Finite element methods for surface PDEs

In this article we consider finite element methods for approximating the solution of partial differential equations on surfaces. We focus on surface finite elements on triangulated surfaces, implicit surface methods using level set descriptions of the surface, unfitted finite element methods and diffuse interface methods. In order to formulate the methods we present the necessary geometric analysis and, in the context of evolving surfaces, the necessary transport formulae. A wide variety of equations and applications are covered. Some ideas of the numerical analysis are presented along with illustrative numerical examples

Weak solutions to problems involving inviscid fluids

We consider an abstract functional-differential equation derived from the pressure-less Euler system with variable coefficients that includes several systems of partial differential equations arising in the fluid mechanics. Using the method of convex integration we show the existence of infinitely many weak solutions for prescribed initial data and kinetic energy

Quantum dot formation on a strain-patterned epitaxial thin film

We model the effect of substrate strain patterning on the self-assembly of quantum dots (QDs). When the surface energy is isotropic, we demonstrate that strain patterning via embedded substrate inclusions may result in ordered, self-organized QD arrays. However, for systems with strong cubic surface energy anisotropy, the same patterning does not readily lead to an ordered array of pyramids at long times. We conclude that the form of the surface energy anisotropy strongly influences the manner in which QDs self-assemble into regular arrays.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/87827/2/133102_1.pd

An Ethnography of Brand Piracy in Guatemala

An important dimension of contemporary capitalism is the global spread of intellectual property rights law, drawing new attention by governments and media to the unauthorized copying of fashion brands. In this dissertation, I draw on sixteen months of ethnographic research with small-scale, indigenous Maya garment manufacturers to examine the cultural and moral context of brand piracy in Guatemala. I analyze what practices of copying and imitation, some of which qualify as piracy under national and international law, among Maya manufacturers reveal about two aspects of the social field: first, changing economic and cultural conditions following waves of neoliberal economic and legal reform, and, second, the nonlinear reproduction of forms of moral and legal reckoning at the margins of the global economy and amidst mounting insecurities that include rising violent crime rates and legal impunity for violent crime. I examine how practices of copying and imitation among manufacturers and competitive behavior more generally are evaluated locally in light of kin relations that promote the sharing of knowledge and resources within a somewhat loose property regime and given ideologies of race and nation that encourage class-based solidarity among Maya people. I find that the normative models and business practices evident among these manufacturers parochialize official portraits of progress, business ethics, and development promoted in neoliberal policy agendas and international law. In addition, I analyze significant gaps between what fashion and branding mean in Guatemalan Maya communities and how they are understood in international projects of legal harmonization that are also about re-branding and re-imagining the Guatemalan nation. Neoliberal statecraft following a long internal armed conflict in Guatemala involves policy approaches that amplify the presence of global brands while compounding conditions of social and economic inequality that limit Maya men and women’s access to authorized goods. Meanwhile, Maya people are invited to participate in a modernist vision of citizenship and social progress that encourages a privatized model of indigenous identity mediated by branded commodities and formal market transactions. The brand emerges as a powerful medium through which claims to legitimacy and authority and senses of belonging are negotiated at national and local levels.Anthropolog

On a Cahn--Hilliard--Darcy system for tumour growth with solution dependent source terms

We study the existence of weak solutions to a mixture model for tumour growth that consists of a Cahn--Hilliard--Darcy system coupled with an elliptic reaction-diffusion equation. The Darcy law gives rise to an elliptic equation for the pressure that is coupled to the convective Cahn--Hilliard equation through convective and source terms. Both Dirichlet and Robin boundary conditions are considered for the pressure variable, which allows for the source terms to be dependent on the solution variables.Comment: 18 pages, changed proof from fixed point argument to Galerkin approximatio

Thermodynamics of non-local materials: extra fluxes and internal powers

The most usual formulation of the Laws of Thermodynamics turns out to be suitable for local or simple materials, while for non-local systems there are two different ways: either modify this usual formulation by introducing suitable extra fluxes or express the Laws of Thermodynamics in terms of internal powers directly, as we propose in this paper. The first choice is subject to the criticism that the vector fluxes must be introduced a posteriori in order to obtain the compatibility with the Laws of Thermodynamics. On the contrary, the formulation in terms of internal powers is more general, because it is a priori defined on the basis of the constitutive equations. Besides it allows to highlight, without ambiguity, the contribution of the internal powers in the variation of the thermodynamic potentials. Finally, in this paper, we consider some examples of non-local materials and derive the proper expressions of their internal powers from the power balance laws.Comment: 16 pages, in press on Continuum Mechanics and Thermodynamic

Critical dynamics of an isothermal compressible non-ideal fluid

A pure fluid at its critical point shows a dramatic slow-down in its dynamics, due to a divergence of the order-parameter susceptibility and the coefficient of heat transport. Under isothermal conditions, however, sound waves provide the only possible relaxation mechanism for order-parameter fluctuations. Here we study the critical dynamics of an isothermal, compressible non-ideal fluid via scaling arguments and computer simulations of the corresponding fluctuating hydrodynamics equations. We show that, below a critical dimension of 4, the order-parameter dynamics of an isothermal fluid effectively reduces to "model A," characterized by overdamped sound waves and a divergent bulk viscosity. In contrast, the shear viscosity remains finite above two dimensions. Possible applications of the model are discussed.Comment: 19 pages, 7 figures; v3: minor corrections and clarifications; as published in Phys. Rev.