Multiscale Particle-In-Cell Method: From Fluid to Solid Mechanics

Abstract

In this chapter, a novel multiscale method is presented that is based upon the Particle-InCell (PIC) finite element approach. The particle method, which is primarily used for fluid mechanics applications, is generalized and combined with a homogenization technique. The resulting technique can take us from fluid mechanics simulations seamlessly into solid mechanics. Here, large deformations are dealt with seamlessly, and material deformation is easily followed by the Lagrangian particles in an Eulerian grid. Using this method, solid materials can be modeled at both micro and macro scales where large strains and large displacements are expected. In a multiscale simulation, the continuum material points at the macroscopic scale are history and scale dependent. When conventional Finite Elements are used for multiscale modeling, microscale models are assigned to each integration point; with these integration points usually placed at Gaussian positions. Macroscale stresses are extrapolated from the solution of the microscale model at these points and significant loss of information occurs whenever re-meshing is performed. In addition, in the case of higherorder formulations, the choice of element type becomes critical as it can dictate the performance, efficiency and stability of the multiscale modeling scheme. Higher-order elements that provide better accuracy lead to systems of equations that are significantly larger than the system of equations from linear elements. The PIC method, on the other hand, avoids all of the problems that arise from element layout and topology because every material point carries material history information regardless of the mesh connectivity. Material points are not restricted to Gaussian positions and can be dispersed in the domain randomly, or with a controlled population and dispersion. Therefore, in analogy to Finite Element Method (FEM) mesh refinement, parts of the domain that might experience localization can have a higher number of material points representing the continuum macroscale in more detail. Similar to the FEM multiscale approach, each material point has a microscale model assigned and information passing between the macro and micro scales is carried out in a conventional form using homogenization formulations. In essence, the homogenization formulation is based on finding the solution of two boundary value problems at the micro and macro scales with information passing between scales. Another advantage of using the particle method is that each particle can represent an individual material property, thus in the microscale phase, interfaces can be followed without the