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We present a new approach for the generation of hexahedral finite el- ement meshes for solid bodies in computer-aided design. The key idea is to use a purely combinatorial method, namely a shelling process, to decompose a topological ball with a prescribed surface mesh into combi- natorial cubes, so-called hexahedra. The shelling corresponds to a s...
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Absolute nodal coordinate formulation (ANCF) is a non-incremental nonlinear finite element procedure that has been successfully applied to the large deformation analysis of multibody systems for more than two decades. Although a comprehensive review on ANCF was conducted by Gerstmayr et al. in 2013, significant theoretical developments have been ma...
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... It is unclear how the process can be steered to naturally establish this required structure in general; therefore, degenerate elements (so-called knives, Fig. 30) and inverted elements are common in the result in these cases. Various strategies (with more or less severe negative side effects on quality) have been proposed to modify the quad-mesh to get rid of such self-intersections in advance [Folwell and Mitchell 1999;Kawamura et al. 2008;Müller-Hannemann 2001;Müller-Hannemann 2002]. ...
... An idea of inserting dual sheets in a divide-and-conquer manner was outlined by [Calvo and Idelsohn 2000]. A concrete algorithm for incremental hex-mesh construction based on sequential dual sheet generation is described by [Müller-Hannemann 2001]. The boundary geometry along an entire candidate sheet is assessed in the decision-making process. ...
... It is unclear how the process can be steered to naturally establish this required structure in general; therefore, degenerate elements (so-called knives, Fig. 32) and inverted elements are common in the result in these cases. Various strategies (with more or less severe negative side effects on quality) have been proposed to modify the quad-mesh to get rid of such self-intersections in advance [Folwell and Mitchell 1999;Kawamura et al. 2008;Müller-Hannemann 2001;Müller-Hannemann 2002]. ...
... An idea of inserting dual sheets in a divide-and-conquer manner was outlined by [Calvo and Idelsohn 2000]. A concrete algorithm for incremental hex-mesh construction based on sequential dual sheet generation is described by [Müller-Hannemann 2001]. The boundary geometry along an entire candidate sheet is assessed in the decision-making process. ...
In this article, we provide a detailed survey of techniques for hexahedral mesh generation. We cover the whole spectrum of alternative approaches to mesh generation, as well as post processing algorithms for connectivity editing and mesh optimization. For each technique, we highlight capabilities and limitations, also pointing out the associated unsolved challenges. Recent relaxed approaches, aiming to generate not pure-hex but hex-dominant meshes, are also discussed. The required background, pertaining to geometrical as well as combinatorial aspects, is introduced along the way.
... It is unclear how the process can be steered to naturally establish this required structure in general; therefore, degenerate elements (so-called knives, Fig. 32) and inverted elements are common in the result in these cases. Various strategies (with more or less severe negative side effects on quality) have been proposed to modify the quad-mesh to get rid of such self-intersections in advance [Folwell and Mitchell 1999;Kawamura et al. 2008;Müller-Hannemann 2001;Müller-Hannemann 2002]. ...
... An idea of inserting dual sheets in a divide-and-conquer manner was outlined by [Calvo and Idelsohn 2000]. A concrete algorithm for incremental hex-mesh construction based on sequential dual sheet generation is described by [Müller-Hannemann 2001]. The boundary geometry along an entire candidate sheet is assessed in the decision-making process. ...
In this article, we provide a detailed survey of techniques for hexahedral mesh generation. We cover the whole spectrum of alternative approaches to mesh generation, as well as post processing algorithms for connectivity editing and mesh optimization. For each technique, we highlight capabilities and limitations, also pointing out the associated unsolved challenges. Recent relaxed approaches, aiming to generate not pure-hex but hex-dominant meshes, are also discussed. The required background, pertaining to geometrical as well as combinatorial aspects, is introduced along the way.
... As explained in [153]: "It remains an open problem to give a sufficient condition on singularity graphs compatible to non-degenerate hexahedral meshing." On the other hand, as extensive surveys on hex meshing algorithms [40,154] point out, it is possible in some cases to §1 Perspectives 155 use surface quad mesh for generating a hex mesh, nevertheless, the developed algorithms, e.g., [155,156], typically can deal only with specific topologies, and as already concluded in [157]: "Generating hexahedral meshes of valid topology and practical geometric properties from quadrilateral surface meshes of arbitrary genus and possibly self-intersecting loops of quadrilaterals is unresolved in general." Unfortunately, a similar conclusion stands for generating hex layout from a quad layout, except for some particular cases [13]. ...
