5/05/2015
Level set-assisted modelling of heterogeneous
microstructures
Bernard Sonon
Bertrand François
Badadjida Wintiba
Thierry J. Massart
Introduction
Level set functions & signed distance fields
Inclusion packing
Advanced microstructural features generation
Foam materials
Level sets in woven composite simulations
Perspectives & conclusions
Introduction
Motivation
Materials with heterogeneities are met in many engineering problems
Rock
Metallic foams
Concrete
Hollow foams
How to represent these within an Representative Volume
Element for homogenization ?
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Introduction
Motivation
Even when no ‘randomness’ or with periodicity, the microstructural
geometry can be complex
How to represent these within an Representative Volume
Element for homogenization ?
Introduction
RVE construction for multi scale modeling
How representative of the real material should an RVE morphology be ?
[Rubik, 1974]
[Massart & Selvadurai, 2014]
[Fritzen, 2009; Rycroft, 2006,2009]
are probably not equivalent to
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Introduction
Motivation
Capturing material heterogeneities induces (many) challenges,
among which
1. Obtaining the real geometry from in-situ information or lab-scale
imaging techniques (µCT scan) [Legrain, 2011; Hashemi et al. 2014]
2. Reconstruct or generate geometrically the essential features of
material heterogeneities, based on morphological information
a.
b.
Use of distance fields for RVE generation input (inclusion-based)
Use of distance fields for RVE generation post processing (fiber- or
yarn-based)
3. Use discretisation schemes able to account for material
boundaries in an efficient manner
Introduction
Level set functions & signed distance fields
Inclusion packing
Advanced microstructural features generation
Foam materials
Level sets in woven composite simulations
Perspectives & conclusions
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Level set functions & signed distance fields
Level set function
A level set function
surface (curve in 2D)
is a function of x defining implicitly the
by
The simplest level set is the signed distance field of Φ
Φ
Many functions can be used as level set functions to define surfaces
Signed distances fields can also be used only to measure distance …
Introduction
Level set functions & signed distance fields
Inclusion packing
Advanced microstructural features generation
Foam materials
Level sets in woven composite simulations
Perspectives & conclusions
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Inclusion packing
Random sequential addition
Classical methodologies to generate random microstructures
Random sequential addition (RSA)
(a)
(b)
DEM packing
No overlap
No overlap + maximal neighbour distance
[Cooper, 1988; Sherwood, 1997 ]
[ He et al. 2010
Stroeven et al., 2004]
However …
Heavy computational time (both RSA & DEM)
High volume fractions cannot be reached (RSA)
Third bridging phase not easy to add in microstructures
(both RSA & DEM)
Inclusion packing
DN-RSA packing generator
With the signed distance fields of several inclusions, construct the first
and second neighbor distance functions
and
Minimum of signed distances of all
inclusions
Minimum of signed distances of all
inclusions except the one closest to x
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Inclusion packing
DN-RSA packing generator
Control sequential addition and the resulting packing with
neighboring distances criteria
no overlap
No overlap and not further than d1
from the nearest
keep at least d1 from the n.
incl. but closer than d2
keep closer than d3 from the second
nearest inclusion
Inclusion packing
DN-RSA performance
Performance in 2D implementation (non optimized distance calculation !)
Cost becomes a linear function of the number of inclusions
DN maps adaptive refresh NOT activated
Non Periodic
Periodic
Classical RSA takes about 100 & 200 sec for RVEs equivalent to
the ones depicted …
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Inclusion packing
DN-RSA performance
Performance in 2D implementation (non optimized distance calculation !)
Nearest neighbor criteria can improve packing efficiency
DN maps adaptive refresh NOT activated
No Overlap
No Overlap + close packing
using DN2
Close packing (b) cannot be achieved using classical RSA …
Inclusion packing
DN-RSA packing generator
DN map refresh upon addition
At each inclusion addition the DN maps have to be refreshed, which is the
costly operation
A fast predictor, the signed distance to the smallest enclosing sphere is used
to restrict the refreshed domain
Compulsory when packing small inclusions in large RVEs …
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Inclusion packing
DN-RSA performance
Performance in 2D implementation (non optimized distance calculation !)
