Dagstuhl Seminar Geometric Modeling Benjamin Karer Hans Hagen Motivation Inextensible Elastic Surface Strips Torsion Bending image wikimedia commons original source ehrtde Ribbon Cables ID: 630504
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Slide1
Drall-based Ruled Surface Modeling
Dagstuhl Seminar Geometric Modeling
Benjamin
Karer,
Hans
HagenSlide2
Motivation: Inextensible Elastic Surface Strips
Torsion Bending
image: wikimedia commons
original source: ehrt.de
Ribbon
Cablesimage: wikimedia commons
DNAimage: wikimedia commons
Physically correct simulationMinimum Bending EnergyMathematical Model:GeneralSimpleExact Slide3
Motivation: Continuum Mechanics
Cosserat
Rods
Ribbons
Plates/Shells
not
exact
if
not
near-isotropic
complicated constraints
but:
flexible geometry
Frame fixed to material
Needed
:
Well-
Founded
Energy
Functionals
generality
Efficient
algorithms
simulationSlide4
Motivation: Geometry
Material Frame
Derivative Equations
curvatures
angular velocity of rotation around centerline
Slide5
Motivation: Geometry
Line Geometry
Derivative Equations
Real:
Dual:
Invariants:
point and direction
Slide6
Motivation: Geometry
Kruppa‘s Frame
Derivative Equations
geodesic curvature and geodesic torsion
striction angle
Slide7
Motivation: Geometry
Material Frames
Kruppa‘s
Frame
(natural
parameterization)
complicated constraints
Typically requires planar reference geometry
Implicit centerline
Implicit
centerline
No
developability
Needed
:
Simple
constraints
modeling
Explicit
centerline
definition
tools
Nondevelopable
surfaces
generalityLine Geometry(Blaschke‘s formulation)Slide8
Solution: Drall
We have:
With
Drall d
.
Define
:
Angular velocity
:
Along centerline:
Slide9
Drall
Slide10
Idea:
Bind
to
the
centerlineBind to the
generators
In
the
centerline
:
Use
the drall for a differential geometric description:Coupled FramesConstruction:Centerline frame
Generator frame Coupled via and striction
Slide11
Transformation along
the
centerline
Slide12
Transformation
along
the generators
Slide13
Results
Striction
relates
the
invariants:
We
have and
minimal
complete
system
of
invariants (in the centerline)developability (well-known result
) constant along generators ( not) surface defined in one parameter Slide14
Results
Ansatz
yields
bending energy
for arbitrary shapes and arbitrary width:
Slide15
ExamplesSlide16
ExamplesSlide17
ExamplesSlide18
Examples
Slide19
Error Computation
Drall
Defect
exact
Striction
Defectexact up to
known uncertainty
Slide20
Future Work
Holes
and
varying widthGeneralization to other surfaces and volumesGPU implementationModeling tools for design
Steer invariants by material propertiesSlide21
Achievements
Before
Either
not general or not simple
Complicated constraintsContinuous models, numeric integrationBending energy for developables
Now
Arbitrary ruled surfaces
, , etc.Closed forms for integralsBending energy considers transversal bending Slide22
Summary
– minimal complete system of invariants for ruled surfaces
Bending energy for arbitrary ruled surfaces of arbitrary width
Highly parallelizable computation
GPU
Exact error between surface and analytical solution
(useful for tessellations)