An effective 3D meshing approach for fractured rocks

An effective mesh generation algorithm is proposed to construct mesh representations for arbitrary fractures in 3D rock masses. With the development of advanced imaging techniques, fractures in a rock mass can be clearly captured by a high‐resolution 3D digital image but with a huge data set. To reduce the data size, corresponding mesh substitutes are required in both visualization and numerical analysis. Fractures in rocks are naturally complicated. They may meet at arbitrary angles at junctions, which could derive topological defects, geometric errors or local connectivity flaws on mesh models. A junction weight is proposed and applied to distinguish fracture junctions from surfaces by an adequate threshold. We take account of fracture junctions and generate an initial surface mesh by a simplified centroidal Voronoi diagram. To further repair the initial mesh, an innovative umbrella operation is designed and adopted to correct mesh topology structures and preserve junction geometry features. Constrained with the aforementioned surface mesh of fracture, a tetrahedral mesh is generated and substituted for the 3D image model to be involved in future numerical analysis. Finally, we take two fractured rock samples as application examples to demonstrate the usefulness and capability of the proposed meshing approach. Copyright © 2015 John Wiley & Sons, Ltd.     

Fully‐automated hex‐dominant mesh generation with directionality control via packing rectangular solid cells

A new fully automatic hex‐dominant mesh generation technique of an arbitrary 3D geometric domain is presented herein. The proposed method generates a high‐quality hex‐dominant mesh by: (1) controlling the directionality of the output hex‐dominant mesh; and (2) avoiding ill‐shaped elements induced by nodes located too closely to each other. The proposed method takes a 3D geometric domain as input and creates a hex‐dominant mesh consisting mostly of hexahedral elements, with additional prism and tetrahedral elements. Rectangular solid cells are packed on the boundary of and inside the input domain to obtain ideal node locations for a hex‐dominant mesh. Each cell has a potential energy field that mimics a body‐centred cubic (BCC) structure (seen in natural substances such as NaCl) and the cells are moved to stable positions by a physically based simulation. The simulation mimics the formation of a crystal pattern so that the centres of the cells provide ideal node locations for a hex‐dominant mesh. Via the advancing front method, the centres of the packed cells are then connected to form a tetrahedral mesh, and this is converted to a hex‐dominant mesh by merging some of the tetrahedrons. Copyright © 2003 John Wiley & Sons, Ltd.     

Variational Delaunay approach to the generation of tetrahedral finite element meshes

We describe an algorithm which generates tetrahedral decomposition of a general solid body, whose surface is given as a collection of triangular facets. The principal idea is to modify the constraints in such a way as to make them appear in an unconstrained triangulation of the vertex set à priori. The vertex set positions are randomized to guarantee existence of a unique triangulation which satisfies the Delaunay empty‐sphere property. (Algorithms for robust, parallelized construction of such triangulations are available.) In order to make the boundary of the solid appear as a collection of tetrahedral faces, we iterate two operations, edge flip and edge split with the insertion of additional vertex, until all of the boundary facets are present in the tetrahedral mesh. The outcome of the vertex insertion is another triangulation of the input surfaces, but one which is represented as a subset of the tetrahedral faces. To determine if a constraining facet is present in the unconstrained Delaunay triangulation of the current vertex set, we use the results of Rajan which re‐formulate Delaunay triangulation as a linear programming problem. Copyright © 2001 John Wiley & Sons, Ltd.     

基于黎曼度量参数曲面四边形网格生成方法及UG实现

论文给出了基于黎曼度量的参数曲面网格生成的改进铺砖算法。阐述了曲面自身的黎曼度量,并且运用黎曼度量计算二维参数域上单元节点的位置,从而使映射到三维物理空间的四边形网格形状良好。文中对原有铺砖法相交处理进行了改进,在运用铺砖法的同时调用UG-NX强大的二次开发库函数获取相应的信息,直接在UG-NX模型的表面生成四边形网格。算例表明,该法能在曲面上生成质量好的网格。     

A surface mesh smoothing and untangling method independent of the CAD parameterization

