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Meshfree approximate solution of the Cauchy-Navier equations of elastodynamics

Valtchev, Svilen S. ; Martins, Nuno F. M.

Proceedings of the 3nd International Conference on Control Systems, Mathematical Modeling, Automation and Energy Efficiency (SUMMA), (2021),

In this study we propose a meshfree scheme for the numerical solution of boundary value problems (BVP) for the nonhomogeneous Cauchy-Navier equations of elastodynamics. The method uses the classical approach where first a particular solution for the partial differential equation (PDE) is calculated and then the corresponding homogeneous BVP is solved for the homogeneous part of the total solution. In particular, we approximate each component of the source term of the nonhomogeneous PDE by superposition of plane wave functions with different frequencies and directions of propagation. Using these expansions, a particular solution for the PDE is derived in the form of a linear combination of elastic P- and S-waves, at no extra computational cost. In the second step of the scheme, we solve the corresponding homogeneous BVP using the classical method of fundamental solutions (MFS), with shape functions given by the Kupradze tensor. The accuracy and the convergence of the proposed technique is illustrated for a Dirichlet BVP, posed in a 2D multiply connected domain, bounded by polygonal and parametric curves.