Numerical simulations of a transient injection flow at low Mach number regime

In this paper, a transient injection flow at low Mach number regime is investigated. Three different methods are used and analyzed. Two of them are based on asymptotic models of the Navier–Stokes equations valid for small Mach numbers, whereas the other is based on the full compressible Navier–Stokes equations, with particular care given to the discretization at low Mach numbers. Numerical solutions are computed both with or without the gravity force. Finally, the performance of the solvers in terms of CPU‐time consumption is investigated, and the sensitivity of the solution to some parameters, which affect CPU time is also performed. Copyright © 2008 John Wiley & Sons, Ltd.     

Lattice Boltzmann simulation of dispersion in two‐dimensional tidal flows

A lattice Boltzmann method (LBM) is applied to problems of dispersion in two‐dimensional water flows. The water flow is modelled by shallow water equations. A two‐distribution lattice Boltzmann equation algorithm is presented to solve the pollutant transport problem within the framework of shallow water flow. One distribution models the shallow water flow. The other distribution models the pollutant transport. Flow characteristics and concentration profiles of dispersive species are obtained at various flow regimes. For fast water flow, the concentration profiles are highly affected by the flow advection and become completely different from those at slow water flow. Numerical results are presented for pollutant transport in bounded and open channel flows. The proposed LBM is also used to simulate a pollution event in the Strait of Gibraltar. The obtained results indicate that the present method is useful for the investigation of transport phenomena by shallow water flows in complex geometries and practical flow problems. Copyright © 2008 John Wiley & Sons, Ltd.     

PVU and wave-particle splitting schemes for Euler equations of gas dynamics

A new way of flux-splitting, termed as the wave-particle splitting is presented for developing upwind methods for solving Euler equations of gas dynamics. Based on this splitting, two new upwind methods termed as Acoustic Flux Vector Splitting (AFVS) and Acoustic Flux Difference Splitting (AFDS) methods are developed. A new Boltzmann scheme, which closely resembles the wave-particle splitting, is developed using the kinetic theory of gases. This method, termed as Peculiar Velocity based Upwind (PVU) method, uses the concept of peculiar velocity for upwinding. A special feature of all these methods is that the unidirectional and multidirectional parts of the flux vector are treated separately. Extensive computations done using these schemes demonstrate the soundness of the ideas.     

A computational stream function method for two‐dimensional incompressible viscous flows

This work concerns the development of a numerical method based on the stream function formulation of the Navier–Stokes equations to simulate two‐dimensional—plane or axisymmetric—viscous flows. The main features of the proposed method are: the use of the high order finite‐difference compact method for the discretization of the stream function equation, the implicit pseudo‐transient Newton–Krylov‐multigrid matrix free method for the stationary stream function equation and the fourth order Runge–Kutta method for the integration of non‐stationary flows. Copyright © 2005 John Wiley & Sons, Ltd.     

A wideband fast multipole algorithm for two‐dimensional volume integral equations

This paper describes a wideband fast multipole algorithm (FMA) for the computation of two‐dimensional volume integral equations. Our previous paper presented the wideband FMA by switching between the diagonal and non‐diagonal forms according to cell size and required accuracy. In order to improve the efficiency of the algorithm, we use interpolation and filtering techniques. Moreover, we introduce a simple and efficient way to store sequences of the special functions and their discrete Fourier transforms. Numerical examples show that the computational and memory complexities are reduced from O(N²) to O(N), where N is the number of square elements followed by the discretization of the volume integral equations. The computation results show very good agreement with the analytical solutions. We present some numerical results for the computation of scattering from a cylindrical object with sharp edges and a Gaussian‐like inhomogeneous cylinder. Copyright © 2008 John Wiley & Sons, Ltd.     

A coupled BEM and arbitrary Lagrangian–Eulerian FEM model for the solution of two‐dimensional laminar flows in external flow fields

This paper describes a new computational model developed to solve two‐dimensional incompressible viscous flow problems in external flow fields. The model based on the Navier–Stokes equations in primitive variables is able to solve the infinite boundary value problems by extracting the boundary effects on a specified finite computational domain, using the pressure projection method. The external flow field is simulated using the boundary element method by solving a pressure Poisson equation that assumes the pressure as zero at the infinite boundary. The momentum equation of the flow motion is solved using the three‐step finite element method. The arbitrary Lagrangian–Eulerian method is incorporated into the model, to solve the moving boundary problems. The present model is applied to simulate various external flow problems like flow across circular cylinder, acceleration and deceleration of the circular cylinder moving in a still fluid and vibration of the circular cylinder induced by the vortex shedding. The simulation results are found to be very reasonable and satisfactory. Copyright © 2001 John Wiley & Sons, Ltd.     