Effect of adaptive refresh of DN maps
(a)
(b)
(c)
(d)
Circular inclusions
5-sided inclusions
10-sided inclusions
20-sided inclusions
No adaptive refresh
Adaptive refresh
Inclusion packing
DN-RSA performance
Performance in 2D implementation (non optimized distance calculation !)
Effect of inclusion size distributions
1089 Arbitrary shaped inclusions
566 Circular inclusions
Generation time = 35 sec
Generation time = 15 sec
200² grid for DN maps evaluations
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Inclusion packing
DN-RSA in 3D
3D implementation
Addition of criteria with respect to
Distance calculation by Optimized Brute Force Approach (OBFA)
Only the distance calculation is ‘optimized’ (but still ‘brute force’, rest of
implementation is full MatLab)
Generation = 30 sec
Generation = 156 sec
Generation = 1036 sec
Inclusion packing
DN-RSA in 3D
3D implementation
Computation time 103sec)
Critical aspect of distance calculations (optimized)
Development of optimized calculation based on Fast Marching Methods
Brute force – total
Brute force – distance
‘Exact’ Fast Marching – total
‘Exact’ Fast Marching – distance
Computation time 103sec)
Inclusion number
Inclusion volume fraction
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Inclusion packing
DN-RSA in 3D
3D implementation – most complex examples
~ 3500 inclusions
~ 450 sec. of distance computation (rest of processing is MatLab which can
be easily further optimized)
Introduction
Level set functions & signed distance fields
Inclusion packing
Advanced microstructural features generation
Foam materials
Level sets in woven composite simulations
Perspectives & conclusions
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Advanced morphological features
Generalized tessellation
‘Voronoï-like’ level set function
vanishes at equal distance between two
neighboring inclusions
Allows extracting generalized tessellation for arbitrary shaped inclusions
Advanced morphological features
Generalized tessellation
Tessellation with widely distributed grain size distributions
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Advanced morphological features
Generalized tessellation
Partial selective tessellation
Given two population A and B one may want to modify only population A
Use Euclidean distance for A
Use distance defined as -
inside inclusions and +
outside for B
Extract contour of
Advanced morphological features
Coating and bridging
Morphing for intermediate tessellation
Extract interfaces from
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Advanced morphological features
Coating and bridging
Coating on inclusions
Coating of width k obtained by offset on function
White curves : DN1(x) = 0
Black curves : DN1(x) – k = 0
Blue : inclusions
Red : coating
Grey : matrix
Advanced morphological features
Coating and bridging
Bridging level set function
Bridging obtained from
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Advanced morphological features
Coating and bridging
Controlled combination of bridging and coating
Combination of
With
and offset
controlling the importance of bridges vs. coating
Advanced morphological features
Coating and bridging
3D extension of bridged and coated inclusions
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Applications
XFEM - Anisotropic fluid transport
Z cut
X cut
Y cut
Applications
XFEM - Anisotropic fluid transport
Flow under vertical pressure gradient
kx / ky= 0.997
kz / ky= 1.224
kz / kx= 1.228
Flow under horizontal pressure gradient
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Applications
XFEM – Unilaterial coupling to fluid transport
Stress dependent permeability evolution in rocks
E
v
c
phi
s1
k0
Gpa
/
MPa
°
GPa
µm/s
B
/
Grain
60
0.1
/
/
/
1
0
Matrix
30
0.3
40
17
1
1
50000
Plastic volume deformation permeability dependence : k = k0.(1 + B.εplast,vol)
Applications
XFEM – Unilaterial coupling to fluid transport
Stress dependent permeability evolution in rocks
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Introduction
Level set functions & signed distance fields
Inclusion packing
Advanced microstructural features generation
Foam materials
Level sets in woven composite simulations
Perspectives & conclusions
Extension towards cellular materials
Closed foams morphologies
Closed foams
‘Voronoï-like’ tessellation level set function to produce the walls (faces) of
closed foams cells
where controls
the wall thickness
RVEs with variable cell sizes, curved cell walls with non constant thickness
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Extension towards cellular materials
Open foam morphologies
Open foams
Extract the edges of the tessellation cells by means of an ad-hoc function
