A cut‐cell non‐conforming Cartesian mesh method for compressible and incompressible flow

This paper details a multigrid‐accelerated cut‐cell non‐conforming Cartesian mesh methodology for the modelling of inviscid compressible and incompressible flow. This is done via a single equation set that describes sub‐, trans‐, and supersonic flows. Cut‐cell technology is developed to furnish body‐fitted meshes with an overlapping mesh as starting point, and in a manner which is insensitive to surface definition inconsistencies. Spatial discretization is effected via an edge‐based vertex‐centred finite volume method. An alternative dual‐mesh construction strategy, similar to the cell‐centred method, is developed. Incompressibility is dealt with via an artificial compressibility algorithm, and stabilization achieved with artificial dissipation. In compressible flow, shocks are captured via pressure switch‐activated upwinding. The solution process is accelerated with full approximation storage (FAS) multigrid where coarse meshes are generated automatically via a volume agglomeration methodology. This is the first time that the proposed discretization and solution methods are employed to solve a single compressible–incompressible equation set on cut‐cell Cartesian meshes. The developed technology is validated by numerical experiments. The standard discretization and alternative methods were found equivalent in accuracy and computational cost. The multigrid implementation achieved decreases in CPU time of up to one order of magnitude. Copyright © 2007 John Wiley & Sons, Ltd.     

Non‐conservative and conservative formulations of characteristics‐based numerical reconstructions for incompressible flows

An investigation of characteristics‐based (CB) schemes for solving the incompressible Navier–Stokes equations in conjunction with the artificial‐compressibility approach, is presented. Both non‐conservative and conservative CB numerical reconstructions are derived and their accuracy and convergence properties are assessed analytically and numerically. We demonstrate by means of eigenvalue analysis that there are differences in the spectral characteristics of these formulations that result in different convergence properties. Numerical tests for two‐ and three‐dimensional flows reveal that the two formulations provide similar accuracy but the non‐conservative formulation converges faster. Copyright © 2005 John Wiley & Sons, Ltd.     

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 decoupled augmented IIM strategy for incompressible two‐phase flows with interfaces on irregular domains

A decoupled augmented immersed interface method for solving incompressible two‐phase flows involving both irregular domains and interfaces is presented. In order to impose the prescribed velocity at the boundary of the irregular domain, singular force as one set of augmented variables is introduced. The velocity components at the two‐fluid interface as another set of augmented variables are introduced to satisfy the continuity condition of the velocity across the interface so that the jump conditions for the velocity and pressure are decoupled across the interface. The augmented variables and/or the forces along the interface/boundary are related to the jumps in both pressure and velocity and the jumps in their derivatives across the interface/boundary and applied to the fluid through jump conditions. The resulting augmented equation is a couple system of these two sets of augmented variables, and the direct application of the GMRES is impractical due to larger iterations. In this work, the novel decoupling of two sets of the augmented variables is proposed, and the decoupled augmented equation is then solved by the LU or the GMRES method. The Stokes equations are discretized via the finite difference method with the incorporation of jump contributions on a staggered Cartesian grid and solved by the conjugate gradient Uzawa‐type method. The numerical results show that second‐order accuracy for the velocity is confirmed. The present method has also been applied to solve for incompressible two‐phase Navier–Stokes flow with interfaces on irregular domains. Copyright © 2011 John Wiley & Sons, Ltd.     

A hybrid element formulation for the three‐dimensional penalty finite element analysis of incompressible fluids

This work presents a hybrid element formulation for the three‐dimensional penalty finite element analysis of incompressible Newtonian fluids. The formulation is based on a mixed variational statement in which velocity and stresses are treated as independent field variables. The main advantage of this formulation is that it bypasses the use of ad hoc techniques such as selective reduced integration that are commonly used in penalty‐based finite element formulations, and directly yields high accuracy for the velocity and stress fields without the need to carry out smoothing. In addition, since the stress degrees of freedom are condensed out at an element level, the cost of solving for the global degrees of freedom is the same as in a standard penalty finite element method, although the gain in accuracy for both the velocity and stress (including the pressure) fields is quite significant. Copyright © 2007 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.     

An L∞–Lp mesh‐adaptive method for computing unsteady bi‐fluid flows

This paper discusses the contribution of mesh adaptation to high‐order convergence of unsteady multi‐fluid flow simulations on complex geometries. The mesh adaptation relies on a metric‐based method controlling the L ^p‐norm of the interpolation error and on a mesh generation algorithm based on an anisotropic Delaunay kernel. The mesh‐adaptive time advancing is achieved, thanks to a transient fixed‐point algorithm to predict the solution evolution coupled with a metric intersection in the time procedure. In the time direction, we enforce the equidistribution of the error, i.e. the error minimization in L ^∞ norm. This adaptive approach is applied to an incompressible Navier–Stokes model combined with a level set formulation discretized on triangular and tetrahedral meshes. Applications to interface flows under gravity are performed to evaluate the performance of this method for this class of discontinuous flows. Copyright © 2010 John Wiley & Sons, Ltd.     

