The Finite Cell Method: High order simulation of complex ... · • Macroscopic mechanical...
Transcript of The Finite Cell Method: High order simulation of complex ... · • Macroscopic mechanical...
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Computation in Engineering
CSE 09
6/2009
The Finite Cell Method: High order simulation of complex structures without meshing
E. Rank, A. Düster, D. Schillinger, Z. Yang
Fakultät für Bauingenieur und Vermessungswesen
Technische Universität München, Germany
This work has partially been supported from the Excellence Initiative by the German Government, the German Science Foundation (DFG) and SIEMENS AG
U. Rüde, N. Thürey, T. Pohl , S. Donath, D. Bartuschat, C. Körner
Technische Fakultät der Universität ErlangenNürnberg, Germany
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Computation in Engineering
CSE 09
6/2009
From: Wikipedia
Metal foams
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Computation in Engineering
CSE 09
6/2009
Goal: Virtual material designObtain mechanical properties from process parameters of foam generation
Step 1: Simulation of foam generation
(Lattice Boltzmann Method)
Step 2: (Microscopic) structural simulation
to obtain mechanical properties
(Finite Cell Method)
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Computation in Engineering
CSE 09
6/2009
Overview
• Motivation
• Foam generation by the Lattice Boltzmann method
• High order finite cell method: Basic principles
• Accuracy considerations
• Aplication to porous structures
• Summary
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Computation in Engineering
CSE 09
6/2009
0=⋅∇ u‚
uPuutu ����
210
)( ∇+∇−=∇+∂∂ νρ
Partial differential equations (Navier–Stokes)
e.g. Finite Difference /Finite Volume equations
Discretization
Discrete model(LGCA or LBM)
Multi–scale analysis
LatticeBoltzmann simulation
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Computation in Engineering
CSE 09
6/2009
The Boltzmann equation:
homogenous "fluid" discrete "particles"
),( ffQxf
tf =
∂∂+
∂∂
8Øξ
: time
: spatial coordinate
: particle velocity
: Probability to find a particle with velocity at time and position
: Collision operator
tx�
ξ�
f
Qξ�
t xH
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Computation in Engineering
CSE 09
6/2009
• Simulate multi–particle system behaviour throughartificial microworlds of particles residing on a computational lattice
• Approximate Boltzmann equation on a lattice with FDmethod
continuum lattice
( ) ( ) ( ) ( )( ) ieq
iiiii Ftxftxftxftttexf +−−=−++ ,,1
,,
τδδ
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Computation in Engineering
CSE 09
6/2009
Extension to free surface flowC. Körner, M. Thies, T. Hofmann, N. Thürey, and U. Rüde:
Journal of Statistical Physics, Vol 121, 2005
• Two phases: Gas and liquid
• Surface capturing method
• Volume of fluid approach
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Computation in Engineering
CSE 09
6/2009
C. Körner, M. Thies, T. Hofmann, N. Thürey, and U. Rüde:
Journal of Statistical Physics, Vol 121, 2005
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Computation in Engineering
CSE 09
6/2009
Idea:
• ‚Virtual‘ material tests to predict macroscopic mechanical properties (e.g. tension and shear test)
Problem:• Mesh generation!
Our approach:• Finite Cell Method for
meshfree structural simulation
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Computation in Engineering
CSE 09
6/2009
Overview
• Motivation
• Foam generation by the Lattice Boltzmann method
• High order finite cell method: Basic principles
• Accuracy considerations
• Aplication to porous structures
• Summary
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Computation in Engineering
CSE 09
6/2009
Finite Cell Method (FCM): The basic formulation
exact solution: exact solution:
‘zero extension’:
+
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Computation in Engineering
CSE 09
6/2009
Discontinuous integrand
• High order integration schemes
• Subcell integration
• Local integration mesh
• A ‘cell’ is a finite element on the extended domain which needs yet not conform with the boundary of the original domain
• Discretization by finite element shape functions on ‘cells’
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Computation in Engineering
CSE 09
6/2009
Overview
• Motivation
• Foam generation by the Lattice Boltzmann method
• High order finite cell method: Basic principles
• Accuracy considerations
• Aplication to porous structures
• Summary
![Page 15: The Finite Cell Method: High order simulation of complex ... · • Macroscopic mechanical properties of porous media, e.g. foam like structures are determined by their microscopic](https://reader034.fdocuments.us/reader034/viewer/2022050717/5e483466d0f430347110967d/html5/thumbnails/15.jpg)
Computation in Engineering
CSE 09
6/2009
Assumptions:
• The structure consists of (homogeneous or inhomogeneous) material with (uniformly or randomly) distributed voids on the microscale
• Singularities do not play a dominant role on the microscale
Question:
• What is, for a given desired energy norm error, the ratio of degrees of freedom between FCM and classical FEM (hextension) ?
