Adaptive variational curve smoothing based on level set method

This paper presents an adaptive method for variational curve smoothing based on level set implementation. A suitable cost functional is minimized via solving the derived Euler–Lagrangian equation, of which the discretization is conducted on unstructured triangular meshes by employing a simple and effective finite volume scheme. Through adaptive refinement of the mesh, the geometry features of the given curve can be well resolved in a cost-effective way. Various numerical experiments demonstrate the effectiveness and efficiency of the proposed approach.     

CONVERGENCE OF A MIXED FINITE ELEMENT FOR THE STOKES PROBLEM ON ANISOTROPIC MESHES

The main aim of this paper is to study the convergence properties of a low order mixed finite element for the Stokes problem under anisotropic meshes. We discuss the anisotropic convergence and superconvergence independent of the aspect ratio. Without the shape regularity assumption and inverse assumption on the meshes, the optimal error estimates and natural superconvergence at central points are obtained. The global superconvergence for the gradient of the velocity and the pressure is derived with the aid of a suitable postprocessing method. Furthermore, we develop a simple method to obtain the superclose properties which improves the results of the previous works .     

An implicit finite volume method for the solution of 3D low Mach number viscous flows using a local preconditioning technique

This paper presents a cell-centered high order finite volume scheme for the solution of the three-dimensional (3D) Navier–Stokes equations with low Mach number. The system of non-linear equations is solved by means of a fully implicit pseudo-transient scheme. Each pseudo-time step is solved by a Newton-GMRes procedure. A local preconditioning technique is used to scale the speed of sound and to improve the system condition number for low Mach number and low cell Reynolds number. This preconditioning is applied to the AUSM+up flux vector splitting function. The method is tested on 2D and 3D low Mach number laminar flows.     

An eigenvector‐based linear reconstruction scheme for the shallow‐water equations on two‐dimensional unstructured meshes

This paper presents a new approach to MUSCL reconstruction for solving the shallow‐water equations on two‐dimensional unstructured meshes. The approach takes advantage of the particular structure of the shallow‐water equations. Indeed, their hyperbolic nature allows the flow variables to be expressed as a linear combination of the eigenvectors of the system. The particularity of the shallow‐water equations is that the coefficients of this combination only depend upon the water depth. Reconstructing only the water depth with second‐order accuracy and using only a first‐order reconstruction for the flow velocity proves to be as accurate as the classical MUSCL approach. The method also appears to be more robust in cases with very strong depth gradients such as the propagation of a wave on a dry bed. Since only one reconstruction is needed (against three reconstructions in the MUSCL approach) the EVR method is shown to be 1.4–5 times as fast as the classical MUSCL scheme, depending on the computational application. Copyright © 2006 John Wiley & Sons, Ltd.     

A study of the recirculating flow in planar,symmetrical branching channels

A numerical method is employed to examine the flow in symmetrical, two‐dimensional branches of Y shape and Tee shape. The methodology is based on a pressure‐correction procedure within the frame of unstructured grids. Specified pressures are imposed at the outlets of the two branches. The area ratio of the branch is allowed to vary in the range of 2–3. Separation of the flow in the bifurcating region is inevitable. With equal outlet pressures, symmetrical flow patterns prevail except for the Y type branch under the conditions of high Reynolds numbers and large area ratios. This implies that the Y‐branch flow is more sensitive to small disturbances. It is shown that with a slightly higher pressure imposed on one of the two branches the structure of the recirculating flow for the Y type is greatly affected and the flow rate is reduced dramatically in the high‐pressure branch channel. In contrast, the influence on the Tee type branch is much lower since the flow behaves like a jet impinging on a confined duct. Copyright © 2005 John Wiley & Sons, Ltd.     

Multiscale Basis Functions for Singular Perturbation on Adaptively Graded Meshes

We apply the multiscale basis functions for the singularly perturbed reaction-diffusion problem on adaptively graded meshes, which can provide a good balance between the numerical accuracy and computational cost. The multiscale space is built through standard finite element basis functions enriched with multiscale basis functions. The multiscale basis functions have abilities to capture originally perturbed information in the local problem, as a result, our method is capable of reducing the boundary layer errors remarkably on graded meshes, where the layer-adapted meshes are generated by a given parameter. Through numerical experiments we demonstrate that the multiscale method can acquire second order convergence in the L² norm and first order convergence in the energy norm on graded meshes, which is independent of ε. In contrast with the conventional methods, our method is much more accurate and effective.     

Error estimation using the error transport equation for finite-volume methods and arbitrary meshes

We present a single grid error estimation technique based on the derivation of a continuous equation for the discretization error. It is developed in the context of finite-volume methods for arbitrary meshes. The key issue of the evaluation of the source term is addressed through the use of a reconstruction operator. Using a higher order accurate evaluation of this term and solving the error equation with the same numerical methods and on the same computational grid as the primal problem leads to a higher order accurate error prediction. The methodology is presented in detail and its properties of asymptotic exactness and superconvergence are illustrated on several cases, including an application of practical engineering complexity. Also presented is the derivation of a powerful criterium for driving any adaptive procedure.     

An efficient implicit unstructured finite volume solver for generalised Newtonian fluids

An implicit finite volume solver is developed for the steady-state solution of generalised Newtonian fluids on unstructured meshes in 2D. The pseudo-compressibility technique is employed to couple the continuity and momentum equations by transforming the governing equations into a hyperbolic system. A second-order accurate spatial discretisation is provided by performing a least-squares gradient reconstruction within each control volume of unstructured meshes. A central flux function is used for the convective terms and a solution jump term is added to the averaged component for the viscous terms. Global implicit time-stepping using successive evolution–relaxation is utilised to accelerate the convergence to steady-state solutions. The performance of our flow solver is examined for power-law and Carreau–Yasuda non-Newtonian fluids in different geometries. The effects of model parameters and Reynolds number are studied on the convergence rate and flow features. Our results verify second-order accuracy of the discretisation and also fast and efficient convergence to the steady-state solution for a wide range of flow variables.     

ADAPTIVE SOLUTIONS FOR UNSTEADY LAMINAR FLOWS ON UNSTRUCTURED GRIDS

An adaptive finite volume method for the simulation of time-dependent, viscous flow is presented. The Navier–Stokes equations are discretized by central schemes on unstructured grids and solved by an explicit Runge–Kutta method. The essential topics of the present study are a new concept for a local Runge–Kutta time-stepping scheme, called multisequence Runge–Kutta, which reduces the severe stability restriction in unsteady problems, a common grid generation and adaptation procedure and the application of dynamic grids for capturing moving flow structures. Results are presented for laminar, separated flow around an aerofoil with a flap.     

Parallelizing FLITE3D—a multigrid finite element Euler solver

FLITE3D is a multigrid Euler solver. It is used extensively by British Aerospace in aircraft design and simulation. This paper presents experiences in parallelizing this industrial code. Owing to the employment of an agglomeration‐based multigrid technique, the communication overhead on the coarser meshes could readily erode any gain from the use of parallel computers. The parallelization of the code therefore required careful design and implementation. The strategy adopted in the parallelization of the code, including the use of data structures and communication primitives, is described. Numerical results are presented to demonstrate the efficiency of the resulting parallel code. Copyright © 2001 John Wiley & Sons, Ltd.