|Computational/Applied Math Seminar - SPRING 2014 - Wed 3:35-4:20 Ayres Hall 123|
|Jan 15||Ignacio Tomas, U Maryland||
From Micropolar Navier-Stokes equations to Ferrofluids: Analysis and Numerics
The Micropolar Navier-Stokes Equations (MNSE), is a system of nonlinear parabolic partial differential equations coupling linear velocity, pressure and angular velocity, i.e.: material particles have both translational and rotational degrees of freedom. The MNSE is a central component of the Rosensweig model for ferrofluids, describing the linear velocity, angular velocity, and magnetization inside the ferrofluids, while subject to distributed magnetic forces and torques.
We present the basic PDE results for the MNSE (energy estimates and existence theorems), together with a first order semi-implicit fully-discrete scheme which decouples the computation of the linear and angular velocities. Similarly, for the Rosensweig model we present the basic PDE results, together with an fully-discrete scheme combining Continuous Galerkin and Discontinuous Galerkin techniques in order to guarantee discrete energy stability. Finally, we demonstrate the capabilities of the Rosensweig model and its numerical implementation with some numerical simulations in the context of ferrofluid pumping by means of external magnetic fields.
|Jan 22||Travis Thompson, JICS, UTK||
Dispersion and superconvergence analysis of continuous finite element approximations of the linear transport equation
An analysis of the dispersion error and some superconvergence properties of continuous finite element approximations of the linear transport equation will be presented. An new tool based on an integral commutator is proposed to carry out this analysis. A superconvergence result for odd degree polynomial approximations is established on centro-symmetric meshes and a fourth-order super-convergence is obtained for the piecewise linear approximation on some particular meshes.
|Jan 29||CANCELLED due to ice||rescheduled for Apr. 9, see below|
|TUE Feb 4, in A404||Timothy Truster , CEE, UTK||
Stabilized Discontinuous Galerkin Method for Modeling Material and Structural Interfaces
A framework is presented for deriving a Discontinuous Galerkin interface method from an underlying Lagrange multiplier formulation. The approach hinges upon the use of the Variational Multiscale method to derive stabilizing terms that enable the condensation of the Lagrange multipliers at the interface. In the process, explicit expressions arise for the numerical flux and penalty parameter that account for variation in the material properties and element geometry. The framework has been applied in the area of solid mechanics problems to accurately and efficiently tie together nonconforming distorted meshes and heterogeneous materials. The numerical flux terms also provide a natural mechanism to embed constitutive models for interfacial response. As motivating examples, the extension of the method and associated numerical results are shown for capturing delamination in composite materials and for incorporating physics-based friction models into simulations of bolted mechanical joints.
|Feb 12||CANCELED due to (expected) snow||
rescheduled for Mar. 5, see below
|Feb 19||Enrique Otarola, U Maryland||
A PDE approach to numerical fractional difussion
We study PDE solution techniques for problems involving fractional powers of symmetric coercive elliptic operators in a bounded domain with Dirichlet boundary conditions. These operators can be realized as the Dirichlet to Neumann map of a degenerate/singular elliptic problem posed on a semi-infinite cylinder, which we analyze in the framework of weighted Sobolev spaces. Motivated by the rapid decay of the solution of this problem, we propose a truncation that is suitable for numerical approximation. We discretize this truncation using first order tensor product finite elements. We derive a priori error estimates in weighted Sobolev spaces, which exhibit optimal regularity but suboptimal order for quasi-uniform meshes and quasi-optimal order-regularity for anisotropic meshes.
We also present some results on efficient computational techniques to solve problems involving fractional powers of the fractional Laplace operator: adaptivity and multilevel methods. As a first step towards adaptivity, we present a computable a posteriori error estimator, which exhibits built-in flux equilibration and is equivalent to the energy error up to data oscillation. We design a simple adaptive strategy, which reduces error and data oscillation. We also discuss a V -cycle multilevel method with a line smoother, together with a nearly uniform convergence result on anisotropic tensor product discretizations. Numerical experiments reveal a competitive performance of this method.
