Technische Universität MünchenComputation in Engineering
To mesh or not to mesh:that is the question
Stefan Kollmannsberger
Computation in Engineering
Technische Universität München
Technische Universität MünchenComputation in Engineering
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Ordinarius: Prof. Dr.rer.nat. Ernst Rank
Numerical Mechanics:Dr.-Ing. Stefan Kollmannsberger, Tino Bog, Nils Zander, Quanji Cai, Martin Schlaffer, Christian Sorger, Hagen Wille,
Efficient Algorithms:Dr.rer.nat. Ralf Mundani, Jérôme Frisch, Jovana Knesevic, Dominik Schillinger, Vasco Varduhn, Matthias Flurl
Bavarian Graduate School of Science and EngineeringDr.-Ing. Martin Ruess
blue: working on FCM red: Mesh Generation
Computation in Engineering
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Statement:
as stated orally during the talk, all presented work is the result of a common effort of the research group and may not only be attributed to the presentor.
Statement included for the online version of the presentation onTuesday, 20th of October 2011
Computation in Engineering
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Outline
Motivation
- the „traditional“ approach incl. mesh generation - modeling with Isogeometric Analysis- modeling with Finite Cells
what am I doing here?
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Motivation
Michael Hardwick and Robert Clay, Sandia National Laboratories, as published in:
J. Austin Cottrell, Thomas J.R. Hughes, Y. Bazilevs. Isogeometric Analysis. Wiley, 2008, ISBN 978-0-470-74873-2
mesh generation
takes more time than
computation
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In general, the problem is…
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and the major assumption is:
we start with a clean! geometry
be „analysis aware“ or fix it-> healing is an important step
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Motivation
Michael Hardwick and Robert Clay, Sandia National Laboratories, as published in:
J. Austin Cottrell, Thomas J.R. Hughes, Y. Bazilevs. Isogeometric Analysis. Wiley, 2008, ISBN 978-0-470-74873-2
mesh generation
takes more time than
computation
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Historic “torch relay” at our chair
Rank, Schweingruber, Halfmann, Scholz,
Kollmannsberger, Sorger….
funded by SOFiSTiK
a) better mesh generation (i.e. for high order solid shells)
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a) better mesh generation (i.e. for high order solid shells)
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a) better mesh generation (i.e. for high order solid shells)
how do we generate a mesh?
TopologyGeometry
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topological components of mesh generation
Fixed points Fixed lines
Holes
Reference edges
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Topology: Pre-processing
1. Raw data 2. Initial geometry
3. Closed polygons 4. Edge division
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Topology: recursive subdivision
recursive subdivision:
Divide
Chop
original idea: R. E. Bank, et. al 1983
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Topology: recursive subdivision
triangle conversion:
Variant 1: 4 triangles 4 quadrilaterals Variant 2: 2 triangles 4 quadrilaterals
Variant 3a: 1 triangles 3 quadrilaterals Variant 3b: 1 triangles 2 quadrilaterals
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Topology: recursive subdivision
relaxation:
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Geometry: mapping
is a mapping describing the true geometry i.e. B-splines, NURBS…
do this for n regions…
+ adaptivity+ respect the matrix of the mapping in the meshing
without metric with metric
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a) better mesh generation… for the industry…
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a) better mesh generation (i.e. for high order solid shells)
ship body
is a mapping describing the true geometry i.e. B-splines, NURBS…
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a) better mesh generation: examples
ship bodymechanical spring for a chiseling tool
wing violin
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a) better mesh generation: examples
high order mesh of a wind turbine
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a) better mesh generation: examples
hexahedral mesh of an aorta for “classical” high order FEM
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b) ?
but what if we don’t wantto take this (painful) step?
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b) Isogeometric Analysis
originally published in/by:J. Austin Cottrell, Thomas J.R. Hughes, Y. Bazilevs. Isogeometric Analysis. Wiley, 2008
Take the geometry from CAD and directly compute on it
but one needs a conforming, hexahedral decomposition
the difference to before?
