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<p>Interfaces play a dominant role in governing the response of many biological systems
and they pose many challenges to traditional finite element. For sharp-interface model,
traditional finite element methods necessitate the finite element mesh to align with
surfaces of discontinuities. Diffuse-interface model replaces the sharp interface
with continuous variations of an order parameter resulting in significant computational
effort. To overcome these difficulties, we focus on developing a computationally efficient
spline-based finite element method for interface problems.</p><p>A key challenge while
employing B-spline basis functions in finite-element methods is the robust imposition
of Dirichlet boundary conditions. We begin by examining weak enforcement of such conditions
for B-spline basis functions, with application to both second- and fourth-order problems
based on Nitsche's approach. The use of spline-based finite elements is further examined
along with a Nitsche technique for enforcing constraints on an embedded interface.
We show that how the choice of weights and stabilization parameters in the Nitsche
consistency terms has a great influence on the accuracy and robustness of the method.
In the presence of curved interface, to obtain optimal rates of convergence we employ
a hierarchical local refinement approach to improve the geometrical representation
of interface. </p><p>In multiple dimensions, a spline basis is obtained as a tensor
product of the one-dimensional basis. This necessitates a rectangular grid that cannot
be refined locally in regions of embedded interfaces. To address this issue, we develop
an adaptive spline-based finite element method that employs hierarchical refinement
and coarsening techniques. The process of refinement and coarsening guarantees linear
independence and remains the regularity of the basis functions. We further propose
an efficient data transfer algorithm during both refinement and coarsening which yields
to accurate results.</p><p>The adaptive approach is applied to vesicle modeling which
allows three-dimensional simulation to proceed efficiently. In this work, we employ
a continuum approach to model the evolution of microdomains on the surface of Giant
Unilamellar Vesicles. The chemical energy is described by a Cahn-Hilliard type density
functional that characterizes the line energy between domains of different species.
The generalized Canham-Helfrich-Evans model provides a description of the mechanical
energy of the vesicle membrane. This coupled model is cast in a diffuse-interface
form using the phase-field framework. The effect of coupling is seen through several
numerical examples of domain formation coupled to vesicle shape changes.</p>
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