Solving partial differential equations in computational mechanics via nonlocal numerical approaches

The work addresses solving partial differential equations (PDE)s for a continuum solid body using a nonlocal formulation. Second order spatial derivatives are of concern as they enable to deal with many dynamic problems in physics. The authors introduce a nonlocal finite difference (FD) scheme to show its applicability to computational mechanics. In the theoretical part of the work exemplary solutions of the second order ordinary derivatives are discussed. The capability of numerical dispersion reduction, which is identified for the proposed approach, is demonstrated for the crude spatial domain discretization. Based on the provided nonlocal FD scheme, discrete forms for the two types of PDEs are derived to indicate potential areas of practical applications for the presented method. Elastic wave equation and thermal diffusion equation are taken into account. Moreover, for the former case, the peridynamics - which is also a nonlocal approach - is also used to derive an alternative discrete formulation. The stability conditions for the provided numerical schemes are determined using von Neumann stability analysis. The limits for the allowed simulation time steps are found. The second part of the paper is devoted to the practical applications of the elaborated nonlocal formulations. There are studied selected thermal and mechanical properties of a gas foil bearing (GFB) component. A rod-like structural part of GFB made of superalloy Inconel 625 is analyzed. A one-dimensional model is built to determine temperature and displacement distributions via transient simulations. Again, reduced numerical dispersion is confirmed for the model with a crude mesh.