Geometrical evolution laws are widely used in continuum modeling of surface and interface motion in materials science. In this article, we first give a brief review of various kinds of geometrical evolution laws and their variational derivations, with an emphasis on strong anisotropy. We then survey some of the finite element based numerical methods for simulating the motion of interfaces focusing on the field of thin film growth. We discuss the finite element method applied to front-tracking, phase-field and level-set methods. We describe various applications of these geometrical evolution laws to materials science problems, and in particular, the growth and morphologies of thin crystalline films.

We present an energy absorbing non-reflecting boundary condition of Clayton-Engquist type for the elastic wave equation together with a discretization which is stable for any ratio of compressional to shear wave speed. We prove stability for a second-order accurate finite-difference discretization of the elastic wave equation in three space dimensions together with a discretization of the proposed non-reflecting boundary condition. The stability proof is based on a discrete energy estimate and is valid for heterogeneous materials. The proof includes all six boundaries of the computational domain where special discretizations are needed at the edges and corners. The stability proof holds also when a free surface boundary condition is imposed on some sides of the computational domain.

Stochastic collocation methods as a promising approach for solving stochastic partial differential equations have been developed rapidly in recent years. Similar to Monte Carlo methods, the stochastic collocation methods are non-intrusive in that they can be implemented via repetitive execution of an existing deterministic solver without modifying it. The choice of collocation points leads to a variety of stochastic collocation methods including tensor product method, Smolyak method, Stroud 2 or 3 cubature method, and adaptive Stroud method. Another type of collocation method, the probabilistic collocation method (PCM), has also been proposed and applied to flow in porous media. In this paper, we discuss these methods in terms of their accuracy, efficiency, and applicable range for flow in spatially correlated random fields. These methods are compared in details under different conditions of spatial variability and correlation length. This study reveals that the Smolyak method and the PCM outperform other stochastic collocation methods in terms of accuracy and efficiency. The random dimensionality in approximating input random fields plays a crucial role in the performance of the stochastic collocation methods. Our numerical experiments indicate that the required random dimensionality increases slightly with the decrease of correlation scale and moderately from one to multiple physical dimensions.

In this study an incompressible flow with non-Newtonian fluids in the reverse roll coating process was investigated. Non-Newtonian behavior of the coating fluid was accounted by using power law model with power index, n, ranging from 0.8 to 1.2. Effect of roll speed ratio (V₂/V₁) of the panel roll to the applicator roll and gap distance on the coating film thickness were also investigated. Numerical results were in good agreement with those of experimental data within 15%-20%. Results indicated that the film thickness ratios are function of power-law index, roll speed ration and ratio of roll radius to gap distance. The equations for film thickness ratio were obtained from numerical results as shown below:
where t₁, t₂ and t₃ are inlet film, transferred film and leakage film thickness, respectively, n is power-law index, V₁ and V₂ are roll speed of applicator roll and panel roll, R_m is average radius of two rolls and H₀ is gap distance. The correlations are accurate within 10% for 0.8≤n≤1.2, 0.5≤V₂/V₁≤2, and 7.7×10⁻⁵≤ H₀/R_m ≤1.54×10⁻⁴.

We study the growth of epitaxial thin films on pre-patterned substrates that influence the surface diffusion of subsequently deposited material using a kinetic Monte Carlo algorithm that combines the use of inverted lists with rejection. The resulting algorithm is well adapted to systems with spatially heterogeneous hopping rates. To evaluate the algorithm's performance we compare it with an efficient, binary-tree based algorithm. A key finding is that the relative performance of the inverted list algorithm improves with increasing system size.

This paper presents a systematic method to derive Beam Propagation Models for optical waveguides. The technique is based on the use of the symbolic calculus rules for pseudodifferential operators. The cases of straight and bent optical waveguides are successively considered.

