Homogenization Method for Elastic Materials

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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Outline

1 Introduction

2 Description of the geometry

3 Problem setting

4 Implementation & Results

5 Bandgaps

6 Conclusion



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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.



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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.


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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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)


Introd ction Description of the geometr Problem setting Implementation & Res lts Bandgaps Concl sion Acknowledgement


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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.




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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)




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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)




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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)




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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).




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Results I

(a) Heterogeneous material (b) Homogenized material

Figure: Magnitude values of the displacement for considered materials (u, u).




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Results II

Figure: L2 norm of displacements u, u.




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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.




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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)




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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.




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Shape optimization I


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)



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Shape optimization II

(a) (b)

Figure: Admissible designs




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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.




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Acknowledgement

AcknowledgementThe work has been supported by the project FRVS 570/2007/G1.




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Homogenization Method for Elastic Materials

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