Unconstrained plastering—Hexahedral mesh generation via advancing‐front geometry decomposition

The generation of all‐hexahedral finite element meshes has been an area of ongoing research for the past two decades and remains an open problem. Unconstrained plastering is a new method for generating all‐hexahedral finite element meshes on arbitrary volumetric geometries. Starting from an unmeshed volume boundary, unconstrained plastering generates the interior mesh topology without the constraints of a pre‐defined boundary mesh. Using advancing fronts, unconstrained plastering forms partially defined hexahedral dual sheets by decomposing the geometry into simple shapes, each of which can be meshed with simple meshing primitives. By breaking from the tradition of previous advancing‐front algorithms, which start from pre‐meshed boundary surfaces, unconstrained plastering demonstrates that for the tested geometries, high quality, boundary aligned, orientation insensitive, all‐hexahedral meshes can be generated automatically without pre‐meshing the boundary. Examples are given for meshes from both solid mechanics and geotechnical applications. Copyright © 2009 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.     

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.     

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.     

Automatic generation of transitional meshes

Advances in commercial computer‐aided design software have made finite element analysis with three‐dimensional solid finite elements routinely available. Since these analyses usually confine themselves to those geometrical objects for which particular CAD systems can produce finite element meshes, expanding the capability of analyses becomes an issue of expanding the capability of generating meshes. This paper presents a method for stitching together two three‐dimensional meshes with diverse elements that can include tetrahedral, pentahedral and hexahedral solid finite elements. The stitching produces a mesh that coincides with the edges which already exist on the portion of boundaries that will be joined. Moreover, the transitional mesh does not introduce new edges on these boundaries. Since the boundaries of the regions to be stitched together can have a mixture of triangles and quadrilaterals, tetrahedral and pyramidal elements provide the transitional elements required to honor these constraints. On these boundaries a pyramidal element shares its base face with the quadrilateral faces of hexahedra and pentahedra. Tetrahedral elements share a face with the triangles on the boundary. Tetrahedra populate the remaining interior of the transitional region. Copyright © 2001 John Wiley & Sons, Ltd.     

Quality improvement methods for hexahedral element meshes adaptively generated using grid‐based algorithm

The quality of finite element meshes is one of the key factors that affects the accuracy and reliability of numerical simulation results of many science and engineering problems. In order to solve the problem wherein the surface elements of the mesh generated by the grid‐based method have poor quality, this paper studied mesh quality improvement methods, including node position smoothing and topological optimization. A curvature‐based Laplacian scheme was used for smoothing of nodes on the C‐edges, which combined the normal component with the tangential component of the Laplacian operator at the curved boundary. A projection‐based Laplacian algorithm for smoothing the remaining boundary nodes was established. The deviation of the newly smoothed node from the practical surface of the solid model was solved. A node‐ and area‐weighted combination method was proposed for smoothing of interior nodes. Five element‐inserting modes, three element‐collapsing modes and three mixed modes for topological optimization were newly established. The rules for harmonious application and conformity problem of each mode, especially the mixed mode, were provided. Finally, several examples were given to demonstrate the practicability and validity of the mesh quality improvement methods presented in this paper. Copyright © 2011 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.     

The tetrahedral finite cell method: Higher‐order immersogeometric analysis on adaptive non‐boundary‐fitted meshes

The finite cell method (FCM) is an immersed domain finite element method that combines higher‐order non‐boundary‐fitted meshes, weak enforcement of Dirichlet boundary conditions, and adaptive quadrature based on recursive subdivision. Because of its ability to improve the geometric resolution of intersected elements, it can be characterized as an immersogeometric method. In this paper, we extend the FCM, so far only used with Cartesian hexahedral elements, to higher‐order non‐boundary‐fitted tetrahedral meshes, based on a reformulation of the octree‐based subdivision algorithm for tetrahedral elements. We show that the resulting TetFCM scheme is fully accurate in an immersogeometric sense, that is, the solution fields achieve optimal and exponential rates of convergence for h‐refinement and p‐refinement, if the immersed geometry is resolved with sufficient accuracy. TetFCM can leverage the natural ability of tetrahedral elements for local mesh refinement in three dimensions. Its suitability for problems with sharp gradients and highly localized features is illustrated by the immersogeometric phase‐field fracture analysis of a human femur bone. Copyright © 2016 John Wiley & Sons, Ltd.     

Variational generation of prismatic boundary‐layer meshes for biomedical computing

Boundary‐layer meshes are important for numerical simulations in computational fluid dynamics, including computational biofluid dynamics of air flow in lungs and blood flow in hearts. Generating boundary‐layer meshes is challenging for complex biological geometries. In this paper, we propose a novel technique for generating prismatic boundary‐layer meshes for such complex geometries. Our method computes a feature size of the geometry, adapts the surface mesh based on the feature size, and then generates the prismatic layers by propagating the triangulated surface using the face‐offsetting method. We derive a new variational method to optimize the prismatic layers to improve the triangle shapes and edge orthogonality of the prismatic elements and also introduce simple and effective measures to guarantee the validity of the mesh. Coupled with a high‐quality tetrahedral mesh generator for the interior of the domain, our method generates high‐quality hybrid meshes for accurate and efficient numerical simulations. We present comparative study to demonstrate the robustness and quality of our method for complex biomedical geometries. Copyright © 2009 John Wiley & Sons, Ltd.     

