lecture 2b: geometric modeling and visualization bem/fem...
TRANSCRIPT
![Page 1: Lecture 2b: Geometric Modeling and Visualization BEM/FEM …bajaj/cs384R07/lectures/2007_09_04... · 2007. 9. 18. · Lofting III : Non-Linear Boundary Elements Input contours G2](https://reader033.vdocuments.net/reader033/viewer/2022060721/60811d0f765cf350b64b4a13/html5/thumbnails/1.jpg)
Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin September 2006
Lecture 2b: Geometric Modeling and Visualization
BEM/FEM Domain Models
Chandrajit Bajaj
http://www.cs.utexas.edu/~bajaj
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Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin September 2006
Linear Interpolation on a line segment
p0 p p1
The Barycentric coordinates α = (α0 α1) for any point p on linesegment <p0 p1>, are given by
)),(
),(,),(
),((
10
0
10
1
ppdist
ppdist
ppdist
ppdist=!
which yields p = α0 p0 + α1 p1
and fp = α0 f0 + α1 f1
ff1f0 fp
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Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin September 2006
Linear interpolation over a triangle
p0
p1 p p2
For a triangle p0,p1,p2, the Barycentric coordinatesα = (α0 α1 α2) for point p,
)),,(
),,(,),,(
),,(,),,(
),,((
210
10
210
20
210
21
ppparea
ppparea
ppparea
ppparea
ppparea
ppparea=!
!=2
0
iipp " !=
2
0
ii fpfp "
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Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin September 2006
Non-Linear Algebraic Curve and Surface FiniteElements ?
a200
a020 a002
a101a110
a011
The conic curve interpolant is the zero of the bivariate quadraticpolynomial interpolant over the triangle
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Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin September 2006
A-spline segment over BB basis
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Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin September 2006
Regular A-spline Segments
If B0(s), B1(s), … has one signchange, then the curve is
(a) D1 - regular curve.
(b) D2 - regular curve.
(c) D3 - regular curve.
(d) D4 - regular curve.
For a given discriminating family D(R, R1,R2), let f(x, y) be a bivariate polynomial .If the curve f(x, y) = 0 intersects witheach curve in D(R, R1, R2) only once inthe interior of R, we say the curve f = 0is regular(or A-spline segment) withrespect to D(R, R1, R2).
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Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin September 2006
Examples of Discriminating Curve Families
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Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin September 2006
Constructing Scaffolds
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Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin September 2006
Input:
G1 / D4 curves:
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Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin September 2006
Spline Surfaces of Revolution
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Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin September 2006
Lofting III : Non-Linear Boundary Elements
Input contours G2 / D4 curves
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Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin September 2006
Linear interpolant over a tetrahedron
Linear Interpolation within a• Tetrahedron (p0,p1,p2,p3)
α = αi are the barycentric coordinates of p
p3
pp0 p2
p1
!=3
0
iipp "
fp0
fp1
fp3
fp2!=3
0
ii fpfp "
fp
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Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin September 2006
Other 3D Finite Elements (contd)
• Unit Prism (p1,p2,p3,p4,p5,p6)
p1p2 p3
p p4
p5 p6
))(1()(6
4
3
3
1
!! ""+=iiiiptptp ##
Note: nonlinear
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Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin September 2006
Other 3D Finite elements
• Unit Pyramid (p0,p1,p2,p3,p4)
p0
p1 p2 p p3
p4
)))1()(1())1(()(1( 43210 pssptpssptuupp !+!+!+!+=
Note:nonlinear
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Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin September 2006
Other 3D Finite Elements
• Unit Cube (p1,p2,p3,p4,p5,p6,p7,p8)– Tensor in all 3 dimensions
p1 p2p3 p4
pp5 p6
p7 p8
)))1()(1())1(()(1(
)))1()(1())1(((
8765
4321
pssptpssptu
pssptpssptup
!+!+!+!
+!+!+!+= Trilinearinterpolant
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Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin September 2006
Topology of Zero-Sets of a Tri-linear Function
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Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin September 2006
Non-linear finite elements-3d
s Non linearTransformationof mesh
UVS space XYZ spacev u
x
y
z
))()1(()1)()1(( svuvusvuvu
z
y
x
543210 pppppp ++!!+!++!!=
"""
#
$
%%%
&
'
•Irregular prism–Irregular prisms may be used to represent data.
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Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin September 2006
Hermite interpolation
f0’ f1’
f0 f1
C1 Interpolant
f(t) = f 0H30(t) + f 00H
31(t) + f 1H
32(t) + f 01H
33(t)
0 1
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Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin September 2006
• Define functions and gradients on the edgesof a prism
• Define functions and gradients on the faces ofa prism
• Define functions on a volume
• Blending
Incremental Basis Construction
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Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin September 2006
Hermite Interpolant on Prism Edges
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Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin September 2006
Hermite Interpolation on Prism Faces
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Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin September 2006
•The function F is C1 over ∑ andinterpolates C1 (Hermite) data
•The interpolant has quadratic precision
Shell Elements (contd)
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Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin September 2006
Side Vertex Interpolation
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Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin September 2006
C^1 Shell Elements
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Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin September 2006
C^1 Quad Shell Surfaces can be built in a similar way, bydefining functions over a cube
C^1 Shell Elements within a Cube
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Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin September 2006
Shell Finite Element Models
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Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin September 2006
Also see my algebraic curve/surface spline lectures 7 and 8 from
http://www.cs.utexas.edu/~bajaj/graphics07/cs354/syllabus.html