A simple quality triangulation algorithm for complex geometries

This paper presents a simple algorithm for quality triangulation in domains with complex geometries. Based on the fact that the equilateral triangles (regular meshes) are ideal for numerical computations in computational fluids dynamics (CFD) analysis, the proposed algorithm starts with an initial equilateral triangle mesh covering the whole domain. Nodes close to the boundary edges satisfy the so‐called non‐encroaching criterion, the distance from any inserted node to any boundary vertices and the midpoints of any boundary edge is greater than a given characteristic length. Both nearly uniform and non‐uniform triangle meshes can be generated using a mesh size reduction technique. Local refinement is achieved by using transition layers. More regular meshes can be generated in the interior of the domain and all angles of the triangle mesh produced by this algorithm are proven to be bounded in a reasonable range (19.5–141°). Copyright © 2010 John Wiley & Sons, Ltd.     

A posteriori bounds for linear functional outputs of Crouzeix–Raviart finite element discretizations of the incompressible Stokes problem

A finite element technique is presented for the efficient generation of lower and upper bounds to outputs which are linear functionals of the solutions to the incompressible Stokes equations in two space dimensions. The finite element discretization is effected by Crouzeix–Raviart elements, the discontinuous pressure approximation of which is central to this approach. The bounds are based upon the construction of an augmented Lagrangian: the objective is a quadratic ‘energy’ reformulation of the desired output, the constraints are the finite element equilibrium equations (including the incompressibility constraint), and the inter‐sub‐domain continuity conditions on velocity. Appealing to the dual max–min problem for appropriately chosen candidate Lagrange multipliers then yields inexpensive bounds for the output associated with a fine‐mesh discretization. The Lagrange multipliers are generated by exploiting an associated coarse‐mesh approximation. In addition to the requisite coarse‐mesh calculations, the bound technique requires the solution of only local sub‐domain Stokes problems on the fine mesh. The method is illustrated for the Stokes equations, in which the outputs of interest are the flow rate past and the lift force on a body immersed in a channel. Copyright © 2000 John Wiley & Sons, Ltd.     

A nodal integral method for quadrilateral elements

Nodal integral methods (NIMs) have been developed and successfully used to numerically solve several problems in science and engineering. The fact that accurate solutions can be obtained on relatively coarse mesh sizes, makes NIMs a powerful numerical scheme to solve partial differential equations. However, transverse integration procedure, a step required in the NIMs, limits its applications to brick‐like cells, and thus hinders its application to complex geometries. To fully exploit the potential of this powerful approach, abovementioned limitation is relaxed in this work by first using algebraic transformation to map the arbitrarily shaped quadrilaterals, used to mesh the arbitrarily shaped domain, into rectangles. The governing equations are also transformed. The transformed equations are then solved using the standard NIM. The scheme is developed for the Poisson equation as well as for the time‐dependent convection–diffusion equation. The approach developed here is validated by solving several benchmark problems. Results show that the NIM coupled with an algebraic transformation retains the coarse mesh properties of the original NIM. Copyright © 2008 John Wiley & Sons, Ltd.     

二维自适应网格生成的改进AFT与背景网格法

在基于重生成的自适应有限元网格生成算法研究中,将推进波前法（AFT）与背景网格法结合并提出改进方法,有效地解决了网格生成和单元尺寸计算这两个关键问题。改进的AFT方法,将前沿区分为活跃前沿和非活跃前沿两类,在选取目标前沿时既考虑前沿尺寸又考虑前沿分类。改进的背景网格法,利用结构化栅格对背景网格进行管理,在栅格中直接存放背景网格中的单元,既提高了新单元尺寸的计算速度,又从数值上保证了新生成网格中单元之间尺寸合理过渡。     

Coupled solution of the species conservation equations using unstructured finite‐volume method

A coupled solver was developed to solve the species conservation equations on an unstructured mesh with implicit spatial as well as species‐to‐species coupling. First, the computational domain was decomposed into sub‐domains comprised of geometrically contiguous cells—a process similar to additive Schwarz decomposition. This was done using the binary spatial partitioning algorithm. Following this step, for each sub‐domain, the discretized equations were developed using the finite‐volume method, and solved using an iterative solver based on Krylov sub‐space iterations, that is, the pre‐conditioned generalized minimum residual solver. Overall (outer) iterations were then performed to treat explicitness at sub‐domain interfaces and nonlinearities in the governing equations. The solver is demonstrated for both two‐dimensional and three‐dimensional geometries for laminar methane–air flame calculations with 6 species and 2 reaction steps, and for catalytic methane–air combustion with 19 species and 24 reaction steps. It was found that the best performance is manifested for sub‐domain size of 2000 cells or more, the exact number depending on the problem at hand. The overall gain in computational efficiency was found to be a factor of 2–5 over the block (coupled) Gauss–Seidel procedure. All calculations were performed on a single processor machine. The largest calculations were performed for about 355 000 cells (4.6 million unknowns) and required 900 MB of peak runtime memory and 19 h of CPU on a single processor. Copyright © 2009 John Wiley & Sons, Ltd.     

