Step Function Representation of Solid Models and Application to Mesh Free Engineering Analysis

[+] Author and Article Information
Ashok V. Kumar, Jongho Lee

Department of Mechanical and Aerospace Engineering,  University of Florida, Gainesville, FL 32611

J. Mech. Des 128(1), 46-56 (Apr 13, 2005) (11 pages) doi:10.1115/1.2121743 History: Received June 23, 2004; Revised March 03, 2005; Accepted April 13, 2005

Numerical methods for solving boundary value problems that do not require generation of mesh to approximate the analysis domain have been referred to as mesh-free methods. While many of these are “mesh less” methods that do not have connectivity between nodes, a subset of these methods uses a structured mesh or grid for the analysis that does not conform to the geometry of the domain of analysis. Instead the geometry is represented using implicit equations. In this paper we present a method for constructing step functions of solids whose boundaries are represented using implicit equations. Step functions can be used to compute volume integrals over the solid that are needed for mesh free analysis. The step function of the solid has a unit value within the solid and zero outside. A level set of this step function can then be defined as the boundary of the solid. Boolean operators are defined in this paper that enable step functions of half-spaces and primitives to be combined to construct a single step function for more complex solids. Application of step functions to analysis using nonconforming mesh is illustrated.

Copyright © 2006 by American Society of Mechanical Engineers
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Figure 1

Quadratic nine-node element in two different coordinate systems

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Figure 2

Density distribution within a nine-node quadrilateral element

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Figure 3

Node numbering for 8-node and 18-node hexahedral element

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Figure 4

Solid primitive created by extruding ellipse

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Figure 5

Mapping from cylindrical to cartesian coordinates

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Figure 6

Solid primitive created by revolving a circle

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Figure 7

Sweep trajectory and coordinate systems used for a sweep element

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Figure 8

Primitive created by sweeping a circle along a parametric curve

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Figure 9

Constructive solid geometry tree

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Figure 10

CSG tree of solids created using step functions

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Figure 11

Comparison between ordinary and regularized Boolean operations

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Figure 12

Approximation errors in Boolean result

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Figure 13

Analysis using nonconforming mesh

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Figure 14

Plate with mesh and boundary condition

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Figure 15

Normal stress computed using nonconforming grid



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