In practical flow simulations the time step size is usually chosen based on considerations of accuracy and stability. The maximum time step size that can be used is oftentimes restricted
by stability, which is especially the case as Reynolds number becomes large. With the often-used semi-implicit schemes, in which the nonlinear term of the Navier-Stokes
equations is treated explicitly and the viscous term implicitly, at high Reynolds numbers the maximum allowable time step size dictated by the CFL number can be orders of magnitude
smaller than the Kolmogorov time scale, which is the smallest time scale in turbulence. Therefore, the time step sizes with semi-implicit type schemes widely employed in current
flow simulations can be overly small to be computationally efficient. On the other hand, fully implicit schemes exhibit favorable stability properties, but entail the iterative solution of
nonlinear algebraic equations, rendering the overall approach inefficient.
In this talk I will discuss a linearly implicit numerical scheme for incompressible Navier-Stokes equations that allows the use of large time step sizes and is also
computationally efficient. The time step size with the new scheme can be one to two orders of magnitude larger than the maximum allowable time step size with semi-implicit type
schemes. Some notable features of the scheme include: (1) it is a splitting type scheme, decoupling pressure and velocity solves; (2) it requires solving only a linear
convection-diffusion equation at each time step, involving no nonlinear algebraic solves. Several flow problems will be used to compare the time step sizes and computational cost
between the new scheme and the semi-implicit scheme.
This is a joint work with J. Shen
The difficulties in computer modeling of human blood are largely due to modeling the motion
of multiple, interacting, highly deformable red blood cells.
A red blood cell has the form of a biconcave disc with a diameter in the
nanometer range enclosed by a membrane. The membrane can be modeled as a nonlinear
viscoelastic thin shell, where the nonlinearity is caused by large displacements (assuming
small strains).
The usual approach to a numerical solution is to derive the differential balance equations
starting from integral balance equations, and then to discretize these equations and solve
them numerically. In particular, the description of curvature and curvature related terms
may become very complicated while dealing with problems involving moving surfaces and
we need a fair amount of tensor calculus.
Alternatively, the equations can be formulated in terms of exterior differential forms using
the method of moving frames. Then the model equations are derived from the first variation
of the free energy functional of the problem, which has in our case a geometric part and a
part related to the in-plane strain. This work was done by Tu and Ou-Yang.
If we follow this approach, the model equations come with a rich exterior calculus structure
which can be translated into discrete exterior calculus later.
In this talk I will first summarize the model equations for red blood cells using shell theory
and then briefly present the moving frame method. I will compare the Euler-Lagrange
equations with the force and momentum balance equations, and identify geometric terms,
bending, and in-plane strain terms. I will discuss which terms and properties need to be
conserved and make some comments on how they translate into discrete exterior calculus.
Multiphase flows are characterized by the interaction between different media. In this talk, I will focus on Fluid-Structure Interaction problems, which are characterized by the interaction between a deformable solid structure and an internal or surrounding fluid flow. Solid and fluid mechanics have been studied for a long time, but the coupling between the two presents unexpected intriguing challenges which have yet to be fully understood. The main mathematical challenges involved in the study of fluid-structure interactions are the following:
In this talk I will present a numerical method for simulating fluid-particle interactions in a two-dimensional channel filled with an Oldroyd-B fluid. The fluid-particle interaction is a two-phase flow problem. Recently we have combined a fictitious domain/distributed Lagrange multiplier method for the fluid-particle problem with a factorization approach via an operator splitting technique. The new scheme preserves the positive definiteness of the conformation tensor at the discrete level. I will show numerical validations for the method and also the simulations for fluid-particle interactions for the cases of one particle, two particles, and multiple particles.
PyDEC is our Python software package for Discretization of Exterior Calculus in which we implemented a primal-dual version of discrete exterior calculus (the DEC of my thesis) as well as lowest order finite element exterior calculus via Whitney forms. A fast implementation of the Whitney inner product in PyDEC provides a way to construct stiffness and mass matrices of finite element exterior calculus conveniently. I'll describe and then show the code and demo of 3 applications. The first application is our recent work on optimal homology where the only feature of PyDEC used is the computation of boundary matrices. Second application will be Darcy flow in a primal-dual DEC framework. I'll develop it in stages, from vector calculus, to exterior calculus, to DEC, to the linear system including a careful development of the boundary conditions. The third application will be the curl-curl resonant cavity problem in which Whitney forms are used for stiffness and mass matrices as well as visualization of forms as vector fields. The first two applications are the content of two of our papers and the curl-curl problem is a problem popularized by Doug Arnold. The goal of my talk is not to stress the new developments of our homology or Darcy flow work, but to show off the capabilities of PyDEC. PyDEC is joint work with N. Bell (Nvidia) and a paper about it is in preparation. The optimal homology theory is joint work with T. Dey (OSU) and B. Krishnamoorthy (WSU) and details are in a paper accepted at STOC10 and available as e-print on arXiv. The Darcy flow work is joint work with K. Nakshatrala (TAMU) and J. Chaudhry (Illinois) and is the content of an e-print on arXiv.
I will give a "learning-style" talk on how to use differential forms to describe the geometry of surfaces, including curvature, geodesics, etc.
In this talk, we introduce a geometric derivation of the discrete equations of motion for fluid dynamics from first principles in a purely Eulerian form. Our approach approximates the group of volume-preserving diffeomorphisms using a finite dimensional Lie group, and associated discrete Euler equations are derived from a variational principle with non-holonomic constraints. The resulting discrete equations of motion yield a structure-preserving time integrator with good long-term energy behavior and for which an exact discrete Kelvin's circulation theorem holds. This is joint work with D. Pavlov, P. Mullen, J.E. Marsden, M. Desbrun (Caltech), and E. Kanso (USC).
In this talk, I will present resent progresses we have made for two projects we have been working on.
One project is the study of the blood clot formation in veins. I will present a multiscale model of blood thrombus formation by integrating a detailed tissue factor pathway submodel of blood coagulation, a blood flow submodel and a stochastic discrete cellular Potts submodel. Surface reactions of the extrinsic coagulation pathway on membranes of platelets are studied under different flow conditions. The model is combined with experimental approach and image analysis to study the role of factor VII and protein C. It is shown that low levels of FVII in blood result in a significant delay in thrombin production demonstrating that FVII plays an active role in promoting thrombus development at an early stage. In addition, I will discuss a new subcellular element method for simulating cellular components of the blood and their interactions.
The other project is the study of the swarming pattern formation for Pseudomonas aeruginosa. We develop a thin liquid film submodel which is coupled with the cell-based discrete submodel to study population-dependent surface swarming of cells. One focus of the study is to analyze the fractal structure developed by the swarms due to the liquid film surface tension driven instability caused the surfactant secreted by the cells.
The Nature is rich in fluid-deformable-structure-interaction problems. It has become apparent that the immersed boundary (IB) method is a practical and important method for numerically solving problems of this type. In this talk we will discuss the fundamentals of the IB method (formulation, derivation, and algorithms). Some recent applications in both two and three dimensions will be briefly introduced at the end.