University of Minnesota
http://www.umn.edu/

Mathematics of Materials and Macromolecules: Multiple Scales, Disorder, and Singularities, September 2004 - June 2005

## IMA Tutorial/Workshop:

March 28-30, 2005

Gerard Awanou (IMA, University of Minnesota)

Trivariate Spline Approximations of 3D Navier-Stokes Equations

The numerical analysis of the Navier-Stokes equations is very important. Among many reasons it plays an important role in the study of the cardiovascular system. For example, blood flow in large arteries can be modeled by the Navier-Stokes equations.

There are many computational methods available in the literature for the numerical solution of the 3D Navier-Stokes equations. New and more efficient methods are being developed to increase the power of computational flow simulations. To achieve significant improvements for the quality of computer simulations for real-life problems is not only dependent on the continuously increasing computing power, but also the approximation power of the numerical methods.

We propose to use trivariate spline functions for the numerical solution of 3D Navier-Stokes equations. This approach also captures the smoothness of the solution inside the domain.

Elena Dimitrova (Virginia Bioinformatics Institute (0477), Virginia Tech)

A Graph-theoretic Method for the Discretization of Gene Expression Measurements

The paper introduces a method for the discretization of experimental data into a finite number of states. While it is of interest in various fields, this method is particularly useful in bioinformatics for reverse engineering of gene regulatory networks built from gene expression data. Many of these applications require discrete data, but gene expression measurements are continuous. Statistical methods for discretization are not applicable due to the prohibitive cost of obtaining sample sets of sufficient size. We have developed a new method of discretizing the variables of a network into the same optimal number of states while at the same time maintaining high information content. We employ a graph-theoretic clustering method to affect the discretization of gene expression measurements. Our C++ program takes as an input one or more time series of gene expression data and discretizes these values into a number of states that best fits the data. The method is being validated by incorporating it into the recently published computational algebra approach to the reverse engineering of gene regulatory networks by Laubenbacher and Stigler.

Qiang Du (Department of Mathematics, Pennsylvania State University)

Phase Field Modeling and Simulation of Cell Membranes

Recently, we have produced a series of works on the phase field modeling and simulation of vesicle bio-membranes formed by lipid bilayers. We have considered both the shape deformation of vesicles minimizing the elastic bending energy with volume and surface area constraints and those moving in an incompressible viscous fluid. Rigorous mathematical analysis have been carried out along with extensive numerical experiments. We have also developed useful computational techniques for detecting the topological changes within a broad phase field framework.

References:

1. A Phase Field Approach in the Numerical Study of the Elastic Bending Energy for Vesicle Membranes, Q. Du, C. Liu and X. Wang, J. Computational Physics, 198, pp450-468, 2004
2. Retrieving topological information for phase field models, Q. Du, C. Liu and X. Wang, 2004, to appear in SIAM J. Appl. Math
3. Phase field modeling of the spontaneous curvature effect in cell membranes, Q. Du C. Liu, R. Ryham and X. Wang, 2005, to appear in CPAA
4. A phase field formulation of the Willmore problem. Q. Du, C. Liu, R. Ryham and X. Wang, 2005, to appear in Nonlinearity

Changfeng Gui (Department of Mathematics, University of Connecticut)

Level Set Evolution without Re-initialization: A New Variational Formulation

Level set methods have been a powerful tool and extensively used in image processing and computer vision. Periodically reinitializing the level set function to a signed distance function during the curve evolution is a critical numerical remedy to maintain stable curve evolution and ensure usable results. However, most of the level set methods are plagued with such questions as when and how to reinitialize the level set function to a signed distance function.

I will present a variational level set formulation of active contours without reinitialization. The evolution of the level set function is the gradient flow of an energy functional. We define an energy functional with two terms: the external energy term and the internal energy term. The external energy characterizes the conformity of the zero level set and the object boundaries, while the internal energy penalizes the deviation of the level set function from a signed distance function. As a result of introducing such internal energy term, the reinitialization procedure is eliminated, and the level set function converges very well in the whole image domain, with zero level set converging to the desired object boundaries. Moreover, the proposed variational level set formulation has at least the following practical advantages over the traditional level set formulations. Firstly, significantly larger time step can be used in our method for numerically solving the evolution PDE, and therefore the evolution is faster than the traditional level set methods. Secondly, the curve evolution is less sensitive to the location of the initial contour. Thirdly, the evolving curve can be stopped at quite weak object boundaries. Our method has been successfully applied to synthetic and real images of different modalities, including optical, MR, and ultrasound images.

Viet Ha Hoang (Department of Applied Mathematics and Theoretical Physics, CMS, United Kingdom University of Cambridge)

High-dimensional Finite Elements for Elliptic Problems with Multiple Scales

Joint work with Christoph Schwab.

Elliptic homogenization problems in a d dimensional domain with n+1 separated scales are reduced to elliptic one-scale problems in dimension (n+1)d. These one-scale problems are discretized by a sparse tensor product finite element method (FEM). We prove that this sparse FEM has accuracy, work and memory requirement comparable to standard FEM for single scale problems in while it gives numerical approximations of the correct homogenized limit as well as of all first order correctors, throughout the physical domain with performance independent of the physical problem's scale parameters. Numerical examples for model diffusion problems with two and three scales confirm our results.

