Empirical versus Direct sensitivity computation

Transcription

1 Empirical versus Direct sensitivity computation Application to the shallow water equations C. Delenne P. Finaud Guyot V. Guinot B. Cappelaere HydroSciences Montpellier UMR 5569 CNRS IRD UM1 UM2 Simhydro 2010, 2 4 June 2010, Sophia Antipolis

2 2D Shallow Water Equations (SWEs) 2 U t + F x x + F y y = S U = h q S = 0 gh (S 0,x S f,x ) r gh (S 0,y S f,y ) F x = q q 2 /h + gh 2 /2 F y = r qr/h qr/h r 2 /h + gh 2 /2

3 Empirical sensitivity computation 3 Compute the solution for a given value of a parameter ψ: ψ 0 Compute the solution with a small variation dψ of the parameter ψ U(ψ 0 ) U(ψ 0 +dψ) U(ψ 0 +dψ) U(ψ 0 ) dψ Compute the difference between the two solutions divided by the parameter variation

4 Empirical sensitivity computation 3 Compute the solution for a given value of a parameter ψ: ψ 0 Compute the solution with a small variation dψ of the parameter ψ U(ψ 0 ) U(ψ 0 +dψ) U(ψ 0 +dψ) U(ψ 0 ) NB: dψ = εδψ with ε(x, y, t) = 1 where ψ is modified dψ Compute the difference between the two solutions divided by the parameter variation

5 Direct sensitivity computation 4 The perturbation dψ yields a variation du introduce dψ and du in the Shallow Water equations: (U +du) t + F x (U +du, ψ +dψ) x + F y (U +du, ψ +dψ) y perform a first order series expansion... = S (U +du, ψ +dψ)

6 2D sensitivity equations 5 s t + G x x + G y y = Q s U ψ = h/ ψ q/ ψ r/ ψ = η θ ρ G x = F x U s, G y = F y U s Q = S U s + S ψ ε x ( ) Fx ψ ε y ( ) Fy ψ ε Guinot et al. Adv. in Water Resources (31), 2009

7 Agenda Empirical vs Direct approaches 6 3 examples of sensitivity analysis (SA): 1D steady flow on a sloping bed (backwater curve) 1D transient flow with discontinuous solution (dambreak) 2D flow in a roundabout

8 1D Steady flow on a sloping bed 7 SA with respect to the friction coefficient n Reference solution: The backwater curve and its derivative with respect to n (discretization using Explicit Euler method) dh dx = S 0 S f 1 Fr 2 dη dx = η h ( S0 S f 1 Fr 2 ) + ε nm n M ( S0 S f 1 Fr 2 )!'()*!"#&!!"+, -./ 0 1# 2!$"!"#$% +,-,.,/0,!#" %&'&(&)*&!"#$ 1!()*!" 3!+,2 Steady state => q=constant and θ=0

9 1D Steady flow on a sloping bed SA with respect to the friction coefficient n 8 Numerical flow solution Resolution of the 1d SWEs with HLLC Riemann solver Transient simulation until the steady state is reached

10 1D Steady flow on a sloping bed SA with respect to the friction coefficient n 8 Numerical flow solution Resolution of the 1d SWEs with HLLC Riemann solver Transient simulation until the steady state is reached!'()*!"#&!"#$% +,-,.,/0, 12),.3045!2! !%#""%!%!"#$$$ &'(')'*+',-.')/+01!"#$ 6!()* 8!3.7 Incorrect estimation of q due to the balancing process in the HLLC (flux at interfaces and values at cells)

11 1D Steady flow on a sloping bed SA with respect to the friction coefficient n 9 Sensitivity solution:!!"%& '() * +#,!$"!#"./0/1/23/ 4& !"!$" -!%&,!#" %&'&(&)*& +,(&*-!".!/01 Equivalent results for η Delenne et al. Num. Meth. in Fluids, in revision

12 1D Steady flow on a sloping bed SA with respect to the friction coefficient n 9 Sensitivity solution:!!"%& '() * +#,!$"!#"./0/1/23/ 4& ! (m 7/3 ) %"#$ %" &'(')'*+',-)'+. /01-)-+23!"!$" -!%&,!"#$ 4%506!#"!" %&'&(&)*& +,(&*-.!/01 Better results for the Direct approach (θ=0 except downstream) Equivalent results for η Delenne et al. Num. Meth. in Fluids, in revision

13 1D Discontinuous flow: Dambreak SA with respect to the initial water depth in the dam h up 10 Initial condition problem q(x, 0) = 0 h(x, 0) = { h up x<x 0 h ds x x 0 Flow solution: Water elevation z Unit discharge q #%" #%$ &#'() h up +,-./ ( ( '!() $ *+,!&"!%" #"!$" #$!" * $!" *#'()!#" -./01234/0 56)7834/0 9 " 9!(),

