Homogenization method for elastic materials

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2 Outline 1 Introduction 2 Description of the geometry 3 Problem setting 4 Implementation & Results 5 Bandgaps 6 Conclusion

3 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.

4 Description of the geometry Geometry N N cells, cell size ε, domain Ω ε 1 - elastic material 1, domain Ω ε 2 - elastic material 2, reference cell Y = [0, 1[ 3. Figure: Geometry of the lattice Coordinates system (x 1, x 2 ) macro coordinates, (y 1, y 2 ) micro coordinates, (x, y) represents ε [ x ε] + εy.

5 State equations State equations deflection of the loaded lattice, material coefficients c ε ijkh(x) = c ijkh ( x ε ), (1) classical sense formulation ( ) c ε ijkh(x) uε k = f i v Ω, x j x h u ε (x) = 0 na Ω. (2)

6 State equations Weak formulation Find u ε H 1 0(Ω) such that c ε mnkle kl (u ε )e mn (Φ) = Ω Ω f Φ Φ H 1 0(Ω). (3) Cauchy tensor e kl (v) = 1 2 ( vk x l + v ) l, (4) x k H 1 0(Ω) is the Sobolev space H 1 (Ω) with compact support.

7 Homogenization method I State equations for the homogenized material ( ) c x ijkh(x) u k = f i j x h in Ω, u (x) = 0 on Ω. (5) homogeneous coefficients (effective parameters) c ijkh = c average ijkh c corrector ijkh, (6) integral average of heterogeneous material coefficients c average ijkh = 1 c ijkh (y) dy. (7) Y Y

8 Homogenization method II Corrector coefficients c corrector ijkh = 1 Y auxiliary functions χ kh c ε ijkhe ij (χ ij )e kh (v) dy = Y Y Y c ijlm (y) χkh l dy, (8) y m c ε lmkhe kh (v) dy v W 1 per (Y ), (9) where Wper 1 (Y ) is the space of Y-periodic functions with a zero integral average { Wper 1 (Y ) = v } v H 1 (Y ), v i dy = 0, i = 1, 2. (10) Y

9 Discretization Discretization triangular mesh, finite elements method, mass and force matrix K ε = e K ε e, f ε = e f ε e, (11) state equation - heterogeneous material K ε u ε = f, (12) state equation - homogenized material K u = f. (13)

10 Implementation Computation follows in four steps computation of u ε, solution to (3), computation of the auxiliary functions (9), computation of effective parameters cijkh, computation of u (5).

11 Results I (a) Heterogeneous material (b) Homogenized material Figure: Magnitude values of the displacement for considered materials (u ε, u ).

12 Results II Figure: L 2 norm of displacements u ε, u.

13 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.

14 Bandgaps II Weak formulation ω 2 Ω r ε u ε Φ c ε mnkle kl (u ε )e mn (Φ) = f Φ Φ H 1 0(Ω). (14) Ω Ω the mass density r ε, scaling ε 2 = strong heterogeneity in the relations for the material coefficients, ω is the angular frequency, for ω = 0 we get exactly the previous case, for ω different from the resonance values - unique solution u ε H 1 0(Ω). Discretization (K ε ω 2 M)u ε = f. (15)

15 Conclusion Summary comparison of the real heterogeneous material with the homogeneous material, under certain circumstances good approximation. Further goals shape optimization, objective function: larger bandgaps.

16 Shape optimization I Closed B-spline of order k = 4 cubic polynomials, design curves - material interfaces, n j + 1 is the amount of control points, control points d j i, j = 1, 2, i = 0,..., n j, N ik are basis functions, formula for the B-spline curves Figure: Initial design n j X j (t) = d j i N i4(t) t [ t 0, t nj +1], i=0 T = (t 0, t 1,..., t nj, t 0, t 1, t 2, t 3 ). (16)

17 Shape optimization II (a) Figure: Admissible designs (b)

18 Literature D. Cioranescu, P. Donato, An Introduction to Homogenization, Oxford University Press, Ávila, A., Griso, G., Miara, B., Rohan, E., submitted. Multi-scale modelling of 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 material parameters in modelling of vibrations of heterogeneous materials, Computational mechanics 2006, pages

19 Acknowledgement Acknowledgement The work has been supported by the project FRVŠ 570/2007/G1.

20 Thank you for you patience...