Adaptive triangular–quadrilateral mesh generation

In this paper, we begin by recalling an adaptive mesh generation method governed by isotropic and anisotropic discrete metric maps, by means of the generation of a unit mesh with respect to a Riemannian structure. We propose then an automatic triangular to quadrilateral mesh conversion scheme, which generalizes the standard case to the anisotropic context. In addition, we introduce an optimal vertex smoothing procedure. Application test examples, in particular a CFD test, are given to demonstrate the efficiency of the proposed method. © 1998 John Wiley & Sons, Ltd.     

Automated refinement of conformal quadrilateral and hexahedral meshes

Conformal refinement using a shrink and connect strategy, known as pillowing or buffer insertion, contracts and reconnects contiguous elements of an all‐quadrilateral or an all‐hexahedral mesh in order to locally increase vertex density without introducing hanging nodes or non‐cubical elements. Using layers as shrink sets, the present method automates the anisotropic refinement of such meshes according to a prescribed size map expressed as a Riemannian metric field. An anisotropic smoother further enhances vertex clustering to capture the features of the metric. Both two‐ and three‐dimensional test cases with analytic control metrics confirm the feasibility of the present approach and explore strategies to minimize the trade‐off between element shape quality and size conformity. Additional examples using discrete metric maps illustrate possible practical applications. Although local vertex removal and reconnection capabilities have yet to be developed, the present refinement method is a step towards an automated tool for conformal adaptation of all‐quadrilateral and all‐hexahedral meshes. Copyright © 2004 John Wiley & Sons, Ltd.     

Isotropic conforming refinement of quadrilateral and hexahedral meshes using two‐refinement templates

This paper presents three automated algorithms for isotropic, conforming refinement of all‐quadrilateral and all‐hexahedral meshes. These algorithms are based on the two‐refinement templates introduced by Schneiders. However, we introduce a novel technique to choose the appropriate refinement template locally. This enables efficient implementation of the proposed algorithms, even in parallel. The proposed algorithms can handle refined regions of complicated geometry and do not suffer from concavity restrictions associated with the three‐refinement methods currently dominating the literature. Several application examples show the strength of these new algorithms in problems that require fast dynamic remeshing such as computational fluid dynamics and computer graphics. Copyright © 2011 John Wiley & Sons, Ltd.     

Octree‐based reasonable‐quality hexahedral mesh generation using a new set of refinement templates

An octree‐based mesh generation method is proposed to create reasonable‐quality, geometry‐adapted unstructured hexahedral meshes automatically from triangulated surface models without any sharp geometrical features. A new, easy‐to‐implement, easy‐to‐understand set of refinement templates is developed to perform local mesh refinement efficiently even for concave refinement domains without creating hanging nodes. A buffer layer is inserted on an octree core mesh to improve the mesh quality significantly. Laplacian‐like smoothing, angle‐based smoothing and local optimization‐based untangling methods are used with certain restrictions to further improve the mesh quality. Several examples are shown to demonstrate the capability of our hexahedral mesh generation method for complex geometries. Copyright © 2008 John Wiley & Sons, Ltd.     

Tetrahedral mesh generation using Delaunay refinement with non‐standard quality measures

This paper studies the practical performance of Delaunay refinement tetrahedral mesh generation algorithms. By using non‐standard quality measures to drive refinement, we show that sliver tetrahedra can be eliminated from constrained Delaunay tetrahedralizations solely by refinement. Despite the fact that quality guarantees cannot be proven, the algorithm can consistently generate meshes with dihedral angles between 18^circ and 154^°. Using a fairer quality measure targeting every type of bad tetrahedron, dihedral angles between 14^° and 154^° can be obtained. The number of vertices inserted to achieve quality meshes is comparable to that needed when driving refinement with the standard circumradius‐to‐shortest‐edge ratio. We also study the use of mesh improvement techniques on Delaunay refined meshes and observe that the minimum dihedral angle can generally be pushed above 20^°, regardless of the quality measure used to drive refinement. The algorithm presented in this paper can accept geometric domains whose boundaries are piecewise smooth. Copyright © 2011 John Wiley & Sons, Ltd.     

