Robust and provably second‐order explicit–explicit and implicit–explicit staggered time‐integrators for highly non‐linear compressible fluid–structure interaction problems

An explicit–explicit staggered time‐integration algorithm and an implicit–explicit counterpart are presented for the solution of non‐linear transient fluid–structure interaction problems in the Arbitrary Lagrangian–Eulerian (ALE) setting. In the explicit–explicit case where the usually desirable simultaneous updating of the fluid and structural states is both natural and trivial, staggering is shown to improve numerical stability. Using rigorous ALE extensions of the two‐stage explicit Runge–Kutta and three‐point backward difference methods for the fluid, and in both cases the explicit central difference scheme for the structure, second‐order time‐accuracy is achieved for the coupled explicit–explicit and implicit–explicit fluid–structure time‐integration methods, respectively, via suitable predictors and careful stagings of the computational steps. The robustness of both methods and their proven second‐order time‐accuracy are verified for sample application problems. Their potential for the solution of highly non‐linear fluid–structure interaction problems is demonstrated and validated with the simulation of the dynamic collapse of a cylindrical shell submerged in water. The obtained numerical results demonstrate that, even for fluid–structure applications with strong added mass effects, a carefully designed staggered and subiteration‐free time‐integrator can achieve numerical stability and robustness with respect to the slenderness of the structure, as long as the fluid is justifiably modeled as a compressible medium. Copyright © 2010 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.     

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.     

On the equivalence of the time domain differential quadrature method and the dissipative Runge–Kutta collocation method

Numerical solutions for initial value problems can be evaluated accurately and efficiently by the differential quadrature method. Unconditionally stable higher order accurate time step integration algorithms can be constructed systematically from this framework. It has been observed that highly accurate numerical results can also be obtained for non‐linear problems. In this paper, it is shown that the algorithms are in fact related to the well‐established implicit Runge–Kutta methods. Through this relation, new implicit Runge–Kutta methods with controllable numerical dissipation are derived. Among them, the non‐dissipative and asymptotically annihilating algorithms correspond to the Gauss methods and the Radau IIA methods, respectively. Other dissipative algorithms between these two extreme cases are shown to be B‐stable (or algebraically stable) as well and the order of accuracy is the same as the corresponding Radau IIA method. Through the equivalence, it can be inferred that the differential quadrature method also enjoys the same stability and accuracy properties. Copyright © 2001 John Wiley & Sons, Ltd.     

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.     

An alternative approach for numerical solutions of the Navier–Stokes equations

Conventional approaches for solving the Navier–Stokes equations of incompressible fluid dynamics are the primitive‐variable approach and the vorticity–velocity approach. In this paper, an alternative approach is presented. In this approach, pressure and one of the velocity components are eliminated from the governing equations. The result is one higher‐order partial differential equation with one unknown for two‐dimensional problems or two higher‐order partial differential equations with two unknowns for three‐dimensional problems. A meshless collocation method based on radial basis functions for solving the Navier–Stokes equations using this approach is presented. The proposed method is used to solve a two‐ and a three‐dimensional test problem of which exact solutions are known. It is found that, with appropriate values of the method parameters, solutions of satisfactory accuracy can be obtained. Copyright © 2006 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.     

Fully implicit Lagrange–Newton–Krylov–Schwarz algorithms for boundary control of unsteady incompressible flows

We develop a parallel fully implicit domain decomposition algorithm for solving optimization problems constrained by time‐dependent nonlinear partial differential equations. In particular, we study the boundary control of unsteady incompressible Navier–Stokes equations. After an implicit discretization in time, a fully coupled sparse nonlinear optimization problem needs to be solved at each time step. The class of full space Lagrange–Newton–Krylov–Schwarz algorithms is used to solve the sequence of optimization problems. Among optimization algorithms, the fully implicit full space approach is considered to be the easiest to formulate and the hardest to solve. We show that Lagrange–Newton–Krylov–Schwarz, with a one‐level restricted additive Schwarz preconditioner, is an efficient class of methods for solving these hard problems. To demonstrate the scalability and robustness of the algorithm, we consider several problems with a wide range of Reynolds numbers and time step sizes, and we present numerical results for large‐scale calculations involving several million unknowns obtained on machines with more than 1000 processors. Copyright © 2012 John Wiley & Sons, Ltd.     

