Riemann solvers for water hammer simulations by Godunov method

The water hammer phenomenon can be described by a 2×2 system of hyperbolic partial differential equations (PDEs). Numerical solution of these PDEs using finite‐volume schemes is investigated herein. The underlying concept of the Godunov scheme is the Riemann problem, that must be solved to provide fluxes between the computational cells. The presence of the kinetic terms in the momentum equation determines the existence of shock and rarefaction waves, which influence the design of the Riemann solver. Approximation of the expressions for the Riemann invariants and jump relationships can be used to derive first‐ and second‐order approximate, non‐iterative solvers. The first‐order approximate solver is almost 2000 times faster than the exact one, but gives inaccurate predictions when the densities and celerities are low. The second‐order approximate solver gives very accurate solutions, and is 300 times faster than the exact, iterative one. Detailed indications are provided in the appendices for the practical implementation of the Riemann solvers described herein. Copyright © 2000 John Wiley & Sons, Ltd.     

Hybrid finite element/volume method for shallow water equations

A hybrid numerical scheme based on finite element and finite volume methods is developed to solve shallow water equations. In the recent past, we introduced a series of hybrid methods to solve incompressible low and high Reynolds number flows for single and two‐fluid flow problems. The present work extends the application of hybrid method to shallow water equations. In our hybrid shallow water flow solver, we write the governing equations in non‐conservation form and solve the non‐linear wave equation using finite element method with linear interpolation functions in space. On the other hand, the momentum equation is solved with highly accurate cell‐center finite volume method. Our hybrid numerical scheme is truly a segregated method with primitive variables stored and solved at both node and element centers. To enhance the stability of the hybrid method around discontinuities, we introduce a new shock capturing which will act only around sharp interfaces without sacrificing the accuracy elsewhere. Matrix‐free GMRES iterative solvers are used to solve both the wave and momentum equations in finite element and finite volume schemes. Several test problems are presented to demonstrate the robustness and applicability of the numerical method. Copyright © 2010 John Wiley & Sons, Ltd.     

A three‐dimensional hybrid meshfree‐Cartesian scheme for fluid–body interaction

A numerical method based on a hybrid meshfree‐Cartesian grid is developed for solving three‐dimensional fluid–solid interaction (FSI) problems involving solid bodies undergoing large motion. The body is discretized and enveloped by a cloud of meshfree nodes. The motion of the body is tracked by convecting the meshfree nodes against a background of Cartesian grid points. Spatial discretization of second‐order accuracy is accomplished by the combination of a generalized finite difference (GFD) method and conventional finite difference (FD) method, which are applied to the meshfree and Cartesian nodes, respectively. Error minimization in GFD is carried out by singular value decomposition. The discretized equations are integrated in time via a second‐order fractional step projection method. A time‐implicit iterative procedure is employed to compute the new/evolving position of the immersed bodies together with the dynamically coupled solution of the flow field. The present method is applied on problems of free falling spheres and tori in quiescent flow and freely rotating spheres in simple shear flow. Good agreement with published results shows the ability of the present hybrid meshfree‐Cartesian grid scheme to achieve good accuracy. An application of the method to the self‐induced propulsion of a deforming fish‐like swimmer further demonstrates the capability and potential of the present approach for solving complex FSI problems in 3D. Copyright © 2011 John Wiley & Sons, Ltd.     

Unstructured,curved elements for the two‐dimensional high order discontinuous control‐volume/finite‐element method

Quadrilateral and triangular elements with curved edges are developed in the framework of spectral, discontinuous, hybrid control‐volume/finite‐element method for elliptic problems. In order to accommodate hybrid meshes, encompassing both triangular and quadrilateral elements, one single mapping is used. The scheme is applied to two‐dimensional problems with discontinuous, anisotropic diffusion coefficients, and the exponential convergence of the method is verified in the presence of curved geometries. Copyright © 2014 John Wiley & Sons, Ltd.     

Time‐domain hybrid formulations for wave simulations in three‐dimensional PML‐truncated heterogeneous media

We are concerned with the numerical simulation of wave motion in arbitrarily heterogeneous, elastic, perfectly‐matched‐layer‐(PML)‐truncated media. We extend in three dimensions a recently developed two‐dimensional formulation, by treating the PML via an unsplit‐field, but mixed‐field, displacement‐stress formulation, which is then coupled to a standard displacement‐only formulation for the interior domain, thus leading to a computationally cost‐efficient hybrid scheme. The hybrid treatment leads to, at most, third‐order in time semi‐discrete forms. The formulation is flexible enough to accommodate the standard PML, as well as the multi‐axial PML. We discuss several time‐marching schemes, which can be used à la carte, depending on the application: (a) an extended Newmark scheme for third‐order in time, either unsymmetric or fully symmetric semi‐discrete forms; (b) a standard implicit Newmark for the second‐order, unsymmetric semi‐discrete forms; and (c) an explicit Runge–Kutta scheme for a first‐order in time unsymmetric system. The latter is well‐suited for large‐scale problems on parallel architectures, while the second‐order treatment is particularly attractive for ready incorporation in existing codes written originally for finite domains. We compare the schemes and report numerical results demonstrating stability and efficacy of the proposed formulations. Copyright © 2014 John Wiley & Sons, Ltd.     

