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

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

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 "

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

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

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).

Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin September 2006

Examples of Discriminating Curve Families

Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin September 2006

Constructing Scaffolds

Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin September 2006

Input:

G1 / D4 curves:

Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin September 2006

Spline Surfaces of Revolution

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

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

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

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

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

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

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.

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

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

Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin September 2006

Hermite Interpolant on Prism Edges

Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin September 2006

Hermite Interpolation on Prism Faces

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)

Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin September 2006

Side Vertex Interpolation

Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin September 2006

C^1 Shell Elements

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

Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin September 2006

Shell Finite Element Models

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