The particle finite element method: a powerful tool to solve incompressible flows with free‐surfaces and breaking waves

Particle Methods are those in which the problem is represented by a discrete number of particles. Each particle moves accordingly with its own mass and the external/internal forces applied to it. Particle Methods may be used for both, discrete and continuous problems. In this paper, a Particle Method is used to solve the continuous fluid mechanics equations. To evaluate the external applied forces on each particle, the incompressible Navier–Stokes equations using a Lagrangian formulation are solved at each time step. The interpolation functions are those used in the Meshless Finite Element Method and the time integration is introduced by an implicit fractional‐step method. In this manner classical stabilization terms used in the momentum equations are unnecessary due to lack of convective terms in the Lagrangian formulation. Once the forces are evaluated, the particles move independently of the mesh. All the information is transmitted by the particles. Fluid–structure interaction problems including free‐fluid‐surfaces, breaking waves and fluid particle separation may be easily solved with this methodology. Copyright © 2004 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.     

Remarks on the interpretation of current non‐linear finite element analyses as differential–algebraic equations

For the numerical solution of materially non‐linear problems like in computational plasticity or viscoplasticity the finite element discretization in space is usually coupled with point‐wise defined evolution equations characterizing the material behaviour. The interpretation of such systems as differential–algebraic equations (DAE) allows modern‐day integration algorithms from Numerical Mathematics to be efficiently applied. Especially, the application of diagonally implicit Runge–Kutta methods (DIRK) together with a Multilevel‐Newton method preserves the algorithmic structure of current finite element implementations which are based on the principle of virtual displacements and on backward Euler schemes for the local time integration. Moreover, the notion of the consistent tangent operator becomes more obvious in this context. The quadratical order of convergence of the Multilevel‐Newton algorithm is usually validated by numerical studies. However, an analytical proof of this second order convergence has already been given by authors in the field of non‐linear electrical networks. We show that this proof can be applied in the current context based on the DAE interpretation mentioned above. We finally compare the proposed procedure to several well‐known stress algorithms and show that the inclusion of a step‐size control based on local error estimations merely requires a small extra time‐investment. Copyright © 2001 John Wiley & Sons, Ltd.     

On a high‐order Newton linearization method for solving the incompressible Navier–Stokes equations

The present study aims to accelerate the non‐linear convergence to incompressible Navier–Stokes solution by developing a high‐order Newton linearization method in non‐staggered grids. For the sake of accuracy, the linearized convection–diffusion–reaction finite‐difference equation is solved line‐by‐line using the nodally exact one‐dimensional scheme. The matrix size is reduced and, at the same time, the CPU time is considerably saved owing to the reduction of stencil points. This Newton linearization method is computationally efficient and is demonstrated to outperform the classical Newton method through computational exercises. Copyright © 2005 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.     

A rheological interface model and its space–time finite element formulation for fluid–structure interaction

This contribution discusses extended physical interface models for fluid–structure interaction problems and investigates their phenomenological effects on the behavior of coupled systems by numerical simulation. Besides the various types of friction at the fluid–structure interface the most interesting phenomena are related to effects due to additional interface stiffness and damping. The paper introduces extended models at the fluid–structure interface on the basis of rheological devices (Hooke, Newton, Kelvin, Maxwell, Zener). The interface is decomposed into a Lagrangian layer for the solid‐like part and an Eulerian layer for the fluid‐like part. The mechanical model for fluid–structure interaction is based on the equations of rigid body dynamics for the structural part and the incompressible Navier–Stokes equations for viscous flow. The resulting weighted residual form uses the interface velocity and interface tractions in both layers in addition to the field variables for fluid and structure. The weak formulation of the whole coupled system is discretized using space–time finite elements with a discontinuous Galerkin method for time‐integration leading to a monolithic algebraic system. The deforming fluid domain is taken into account by deformable space–time finite elements and a pseudo‐structure approach for mesh motion. The sensitivity of coupled systems to modification of the interface model and its parameters is investigated by numerical simulation of flow induced vibrations of a spring supported fluid‐immersed cylinder. It is shown that the presented rheological interface model allows to influence flow‐induced vibrations. Copyright © 2010 John Wiley & Sons, Ltd.     

A high‐order accurate particle‐in‐cell method

We propose the use of high‐order weighted essentially non‐oscillatory interpolation and moving‐least‐squares approximation schemes alongside high‐order time integration to enable high‐order accurate particle‐in‐cell methods. The key insight is to view the unstructured set of particles as the underlying representation of the continuous fields; the grid used to evaluate integro–differential coupling terms is purely auxiliary. We also include a novel regularization term to avoid the accumulation of noise in the particle samples without harming the convergence rate. We include numerical examples for several model problems: advection–diffusion, shallow water, and incompressible Navier–Stokes in vorticity formulation. The implementation demonstrates fourth‐order convergence, shows very low numerical dissipation, and is competitive with high‐order Eulerian schemes. Copyright © 2012 John Wiley & Sons, Ltd.     

