Homogenization Theory

Transcription

1 Homogenization Theory Sabine Attinger

2 Lecture: Homogenization Tuesday Wednesday Thursday August 15 August 16 August 17 Lecture Block 1 Motivation Basic Ideas Elliptic Equations Calculation of Effective Coefficients Elliptic Equations Numerical Homogenization Lecture Block 2 Elliptic Equations Derivation of Homogenized Equations Elliptic Equations: HT in comparison with other Upscaling Methods HT of other Equations Exercises Exercises e.g. Advection- Diffusion-Equ.

3 A Definition Homogenization Theory is concerned with the analysis of Partial Differential Equations (PDEs) with rapidly oscillating coefficients (1.1) Α ε ε u = f where Α ε u ε ε a differential operator the solution a nondimensional parameter associated with the oscillations

4 Example Steady flow through a saturated porous medium ( x) φ( x) f K = (1.2) φ( x) K( x) f with pressure head conductivity source/sink term L l

5 Example 2 scales: observation scale oscillations on scale l L and conductivity rewriting equation (1.2) in dimensionless variables x x x L 1 xˆ = xˆ L l L l ε ˆ xˆ ˆ xˆ K φ xˆ, = A ε φ ε = ε ε ( xˆ ) f ( xˆ )

6 Other examples C 3D Richards/Pressure Equation ψ t p = q ± ρ q 3D Darcy Eqaution ( ) ( z) = K ψ ψ p p + pressure/head capacity Darcy flow sink terms conductivity

7 Other examples θ c = q c + D c ± t i i i ρ ( c,...) i concentration Transport velocity porosity Molecular diffusion Machanical dispersion Chemical reactions...

8 What is the problem? Heterogeneities may cause large computational problems (3D) Area Typical aquifer heterogeneities Numerical resolution Total number of grid cells 100m x 100m x10m l 1m h l 5 Is it possible to reduce the computational resolution with tolerable errors?

9 What is the problem? Heterogeneities may cause ill-conditioned (stiff) numerical problems Convergence estimates for the Method of Finite Elements yield u uh C h ε Is it possible to formulate a well-conditioned numerical problem?

10 Basic Ideas Is there an equivalent homogeneous aquifer? Interactive GroundWater (IGW), by Dr. Li du/igw/ Flow and Transport through a Complex Aquifer System

11 Basic Ideas Large Scale Flow Model with effective conductivity r K( x) K 0 fine grid model large grid model ~ ) ( ) ε ε K + K ( x) φ ( x = f ( eff K ) φ x) = f 0 (

12 Basic Ideas We want to derive an effective, homogeneous model, where the heterogeneity is no longer seen Consider the limit ε 0 l l ε= L 0 L

13 Basic Ideas A heterogeneous medium is similar to a periodic field. Unit cell: Two distinct length scales: l L

14 Basic Ideas We want to derive an effective, homogeneous model, where the heterogeneity is no longer seen Consider the limit...

15 Questions 1. Convergence to a limit=homogenized solution: u 0 Is there a limit, as ε? In which sense should we understand the convergence (i.e. in which norm)? What is the convergence rate?

16 Questions 2. Derivation of the homogenized solution: What kind of equation does the limit satisfy? Suppose that the limiting equation is of the following form A Au = Is the operator of the same type as? f u A ε Example: ~ ) ) ( eff ) ε ε K + K) ( φx) x = φ ( xf = f 0 (

17 Questions 3. Calculation and properties of : A How can we compute homogenized operators / effective coefficients? How do the properties of the homogenized equation compare with those of the fine scale problem? How do the effective coefficients depend on the fine scale problem? Example: ( eff K ) φ x = f ) 0 (

18 Questions 4. Comparison with other upscaling methods How does Homogenization compare e.g. to - Stochastic Modelling (Ensemble Averaging) - Volume Averaging?

19 Questions 5. Numerical Homogenization: Can we design and implement efficient algorithms for problem (1.1) based on the method of homogenization? Can we calculate the homogenized equation in a computationally efficient manner?

