Optimization of a regularized distortion measure to generate curved high‐order unstructured tetrahedral meshes

We present a robust method for generating high‐order nodal tetrahedral curved meshes. The approach consists of modifying an initial linear mesh by first, introducing high‐order nodes, second, displacing the boundary nodes to ensure that they are on the computer‐aided design surface, and third, smoothing and untangling the mesh obtained after the displacement of the boundary nodes to produce a valid curved high‐order mesh. The smoothing algorithm is based on the optimization of a regularized measure of the mesh distortion relative to the original linear mesh. This means that whenever possible, the resulting mesh preserves the geometrical features of the initial linear mesh such as shape, stretching, and size. We present several examples to illustrate the performance of the proposed algorithm. Furthermore, the examples show that the implementation of the optimization problem is robust and capable of handling situations in which the mesh before optimization contains a large number of invalid elements. We consider cases with polynomial approximations up to degree ten, large deformations of the curved boundaries, concave boundaries, and highly stretched boundary layer elements. The meshes obtained are suitable for high‐order finite element analyses. Copyright © 2015 John Wiley & Sons, Ltd.     

Mesh improvement by minimizing a weighted sum of squared element volumes

We describe a new mesh smoothing method that consists of minimizing the sum of squared element volumes over the free vertex positions. To the extent permitted by the fixed vertices and mesh topology, the resulting mesh elements have uniformly distributed volumes. In the case of a triangulation, uniform volume implies well‐shaped triangles. If a graded mesh is required, the element volumes may be weighted by centroidal values of a sizing function, resulting in a mesh that conforms to the required vertex density. The method has both a local and a global implementation. In addition to smoothing, the method serves as a simple parameter‐free means of untangling a mesh with inverted elements. It applies to all types of meshes, but we present test results here only for planar triangle meshes. Our test results show the new method to be fast, superior in uniformity or conformity to a sizing function, and among the best methods in terms of triangle shape quality. We also present a new angle‐based method that is simpler and more effective than alternatives. This method is directly aimed at producing well‐shaped triangles and is particularly effective when combined with the volume‐based method. It also generalizes to anisotropic mesh smoothing. Copyright © 2014 John Wiley & Sons, Ltd.     

A distortion measure to validate and generate curved high‐order meshes on CAD surfaces with independence of parameterization

A framework to validate and generate curved nodal high‐order meshes on Computer‐Aided Design (CAD) surfaces is presented. The proposed framework is of major interest to generate meshes suitable for thin‐shell and 3D finite element analysis with unstructured high‐order methods. First, we define a distortion (quality) measure for high‐order meshes on parameterized surfaces that we prove to be independent of the surface parameterization. Second, we derive a smoothing and untangling procedure based on the minimization of a regularization of the proposed distortion measure. The minimization is performed in terms of the parametric coordinates of the nodes to enforce that the nodes slide on the surfaces. Moreover, the proposed algorithm repairs invalid curved meshes (untangling), deals with arbitrary polynomial degrees (high‐order), and handles with low‐quality CAD parameterizations (independence of parameterization). Third, we use the optimization procedure to generate curved nodal high‐order surface meshes by means of an a posteriori approach. Given a linear mesh, we increase the polynomial degree of the elements, curve them to match the geometry, and optimize the location of the nodes to ensure mesh validity. Finally, we present several examples to demonstrate the features of the optimization procedure, and to illustrate the surface mesh generation process. Copyright © 2015 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.     

A minimal element distortion strategy for computational mesh dynamics

Mesh motion strategy is one of the key points in many fluid–structure interaction (FSI) problems. Due to the increasing application of FSI to solve the current challenging engineering problems, this topic has become of great interest. There are several different strategies to solve this problem, some of them use a discrete and lumped spring–mass system to propagate the boundary motion into the volume mesh, and many others use an elastostatic problem to deform the mesh. In all these strategies there is always risk of producing an invalid mesh, i.e. a mesh with some elements inverted. Normally this condition is irreversible and once an invalid mesh is obtained it is difficult to continue. In this paper the mesh motion strategy is defined as an optimization problem. By its definition this strategy can be classified as a particular case of an elastostatic problem where the material constitutive law is defined in terms of the minimization of certain energy functional that takes into account the degree of element distortion. Some advantages of this strategy are its natural tendency to high quality meshes, its robustness and its straightforward extension to 3D problems. Several examples included in this paper show these capabilities. Even though this strategy seems to be very robust it is not able to recover a valid mesh starting from an invalid one. This improvement is left for future work. Copyright © 2006 John Wiley & Sons, Ltd.     