Stable and time-dissipative finite element methods for the incompressible Navier–Stokes equations in advection dominated flows

This paper examines a new Galerkin method with scaled bubble functions which replicates the exact artificial diffusion methods in the case of 1-D scalar advection–diffusion and that leads to non-oscillatory solutions as the streamline upwinding algorithms for 2-D scalar advection–diffusion and incompressible Navier–Stokes. This method retains the satisfaction of the Babuska–Brezzi condition and, thus, leads to optimal performance in the incompressible limit. This method, when, combined with the recently proposed linear unconditionally stable algorithms of Simo and Armero (1993), yields a method for solution of the incompressible Navier–Stokes equations ideal for either diffusive or advection-dominated flows. Examples from scalar advection–diffusion and the solution of the incompressible Navier–Stokes equations are presented.     

A stabilized Runge–Kutta,Taylor smoothed particle hydrodynamics algorithm for large deformation problems in dynamics

Numerical simulation of large deformation and failure problems present a series of difficulties when solved using mesh based methods. Meshless methods present an interesting alternative that has been explored in the past years by researchers. Here we propose a Runge–Kutta Taylor SPH model based on formulating the dynamic problem as a set of first‐order PDEs. Two sets of nodes are used for time steps n and n + 1 ∕ 2, resulting on avoiding the classical tensile instability of some other SPH formulations. To improve the accuracy and stability of the algorithm, the Taylor expansion in time of the advective terms is combined with a Runge–Kutta integration of the sources. Finally, as boundaries change during the process, a free surface detection algorithm is introduced. Copyright © 2012 John Wiley & Sons, Ltd.     

Numerical simulation of flow over delta wing with trailing‐edge jet at high angle of attack

The incompressible Reynolds‐averaged Navier–Stokes equations, together with a modified mixing length algebraic turbulence model, are solved to simulate the flow over a delta wing with trailing‐edge jets at high angles of attack. An artificial compressibility method and a Beam–Warming implicit approximate factorization scheme are employed to discretize the equations. The computed results indicate that trailing‐edge jets tend to not only decrease the pressure but also increase the velocity at the cores of the streamwise primary vortices. This moves the vortex breakdown locations aft and stabilizes the vortical flow. As a result, the jets lead to a decreased pressure on the upper wing surface and an increased pressure on the lower wing surface, thereby increasing the overall lift. Computations further show that as the exit area of the trailing‐edge jets is enlarged, or, as the jets are deflected downward, the above‐mentioned effects become more pronounced. Copyright © 2004 John Wiley & Sons, Ltd.     

Comparison of implicit and explicit hybridizable discontinuous Galerkin methods for the acoustic wave equation

We describe implicit and explicit formulations of the hybridizable discontinuous Galerkin method for the acoustic wave equation based on state‐of‐the‐art numerical software and quantify their efficiency for realistic application settings. In the explicit scheme, the trace of the acoustic pressure is computed from the solution on the two elements adjacent to the face at the old time step. Tensor product shape functions for quadrilaterals and hexahedra evaluated with sum factorization are used to ensure low operation counts. For applying the inverse mass matrix of Lagrangian shape functions with full Gaussian quadrature, a new tensorial technique is proposed. As time propagators, diagonally implicit and explicit Runge–Kutta methods are used, respectively. We find that the computing time per time step is 25 to 200 times lower for the explicit scheme, with an increasing gap in three spatial dimensions and for higher element degrees. Our experiments on realistic 3D wave propagation with variable material parameters in a photoacoustic imaging setting show an improvement of two orders of magnitude in terms of time to solution, despite stability restrictions on the time step of the explicit scheme. Operation counts and a performance model to predict performance on other computer systems accompany our results. Copyright © 2015 John Wiley & Sons, Ltd.     

A bridge between projection methods and SIMPLE type methods for incompressible Navier–Stokes equations

A bridge is built between projection methods and SIMPLE type methods (Semi‐Implicit Method for Pressure‐Linked Equation). A general second‐order accurate projection method is developed for the simulation of incompressible unsteady flows by employing a non‐linear update of pressure term as Θⁿ?pⁿ⁺¹+(I?Θⁿ)?pⁿ, where Θⁿ is a coefficient matrix, which may depend on the grid size, time step and even velocity. It includes three‐ and four‐step projection methods. The standard SIMPLE method is written in a concise formula for steady and unsteady flows. It is proven that SIMPLE type methods have second‐order temporal accuracy for unsteady flows. The classical second‐order projection method and SIMPLE type methods are united within the framework of the general second‐order projection formula. Two iteration algorithms of SIMPLE type methods for unsteady flows are described and discussed. In addition, detailed formulae are provided for general projection methods by using the Runge–Kutta technique to update the convective term and Crank–Nicholson scheme for the diffusion term. Copyright © 2007 John Wiley & Sons, Ltd.