Comparing RBF‐FD approximations based on stabilized Gaussians and on polyharmonic splines with polynomials

In this work, we are concerned with radial basis function–generated finite difference (RBF‐FD) approximations. Numerical error estimates are presented for stabilized flat Gaussians (RBF(SGA)‐FD) and polyharmonic splines with supplementary polynomials (RBF(PHS)‐FD) using some analytical solutions of the Poisson equation in a square domain. Both structured and unstructured point clouds are employed for evaluating the influence of cloud refinement, size of local supports, and maximal permissible degree of the polynomials in RBF(PHS)‐FD. High order of accuracy was attained with both RBF(SGA)‐FD and RBF(PHS)‐FD especially for unstructured clouds. Absolute errors in the first and second derivatives were also estimated at all points of the domain using one of the analytical solutions. For RBF(SGA)‐FD, this test showed the occurrence of improprieties of some decentered supports localized on boundary neighborhoods. This phenomenon was not observed with RBF(PHS)‐FD.     

Numerical modelling of tidal flows in complex estuaries including turbulence: an unstructured finite volume solver and experimental validation

In this paper an unstructured finite volume model for quasi‐2D tidal flow with wet–dry fronts and turbulence modelling is presented, and applied to the Crouch–Roach estuarine system (Essex, U.K.). Two depth averaged turbulence models, a mixing length model and a k–ε model, are used in the numerical computations. An additional limiter to the production of turbulence due to bed friction is introduced in order to improve the performance and numerical stability of the model near wet–dry fronts. In addition to a first‐order and a second‐order schemes, an hybrid second‐order/first‐order upwind scheme which improves the accuracy of the first‐order scheme while maintaining a good numerical stability is used to discretize the convective flux. Numerical results are compared with observed current speed and water level data, with particular reference to the ability of the model to reproduce shallow water tidal harmonics. Copyright © 2006 John Wiley & Sons, Ltd.     

Slip flow and heat transfer in rectangular and circular microchannels using hybrid FE/FV method

Rarefied gas flows typically encountered in MEMS systems are numerically investigated in this study. Fluid flow and heat transfer in rectangular and circular microchannels within the slip flow regime are studied in detail by our recently developed implicit, incompressible, hybrid (finite element/finite volume) flow solver. The hybrid flow solver methodology is based on the pressure correction or projection method, which involves a fractional step approach to obtain an intermediate velocity field by solving the original momentum equations with the matrix‐free, implicit, cell‐centered finite volume method. The Poisson equation resulting from the fractional step approach is then solved by node based Galerkin finite element method for an auxiliary variable, which is closely related to pressure and is used to update the velocity field and pressure field. The hybrid flow solver has been extended for applications in MEMS by incorporating first order slip flow boundary conditions. Extended inlet boundary conditions are used for rectangular microchannels, whereas classical inlet boundary conditions are used for circular microchannels to emphasize on the entrance region singularity. In this study, rarefaction effects characterized by Knudsen number (Kn) in the range of 0 ⩽ Kn ⩽ 0.1 are numerically investigated for rectangular and circular microchannels with constant wall temperature. Extensive validations of our hybrid code are performed with available analytical solutions and experimental data for fully developed velocity profiles, friction factors, and Nusselt numbers. The influence of rarefaction on rectangular microchannels with aspect ratios between 0 and 1 is thoroughly investigated. Friction coefficients are found to be decreasing with increasing Knudsen number for both rectangular and circular microchannels. The reduction in the friction coefficients is more pronounced for rectangular microchannels with smaller aspect ratios. Effects of rarefaction and gas‐wall surface interaction parameter on heat transfer are analyzed for rectangular and circular microchannels. For most engineering applications, heat transfer is decreased with rarefaction. However, for fluids with very large Prandtl numbers, velocity slip dominates the temperature jump resulting in an increase in heat transfer with rarefaction. Depending on the gas‐wall surface interaction properties, extreme reductions in the Nusselt number can occur. Present results confirm the existence of a transition point below and above wherein heat transfer enhancement and reduction can occur. Copyright © 2011 John Wiley & Sons, Ltd.     