复杂区域非结构化四边形网格全自动生成方法

针对多介质平面复杂区域，本文建议了一种基子推进网阵法的全四边形网格自动生成方法。其特点是对不规则区域适应能力强，考虑了节理单元自动生成。文中提出了网格质量改进方法，并给出了三个工程算例。     

Dynamic parallel Galerkin domain decomposition procedures with grid modification for parabolic equation

Dynamic parallel Galerkin domain decomposition procedures with grid modification for semi‐linear parabolic equation are given. These procedures allow one to apply different domain decompositions, different grids, and interpolation polynomials on the sub‐domains at different time levels when necessary, in order to capture time‐changing localized phenomena, such as, propagating fronts or moving layers. They rely on an implicit Galerkin method in the sub‐domains and simple explicit flux calculation on the inter‐domain boundaries by integral mean method to predict the inner‐boundary conditions. Thus, the parallelism can be achieved by these procedures. These procedures are conservative both in the sub‐domains and across inter‐boundaries. The explicit nature of the flux prediction induces a time step limitation that is necessary to preserve stability, but this constraint is less severe than that for a fully explicit method. Stability and convergence analysis in L²‐norm are derived for these procedures. The experimental results are presented to confirm the theoretical results. Copyright © 2010 John Wiley & Sons, Ltd.     

Adaptive non‐conformal mesh refinement and extended finite element method for viscous flow inside complex moving geometries

Two different techniques to analyze non‐Newtonian viscous flow in complex geometries with internal moving parts and narrow gaps are compared. The first technique is a non‐conforming mesh refinement approach based on the fictitious domain method (FDM), and the second one is the extended finite element method (XFEM). The refinement technique uses one fixed reference mesh, and to impose continuity across non‐conforming regions, constraints using Lagrangian multipliers are used. The size of elements locally in the high shear rate regions is reduced to increase accuracy. FDM is shown to have limitations; therefore, XFEM is applied to decouple the fluid from the internal moving rigid bodies. In XFEM, the discontinuous field variables are captured by using virtual degrees of freedom that serve as enrichment and by applying special integration over the intersected elements. The accuracy of the two methods is demonstrated by direct comparison with results of a boundary‐fitted mesh applied to a two‐dimensional cross section of a twin‐screw extruder. Compared with non‐conforming FDM, XFEM shows a considerable improvement in accuracy around the rigid body, especially in the narrow gap regions. Copyright © 2011 John Wiley & Sons, Ltd.     

Exact pressure integrations on submerged bodies in waves using a quadtree adaptive mesh algorithm

The development of an adaptive free surface, mesh cutting, methodology, in order to analytically integrate pressures on varying wet parts of partially submerged surfaces in the presence of waves, is presented. Given a function of free‐surface elevation, the algorithm checks for the intersection of the body with the free surface and, based on user‐defined parameters, modifies the initial mesh, by subdividing the elements where necessary and eliminating others, via a quadtree approach. Redundant sub‐divisions, generated in the quad‐division process, are partially eliminated, but the quadrilateral nature of the elements is always kept. The free‐surface function must be single‐valued and its definition domain simply connected. Hydrostatic and Froude–Krylov forces are computed exactly on each panel by means of analytical formulations, which are derived and presented, based on the theory of linear gravity waves and from applying Green's theorem. Copyright © 2014 John Wiley & Sons, Ltd.     

Adaptive mesh refinement‐enhanced local DFD method and its application to solve Navier–Stokes equations

Recently, the domain‐free discretization (DFD) method was presented to efficiently solve problems with complex geometries without introducing the coordinate transformation. In order to exploit the high performance of the DFD method, in this paper, the local DFD method with the use of Cartesian mesh is presented, where the physical domain is covered by a Cartesian mesh and the local DFD method is applied for numerical discretization. In order to further improve the efficiency of the solver, the newly developed solution‐based adaptive mesh refinement (AMR) technique is also introduced. The proposed methods are then applied to the simulation of natural convection in concentric annuli between a square outer cylinder and a circular inner cylinder. Numerical experiments show that the present numerical results agree very well with available data in the literature, and AMR‐enhanced local DFD method is an effective tool for the computation of flow problems. Copyright © 2005 John Wiley & Sons, Ltd.     