... Simplicial complexes are combinatorial structures frequently used in geometrical applications because of their flexibility for modeling objects from different spatial dimensions. The presence of one of their combinatorial properties, known as shellability, has proved to be useful in practical situations (see, for example the works of Herlihy (2010) and Müller-Hannemann (2001)). The concept also appears in graph theory where, through the Stanley-Reisner correspondence, a simplicial complex may be associated to a graph (Van Tuyl & Villarreal, 2008). ...
La escalonabilidad* de grafos es un problema en NP del que se desconoce su inclusión en las clases de complejidad P o NP-completa. Con el fin de comprender su comportamiento computacional en el caso particular de los grafos bipartitos, podría ser de utilidad disponer de un método eficiente para generar y analizar instancias escalonables. La literatura reporta un experimento secuencial, y de costo exponencial, diseñado para determinar la escalonabilidad de un conjunto de instancias. En el presente trabajo, y con el fin de mejorar el desempeño experimento mencionado, proponemos tres alternativas utilizando Apache Spark: una multinúcleo, otra multinodo y otra completamente paralela. Además, comparamos el tiempo de ejecución de cada una de ellas respecto a la versión original en grafos bipartitos aleatorios con 10,12,15,20 y 50 vértices, y obtuvimos aceleraciones (speedups) entre 1.37 y 1.67 para la versión multinúcleo, entre 2.34 y 3.56 para la versión multinodo, y entre 2.37 y 3.12 para la versión completamente paralela. Los resultados sugieren que la paralelización del experimento podría mitigar los enormes tiempos de ejecución del enfoque original.
... The material orientation employed for the composite rods, the over-brad sleeve (i.e. shell), and the assembly [38][39][40]. ...
... On a closed surface, all STC chords are loops. Müller-Hannemann [13,14], Folwell and Mitchel [15] proposed to use curve contraction method for hexahedral mesh generation from quadrilateral meshes without self-intersecting STC loops. Erickson [12] gave a constructive proof which leads to an algorithm to produce a hex-mesh from an extendable quad-mesh. ...
... However, generating hexahedral meshes of valid topology and practical geometric properties from quadrilateral surface meshes of arbitrary genus and possibly self-intersecting loops of quadrilaterals is unresolved in general. In [23] Müller-Hannemann proposes a combinatorial method for the generation of hexahedral complexes from quad meshes that are of genus zero and do not contain self-intersecting quad loops. At a first glance, this appears to be a serious restriction of the class of hex-meshable surface meshes. ...
... In a subsequent step, the geometric embedding of the mesh is computed using the Mesquite library [9] which offers methods to optimize the geometry of hex meshes. While the method works appropriately on almost convex objects, the author suggests to decompose non-convex objects into convex ones and remesh the interfaces accordingly [23,22]. However, cutting and remeshing the surface meshes is generally not desirable since it either requires the user to know the interior topology of the decomposition and/or might compromise global topological properties of the initial surface quadrangulation. ...
... In this paper, we present an extension to the method described in [23] that allows us to automatically generate surface conforming hexahedral meshes for non-convex objects of genus zero. For this, we first analyze how the order in which the dual cycles are eliminated from the dual graph influences the resulting mesh's topology and propose some practical guidelines for many common mesh configurations in Section 4. Furthermore, for the handling of concave objects, we introduce the notion of concave dual cycles and the significance of their elimination from the dual graph in Section 5. From this analysis we derive a set of rules that, applied to the heuristic that determines the elimination order of the dual cycles, improves the topological quality of the resulting mesh. ...
A purely topological approach for the generation of hexahedral meshes from quadrilateral surface meshes of genus zero has been proposed by M. Müller-Hannemann: in a first stage, the input surface mesh is reduced to a single hexahedron by successively eliminating loops from the dual graph of the quad mesh; in the second stage, the hexahedral mesh is constructed by extruding a layer of hexahedra for each dual loop from the first stage in reverse elimination order. In this paper, we introduce several techniques to extend the scope of target shapes of the approach and significantly improve the quality of the generated hexahedral meshes. While the original method can only handle “almost convex” objects and requires mesh surgery and remeshing in case of concave geometry, we propose a method to overcome this issue by introducing the notion of concave dual loops. Furthermore, we analyze and improve the heuristic to determine the elimination order for the dual loops such that the inordinate introduction of interior singular edges, i.e. edges of degree other than four in the hexahedral mesh, can be avoided in many cases.
... In recent decades, different approaches have been followed to generate a full hexahedral mesh for any geometric 3D object. Starting from a premeshed boundary surface some authors have proposed pure geometric approaches like plastering[2]or H-morph[3]algorithms, while other authors have followed a pure topological approach proposing different algorithms[4][5][6][7][8][9]with limited success. Some unfilled cavities can remain or inverted cells, and negative Jacobian cells can be generated inside the mesh. ...
