Optimal shapes in mechanics of Darcian flows

Anvar R.Kacimov

Department of Soil and Water Sciences, Sultan Qaboos University,

Al-Khod 123, PO Box 32, Sultanate of Oman

anvar@squ.edu.om

Overview of analytical solutions of optimal shape design problems for saturated Darcian flows in homogeneous porous media is presented. The statements of problems and model description are given for seepage from an unlined soil channel and under a concrete dam. Optimization of other hydrotechnical constructions is briefly described.

Queen Dido of Carthage solved the first isoperimetric problems with an area of a plain figure as a criterion and the figure perimeter as a constraint, without any physical field in the enclosed area. Rooted in the Dido trick, optimal shape design (OSD) problems are now understood as control of a partial differential equation, which describes a physical process within a domain, by variation of the domain boundary. Optimal shapes have been found in aerodynamics, mechanics of elastic bodies, heat transfer theory, to mention only few applications, where the continuum approach is applicable. As is well known, in a domain with fixed or free boundaries certain functionals (e.g. energy) attain a maximum or minimum, which is presumed to reflect the optimality of Creator as Leibniz coined in . The realization of wishful fantasies by OSD is a pinnacle of the Promethean paradigm. Indeed, if in a humble pursuit the hidden optimality of Einstein's "rafiniert Herrgott" is revealed by a scientist, a constructive approach is creative, the object is designed and manufactured in congruity with criteria and constraints, which are set by an engineer, albeit the state equation remaining as an often ill-favoured restriction of Nature. Consequently, the criteria and constraints may become exorbitant, i.e. the optimum is not realizable within the scope of the posited model. Tralatitiosly, OSD - if set in the framework of boundary-value problems, which solutions are sought by available mathematical routines- culminates as an esemplastic link between prediction of the response of a fixed object placed into a given physical field and control of the field by handling the boundary conditions, including the shape.

In this paper we overview recent OSD results in subsurface mechanics of steady, 2-D, Darcian seepage of a single-phase fluid in a homogeneous, isotropic, porous medium. The assumptions on the fluid and skeleton bring about the Laplace equation for the hydraulic head, which serves as a state variable. It makes seepage similar to other branches in mechanics of ideal fluids, in particular, it allows one to implement the theory of analytic functions.