### BILAYER DIFFERENCE SCHEME OF A NUMERICAL SOLUTION TO TWO-DIMENSIONAL DYNAMIC PROBLEMS OF ELASTICITY

Pages 104 - 111

Numerical modeling of dynamic problems of the theory of elasticity remains a relevant task.

A complex network of waves that propagate within solid bodies, including longitudinal, transverse,

conical and surface Rayleigh waves, etc., prevents the separation of wave fronts for modeling purposes.

Therefore, it is required to apply the so-called "pass-through analysis".

The method applied to resolve dynamic problems of the two-dimensional theory of elasticity

employs finite elements to approximate computational domains of complex shapes, whereby the

software calculates the speed and voltage in the medium at each step. Preset boundary conditions

are satisfied precisely.

The resulting method is classified as explicit bilayer difference schemes that form special

relationships at the boundary points.

The method is based on an implicit bilayer time-difference scheme based on a system of

dynamic equations of the theory of elasticity of the first order, which is converted into an explicit

scheme with the help of a Taylor series in time, while basic relations are resolved with the help of

the Galerkin method. The author demonstrates that the speed and voltage are calculated with the

same accuracy as the one provided by the classical finite element method, whereby determination

of stresses has to act as a numerically differentiating displacement.

The author identifies the relations needed to calculate both the internal points of the computational

domain and the boundary points. The author has also analyzed the accuracy and convergence

of the resulting method having completed a numerical simulation of the well-known problem

of diffraction of a longitudinal wave speed in a circular aperture. The problem has an analytical

solution.

DOI: 10.22227/1997-0935.2012.8.104 - 111

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- Klifton R.Dzh. Raznostnyy metod v ploskikh zadachakh dinamicheskoy uprugosti [Difference Method for Plane Problems of Dynamic Elasticity]. Mekhanika [Mechanics]. 1968, no. 1 (107), pp. 103—122.
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- Clifnon R.J. A Difference Method for Plane Problems in Dynamic Elasticity. Quart. Appl. Mfth. 1967, vol. 25, no. 1, pp. 97—116.