a general survey of hexahedral mesh generation

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A general survey of hexahedral mesh generation Sudarsan N.S Acharya 19. Juli 2004 Universit¨ at Erlangen–N¨ urnberg

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A General Survey of Hexahedral Mesh Generation

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  • A general survey of hexahedral mesh generation

    Sudarsan N.S Acharya

    19. Juli 2004

    Universitat ErlangenNurnberg

  • Introduction 1

    Talk outline

    Introduction

    Why hexahedral meshes ?

    Semiautomatic hexahedral generation methods

    Automatic hexahedral generation methods

    Specific advantages of hexahedral meshes

  • Introduction 2

    Introduction

    In the field of computer aided design, a model is approximately generatedas a coarse mesh of 3-dimensional curved polygons. Further analysis of thevarious technical properties of the model is performed by means of numericalmethods such as the finite element method. To this end, a finer mesh ofthe model is required. Such a mesh consists of elements such as triangles,quadrilaterals in 2-D and tetrahedra, hexahedra in 3-D. The elementsof a mesh should satisfy certain criteria such as angles, confirmity etc.Automatic generation of such meshes, especially hexahedral meshes havefound growing popularity in modern industry. In this talk, a brief survey ofhexahedral meshes, applications and generation techniques are discussed.

  • Hexahedral meshes 3

    Hexahedral meshesHexahedral or more famously called as the brick element has 8 nodes

    and 6 quadrilateral faces. Hexahedral meshes have found growng popularitybecause of the following reasons.

    A hexahedral meshing of a model results in fewer elements.

    Complexity of mesh optimisation for hexahedral meshes is an openquestion.

    Hexahedrans approximate complex domians to close accuracy.

    Better numerical behaviour is achieved in problems such as stress analysis.

    Meshes comprised of hexahedrons are easier to visualize than meshescomprised of tetrahedrons.

  • Generation methods 4

    Hexahedral meshes - Semiautomatic generation methods

    Hexahedral meshes can be generated by one of the following threemethods.

    Structured - rigid topological framework

    Multiblock - unstructured collection of structured blocks

    Unstructured - no underlying structure

  • Structured grid generation 5

    Hexahedral meshes - Structured grid generation

    Hexahedral meshes can be generated by one of the following threemethods.

    Structured grids are built with a repeating geometric and topologicalstructure.

    Structured grids could easily be generated with hexahedral elements.

    Structured grids are computationally efficient because of simple compu-tational connectivity of the grid Multigrid methods

    Structured grid applications are very common in CFD.

  • Structured grid generation 6

    Hexahedral mesh - Structured grid generation

  • Multiblock grid generation 7

    Hexahedral meshes - Multiblock grid generation

    Multiblock method is an early approach to creation of hexahedral meshesand is based on a mapped meshing.

    The domain to be discretised is divided into blocks which are thenmeshed separately using techniques such as parametric space mappingetc.

    Block division is unstructured Global meshing is unstructured.

    Multiblock grid generation is used in aerodynamic applications.

  • Multiblock grid generation 8

    Hexahedral mesh - Multiblock grid generation

  • Mapping transformation method 9

    Mapping transformation method

    Mapping transformation method is a commonly used method in hexahe-dral mesh generation. In case of a 2-D surface the following steps are tobe observed.

    Typical mapped meshes are created using a rectangular primitive.

    An initial cartesian mesh of the surface is made.

    Transformation equations are used to fill the surface with rows andcolumns of interior nodes, using the rectangular primitives.

    The two dimensional transformation functions interpolate each of theinterior nodes using the four corner and four boundary nodes from therectangular primitive.

  • Mapping transformation method 10

    The interior nodes are created according to the mesh density.

    The mesh connectivity is created by connecting the created nodes inrows and columns .

    This technique is not automatic, the object must be decomposed intoregions that map well in parametric space .

    Recent algorithms to automate the procedure of geometry splitting Medial-axes technique.

  • Mapping transformation method 11

    Mapping transformation method - Rectangular primitive

    The interior node is interpolated by the eight surrounding nodes.

  • Automatic generation 12

    Hexahedral meshes - Automatic generation methods

    Researchers continue to further automate the meshing process forhexahedral elements, but no algorithm has been developed to quicklygenerate a high quality mesh on all general geometries. Some automaticmesh generators are,

    Object based decomposition

    Skeleton based decomposition

    Advancing front techniques

    Connectivity based theories (Whisker weaving algorithm).

  • Object based decomposition 13

    Object based decomposition

    This algorithm works on the fact that solid models are created fromgeometric primitives.

    Set of primitives such as triangles, rectangles are preprogrammed intothe algorithm.

    An appropriate mesh is applied on a primitive on global level.

    Disadvantages Mesh overlapping, poor flexibility in modeling.

    Example ICEM AUTOHEXA.

  • Object base decomposition 14

    Object based decomposition - Example

  • Skeleton based decomposition 15

    Skeleton based decompoistion

    This algorithm is based on the medial axis representation of the model.

    Medial axis for a surface Locus of the center of the circle of maximaldiameter that can be inscribed, as it rolls around the regions interior.

    Using medial axis transform a complex region can be subdivided intosubregions.

    These subregions are meshed by the mapping transformation method.

    Advantage Works for arbitrarily shaped domains.

    Disadvantage Formulation of the medial axis and recognition ofsubregions, when medial axis/surfaces meet.

  • Advantages 16

    Specific advantage of hexahedral meshes

    The eight-node hexahedral element is linear (p = 1), with a linear strainvariation displacement mode accurate results in solid mechanicsapplications.

    The reaction of hexahedral elements to the application of body loadsmore precisely corresponds to loads under real world conditions.

    Visualising a hexahedral mesh is easier than of a tetrahedral one.

    For elastoplastic problems tetrahedrans do not converge, buthexahedrans do.

  • Advantages 17

    Hexahedral mesh - A shaft with a flange

  • Advantages 18

    Hexahedral mesh - A cylinder with protrusions