20 ε Two-scale Expansion Idea: is a small parameter in (1.1), thus it is natural to expand in a power series in ε ε u all terms depend explicitly on both xˆ and ˆx / ε u ε x ε x ε x ε 2 ( x) = u x ε u x, ε u x,... 0,

21 Two-scale Expansion assumption of x and y as independent variables (only for problems with scale separation or for y-periodic problems) l << L u ε ( ) ( ( ) 2 x = u x ε ( y) 0 x + ε u x, yu + ( xε ) u ( x, y)... 0 u, It is one of the basic assumptions in homogenization theory, that the solution can be expanded like this. The convergence of the expansion has in principle to be proved!!

22 Two-Scale Expansion Counterexample for scale separation: l???? L

23 Two-Scale Expansion Example: Propagation of a wave with wavelength in a heterogeneous porous medium L L Fluctuations due to the heterogeneity L L has to be large compared to l

24 Two-Scale Expansion Example:

25 Exercise 1 How do spatial derivatives of a two-scale function look like? x x +..., + x... Example:

26 Two-Scale Expansion Partial derivatives then become x x 1 + ε y, x 1 + ε y Procedure: 1.Insert the two-scale ansatz into the fine scale problem (1.1) 2.Group the terms in orders of ε 3.Take the limit ε 0 4.Solve resulting equations for u, u,

27 Comparison with Volume Averaging Averaging volume Averaging volume Averaging Volume=REV: small compared to the macroscopic Volume large enough to contain all information about heterogeneities

28 Volume Averaging Introduction of a Filter Function (= Spatial Average Function) as moving average 1 d φ( x) d y FV ( x y) φ( y) V V V

29 Comparison with Volume Averaging Starting Point Volume Average ( x) φ( x) f ( x) K = V K ( x) φ( x) = f ( x) K( x) φ( x) V? K VolAve φ ( x) = f ( x)

30 Comparison with Stochastic Theories Assumptions: Spatially heterogeneous medium properties are modelled as random space function or stochastic process governing differential equations with dependent variables become stochastic PDE s. Procedure: Specific aquifer is considered as one realisation out of the ensemble of all possible realisations. Average over all realizations

31 Comparison with Stochastic Theories Averages over the ensemble first of all describe statistical properties of the formation. Their predictive value with respect to a particular (deterministic) geologic formation might be very limited. If deviations from the mean are small the mean value is characteristic or predictive also for a single deterministic realisation (ergodicity assumption).

32 Example: Stochastic Theory Different realizations of a catchment zone Mean catchment zone 1200 x_2 [m] 700 Risk Assessment x_1 [m] Stauffer et al., WRR, 2002

33 Comparison with Stochastic Theory Starting Point ( x) φ( x) f ( x) K = Ensemble Average K ( x) φ( x) = f ( x) K( x) φ( x) ens K φ = ( x) f ( x)

34 Comparison of methods Volume Average Ensemble Average Volume Averaging and ensemble Averaging becomes equivalent if ergodicity holds l REV Ergodicity: f = P( f ) df = REV f dv

35 Comparison of methods Local periodicity Stationarity In stochastic theory we have the requirement of local stationarity. This is equivalent to local periodicity in periodic media. l Stationarity: REV Periodicity: ( x) = f ( x l) f = f + ( x) = f ( x l) f + REV REV f dv = f REV + l dv

36 Numerical Homogenization Introduction of mathematical norms to measure if e.g. errors numerical errors Homogenization errors are limited by an upper finite bound

37 Two-Scale Expansion Example:

38 Numerical Homogenization Example:

39 Summary - Block 1 What is the problem with two-scale equations like ˆ xˆ ˆ xˆ K φ xˆ, = A ε φ ε = ε ε ( xˆ ) f ( xˆ )? Introduction of the main questions: 1. Derivation of homogenized equations 2. Calculation of coefficients 3. Comparison to other upscaling methods 4. Numerical Homogenization