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BILAYER DIFFERENCE SCHEME OF A NUMERICAL SOLUTION TO TWO-DIMENSIONAL DYNAMIC PROBLEMS OF ELASTICITY
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.
BILAYER DIFFERENCE SCHEME OF A NUMERICAL SOLUTION TO TWO-DIMENSIONAL DYNAMIC PROBLEMS OF ELASTICITY
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.
BILAYER DIFFERENCE SCHEME OF A NUMERICAL SOLUTION TO TWO-DIMENSIONAL DYNAMIC PROBLEMS OF ELASTICITY
Nemchinov Vladimir Valentinovich (author)
2012
Article (Journal)
Electronic Resource
Unknown
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