The generation of triangular meshes for NURBS‐enhanced FEM

This paper presents the first method that enables the fully automatic generation of triangular meshes suitable for the so‐called non‐uniform rational B‐spline (NURBS)‐enhanced finite element method (NEFEM). The meshes generated with the proposed approach account for the computer‐aided design boundary representation of the domain given by NURBS curves. The characteristic element size is completely independent of the geometric complexity and of the presence of very small geometric features. The proposed strategy allows to circumvent the time‐consuming process of de‐featuring complex geometric models before a finite element mesh suitable for the analysis can be produced. A generalisation of the original definition of a NEFEM element is also proposed, enabling to treat more complicated elements with an edge defined by several NURBS curves or more than one edge defined by different NURBS. Three examples of increasing difficulty demonstrate the applicability of the proposed approach and illustrate the advantages compared with those of traditional finite element mesh generators. Finally, a simulation of an electromagnetic scattering problem is considered to show the applicability of the generated meshes for finite element analysis. ©2016 The Authors. International Journal for Numerical Methods in Engineering published by John Wiley & Sons Ltd.     

Direct modification for non‐conforming elements with drilling DOF

A unified approach to eliminate the undesirable lockings in distorted meshes for both non‐conforming quadrilateral membrane and hexahedral elements with drilling (or rotational) degrees of freedom is presented. The direct modification scheme is utilized in the formulation of non‐conforming modes to improve the general behaviour of isoparametric‐based elements. It is shown that the direct modification is very effective in eliminating the locking and this improvement of element behaviour may be doubled if the selective integration scheme is used simultaneously. To verify the validity of elements formulated by the proposed schemes and to evaluate their effectiveness, several numerical tests are carried out. The combined use of additional non‐conforming modes with the direct modification method and the selective integration technique plays an important role to improve the behaviour of elements, especially for distorted mesh cases. Test results also show that the results for both non‐conforming quadrilateral membrane and hexahedral elements obtained by the proposed schemes are equally satisfactory. Copyright © 2002 John Wiley & Sons, Ltd.     

Automatic generation of multiblock decompositions of surfaces

Multiblock‐structured meshes have significant advantages over fully unstructured meshes in numerical simulation, but automatically generating these meshes is considerably more difficult. A method is described herein for automatically generating high‐quality multiblock decompositions of surfaces with boundaries. Controllability and flexibility are useful capabilities of the method. Additional alignment constraints for forcing the appearance of particular features in the decomposition can be easily handled. Also, adjustments are made according to input metric tensor fields that describe target element size properties. The general solution strategy is based around using a four‐way symmetry vector‐field, called a cross‐field, to describe the local mesh orientation on a triangulation of the surface. Initialisation is performed by propagating the boundary alignment constraints to the interior in a fast marching method. This is similar in a way to an advancing‐front or paving method but much more straightforward and flexible because mesh connectivity does not have to be managed in the cross‐field. Multiblock decompositions are generated by tracing the separatrices of the cross‐field to partition the surface into quadrilateral blocks with square corners. The final task of meshing the decomposition requires solving an integer programming problem for block division numbers. Copyright © 2015 John Wiley & Sons, Ltd.     

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.     

Three‐dimensional grayscale‐free topology optimization using a level‐set based r‐refinement method

In this paper, we propose a three‐dimensional (3D) grayscale‐free topology optimization method using a conforming mesh to the structural boundary, which is represented by the level‐set method. The conforming mesh is generated in an r‐refinement manner; that is, it is generated by moving the nodes of the Eulerian mesh that maintains the level‐set function. Although the r‐refinement approach for the conforming mesh generation has many benefits from an implementation aspect, it has been considered as a difficult task to stably generate 3D conforming meshes in the r‐refinement manner. To resolve this task, we propose a new level‐set based r‐refinement method. Its main novelty is a procedure for minimizing the number of the collapsed elements whose nodes are moved to the structural boundary in the conforming mesh; in addition, we propose a new procedure for improving the quality of the conforming mesh, which is inspired by Laplacian smoothing. Because of these novelties, the proposed r‐refinement method can generate 3D conforming meshes at a satisfactory level, and 3D grayscale‐free topology optimization is realized. The usefulness of the proposed 3D grayscale‐free topology optimization method is confirmed through several numerical examples. Copyright © 2017 John Wiley & Sons, Ltd.     

