An adaptive meshfree method for phase-field models of biomembranes. Part II: A Lagrangian approach for membranes in viscous fluids

We present a Lagrangian phase-field method to study the low Reynolds number dynamics of vesicles embedded in a viscous fluid. In contrast to previous approaches, where the field variables are the phase-field and the fluid velocity, here we exploit the fact that the phasefield tracks a material interface to reformulate the problem in terms of the Lagrangian motion of a background medium, containing both the biomembrane and the fluid. We discretize the equations in space with maximum-entropy approximants, carefully shown to perform well in phase-field models of biomembranes in a companion paper. The proposed formulation is variational, lending itself to implicit time-stepping algorithms based on minimization of a time-incremental energy, which are automatically nonlinearly stable. The proposed method deals with two of the major challenges in the numerical treatment of coupled fluid/phase-field models of biomembranes, namely the adaptivity of the grid to resolve the sharp features of the phase-field, and the stiffness of the equations, leading to very small time-steps. In our method, local refinement follows the features of the phasefield as both are advected by the Lagrangian motion, and large time-steps can be robustly chosen in the variational time-stepping algorithm, which also lends itself to time adaptivity. The method is presented in the axisymmetric setting, but it can be directly extended to 3D. We present a Lagrangian phase-field method to study the low Reynolds number dynamics of vesicles embedded in a viscous fluid. In contrast to previous approaches, where the field variables are the phase-field and the fluid velocity, here we exploit the fact that the phase-field tracks a material interface to reformulate the problem in terms of the Lagrangian motion of a background medium, containing both the biomembrane and the fluid. We discretize the equations in space with maximum-entropy approximants, carefully shown to perform well in phase-field models of biomembranes in a companion paper. The proposed formulation is variational, lending itself to implicit time-stepping algorithms based on minimization of a time-incremental energy, which are automatically nonlinearly stable. The proposed method deals with two of the major challenges in the numerical treatment of coupled fluid/phase-field models of biomembranes, namely the adaptivity of the grid to resolve the sharp features of the phase-field, and the stiffness of the equations, leading to very small time-steps. In our method, local refinement follows the features of the phase-field as both are advected by the Lagrangian motion, and large time-steps can be robustly chosen in the variational time-stepping algorithm, which also lends itself to time adaptivity. The method is presented in the axisymmetric setting, but it can be directly extended to 3D

Second order convex maximum entropy approximants with applications to high order PDE

We present a new approach for second order maximum entropy (max-ent) meshfree approximants that produces positive and smooth basis functions of uniform aspect ratio even for non-uniform node sets, and prescribes robustly feasible constraints for the entropy maximization program defining the approximants. We examine the performance of the proposed approximation scheme in the numerical solution by a direct Galerkin method of a number of partial differential equations (PDEs), including structural vibrations, elliptic second order PDEs, and fourth order PDEs for Kirchhoff-Love thin shells and for a phase field model describing the mechanics of biomembranes. The examples highlight the ability of the method to deal with non-uniform node distributions, and the high accuracy of the solutions. Surprisingly, the first order meshfree max-ent approximants with large supports are competitive when compared to the proposed second order approach in all the tested examples, even in the higher order PDEs.Peer ReviewedPostprint (author's final draft

Efficient coarse-grained brownian dynamics simulations for dna and lipid bilayer membrane with hydrodynamic interactions

The coarse-grained molecular dynamics (CGMD) or Brownian dynamics (BD) simulation is a particle-based approach that has been applied to a wide range of biological problems that involve interactions with surrounding fluid molecules or the so-called hydrodynamic interactions (HIs). From simple biological systems such as a single DNA macromolecule to large and complicated systems, for instances, vesicles and red blood cells (RBCs), the numerical results have shown outstanding agreements with experiments and continuum modeling by adopting Stokesian dynamics and explicit solvent model. Finally, when combined with fast algorithms such as the fast multipole method (FMM) which has nearly optimal complexity in the total number of CG particles, the resulting method is parallelizable, scalable to large systems, and stable for large time step size, thus making the long-time large-scale BD simulation within practical reach. This will be useful for the study of a large collection of molecules or cells immersed in the fluids. This dissertation can be divided into three main subjects: (1) An efficient algorithm is proposed to simulate the motion of a single DNA molecule in linear flows. The algorithm utilizes the integrating factor method to cope with the effect of the linear flow of the surrounding fluid and applies the Metropolis method (MM) in [N. Bou-Rabee, A. Donev, and E. Vanden-Eijnden, Multiscale Model. Simul. 12, 781 (2014)] to achieve more efficient BD simulation. More importantly, this proposed method permits much larger time step size than methods in previous literature while still maintaining the stability of the BD simulation, which is advantageous for long-time BD simulation. The numerical results on λ-DNA agree very well with both experimental data and previous simulation results. (2) Lipid bilayer membranes have been extensively studied by CGMD simulations. Numerical efficiencies have been reported in the cases of aggressive coarse-graining, where several lipids are coarse-grained into a particle of size 4 ~ 6 nm so that there is only one particle in the thickness direction. In [H. Yuan et al., Phys. Rev. E, 82, 011905 (2010)], Yuan et al. proposed a pair-potential between these one-particle-thick coarse-grained lipid particles to capture the mechanical properties of a lipid bilayer membrane, such as gel-fluid-gas phase transitions of lipids, diffusion, and bending rigidity. This dissertation provides a detailed implementation of this interaction potential in LAMMPS to simulate large-scale lipid systems such as a giant unilamellar vesicle (GUV) and RBCs. Moreover, this work also considers the effect of cytoskeleton on the lipid membrane dynamics as a model for RBC dynamics, and incorporates coarse-grained water molecules to account for hydrodynamic interactions. (3) An action field method for lipid bilayer membrane model is introduced where several lipid molecules are represented by a Janus particle with corresponding orientation pointing from lipid head to lipid tail. With this level of coarse-grained modeling, as the preliminary setup, the lipid tails occupy a half sphere and the lipid heads take the other half. An action field is induced from lipid-lipid interactions and exists everywhere in the computational domain. Therefore, a hydrophobic attraction energy can be described from utilizing the variational approach and its minimizer with respect to the action field is the so-called screened Laplace equation. For the numerical method, the well-known integral equation method (IEM) has great capability to solve exterior screened Laplace equation with Dirichlet boundary conditions. Finally, one then can obtain the lipid dynamics to validate the self-assembly property and other physical properties of lipid bilayer membrane. This approach combines continuum modeling with CGMD and gives a different perspective to the membrane energy model from the traditional Helfrich membrane free energy