A finite element immersed boundary method for fluid flow around moving objects

This paper presents the extension of a recently proposed immersed boundary method to the solution of the flow around moving objects. Solving the flow around objects with complex shapes may involve extensive meshing work that has to be repeated each time a change in the geometry is needed. Mesh generation and solution interpolation between successive grids may be costly and introduce errors if the geometry changes significantly during the course of the computation. These drawbacks are avoided when the solution algorithm can tackle grids that do not fit the shape of immersed objects. This work presents an extension of our recently developed finite element Immersed Boundary (IB) method to transient applications involving the movement of immersed fluid/solid interfaces. As for the fixed solid boundary case, the method produces solutions of the flow satisfying accurately boundary conditions imposed on the surface of immersed bodies. The proposed algorithm enriches the finite element discretization of interface elements with additional degrees of freedom, the latter being eliminated at element level. The boundary of immersed objects is defined using a time dependent level-set function. Solutions are shown for various flow problems and the accuracy of the present approach is measured with respect to solutions on body-conforming meshes.     

Incompressible moving boundary flows with the finite volume particle method

Mesh-free methods offer the potential for greatly simplified modeling of flow with moving walls and phase interfaces. The finite volume particle method (FVPM) is a mesh-free technique based on interparticle fluxes which are exactly analogous to intercell fluxes in the mesh-based finite volume method. Consequently, the method inherits many of the desirable properties of the classical finite volume method, including implicit conservation and a natural introduction of boundary conditions via appropriate flux terms. In this paper, we describe the extension of FVPM to incompressible viscous flow with moving boundaries. An arbitrary Lagrangian–Eulerian approach is used, in conjunction with the mesh-free discretisation, to facilitate a straightforward treatment of moving bodies. Non-uniform particle distribution is used to concentrate computational effort in regions of high gradients. The underlying method for viscous incompressible flow is validated for a lid-driven cavity problem at Reynolds numbers of 100 and 1000. To validate the simulation of moving boundaries, flow around a translating cylinder at Reynolds numbers of 20, 40 and 100 is modeled. Results for pressure distribution, surface forces and vortex shedding frequency are in good agreement with reference data from the literature and with FVPM results for an equivalent flow around a stationary cylinder. These results establish the capability of FVPM to simulate large wall motions accurately in an entirely mesh-free framework.     

Accurate shock-capturing finite volume method for advection-dominated flow and pollution transport

This paper describes an accurate shock-capturing finite volume numerical method to solve a two-dimensional flow and solute transport problem in shallow water. Hydrodynamic and advection-diffusion equations are simultaneously solved by means of a Strang operator-splitting approach. The advective part is solved in time by a third-order TDV Runge-Kutta method and in space by a second-order WAF method coupled with a fifth-order WENO reconstruction. The diffusion part is solved in time and space by a second-order accurate method. Thus the overall accuracy is second-order both in time and space. Nevertheless the Strang splitting approach allows the advective part of the equations to be solved with a reconstruction of high order, where at lower orders it shows excessive numerical diffusion and damping, especially for very long time simulations. Very good results have been obtained applying the model to standard long time numerical tests.     

Simulation of natural convection with the collocated clustered finite volume scheme

This paper presents numerical results obtained in the case of natural convection within non constant fluid density, using the collocated clustered finite volume (CCFV) scheme. The continuous equations are first given in a dimensionless form. Then we present the finite volume scheme with the principles and the spatial discretization used. Analytical tests illustrate the numerical behavior of this scheme according to the type of grid, of the pressure stabilization method and check the robustness of this scheme. Next, the results obtained on the square thermally driven cavity under large temperature differences show that the CCFV scheme accurately fits the reference results.     

A finite volume method to solve the 3D Navier–Stokes equations on unstructured collocated meshes

A new method to solve the Navier–Stokes equations for incompressible viscous flows and the transport of a scalar quantity is proposed. This method is based upon a fractional time step scheme and the finite volume method on unstructured meshes. The governing equations are discretized using a collocated, cell-centered arrangement of velocity and pressure. The solution variables are stored at the cell-circumcenters. Theoretical results and numerical properties of the scheme are provided. Predictions of lid-driven cavity flow, flows past a cylinder and heat transport in a cylinder are performed to validate the method.     

