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Homogenisation theory for partial differential equations

Homogenisation theory for partial differential equations. An introduction to homogenisation. Yves van Gennip, CASA Seminar Wednesday 26 January 2005. →. G.A. Pavliotis – Homogenization theory for partial differential equations http://www.ma.ic.ac.uk/~pavl/homogenization.html.

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Homogenisation theory for partial differential equations

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  1. Homogenisation theory for partial differential equations An introduction to homogenisation Yves van Gennip, CASA Seminar Wednesday 26 January 2005 → G.A. Pavliotis – Homogenization theory for partial differential equations http://www.ma.ic.ac.uk/~pavl/homogenization.html

  2. Overview of my talk • What is homogenisation? • Homogenisation applied to steady state heat conduction • One dimensional case • Some properties of the homogenised coefficients

  3. What is homogenisation? • Problem with two time or length scales: slow/macroscopic and fast/microscopic • Treat these scales as independent variables • Derive a homogenised problem: depends only on slow scale and still has the relevant macroscopic structure

  4. Steady heat conduction

  5. Assumptions and ansatz

  6. Treat x and y as independent

  7. Insert expansion

  8. Lemma

  9. Sub problem (1)

  10. Sub problem (2)

  11. Sub problem (3)

  12. Back to heat conducting

  13. Solve equation

  14. Some remarks at this point • The cell problem satisfies the solvability condition. • Unique first order corrector field if we demand zero average over Y. • Function undetermined at this point, but not needed here.

  15. Third equation

  16. Summary of homogenisation • Multiple scales expansion ansatz • Derive equations for , and . • First equation independent of y. • Second equation gives cell problem. • Third equation gives homogenised equation.

  17. One dimensional case

  18. Effective coefficient in 1D

  19. Bounds

  20. Recap • Homogenised problem for heat conduction. • The effective coefficients in the one dimensional case. • Now: more general properties of the coefficients.

  21. Cell problem rewritten

  22. Effective coefficients rewritten

  23. Uniform ellipticity

  24. Symmetry

  25. Recap • Variational formulation for cell problem and effective coefficients rewritten. • Homogenisation preserves positive definiteness and symmetry. • It does not preserve isotropy.

  26. Conclusions • Homogenisation: look at macro scale structure. • Get cell problem, homogenised equation and effective coefficients. • In one dimension we calculated the coefficient. • Homogenisation preserves some properties, not all.

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