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Computation in Engineering
CSE 09
6/2009
Model problem for investigating the energy error per unit pore
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Computation in Engineering
CSE 09
6/2009
hextension
1% by 100000 dof
6% by 3000 dof3% by 12000 dof
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Computation in Engineering
CSE 09
6/2009
Finite Cell Method: 4 cells, pextension
1% by 500 dof
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Computation in Engineering
CSE 09
6/2009
von Mises stress along the cutlines
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Computation in Engineering
CSE 09
6/2009
Finite Cell method Reference Solution (pFEM)
von Mises contours
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Computation in Engineering
CSE 09
6/2009
Tensile plate with many inclusions
Finite element mesh Finite cell mesh
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Computation in Engineering
CSE 09
6/2009
Tensile plate with many inclusions
Error in von Mises stresses nowhere larger than 1%
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Computation in Engineering
CSE 09
6/2009
Overview
• Motivation
• Foam generation by the Lattice Boltzmann method
• High order finite cell method: Basic principles
• Accuracy considerations
• Aplication to porous structures
• Summary
![Page 24: The Finite Cell Method: High order simulation of complex ... · • Macroscopic mechanical properties of porous media, e.g. foam like structures are determined by their microscopic](https://reader034.fdocuments.us/reader034/viewer/2022050717/5e483466d0f430347110967d/html5/thumbnails/24.jpg)
Computation in Engineering
CSE 09
6/2009
A 3Dexample: Xray MicroCT examination of human bone
Baruffaldi, Perilli, Bn_326, from: The BEL Repository, http: //www.techno.ior.it/VRLABInstituti Ortopedici Rizzoli, Bologna, Italy
Human bone biopsy, taken from the femoral neck
Geometry:
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Computation in Engineering
CSE 09
6/2009
190 x 176 x 71 voxel
Voxel model
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Computation in Engineering
CSE 09
6/2009
2412 hexahedrals (cells)
Finite Cells
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Computation in Engineering
CSE 09
6/2009
190 x 176 x 71 voxel
Voxel model
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Computation in Engineering
CSE 09
6/2009
Reference solution based on the FEM (Abaqus, Netgen)
Netgen (Schöberl, www.hpfem.jku.at/netgen)
23,450 – 2,129,547
quadratic tetrahedral (10noded) elements
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Computation in Engineering
CSE 09
6/2009
Mesh generator: Netgen (Schöberl, www.hpfem.jku.at/netgen)23,450 – 2,129,547 tets
*Intel® Xeon® E5405 (2,00 GHz, 1.333 MHz, 2 x 6 MB, QuadCore)
FCM: 2412 cells
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Computation in Engineering
CSE 09
6/2009
From:
C. Körner, M. Thies, T. Hofmann, N. Thürey, and U. Rüde:
Journal of Statistical Physics, Vol 121, 2005
Foam generation with LBM
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Computation in Engineering
CSE 09
6/2009
Uniaxial tension test
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Computation in Engineering
CSE 09
6/2009
Structural analysis with FCM
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Computation in Engineering
CSE 09
6/2009
Strain Energy as measure for Young‘s modulus
Computational effort for p=6:
• 770 sec on Intel quadcoreprocessor workstation
• No time for meshing
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Computation in Engineering
CSE 09
6/2009
Summary• Macroscopic mechanical properties of porous media, e.g. foam like
structures are determined by their microscopic geometric structure
• The Lattice Boltzmann Method with free surface flow allows simulation of the formation of the microscopic structure of metal foams
• With the High order Finite Cell Method geometrically complex structures can be simulated without meshing
Future work• More investigations to verification and validation necessary
• Extension of the method to obtain nonlinear material properties
• Combination with stochastic analysis
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Computation in Engineering
CSE 09
6/2009
![Page 36: The Finite Cell Method: High order simulation of complex ... · • Macroscopic mechanical properties of porous media, e.g. foam like structures are determined by their microscopic](https://reader034.fdocuments.us/reader034/viewer/2022050717/5e483466d0f430347110967d/html5/thumbnails/36.jpg)
Computation in Engineering
CSE 09
6/2009
Macroscopicquantities
Particle distribution functions
Discrete model
(ensembleaveraging)
Multiplerelaxationtime latticeBoltzmann equation
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Computation in Engineering
CSE 09
6/2009
linear elastic material behaviour assumed
Finite Cells
190 x 176 x 71 voxel
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Computation in Engineering
CSE 09
6/2009
A 3Dexample: Xray MicroCT examination of human bone
Baruffaldi, Perilli, Bn_326, from: The BEL Repository, http: //www.techno.ior.it/VRLABInstituti Ortopedici Rizzoli, Bologna, Italy
Human bone biopsy, taken from the femoral neck
Geometry:
![Page 39: The Finite Cell Method: High order simulation of complex ... · • Macroscopic mechanical properties of porous media, e.g. foam like structures are determined by their microscopic](https://reader034.fdocuments.us/reader034/viewer/2022050717/5e483466d0f430347110967d/html5/thumbnails/39.jpg)
Computation in Engineering
CSE 09
6/2009
• Embedded domain method
• Fictitious domain method
• Meshfree/meshless methods
• Level set methods
in general for low order approximations, only.
Also similarities to
• Extended/enhanced finite element methods
• Generalized finite element methods
• Unfitted discontinuous Galerkin method
Some related methods