Application of the level set method to the finite element solution of two‐phase flows

This paper presents a method to solve two‐phase flows using the finite element method. On one hand, the algorithm used to solve the Navier–Stokes equations provides the neccessary stabilization for using the efficient and accurate three‐node triangles for both the velocity and pressure fields. On the other hand, the interface position is described by the zero‐level set of an indicator function. To maintain accuracy, even for large‐density ratios, the pseudoconcentration function is corrected at the end of each time step using an algorithm successfully used in the finite difference context. Coupling of both problems is solved in a staggered way. As demonstrated by the solution of a number of numerical tests, the procedure allows dealing with problems involving two interacting fluids with a large‐density ratio. Copyright © 2001 John Wiley & Sons, Ltd.     

An unconditionally stable three level finite difference scheme for solving parabolic two‐step micro heat transport equations in a three‐dimensional double‐layered thin film

Heat transport at the microscale is of vital importance in microtechnology applications. The heat transport equations are parabolic two‐step equations, which are different from the traditional heat diffusion equation. In this study, we develop a three‐level finite difference scheme for solving the micro heat transport equations in a three‐dimensional double‐layered thin film. It is shown by the discrete energy method that the scheme is unconditionally stable. Numerical results for thermal analysis of a gold layer on a chromium padding layer are obtained. Copyright © 2003 John Wiley & Sons, Ltd.     

A posteriori finite element bounds to linear functional outputs of the three‐dimensional Navier–Stokes equations

An implicit a posteriori finite element error estimation method is presented to inexpensively calculate lower and upper bounds for a linear functional output of the numerical solutions to the three‐dimensional Navier–Stokes (N–S) equations. The novelty of this research is to utilize an augmented Lagrangian based on a coarse mesh linearization of the N–S equations and the finite element tearing and interconnecting (FETI) procedure. The latter approach extends the a posteriori bound method to the three‐dimensional Crouzeix–Raviart space for N–S problems. The computational advantage of the bound procedure is that a single coupled non‐symmetric large problem can be decomposed into several uncoupled symmetric small problems. A simple model problem, which is selected here to illustrate the procedure, is to find the lower and upper bounds of the average velocity of a pressure driven, incompressible, steady Newtonian fluid flow moving at low Reynolds numbers through an endless square channel which has an array of rectangular obstacles. Numerical results show that the bounds for this output are rigorous, i.e. always in the asymptotic certainty regime, that they are sharp and that the required computational resources decrease significantly. Parallel implementation on a Beowulf cluster is also reported. Copyright © 2004 John Wiley & Sons, Ltd.     

A two‐grid method for expanded mixed finite‐element solution of semilinear reaction–diffusion equations

We present a scheme for solving two‐dimensional semilinear reaction–diffusion equations using an expanded mixed finite element method. To linearize the mixed‐method equations, we use a two‐grid algorithm based on the Newton iteration method. The solution of a non‐linear system on the fine space is reduced to the solution of two small (one linear and one non‐linear) systems on the coarse space and a linear system on the fine space. It is shown that the coarse grid can be much coarser than the fine grid and achieve asymptotically optimal approximation as long as the mesh sizes satisfy H=O(h^1/3). As a result, solving such a large class of non‐linear equation will not be much more difficult than solving one single linearized equation. Copyright © 2003 John Wiley & Sons, Ltd.     

An algorithm for modelling the interaction of a flexible rod with a two‐dimensional high‐speed flow

We present an algorithm for modelling coupled dynamic interactions of a very thin flexible structure immersed in a high‐speed flow. The modelling approach is based on combining an Eulerian finite volume formulation for the fluid flow and a Lagrangian large‐deformation formulation for the dynamic response of the structure. The coupling between the fluid and the solid response is achieved via an approach based on extrapolation and velocity reconstruction inspired in the Ghost Fluid Method. The algorithm presented does not assume the existence of a region exterior to the fluid domain as it was previously proposed and, thus, enables the consideration of very thin open boundaries and structures where the flow may be relevant on both sides of the interface. We demonstrate the accuracy of the method and its ability to describe disparate flow conditions across a fixed thin rigid interface without pollution of the flow field across the solid interface by comparing with analytical solutions of compressible flows. We also demonstrate the versatility and robustness of the method in a complex fluid–structure interaction problem corresponding to the transient supersonic flow past a highly flexible structure. Copyright © 2005 John Wiley & Sons, Ltd.     

