A Constrained Cauchy-Born Elasticity Accelerated Multigrid Method for Nanoindentation

We introduce a new multigrid method to study the lattice statics model arising from nanoindentation.A constrained Cauchy-Born elasticity model is used as the coarse-grid operator.This method accelerates the relaxation process and considerably reduces the computational cost.In particular,it saves quite a bit when dislocations nucleate and move,as demonstrated by the simulation results.

A Coupled Immersed Interface and Level Set Method for Three-Dimensional Interfacial Flows with Insoluble Surfactant

In this paper,a numerical method is presented for simulating the 3D interfacial flows with insoluble surfactant.The numerical scheme consists of a 3D immersed interface method(IIM)for solving Stokes equations with jumps across the interface and a 3D level-set method for solving the surfactant convection-diffusion equation along a moving and deforming interface.The 3D IIM Poisson solver modifies the one in the literature by assuming that the jump conditions of the solution and the flux are implicitly given at the grid points in a small neighborhood of the interface.This assumption is convenient in conjunction with the level-set techniques.It allows standard Lagrangian interpolation for quantities at the projection points on the interface.The interface jump relations are re-derived accordingly.A novel rotational procedure is given to generate smooth local coordinate systems and make effective interpolation.Numerical examples demonstrate that the IIM Poisson solver and the Stokes solver achieve second-order accuracy.A 3D drop with insoluble surfactant under shear flow is investigated numerically by studying the influences of different physical parameters on the drop deformation.

A Characterization of the Ejiri Torus in S^5

We conjecture that a Willmore torus having Willmore functional between 2π2 and 2π2 √3 is either conformally equivalent to the Clifford torus, or conformally equivalent to the Ejiri torus. Ejiri＇s torus in S5 is the first example of Willmore surface which is not conformally equivalent to any minimal surface in any real space form. Li and Vrancken classified all Willmore surfaces of tensor product in S n by reducing them into elastic curves in S3, and the Ejiri torus appeared as a special example. In this paper, we first prove that among all Willmore tori of tensor product, the Willmore functional of the Ejiri torus in S5 attains the minimum 2π2 √3, which indicates our conjecture holds true for Wilhnore surfaces of tensor product. Then we show that all Willmore tori of tensor product are unstable when the co-dimension is big enough. We also show that the Ejiri torus is unstable even in S5. Moreover, similar to Li and Vrancken, we classify all constrained Wilhnore surfaces of tensor product by reducing them with elastic curves in S3. All constrained Willmore tori obtained this way are also shown to bc unstable when the co-dimension is big enough.