A two-layer mesh method for discrete element simulation of gas-particle systems with arbitrarily polyhedral mesh

Inter-phase momentum coupling for particle flows is usually achieved by means of direct numerical simulation (DNS) or point source method (PSM). DNS requires the mesh size of the continuous phase to be much smaller than the size of the smallest particle in the system, whereas PSM requires the mesh size of the continuous phase to be much larger than the particle size. However, for applications where mesh sizes are similar to the size of particles in the system, neither DNS nor PSM is suitable. In order to overcome the dependence of mesh on particle sizes associated with DNS or PSM, a two-layer mesh method (TMM) is proposed. TMM involves the use of a coarse mesh to track the movement of particle clouds and a fine mesh for the continuous phase, with mesh interpolation for information exchange between the coarse and fine mesh Numerical tests of different interpolation methods show that a conservative interpolation scheme of the second order yields the most accurate results. Numerical simulations of a fluidized bed show that there is a good agreement between predictions using TMM with a second-order interpolation scheme and the experimental results, as well as predictions obtained with PSM. Copyright © 2015 John Wiley & Sons, Ltd.     

Applications of mesh smoothing: copy,morph, and sweep on unstructured quadrilateral meshes

Mesh smoothing is demonstrated to be an effective means of copying, morphing, and sweeping unstructured quadrilateral surface meshes from a source surface to a target surface. Construction of the smoother in a particular way guarantees that the target mesh will be a ‘copy’ of the source mesh, provided the boundary data of the target surface is a rigid body rotation, translation, and/or uniform scaling of the original source boundary data and provided the proper boundary node correspondence between source and target has been selected. Copying is not restricted to any particular smoother, but can be based on any locally elliptic second‐order operator. When the bounding loops are more general than rigid body transformations the method generates high‐quality, ‘morphed’ meshes. Mesh sweeping, if viewed as a morphing of the source surface to a set of target surfaces, can be effectively performed via this smoothing algorithm. Published in 1999 by John Wiley & Sons, Ltd. This article is a U.S. government work and is in the public domain in the United States.     

An HR-method of mesh refinement for boundary element method

This paper describes a mesh refinement technique for boundary element method in which the number of elements, the size of elements and the element end location are determined iteratively in order to obtain a user specified accuracy. The method uses L₁ norm as a measure of error in the density function and a grading function that ensures that error over each element is the same. The use of grading function along with L₁ norm makes the mesh refinement technique applicable to Direct and Indirect boundary element method formulation for a variety of boundary element method applications. Numerical problems in elastostatics, fracture mechanics, and bending of plate solved using Direct and Indirect method in which the density functions are approximated by Linear Lagrange, Quadratic Lagrange or Cubic Hermite polynomials validate the effectiveness of the proposed mesh refinement technique. © 1998 John Wiley & Sons, Ltd.     

Adaptive mesh refinement and coarsening for cohesive zone modeling of dynamic fracture

Adaptive mesh refinement and coarsening schemes are proposed for efficient computational simulation of dynamic cohesive fracture. The adaptive mesh refinement consists of a sequence of edge‐split operators, whereas the adaptive mesh coarsening is based on a sequence of vertex‐removal (or edge‐collapse) operators. Nodal perturbation and edge‐swap operators are also employed around the crack tip region to improve crack geometry representation, and cohesive surface elements are adaptively inserted whenever and wherever they are needed by means of an extrinsic cohesive zone model approach. Such adaptive mesh modification events are maintained in conjunction with a topological data structure (TopS). The so‐called PPR potential‐based cohesive model (J. Mech. Phys. Solids 2009; 57 :891–908) is utilized for the constitutive relationship of the cohesive zone model. The examples investigated include mode I fracture, mixed‐mode fracture and crack branching problems. The computational results using mesh adaptivity (refinement and coarsening) are consistent with the results using uniform mesh refinement. The present approach significantly reduces computational cost while exhibiting a multiscale effect that captures both global macro‐crack and local micro‐cracks. Copyright © 2012 John Wiley & Sons, Ltd.     

Simulation of two‐ and three‐dimensional viscoplastic flows using adaptive mesh refinement

This paper presents a finite element solver for the simulation of steady non‐Newtonian flow problems, using a regularized Bingham model, with adaptive mesh refinement capabilities. The solver is based on a stabilized formulation derived from the variational multiscale framework. This choice allows the introduction of an a posteriori error indicator based on the small scale part of the solution, which is used to drive a mesh refinement procedure based on element subdivision. This approach applied to the solution of a series of benchmark examples, which allow us to validate the formulation and assess its capabilities to model 2D and 3D non‐Newtonian flows.     