A method to optimize triangular and quadrilateral meshes on parameterized surfaces is proposed. The optimization procedure relocates the nodes on the surface to improve the quality (smooth) and ensures that the elements are not inverted (untangle). We detail how to express any measure for planar elements in terms of the parametric coordinates of the nodes. The extended measures can be used to check the quality and validity of a surface mesh. Then, we detail how to optimize any Jacobian-based distortion measure to obtain smoothed and untangled meshes with the nodes on the surface. We prove that this method is independent of the surface parameterization. Thus, it can optimize meshes on CAD surfaces defined by low-quality parameterizations. The examples show that the method can optimize meshes composed by a large number of inverted elements. Finally, the method can be extended to obtain high-order meshes with the nodes on the CAD surfaces.     

A new approach to free-form surface mesh control in a CAD environment

An adaptive process controlling the position of nodes on a surface mesh is presented. The control can depend on one (or more) criterion(ria) about element quality. The mesh is attached, through the concept of classification, to a geometric model issued by a computer aided design software. Thus, the surface domain is described by entities currently available in such systems, i.e. any free-form patches like Non-Uniform Rational B-Spline or Bézier patches can be used, even if they are restricted. Multi-connected surface domains can be treated using the same geometrical definition. The method described allows nodes to slide on a patch or jump from a patch onto another one. Such movements greatly improve the mesh quality with regard to a chosen criterion. Problems occurring with patch-by-patch meshing techniques when surfaces patches exhibit significant size differences are then overcome. The adaptation technique can be made independent of CAD data structures and meshing techniques, hence it constitutes the basis of a mesh management module.     

Variational surface reconstruction based on Delaunay triangulation and graph cut

A novel method for surface reconstruction from an unorganized point set is presented. An energy functional based on a weighted minimal surface model is proposed for surface reconstruction, which is efficiently minimized by graph cut methods. By solving the minimization problem on the graph dual to a Delaunay‐based tetrahedral mesh, the advantages of explicit and implicit methods for surface reconstruction are well integrated. A triangular surface mesh homeomorphic to the original surface can be extracted directly from the tetrahedral mesh provided a sufficient sampling density exists. Difficult cases involving undersampling, non‐uniformity, noises and topological complexities can be handled effectively as well. Furthermore, for the first time, multi‐phase surface reconstruction is realized based on the graph cut methods. Various examples are included for demonstrating the efficiency and effectiveness of the proposed method. Copyright © 2010 John Wiley & Sons, Ltd.     

An immersed boundary element method for level‐set based topology optimization

In this paper, we propose a new BEM for level‐set based topology optimization. In the proposed BEM, the nodal coordinates of the boundary element are replaced with the nodal level‐set function and the nodal coordinates of the Eulerian mesh that maintains the level‐set function. Because this replacement causes the nodal coordinates of the boundary element to disappear, the boundary element mesh appears to be immersed in the Eulerian mesh. Therefore, we call the proposed BEM an immersed BEM. The relationship between the nodal coordinates of the boundary element and the nodal level‐set function of the Eulerian mesh is clearly represented, and therefore, the sensitivities with respect to the nodal level‐set function are strictly derived in the immersed BEM. Furthermore, the immersed BEM completely eliminates grayscale elements that are known to cause numerical difficulties in topology optimization. By using the immersed BEM, we construct a concrete topology optimization method for solving the minimum compliance problem. We provide some numerical examples and discuss the usefulness of the constructed optimization method on the basis of the obtained results. Copyright © 2012 John Wiley & Sons, Ltd.     

Adaptive generation of hexahedral element meshes for finite element analysis of metal plastic forming process

A method for the adaptive generation of hexahedral element mesh based on the geometric features of solid model is proposed. The first step is to construct the refinement information fields of source points and the corresponding ones of elements according to the surface curvature of the analyzed solid model. A thickness refinement criterion is then used to construct the thickness-based refinement information field of elements from digital topology. The second step is to generate a core mesh through removing all the undesired elements using even and odd parity rules. Then the core mesh is magnified in an inside–out manner method through a surface node projection process using the closest position approach. Finally, in order to match the mesh to the characteristic boundary of the solid model, a threading method is proposed and applied. The present method was applied in the mesh construction of different engineering problems. The resulting meshes are well-shaped and capture all the geometric features of the original solid models.     