A fast and practical method to pack spheres for mesh generation

Sphere packing is an attractive way to generate high quality mesh. Several algorithms have been proposed in this topic, however these algorithms are not sufficiently fast for large scale problems. The paper presents an efficient sphere packing algorithm which is much faster and appears to be the most practical among all sphere packing methods presented so far for mesh generation. The algorithm packs spheres inside a domain using advancing front method. High efficiency has resulted from a concept of 4R measure, which localizes all the computations involved in the whole sphere packing process.     

A local mesh refinement approach for large‐eddy simulations of turbulent flows

In this paper, a local mesh refinement (LMR) scheme on Cartesian grids for large‐eddy simulations is presented. The approach improves the calculation of ghost cell pressures and velocities and combines LMR with high‐order interpolation schemes at the LMR interface and throughout the rest of the computational domain to ensure smooth and accurate transition of variables between grids of different resolution. The approach is validated for turbulent channel flow and flow over a matrix of wall‐mounted cubes for which reliable numerical and experimental data are available. Comparisons of predicted first‐order and second‐order turbulence statistics with the validation data demonstrated a convincing agreement. Importantly, it is shown that mean streamwise velocities and fluctuating turbulence quantities transition smoothly across coarse‐to‐fine and fine‐to‐coarse interfaces. © 2016 The Authors International Journal for Numerical Methods in Fluids Published by John Wiley & Sons Ltd     

GENSMAC3D: a numerical method for solving unsteady three‐dimensional free surface flows

A numerical method for solving three‐dimensional free surface flows is presented. The technique is an extension of the GENSMAC code for calculating free surface flows in two dimensions. As in GENSMAC, the full Navier–Stokes equations are solved by a finite difference method; the fluid surface is represented by a piecewise linear surface composed of quadrilaterals and triangles containing marker particles on their vertices; the stress conditions on the free surface are accurately imposed; the conjugate gradient method is employed for solving the discrete Poisson equation arising from a velocity update; and an automatic time step routine is used for calculating the time step at every cycle. A program implementing these features has been interfaced with a solid modelling routine defining the flow domain. A user‐friendly input data file is employed to allow almost any arbitrary three‐dimensional shape to be described. The visualization of the results is performed using computer graphic structures such as phong shade, flat and parallel surfaces. Results demonstrating the applicability of this new technique for solving complex free surface flows, such as cavity filling and jet buckling, are presented. Copyright © 2001 John Wiley & Sons, Ltd.     

适应复杂外形的三维粘性混合网格生成算法

提出一类适应复杂外形的粘性混合网格生成算法。表面网格由前沿推进三角形曲面网格程序获得,边界层布置各向异性的三棱柱体网格,远物面区域采用Delaunay方法生成四面体网格。针对模型的复杂几何特征,综合采用了各种网格处理技术,以保证边界层网格的质量,并避免算法失效问题。网格实例及计算结果表明了本文算法的实用性及和效性。     

Automatic unstructured surface mesh generation for complex configurations

In this paper a new object‐oriented (OO) approach is presented for automatic parallel advancing front based surface mesh generation and adaptive remeshing for complex configurations. Based on the ST++‐system the advantages of the OO design and implementation compared to the traditional structural approach are described. Algorithmic enhancements to the advancing front method are explained, enabling a robust NURBS based triangulation process directly on B‐rep CAD data. The message passing (MPI) parallelization strategy together with the achievable performance improvements are demonstrated. With the outlined parallel geometry analysis/rasterization a powerful method is described to derive automatically a well suited mesh size specification without any user‐interaction from scratch. The application of this method to a complex ‘real world’ example finishes this paper. Copyright © 2004 John Wiley & Sons, Ltd.     

On the immersed boundary‐lattice Boltzmann simulations of incompressible flows with freely moving objects

For simulating freely moving problems, conventional immersed boundary‐lattice Boltzmann methods encounter two major difficulties of an extremely large flow domain and the incompressible limit. To remove these two difficulties, this work proposes an immersed boundary‐lattice Boltzmann flux solver (IB‐LBFS) in the arbitrary Lagragian–Eulerian (ALE) coordinates and establishes a dynamic similarity theory. In the ALE‐based IB‐LBFS, the flow filed is obtained by using the LBFS on a moving Cartesian mesh, and the no‐slip boundary condition is implemented by using the boundary condition‐enforced immersed boundary method. The velocity of the Cartesian mesh is set the same as the translational velocity of the freely moving object so that there is no relative motion between the plate center and the mesh. This enables the ALE‐based IB‐LBFS to study flows with a freely moving object in a large open flow domain. By normalizing the governing equations for the flow domain and the motion of rigid body, six non‐dimensional parameters are derived and maintained to be the same in both physical systems and the lattice Boltzmann framework. This similarity algorithm enables the lattice Boltzmann equation‐based solver to study a general freely moving problem within the incompressible limit. The proposed solver and dynamic similarity theory have been successfully validated by simulating the flow around an in‐line oscillating cylinder, single particle sedimentation, and flows with a freely falling plate. The obtained results agree well with both numerical and experimental data. Copyright © 2016 John Wiley & Sons, Ltd.     