A displacement‐based finite element formulation for general polyhedra using harmonic shape functions

A displacement‐based continuous‐Galerkin finite element formulation for general polyhedra is presented for applications in nonlinear solid mechanics. The polyhedra can have an arbitrary number of vertices or faces. The faces of the polyhedra can have an arbitrary number of edges and can be nonplanar. The polyhedra can be nonconvex with only the mild restriction of star convexity with respect to the vertex‐averaged centroid. Conforming shape functions are constructed using harmonic functions defined on the undeformed configuration, thus requiring the use of a total‐Lagrangian finite element formulation in large deformation applications. For nonlinear applications with computationally intensive constitutive models, it is important to minimize the number of element quadrature points. For this reason, an integration scheme is adopted in which the number of quadrature points is equal to the number of vertices. As a first step toward verifying the element behavior in the general nonlinear setting, several linear verification examples are presented using both random Voronoi meshes and distorted hexahedral meshes. For the hexahedral meshes, results for the polyhedral formulation are compared to those of the standard trilinear hexahedral formulation. The element behavior in the nearly incompressible regime is also examined. Copyright © 2013 John Wiley & Sons, Ltd.     

Q‐Morph: an indirect approach to advancing front quad meshing

Q‐Morph is a new algorithm for generating all‐quadrilateral meshes on bounded three‐dimensional surfaces. After first triangulating the surface, the triangles are systematically transformed to create an all‐quadrilateral mesh. An advancing front algorithm determines the sequence of triangle transformations. Quadrilaterals are formed by using existing edges in the triangulation, by inserting additional nodes, or by performing local transformations to the triangles. A method typically used for recovering the boundary of a Delaunay mesh is used on interior triangles to recover quadrilateral edges. Any number of triangles may be merged to form a single quadrilateral. Topological clean‐up and smoothing are used to improve final element quality. Q‐Morph generates well‐aligned rows of quadrilaterals parallel to the boundary of the domain while maintaining a limited number of irregular internal nodes. The proposed method also offers the advantage of avoiding expensive intersection calculations commonly associated with advancing front procedures. A series of examples of Q‐Morph meshes are also presented to demonstrate the versatility of the proposed method. Copyright © 1999 John Wiley & Sons, Ltd.     

An embedded mesh method for treating overlapping finite element meshes

A new technique for treating the mechanical interactions of overlapping finite element meshes is presented. Such methods can be useful for numerous applications, for example, fluid–solid interaction with a superposed meshed solid on an Eulerian background fluid grid. In this work, we consider the interaction of two elastic domains: one mesh is the foreground and defines the surface of interaction, the other is a background mesh and is often a structured grid. Many of the previously proposed methods employ surface defined Lagrange multipliers or penalties to enforce the boundary constraints. It has become apparent that these methods will cause mesh locking under certain conditions. Appropriately applied, the Nitsche method can overcome this locking, but, in its canonical form, is generally not applicable to non‐linear materials such as hyperelastics. The relationship between interior point penalty, discontinuous Galerkin and Nitsche's method is well known. Based on this relationship, a nonlinear theory analogous to the Nitsche method is proposed to treat nonlinear materials in an embedded mesh. Here, a discontinuous Galerkin derivative based on a lifting of the interface surface integrals provides a consistent treatment for non‐linear materials and demonstrates good behavior in example problems. Published 2012. This article is a US Government work and is in the public domain in the USA.     

The generation of hexahedral meshes for assembly geometry: survey and progress

The finite element method is being used today to model component assemblies in a wide variety of application areas, including structural mechanics, fluid simulations, and others. Generating hexahedral meshes for these assemblies usually requires the use of geometry decomposition, with different meshing algorithms applied to different regions. While the primary motivation for this approach remains the lack of an automatic, reliable all‐hexahedral meshing algorithm, requirements in mesh quality and mesh configuration for typical analyses are also factors. For these reasons, this approach is also sometimes required when producing other types of unstructured meshes. This paper will review progress to date in automating many parts of the hex meshing process, which has halved the time to produce all‐hex meshes for large assemblies. Particular issues which have been exposed due to this progress will also be discussed, along with their applicability to the general unstructured meshing problem. Published in 2001 by John Wiley & Sons, Ltd.     

Effective volume‐fraction optimization for thermal stress reduction in FGMs using irregular h‐refinement

The volume fraction of constituent particles in functionally graded materials (FGM) varies continuously and functionally, and its optimal tailoring could be made by the numerical optimization incorporated with the finite element method. In such a case, the mesh density in finite element discretization of the volume fraction field influences the final design quality such that the further objective function reduction requires the refinement of volume fraction meshes. However, the uniform refinement of the volume fraction mesh is not effective, from the numerical point of view, particularly when the finite difference scheme is employed. This numerical inefficiency could be resolved by locally increasing the mesh density only where more volume‐fraction flexibility (i.e. more mesh density) is required. In this paper, we propose an effective volume‐fraction optimization procedure by applying the irregular h‐refinement to the volume fraction discretization. Copyright © 2003 John Wiley & Sons, Ltd.