×

An optimal design problem for submerged bodies. (English) Zbl 0622.76022

The problem of finding the shape of a smooth body submerged in a fluid of finite depth which minimizes added mass or damping is considered. The optimal configuration is sought in a suitably constrained class so as to be physically meaningful and for which the mathematical problem of a submerged body with linearized free surface condition is uniquely solvable. The problem is formulated as a constrained optimization problem whose cost functional (e.g. added mass) is a domain functional. Continuity of the solution of the boundary value problem with respect to variations of the boundary is established in an appropriate function space setting and this is used to establish existence of an optimal solution. A variational inequality is derived for the optimal shape and it is shown how finite dimensional approximate solutions may be found.

MSC:

76B99 Incompressible inviscid fluids
49K20 Optimality conditions for problems involving partial differential equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Angell, The three dimensional inverse scattering problem for acoustic waves, J. Diff. Equations 46 pp 46– (1982) · Zbl 0496.65069 · doi:10.1016/0022-0396(82)90108-5
[2] Begis, Application de la méthode des éléments finis à l’approximation d’un domaine optimal, Appl. Math. and Opt. 2 pp 130– (1975) · Zbl 0323.90063 · doi:10.1007/BF01447854
[3] Bergman, Kernel Functions and Elliptic Differential Equations in Mathematical Physics (1953) · Zbl 0053.39003
[4] Cea, Optimization: théorie et algorithms (1971)
[5] Cea, Lecture Notes in Computer Science 3 pp 92– (1975)
[6] Cea, Lecture Notes in Computer Science 11 (1974)
[7] Chenais, On the existence of a solution in a domain identification problem, J. Math. Anal. Appl. 52 pp 189– (1975) · Zbl 0317.49005 · doi:10.1016/0022-247X(75)90091-8
[8] Colton, Integral Equation Methods in Scattering Theory (1983) · Zbl 0522.35001
[9] Garabedian, Partial Differential Equations (1964)
[10] Günter, Potential Theory and its Applications to Basic Problems of Mathematical Physics (1967)
[11] Hadamard, Memoire sur le problème d’analyse relatif à l’equilibre des plaques élastiques encastrées, Memoires présentés par divers savants à l’Académie des Sciences 33 (1908)
[12] Hulme, Some applications of Maz’ja’s uniqueness theorem to a class of linear water wave problems, Math. Proc. Camb. Phil. Soc. 95 pp 165– (1984) · Zbl 0546.76031 · doi:10.1017/S0305004100061417
[13] John, On the motion of floating bodies II, Comm. Proc. Appl. Math. 3 pp 45– (1950) · doi:10.1002/cpa.3160030106
[14] Kirsch , A. A numerical method for an inverse scattering problem 1982
[15] Kleinman , R. E. On the mathematical theory of the motion of floating bodies - an update.
[16] Kleinman, Boundary integral equations for the three dimensional Helmholtz equation, SIAM Review 16 pp 214– (1974) · Zbl 0253.35023 · doi:10.1137/1016029
[17] Lions , J. L. Some Aspects of the Optimal Control of Distributed Parameter Systems. 1972 · Zbl 0275.49001
[18] Marrocco, Optimum design with Lagrangian finite elements, Comp. Meth. App. Mech. Eng. 15 pp 277– (1978) · doi:10.1016/0045-7825(78)90045-2
[19] Maz’ja, Solvability of the problem on the oscillations of a fluid containing a submerged body, J. Soviet Math. 10 pp 86– (1978) · doi:10.1007/BF01109726
[20] Miranda, Partial Differential Equatioans of Elliptic Type (1970)
[21] Oden, Theory of variational inequalities with applications to problems of flow through porous media. Intl, J. Eng. Sci. 18 pp 1173– (1980) · Zbl 0444.76069 · doi:10.1016/0020-7225(80)90111-1
[22] Pironneau, On optimum profiles in Stokes flow, J. Fluid Mech. 59 pp 117– (1973) · Zbl 0274.76022 · doi:10.1017/S002211207300145X
[23] Pironneau, On optimal design in fluid mechanics, J. Fluid Mech. 64 pp 97– (1974) · Zbl 0281.76020 · doi:10.1017/S0022112074002023
[24] Pironneau, Springer Series in Computational Physics (1984)
[25] Ursell , F. Newcastle upon Tyne, England 1984
[26] Wehausen, Handbuch der Physik IX pp 446– (1960)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.