A discrete splitting finite element method for numerical simulations of incompressible Navier–Stokes flows

The presence of the pressure and the convection terms in incompressible Navier–Stokes equations makes their numerical simulation a challenging task. The indefinite system as a consequence of the absence of the pressure in continuity equation is ill‐conditioned. This difficulty has been overcome by various splitting techniques, but these techniques incur the ambiguity of numerical boundary conditions for the pressure as well as for the intermediate velocity (whenever introduced). We present a new and straightforward discrete splitting technique which never resorts to numerical boundary conditions. The non‐linear convection term can be treated by four different approaches, and here we present a new linear implicit time scheme. These two new techniques are implemented with a finite element method and numerical verifications are made. Copyright © 2005 John Wiley & Sons, Ltd.     

Free‐slip boundary conditions for simulating free‐surface incompressible flows through the particle finite element method

The possibility of using free‐slip conditions within the context of the particle finite element method (PFEM) is investigated. For high Reynolds number engineering applications in which tangential effects at the fluid–solid boundaries are not of primary interest, the use of free‐slip conditions can alleviate the need for very fine boundary layer meshes. Two novel ways for the imposition of free‐slip conditions in the framework of the PFEM are presented. The proposed approach emphasizes robustness and simplicity, while retaining a sufficient level of generality. These two methods are then tested in the case of dam break and sloshing problems, and their respective advantages and drawbacks are discussed. It is also shown how the use of free‐slip conditions can indirectly improve mass conservation properties of the PFEM, even when coarse meshes are employed. Copyright © 2016 John Wiley & Sons, Ltd.     

A monolithic geometric multigrid solver for fluid‐structure interactions in ALE formulation

We present a monolithic geometric multigrid solver for fluid‐structure interaction problems in Arbitrary Lagrangian Eulerian coordinates. The coupled dynamics of an incompressible fluid with nonlinear hyperelastic solids gives rise to very large and ill‐conditioned systems of algebraic equations. Direct solvers usually are out of question because of memory limitations, and standard coupled iterative solvers are seriously affected by the bad condition number of the system matrices. The use of partitioned preconditioners in Krylov subspace iterations is an option, but the convergence will be limited by the outer partitioning. Our proposed solver is based on a Newton linearization of the fully monolithic system of equations, discretized by a Galerkin finite element method. Approximation of the linearized systems is based on a monolithic generalized minimal residual method iteration, preconditioned by a geometric multigrid solver. The special character of fluid‐structure interactions is accounted for by a partitioned scheme within the multigrid smoother only. Here, fluid and solid field are segregated as Dirichlet–Neumann coupling. We demonstrate the efficiency of the multigrid iteration by analyzing 2d and 3d benchmark problems. While 2d problems are well manageable with available direct solvers, challenging 3d problems highly benefit from the resulting multigrid solver. Copyright © 2015 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.     

A Jacobian‐free‐based IIM for incompressible flows involving moving interfaces with Dirichlet boundary conditions

In this paper, a finite difference marker‐and‐cell (MAC) scheme is presented for the steady Stokes equations with moving interfaces and Dirichlet boundary condition. The moving interfaces are represented by Lagrangian control points and their position is updated implicitly using a Jacobian‐free approach within each time step. The forces at the moving interfaces are calculated from the position of the interfaces and interpolated using cubic splines and then applied to the fluid through the related jump conditions. The proposed Jacobian‐free Newton–generalized minimum residual (GMRES) method avoids the need to form and store the matrix explicitly in the computation of the inverse of the Jacobian and betters numerical stability. The Stokes equations are discretized on a MAC grid via a second‐order finite difference scheme with the incorporation of jump contributions and the resulting saddle point system is solved by the conjugate gradient Uzawa‐type method. Numerical results demonstrate very well the accuracy and effectiveness of the proposed method. The present algorithm has been applied to solve incompressible Navier–Stokes flows with moving interfaces. Copyright © 2010 John Wiley & Sons, Ltd.     

A new fast hybrid adaptive grid generation technique for arbitrary two‐dimensional domains

This paper describes a new fast hybrid adaptive grid generation technique for arbitrary two‐dimensional domains. This technique is based on a Cartesian background grid with square elements and quadtree decomposition. A new algorithm is introduced for the distribution of boundary points based on the curvature of the domain boundaries. The quadtree decomposition is governed either by the distribution of the boundary points or by a size function when a solution‐based adaptive grid is desired. The resulting grid is quaddominant and ready for the application of finite element, multi‐grid, or line‐relaxation methods. All the internal angles in the final grid have a lower bound of 45° and an upper bound of 135°. Although our main interest is in grid generation for unsteady flow simulations, the technique presented in this paper can be employed in many other fields. Several application examples are provided to illustrate the main features of this new approach. Copyright © 2010 John Wiley & Sons, Ltd.     

