Институт вычислительной математики
и математической геофизики

The International Conference on Computational Mathematics


Numerical solution of differential and integral equations

A difference method for solving initial boundary value problems for the differential equation of crack pattern formation

Voevodin A.F., Katyshev V.

Lavrentyev Institute of Hydrodynamics of SB RAS (Novosibirsk),
Novosibirsk State University (Novosibirsk)

We propose a computational method for realizing a mathematical model of the formation of crack patterns in solids subjected to volumetric cooling by convection. The model assumes that cracks are uniformly distributed and that pieces of the solid separated by cracks have square shapes. Crack widths and the sizes of solid pieces are influenced by the cooling effect, which is provided by a fluid or air flow along the cracks resulting from pressure differences. We assume that crack widths are much smaller than the size of the solid. Under our assumptions, the mathematical model of crack growth reduces to an initial boundary value problem for the heat equation with a nonlocal, and moreover, a nonlinear boundary condition. Problems of this kind fall outside of the scope of the classical theory of partial differential equations, thus in this paper we investigate the question of existence of solutions, proving a theorem of existence and uniqueness of a weak solution. To approximate the heat equation, we use the Crank--Nicholson difference method, which is of the second order of approximation by time and space. Approximating also the boundary conditions to the second order by time and space, we obtain a system of nonlinear difference equations, which is solved by iterations. To estimate the efficiency of the proposed method, we have run test computations and compared their results with those of the paper A.~Bejan, Y.~Ikegami, G.~A.~Ledezma, {it Constructal theory of natural crack pattern formation for fastest cooling,} Int. J. Heat Mass Transfer, 41, no.~13 (1998), 1945-1954".

Note. Abstracts are published in author's edition

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