14 1D Discontinuous flow: Dambreak 11 SA with respect to the initial water depth in the dam h up Empirical method: compute the hydraulic solution with 2 slightly different values of h up sensitivity of h with respect to h up!!%3'!# ()*+,-./*+ 0&1.2./*+!# sensitivity of q with respect to h up!!$%&'(!" $ " $!%&'!" ) " )!$%(

15 1D Discontinuous flow: Dambreak 11 SA with respect to the initial water depth in the dam h up Empirical method: compute the hydraulic solution with 2 slightly different values of h up sensitivity of h with respect to h up!!%3'!# ()*+,-./*+ 0&1.2./*+!# sensitivity of q with respect to h up!!$%&'(!" $ " $!%&'!" ) " )!$%( Problem: the shock velocity depends on h up

16 1D Discontinuous flow 12 SA with respect to the initial water depth in the dam h up Empirical method: compute the hydraulic solution with 2 slightly different values of h up h h up h up +dh up h dh up 1

17 1D Discontinuous flow 12 SA with respect to the initial water depth in the dam h up Empirical method: compute the hydraulic solution with 2 slightly different values of h up h h up h up +dh up Problem across the shock: Δh remains high when dh up tends to 0 h dh up 1

18 1D Discontinuous flow 13 SA with respect to the initial water depth in the dam h up Direct approach: Sensitivity equations defined for continuous solutions Modification: extra source term R applied only in case of shock s t + G x = Q + Rδ s In case of a Riemann problem, R can be simplified into R = c s ψ (U L U R ) U L, U R : left and right states of the Riemann problem c s : shock speed

19 1D Discontinuous flow 14 SA with respect to the initial water depth in the dam h up Direct approach: Solve the sensitivity equations (including R) with the following initial conditions: η(x, 0) = θ(x, 0) = 0 { 1 x<x 0 0 x x 0!!032!!%&()'!# $%&'()*+&',*-.+)!#!" / " /!012!" $ " $!%&' Delenne et al. CR Mécanique, 2008

20 2D flow in a roundabout SA with respect to the boundary condition in street C: h C 15 q up q up A B D h D C h C 0m 10m

21 2D flow in a roundabout 16 SA with respect to the boundary condition in street C: h C Flow solution: Solve the 2D SWEs with the HLLC Riemann solver water depth h discharge (q,r)

22 2D flow in a roundabout 16 SA with respect to the boundary condition in street C: h C Flow solution: Solve the 2D SWEs with the HLLC Riemann solver water depth h discharge (q,r)

23 2D flow in a roundabout SA with respect to the boundary condition in street C: h C 17 Sensitivity η of h with respect to hc Empirical Direct Guinot et al. Adv. in Water Resources (32), 2009

24 2D flow in a roundabout SA with respect to the boundary condition in street C: h C 18 Sensitivity θ of q with respect to hc Empirical Direct Guinot et al. Adv. in Water Resources (31), 2009

25 2D flow in a roundabout SA with respect to the boundary condition in street C: h C 18 Sensitivity θ of q with respect to hc Empirical Direct Guinot et al. Adv. in Water Resources (31), 2009

26 2D flow in a roundabout SA with respect to the boundary condition in street C: h C 18 Sensitivity θ of q with respect to hc Empirical Direct Guinot et al. Adv. in Water Resources (31), 2009

27 Conclusion 19 Empirical Applicable even if the model equations are not known vs Direct Requires the model equations and their derivative / ψ Two simulations One simulation Problem with discontinuous flows (shock, jump...) Possible discontinuous flows Propagates the artifacts of flow solution to the sensitivity Eliminates those artifacts

28 Conclusion 19 Empirical Applicable even if the model equations are not known vs Direct Requires the model equations and their derivative / ψ Two simulations One simulation Problem with discontinuous flows (shock, jump...) Possible discontinuous flows Propagates the artifacts of flow solution to the sensitivity Eliminates those artifacts

29 Conclusion 19 Empirical Applicable even if the model equations are not known vs Direct Requires the model equations and their derivative / ψ Two simulations One simulation Problem with discontinuous flows (shock, jump...) Possible discontinuous flows Propagates the artifacts of flow solution to the sensitivity Eliminates those artifacts

30 Conclusion 19 Empirical Applicable even if the model equations are not known vs Direct Requires the model equations and their derivative / ψ Two simulations One simulation Problem with discontinuous flows (shock, jump...) Possible discontinuous flows Propagates the artifacts of flow solution to the sensitivity Eliminates those artifacts

31 Conclusion 19 Empirical Applicable even if the model equations are not known vs Direct Requires the model equations and their derivative / ψ Two simulations One simulation Problem with discontinuous flows (shock, jump...) Possible discontinuous flows Propagates the artifacts of flow solution to the sensitivity Eliminates those artifacts

32 Empirical versus Direct sensitivity computation Application to the shallow water equations C. Delenne P. Finaud Guyot V. Guinot B. Cappelaere HydroSciences Montpellier UMR 5569 CNRS IRD UM1 UM2 Simhydro 2010, 2 4 June 2010, Sophia Antipolis