homogenization method for elastic materials
TRANSCRIPT
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8/3/2019 Homogenization Method for Elastic Materials
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Introduction Description of the geometry Problem setting Implementation & Results Bandgaps Conclusion Acknowledgement
Homogenization method for elastic materials
Frantisek SEIFRT
Frantisek SEIFRT Homogenization method for elastic materials
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Introduction Description of the geometry Problem setting Implementation & Results Bandgaps Conclusion Acknowledgement
Outline
1 Introduction
2 Description of the geometry
3 Problem setting
4 Implementation & Results
5 Bandgaps
6 Conclusion
Frantisek SEIFRT Homogenization method for elastic materials
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Introduction Description of the geometry Problem setting Implementation & Results Bandgaps Conclusion Acknowledgement
Introduction
Introduction
study of the homogenization method applied on elastic materials,
G. Nguetseng (1989), G. Allaire,
D. Cioranescu, P. Donato.
Homogenization method
simplifies description of behavior of heterogeneous materials,
replacement by the homogenized, fictive material,homogenized material should be a good approximation of the original het.material.
Frantisek SEIFRT Homogenization method for elastic materials
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Introduction Description of the geometry Problem setting Implementation & Results Bandgaps Conclusion Acknowledgement
Description of the geometry
Figure: Geometry of the lattice
Geometry
N N cells, cell size ,
domain 1 - elastic material 1,
domain 2 - elastic material 2,
reference cell Y = [0, 1[3.
Coordinates system(x1, x2) macro coordinates,
(y1, y2) micro coordinates,
(x, y) represents
x
+ y.
Frantisek SEIFRT Homogenization method for elastic materials
I d i D i i f h P bl i I l i & R l B d C l i A k l d
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Introduction Description of the geometry Problem setting Implementation & Results Bandgaps Conclusion Acknowledgement
State equations
State equations
deflection of the loaded lattice,
material coefficients
cijkh(x) = cijkh
x
, (1)
classical sense formulation
xj
c
ijkh(x)
uk
xh
= fi v ,u(x) = 0 na .
(2)
Frantisek SEIFRT Homogenization method for elastic materials
Introd ction Description of the geometr Problem setting Implementation & Res lts Bandgaps Concl sion Acknowledgement
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Introduction Description of the geometry Problem setting Implementation & Results Bandgaps Conclusion Acknowledgement
State equations
Weak formulation
Find u H10() such that
c
mnklekl(u
)emn() =
f H10().
(3)
Cauchy tensor
ekl(v) =
1
2vk
xl +
vl
xk
, (4)
H10() is the Sobolev space H1() with compact support.
Frantisek SEIFRT Homogenization method for elastic materials
Introduction Description of the geometry Problem setting Implementation & Results Bandgaps Conclusion Acknowledgement
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Introduction Description of the geometry Problem setting Implementation & Results Bandgaps Conclusion Acknowledgement
Homogenization method I
State equations for the homogenized material
xj
cijkh(x)
ukxh
= fi in ,
u
(x
) =0
on
.
(5)
homogeneous coefficients (effective parameters)
cijkh = caverageijkh c
correctorijkh , (6)
integral average of heterogeneous material coefficients
caverageijkh =
1
|Y|
Y
cijkh(y) dy. (7)
Frantisek SEIFRT Homogenization method for elastic materials
Introduction Description of the geometry Problem setting Implementation & Results Bandgaps Conclusion Acknowledgement
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Introduction Description of the geometry Problem setting Implementation & Results Bandgaps Conclusion Acknowledgement
Homogenization method II
Corrector coefficients
ccorrectorijkh =1
|Y|
Y
cijlm(y)khlym
dy, (8)
auxiliary functions
kh
Y
cijkheij(ij)ekh(v) dy =
Y
clmkhekh(v) dy v W1per(Y), (9)
where W1per(Y) is the space of Y-periodic functions with a zero integral
average
W1per(Y) =
vv H1(Y),
Y
vi dy = 0, i = 1, 2
. (10)
Frantisek SEIFRT Homogenization method for elastic materials
Introduction Description of the geometry Problem setting Implementation & Results Bandgaps Conclusion Acknowledgement
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Introduction Description of the geometry Problem setting Implementation & Results Bandgaps Conclusion Acknowledgement
Discretization
Discretization
triangular mesh, finite elements method,
mass and force matrix
K
=e
K
e, f
=e
f
e, (11)
state equation - heterogeneous material
Ku = f, (12)
state equation - homogenized material
Ku = f. (13)
Frantisek SEIFRT Homogenization method for elastic materials
Introduction Description of the geometry Problem setting Implementation & Results Bandgaps Conclusion Acknowledgement
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p g y g p g p g
Implementation
Computation follows in four steps
computation of u
, solution to (3),computation of the auxiliary functions (9),
computation of effective parameters cijkh,
computation of u (5).
