4,610 research outputs found
An isogeometric finite element formulation for phase transitions on deforming surfaces
This paper presents a general theory and isogeometric finite element
implementation for studying mass conserving phase transitions on deforming
surfaces. The mathematical problem is governed by two coupled fourth-order
nonlinear partial differential equations (PDEs) that live on an evolving
two-dimensional manifold. For the phase transitions, the PDE is the
Cahn-Hilliard equation for curved surfaces, which can be derived from surface
mass balance in the framework of irreversible thermodynamics. For the surface
deformation, the PDE is the (vector-valued) Kirchhoff-Love thin shell equation.
Both PDEs can be efficiently discretized using -continuous interpolations
without derivative degrees-of-freedom (dofs). Structured NURBS and unstructured
spline spaces with pointwise -continuity are utilized for these
interpolations. The resulting finite element formulation is discretized in time
by the generalized- scheme with adaptive time-stepping, and it is fully
linearized within a monolithic Newton-Raphson approach. A curvilinear surface
parameterization is used throughout the formulation to admit general surface
shapes and deformations. The behavior of the coupled system is illustrated by
several numerical examples exhibiting phase transitions on deforming spheres,
tori and double-tori.Comment: fixed typos, extended literature review, added clarifying notes to
the text, added supplementary movie file
Error control for the FEM approximation of an upscaled thermo-diffusion system with Smoluchowski interactions
We analyze a coupled system of evolution equations that describes the effect
of thermal gradients on the motion and deposition of populations of
colloidal species diffusing and interacting together through Smoluchowski
production terms. This class of systems is particularly useful in studying drug
delivery, contaminant transportin complex media, as well as heat shocks
thorough permeable media. The particularity lies in the modeling of the
nonlinear and nonlocal coupling between diffusion and thermal conduction. We
investigate the semidiscrete as well as the fully discrete em a priori error
analysis of the finite elements approximation of the weak solution to a
thermo-diffusion reaction system posed in a macroscopic domain. The
mathematical techniques include energy-like estimates and compactness
arguments
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