An AMG Preconditioner for Solving the Navier-Stokes Equations with a Moving Mesh Finite Element Method

AMG preconditioners are typically designed for partial differential equation solvers and divergence-interpolation in a moving mesh strategy. Here we introduce an AMG preconditioner to solve the unsteady Navier-Stokes equations by a moving mesh finite element method. A $4P1$ − $P1$ element pair is selected based on the data structure of the hierarchy geometry tree and two-layer nested meshes in the velocity and pressure. Numerical experiments show the efficiency of our approach.     

A multiobjective mesh optimization framework for mesh quality improvement and mesh untangling

We propose a multiobjective mesh optimization framework for mesh quality improvement and mesh untangling. Our framework combines two or more competing objective functions into a single objective function to be solved using one of various multiobjective optimization methods. Methods within our framework are able to optimize various aspects of the mesh such as the element shape, element size, associated PDE interpolation error, and number of inverted elements, but the improvement is not limited to these categories. The strength of our multiobjective mesh optimization framework lies in its ability to be extended to simultaneously optimize any aspects of the mesh and to optimize meshes with different element types. We propose the exponential sum, objective product, and equal sum multiobjective mesh optimization methods within our framework; these methods do not require articulation of preferences. However, the solutions obtained satisfy a sufficient condition of weak Pareto optimality. Experimental results show that our multiobjective mesh optimization methods are able to simultaneously optimize two or more aspects of the mesh and also are able to improve mesh qualities while eliminating inverted elements. We successfully apply our methods to real‐world applications such as hydrocephalus treatment and shape optimization. Copyright © 2013 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.     

Reduced order mesh optimisation using proper orthogonal decomposition and a modified cuckoo search

A new mesh optimisation scheme, reduced order mesh optimisation, is introduced. The technique uses proper orthogonal decomposition to reduce the number of dimensions in a mesh optimisation problem. This reduction in dimensionality allows the expression of the optimisation problem globally rather than the more traditional local mesh optimisation or smoothing algorithms. To perform the optimisation, the recently developed gradient‐free technique modified cuckoo search is applied. The effectiveness of the algorithm is shown by considering the problem of optimising meshes for use in co‐volume techniques. Co‐volume techniques require the existence of two mutually orthogonal meshes; this is achieved by utilising the Delaunay–Voronoi dual. A combination of considering the problem globally and the use of a gradient‐free technique results in a scheme that significantly outperforms previous methods in solving this particular problem. Although the examples presented in this paper are specific to optimising dual meshes, the technique is general and can be simply modified to any mesh optimisation problem. Copyright © 2012 John Wiley & Sons, Ltd.     

A simple mesh deformation technique for fluid–structure interaction based on a submesh approach

This paper describes an improvement in techniques currently used for mesh deformations in fluid–structure calculations in which large body motions are encountered. The proposed approach moving submesh approach (MSA) is based on the assumption of a pseudo-material deformation applied on a triangular coarse mesh to significantly reduce the CPU time. The computation mesh is then updated using an interpolation technique similar to the finite element method. This method may be applied on structured as well as on unstructured meshes. An extension to complex boundaries undergoing large rigid-body motions is proposed combining the MSA and an encapsulation box. The influence of the coarse mesh on the quality mesh is discussed. Copyright © 2008 John Wiley & Sons, Ltd.     

Tetrahedral mesh improvement using swapping and smoothing

Automatic mesh generation and adaptive refinement methods for complex three-dimensional domains have proven to be very successful tools for the efficient solution of complex applications problems. These methods can, however, produce poorly shaped elements that cause the numerical solution to be less accurate and more difficult to compute. Fortunately, the shape of the elements can be improved through several mechanisms, including face- and edge-swapping techniques, which change local connectivity, and optimization-based mesh smoothing methods, which adjust mesh point location. We consider several criteria for each of these two methods and compare the quality of several meshes obtained by using different combinations of swapping and smoothing. Computational experiments show that swapping is critical to the improvement of general mesh quality and that optimization-based smoothing is highly effective in eliminating very small and very large angles. High-quality meshes are obtained in a computationally efficient manner by using optimization-based smoothing to improve only the worst elements and a smart variant of Laplacian smoothing on the remaining elements. Based on our experiments, we offer several recommendations for the improvement of tetrahedral meshes. © 1997 John Wiley & Sons, Ltd.     

