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 $C^1$-continuous interpolations without derivative degrees-of-freedom (dofs). Structured NURBS and unstructured spline spaces with pointwise $C^1$-continuity are utilized for these interpolations. The resulting finite element formulation is discretized in time by the generalized-$\alpha$ 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

Efficient isogeometric thin shell formulations for soft biological materials

This paper presents three different constitutive approaches to model thin rotation-free shells based on the Kirchhoff-Love hypothesis. One approach is based on numerical integration through the shell thickness while the other two approaches do not need any numerical integration and so they are computationally more efficient. The formulation is designed for large deformations and allows for geometrical and material nonlinearities, which makes it very suitable for the modeling of soft tissues. Furthermore, six different isotropic and anisotropic material models, which are commonly used to model soft biological materials, are examined for the three proposed constitutive approaches. Following an isogeometric approach, NURBS-based finite elements are used for the discretization of the shell surface. Several numerical examples are investigated to demonstrate the capabilities of the formulation. Those include the contact simulation during balloon angioplasty.Comment: Typos are removed. Remark 3.4 is added. Eq. (18) in the previous version is removed. Thus, the equations get renumbered. Example 5.5 is updated. Minor typos in Eqs. (17), (80), (145) and (146), are corrected. They do not affect the result

Heart Valve Mathematical Models

Nearly 100,000 heart valve replacements or repairs are performed in the US every year. Mathematical models of heart valves are used to improve artificial valve design and to guide surgeons performing valve-repairing surgeries. Models can be used to define the geometry of a valve, predict blood flow dynamics, or demonstrate operating mechanisms of the valve. In this thesis we reviewed features that are typically considered when developing a model of a heart valve. The main modeling methods include representing a heart valve using lumped parameters, finite elements, or isogeometric elements. Examples of a lumped-parameter model and isogeometric analysis are explored. First, we developed a simulation for the lumped-parameter model of Virag and Lulić, and we demonstrated its ability to capture the dynamical behavior of blood pressures, volumes, and flows in the aortic valve region. A Newton-Krylov method was used to estimate periodic solution trajectories, which provide a basis for examining the response to perturbations about initial conditions. Next, an isogeometric model of a heart valve was constructed based on NURBS geometry. The mechanical stiffness of the valve was computed. We discussed how the isogeometric representation could be used in a more complex fluid-structure interaction model to measure surface shear and estimate fatigue failure

A finite membrane element formulation for surfactants

Surfactants play an important role in various physiological and biomechanical applications. An example is the respiratory system, where pulmonary surfactants facilitate the breathing and reduce the possibility of airway blocking by lowering the surface tension when the lung volume decreases during exhalation. This function is due to the dynamic surface tension of pulmonary surfactants, which depends on the concentration of surfactants spread on the liquid layer lining the interior surface of the airways and alveoli. Here, a finite membrane element formulation for liquids is introduced that allows for the dynamics of concentration-dependent surface tension, as is the particular case for pulmonary surfactants. A straightforward approach is suggested to model the contact line between liquid drops/menisci and planar solid substrates, which allows the presented framework to be easily used for drop shape analysis. It is further shown how line tension can be taken into account. Following an isogeometric approach, NURBS-based finite elements are used for the discretization of the membrane surface. The capabilities of the presented computational model is demonstrated by different numerical examples - such as the simulation of liquid films, constrained and unconstrained sessile drops, pendant drops and liquid bridges - and the results are compared with experimental data.Comment: Some typos are removed. Eqs. 13 and 105 are modified. Eqs. 64 and 73 are added; thus, the rest of equations is renumbered. All the numerical experiments are repeated. The example of Sec. 6.3 is slightly modifie