Adjoint Sensitivity Analysis for Shallow-Water Wave Control
Publication: Journal of Engineering Mechanics
Volume 126, Issue 9
Abstract
An adjoint sensitivity method based on the shallow-water equations is developed for water wave control in river and estuarine systems. The method is used to compute the gradient of a user-defined objective function in the N-dimensional parameter space consisting of system control settings with just one solution of the basic problem and one solution of the associated adjoint problem. Characteristic equations are derived for the adjoint problem and a new formalism is proposed for the sensitivity of shallow-water flow to boundary changes in depth and discharge. New adjoint boundary conditions are developed for river and estuarine forecasting models with open-water inflow and outflow sections. This gives rise to new expressions for sensitivities at these sections. Characteristic analysis of the adjoint and basic problems shows that sensitivities propagate in the reverse time direction along the characteristic paths of the basic problem. The Riemann variables of the adjoint problem are shown to precisely describe the sensitivity of the objective function to changes in depth and discharge at system boundaries. The method is extended to two space dimensions by bicharacteristic analysis.
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References
1.
Abbott, M. B. (1997). “Range of tidal flow modeling.”J. Hydr. Engrg., ASCE, 123(4), 257–277.
2.
Cacuci, D. G. (1981a). “Sensitivity theory for nonlinear systems. I. Nonlinear functional analysis approach.” J. Math Phys., 22(12), 2794–2802.
3.
Cacuci, D. G. (1981b). “Sensitivity theory for nonlinear systems. II. Extensions to additional classes of responses.” J. Math. Phys., 22(12), 2803–2812.
4.
Chaudhry, M. H. (1993). Open-channel flow, Prentice-Hall, Englewood Cliffs, N.J.
5.
Cheng, R. T., Casulli, V., and Gartner, J. W. (1993). “Tidal, residual, intertidal mudflat (TRIM) model and its application to San Francisco Bay, California.” Estuarine, Coast. and Shelf Sci., 36, 235–280.
6.
Chertok, D. L., and Lardner, R. W. (1996). “Variational data assimilation for a nonlinear hydraulic model.” Appl. Math. Modeling, 20, 675– 682.
7.
Das, S. K., and Lardner, R. W. (1991). “On the estimation of parameters of hydraulic models by assimilation of periodic tidal data.” J Geophys. Res., 96(C8), 15187–15196.
8.
Das, S. K., and Lardner, R. W. (1993). “Variational parameter estimation for a two-dimensional numerical tidal model.” Int. J. Numer. Methods in Fluids, 15, 313–327.
9.
Engquist, B., and Majda, A. (1977). “Absorbing boundary conditions for the numerical simulation of waves.” Math. Comp., 31, 629–651.
10.
Hall, M. C. G., and Cacuci, D. G. (1983). “Physical interpretation of the adjoint functions for sensitivity analysis of atmospheric models.” J. Atmospheric Sci., 40, 2537–2546.
11.
Hall, M. C. G., Cacuci, D. G., and Schlesinger, M. E. (1982). “Sensitivity analysis of a radiative-convective model by the adjoint method.” J. Atmospheric Sci., 39, 2038–2050.
12.
Katopodes, N. D., and Piasecki, M. (1996). “Site and size optimization of contaminant sources in surface water systems.”J. Envir. Engrg., ASCE, 122(10), 917–923.
13.
Katopodes, N. D., and Strelkoff, T. (1979). “Two-dimensional shallow water-wave models.”J. Engrg. Mech. Div., ASCE, 105(2), 317–334.
14.
Lardner, R. W. (1993). “Optimal control of open boundary conditions for a numerical tidal model.” Comp. Methods in Appl. Mech. and Engrg., 102, 367–387.
15.
Lardner, R. W., Al-Rabeh, A. H., and Gunay, N. (1993). “Optimal estimation of parameters for a two-dimensional hydrodynamical model of the Arabian gulf.” J. Geophys. Res., 98(C10), 18229–18224.
16.
Le Dimet, F.-X., and Talagrand, O. (1986). “Variational algorithms for analysis and assimilation of meteorological observations: Theoretical aspects.” Tellus, 38A, 97–110.
17.
Lewis, J. M., and Derber, J. C. (1985). “The use of adjoint equations to solve a variational adjustment problem with advective constraints.” Tellus, 37A, 309–322.
18.
Liggett, J. A., and Chen, L.-C. (1994). “Inverse transient analysis in pipe networks.”J. Hydr. Engrg., ASCE, 120(8), 934–955.
19.
Marchuk, G. I. (1995). Adjoint equations and analysis of complex systems, Kluwer Academic, Boston.
