Numerical systems analysis of multicomponent distributed systems
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SYSTEMS ANALYSIS NUMERICAL SYSTEMS ANALYSIS OF MULTICOMPONENT DISTRIBUTED SYSTEMS I. V. Sergienkoa† and V. S. Deinekaa‡
UDC 519.6
Abstract. The paper presents the results in the construction of high-accuracy computational algorithms for the classes of partial derivative problems with discontinuous solutions, including ill-posed and eigenvalue problems. The optimal control in complex distributed systems is investigated. On the basis of the optimal control theory, explicit expressions are obtained for gradients of residual functionals to identify different parameters of multicomponent distributed systems. The possibility of using pseudoinverse matrices to solve some linear inverse problems in a finite number of arithmetic operations is considered. Keywords: multicomponent distributed systems, numerical modeling, optimal control, parameter identification. INTRODUCTION The recent achievements in calculus mathematics, programming, theory of optimal control of various distributed systems, theory of numerical processing of complex geometric objects, as well as available supercomputers allow creating powerful information technologies to analyze the state and to predict the behavior of complex objects in power and mechanical engineering and to solve problems in extending their lifetime, medium conservancy, etc. The structure of various complex objects is multicomponent, their components differ in the mechanical, physical, and other characteristics. Multicomponent bodies often contain technological or natural interlayers, inclusions whose characteristics considerably differ from the corresponding characteristics of other components and may substantially influence the strain of the whole body, formation of its temperature field, fluid motion, migration of toxic chemical elements, etc. NUMERICAL MODELING OF PROCESSES IN MULTICOMPONENT MEDIA The developments at the V. M. Glushkov Institute of Cybernetics of the NAS of Ukraine and other organizations suggest that to take into account the influence of the above interlayers/cracks on, for example, the thermostressed state, the fluid motion both along the whole body under study and near the interlayers, so-called conjugation (interface) conditions (i.e., differential equations with respect to the mid-surfaces of the inclusions) can be used. A feature of such mathematical models is new classes of boundary-value, initial–boundary-value partial differential problems with discontinuous solutions on specified arbitrarily located surfaces. To determine the natural frequencies and shapes of such bodies, spectral problems for second- and fourth-order operators with discontinuous eigenfunctions are obtained. For classical generalized problems in weak formulations, a construction technique is developed; for second- and fourth-order elliptic problems (rod systems and compound thin plates), the corresponding energy functionals defined on classes of discontinuous functions are obtained; and the existence and uniqueness theorems for generalized discontinuous solutions are proved [1–5]. a
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