Adaptive mesh generation based on multi‐resolution quadtree representation

An advanced CAD model is required for efficient, real‐time adaptive generation of FE meshes. In this paper, a discrete level of detail (LOD) method for reconstructing progressive multiresolution models is proposed. With this approach, the model is reconstructed a priori so that any level of detail can be accessed directly, in real time, according to application requirements. The mesh is generated adaptively according to geometrical or analysis error indicators, where even at lower levels of resolution, critical areas are preserved. The method has been extended to progressive time and geometrical models for simulation and is demonstrated by several examples. Copyright © 2000 John Wiley & Sons, Ltd.     

Automatic image‐based stress analysis by the scaled boundary finite element method

Digital imaging technologies such as X‐ray scans and ultrasound provide a convenient and non‐invasive way to capture high‐resolution images. The colour intensity of digital images provides information on the geometrical features and material distribution which can be utilised for stress analysis. The proposed approach employs an automatic and robust algorithm to generate quadtree (2D) or octree (3D) meshes from digital images. The use of polygonal elements (2D) or polyhedral elements (3D) constructed by the scaled boundary finite element method avoids the issue of hanging nodes (mesh incompatibility) commonly encountered by finite elements on quadtree or octree meshes. The computational effort is reduced by considering the small number of cell patterns occurring in a quadtree or an octree mesh. Examples with analytical solutions in 2D and 3D are provided to show the validity of the approach. Other examples including the analysis of 2D and 3D microstructures of concrete specimens as well as of a domain containing multiple spherical holes are presented to demonstrate the versatility and the simplicity of the proposed technique. Copyright © 2016 John Wiley & Sons, Ltd.     

Automatic three-dimensional finite element mesh generation via modified ray casting

Three-dimensional (3-D) finite element mesh generation has been the target of automation due to the complexities associated with generating and visualizing the mesh. A fully automatic 3-D mesh generation method is developed. The method is capable of meshing CSG solid models. It is based on modifying the classical ray-casting technique to meet the requirements of mesh generation. The modifications include the utilization of the element size in the casting process, the utilization of 3-D space box enclosures, and the casting of ray segments (rays with finite length). The method begins by casting ray segments into the solid. Based on the intersections between the segments and the solid boundary, the solid is discretized into cells arranged in a structure. The cell structure stores neighbourhood relations between its cells. Each cell is meshed with valid finite elements. Mesh continuity between cells is achieved via the neighbourhood relations. The last step is to process the boundary elements to represent closely the boundary. The method has been tested and applied to a number of solid models. Sample examples are presented.     

A bubble‐inspired algorithm for finite element mesh partitioning

This paper presents a bubble‐inspired algorithm for partitioning finite element mesh into subdomains. Differing from previous diffusion BUBBLE and Center‐oriented Bubble methods, the newly proposed algorithm employs the physics of real bubbles, including nucleation, spherical growth, bubble–bubble collision, reaching critical state, and the final competing growth. The realization of foaming process of real bubbles in the algorithm enables us to create partitions with good shape without having to specify large number of artificial controls. The minimum edge cut is simply achieved by increasing the volume of each bubble in the most energy efficient way. Moreover, the order, in which an element is gathered into a bubble, delivers the minimum number of surface cells at every gathering step; thus, the optimal numbering of elements in each subdomain has naturally achieved. Because finite element solvers, such as multifrontal method, must loop over all elements in the local subdomain condensation phase and the global interface solution phase, these two features have a huge payback in terms of solver efficiency. Experiments have been conducted on various structured and unstructured meshes. The obtained results are consistently better than the classical kMetis library in terms of the edge cut, partition shape, and partition connectivity. Copyright © 2012 John Wiley & Sons, Ltd.     

A compact adjacency‐based topological data structure for finite element mesh representation

This paper presents a novel compact adjacency‐based topological data structure for finite element mesh representation. The proposed data structure is designed to support, under the same framework, both two‐ and three‐dimensional meshes, with any type of elements defined by templates of ordered nodes. When compared to other proposals, our data structure reduces the required storage space while being ‘complete’, in the sense that it preserves the ability to retrieve all topological adjacency relationships in constant time or in time proportional to the number of retrieved entities. Element and node are the only entities explicitly represented. Other topological entities, which include facet, edge, and vertex, are implicitly represented. In order to simplify accessing topological adjacency relationships, we also define and implicitly represent oriented entities, associated to the use of facets, edges, and vertices by an element. All implicit entities are represented by concrete types, being handled as values, which avoid usual problems encountered in other reduced data structures when performing operations such as entity enumeration and attribute attachment. We also extend the data structure with the use of ‘reverse indices’, which improves performance for extracting adjacency relationships while maintaining storage space within reasonable limits. The data structure effectiveness is demonstrated by two different applications: for supporting fragmentation simulation and for supporting volume rendering algorithms. Copyright © 2005 John Wiley & Sons, Ltd.     