Finite element solution of free‐surface ship‐wave problems

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

Finite element analysis of time‐dependent semi‐infinite wave‐guides with high‐order boundary treatment

A new finite element (FE) scheme is proposed for the solution of time‐dependent semi‐infinite wave‐guide problems, in dispersive or non‐dispersive media. The semi‐infinite domain is truncated via an artificial boundary ??, and a high‐order non‐reflecting boundary condition (NRBC), based on the Higdon non‐reflecting operators, is developed and applied on ??. The new NRBC does not involve any high derivatives beyond second order, but its order of accuracy is as high as one desires. It involves some parameters which are chosen automatically as a pre‐process. A C⁰ semi‐discrete FE formulation incorporating this NRBC is constructed for the problem in the finite domain bounded by ??. Augmented and split versions of this FE formulation are proposed. The semi‐discrete system of equations is solved by the Newmark time‐integration scheme. Numerical examples concerning dispersive waves in a semi‐infinite wave guide are used to demonstrate the performance of the new method. Copyright © 2003 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.     

FLEXMG: A new library of multigrid preconditioners for a spectral/finite element incompressible flow solver

A new library called FLEX MG has been developed for a spectral/finite element incompressible flow solver called SFELES. FLEX MG allows the use of various types of iterative solvers preconditioned by algebraic multigrid methods. Two families of algebraic multigrid preconditioners have been implemented, namely smooth aggregation‐type and non‐nested finite element‐type. Unlike pure gridless multigrid, both of these families use the information contained in the initial fine mesh. A hierarchy of coarse meshes is also needed for the non‐nested finite element‐type multigrid so that our approaches can be considered as hybrid. Our aggregation‐type multigrid is smoothed with either a constant or a linear least‐square fitting function, whereas the non‐nested finite element‐type multigrid is already smooth by construction. All these multigrid preconditioners are tested as stand‐alone solvers or coupled with a GMRES method. After analyzing the accuracy of the solutions obtained with our solvers on a typical test case in fluid mechanics, their performance in terms of convergence rate, computational speed and memory consumption is compared with the performance of a direct sparse LU solver as a reference. Finally, the importance of using smooth interpolation operators is also underlined in the study. Copyright © 2010 John Wiley & Sons, Ltd.     

Lower bound static approach for the yield design of thick plates

The present work addresses the lower bound limit analysis (or yield design) of thick plates under shear‐bending interaction. Equilibrium finite elements are used to discretize the bending moment and the shear force fields. Different strength criteria, formulated in the five‐dimensional space of bending moment and shear force, are considered, one of them taking into account the interaction between bending and shear resistances. The criteria are chosen to be sufficiently simple so that the resulting optimization problem can be formulated as a second‐order cone programming problem (SOCP), which is solved by the dedicated solver MOSEK . The efficiency of the proposed finite element is illustrated by means of numerical examples on different plate geometries, for which the thin plate solutions as well as the pure shear solutions are accurately obtained as two different limit cases of the plate slenderness ratio. In particular, the proposed element exhibits a good behavior in the thin plate limit. Copyright © 2014 John Wiley & Sons, Ltd.     

A dissection solver with kernel detection for symmetric finite element matrices on shared memory computers

A direct solver for symmetric sparse matrices from finite element problems is presented. The solver is supposed to work as a local solver of domain decomposition methods for hybrid parallelization on cluster systems of multi‐core CPUs, and then it is required to run on shared memory computers and to have an ability of kernel detection. Symmetric pivoting with a given threshold factorizes a matrix with a decomposition introduced by a nested bisection and selects suspicious null pivots from the threshold. The Schur complement constructed from the suspicious null pivots is examined by a factorization with 1 × 1 and 2 × 2 pivoting and by a robust kernel detection algorithm based on measurement of residuals with orthogonal projections onto supposed image spaces. A static data structure from the nested bisection and a block sub‐structure for Schur complements at all bisection levels can use level 3 BLAS routines efficiently. Asynchronous task execution for each block can reduce idle time of processors drastically, and as a result, the solver has high parallel efficiency. Competitive performance of the developed solver to Intel Pardiso on shared memory computers is shown by numerical experiments. Copyright © 2014 John Wiley & Sons, Ltd.     

High‐order Higdon‐like boundary conditions for exterior transient wave problems

Recently developed non‐reflecting boundary conditions are applied for exterior time‐dependent wave problems in unbounded domains. The linear time‐dependent wave equation, with or without a dispersive term, is considered in an infinite domain. The infinite domain is truncated via an artificial boundary ??, and a high‐order non‐reflecting boundary condition (NRBC) is imposed on ??. Then the problem is solved numerically in the finite domain bounded by ??. The new boundary scheme is based on a reformulation of the sequence of NRBCs proposed by Higdon. We consider here two reformulations: one that involves high‐order derivatives with a special discretization scheme, and another that does not involve any high derivatives beyond second order. The latter formulation is made possible by introducing special auxiliary variables on ??. In both formulations the new NRBCs can easily be used up to any desired order. They can be incorporated in a finite element or a finite difference scheme; in the present paper the latter is used. In contrast to previous papers using similar formulations, here the method is applied to a fully exterior two‐dimensional problem, with a rectangular boundary. Numerical examples in infinite domains are used to demonstrate the performance and advantages of the new method. In the auxiliary‐variable formulation long‐time corner instability is observed, that requires special treatment of the corners (not addressed in this paper). No such difficulties arise in the high‐derivative formulation. Published in 2005 by John Wiley & Sons, Ltd.     