A meshfree‐enriched finite element method for compressible and near‐incompressible elasticity

In this paper, a two‐dimensional displacement‐based meshfree‐enriched FEM (ME‐FEM) is presented for the linear analysis of compressible and near‐incompressible planar elasticity. The ME‐FEM element is established by injecting a first‐order convex meshfree approximation into a low‐order finite element with an additional node. The convex meshfree approximation is constructed using the generalized meshfree approximation method and it possesses the Kronecker‐delta property on the element boundaries. The gradient matrix of ME‐FEM element satisfies the integration constraint for nodal integration and the resultant ME‐FEM formulation is shown to pass the constant stress test for the compressible media. The ME‐FEM interpolation is an element‐wise meshfree interpolation and is proven to be discrete divergence‐free in the incompressible limit. To prevent possible pressure oscillation in the near‐incompressible problems, an area‐weighted strain smoothing scheme incorporated with the divergence‐free ME‐FEM interpolation is introduced to provide the smoothing on strains and pressure. With this smoothed strain field, the discrete equations are derived based on a modified Hu–Washizu variational principle. Several numerical examples are presented to demonstrate the effectiveness of the proposed method for the compressible and near‐incompressible problems. Copyright © 2012 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.     

A nested iterative scheme for computation of incompressible flows in long domains

We present an effective preconditioning technique for solving the nonsymmetric linear systems encountered in computation of incompressible flows in long domains. The application category we focus on is arterial fluid mechanics. These linear systems are solved using a nested iterative scheme with an outer Richardson scheme and an inner iteration that is handled via a Krylov subspace method. Test computations that demonstrate the robustness of our nested scheme are presented.     

Generalized Robin–Neumann explicit coupling schemes for incompressible fluid‐structure interaction: Stability analysis and numerics

We introduce a new class of explicit coupling schemes for the numerical solution of fluid‐structure interaction problems involving a viscous incompressible fluid and an elastic structure. These methods generalize the arguments reported in [Comput. Methods Appl. Mech. Engrg., 267:566–593, 2013, Numer. Math., 123(1):21–65, 2013] to the case of the coupling with thick‐walled structures. The basic idea lies in the derivation of an intrinsic interface Robin consistency at the space semi‐discrete level, using a lumped‐mass approximation in the structure. The fluid–solid splitting is then performed through appropriate extrapolations of the solid velocity and stress on the interface. Based on these methods, a new, parameter‐free, Robin–Neumann iterative procedure is also proposed for the partitioned solution of implicit coupling. A priori energy estimates, guaranteeing the stability of the schemes and the convergence of the iterative procedure, are established within a representative linear setting. The accuracy and performance of the methods are illustrated in several numerical examples. Copyright © 2014 John Wiley & Sons, Ltd.     

A monolithic approach for interaction of incompressible viscous fluid and an elastic body based on fluid pressure Poisson equation

This paper describes a new monolithic approach based on the fluid pressure Poisson equation (PPE) to solve an interaction problem of incompressible viscous fluid and an elastic body. The PPE is derived so as to be consistent with the coupled equation system for the fluid‐structure interaction (FSI). Based on this approach, we develop two kinds of efficient monolithic methods. In both methods, the fluid pressure is derived implicitly so as to satisfy the incompressibility constraint, and all other unknown variables are derived fully explicitly or partially explicitly. The coefficient matrix of the PPE for the FSI becomes symmetric and positive definite and its condition is insensitive to inhomogeneity of material properties. The arbitrary Lagrangian–Eulerian (ALE) method is employed for the fluid part in order to take into account the deformable fluid‐structure interface. To demonstrate fundamental performances of the proposed approach, the developed two monolithic methods are applied to evaluate the added mass and the added damping of a circular cylinder as well as to simulate the vibration of a rectangular cylinder induced by vortex shedding. Copyright © 2005 John Wiley & Sons, Ltd.     

A unified dual‐primal finite element tearing and interconnecting approach for incompressible Stokes equations

A unified framework of dual‐primal finite element tearing and interconnecting (FETI‐DP) algorithms is proposed for solving the system of linear equations arising from the mixed finite element approximation of incompressible Stokes equations. A distinctive feature of this framework is that it allows using both continuous and discontinuous pressures in the algorithm, whereas previous FETI‐DP methods only apply to discontinuous pressures. A preconditioned conjugate gradient method is used in the algorithm with either a lumped or a Dirichlet preconditioner, and scalable convergence rates are proved. This framework is also used to describe several previously developed FETI‐DP algorithms and greatly simplifies their analysis. Numerical experiments of solving a two‐dimensional incompressible Stokes problem demonstrate the performances of the discussed FETI‐DP algorithms represented under the same framework.Copyright © 2012 John Wiley & Sons, Ltd.