Plateau borders in liquid foams form at intersection of three films and
exhibit a triangular shape
Combine the three first neighbor functions
with a function vanishing at equal distance of 3 closest inclusions
Mono-sized spheres
Variable-sized spheres
Arbitrary shaped inclusions
Extension towards cellular materials
Open foam morphologies
Open foams
Plateau borders defined by
Parameter controls thickness of
extracted borders
Plateau borders produced are largest
that could be completely contained
in closed cell walls produced by
with parameter
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Extension towards cellular materials
Open foam morphologies
Open foams
Mono-sized spheres
Straight Plateau borders
Variable-sized spheres
Curved Plateau borders
Arbitrary shaped inclusions
Complex Plateau borders
Extension towards cellular materials
Open foam morphologies
Open foams
Extraction from a single level set of
leads to jagged sharp edges
Requires the use of local ‘dedoubled’ level sets used in slicing operations
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Extension towards cellular materials
Open foam morphologies
Open/closed intermediates
Linear combination of
with
and
as a pore closure index
Extension towards cellular materials
Open foam morphologies
Concavity of Plateau borders
Concavity of Plateau borders by substracting a non constant negative
function vanishing on the Plateau border sharp edges to alter curvature
controlling the obtained concavity
t level set of
t level set of
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Extension towards cellular materials
Open foam morphologies
Orientation sensitivity
Random field T(x) between tmin and tmax
Global field A(x) according to wall orientation to mimic anisotropic growth
of the foam microstructure
Walls normal to a chosen orientation n open while closing others
Top view
side view
Extension towards cellular materials
Open foam morphologies
Hollow foams
Obtained by simple offset of the previously defined functions
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Introduction
Level set functions & signed distance fields
Inclusion packing
Advanced microstructural features generation
Foam materials
Level sets in woven composite simulations
Perspectives & conclusions
Woven Composites
Pre-processing for RVE generation
Weave tensioning & contact treatment
Starting from a discretized wire representation
Geometrical phenomenological iterative procedure
Circular cross-section , No friction , No bending stiffness , No mechanical
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Woven Composites
Pre-processing for RVE generation
‘Geometrical’ tensioning
Operation A
Move each vertex to the middle of the segment linking its 2 neighbors.
The magnitude of displacement depends on both segment angle and length.
Conditions are required on first and last vertex of a yarns.
Leads to yarns inter-penetrations.
Reaches asymptotically a straight configuration.
Side view on wires figuring yarns
Woven Composites
Pre-processing for RVE generation
Contact – resolving interpenetration
Operation B
Detect contact using distance from vertex to a polyline, taking into account yarn radii.
Resolve contact by moving each yarn by the half of the inter-penetration distance.
Use gradient of the yarns distance to get the direction of the displacement required.
A
B
Resolves exactly contacts for 2-yarns situations
For 2 yarns equally discretized, this fulfills the local balance between
tension and contact forces
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Woven Composites
Pre-processing for RVE generation
Implicit control on relative tensions (qualitative)
Equilibrium situation depends on the discretization.
Straight yarns cannot be obtained
Balance the displacement of operation B by coefficients p and 1-p defined for each
yarns type pair.
A
p = 0.5
B
p = 0.05
Woven Composites
Level set post-processing in RVE generation
Need for post-processing after wire-based RVE generation
Wire representation with discretization induces spurious yarn overlaps at multi-yarns
contact loci that need to be suppressed for finite element discretization
Introduce a clearance between yarns at contact zones.
Alter yarn section shape to mimic deformation at contact loci.
Can be coupled to any existing RVE generation procedure (wiseTeX, …)
Provide input for XFEM.
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Woven Composites
Level set post-processing in RVE generation
Suppression of interpenetrations
Distance field to yarn i DSi(x)
Distance to all yarns except i DOi(x) = miny( DSy(x) ) with y ≠ i
DSi(x)- DOi(x) takes negative value at any point closer to yarn i than to others.
