Classical dynamical density functional theory (DDFT) is one of the cornerstones of modern
statistical mechanics. It is an extension of the highly successful method of classical density …

This paper outlines a rigorous variational-based multilevel Global–Local formulation for
ductile fracture. Here, a phase-field formulation is used to resolve failure mechanisms by …

The prediction of crack initiation and propagation in ductile failure processes are
challenging tasks for the design and fabrication of metallic materials and structures on a …

The phase-field crystal equation is a sixth-order nonlinear parabolic equation and can be
applied to simulate various phenomena such as epitaxial growth, material hardness, and …

A novel adaptive numerical approach is developed for an accurate and computationally
efficient phase-field modelling of dendritic solidification. The adaptivity is based on the …

We consider numerical approximations for the modified phase field crystal equation in this
paper. The model is a nonlinear damped wave equation that includes both diffusive …

In this paper, we present a high-order accurate compact scheme for the phase field crystal
model in two-and three-dimensional spaces. The proposed scheme is derived by combining …

In this paper we analyze and implement a second-order-in-time numerical scheme for the
three-dimensional phase field crystal (PFC) equation. The numerical scheme was proposed …

We present temporally first-and second-order accurate methods for the Swift–Hohenberg
(SH) equation with quadratic–cubic nonlinearity. In order to handle the nonconvex …

Abstract The Swift–Hohenberg (SH) equation has been widely used as a model for the study
of pattern formation. The SH equation is a fourth-order nonlinear partial differential equation …