An adaptive mesh refinement method for solution of the transported PDF equation

This paper presents the results of an investigation into a possible alternative to Monte Carlo methods for solving the transported probability density function (PDF) equation for scalars (compositions). The method uses a finite‐volume approach combined with adaptive mesh refinement (AMR) in a multi‐dimensional compositional space. Comparisons are made between the new method and Monte Carlo solutions for analytical test cases involving the reaction of two or three chemical species. These tests demonstrate the potential of the new method in terms of both accuracy and run time. Additional test cases involving various models for molecular mixing were also conducted with similar conclusions. Copyright © 2009 John Wiley & Sons, Ltd.     

On the incompressible constraint of the 4-node quadrilateral element

The standard plane 4-node element is written as the summation of a constant gradient matrix, usually obtained from underintegration, and a stabilization matrix. The split is based on a Taylor series expansion of element basis functions. In the incompressible limit, the ‘locking’-effect of the quadrilateral is traced back to the stabilization matrix which reflects the incomplete higher-order term in the Taylor series. The incompressibility condition is formulated in a weak sense so that the element displacement field is divergence-free when integrated over the element volume. The resulting algebraic constraint is shown to coincide with a particular eigenvector of the constant gradient matrix which is obtained from the first-order terms of the Taylor series. The corresponding eigenvalue enforces incompressibility implicitly by means of a penalty-constraint. Analytical expressions for that constant-dilatation eigenpair are derived for arbitrary element geometries. It is shown how the incompressible constraint carries over to the element stiffness matrix if the element stabilization is performed in a particular manner. For several classical and recent elements, the eigensystems are analysed numerically. It is shown that most of the formulations reflect the incompressible constraint identically. In the incompressible limit, the numerical accuracies of the elements are compared.     

A multi‐level adaptive mesh refinement method for level set simulations of multiphase flow on unstructured meshes

An adaptive mesh refinement (AMR) technique is proposed for level set simulations of incompressible multiphase flows. The present AMR technique is implemented for two‐dimensional/three‐dimensional unstructured meshes and extended to multi‐level refinement. Smooth variation of the element size is guaranteed near the interface region with the use of multi‐level refinement. A Courant–Friedrich–Lewy condition for zone adaption frequency is newly introduced to obtain a mass‐conservative solution of incompressible multiphase flows. Finite elements around the interface are dynamically refined using the classical element subdivision method. Accordingly, finite element method is employed to solve the problems governed by the incompressible Navier–Stokes equations, using the level set method for dynamically updated meshes. The accuracy of the adaptive solutions is found to be comparable with that of non‐adaptive solutions only if a similar mesh resolution near the interface is provided. Because of the substantial reduction in the total number of nodes, the adaptive simulations with two‐level refinement used to solve the incompressible Navier–Stokes equations with a free surface are about four times faster than the non‐adaptive ones. Further, the overhead of the present AMR procedure is found to be very small, as compared with the total CPU time for an adaptive simulation. Copyright © 2016 John Wiley & Sons, Ltd.     

An improved discrete Kirchhoff quadrilateral element based on third‐order zigzag theory for static analysis of composite and sandwich plates

A new improved discrete Kirchhoff quadrilateral element based on the third‐order zigzag theory is developed for the static analysis of composite and sandwich plates. The element has seven degrees of freedom per node, namely, the three displacements, two rotations and two transverse shear strain components at the mid‐surface. The usual requirement of C¹ continuity of interpolation functions of the deflection in the third‐order zigzag theory is circumvented by employing the improved discrete Kirchhoff constraint technique. The element is free from the shear locking. The finite element formulation and the computer program are validated by comparing the results for simply supported plate with the analytical Navier solution of the zigzag theory. Comparison of the present results with those using other available elements based on zigzag theories for composite and sandwich plates establishes the superiority of the present element in respect of simplicity, accuracy and computational efficiency. The accuracy of the zigzag theory is assessed by comparing the finite element results of the square all‐round clamped composite plates with the converged three‐dimensional finite element solution obtained using ABAQUS. The comparisons also establish the superiority of the zigzag theory over the smeared third‐order theory having the same number of degrees of freedom. Copyright © 2006 John Wiley & Sons, Ltd.     