Depending upon the numerical approximation method that may be implemented, hexahedral meshes are frequently preferred to tetrahedral
meshes. Because of the layered structure of hexahedral meshes, the automatic generation of hexahedral meshes for arbitrary
geometries is still an open problem. This layered structure usually requires topological modifications to propagate globally,
thus preventing the general development of meshing algorithms such as Delaunay’s algorithm for tetrahedral meshes or the advancing-front
algorithm based on local decisions. To automatically produce an acceptable hexahedral mesh, we claim that both global geometric
and global topological information must be taken into account in the mesh generation process. In this work, we propose a theoretical
classification of the layers or sheets participating in the geometry capture procedure. These sheets are called fundamental,
or fun-sheets for short, and make the connection between the global layered structure of hexahedral meshes and the geometric surfaces that
are captured during the meshing process. Moreover, we propose a first generation algorithm based on fun-sheets to deal with
3D geometries having 3- and 4-valent vertices.
KeywordsHexahedral mesh–Block-structured mesh–Dual mesh–Fundamental mesh–Mesh generation
... Hexahedral meshes have received a lot of attention because of several desirable properties: they usually enable building meshes with fewer elements and exhibit better numerical behavior in various problems [Benzley et al. 1995]. Although the automatic generation of hexahedral meshes tends to be more difficult than that of tetrahedral meshes, there are now a number of algorithms [Blacker 1996;Muller-Hannemann 2001;Staten et al. 2005] that can generate high-quality hexahedral meshes (see Figure 4.1) with little or no user intervention. The main field of application of unstructured hexahedral meshes is scientific simulations -the graphics community often resorts to ray-traced structured voxel grids or "onion peeled" transparent surfaces when dealing with volume data. ...
... This produces meshes with very poor coherence. On the other hand, hexahedral meshes tend to have very coherent layouts, because they are mainly generated using advancing-front techniques [Blacker 1996;Muller-Hannemann 2001;Staten et al. 2005]. Figure 4.2 compares the layout coherence of typical tetrahedral and hexahedral meshes. ...
A decade ago, 3D content was restricted to a few applications – mainly games, 3D graphics andscientific simulations. Nowadays, thanks to the development cheap and efficient specialized renderingdevices, 3D objects are ubiquitous. Virtually all devices with a display – from a large visualizationclusters to smart phones – now integrate 3D rendering capabilities. Therefore, 3D applications arenow far more diverse than a few years ago, and include for example real-time virtual and augmentedreality, as well as 3D virtual worlds. In this context, there is an ever increasing need for efficient toolsto transmit and visualize 3D content.In addition, the size of 3D meshes always increases with accuracy of representation. On one hand,recent 3D scanners are able to digitalize real-world objects with a precision of a few micrometers, andgenerate meshes with several hundred million elements. On the other hand, numerical simulationsalways require finer meshes for better accuracy, and massively parallel simulation methods now generatemeshes with billions of elements. In this context, 3D data compression – in particular 3D meshcompression – services are of strategic importance.The previous decade has seen the development of many efficient methods for encoding polygonalmeshes. However, these techniques are no longer adapted to the current context, because they supposethat encoding and decoding are symmetric processes that take place on the same kind of hardware.In contrast, remote 3D content will typically be created, compressed and served by high-performancemachines, while exploitation (e.g. visualization) will be carried out remotely on smaller – possiblyhand held – devices that cannot handle large meshes as a whole. This makes mesh compression anintrinsically asymmetric process.Our objective in this dissertation is to address the compression of these large meshes. In particularwe study random-accessible compression schemes, that consider mesh compression as an asymmetricproblem where the compressor is an off-line process and has access to a large amount of resources,while decompression is a time-critical process with limited resources. We design such a compressionscheme and apply it to interactive visualization.In addition, we propose a streaming compression algorithm that targets the very large hexahedralmeshes that are common in the context of scientific numerical simulation. Using this scheme, we areable to compress meshes of 50 million hexahedra in less than two minutes using a few megabytes ofmemory.Independently from these two specific algorithms, we develop a generic theoretical framework toaddress mesh geometry compression. This framework can be used to derive geometry compressionschemes for any mesh compression algorithm based on a predictive paradigm – which is the case of thelarge majority of compression schemes. Using this framework, we derive new geometry compressionschemes that are compatible with existing mesh compression algorithms but improve compressionratios – by approximately 9% on average. We also prove the optimality of some other schemes underusual smoothness assumptions.
