# Институт вычислительной математики

и математической геофизики

# The International Conference on Computational Mathematics

ICCM-2004

June, 21-25, 2004
Novosibirsk, Academgorodok

## Abstracts

*Numerical solution of differential and integral equations*

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

**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*

© 1996-2000, Siberian Branch of Russian Academy of Sciences, Novosibirsk

Last update: 06-Jul-2012 (11:52:06)