurate results while reducing the computational cost by several orders of magnitude. For the full three‐dimensional problem, a forward solve using a reduced model that has high fidelity over the input parameter space is more than two million times faster than the full‐order finite element model, making tractable the solution of the statistical inverse problem that would otherwise require many years of CPU time. Copyright © 2009 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 thermo‐hydro‐mechanical model for multiphase geomaterials in dynamics with application to strain localization simulation

In this paper, the non‐isothermal elasto‐plastic behaviour of multiphase geomaterials in dynamics is investigated with a thermo‐hydro‐mechanical model of porous media. The supporting mathematical model is based on averaging procedures within the hybrid mixture theory. A computationally efficient reduced formulation of the macroscopic balance equations that neglects the relative acceleration of the fluids, and the convective terms is adopted. The modified effective stress state is limited by the Drucker–Prager yield surface. Small strains and dynamic loading conditions are assumed. The standard Galerkin procedure of the finite element method is applied to discretize the governing equations in space, while the generalized Newmark scheme is used for the time discretization. The final non‐linear set of equations is solved by the Newton method with a monolithic approach. Coupled dynamic analyses of strain localization in globally undrained samples of dense and medium dense sands are presented as examples. Vapour pressure below the saturation water pressure (cavitation) develops at localization in case of dense sands, as experimentally observed. A numerical study of the regularization properties of the finite element model is shown and discussed. A non‐isothermal case of incipient strain localization induced by temperature increase where evaporation takes place is also analysed. Copyright © 2016 John Wiley & Sons, Ltd.     

Approximation of Cahn–Hilliard diffuse interface models using parallel adaptive mesh refinement and coarsening with C1 elements

A variational formulation and C¹ finite element scheme with adaptive mesh refinement and coarsening are developed for phase‐separation processes described by the Cahn–Hilliard diffuse interface model of transport in a mixture or alloy. The adaptive scheme is guided by a Laplacian jump indicator based on the corresponding term arising from the weak formulation of the fourth‐order non‐linear problem, and is implemented in a parallel solution framework. It is then applied to resolve complex evolving interfacial solution behavior for 2D and 3D simulations of the classic spinodal decomposition problem from a random initial mixture and to other phase‐transformation applications of interest. Simulation results and adaptive performance are discussed. The scheme permits efficient, robust multiscale resolution and interface characterization. Copyright © 2008 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.     

A finite‐volume particle‐in‐cell method for the numerical treatment of Maxwell‐Lorentz equations on boundary‐fitted meshes

A new conceptual framework solving numerically the time‐dependent Maxwell–Lorentz equations on a non‐rectangular quadrilateral mesh in two space dimensions is presented. Beyond a short review of the applied particle treatment based on the particle‐in‐cell method, a finite‐volume scheme for the numerical approximation of the Maxwell equations is introduced using non‐rectangular quadrilateral grid arrangements. The coupling of a high‐resolution FV Maxwell solver with the PIC method is a new approach in the context of self‐consistent charged particle simulation in electromagnetic fields. Furthermore, first simulation results of the time‐dependent behaviour of an externally applied‐B ion diode developed at the Forschungszentrum in Karlsruhe are presented. Copyright © 1999 John Wiley & Sons, Ltd.     

Numerical analysis of a new Eulerian–Lagrangian finite element method applied to steady‐state hot rolling processes

A finite element code for steady‐state hot rolling processes of rigid–visco‐plastic materials under plane–strain conditions was developed in a mixed Eulerian–Lagrangian framework. This special set up allows for a direct calculation of the local deformations occurring at the free surfaces outside the contact region between the strip and the work roll. It further simplifies the implementation of displacement boundary conditions, such as the impenetrability condition. When applied to different practical hot rolling situations, ranging from thick slab to ultra‐thin strip rolling, the velocity–displacement based model (briefly denoted as vu‐model) in this mixed Eulerian–Lagrangian reference system proves to be a robust and efficient method. The vu‐model is validated against a solely velocity‐based model (vv‐model) and against elementary methods based on the Kármán–Siebel and Orowan differential equations. The latter methods, when calibrated, are known to be in line with experimental results for homogeneous deformation cases. For a massive deformation it is further validated against the commercial finite‐element software package Abaqus/Explicit. It is shown that the results obtained with the vu‐model are in excellent agreement with the predictions of the vv‐model and that the vu‐model is even more robust than its vv‐counterpart. Throughout the study we assumed a rigid cylindrical work roll; only for the homogeneous test case, we also investigated the effect of an elastically deformable work roll within the frame of the Jortner Green's function method. The new modelling approach combines the advantages of conventional Eulerian and Lagrangian modelling concepts and can be extended to three dimensions in a straightforward manner. Copyright © 2004 John Wiley & Sons, Ltd.     

A finite deformation mortar contact formulation using a primal–dual active set strategy

In recent years, nonconforming domain decomposition techniques and, in particular, the mortar method have become popular in developing new contact algorithms. Here, we present an approach for 2D frictionless multibody contact based on a mortar formulation and using a primal–dual active set strategy for contact constraint enforcement. We consider linear and higher‐order (quadratic) interpolations throughout this work. So‐called dual Lagrange multipliers are introduced for the contact pressure but can be eliminated from the global system of equations by static condensation, thus avoiding an increase in system size. For this type of contact formulation, we provide a full linearization of both contact forces and normal (non‐penetration) and tangential (frictionless sliding) contact constraints in the finite deformation frame. The necessity of such a linearization in order to obtain a consistent Newton scheme is demonstrated. By further interpreting the active set search as a semi‐smooth Newton method, contact nonlinearity and geometrical and material nonlinearity can be resolved within one single iterative scheme. This yields a robust and highly efficient algorithm for frictionless finite deformation contact problems. Numerical examples illustrate the efficiency of our method and the high quality of results. Copyright © 2009 John Wiley & Sons, Ltd.