If no interpenetration, its 0 level is a closed curve all points closer to it than to other yarns
If interpenetration, its 0 level set is a closed surface that cuts the interpenetration in two, i.e.
is locally the locus of point equidistant from boundaries of both yarns
Intersection between the volume bounded by the 0 level set of DSi(x)- DOi(x) and yarn i is
given by max( DSi(x) , DSi(x) – DOi(x)) matches original yarn less half of the
interpenetration
Woven Composites
Level set post-processing in RVE generation
Suppression of interpenetrations & volume fraction control
A clearance at contacts can be added to simplify meshing procedures using the function
max( DSi(x) , DSi(x) – DOi(x) + c)
The same principle may be used to further update the reinforcement volume fraction
(compensate for fraction lost by cutting or increase it further to control the volume fraction)
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Woven Composites
Post-processing in RVE generation
Examples
Impose targeted volume fractions of reinforcement
Yarns transverse deformations at contacts
a
b
a
c
b
c
Woven Composites
Post-processing in RVE generation
Examples with existing RVE generator WiseTeX
Generalised yarns cross sections, but with residual interpenetrations
36% reinforcement
2% overlapping
Level set post-processing
36% reinforcement , no interpenetration
Increase to 45% reinforcement
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Woven Composites
Use of level sets in XFEM enrichment
Introduce displacement gradient jump at interfaces
Allow non conforming meshes with strain jumps inside elements
Ψy(x) = Nj(x).|Ojy| – |Nj(x).Ojy|
DSoy(x) = mino( DSo(x) ) , with o ≠ y
Oy(x) = max( DSy(x) , DSy(x) – DSoy(x) + c )
Woven Composites
XFEM simulation of material behaviour
Compare different weaving schemes
3D
2.5D a
2.5D b
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Woven Composites
XFEM simulation of material behaviour
Interfacial tangential stresses
3D
2.5D a
2.5D b
Introduction
Level set functions & signed distance fields
Inclusion packing
Advanced microstructural features generation
Foam materials
Level sets in woven composite simulations
Perspectives & conclusions
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Summary & Outlook
Integration of level sets-based tools
Summary
Level set-based tools for geometry generation were used for complex RVES
Such techniques can be seamlessly combined with XFEM principles
The combination of techniques can be applied to heterogeneous problems
These techniques can be further integrated with level set-enhanced image-based
modelling [ Legrain et al., 2011]
Work in progress …
Comparisons to experimental data based on morphological operators
Closer control on pore space size distributions
Conforming meshes could be produced using implicit geometries meshing tools
available in literature [ Persson et al., 2006]
Summary & Outlook
Integration of level sets-based tools
Under development
Compare generated RVEs with real CT scan-based data and simulations
XRµCT with mechanical simulation on sand-bentonite mixture
[ Hashemi et al., 2014]
Explicit representation of pore space & capillary effects
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Summary & Outlook
References
References
B. Sonon, B. François, T.J. Massart, A unified level set based methodology for fast generation of complex
microstructural multi-phase RVEs, Comp. Meth. Appl. Mech. Engng., 223:103-122, 2012
B. Sonon, B. François, A.P.S. Selvadurai, T.J. Massart, XFEM modelling of degradation-permeability
coupling in complex geomaterials, International Conference on Fracture (ICF13), Beijing, China, 2013
B. Sonon, T.J. Massart, A Level set-based Representative Volume Element generator and XFEM
Simulations for textile and 3D-reinforced composites, Materials, 6(12), 2013
B. Sonon, On advanced techniques for generation and discretization of the microstructure of complex
heterogeneous materials, PhD Thesis, Université Libre de Bruxelles (ULB), 2014
B. Sonon, B. François, T.J. Massart, An advanced approach for the generation of cellular materials
Representative Volume Elements using distance fields and level sets, Revised version, 2015
B.Sonon, B. François, P. Berke, A.P.S. Selvadurai, T.J. Massart, A level set-based integrated methodology
for the upscaling of fluid transport properties in heterogenous geomaterials, In Preparation
B.Sonon, B. François, P. Berke, T.J. Massart, Computational analysis of microstructural effects in closedcell foams using implicit geometries, In Preparation
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