Computations of Asymptotic Scaling for the Kohn-Müller and Aviles-Giga Functionals

Transcription

1 Computations of Asymptotic Scaling for the Kohn-Müller and Aviles-Giga Functionals B.K. Muite Acknowledgements John Ball, Robert Kohn, Petr Plechá c Mathematical Institute University of Oxford Present Address: University of Michigan muite@umich.edu muite 23 May 2

2 Outline The Kohn-Müller model The model The conjecture Numerical results Conclusion The Aviles-Giga model The model The conjecture Numerical results Conclusion Outlook

3 Section : Computations of the Kohn-Müller model The sharp interface Kohn-Müller model The smooth interface Kohn-Müller model Computational results

4 Simplified model which may capture twinning at a boundary Twinning in Copper Aluminum Nickel (picture by Chu and James)

5 The Kohn-Müller model Conjectured correspondence ( 2 γ 2 w y wx 2 ) + ɛ 2 2 w xxdxdy 2 Ω γ 2 w y 2 dxdy + 2ɛ 3 [w x] 3 dh. Ω J w x Smooth interface 5 2 w y Ω ( w 2 x ) 2 + ɛ 2 2 w 2 xxdx Equation of motion ρw tt β w t = 5(3w 2 x )w xx + 5w yy ɛ 2 w xxxx

6 Numerical Results for the Kohn-Müller model (a) A plot showing the evolution of the different energy of ɛ2 w 2 xx. 2 (b) Initial Iterate Contour plot components. A local minimum for the Kohn-Müller energy functional with ρ =, β =. and ɛ =.36.

7 Numerical Results for the Kohn-Müller model (c) Final iterate. Colours show the function, w. (d) Contour plot of 5 w 2 2 y in the final iterate. A local minimum for the Kohn-Müller energy with ρ =, β =. and ɛ =.36.

8 Numerical Results for the Kohn-Müller model (e) Contour plot of 5 4 (w 2 x ) 2 in the final iterate. (f) Contour plot of ɛ2 2 w 2 xx in the final iterate. A local minimum for the Kohn-Müller energy functional with ρ =, β =. and ɛ =.36.

9 Numerical Results for the Kohn-Müller model (g) A plot showing the evolution of the different energy of ɛ2 w 2 xx. 2 (h) Initial Iterate Contour plot components. A local minimum for the Kohn-Müller energy functional with ρ =, β =. and ɛ =..

10 Numerical Results for the Kohn-Müller model (i) Final iterate. Colours show the function, w. (j) Contour plot of final iterate. 5 2 w 2 y in the A local minimum for the Kohn-Müller energy with ρ =, β =. and ɛ =..

11 Numerical Results for the Kohn-Müller model (k) Contour plot of 5 (w 2 4 x ) 2 (l) Contour plot of ɛ2 w 2 xx 2 in the in the final iterate. final iterate. A local minimum for the Kohn-Müller energy functional with ρ =, β =. and ɛ =..

12 Scaling law predicted by Kohn and Müller for the total energy!"!!" " +,-./*&%'() *2-'.3&*%&%'() 4*2-'.3&*%&%'() 5&-%'6.73./*%&%'() * %&%'()!"!!!"!$ *!"!#!"!$!"!!! Energy scaling, reference line drawn for ease of comparison. For small ɛ good agreement with theoretical predictions.

13 Section 2: Computations of the Aviles-Giga model The model The conjecture Computational results

14 The model The energy to minimise 4ɛ (w 2 x + w 2 y ) 2 + ɛ 2 ( w)2 dxdy limiting solution as ɛ, and minimise Remarks: J w w 2 x + w 2 y = 2 2 [ w] 3 dh Assume a one dimensional interface 2 The above assumption leads to equipartition of energy

15 Points which need to be demonstrated for the conjecture to hold The Γ limit is infinite unless w xx + w yy = almost everywhere The proposed sharp interface limit w /3 is correct The asymptotic energy lives on a one dimensional defect set, and lower dimensional singularities carry no energy DeSimone, Kohn, Müller and Otto

16 Equation of motion The energy to minimise 4 (w 2 x + w 2 y ) 2 + ɛ2 2 ( w)2 dxdy Viscoelastic dynamics ρw tt β w t [ [ ] = w x (wx 2 + wy 2 ) ]x + w y (wx 2 + wy 2 ) y ɛ2 2 w

17 Numerical Results for the Aviles-Giga model energy 5 5 total energy strain energy interfacial energy kinetic energy time (m) A plot showing the evolution of the different energy components. function..5 y x.5 (n) Initial Iterate x Numerically computed minimizer for the Aviles-Giga energy functional with ρ =, β =., ɛ =.5, w(y = ±) = and w yy (y = ±) =.

