Peculiarities of inverse boundary-value
problems
Russian Scientific
Schools
history
and achievements
N.B.Ilyinsky
Kazan State University
Chebotarev Institute of Mathematics and Mechanics
Russia
History and development of the theory of inverse boundary-value problems is presented.š Inherently different nature of these problems as compared with many other inverse ill-posed and free-boundary problems is elucidated. The pioneering role of G.G.Tumashev and M.T.Nugin in the creation and explication of the theory if inverse boundary-value problems for analytic functions and applications in fluid mechanics is elaborated. The achievements of theš theory proponents and followers are overviewed based on published monographs and review papers. The results of the last decade in the field of inverse boundary-problems in aerohydrodynamics are discussed.
Retrospectively, the history of origin and development of the theory of
inverse boundary-value problems is undoubtedly evincing that this theory for
analytic functions, as a branch of mathematics, was created in Kazan State
University (KSU). The founders of the theory, who contributed tremendously in
its success, were G.G.Tumashev and M.T.Nughin. Later, their numerous students
and disciples in KSU and other universities and scientific institutes of Russia
achieved significant progress both in the theory and applications.
We note
that in the last half of the 20-th century several inverse problems were
stated, which can be divided into two classes, inverse ill-posed problems
(IIPP) and inverse boundary-value problems (IBVP).
Along with
theoretical studies of IBVPA practical application were pursued. Hydrodynamic
analysis of the intra-blade zone of a vane šëò 050.01.0 of a submersible pump
at different pumping regimes was done. By the help of IBVPAš hydrofoil arrays on axisymmetric streamsurfaces
were studied, design of modified blade shapes was conducted.
A problem of design of a zero-moment airfoil in
an unseparated flow at a given attack angle is stated and solved based on a
give velocity distribution along the contour. Zero-moment airfoils are those
which posses zero moment with a zero lift. Numeric-analytic method of calculation
of these airfoils in terms of the model of ideal fluids is based on IBVPA.
Unseparated regimes are ensuredš for
zero-moment airfoil by specifying hydrodynamically reasonable velocity distributions.
Examples of design of these airfoils are presented.
The results above
confirm that methods have been developed to solve very diverse IBVPA. Explicit
analytical solutions possible in many case are the advantage of the method.
Despite their apparent simplicity, considerable problems arise when the methods
are implemented numerically.
The first group of problems is related to the
method of isolation of singularities in analytical solutions and determination
of solvability conditions in the form tractable in numerical codes and applications
of the method of quasisolutions as a tool in satisfying theš conditions. The algorithms of numerical
computations based on analytical solutions were described and all the listed
problems were illustrated for different IBVPA. The second group of problems in
numerical codes is essential in those IBVPA when compressibility and viscosity
of the fluid are important and whenš
solvability conditions require iteration procedures. Finally, the third
group of problems is relevant to numerical solutions of IBVPA for hydrofoil cascades,
especially ones with a small period or two-array cascades set on axisymmetric
streamsurfaces. Analytic tricks and numerical techniques obviating the problems
were elaborated.
Thus the
following results in IBVPA can be recapped
ž
A
class of boundary value problems in aerohydrodynamics has been stated and
solved: problems of design and calculation of airfoils inš a plane unseparated flow of an ideal fluid
as well for a compressible Chaplygin-type gas at subsonic regimes and with viscosity
considered within the scope of the BL theory; the class is unified by the
object of investigation (airfoils and hydrofoil cascades), by the methods
(direct, inverse and mixed boundary-value problems for analytic functions) and
the objective (improvement of aerodynamic performance of airfoils including
their optimization by various parameters).
ž
The
advantages of numerical-analytical design of hydrodynamic objects by the methodology
of IBVPA is shown in comparisons with methods requiring multiple numerical
solution of direct problems
ž
A
number of IBVPA for contours with devices controlling the external flow is
studied. A new class of optimization problems is solved, variational IBVPA
ž
Novel
mathematical and aerodynamic problems are formulated
ž
Practical
applications of IBVPA are continued.
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