Frantisek SEIFRT Homogenization method for elastic materials
Introduction Description of the geometry Problem setting Implementation & Results Bandgaps Conclusion Acknowledgement
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Results I
(a) Heterogeneous material (b) Homogenized material
Figure: Magnitude values of the displacement for considered materials (u, u).
Frantisek SEIFRT Homogenization method for elastic materials
Introduction Description of the geometry Problem setting Implementation & Results Bandgaps Conclusion Acknowledgement
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Results II
Figure: L2 norm of displacements u, u.
Frantisek SEIFRT Homogenization method for elastic materials
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Bandgaps I
Bandgaps
Material with a periodic structure can exhibit acoustic bandgaps.Bandgaps = frequency ranges for which elastic or acoustic waves cannot
propagate.
Possible applications
frequency filters,
vibration dampers,
waveguides.
Frantisek SEIFRT Homogenization method for elastic materials
Introduction Description of the geometry Problem setting Implementation & Results Bandgaps Conclusion Acknowledgement
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Bandgaps II
Weak formulation
2
ru
cmnklekl(u)emn() =
f H10(). (14)
the mass densityr
,scaling 2 = strong heterogeneity in the relations for the materialcoefficients,
is the angular frequency,
for = 0 we get exactly the previous case,
for different from the resonance values - unique solution u
H10().
Discretization
(K 2M)u = f. (15)
Frantisek SEIFRT Homogenization method for elastic materials
Introduction Description of the geometry Problem setting Implementation & Results Bandgaps Conclusion Acknowledgement
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Conclusion
Summary
comparison of the real heterogeneous material with the homogeneousmaterial,
under certain circumstances good approximation.
Further goals
shape optimization,
objective function: larger bandgaps.
Frantisek SEIFRT Homogenization method for elastic materials
Introduction Description of the geometry Problem setting Implementation & Results Bandgaps Conclusion Acknowledgement
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Shape optimization I
Frantisek SEIFRT Homogenization method for elastic materials
Figure: Initial design
Closed B-spline of order k = 4
cubic polynomials,
design curves - material interfaces,
nj + 1 is the amount of control points,
control points dji,j = 1, 2, i = 0, . . . , nj ,
Nik are basis functions,
formula for the B-spline curves
Xj
(t) =
nj
i=0
d
j
iNi4(t) t
t0, tnj+1,
T = (t0, t1, . . . , tnj, t0, t1, t2, t3).
(16)
Introduction Description of the geometry Problem setting Implementation & Results Bandgaps Conclusion Acknowledgement
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Shape optimization II
(a) (b)
Figure: Admissible designs
Frantisek SEIFRT Homogenization method for elastic materials
Introduction Description of the geometry Problem setting Implementation & Results Bandgaps Conclusion Acknowledgement
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Literature
D. Cioranescu, P. Donato, An Introduction to Homogenization, OxfordUniversity Press, 1999.
Avila, A., Griso, G., Miara, B., Rohan, E., submitted. Multi-scale modellingof elastic waves, Theoretical justification and numerical simulation of band
gaps. Multiscale Modeling & Simulation, SIAM journal.
F. Seifrt, E. Rohan, B. Miara, Influence of the scale and materialparameters in modelling of vibrations of heterogeneous materials,Computational mechanics 2006, pages 535-542.
Frantisek SEIFRT Homogenization method for elastic materials
Introduction Description of the geometry Problem setting Implementation & Results Bandgaps Conclusion Acknowledgement
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Acknowledgement
AcknowledgementThe work has been supported by the project FRVS 570/2007/G1.
Frantisek SEIFRT Homogenization method for elastic materials
Introduction Description of the geometry Problem setting Implementation & Results Bandgaps Conclusion Acknowledgement
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Frantisek SEIFRT Homogenization method for elastic materials