Adaptive isogeometrische dreidimensionale Phasenfeldmodellierung zur morphologischen Evolution von Lipid-Doppelschichten in Wechselwirkung mit Strömungsfeldern und elektrischen Feldern
Zusammenfassung der Projektergebnisse
In this project, we presented a robust computational framework based on isogeometric analysis and phase-field method for modeling morphological evolution of single- and multi-phase biomembranes with chemo-mechanical, electro-mechanical and hydrodynamical interactions. As the first step, we studied the numerical properties of IGA for phase-field approximations of mean curvature flow and Willmore flow problems, which are two prominent geometric partial differential equations used for modeling the dynamics of biomembranes and capillary interfaces. This was followed by investigating a phase-field model for the static deformation of a vesicle membrane under the elastic bending energy, with prescribed bulk volume and surface area. We solved this problem using IGA and compared different approaches for imposing constraints including a penalty formulation and a modified augmented Lagrange multiplier approach. The results were then compared and verified against the sharp interface model of biomembranes statics. Next, we extended our formulation to a general chemo-mechanical phase-field model of multi-component vesicles which accounted for simultaneous shape change and concentration evolution. We studied the effects of line tension and bending rigidity on the vesicle equilibrium shapes, and made some qualitative comparisons with the experimental results. Subsequently, we developed a three-dimensional isogeometric analysis formulation for the phase-field constrained optimization problem of morphological evolution of vesicles in electrical fields. We proposed an efficient staggered scheme for solving governing time-dependent, nonlinear, highorder PDEs. We used the modified augmented Lagrange multiplier (mALM) approach, developed earlier for studying bending deformation of vesicles, to satisfy the geometric constraints of the model while maintaining numerical stability and a relatively large time step size. This approach guaranteed the satisfaction of the constraints at each time step over the entire temporal domain. The effect of the flexoelectricity and the conductivity ratio of the electrolyte on vesicle equilibrium shape were studied through several 3D numerical examples. We captured sphere-oblate and sphere-prolate shape transitions under varying conductivity ratio, which agreed very well with experimental observations. As the final part of the project, we focused on studying the dynamical behavior of inextensible vesicles under external fluid flows with inertial forces. We considered a phase-field model for the coupled fluidvesicle problem. The model uses a Lagrange multiplier method for enforcing global area and volume constraints while it relies on an additional partial differential equation (PDE) for enforcing local inextensibility condition. In order to solve the system of PDEs in the model, we presented an implicit, fully monolithic scheme based on the generalized-α time integration method. We treated Navier-Stokes equations using a finite element formulation based on residual-based variational multiscale method. For the rest of PDEs, we employed a standard Galerkin method. Compared to earlier works in the literature, our computational framework reduces the number of equations to be solved by leveraging high continuity of NURBS functions. We implemented the algorithm both for 2D and 3D problems and introduced a resistive immersed surface method into the formulation to handle different geometries of the domain in a weak sense using a diffuse–interface approach. We solved several two-dimensional numerical examples which simulate the vesicle dynamics in a quiescent fluid, in a shear flow with tank-treading and tumbling motions, and in plane Poiseuille flow with and without obstructions. We further investigated the problem of a vesicle passing through a constriction in 3D where the model resembles the situation that a vesicle undergoes while moving inside a stenosed microchannel. In all numerical examples, we found a very good qualitative/quantitative agreement with previous numerical studies and experimental observations.
Projektbezogene Publikationen (Auswahl)
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Isogeometric analysis for phase-field models of geometric PDEs and high-order PDEs on stationary and evolving surfaces. Computer Methods in Applied Mechanics and Engineering, 351, 599-642.
Valizadeh, Navid & Rabczuk, Timon
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On the use of local maximum entropy approximants for Cahn–Hilliard phase-field models in 2D domains and on surfaces. Computer Methods in Applied Mechanics and Engineering, 346, 1-24.
Amiri, Fatemeh; Ziaei-Rad, Saeed; Valizadeh, Navid & Rabczuk, Timon
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Isogeometric analysis of insoluble surfactant spreading on a thin film. Computer Methods in Applied Mechanics and Engineering, 370, 113272.
Medina, David; Valizadeh, Navid; Samaniego, Esteban; Jerves, Alex X. & Rabczuk, Timon
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Isogeometric analysis for a phase-field constrained optimization problem of morphological evolution of vesicles in electrical fields. Computer Methods in Applied Mechanics and Engineering, 377, 113669.
Ashour, Mohammed; Valizadeh, Navid & Rabczuk, Timon
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An isogeometric phase–field based shape and topology optimization for flexoelectric structures. Computer Methods in Applied Mechanics and Engineering, 391, 114564.
López, Jorge; Valizadeh, Navid & Rabczuk, Timon
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Isogeometric analysis of hydrodynamics of vesicles using a monolithic phase-field approach. Computer Methods in Applied Mechanics and Engineering, 388, 114191.
Valizadeh, Navid & Rabczuk, Timon