18 Numerical Results for the Aviles-Giga model function.5.5 y x function.5 y x (o) Final Iterate (p) Viscosity solution assuming equipartition of energy Numerically computed minimizer for the Aviles-Giga energy functional with ρ =, β =., ɛ =.5, w(y = ±) = and w yy (y = ±) =.

19 Numerical Results for the Aviles-Giga model * # $%&.2 $%+ $%*+.5.5 " $ y!$%+ $%*.5.!* # $ $%& $%' $%( $%) *! $%$ x (q) Contour plot of interface (r) Contour plot of (w 4 x 2 +wy 2 energy term ɛ2 2 ( w)2 in the final ) 2 in the final iterate iterate..5 Numerically computed minimizer for the Aviles-Giga energy functional with ρ =, β =., ɛ =.5, w(y = ±) = and w yy (y = ±) =.

20 Numerical Results for the Aviles-Giga model energy 2 total energy strain energy interfacial energy kinetic energy time function.5 y.5 x.5 (s) A plot showing the evolution of the different energy components. (t) Initial Iterate Numerically computed minimizer for the Aviles-Giga energy functional with ρ =, β =., ɛ =., w(y = ±) =.5 sin(2πx) and w yy (y = ±) =.

21 Numerical Results for the Aviles-Giga model 2.8 function y x.5.2 (u) Final Iterate Numerically computed minimizer for the Aviles-Giga energy functional with ρ =, β =., ɛ =., w(y = ±) =.5 sin(2πx) and w yy (y = ±) =.

22 Numerical Results for the Aviles-Giga model y.25.2 y x x (v) Contour plot of interface (w) Contour plot of (w 4 x 2 +wy 2 energy term ɛ2 2 ( w)2 in the final ) 2 in the final iterate iterate..2 Numerically computed minimizer for the Aviles-Giga energy functional with ρ =, β =., ɛ =.5, w(y = ±) = and w yy (y = ±) =.

23 Numerical Results for the Aviles-Giga model 3 energy 5 total energy strain energy interfacial energy kinetic energy time x 4 (x) A plot showing the evolution of the different energy components. (y) Initial Iterate Numerically computed minimizer for the Aviles-Giga energy functional with ρ =, β =., ɛ =.25, w(y = ±) =.5 sin(2πx) and w yy (y = ±) =.

24 Numerical Results for the Aviles-Giga model 3 (z) Final Iterate Numerically computed minimizer for the Aviles-Giga energy functional with ρ =, β =., ɛ =.25, w(y = ±) =.5 sin(2πx) and w yy (y = ±) =.

25 Numerical Results for the Aviles-Giga model y y x x () Contour plot of interface energy term ɛ2 2 ( w)2 in the final ) 2 in the final iterate () Contour plot of (w 4 x 2 + wy 2 iterate..5 Numerically computed minimizer for the Aviles-Giga energy functional with ρ =, β =., ɛ =.25, w(y = ±) = and w yy (y = ±) =.

26 Result Summary w(y = ±) ɛ 4 (w x 2 + wy 2 ) 2 ɛ 2 2 ( w)2 Total Energy ɛ sin(2πx) sin(2πx) sin(2πx) sin(2πx) sin(2πx) Prediction that (Total Energy)/ɛ= 2 2/3.943 if Aviles-Giga conjecture holds and zero boundary conditions.

27 Conclusions Simulations can be a useful tool to test scaling assumptions For the Kohn-Müller model, the simulations are in agreement with the model For the Aviles-Giga model, the conjecture may not hold when non-zero boundary conditions are applied

28 References Aviles and Giga, 999 Conti, DeSimone and Müller, 25 DeSimone, Kohn, Müller and Otto, 2 Jin and Kohn, 2 Kohn and Müller, 992, 994 Lorent, 29 Muite, 2 Schreiber, 994

29 Acknowledgements The Marie Curie Research Training Network MULTIMAT EPSRC grant, OxMOS New Frontiers in the Mathematics of Solids EPSRC grant, OxPDE Oxford Center for Nonlinear Partial Differential Equations