->in order to have to generate a meshone draws the (coarsest) mesh and refines it
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c) FCM: avoid mesh generation, generate grids instead
foambone
porous media
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Finite Cell Method
(Parvizian, Düster, Rank 2007, Düster, Parvizian, Yang, Rank 2008)
A fictitious domain method with high order polynomial basis functions
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Finite Cell Method
(Parvizian, Düster, Rank 2007, Düster, Parvizian, Yang, Rank 2008)
+ =
A fictitious domain method with high order polynomial basis functions
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Finite Cell Method
(Parvizian, Düster, Rank 2007, Düster, Parvizian, Yang, Rank 2008)
+ =
α=0
α=1
α=1
α=0
A fictitious domain method with high order polynomial basis functions
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Finite Cell Method: determination of α
discontinous „indicator function“
α=1
α=0
1. Inside or outside? 2. adaptive integration
even =outside-> α =0
odd=inside-> α=1
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FCM: Why does it work?
D. Schillinger, A. Düster, E. Rank: hpd adaptive Finite Cell Method for Geometrically Nonlinear Problems of solid mechanics, submitted to: IJNME.
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FCM: Why does it work?
- Smooth extension of solution fields
- Best approximation property to strain energy +penalization of fictitious domain
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Finite Cell Method: how does it work…
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Finite Cell Method: how does it work…
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Finite Cell Method: how does it work…
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Finite Cell Method: how does it work…
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Finite Cell Method: thick solid shells
Geometry:
Boundary conditions:
vertical shell weight:
Material:
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Finite Cell Method: thick solid shells
txp_fig
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Finite Cells in Biomechanics: Computational Steering
Joint project with:
R. Westermann (TUM-IN)
R. Burgkart (TUM Klinikum rechts der Isar)
A.Düster (TUHH)
J. Parvizian (Univ. Isfahan)
Z. Yosibash (Univ. Beer Sheva)
Scientific staff:
Ch. Dick, S. Kollmannsberger, M. Ruess, Z. Yang
Funding:
IGSSE, TUM-IAS, Humboldt-Foundation, SIEMENS
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Finite Cells in Biomechanics
CT data (Hounsfield Unit)CT scans
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Finite Cells in Biomechanics
Resolution of CT scan: Δx = Δy = 0.78125 mm, Δz=0.75 mm
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Finite Cells in Biomechanics
-> A “cell” is a “finite element”
-> A finite element consists of n voxelsprecompute
one cell one voxel
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Finite Cells in Biomechanics
Fz =1500N, uz = 0.45mm
FCMCT data
Yosibash et al. 2007
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Finite Cells in Biomechanics
Fz =1500N, uz = 0.45mm
Convergence of Fz
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Finite Cells in Biomechanics
Displacements von Mises stress Fz =1500N, uz = 0.45mm
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Finite Cells in Biomechanics
von Mises stress
A A
A-A
Fz =1500N, uz = 0.45mm
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Finite Cells in Biomechanics
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back to Isogeometric Analysis
originally published in/by:J. Austin Cottrell, Thomas J.R. Hughes, Y. Bazilevs. Isogeometric Analysis. Wiley, 2008
Take the geometry from CAD and directly compute on it
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reconsider: topology in Isogeometric Analysis
what kind of topology are we provided with?
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Remarks on Isogeometric Analysis
A CAD modeler:
the CAD modeler provides trimmed surfaces/volumes
-> We should compute with them!
one possible approach is IGA + FCM
Isotopological Analysis?
A CAD modeler does not think in elements:
Original idea of IGA: compute with what the CAD modeler provides
draw cylinderdraw volume trim volume at cylinder
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topology/geometry in Isogeometric Analysis
(Rank, Kollmannsberger, Sorger, Düster, 2011)
NURBS model
a) FEM: generate and compute c) IGA/FCM: only compute
b) IGA:
generate new mesh and compute:
alternatively: convert mesh and compute
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Vigoni project proposal
(Kollmannsberger, Reali, Auricchio, Rank 2011)
- compute on trimmed surfaces
- enforce conformity across trimmed patches in a weak sense
Technische Universität MünchenComputation in Engineering
To mesh or not to mesh:that is the question
Stefan Kollmannsberger
Computation in Engineering
Technische Universität München
Technische Universität MünchenComputation in Engineering
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b) remarks to Isogeometric Analysis
-> in order to have to generate a meshone draws the (coarsest) mesh and refines it
issue 1: The geometry is the mesh is the discretization“Analysis aware modeling”
E. Cohen, T. Martin, R.M. Kirby, T. Lyche, R.F. Riesenfeld: Analysis-aware modeling: Understanding quality considerations in modeling for isogeometric analysis, Comput. Methods Appl. Mechn. Engrg: 199:2010.