Coarsening unstructured meshes by edge contraction

A new unstructured mesh coarsening algorithm has been developed for use in conjunction with multilevel methods. The algorithm preserves geometrical and topological features of the domain, and retains a maximal independent set of interior vertices to produce good coarse mesh quality. In anisotropic meshes, vertex selection is designed to retain the structure of the anisotropic mesh while reducing cell aspect ratio. Vertices are removed incrementally by contracting edges to zero length. Each vertex is removed by contracting the edge that maximizes the minimum sine of the dihedral angles of cells affected by the edge contraction. Rarely, a vertex slated for removal from the mesh cannot be removed; the success rate for vertex removal is typically 99.9% or more. For two‐dimensional meshes, both isotropic and anisotropic, the new approach is an unqualified success, removing all rejected vertices and producing output meshes of high quality; mesh quality degrades only when most vertices lie on the boundary. Three‐dimensional isotropic meshes are also coarsened successfully, provided that there is no difficulty distinguishing corners in the geometry from coarsely‐resolved curved surfaces; sophisticated discrete computational geometry techniques appear necessary to make that distinction. Three‐dimensional anisotropic cases are still problematic because of tight constraints on legal mesh connectivity. More work is required to either improve edge contraction choices or to develop an alternative strategy for mesh coarsening for three‐dimensional anisotropic meshes. Copyright © 2003 John Wiley & Sons, Ltd.     

Crack growth analysis using mesh superposition technique and X‐FEM

In this paper, a new approach to crack propagation analysis for large scale or complicated geometry structures is presented. That is the connection of mesh superposition technique and the extended finite element method. The former is a technique that increases the accuracy of analysis locally by superimposing additional mesh of higher resolution onto the mesh that represents the rough deformation of the structure. In this technique, the boundaries and nodes of both meshes do not have to coincide with each other. In the latter technique, the discontinuity across the crack segment and singularity around the crack tips are represented in the approximation by enriching the nodal degrees using partition of unity condition. This technique does not require meshes to conform to the crack geometry and this enables crack propagation procedure with no re‐meshing process by connecting both techniques, it becomes possible to analyse crack growth models of large scale and complicated shape with highly flexible modelling and no remeshing process. By using this approach, some numerical examples are analysed and appropriate results are obtained. Copyright © 2007 John Wiley & Sons, Ltd.     

Achieving finite element mesh quality via optimization of the Jacobian matrix norm and associated quantities. Part II—A framework for volume mesh optimization and the condition number of the Jacobian matrix

Three‐dimensional unstructured tetrahedral and hexahedral finite element mesh optimization is studied from a theoretical perspective and by computer experiments to determine what objective functions are most effective in attaining valid, high‐quality meshes. The approach uses matrices and matrix norms to extend the work in Part I to build suitable 3D objective functions. Because certain matrix norm identities which hold for 2×2 matrices do not hold for 3×3 matrices, significant differences arise between surface and volume mesh optimization objective functions. It is shown, for example, that the equality in two dimensions of the smoothness and condition number of the Jacobian matrix objective functions does not extend to three dimensions and further, that the equality of the Oddy and condition number of the metric tensor objective functions in two dimensions also fails to extend to three dimensions. Matrix norm identities are used to systematically construct dimensionally homogeneous groups of objective functions. The concept of an ideal minimizing matrix is introduced for both hexahedral and tetrahedral elements. Non‐dimensional objective functions having barriers are emphasized as the most logical choice for mesh optimization. The performance of a number of objective functions in improving mesh quality was assessed on a suite of realistic test problems, focusing particularly on all‐hexahedral ‘whisker‐weaved’ meshes. Performance is investigated on both structured and unstructured meshes and on both hexahedral and tetrahedral meshes. Although several objective functions are competitive, the condition number objective function is particularly attractive. The objective functions are closely related to mesh quality measures. To illustrate, it is shown that the condition number metric can be viewed as a new tetrahedral element quality measure. Published in 2000 by John Wiley & Sons, Ltd.     