A finite volume method parallelization for the simulation of free surface shallow water flows

We construct a parallel algorithm, suitable for distributed memory architectures, of an explicit shock-capturing finite volume method for solving the two-dimensional shallow water equations. The finite volume method is based on the very popular approximate Riemann solver of Roe and is extended to second order spatial accuracy by an appropriate TVD technique. The parallel code is applied to distributed memory architectures using domain decomposition techniques and we investigate its performance on a grid computer and on a Distributed Shared Memory supercomputer. The effectiveness of the parallel algorithm is considered for specific benchmark test cases. The performance of the realization measured in terms of execution time and speedup factors reveals the efficiency of the implementation.     

Simulation for viscoelastic flow by a finite volume/element method

Stability of a second-order finite element/finite volume (FE/FV) hybrid scheme is investigated on the basis of flows with increasing Weissenberg number. FEs are used to discretise the balances of mass and momentum. For the stress equation a FV method is used, based on the recent development with fluctuation distribution schemes for pure convection problems. Examples considered include a start-up channel flow, flow past a cylinder and the non-smooth 4:1 contraction flow for an Oldroyd-B fluid. A considerable gain in efficiency per time step can be obtained compared to an alternative pure FE implementation. A distribution based on the flux terms is unstable for higher Weissenberg numbers, and this is also true for a distribution based on source terms alone. The instability is identified as being caused by the interaction of the balance equations and stress equation. A combination of distribution schemes based on flux and source terms, however, gives a considerable improvement to the hybrid FE/FV implementation. With respect to limiting Weissenberg number attenuation, the hybrid scheme is more stable than the pure FE alternative for the smooth flow past a cylinder, but less so for the non-smooth contraction flow. The influence of additional strain-rate stabilisation techniques is also analysed and found to be beneficial.     

Numerical simulation of low-Reynolds number flows past rectangular cylinders based on adaptive finite element and finite volume methods

The flow past rectangular cylinders has been investigated by two different numerical techniques, an adaptive finite-element (AFEM) and a finite-volume method (FVM). A square and a rectangular cylinder with width-to-height equal to 5 are taken into account. 2D computations have been performed for different Reynolds numbers in order to consider different flow regimes, i.e. the stationary, the periodic and the turbulent flow. The comparison between the two methods regarded both the reliability of the computed solutions and the overall resulting efficiency of the methods. Velocity profiles and integral parameters such as Strouhal number, drag coefficient and recirculation length have been compared. A good agreement between the adaptive FEM and the FVM computations, as well as with the available literature results, has been found. The computational effort has been evaluated in terms of used degrees of freedom in space and time and human resources employed to reach the mesh and timestep-length independence of the solutions. Relevant outcomes of this work are the cross validation of an adaptive FE method and a popular open source FV code.     

A finite element method for flow separation

The finite element model has been developed in order to solve separation pattern of the flow past an obstruction in a two-dimensional flow field. The Helmholtz-Poisson form of the Reynolds equations are solved alternately until a stable flow separation in the neighbourhood of the obstruction is obtained. In order to check the results of the finite element model, an experimental separation pattern using Pitot-tube measurements has been conducted. The computed and the experimental flow separation patterns show a good agreement.     

Unsteady flow computations for flow past multiple moving boundaries using LSKUM

We present the latest developments in the least squares kinetic upwind method (LSKUM), a kinetic theory based grid free approach for the solution of Euler equations. A single step higher order scheme through modified CIR splitting is presented. A new weighted least squares method has been used in the present work which simplifies the 2-D formulae to an equivalent 1-D form. This is achieved through diagonalisation of the least squares matrix through suitable choices of the weights. All these developments have been extended to problems with moving nodes and boundaries. A 2-D unsteady Euler code has been developed incorporating all the above ideas along with the well known dual time stepping procedure. The code has been verified and validated for the standard test case AGARD CT(5) which corresponds to unsteady flow past oscillating NACA0012 airfoil pitching about quarter chord. Good comparisons with the experimental values have been obtained. In order to demonstrate the ability of the method to handle multiple moving bodies we have computed unsteady flow past two oscillating NACA0012 airfoils one behind the other. Some interesting results are presented for this case.     