Adaptive mesh refinement applied to the scalar transported PDF equation in a turbulent jet

This paper describes a novel solution method for the transported probability density function (PDF) equation for scalars (compositions). In contrast to conventional solution methods based on the Monte Carlo approach, we use a finite‐volume method combined with adaptive mesh refinement (AMR) applied in both physical and compositional space. The obvious advantage of this over a uniform grid is that fine meshes are only used where the solution requires high resolution. The efficiency of the method is demonstrated by a number of tests involving a turbulent jet flow with up to two scalars (both reacting and non‐reacting). We find that the AMR calculation can be at a fraction of the computer cost of a uniform grid calculation with the same accuracy. Copyright © 2010 John Wiley & Sons, Ltd.     

Tetrahedral mesh improvement via optimization of the element condition number

We present a new shape measure for tetrahedral elements that is optimal in that it gives the distance of a tetrahedron from the set of inverted elements. This measure is constructed from the condition number of the linear transformation between a unit equilateral tetrahedron and any tetrahedron with positive volume. Using this shape measure, we formulate two optimization objective functions that are differentiated by their goal: the first seeks to improve the average quality of the tetrahedral mesh; the second aims to improve the worst‐quality element in the mesh. We review the optimization techniques used with each objective function and present experimental results that demonstrate the effectiveness of the mesh improvement methods. We show that a combined optimization approach that uses both objective functions obtains the best‐quality meshes for several complex geometries. Copyright © 2001 John Wiley & Sons, Ltd.     

A fixed‐mesh finite element thermally coupled flow formulation for the numerical analysis of melting processes

In the framework of the finite element method, a temperature‐based thermally coupled flow formulation including phase‐change effects is proposed to study melting processes. The governing equations of the problem, written in terms of its primitive variables, are solved using a generalized streamline operator technique that enables the use of equal interpolation functions for the unknowns: velocity, pressure and temperature. Moreover, a unique fixed finite element mesh is used to avoid the difficulties related to moving meshes. This methodology is applied and assessed in the numerical analysis of a benchmark problem known as the melting process of gallium in a differentially heated recipient using distinct geometric aspect ratios. Copyright © 2001 John Wiley & Sons, Ltd.     

Mismatching refinement with domain decomposition for the analysis of steady‐state metal forming process

The finite element analysis of three‐dimensional metal forming processes is generally subject to large computational burden due to its non‐linearity. For economic computation, the mismatching refinement, an efficient domain decomposition method with different mesh density for each subdomain, is developed in the present study. A modified velocity alternating scheme for the interface treatment is proposed in order to obtain good convergence and accuracy in the mismatching refinement. As a numerical example, the analysis of the axisymmetric extrusion processes is carried out. The results are discussed for the various velocity update schemes and for the variation of the length of overlapped region. The three‐dimensional extrusion processes for a rectangular section and an E‐section are analysed in order to verify the effectiveness of the proposed method. Comparing the results with those of the conventional method of full region analysis, the accuracy and the computational efficiency of the proposed method are then discussed. Copyright © 2000 John Wiley & Sons, Ltd.     

On the central difference algorithm in discrete element modelling of impact

The paper provides a comprehensive analysis of the accuracy and stability of the central difference scheme when applied to simulate a simple impact problem in the context of the discrete element approach. It is revealed that the algorithm exhibits some different behaviour due to the inherent non‐linearity/discontinuity of the impact system. Particularly, for an elastic or slightly inelastic impact, the stable/unstable region governing the selection of a maximum time step size is essentially different from that defined by the conventional linear stability criterion, and thus a smaller time step should be employed in order to reduce the possible occurrence of an unexpected numerical instability. Copyright © 2005 John Wiley & Sons, Ltd.     

离散全球格网模型几何均匀性研究

在深入分析离散格网模型系统几何形变的研究现状与不足的基础上,提出了通过研究格网中单个结点与拓扑邻近结点间空间格局,对格网结点空间的均匀性进行评价的新的研究思路.提出了几种新的评价指标,包括邻近点间距离的最大值与最小值比值、邻近点间张角的最大值与最小值比值以及所有结点相互之间边长的标准差.采用上述指标,对6种代表不同类型的全球离散格网模型进行了研究,分析了各种格网模型结点均匀性的变化规律.