An enhanced parallel sub‐domain generation method for mesh partitioning in parallel finite element analysis

This paper describes an optimization and artificial intelligence‐based approach for solving the mesh partitioning problem for explicit parallel dynamic finite element analysis. The Sub‐Domain Generation Method (SGM) (Topping, Khan, Parallel Finite Element Computations. Saxe‐Coburg Publications: Edinburgh, U.K., 1996) is briefly introduced with its virtues and drawbacks. This paper describes the enhancement of the SGM algorithm (ESGM) by the introduction of a new, non‐convex bisection procedure and a new Genetic Algorithm (GA) module, which is better tuned for this particular optimization problem. Example decompositions are given and comparisons made between parallel versions of the ESGM, the SGM and other decomposition methods. The scalability of the ESGM is examined by using a range of examples. Copyright © 2000 John Wiley & Sons, Ltd.     

Achieving finite element mesh quality via optimization of the Jacobian matrix norm and associated quantities. Part I—a framework for surface mesh optimization

Structured mesh quality optimization methods are extended to optimization of unstructured triangular, quadrilateral, and mixed finite element meshes. New interpretations of well‐known nodally based objective functions are made possible using matrices and matrix norms. The matrix perspective also suggests several new objective functions. Particularly significant is the interpretation of the Oddy metric and the smoothness objective functions in terms of the condition number of the metric tensor and Jacobian matrix, respectively. Objective functions are grouped according to dimensionality to form weighted combinations. A simple unconstrained local optimum is computed using a modified Newton iteration. The optimization approach was implemented in the CUBIT mesh generation code and tested on several problems. Results were compared against several standard element‐based quality measures to demonstrate that good mesh quality can be achieved with nodally based objective functions. Published in 2000 by John Wiley & Sons, Ltd.     

Clustered generalized finite element methods for mesh unrefinement,non‐matching and invalid meshes

In spite of significant advancements in automatic mesh generation during the past decade, the construction of quality finite element discretizations on complex three‐dimensional domains is still a difficult and time demanding task. In this paper, the partition of unity framework used in the generalized finite element method (GFEM) is exploited to create a very robust and flexible method capable of using meshes that are unacceptable for the finite element method, while retaining its accuracy and computational efficiency. This is accomplished not by changing the mesh but instead by clustering groups of nodes and elements. The clusters define a modified finite element partition of unity that is constant over part of the clusters. This so‐called clustered partition of unity is then enriched to the desired order using the framework of the GFEM. The proposed generalized finite element method can correctly and efficiently deal with: (i) elements with negative Jacobian; (ii) excessively fine meshes created by automatic mesh generators; (iii) meshes consisting of several sub‐domains with non‐matching interfaces. Under such relaxed requirements for an acceptable mesh, and for correctly defined geometries, today's automated tetrahedral mesh generators can practically guarantee successful volume meshing that can be entirely hidden from the user. A detailed technical discussion of the proposed generalized finite element method with clustering along with numerical experiments and some implementation details are presented. Copyright © 2006 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.     

An extended advancing front technique for closed surfaces mesh generation

An extended advancing front technique (AFT) with shift operations and Riemann metric named as shifting‐AFT is presented for finite element mesh generation on 3D surfaces, especially 3D closed surfaces. Riemann metric is used to govern the size and shape of the triangles in the parametric space. The shift operators are employed to insert a floating space between real space and parametric space during the 2D parametric space mesh generation. In the previous work of closed surface mesh generation, the virtual boundaries are adopted when mapping the closed surfaces into 2D open parametric domains. However, it may cause the mesh quality‐worsening problem. In order to overcome this problem, the AFT kernel is combined with the shift operator in this paper. The shifting‐AFT can generate high‐quality meshes and guarantee convergence in both open and closed surfaces. For the shifting‐AFT, it is not necessary to introduce virtual boundaries while meshing a closed surface; hence, the boundary discretization procedure is largely simplified, and moreover, better‐shaped triangles will be generated because there are no additional interior constraints yielded by virtual boundaries. Comparing with direct methods, the shifting‐AFT avoids costly and unstable 3D geometrical computations in the real space. Some examples presented in this paper have demonstrated the advantages of shift‐AFT in 3D surface mesh generation, especially for the closed surfaces. Copyright © 2007 John Wiley & Sons, Ltd.     