基于"边折叠"的可逆累进网格生成算法的研究

在分析已有累进网格生成算法的基础上,提出一种基于“边折叠”网格化简方法的累进网格生成算法,构造了累进网格的表示新方法。算法消除了累进网格技术中的二义性,具有支持多种网格类型、保持相邻层次细节模型间的平滑过渡等特点。实验表明,算法具有有效性和可靠性。     

Quality improvement of non‐manifold hexahedral meshes for critical feature determination of microstructure materials

This paper describes a novel approach to improve the quality of non‐manifold hexahedral meshes with feature preservation for microstructure materials. In earlier works, we developed an octree‐based isocontouring method to construct unstructured hexahedral meshes for domains with multiple materials by introducing the notion of material change edge to identify the interface between two or more materials. However, quality improvement of non‐manifold hexahedral meshes is still a challenge. In the present algorithm, all the vertices are categorized into seven groups, and then a comprehensive method based on pillowing, geometric flow and optimization techniques is developed for mesh quality improvement. The shrink set in the modified pillowing technique is defined automatically as the boundary of each material region with the exception of local non‐manifolds. In the relaxation‐based smoothing process, non‐manifold points are identified and fixed. Planar boundary curves and interior spatial curves are distinguished, and then regularized using B‐spline interpolation and resampling. Grain boundary surface patches and interior vertices are improved as well. Finally, the optimization method eliminates negative Jacobians of all the vertices. We have applied our algorithms to two beta titanium data sets, and the constructed meshes are validated via a statistics study. Finite element analysis of the 92‐grain titanium is carried out based on the improved mesh, and compared with the direct voxel‐to‐element technique. Copyright © 2009 John Wiley & Sons, Ltd.     

COMBINATION OF ADAPTIVITY AND MESH SMOOTHING FOR THE FINITE ELEMENT ANALYSIS OF SHELLS WITH INTERSECTIONS

A compatible hierarchical adaptive scheme is proposed which allows to control both density and geometrical properties of meshes with four-node linear finite shell elements. The algorithm produces a sequence of meshes with two aims, nearly equal distribution of the local error in each element and a mesh with regular elements, thus internal element angles near 90° and length ratios of adjacent element sides near unity. This goal is achieved in an efficient manner imposing a combination of a local smoothing algorithm with the adaptive mesh generation. New created nodes are positioned on the real shell surface and shell boundaries which may be given e.g. by CAD data. Also the shell directors are determined from the normals on the real geometry. Shell intersections are detected automatically as common curves of two adjacent shell parts. As a shell continuum cannot be assumed for these intersections and thus simple standard adaptive schemes fail, shell intersections have to be treated in a way similar to shell boundaries. For some numerical examples the developed algorithms are demonstrated and the resulting meshes are shown. © 1997 by John Wiley & Sons, Ltd.     

Remeshing for metal forming simulations—Part I: Two‐dimensional quadrilateral remeshing

In this paper, a general framework of practical two‐dimensional quadrilateral remeshing, which includes the determination of remeshing time, automatic quadrilateral mesh generation, and data transfer process, will be formulated. In particular, the current work contains new algorithms of mesh density specification according to the distribution of effective strain‐rate gradients, mesh density smoothing by fast Fourier transform (FFT) and low‐pass filtering techniques, coarsening it by node placement scheme, and a modified Laplacian mesh smoothing technique. The efficiency of the developed remeshing scheme was tested through three practical two‐dimensional metal forming simulations. The results clearly indicate that the algorithms proposed in this study make it possible to simulate two‐dimensional metal forming problems efficiently and automatically. Copyright © 2002 John Wiley & Sons, Ltd.     

A consistent point‐searching algorithm for solution interpolation in unstructured meshes consisting of 4‐node bilinear quadrilateral elements

To translate and transfer solution data between two totally different meshes (i.e. mesh 1 and mesh 2), a consistent point‐searching algorithm for solution interpolation in unstructured meshes consisting of 4‐node bilinear quadrilateral elements is presented in this paper. The proposed algorithm has the following significant advantages: (1) The use of a point‐searching strategy allows a point in one mesh to be accurately related to an element (containing this point) in another mesh. Thus, to translate/transfer the solution of any particular point from mesh 2 to mesh 1, only one element in mesh 2 needs to be inversely mapped. This certainly minimizes the number of elements, to which the inverse mapping is applied. In this regard, the present algorithm is very effective and efficient. (2) Analytical solutions to the local co‐ordinates of any point in a four‐node quadrilateral element, which are derived in a rigorous mathematical manner in the context of this paper, make it possible to carry out an inverse mapping process very effectively and efficiently. (3) The use of consistent interpolation enables the interpolated solution to be compatible with an original solution and, therefore guarantees the interpolated solution of extremely high accuracy. After the mathematical formulations of the algorithm are presented, the algorithm is tested and validated through a challenging problem. The related results from the test problem have demonstrated the generality, accuracy, effectiveness, efficiency and robustness of the proposed consistent point‐searching algorithm. Copyright © 1999 John Wiley & Sons, Ltd.