Frederic Legoll (IMA, University of Minnesota)

Analysis of a Prototypical Multiscale Method Coupling Atomistic and Continuum Mechanics

In order to describe a solid which deforms smoothly in some region, but non smoothly in some other region, many multiscale methods have been recently proposed, that aim at coupling an atomistic model (discrete mechanics) with a macroscopic model (continuum mechanics). We present here a theoretical analysis for such a coupling in a one-dimensional setting. We study both the general case of a convex energy and a specific example of a nonconvex energy, the Lennard-Jones case.

In the latter situation, we prove that the discretization needs to account in an adequate way for the coexistence of a discrete model and a continuous one. Otherwise, spurious discretization effects may appear. We also consider the effect of the discretization of the continuum model on the behaviour of the coupled model.

This work is joint with Xavier Blanc (Paris 6) and Claude Le Bris (CERMICS, ENPC).

Melvin Leok (Department of Mathematics, University of Michigan) http://www.math.lsa.umich.edu/~mleok/

Generalized Galerkin Variational Integrators: Lie Group, Multiscale and Spectral Methods
Slides:  pdf

Geometric mechanics involves the study of Lagrangian and Hamiltonian mechanics using geometric and symmetry techniques. Computational algorithms obtained from a discrete Hamilton's principle yield a discrete analogue of Lagrangian mechanics, and they exhibit excellent structure-preserving properties that can be ascribed to their variational derivation.

We propose a natural generalization of discrete variational mechanics, whereby the discrete action, as opposed to the discrete Lagrangian, is the fundamental object. This is achieved by appropriately choosing a finite dimensional function space to approximate sections of the configuration bundle and numerical quadrature techniques to approximate the action integral.

We will discuss how this general framework allows us to recover high-order Galerkin variational integrators, asynchronous variational integrators, and symplectic-energy-momentum integrators. In addition, we will also introduce generalizations such as high-order symplectic-energy-momentum integrators, Lie group integrators, high-order Euler-Poincare integrators, multiscale variational integrators, and pseudospectral variational integrators.

This framework will be illustrated by an application of Lie group variational integrators to rigid body dynamics wherein the discrete trajectory evolves in the space of 3x3 matrices, while automatically staying on the rotation group, without the use of local coordinates, constraints, or reprojection.

This is joint work with Taeyoung Lee and Harris McClamroch.

Peter Philip (IMA, University of Minnesota) http://www.ima.umn.edu/~philip/homepage/

Numerical Simulation of Heat Transfer in Materials with Anisotropic Thermal Conductivity: A Finite Volume Scheme to Handle Complex Geometries

A finite volume scheme suitable for nonlinear heat transfer in materials with anisotropic thermal conductivity is formulated, focussing on the difficulties arising from the discretization of complex domains which are typical in the simulation of industrially relevant processes. The discretization is based on unstructured constrained Delaunay triangulations of the domain. For simplicity, it is assumed that the thermal conductivity tensor has vanishing off-diagonal entries and that the anisotropy is independent of the temperature. Numerical simulations verify the accuracy of the method in two test cases where a closed-form solution is available. Further results demonstrate the effectiveness of the method in computing the heat transfer in a complex growth apparatus used in crystal growth.

Jie Shen (Department of Mathematics, Purdue University )  http://www.math.purdue.edu/~shen

Numerical Simulations of Drop Pinching Using a Phase-Field Model

The dynamics and pinch-off phenomena of a liquid filament is studied by using a phase field model which describes the motion of mixtures of two incompressible fluids. An efficient and accurate numerical scheme is presented and implemented for the coupled nonlinear system of Navier-Stokes equations and Allen-Cahn phase equation. Detailed numerical simulations for a Newtonian fluid filament falling into another ambient Newtonian fluid are carried out. The dynamical scaling behavior and the pinch-off behavior, as well as the formation of the consequent satellite droplets are investigated.

Igor Tsukerman (Department of Electrical & Computer Engineering , University of Akron)

A New Finite-Difference Calculus and Its Applications

The generic Taylor approximation in standard Finite-Difference (FD) methods often fails to capture with sufficient accuracy many important features of the solution, for example: discontinuities at material interfaces; boundary layers; dominant dipole components near polarized or magnetized spherical particles; electrostatic double layers around colloidal particles, and much more. A new FD calculus of Flexible Local Approximation MEthods (FLAME) replaces the Taylor polynomials with any desired local basis functions, such as exponentials, cylindrical or spherical harmonics, plane waves, and so on. The accuracy of the solution is improved, both quantitatively and qualitatively.

Although the method usually operates on regular Cartesian grids, it is in some cases much more accurate than the Finite Element Method with its complex meshes. While one motivation is to minimize the notorious 'staircase' effect at curved and slanted interface boundaries, the new approach has much broader applications and implications.

Illustrative examples include 3-point schemes of arbitrarily high order for the Schrödinger equation and for a 1D singular equation; fields of finite-size charged, polarized and/or magnetized colloidal particles in free space or in a solvent with or without salt; scattering of electromagnetic waves; plasmon resonances; wave propagation in a photonic crystal. Furthermore, many existing FD schemes are revealed as natural particular cases of the FLAME calculus: low- and high-order Taylor schemes, the Collatz "Mehrstellen" schemes, flux balance / control volume schemes, and a few more special techniques.