Extended discontinuous Galerkin methods for two‐phase flows: the spatial discretization

This work discusses a discontinuous Galerkin (DG) discretization for two‐phase flows. The fluid interface is represented by a level set, and the DG approximation space is adapted such that jumps and kinks in pressure and velocity fields can be approximated sharply. This adaption of the original DG space, which can be performed ‘on‐the‐fly’ for arbitrary interface shapes, is referred to as extended discontinuous Galerkin. By combining this ansatz with a special quadrature technique, one can regain spectral convergence properties for low‐regularity solutions, which is demonstrated by numerical examples. This work focuses on the aspects of spatial discretization, and special emphasis is devoted on how to overcome problems related to quadrature, small cut cells, and condition number of linear systems. Temporal discretization will be discussed in future works. Copyright © 2016 John Wiley & Sons, Ltd.     

On a two‐level finite element method for the incompressible Navier–Stokes equations

We consider the Galerkin finite element method for the incompressible Navier–Stokes equations in two dimensions, where the finite‐dimensional space(s) employed consist of piecewise polynomials enriched with residual‐free bubble functions. To find the bubble part of the solution, a two‐level finite element method (TLFEM) is described and its application to the Navier–Stokes equation is displayed. Numerical solutions employing the TLFEM are presented for three benchmark problems. We compare the numerical solutions using the TLFEM with the numerical solutions using a stabilized method. Copyright © 2001 John Wiley & Sons, Ltd.     

A coupled hp‐finite element scheme for the solution of two‐dimensional electrostrictive materials

As part of the ongoing research within the field of computational analysis for the coupled electro‐magneto‐mechanical response of smart materials, the problem of linearised electrostriction is revisited and analysed for the first time using the framework of hp‐finite elements. The governing equations modelling the physics of the dielectric are suitably modified by introducing a new total Cauchy stress tensor (A. Dorfmann and R.W. Ogden. Nonlinear electroelasticity. Acta Mechanica, 174:167–183, 2005), which includes the electrostrictive effect and a staggered partitioned scheme for the numerical solution of the coupling phenomena. With the purpose of benchmarking numerical results, the problem of an infinite electrostrictive plate with a circular/elliptical dielectric insert is revisited. The presented analytical solution is based on the theoretical framework for two‐dimensional electrostriction proposed by Knops (R.J. Knops. Two‐dimensional electrostriction. Quarterly Journal of Mechanics and Applied Mathematics, 16:377–388, 1963) and uses classical techniques of complex variable analysis. Our presentation, to the best of our knowledge, provides the first correct closed form expression for the solution to the infinite electrostrictive plate with a circular/elliptical dielectric insert, correcting the errors made in previous presentations of this problem. We use this analytical solution to assess the accuracy, efficiency and robustness of the hp‐formulation in the case of nearly incompressible electrostrictive materials. Copyright © 2012 John Wiley & Sons, Ltd.     

A front remeshing technique for a Lagrangian description of moving interfaces in two‐fluid flows

The numerical analysis of two‐fluid flows involves the treatment of a discontinuity that appears at the separating interface. Classical Lagrangian schemes applied to update the front position between two immiscible incompressible fluids have been long recognized to provide a sharp representation of the interface. However, the main drawback of these approaches is the progressive distortion in the distribution of the markers used to identify the material front. To avoid this problem, an interface remeshing algorithm based on the diffuse approximation of the interface curvature is proposed in this work. In addition, the remeshed front is enforced to preserve the global volume. These new aspects are incorporated in an existing fluid dynamics formulation for the analysis of two‐fluid flows problems. The resulting formulation is called in this work as the moving Lagrangian interface remeshing technique (MLIRT). Copyright © 2005 John Wiley & Sons, Ltd.     

