Novel quadratic Bézier triangular and tetrahedral elements using existing mesh generators: Applications to linear nearly incompressible elastostatics and implicit and explicit elastodynamics
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Summary In this paper, we present novel techniques of using quadratic Bézier triangular and tetrahedral elements for elastostatic and implicit/explicit elastodynamic simulations involving nearly incompressible linear elastic materials. A simple linear mapping is proposed for developing finite element meshes with quadratic Bézier triangular/tetrahedral elements from the corresponding quadratic Lagrange elements that can be easily generated using the existing mesh generators. Numerical issues arising in the case of nearly incompressible materials are addressed using the consistent B ‐bar formulation, thus reducing the finite element formulation to one consisting only of displacements. The higher‐order spatial discretization and the nonnegative nature of Bernstein polynomials are shown to yield significant computational benefits. The optimal spatial convergence of the B ‐bar formulation for the quadratic triangular and tetrahedral elements is demonstrated by computing error norms in displacement and stresses. The applicability and computational efficiency of the proposed elements for elastodynamic simulations are demonstrated by studying several numerical examples involving real‐world geometries with complex features. Numerical results obtained with the standard linear triangular and tetrahedral elements are also presented for comparison.Keywords:
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Abstract 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.
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Abstract A method has been developed to generate regular hexahedral meshes automatically from arbitrary solid models by volume decomposition. This method first decomposes a solid model having a complex shape into volumes having simple shapes. Then, shape-specific meshing methods like mapping are applied to generate regular hexahedral meshes from these volumes. Finally, all regular hexahedral meshes of these volumes are combined into a regular hexahedral mesh of the original solid model. Thus the method generates regular hexahedral meshes automatically in a way similar to the way a human does interactively. This is in contrast to the previous methods of automatic hexahedral mesh generation, which try to generate hexahedral meshes from solid models directly.
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Generating high quality, subject-specific, 3D finite element anatomic models with minimal user-intervention remains a challenge. Numerous automated tetrahedral mesh generators are available, but hexahedral meshes are preferred due to their higher accuracy and faster computational time over tetrahedral meshes. Historically, the generation of hexahedral meshes for analysis is a tedious and time-consuming task. Therefore, the utilization of hexahedral meshes is often limited to baseline models that are powerful, but not patient-specific. Once a high quality mesh has been created, it would be ideal if it can be used to create meshes of similar surfaces from different subjects, without disrupting mesh quality.
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For structure analysis with the finite element method (FEM), the hexahedral element is preferable to the tetrahedral one from the viewpoint of accuracy. Previously, we had introduced a label-driven subdivision method for a two-dimensional mesh and showed that the meshes generated by our method were useful for structural analyses. In this study, we extend our two-dimensional algorithm to three-dimensions and verify that the meshes generated by the proposed mesh-subdivision algorithm are useful for structural analyses.
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We present a simple and robust method for generating degenerate hexahedral meshes by using VCAD system for structural analysis. Our method consists of only two steps: First, dual meshes are computed from VCAD models. Second, the positions of new vertices are computed by least-square fitting. Concave elements can be eliminated completely by simple subdivision of hexahedrons. We also demonstrate the results of structural analysis by our meshes and discuss strengths and the limitation of our method.
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Advanced calculation due to progress in computing power and the development of mesh-generating capabilities for analyzing structures have led to the need for a pre-processor that can generate an optimal mesh quickly and accurately. Tetrahedron and hexahedron meshes are generally used in structural analysis ; the former is very adaptable to various structural configurations, and the latter calculates accurate results. Using these features, we have developed an automatic mesh pre-processor that can generate a hexahedron mesh for the part requiring precise results and tetrahedron mesh for the remaining part. To conserve displacement compatibility, a pyramidal mesh is used for the interface between these two meshes. Simulation showed that this method works well for finite-element-method modeling of holed-plate extensions and gearings.
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Unstructured hexahedral volume meshes are of particular interest for visualization and simulation applications. They allow regular tiling of the three-dimensional space and show good numerical behaviour in finite element computations. Beside such appealing properties, volume meshes take huge amount of space when stored in a raw format. We present a technique for encoding connectivity and geometry of unstructured hexahedral volume meshes. For connectivity compression, we extend the idea of coding with degrees as pioneered by Touma and Gotsman (1998) to volume meshes. Hexahedral connectivity is coded as a sequence of edge degrees. This naturally exploits the regularity of typical hexahedral meshes. We achieve compression rates of around 1.5 bits per hexahedron (bph) that go down to 0.18 bph for regular meshes. On our test meshes the average connectivity compression ratio is 1:162.7. For geometry compression, we perform simple parallelogram prediction on uniformly quantized vertices within the side of a hexahedron. Tests show an average geometry compression ratio of 1:3.7 at a quantization level of 16 bits.
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Hexahedral meshes are largely used by the Finite Element Method in a high variety of simulation problems. One of the most common problems of these type of meshes is to achieve an adequate approximation of curved domains; a feature typically found in the shape of organs. This work introduces a set of mixed-elements patterns, which are employed at the surface of target domain, and allow to conserve hexahedra elsewhere. These patterns are meant to be combined with any meshing technique producing a regular or non-regular hexahedral mesh.
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