Meshless numericalmethods applied tomultiphysics andmultiscale problems

N many fields of science and engineering, such as fluid or structural mechanics, and nanotechnology, dynamical systems at different scale need to be simulated, optimized or controlled. They are often described by discretizations of systems of nonlinear partial differential equations yielding high-dimensional discrete phase spaces. For this reason, in recent decades, research has mainly focused on the development of sophisticated analytical and numerical (linear and nonlinear) tools to help understand the overall multiscale system behavior. Various models and numerical methods have been developed to simulate different physical processes at different scales. The choice of these methods will depend largely on the problem, the available computational resources and constitutive equations. Smoothed particle hydrodynamics (SPH) was developed a few decades ago to model inviscid fluid and gas flow dynamics in astrophysical problems. The SPH is an interpolation-based numerical technique that can be used to solve systems of partial differential equations (PDEs) using either Lagrangian or Eulerian descriptions. The nature of SPH method allows to incorporate different physical and chemical effects into the discretized governing equations with relatively small code-development effort. In addition, geometrically complex and/or dynamic boundaries, and interfaces can be handled without undue difficulty. The SPH numerical procedure of calculating state variables (i.e., density, velocity, and gradient of deformation) are computed as a weighted average of values in a local region.