Simon Praetorius

Software concepts and numerical algorithms for a scalable adaptive parallel finite element method

An efficient implementation of an adaptive finite element method on distributed memory systems requires an efficient linear solver. Most solver methods, which show scalability to a large number of processors make use of some geometric information of the mesh. This information has to be provided to the solver in an efficient and solver specific way. We introduce data structures and numerical algorithms which fulfill this task and allow in addition for an user-friendly implementation of a large class of linear solvers. The concepts and algorithms are demonstrated for global matrix solvers and domain decomposition methods for various problems in fluid dynamics, continuum mechanics and materials science. Weak and strong scaling is shown for up to 16.384 processors.

A Continuous Approach to Discrete Ordering on S^2

We consider the classical problem to find optimal distributions of interacting particles on a sphere by solving an evolution problem for a particle density. The higher order surface partial differential equation is an approximation of a surface dynamic density functional theory. We motivate the approach phenomenologically and sketch a derivation of the model starting from an interatomic potential. Different numerical approaches are discussed to solve the evolution problem: (a) an implicit approach to describe the surface using a phase-field description, (b) a parametric finite element approach, and (c) a spectral method based on nonequispaced fast Fourier transforms on the sphere. Results for computed minimal energy configurations are discussed for various particle numbers and are compared with known rigorous asymptotic results. Furthermore extensions to other more complex and evolving surfaces are mentioned.

A mechanism for cell motility by active polar gels

We analyse a generic motility model, with the motility mechanism arising by contractile stress due to the interaction of myosin and actin. A hydrodynamic active polar gel theory is used to model the cytoplasm of a cell and is combined with a Helfrich-type model to account for membrane properties. The overall model allows consideration of the motility without the necessity for local adhesion. Besides a detailed numerical approach together with convergence studies for the highly nonlinear free boundary problem, we also compare the induced flow field of the motile cell with that of classical squirmer models and identify the motile cell as a puller or pusher, depending on the strength of the myosin–actin interactions.

Orientational Order on Surfaces: The Coupling of Topology, Geometry, and Dynamics

We consider the numerical investigation of surface bound orientational order using unit tangential vector fields by means of a gradient flow equation of a weak surface Frank–Oseen energy. The energy is composed of intrinsic and extrinsic contributions, as well as a penalization term to enforce the unity of the vector field. Four different numerical discretizations, namely a discrete exterior calculus approach, a method based on vector spherical harmonics, a surface finite element method, and an approach utilizing an implicit surface description, the diffuse interface method, are described and compared with each other for surfaces with Euler characteristic 2. We demonstrate the influence of geometric properties on realizations of the Poincaré–Hopf theorem and show examples where the energy is decreased by introducing additional orientational defects.

A Navier-Stokes phase-field crystal model for colloidal suspensions

We develop a fully continuous model for colloidal suspensions with hydrodynamic interactions. The Navier-Stokes Phase-Field Crystal model combines ideas of dynamic density functional theory with particulate flow approaches and is derived in detail and related to other dynamic density functional theory approaches with hydrodynamic interactions. The derived system is numerically solved using adaptive finite elements and is used to analyze colloidal crystallization in flowing environments demonstrating a strong coupling in both directions between the crystal shape and the flow field. We further validate the model against other computational approaches for particulate flow systems for various colloidal sedimentation problems.

Development and Analysis of a Block-Preconditioner for the Phase-Field Crystal Equation

We develop a preconditioner for the linear system arising from a finite element discretization of the phase-field crystal (PFC) equation. The PFC model serves as an atomic description of crystalline materials on diffusive time scales and thus offers the opportunity to study long time behavior of materials with atomic details. This requires adaptive time stepping and efficient time-discretization schemes, for which we use an embedded Rosenbrock scheme. To resolve spatial scales of practical relevance, parallel algorithms are also required, which scale to large numbers of processors. The developed preconditioner provides such a tool. It is based on an approximate factorization of the system matrix and can be implemented efficiently. The preconditioner is analyzed in detail and shown to speed up the computation drastically.

Structure and dynamics of interfaces between two coexisting liquid-crystalline phases

A phase-field-crystal model is used to access the structure and thermodynamics of interfaces between two coexisting liquid-crystalline phases in two spatial dimensions. Depending on the model parameters, there is a variety of possible coexistences between two liquid-crystalline phases, including a plastic triangular crystal (PTC). Here, we numerically calculate the profiles for the mean density and for the nematic order tensor across the interface for isotropic-PTC and columnar-PTC (or equivalently smectic-A-PTC) phase coexistence. As a general finding, the width of the interface with respect to the nematic order parameter characterizing the orientational order is larger than the width of the mean-density interface. In approaching the interface from the PTC side, at first, the mean density goes down, and then the nematic order parameter follows. The relative shift in the two profiles can be larger than a full lattice constant of the plastic crystal. Finally, we also present numerical results for the dynamic relaxation of an initial order-parameter profile towards its equilibrium interfacial profile. Our predictions for the interfacial profiles can, in principle, be verified in real-space experiments of colloidal dispersions.

A microscopic field theoretical approach for active systems

We consider a microscopic modeling approach for active systems. The approach extends the phase field crystal (PFC) model and allows us to describe generic properties of active systems within a continuum model. The approach is validated by reproducing results obtained with corresponding agent-based and microscopic phase field models. We consider binary collisions, collective motion and vortex formation. For larger numbers of particles we analyze the coarsening process in active crystals and identify giant number fluctuation in a cluster formation process.

An adaptive finite element multi-mesh approach for interacting deformable objects in flow

Abstract We consider a hydrodynamic multi-phase field problem to model the interaction of deformable objects. The numerical approach considers one phase field variable for each object and allows for an independent adaptive mesh refinement for each variable. Using the special structure of various terms allows interpolating the solution on one mesh onto another without loss of information. Together with a general multi-mesh concept for the other terms speedup by a factor of two can be demonstrated which improves with the number of interacting objects. The general concept is demonstrated on an example describing the interaction of red blood cells in an idealized vessel.

Nematic liquid crystals on curved surfaces: a thin film limit

We consider a thin film limit of a Landau–de Gennes Q-tensor model. In the limiting process, we observe a continuous transition where the normal and tangential parts of the Q-tensor decouple and various intrinsic and extrinsic contributions emerge. The main properties of the thin film model, like uniaxiality and parameter phase space, are preserved in the limiting process. For the derived surface Landau–de Gennes model, we consider an L2-gradient flow. The resulting tensor-valued surface partial differential equation is numerically solved to demonstrate realizations of the tight coupling of elastic and bulk free energy with geometric properties.