A sub‒domain smoothed Galerkin method for solid mechanics problems

A sub?domain smoothed Galerkin method is proposed to integrate the advantages of mesh?free Galerkin method and FEM. Arbitrarily shaped sub?domains are predefined in problems domain with mesh?free nodes. In each sub?domain, based on mesh?free Galerkin weak formulation, the local discrete equation can be obtained by using the moving Kriging interpolation, which is similar to the discretization of the high?order finite elements. Strain smoothing technique is subsequently applied to the nodal integration of sub?domain by dividing the sub?domain into several smoothing cells. Moreover, condensation of DOF can also be introduced into the local discrete equations to improve the computational efficiency. The global governing equations of present method are obtained on the basis of the scheme of FEM by assembling all local discrete equations of the sub?domains. The mesh?free properties of Galerkin method are retained in each sub?domain. Several 2D elastic problems have been solved on the basis of this newly proposed method to validate its computational performance. These numerical examples proved that the newly proposed sub?domain smoothed Galerkin method is a robust technique to solve solid mechanics problems based on its characteristics of high computational efficiency, good accuracy, and convergence. Copyright © 2014 John Wiley & Sons, Ltd.     

Small polyhedron reconnection for mesh improvement and its implementation based on advancing front technique

Local transformation, or topological reconnection, is one of the effective procedures for mesh improvement method, especially for three‐dimensional tetrahedral mesh. The most frequently used local transformations for tetrahedral mesh are so‐called elementary flips, such as 2‐3 flip, 3‐2 flip, 2‐2 flip, and 4‐4 flip. Owing to the reason that these basic transformations simply make a selection from several possible configurations within a relatively small region, the improvement of mesh quality is confined. In order to further improve the quality of mesh, the authors recently suggested a new local transformation operation, small polyhedron reconnection (SPR) operation, which seeks for the optimal tetrahedralization of a polyhedron with a certain number of nodes and faces (typically composed of 20–40 tetrahedral elements). This paper is an implementation of the suggested method. The whole process to improve the mesh quality by SPR operation is presented; in addition, some strategies, similar to those used in advancing front technique, are introduced to speed up the operation. The numerical experiment shows that SPR operation is quite effective in mesh improvement and more suitable than elementary flips when combined with smoothing approach. The operation can be applied to practical problems, gaining high mesh quality with acceptable cost for computational time. Copyright © 2009 John Wiley & Sons, Ltd.     

Universal meshes for smooth surfaces with no boundary in three dimensions

We introduce a method to mesh the boundary Γ of a smooth, open domain in immersed in a mesh of tetrahedra. The mesh follows by mapping a specific collection of triangular faces in the mesh to Γ. Two types of surface meshes follow: (a) a mesh that exactly meshes Γ, and (b) meshes that approximate Γ to any order, by interpolating the map over the selected faces; that is, an isoparametric approximation to Γ. The map we use to deform the faces is the closest point projection to Γ. We formulate conditions for the closest point projection to define a homeomorphism between each face and its image. These are conditions on some of the tetrahedra intersected by the boundary, and they essentially state that each such tetrahedron should (a) have a small enough diameter, and (b) have two of its dihedral angles be acute. We provide explicit upper bounds on the mesh size, and these can be computed on the fly. We showcase the quality of the resulting meshes with several numerical examples. More importantly, all surfaces in these examples were meshed with a single background mesh. This is an important feature for problems in which the geometry evolves or changes, because it could be possible for the background mesh to never change as the geometry does. In this case, the background mesh would be a universal mesh 1 for all these geometries. We expect the method introduced here to be the basis for the construction of universal meshes for domains in three dimensions. Copyright © 2016 John Wiley & Sons, Ltd.     