Gmsh: A 3‐D finite element mesh generator with built‐in pre‐ and post‐processing facilities

Gmsh is an open‐source 3‐D finite element grid generator with a build‐in CAD engine and post‐processor. Its design goal is to provide a fast, light and user‐friendly meshing tool with parametric input and advanced visualization capabilities. This paper presents the overall philosophy, the main design choices and some of the original algorithms implemented in Gmsh. Copyright © 2009 John Wiley & Sons, Ltd.     

An algorithm to generate quadrilateral or triangular element surface meshes in arbitrary domains with applications to crack propagation

A new hybrid algorithm for automatically generating either an all-quadrilateral or an all-triangular element mesh within an arbitrarily shaped domain is described. The input consists of one or more closed loops of straight-line segments that bound the domain. Internal mesh density is inferred from the boundary density using a recursive spatial decomposition (quadtree) procedure. All-triangular element meshes are generated using a boundary contraction procedure. All-quadrilateral element meshes are generated by modifying the boundary contraction procedure to produce a mixed element mesh at half the density of the final mesh and then applying a polygon-splitting procedure. The final meshes exhibit good transitioning properties and are compatible with the given boundary segments which are not altered. The algorithm can support discrete crack growth simulation wherein each step of crack growth results in an arbitrarily shaped region of elements deleted about each crack tip. The algorithm is described and examples of the generated meshes are provided for a representative selection of cracked and uncracked structures.     

A mesh adaptation procedure for periodic domains

In this paper, an original technique is developed in order to build adaptive meshes on periodic domains. The new approach has the important property that it is code‐reused. The procedure is used against three different algorithms, namely, MAdLib ( Int. J. Numer. Meth. Engng 2000; in press), mmg (Proc. 17th Int. Meshing Roundtable, 2008) and the couple Yams (Rapport Technique RT‐0252, 2001) /Ghs3d (Proc. 8th Int. Meshing Roundtable, 1999). None of the latter algorithms needs to be adapted before it is applied to periodic domains. Some examples of adaptation are presented based on analytical, isotropic and anisotropic mesh‐size fields. Periodicity in translation and rotation both are considered. Finally, the mesh adaptation strategy is used in order to reduce the computational cost of a prediction of strain heterogeneity throughout a periodic polycrystalline aggregate deforming by dislocation slip. Copyright © 2011 John Wiley & Sons, Ltd.     

Advances in tetrahedral mesh generation for modelling of three‐dimensional regions with complex,curvilinear crack shapes

Advances in tetrahedral mesh generation for general, three‐dimensional domains with and without cracks are described and validated through extensive studies using a wide range of global geometries and local crack shapes. Automated methods are described for (a) implementing geometrical measures in the vicinity of the crack to identify irregularities and to improve mesh quality and (b) robust node selection on crack surfaces to ensure optimal meshing both locally and globally. The resulting numerical algorithms identify both node coincidence and also local crack surface penetration due to discretization of curved crack surfaces, providing a proven approach for removing inconsistencies. Numerical examples using the resulting 3D mesh generation program to mesh complex 3D domains containing a range of crack shapes and sizes are presented. Quantitative measures of mesh quality clearly show that the element shape and size distributions are excellent, including in regions surrounding crack fronts. Copyright © 2005 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.     

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.     

A partition‐of‐unity‐based finite element method for level sets

Level set methods have recently gained much popularity to capture discontinuities, including their possible propagation. Typically, the partial differential equations that arise in level set methods, in particular the Hamilton–Jacobi equation, are solved by finite difference methods. However, finite difference methods are less suited for irregular domains. Moreover, it seems slightly awkward to use finite differences for the capturing of a discontinuity, while in a subsequent stress analysis finite elements are normally used. For this reason, we here present a finite element approach to solving the governing equations of level set methods. After a review of the governing equations, the initialization of the level sets, the discretization on a finite domain, and the stabilization of the resulting finite element method will be discussed. Special attention will be given to the proper treatment of the internal boundary condition, which is achieved by exploiting the partition‐of‐unity property of finite element shape functions. Finally, a quantitative analysis including accuracy analysis is given for a one‐dimensional example and a qualitative example is given for a two‐dimensional case with a curved discontinuity. Copyright © 2008 John Wiley & Sons, Ltd.     

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.