Uniqueness of a Solution to Problems of Identification of Mathematical Heat-Transfer Models with Lumped Parameters
- PDF / 1,423,906 Bytes
- 20 Pages / 594 x 792 pts Page_size
- 35 Downloads / 184 Views
Journal of Engineering Physics and Thermophysics, Vol. 93, No. 4, July, 2020
HEAT CONDUCTION AND HEAT TRANSFER IN TECHNOLOGICAL PROCESSES UNIQUENESS OF A SOLUTION TO PROBLEMS OF IDENTIFICATION OF MATHEMATICAL HEAT-TRANSFER MODELS WITH LUMPED PARAMETERS A. G. Vikulov and A. V. Nenarokomov
UDC 51.74, 536.37
An analysis has been made of the problem of identification of mathematical heat-transfer models with lumped parameters that have a nonunique solution. The authors have investigated the possibility of restoring conditionally the uniqueness of a solution to such a problem with supplementary conditions for the selection of solutions in the form of linearly independent time functions of temperature that can be obtained in one experiment with relevant designing, Consideration has been given to the applicability of iteration-regularization methods with a vector regularization parameter to identifying a full set of functions of a thermal-state matrix of mathematical models with lumped parameters. On the basis of computational experiments, the capabilities of a variational principle of selection of solutions for conditionally correct identification problems have been demonstrated. Keywords: mathematical models, inverse problems, identification, uniqueness and stability of a solution, regularization, variational method, heat transfer. Introduction. Inverse heat-transfer problems solvable within the framework of mathematical physics, and also retrospective problems of finding the temperature in inverse time and problems of optimum planning and control over thermal regimes, can be ill-posed mathematically if they do not satisfy the requirements of uniqueness and stability of a solution. In identifying thermal processes in structural materials and elements, the problems of optimum designing of nonstationary thermophysical experiments are represented in the form of the operator equation [1] La u = f ,
(1)
where La is the nonlinear operator constructed from the analyzed heat-transfer model, u is the vector of unknown characteristics, and f = ( f i (τ))1N of the vector function of measurements. The model of state and hence the operator La depend on a number of quantities determining experimental conditions, which are combined into a vector ω. Furthermore, the operator La also depends on the measuring scheme of temperature ξ = (N, X), where X = ( X i )1N is the vector of the coordinates of arrangement of temperature-sensitive elements. The vectors ω and ξ together make up the design of an experiment [1] π = (ω, ξ) .
(2)
Optimum designing of an experiment consists in selecting such elements of the design (2) at which the maximum accuracy in determining the characteristics of the process under study is ensured. Searching for the optimum experimental design leads to a solution to the extremum problem [1] π 0 = Arg max Ψ (π ) , π∈Π
(3)
where Ψ(π) of the experiment′s quality criterion characterizing the exactness of the solution to the inverse problem and Π is the set of possible designs. In solving inverse problems on determ
Data Loading...