On the interpolation of normal vectors for triangle meshes

This paper analyzes the problem arising from the need to assign information about the normal vectors to the surface at the nodes of a mesh of triangles. Meshes of triangles do not have normals uniquely defined at the nodes. A widely used technique to compute the normal direction at any given node is to compute the weighted average of the normals of each surrounding triangle. The present study proposes new weighting factors to compute the normal directions at the nodes of the mesh of triangles of a general surface. Previous weights found in the literature used the geometric dimensions of the triangles themselves to design the weighting factors. The new factors are proposed using the triangles’ circumscribed circles dimensions. The new weights provide superior results than the ones obtained by previous best practices for a wide range of surfaces. An advanced framework based on the approachability of smooth surfaces by quadrics is presented and used. This framework helps to understand the improved performance of the presented factors with respect to other factors found in the literature. A comprehensive numerical comparison analysis is performed, and the most precise of all factors is clearly identified. Copyright © 2013 John Wiley & Sons, Ltd.     

Automatic merging of tetrahedral meshes

A generic algorithm is proposed to merge arbitrary solid tetrahedral meshes automatically into one single valid finite element mesh. The intersection segments in the form of distinct nonoverlapping loops between the boundary surfaces of the given solid objects are determined by the robust neighbor tracing technique. Each intersected triangle on the boundary surface will be triangulated to incorporate the intersection segments onto the boundary surface of the objects. The tetrahedra on the boundary surface associated with the intersected triangular facets are each divided into as many tetrahedra as the number of subtriangles on the triangulated facet. There is a natural partition of the boundary surfaces of the solid objects by the intersection loops into a number of zones. Volumes of intersection can now be identified by collected bounding surfaces from the surface patches of the partition. Whereas mesh compatibility has already been established on the boundary of the solid objects, mesh compatibility has yet to be restored on the bounding surfaces of the regions of intersection. Tetrahedra intersected by the cut surfaces are removed, and new tetrahedra can be generated to fill the volumes bounded by the cut surfaces and the portion of cavity boundary connected to the cut surfaces to restore mesh compatibility at the cut surfaces. Upon restoring compatibility on the bounding surfaces of the regions of intersection, the objects are ready to be merged together as all regions of intersection can be detached freely from the objects. All operations, besides the determination of intersections structurally in the form of loops, are virtually topological, and no parameter and tolerance is needed in the entire merging process. Examples are presented to show the steps and the details of the mesh merging procedure. Copyright © 2012 John Wiley & Sons, Ltd.     

Approximate imposition of boundary conditions in immersed boundary methods

We analyze several possibilities to prescribe boundary conditions in the context of immersed boundary methods. As basic approximation technique we consider the finite element method with a mesh that does not match the boundary of the computational domain, and therefore Dirichlet boundary conditions need to be prescribed in an approximate way. As starting variational approach we consider Nitsche's methods, and we then move to two options that yield non‐symmetric problems but that turned out to be robust and efficient. The essential idea is to use the degrees of freedom of certain nodes of the finite element mesh to minimize the difference between the exact and the approximated boundary condition. Copyright © 2009 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.     

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.     

Formulation and implementation of a high-order 3-D domain integral method for the extraction of energy release rates

This article presents a three dimensional (3-D) formulation and implementation of a high-order domain integral method for the computation of energy release rate. The method is derived using surface and domain formulations of the J-integral and the weighted residual method. The J-integral along 3-D crack fronts is approximated by high-order Legendre polynomials. The proposed implementation is tailored for the Generalized/eXtended Finite Element Method and can handle discontinuities arbitrarily located within a finite element mesh. The domain integral calculations are based on the same integration elements used for the computation of the stiffness matrix. Discontinuities of the integrands across crack surfaces and across computational element boundaries are fully accounted for. The proposed method is able to deliver smooth approximations and to capture the boundary layer behavior of the J-integral using tetrahedral meshes. Numerical simulations of mode-I and mixed mode benchmark fracture mechanics examples verify expected convergence rates for the computed energy release rates. The results are also in good agreement with other numerical solutions available in the literature.