A finite point method for incompressible flow problems

A stabilized finite point method (FPM) for the meshless analysis of incompressible fluid flow problems is presented. The stabilization approach is based in the finite increment calculus (FIC) procedure developed by O?ate [14]. An enhanced fractional step procedure allowing the semi-implicit numerical solution of incompressible fluids using the FPM is described. Examples of application of the stabilized FPM to the solution of two incompressible flow problems are presented. Received: 30 June 1999 / Accepted: 21 September 1999     

A fixed-grid model for simulation of a moving body in free surface flows

A two-dimensional computer model is developed to simulate free surface flow interaction with a moving body. The model is based on the cut-cell technique in a fixed-grid system. In this model, a body is approximated by the partial cell treatment (PCT), in which an irregular body is represented by the volumetric fraction of solid in Cartesian cells. The body motion is tracked by Lagrangian method whereas the fluid motion around the body is solved by Eulerian method. The concept of “locally relative stationary (LRS)” is introduced in this study. In the LRS method, a source term is added locally to the conventional continuity equation on body surfaces to take account of body motions, which subsequently affects the computational results of fluid pressure and flow velocity around the body. The LRS method is incorporated into an earlier Reynolds averaged Navier-Stokes (RANS) equations model developed by Lin and Liu [A numerical study of breaking waves in the surf zone. J Fluid Mech 1998;359:239-64]. The new model is capable of simulating generic turbulent free surface flows and their interaction with a moving body or multiple moving bodies. A series of numerical experiments have been conducted to verify the accuracy of the model for simulation of moving body interaction with a free surface flow. These tests include the generation of a solitary wave with the prescribed wave paddle movements, water exit and water impact and entry of a horizontal circular cylinder, fluid sloshing in a horizontally excited tank, and the acceleration/deceleration of an elliptical cylinder near a water surface. Excellent agreements are obtained when numerical results are compared to available analytical, experimental, and other numerical results. The model is a simple-to-implement computational tool for simulating a moving body in turbulent free surface flows.     

A finite volume method for Stokes problems on quadrilateral meshes

We present a finite volume method for Stokes problems using the isoparametric $Q_1$–$Q_0$ element pair on quadrilateral meshes. To offset the lack of the $inf$–$sup$ condition, a jump term of discrete pressure (stabilizing term) is added to the continuity approximation equation. Thus, we establish a stabilized finite volume scheme on quadrilateral meshes. Then, based on some superclose estimates, we derive the optimal error estimates in the $H^1$- and $L_2$-norms for velocity and in the $L_2$-norm for pressure, respectively. Numerical examples are provided to illustrate our theoretical analysis. We emphasize that our work is the first time to propose and analyze a finite volume method for Stoke problems using isoparametric elements on quadrilateral meshes.     

A fast and stable penalty method for rigid body simulation

Two methods have been used extensively to model resting contact for rigid body simulation. The first approach, the penalty method, applies virtual springs to surfaces in contact to minimize interpenetration. This method, as typically implemented, results in oscillatory behavior and considerable penetration. The second approach, based on formulating resting contact as a linear complementarity problem, determines the resting contact forces analytically to prevent interpenetration. The analytical method exhibits expected-case polynomial complexity in the number of contact points, and may fail to find a solution in polynomial time when friction is modeled. We present a fast penalty method that minimizes oscillatory behavior and leads to little penetration during resting contact; our method compares favorably to the analytical method with regard to these two measures, while exhibiting much faster performance both asymptotically and empirically.     

