This chapter surveys recent numerical advances in the phase field method for geometric
surface evolution and related geometric nonlinear partial differential equations (PDEs) …

Semiconductor heteroepitaxy involves a wealth of qualitatively different, competing
phenomena. Examples include three-dimensional island formation, injection of dislocations …

Dewetting is a ubiquitous phenomenon in nature; many different thin films of organic and
inorganic substances (such as liquids, polymers, metals, and semiconductors) share this …

We extend the symmetrized parametric finite method for the evolution of a closed curve
under anisotropic surface diffusion in two dimensions, recently proposed by us [W. Bao, W …

Accurately modeling solid-state dewetting in materials with strong crystalline anisotropy is
an open problem in materials science with importance for both the manufacturing and …

Several crystalline structures are metastable or kinetically frozen out-of-equilibrium states in
the phase space. When the corresponding lifetime is sufficiently long, typical equilibrium …

We study the dynamics of a small solid particle arising from the dewetting of a thin film on a
curved substrate driven by capillarity, where mass transport is controlled by surface …

Materials featuring anomalous suppression of density fluctuations over large length scales
are emerging systems known as disordered hyperuniform. The underlying hidden order …

Layout design is encountered in many fields of engineering and science. Those with
complex constraints are particularly challenging to solve due to the non-uniqueness of the …

Tissue folds are structural motifs critical to organ function. In the intestine, bending of a flat
epithelium into a periodic pattern of folds gives rise to villi, finger-like protrusions that enable …