A new mass lumping scheme for the mixed hybrid finite element method

Diffusion‐type partial differential equation is a common mathematical model in physics. Solved by mixed finite elements, it leads to a system matrix which is not always an M‐matrix. Therefore, the numerical solution may exhibit unphysical results due to oscillations. The criterion necessary to obtain an M‐matrix is discussed in details for triangular, rectangular and tetrahedral elements. It is shown that the system matrix is never an M‐matrix for rectangular elements and can be an M‐matrix for triangular an tetrahedral elements if criteria on the element's shape and on the time step length are fulfilled. A new mass lumping scheme is developed which leads to a less restrictive criterion: the discretization must be weakly acute (all angles less than π/2) and there is no constraint on the time step length. The lumped formulation of mixed hybrid finite element can be applied not only to triangular meshes but also to more general shape elements in two and three dimensions. Numerical experiments show that, compared to the standard mixed hybrid formulation, the lumping scheme avoids (or strongly reduce) oscillations and does not create additional numerical errors. Copyright © 2006 John Wiley & Sons, Ltd.     

Application of the finite volume method and unstructured meshes to linear elasticity

A recent emergence of the finite volume method (FVM) in structural analysis promises a viable alternative to the well‐established finite element solvers. In this paper, the linear stress analysis problem is discretized using the practices usually associated with the FVM in fluid flows. These include the second‐order accurate discretization on control volumes of arbitrary polyhedral shape; segregated solution procedure, in which the displacement components are solved consecutively and iterative solvers for the systems of linear algebraic equations. Special attention is given to the optimization of the discretization practice in order to provide rapid convergence for the segregated solution procedure. The solver is set‐up to work efficiently on parallel distributed memory computer architectures, allowing a fast turn‐around for the mesh sizes expected in an industrial environment. The methodology is validated on two test cases: stress concentration around a circular hole and transient wave propagation in a bar. Finally, the steady and transient stress analysis of a Diesel injector valve seat in 3‐D is presented, together with the set of parallel speed‐up results. Copyright © 2000 John Wiley & Sons, Ltd.     

A TLM method for steady‐state convection–diffusion: Some additions and refinements

A recent paper introduced a novel and efficient scheme, based on the transmission line modelling (TLM) method, for solving steady‐state convection–diffusion problems. This paper shows how this one‐dimensional scheme can be adapted to include reaction and source terms and how it can be implemented with non‐equidistant nodes. It introduces new ways of calculating the necessary model parameters which can improve the accuracy of the scheme, shows how steady‐state solutions can be obtained directly, and compares results with those from two finite difference (FD) methods. While the cost of implementation is higher than for the FD schemes, the new TLM scheme can be significantly more accurate, especially when convection dominates. Copyright © 2008 John Wiley & Sons, Ltd.     

A stable and optimally convergent LaTIn‐CutFEM algorithm for multiple unilateral contact problems

In this paper, we propose a novel unfitted finite element method for the simulation of multiple body contact. The computational mesh is generated independently of the geometry of the interacting solids, which can be arbitrarily complex. The key novelty of the approach is the combination of elements of the CutFEM technology, namely, the enrichment of the solution field via the definition of overlapping fictitious domains with a dedicated penalty‐type regularisation of discrete operators and the LaTIn hybrid‐mixed formulation of complex interface conditions. Furthermore, the novel P1‐P1 discretisation scheme that we propose for the unfitted LaTIn solver is shown to be stable, robust, and optimally convergent with mesh refinement. Finally, the paper introduces a high‐performance 3D level set/CutFEM framework for the versatile and robust solution of contact problems involving multiple bodies of complex geometries, with more than 2 bodies interacting at a single point.     

On the accuracy of finite volume and discontinuous Galerkin discretizations for compressible flow on unstructured grids

This paper presents a comparison between two high‐order methods. The first one is a high‐order finite volume (FV) discretization on unstructured grids that uses a meshfree method (moving least squares (MLS)) in order to construct a piecewise polynomial reconstruction and evaluate the viscous fluxes. The second method is a discontinuous Galerkin (DG) scheme. Numerical examples of inviscid and viscous flows are presented and the solutions are compared. The accuracy of both methods, for the same grid resolution, is similar, although the DG scheme requires a larger number of degrees of freedom than the FV–MLS method. Copyright © 2009 John Wiley & Sons, Ltd.