New results concerning the construction and application of adaptive numerical grids for solving applied problems are presented. The grid generation technique is based on the numerical solu tion of inverted Beltrami and diffusion equations for a monitor metric. The capabilities of the spherical metric tensor as applied to adaptive grid generation are examined in detail. Adaptive hexahedral grids are used to numerically solve a boundary value problem for the three dimensional heat equation with a moving boundary in a continuous medium with discontinuous thermophysical parameters; this problem models the interaction of a thermal wave with a thermocouple embedded in the solid.

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