A hybrid smoothed extended finite element/level set method for modeling equilibrium shapes of nano-inhomogeneities

Abstract

Interfacial energy plays an important role in equilibrium morphologies of nanosized microstructures of solid materials due to the high interface-to-volume ratio, and can no longer be neglected as it does in conventional mechanics analysis. The present work develops an effective numerical approach by means of a hybrid smoothed extended finite element/level set method to model nanoscale inhomogeneities with interfacial energy effect, in which the finite element mesh can be completely independent of the interface geometry. The Gurtin-Murdoch surface elasticity model is used to account for the interface stress effect and the Wachspress interpolants are used for the first time to construct the shape functions in the smoothed extended finite element method. Selected numerical results are presented to study the accuracy and efficiency of the proposed method as well as the equilibrium shapes of misfit particles in elastic solids. The presented results compare very well with those obtained from theoretical solutions and experimental observations, and the computational efficiency of the method is shown to be superior to that of its most advanced competitor.Keywords: Smoothed extended finite element method; Level set method; Wachspress shape function; Interface excess energy; Equilibrium shape. IntroductionRecent advances in nanotechnology exacerbate the need for computational tools that are capable of capturing the effects of interfaces, which play an important role in nanostructured materials due to a characteristically high interface-to-volume ratio. Although atomic level computational tools such as molecular dynamic (MD) and first principle calculations are able to simulate interface effects, these methods are computationally intensive, thus their applications are usually limited to nano-scale samples and nano-second time durations. Many engineering problems, however, occur at much larger spatial and temporal scales. For example, simulating the formation and morphological evolution of precipitates in superalloys involves length scales ranging from several nanometers to tens of micrometers, and the physical processes last up to hours in time. In these cases, it is necessary to use a continuum level model that can capture the interfacial effects of particles at different length and time scales.

Effects of elastic strain energy and interfacial stress on the equilibrium morphology of misfit particles in heterogeneous solids

“…An open source C++ XFEM code is also available for use (Bordas et al, 2007). In the present study, a modified XFEM with strain smoothing technique called SmXFEM (Zhao et al, 2012) was adopted to compute the elastic field in the presence of interface energy effect.…”

“…These advantages could effectively improve the computational efficiency without loss of accuracy of the XFEM at the same time. The readers are referred to the paper (Zhao et al, 2012) and the cited references therein for the detailed numerical implementation of SmXFEM.…”

Equilibrium morphology of misfit particles in elastically stressed solids under chemo-mechanical equilibrium conditions

“…In our previous work (Zhao et al, 2013a;Zhao et al, 2013b), a sharp interface model incorporating the Gurtin-Murdoch interface elasticity effects was developed to study the equilibrium morphology of misfit particles in general elastic anisotropic solids and implemented using a hybrid smoothed XFEM/LSM, in which the chemical free energy was neglected based on the late-stage coarsening assumption, so 6 the particle size consequently remained constant during evolution.…”

Interfacial shear stress optimization in sandwich beams with polymeric core using non-uniform distribution of reinforcing ingredients

“…However, for problems with curved interfaces, using conforming meshes may be cumbersome, especially when such interfaces evolve. Several advanced methods like the Extended Finite Element Method (XFEM) [14,15] and XIGA [16,17] have been developed for capturing material discontinuity using non conforming mesh. Recently Nguyen proposed very simple approaches to handle such discontinuities in IGA for composite delamination using knot insertion for cohesive interfaces [18].…”