Modelling with Partial Differential Equations in Metrology

Transcription

1 Modelling with Partial Differential Eqations in Metrology S. Alonso, M. Bär, H. Groß, M. Henn, G. Lindner, R. Model Modelling & Simlation Grop, Physikalisch-Technische Bndesanstalt (PTB), Berlin 255th PTB. Seminar / Workshop of EURAMET Focs Grop Math & IT

2 Otline Partial Differential Eqations (PDEs) Inverse Methods in PDEs: Scatterometry Mltiscale Methods: Homogenization, Averaging Flow Simlations

3 I. Partial Differential Eqations (PDEs) diffsion eq., heat eq. phase transitions: Ginzbrg-Landa eq. phase separation, thin films: Cahn-Hilliard type eq. electromagnetic fields: Maxwell eqs., Helmholtz eq. hydrodynamics: Navier-Stokes eqs. D t Δ = V t Δ + = Δ + = γ α ε γ 2 / ) ( ( ) ( ) K V M t Δ = / ) ( ) ( ( ) 0 ),,, (,, ),,, ( 2 = Δ t z y x z y x k t z y x

4 PDEs in Metrology Applied mathematics: - Analytical methods, pertrbation theory - Nmerics, discretization schemes FEM, FD, FV etc. - Mltiscale systems: homogenization, averaging Metrology: - PDEs as measrement models - PDEs as qantitative models - PDEs & inverse problems - Uncertainty evalation

5 II. Inverse Problem - Scatterometry Micro- resp. nanostrctre on sbstrate Periodic scattering geometry given by parameters P i Forward problem: Plane waves Geometry (P i ) Scattering efficiencies E i Inverse problem: Scattering efficiencies E i Geometry (P i ) Cooperation with WIAS Berlin (DIPOG-Software), AMTC Dresden

6 Mathematical Model Scatterometry Helmholtz-eqation Δ with k 2 ( x, y) = k( x, y) ( x, y) ( x, y) = ω μ ε ( x, y) +bondary conditions 0 1 μm Lithography mask 2D: geometry

7 Soltion of the Inverse Problem Compare simlated scattering efficiencies E m (h) with measrement data -> Reconstrction: Minimization of fnctional Φ Φ def ( E ( h) ) = Σ ω E ( h) m m M -> profile parameters h characterize profile m m E meas m 2 Test with simlated data: Sccesfl reconstrction Failed reconstrction (too few inpt data) H. Gross et al., Measrement 2006, H. Gross & A. Rathsfeld, Waves in Random & Complex Media,2008.

8 EUV-Scatterometry at PTB Details of setp, model -> talk by M. A. Henn

9 Optimal Simlation Reslts vs. Measred Data Deviations simlated vs. measred efficiencies: %

10 Uncertainties? Measrement noise: < 3 % Systematic errors are important! - Sbstrate model, roghness, incomplete information.. -> model errors T. Germer et al. SPIE Proc H. Groß et al. Meas. Sci. Tech Propagation of ncertainties in simlations: - Monte Carlo method, covariance approach - ncertainties in range < 5 nm ( < 2 %) H. Groß et al. SPIE Proc. 2008, SPIE Proc Improved treatment of random & systematic errors needed -> Talk by M. A. Henn

11 Smmary PDE & Inverse Problems Determination of parameters in PDEs: coefficients, simple geometry from data Comptationally expensive Applications: - Heat condction (see L. Wright talk) - Scatterometry (see M. A. Henn talk).. Otlook: Statistical inverse problems, ncertainty evalation

12 III. PDEs and Mltiscale Systems Mltiscale temporal and/or spatial dynamics Homogenization (averaging in space): Heterogeneos reaction-diffsion systems Simlation of flid flows: Trblence modelling (averaging in time)

13 Electrical Excitation in the Heart Compare: srface electrograms, propagation velocities, electro- & magnetocardiograms

14 Heart Modelling: Bidomain Eqations Cells extracelllar φ e intracelllar φ i Copled contina φ 1 r ( σ e φ e) = χ( Cm + f( φ, n)) t R φ 1 r ( σ i φ i) = χ( Cm + f ( φ, n)) t Rm r n r φ = φ i φ e = g( φ, n) t m copled parabolic & elliptic PDEs for intra- and extracelllar potentials φ i, φ e and ionic channel dynamics n r monodomain approximation: reaction-diffsion eqations

15 Heterogeneos Reaction-Diffsion Systems Model eqations ( D( r) c) R( c, r) t c = + D(r), R(r) describe heterogeneities (scale l het ) Effective models: homogeneity on scale λ >> l het Homogenization Detailed model Effective model

16 Homogenization & Effective Properties Model eqations with heterogeneos reaction and diffsion ( D( r) c) R( c, r) t c = + Averaging, coarse-graining: D e,r e ( D( r) c) R( c, r) t c = + Homogenization ( D c ) R ( c ) t c = e + e Goal: Analytical expressions of effective diffsion constants and reaction rates D 2,R 2 D 1,R 1

17 Binary media: two phases with fractions φ, 1-φ Diffsion and reaction of both phases: D 1, R 1 and D 2, R 2 Example: Random Binary Medim ( ) ( ) ( ) ( ) ( ) φ φ φ φ + = = c R c R c R D d D D D D d D D D e e e e e ) ( 1 1) ( 0 Diffsion Reaction Validation by simlation of heterogeneos media (S. Alonso, R. Kapral, M. Bär, Phys. Rev. Lett. 2009, J. Chem. Phys. 2009, D. Brggeman, Ann. Phys. 1935)

18 IV. Simlating Flid Dynamics Goal: Spplement/ replace experiments Qantitative modelling of reference experiments? Application: Explosion protection, flow measrements RANS Experiment: free jet non-stationary simlation (helim in air) (CFX software package)

19 Navier-Stokes Eqation & Averaging 0 div grad grad) ( = + Δ = + + f p t ν (URANS), ), ( 1 ) ( (RANS) ), ( 1 lim 2 / 2 / 0 +Δ Δ Δ = = t t t t T T d T t dt t T τ τ x U x U velocity field Averaging of velocity fields, e. g. : + closre relations

20 Comparison of Trblence Models Averaged Navier-Stokes eqations Degree of averaging G. Lindner, D. Marks, R. Model, ECCOMAS Proc. (2010).

21 Comparison Velocity Profiles G. Lindner, D. Marks, R. Model, ECCOMAS Proc. (2010).

22 Comparison Experiment Simlation Expt. Siml. propagation velocity o. k., bt profile widths disagree G. Lindner, D. Marks, R. Model, ECCOMAS Proc. (2010).

23 Smmary: Mltiscale Models Homogenization and averaging necessary to redce comptational costs Simlation of detailed models for validation of averaged models Applications: - Flow: hydrodynamics & heat transfer - Explosion protection: hydrodynamics & chemistry - Electromagnetic fields

24 Conclsion PDEs as qantitative models in metrology Inverse problems & PDEs Effective PDE descriptions in mltiscale systems Uncertainty in PDE models: Monte-Carlo methods, statistical approach

25 Qestions? Acknowledgement Inverse Problems: H. Gross, M. A. Henn Mltiscale Modelling: S. Alonso, G. Lindner, R. Model Fnding: BMBF, DFG