Fictitious domain approach for numerical modelling of Navier–Stokes equations

This study investigates a fictitious domain model for the numerical solution of various incompressible viscous flows. It is based on the so‐called Navier–Stokes/Brinkman and energy equations with discontinuous coefficients all over an auxiliary embedding domain. The solid obstacles or walls are taken into account by a penalty technique. Some volumic control terms are directly introduced in the governing equations in order to prescribe immersed boundary conditions. The implicit numerical scheme, which uses an upwind finite volume method on staggered Cartesian grids, is of second‐order accuracy in time and space. A multigrid local mesh refinement is also implemented, using the multi‐level Zoom Flux Interface Correction (FIC) method, in order to increase the precision where it is needed in the domain. At each time step, some iterations of the augmented Lagrangian method combined with a preconditioned Krylov algorithm allow the divergence‐free velocity and pressure fields be solved for. The tested cases concern external steady or unsteady flows around a circular cylinder, heated or not, and the channel flow behind a backward‐facing step. The numerical results are shown in good agreement with other published numerical or experimental data. Copyright © 2000 John Wiley & Sons, Ltd.     

Dilation‐based shock capturing for high‐order methods

In this paper, we introduce a shock‐capturing artificial viscosity technique for high‐order unstructured mesh methods. This artificial viscosity model is based on a non‐dimensional form of the divergence of the velocity. The technique is an extension and improvement of the dilation‐based artificial viscosity methods introduced in Premasuthan et al., 15 and further extended in Nguyen and Peraire 27 . The approach presented has a number attractive properties including non‐dimensional analytical form, sub‐cell resolution, and robustness for complex shock flows on anisotropic meshes. We present extensive numerical results to demonstrate the performance of the proposed approach. Copyright © 2016 John Wiley & Sons, Ltd.     

Stability of algebraic multigrid for Stokes problems

An investigation is made of the performance of algebraic multigrid (AMG) solvers for the discrete Stokes problem. The saddle‐point formulations are based on the direct enforcement of the fundamental conservation laws in discrete spaces and subsequently stabilised with the aid of a regular splitting of the diffusion operator. AMG solvers based on an independent coarsening of the fields (the unknown approach) and also on a common coarsening (the point approach) are investigated. Both mixed‐order and equal‐order interpolations are considered. The dependence of convergence on the ‘degree of coarsening’ is investigated by studying the ‘convergence versus coarsening’ characteristics and their variation with mesh resolution. They show a consistency in shape, which reveals two distinct performance zones, one convergent the other divergent. The transition from the convergent to the divergent zones is discontinuous and occurs at a critical coarsening factor that is largely mesh independent. It signals a breakdown in the stability of the smoothing at the coarser levels of coarse grid approximation. It is shown that the previously observed, mesh‐dependent, scaling of convergence factors, which had suggested inconsistencies in the coarse grid approximation, is not a reliable marker of inconsistency. It is an indirect consequence of the breakdown in the stability of smoothing. For stable smoothing, reduction factors are shown to be largely mesh independent. The ability of mixed‐order interpolation to permit stable smoothing and therefore to deliver mesh‐independent convergence is explained. Two expedient options are suggested for obtaining mesh‐independent convergence for those AMG codes that are based on an equal‐order interpolation. Copyright © 2012 John Wiley & Sons, Ltd.     

High‐order ENO and WENO schemes for unstructured grids

This work describes the implementation and analysis of high‐order accurate schemes applied to high‐speed flows on unstructured grids. The class of essentially non‐oscillatory schemes (ENO), that includes weighted ENO schemes (WENO), is discussed in the paper with regard to the implementation of third‐ and fourth‐order accurate methods. The entire reconstruction process of ENO and WENO schemes is described with emphasis on the stencil selection algorithms. The stencils can be composed by control volumes with any number of edges, e.g. triangles, quadrilaterals and hybrid meshes. In the paper, ENO and WENO schemes are implemented for the solution of the dimensionless, 2‐D Euler equations in a cell centred finite volume context. High‐order flux integration is achieved using Gaussian quadratures. An approximate Riemann solver is used to evaluate the fluxes on the interfaces of the control volumes and a TVD Runge–Kutta scheme provides the time integration of the equations. Such a coupling of all these numerical tools, together with the high‐order interpolation of primitive variables provided by ENO and WENO schemes, leads to the desired order of accuracy expected in the solutions. An adaptive mesh refinement technique provides better resolution in regions with strong flowfield gradients. Results for high‐speed flow simulations are presented with the objective of assessing the implemented capability. Copyright © 2007 John Wiley & Sons, Ltd.