20.
Neuman, S. P. (1980). “A statistical approach to the inverse problem of aquifer hydrology. 3. Improved solution method and added perspective.” Water Resour. Res., 16(2), 331–346.
21.
Neuman, S. P., and Carrera, J. (1985). “Maximum-likelihood adjoint-state finite-element estimation of groundwater parameters under steady- and nonsteady-state conditions.” Appl. Mathematics and Computation, 17, 405–432.
22.
Ounsy, A., Brun, B., and de Credy, F. (1994). “The adjoint sensitivity method, a contribution to the code uncertainty evaluation.” Nuclear Engrg. and Des., 149, 357–364.
23.
Panchang, V. G., and O'Brien, J. J. ( 1989). “On the determination of hydraulic model parameters using the adjoint state formulation.” Modeling marine systems I, A. M. Davies, ed., CRC, Boca Raton, Fla., 5–18.
24.
Panchang, V. G., and Richardson, J. E. (1993). “Inverse adjoint estimation of eddy viscosity for coastal flow models.”J. Hydr. Engrg., ASCE, 119(4), 506–524.
25.
Piasecki, M. ( 1994). “Optimal control of hazardous waste disposal in rivers and estuaries using the adjoint equation solution.” PhD thesis, Dept. of Civ. and Envir. Engrg., University of Michigan, Ann Arbor, Mich.
26.
Piasecki, M., and Katopodes, N. D. (1997a). “Control of contaminant releases in rivers. I: Adjoint sensitivity analysis.”J. Hydr. Engrg., ASCE, 123(6), 486–492.
27.
Piasecki, M., and Katopodes, N. D. (1997b). “Control of contaminant releases in rivers and estuaries. II: Optimal design.”J. Hydr. Engrg., ASCE, 123(6), 493–503.
28.
Press, W. H., Flannery, B. P., Teukolsky, S. A., and Vetterling, W. T. (1990). Numerical recipes, Cambridge University Press, London.
29.
Sanders, B. F. ( 1997). “Control of shallow-water flow using the adjoint sensitivity method.” PhD thesis, Dept. of Civ. and Envir. Engrg., University of Michigan, Ann Arbor, Mich.
30.
Sanders, B. F., and Katopodes, N. D. (1998). “Control of multi-dimensional wave motion in shallow-water.” Proc., 5th Int. Conf. on Estuarine and Coast. Modeling, M. L. Spaulding, ed., 267–278.
31.
Sanders, B. F., and Katopodes, N. D. (1999a). “Active flood hazard mitigation. II: Omnidirectional wave control.”J. Hydr. Engrg., ASCE, 125(10), 1071–1083.
32.
Sanders, B. F., and Katopodes, N. D. (1999b). “Control of canal flow by adjoint sensitivity method.”J. Irrig. and Drain. Engrg., ASCE, 125(5), 287–297.
33.
Steinebach, G. ( 1998). “Using hydrodynamic models in forecast systems for large rivers.” Advances in hydroscience and engineering, K. P. Holz, W. Bechteler, S. S. Y. Wang, and M. Kawahara, eds., Vol. 3.
34.
Sykes, J. F., Wilson, J. L., and Andrews, R. W. (1985). “Sensitivity analysis for steady state groundwater flow using adjoint operators.” Water Resour. Res., 21(3), 359–371.
35.
Thacker, W. C., and Long, R. B. (1988). “Fitting dynamics to data.” J. Geophys. Res., 93(C2), 1227–1240.
36.
Verboom, G. K., Stelling, G. S., and Officier, M. J. ( 1982). “Boundary conditions for the shallow-water equations.” Engineering applications of computational hydraulics, M. B. Abbott and J. A. Cunge, eds., Vol. 1, Pitman Advanced Publishing Program.
37.
Wylie, E. B. (1969). “Control of transient free-surface flow.”J. Hydr. Div., ASCE, 95(1), 347–361.
38.
Yeh, W. W.-G. (1986). “Review of parameter identification procedures in groundwater hydrology: The inverse problem.” Water Resour. Res., 22(2), 95–108.
39.
Yeh, W. W.-G., and Sun, N.-Z. (1990). “Variational sensitivity analysis, data requirements, and parameter identification in a leaky aquifer system.” Water Resour. Res., 26(9), 1927–1938.
Information & Authors
Information
Published In
Journal of Engineering Mechanics
Volume 126 • Issue 9 • September 2000
Pages: 909 - 919
History
Received: Jun 9, 1999
Published online: Sep 1, 2000
Published in print: Sep 2000
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