A finite element based level-set method for multiphase flow applications

A numerical method for simulating incompressible two-dimensional multiphase flow is presented. The method is based on a level-set formulation discretized by a finite-element technique. The treatment of the specific features of this problem, such as surface tension forces acting at the interfaces separating two immiscible fluids, as well as the density and viscosity jumps that in general occur across such interfaces, have been integrated into the finite-element framework. Using a method based on the weak formulation of the Navier-Stokes equations has its advantages. In this formulation, the singular surface tension forces are included through line integrals along the interfaces, which are easily approximated quantities. In addition, differentiation of the discontinuous viscosity is avoided. The discontinuous density and viscosity are included in the finite element integrals. A strategy for the evaluation of integrals with discontinuous integrands has been developed based on a rigorous analysis of the errors associated with the evaluation of such integrals. Numerical tests have been performed. For the case of a rising buoyant bubble the results are in good agreement with results from a front-tracking method. The run presented here is a run including topology changes, where initially separated areas of one fluid merge in different stages due to buoyancy effects. Received: 1 March 1999 / Accepted: 17 June 1999     

A posteriori error analysis for a cut cell finite volume method

We study the solution of a diffusive process in a domain where the diffusion coefficient changes discontinuously across a curved interface. We consider discretizations that use regularly-shaped meshes, so that the interface “cuts” through the cells (elements or volumes) without respecting the regular geometry of the mesh. Consequently, the discontinuity in the diffusion coefficients has a strong impact on the accuracy and convergence of the numerical method. This motivates the derivation of computational error estimates that yield accurate estimates for specified quantities of interest. For this purpose, we adapt the well-known adjoint based a posteriori error analysis technique used for finite element methods. In order to employ this method, we describe a systematic approach to discretizing a cut-cell problem that handles complex geometry in the interface in a natural fashion yet reduces to the well-known Ghost Fluid Method in simple cases. We test the accuracy of the estimates in a series of examples.     

An improved method for bounding stationary measures of finite Markov processes

A new method to compute bounds on stationary results of finite Markov processes in discrete or continuous time is introduced. The method extends previously published approaches using polyhedra of eigenvectors for stochastic matrices with a known lower and upper bound of their elements. Known techniques compute bounds for the elements of the stationary vector with respect to the lower bounds of the matrix elements and another set of bounds with respect to the upper bounds of matrix elements. The resulting bounds are usually not sharp, if lower and upper bounds for the elements are known. The new approach combines lower and upper bounds resulting in sharp bounds which are often much tighter than bounds computed using only one bounding value for the matrix elements.     

A real-time multigrid finite hexahedra method for elasticity simulation using CUDA

We present a multigrid approach for simulating elastic deformable objects in real time on recent NVIDIA GPU architectures. To accurately simulate large deformations we consider the co-rotated strain formulation. Our method is based on a finite element discretization of the deformable object using hexahedra. It draws upon recent work on multigrid schemes for the efficient numerical solution of partial differential equations on such discretizations. Due to the regular shape of the numerical stencil induced by the hexahedral regime, and since we use matrix-free formulations of all multigrid steps, computations and data layout can be restructured to avoid execution divergence of parallel running threads and to enable coalescing of memory accesses into single memory transactions. This enables to effectively exploit the GPU’s parallel processing units and high memory bandwidth via the CUDA parallel programming API. We demonstrate performance gains of up to a factor of 27 and 4 compared to a highly optimized CPU implementation on a single CPU core and 8 CPU cores, respectively. For hexahedral models consisting of as many as 269,000 elements our approach achieves physics-based simulation at 11 time steps per second.     

A finite volume method for solid mechanics incorporating rotational degrees of freedom

A novel finite volume (FV) based discretization method for determining displacement, strain and stress distributions in loaded two dimensional structures with complex geometries is presented. The method incorporates rotation variables in addition to the displacement degrees of freedom employed in earlier FV based structural analysis procedures and conventional displacement based finite element (FE) formulations. The method is used to predict the displacement fields in a number of test cases for which solutions are already available. The effect of mesh refinement upon the accuracy of the solutions predicted by the method is assessed. The results of this assessment indicate that the new method is more accurate than previous FV procedures incorporating displacement variables only, particularly in cases where bending is the predominant mode of deformation, and therefore the new method represents a significant advance in the development of this type of discretization procedure. Interestingly, the results of the accuracy assessment exercise also indicate that the new FV procedure is also more accurate than the equivalent FE formulation incorporating displacement and rotational degrees of freedom.