Random walk method for the two‐ and three‐dimensional Laplace,Poisson and Helmholtz's equations

The random walk method (RWM) is developed here for solving the Laplace, Poisson, and Helmholtz equations in two and three dimensions. The RWM is a local method, i.e. the solution at an arbitrary point can be determined without having to obtain the complete field solution. The method is based on the properties of diffusion processes, the Itô formula, the Dynkin formula, the Feynman–Kac functional, and Monte Carlo simulation. Simplicity, stability, accuracy, and generality are the main features of the proposed method. The RWK is inherently parallel and this fact has been fully exploited in this paper. Extensive numerical results have been presented in order to understand the various parameters involved in the method. Copyright © 2001 John Wiley & Sons, Ltd.     

Multilevel solution of the time‐harmonic Maxwell's equations based on edge elements

A widely used approach for the computation of time‐harmonic electromagnetic fields is based on the well‐known double‐curl equation for either E or H , where edge elements are an appealing choice for finite element discretizations. Yet, the nullspace of the curl ‐operator comprises a considerable part of all spectral modes on the finite element grid. Thus standard multilevel solvers are rendered inefficient, as they essentially hinge on smoothing procedures like Gauss–Seidel relaxation, which cannot provide a satisfactory error reduction for modes with small or even negative eigenvalues. We propose to remedy this situation by an extended multilevel algorithm which relies on corrections in the space of discrete scalar potentials. After every standard V‐cycle with respect to the canonical basis of edge elements, error components in the nullspace are removed by an additional projection step. Furthermore, a simple criterion for the coarsest mesh is derived to guarantee both stability and efficiency of the iterative multilevel solver. For the whole scheme we observe convergence rates independent of the refinement level of the mesh. The sequence of nested meshes required for our multilevel techniques is constructed by adaptive refinement. To this end we have devised an a posteriori error indicator based on stress recovery. Copyright © 1999 John Wiley & Sons, Ltd.     

Implementation of regularized isogeometric boundary element methods for gradient‐based shape optimization in two‐dimensional linear elasticity

The present work addresses shape sensitivity analysis and optimization in two‐dimensional elasticity with a regularized isogeometric boundary element method (IGABEM). Non‐uniform rational B‐splines are used both for the geometry and the basis functions to discretize the regularized boundary integral equations. With the advantage of tight integration of design and analysis, the application of IGABEM in shape optimization reduces the mesh generation/regeneration burden greatly. The work is distinct from the previous literatures in IGABEM shape optimization mainly in two aspects: (1) the structural and sensitivity analysis takes advantage of the regularized form of the boundary integral equations, eliminating completely the need of evaluating strongly singular integrals and jump terms and their shape derivatives, which were the main implementation difficulty in IGABEM, and (2) although based on the same Computer Aided Design (CAD) model, the mesh for structural and shape sensitivity analysis is separated from the geometrical design mesh, thus achieving a balance between less design variables for efficiency and refined mesh for accuracy. This technique was initially used in isogeometric finite element method and was incorporated into the present IGABEM implementation. Copyright © 2016 John Wiley & Sons, Ltd.     

Numerical method for the solution of non‐linear two‐dimensional inverse heat conduction problem using unstructured meshes

A new method of solving multidimensional heat conduction problems is formulated. The developed space marching method allows to determine quickly and exactly unsteady temperature distributions in the construction elements of irregular geometry. The method which is based on temperature measurements at the outer surface, is especially appropriate for determining transient temperature distribution in thick‐wall pressure components. Two examples are included to demonstrate the capabilities of the new approach. Copyright © 2000 John Wiley & Sons, Ltd.     

A reduced order model‐based preconditioner for the efficient solution of transient diffusion equations

This paper presents a novel class of preconditioners for the iterative solution of the sequence of symmetric positive‐definite linear systems arising from the numerical discretization of transient parabolic and self‐adjoint partial differential equations. The preconditioners are obtained by nesting appropriate projections of reduced‐order models into the classical iteration of the preconditioned conjugate gradient (PCG). The main idea is to employ the reduced‐order solver to project the residual associated with the conjugate gradient iterations onto the space spanned by the reduced bases. This approach is particularly appealing for transient systems where the full‐model solution has to be computed at each time step. In these cases, the natural reduced space is the one generated by full‐model solutions at previous time steps. When increasing the size of the projection space, the proposed methodology highly reduces the system conditioning number and the number of PCG iterations at every time step. The cost of the application of the preconditioner linearly increases with the size of the projection basis, and a trade‐off must be found to effectively reduce the PCG computational cost. The quality and efficiency of the proposed approach is finally tested in the solution of groundwater flow models. © 2016 The Authors. International Journal for Numerical Methods in Engineering Published by John Wiley & Sons Ltd.