Xianmin Xu

AN EULERIAN SPACE-TIME FINITE ELEMENT METHOD FOR DIFFUSION PROBLEMS ON EVOLVING SURFACES ∗

In this paper, we study numerical methods for the solution of partial differential equations on evolving surfaces. The evolving hypersurface in

AN EFFICIENT NUMERICAL METHOD FOR CAVITATION IN NONLINEAR ELASTICITY

\Bbb{R}^d

DERIVATION OF THE WENZEL AND CASSIE EQUATIONS FROM A PHASE FIELD MODEL FOR TWO PHASE FLOW ON ROUGH SURFACE

defines a

Analysis of wetting and contact angle hysteresis on chemically patterned surfaces

d

The modified Cassie’s equation and contact angle hysteresis

-dimensional space-time manifold in the space-time continuum

An efficient threshold dynamics method for wetting on rough surfaces

\Bbb{R}^{d+1}

On surface meshes induced by level set functions

. We derive and analyze a variational formulation for a class of diffusion problems on the space-time manifold. For this variational formulation new well-posedness and stability results are derived. The analysis is based on an inf-sup condition and involves some natural, but nonstandard, (anisotropic) function spaces. Based on this formulation a discrete in time variational formulation is introduced that is very suitable as a starting point for a discontinuous Galerkin (DG) space-time finite element discretization. This DG space-time method is explained and results of numerical experiments are presented that illustrate its properties.

Variational method for liquids moving on a substrate

This paper is concerned with the numerical computation of cavitation in nonlinear elasticity. The Crouzeix–Raviart nonconforming finite element method is shown to prevent the degeneration of the mesh provoked by the conventional finite element approximation of this problem. Upon the addition of a suitable stabilizing term to the elastic energy, the method is used to solve cavitation problems in both radially symmetric and non-radially symmetric settings. While the radially symmetric examples serve to illustrate the efficiency of the method, and for validation purposes, the experiments with non-centered and multiple cavities (carried out for the first time) yield novel observations of situations potentially leading to void coalescence.

Theoretical analysis for meniscus rise of a liquid contained between a flexible film and a solid wall

In this paper, the equilibrium behavior of an immiscible two phase fluid on a rough surface is studied from a phase field equation derived from minimizing the total free energy of the system. When the size of the roughness becomes small, we derive the effective boundary condition for the equation by the multiple scale expansion homogenization technique. The Wenzel and Cassie equations for the apparent contact angles on the rough surfaces are then derived from the effective boundary condition. The homogenization results are proved rigorously by the Γ-convergence theory.

Modified Wenzel and Cassie Equations for Wetting on Rough Surfaces

Wetting and contact angle hysteresis on chemically patterned surfaces in two dimensions are analyzed from a stationary phase-field model for immiscible two phase fluids. We first study the sharp-interface limit of the model by the method of matched asymptotic expansions. We then justify the results rigorously by the