Recently Hari Shroff and his collaborators [Nano Letters 5 (2005)] developed a nanoscopic force sensor, but the force which they measured in their single molecular experiment was much lower than the theoretical critical value. In order to fix this problem, we investigate the micromechanics of dsDNA based on the worm-like chain model and flexible hinge model by using Monte Carlo algorithm. The simulation results not only address Hari Shroff's experiment difficulty reasonably, but also provide strong support for flexible hinge mechanism put forward recently by Yan and Marko [Phys. Rev. Lett. 93 (2004)].

The society structure plays an important role in shaping the attitudes, beliefs and public opinion. For studying the role of the society structure in opinion dynamics, we analyze the Sznajd model on small-world network formed by adding shortcuts in a lattice consisting of N nodes arranged in a ring and on two-dimensional (2-D) regular lattice. Through computer simulation, we find that there exists a pseudo-phase transition from the coexistence state for φ<φ_c to the consensus state for φ>φ_c, where φ_c is some threshold for the shortcut density φ, which is dependent of the complex network topology and the dimensionality of complex networks. Our observations indicate the dependence of the opinion dynamics on the complex system topology.

A distributed Lagrangian moving-mesh finite element method is applied to problems involving changes of phase. The algorithm uses a distributed conservation principle to determine nodal mesh velocities, which are then used to move the nodes. The nodal values are obtained from an ALE (Arbitrary Lagrangian-Eulerian) equation, which represents a generalization of the original algorithm presented in Applied Numerical Mathematics, 54:450–469 (2005). Having described the details of the generalized algorithm it is validated on two test cases from the original paper and is then applied to one-phase and, for the first time, two-phase Stefan problems in one and two space dimensions, paying particular attention to the implementation of the interface boundary conditions. Results are presented to demonstrate the accuracy and the effectiveness of the method, including comparisons against analytical solutions where available.

This paper proposes a new technique to speed up the computation of the matrix of spectral collocation discretizations of surface single and double layer operators over a spheroid. The layer densities are approximated by a spectral expansion of spherical harmonics and the spectral collocation method is then used to solve surface integral equations of potential problems in a spheroid. With the proposed technique, the computation cost of collocation matrix entries is reduced from O(M²N⁴) to O(MN⁴), where N² is the number of spherical harmonics (i.e., size of the matrix) and M is the number of one-dimensional integration quadrature points. Numerical results demonstrate the spectral accuracy of the method.

By performing a particular spatial discretization to the nonlinear Schrödinger equation (NLSE), we obtain a non-integrable Hamiltonian system which can be decomposed into three integrable parts (L-L-N splitting). We integrate each part by calculating its phase flow, and develop explicit symplectic integrators of different orders for the original Hamiltonian by composing the phase flows. A 2nd-order reversible constructed symplectic scheme is employed to simulate solitons motion and invariants behavior of the NLSE. The simulation results are compared with a 3rd-order non-symplectic implicit Runge-Kutta method, and the convergence of the formal energy of this symplectic integrator is also verified. The numerical results indicate that the explicit symplectic scheme obtained via L-L-N splitting is an effective numerical tool for solving the NLSE.

We study a continuum paradigm of the lipid bilayer based on minimizing the free energy of a mixture of water and lipid molecules. This paper extends previous work of Blom and Peletier [European J. Appl. Math., 15 (2004), pp. 487-508] in the following ways. (a) It formulates a more general model of the hydrophobic effect to facilitate connections with microscale simulations and first-principles analysis. (b) It clarifies the meaning and role of the model parameters. (c) It outlines a method for determining parameter values so that physically-realistic bilayer density profiles can be obtained, for example for use in macroscale simulations. Points (a)-(c) suggest that the model has potential to robustly connect some micro- and macroscale levels of multiscale blood flow simulations. The mathematical modelling in point (a) is based upon a consideration of the underlying physics of inter-molecular forces. The governing equations thus obtained are minimized by gradient flows via a novel numerical approach; this enables point (b). The numerical results are shown to behave physically in terms of the effect of background concentration, in contrast to the earlier model which is shown here to not display the expected behaviour. A “short-tail” approximation of the lipid molecules also gives an analytical tool which yields critical values of some parameters under certain conditions. Point (c) involves the first quantitative comparison of the numerical data with physical experimental results.