Applied/ACMS

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Applied and Computational Mathematics Seminar


Spring 2024

date speaker title host(s)
Feb 2 Chris Rycroft (UW) The reference map technique for simulating complex materials and multi-body interactions
Feb 9 Scott Weady (Flatiron Institute) Entropy methods in active suspensions Saverio and Laurel
Feb 16 David Saintillan (UC San Diego) TBA Saverio and Tom
Feb 23 Rose Cersonsky (UW) TBA Chris
Mar 1 [4:00pm Colloquium] Per-Gunnar Martinsson (UT Austin) TBA Li
Mar 8
Mar 15 Di Qi (Purdue University) TBA Chen
Mar 22
Mar 29 Spring break
Apr 5 Jinlong Wu (UW) TBA Saverio
Apr 12 Gabriel Zayas-Caban (UW) TBA Li
Apr 19 Tony Kearsley (NIST) TBA Fabien
Apr 26 Malgorzata Peszynska (Oregon State) TBA Fabien

Abstracts

Chris Rycroft (UW–Madison)

Title: The reference map technique for simulating complex materials and multi-body interactions

Conventional computational methods often create a dilemma for fluid–structure interaction problems. Typically, solids are simulated using a Lagrangian approach with grid that moves with the material, whereas fluids are simulated using an Eulerian approach with a fixed spatial grid, requiring some type of interfacial coupling between the two different perspectives. Here, a fully Eulerian method for simulating structures immersed in a fluid will be presented [1]. By introducing a reference map variable to model finite-deformation constitutive relations in the structures on the same grid as the fluid, the interfacial coupling problem is highly simplified. The method is particularly well suited for simulating soft, highly-deformable materials and many-body contact problems [2], and several examples in two and three dimensions [3] will be presented.

  1. K. Kamrin, C. H. Rycroft, and J.-C. Nave, J. Mech. Phys. Solids 60, 1952–1969 (2012). [DOI link]
  2. C. H. Rycroft et al., J. Fluid Mech. 898, A9 (2020). [DOI link]
  3. Y. L. Lin, N. J. Derr, and C. H. Rycroft, Proc. Natl. Acad. Sci. 119, e2105338118 (2022). [DOI link]


Chris Rycroft (UW–Madison)

Title: Entropy methods in active suspensions

Collections of active particles, such as suspensions of E. coli or mixtures of microtubules and molecular motors, can exhibit rich non-equilibrium dynamics due to a combination of activity, hydrodynamic interactions, and steric stresses. Continuum kinetic theories, which characterize the set of particle configurations through a continuous distribution function, provide a powerful framework for analyzing such systems and connecting their micro- to macroscopic dynamics. The probabilistic formulation of kinetic theories leads naturally to a characterization in terms of entropy, whether thermodynamic or information-theoretic. In equilibrium systems, entropy strictly increases and always tends towards steady state. This no longer holds in active systems, however entropy still has a convenient mathematical structure. In this talk, we use entropy methods, specifically variational principles involving the relative entropy functional, to study the nonlinear dynamics and stability of active suspensions in the context of the Doi-Saintillan-Shelley kinetic theory. We first present a class of moment closures that arise as constrained minimizers of the relative entropy, and show these closures preserve the kinetic theory's stability and entropic structure while admitting efficient numerical simulation. We then derive variational bounds on relative entropy fluctuations for apolar active suspensions that are closely related to the moment closures. These bounds provide conditions for global stability and yield estimates of time-averaged order parameters. Finally, we discuss applications of these methods to polar active suspensions.

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