Finally, to show the flexibity of our approach, we consider the discretization of evolution equations with fractional diffusion and fractional time derivative. We show discrete stability estimates which yield a novel energy estimate for evolution problems with a fractional time derivative. We consider a first order semi-implicit fully-discrete scheme: first degree tensor product finite elements in space and a first order discretization in time. We present a priori error estimates for the proposed numerical scheme.
|Mar 5||Xiabong Feng , Math, UTK||
Numerical Differential Calculus: A New Paradigm for Developing Numerical Methods for PDEs
In this talk I shall first present a newly developed (discontinuous Galerkin finite element) differential calculus theory for approximating weak (or distributional) derivatives of broken Sobolev functions. Various properties and calculus rules (such as product and chain rule, integration by parts formula and divergence theorem) for the proposed numerical derivatives will be presented. I shall then discuss how to use those numerical differential calculus machineries to systematically design numerical methods for various linear and nonlinear (including fully nonlinear) PDEs.
|Mar 12||Eirik Endeve , CSM, ORNL||
Discontinuous Galerkin Methods for Simulation of Supernova Neutrino Transport
Explosions of massive stars (i.e., core-collapse supernovae) are the dominant source of heavy elements in the Universe. As multi-messenger events, they are targeted by instruments covering most of the electromagnetic spectrum, as well as by gravitational wave and neutrino detectors. Multi-scale simulations are necessary to understand details of the explosion mechanism, and their connection to the observed signals. To this end, we are developing numerical methods for simulation of supernova neutrino transport. In this work, we aim to develop robust, high-order methods for phase space advection that preserve a maximum principle for the distribution function f (0 ≤ f ≤ 1 for neutrinos), and that are applicable to curvilinear phase space coordinates. Our numerical methods are based on the Runge-Kutta discontinuous Galerkin method. We discuss the physical model, the construction of maximum-principle-satisfying methods for phase space advection, and present numerical results from implementations in spherical and axial symmetry. Our results demonstrate that the method is high-order accurate and that the distribution function preserves the maximum principle. We also discuss challenges of simulating radiation propagating in a strong gravitational field.
|Mar 26||Vasilios Alexiades , Math, UTK||
Parameter identification via sensitivity and optimization
We study a model of a biochemical cascade (triggered by photons in retinal photoreceptors, which constitutes the first stage of vision). The cascade (with multi-stage shutoff of activated rhodopsin) is described by 70 reactions involving 16 primary parameters. A sensitivity analysis suggests that 4 of the parameters affect the response the most. We present an optimization approach to find parameters that result in desired peak and timing of response matching experimental data.
|Apr 9||Cory Hauck , ORNL||
Recent progress on the implementation of entropy-based moment closures
We present recent progress on the implementation of entropy-based moment closures in the context of a simple, linear kinetic equation. The algorithm has two main, coupled components: a second-order kinetic scheme to update the PDE and a Newton-based solver for the dual of the optimization problem that defines the closure. We study in detail the difficulties of solving the dual problem near the boundary of realizable moments, where quadrature formulas are less reliable and the Hessian of the dual objective function is highly ill-conditioned. Extensive numerical experiments are performed to illustrate these difficulties. In cases where the dual problem becomes "too difficult" to solve numerically, we propose a regularization technique to artificially move moments away from the realizable boundary in a way that still preserves local particle concentrations. Results are given for benchmark problems in one and two dimensions. In the latter case, a strategy for parallelization on heterogeneous architectures has been devised in order to reduce the high cost of solving millions of optimization problems.
Cody Lorton , Math, UTK
Yukun Li , Math, UTK
Cody Lorton: An Efficient Numerical Method for Wave Scattering in Random Media
Wave scattering in random media arises in many scientific and engineering field including geoscience, materials science and medical science. Computing quantities of interest for the solutions of such wave problems, especially, in the high frequency case, poses a daunting computational challenge because of sheer amount of computations required to solve those problems. Due to their strong indefiniteness, highly oscillatory nature of solutions, and lack of efficient iterative solvers, standard numerical approaches such as brute force Monte Carlo methods and stochastic Galerkin methods are either too expensive to use or do not work well. In this talk we shall present a newly developed multi-resolution approach for the random Helmholtz problem with large wave numbers. In this approach the original random Helmholtz problem is reduced to a finite number of nearly deterministic and non-homogeneous Helmholtz problems with random source terms, which are discretized by some unconditionally stable discontinuous Galerkin methods. An efficient solver with computational complexity of order O(3N^3/2) is also proposed to solve the resulting algebraic problems. Convergence analysis and numerical experiments will be presented to demonstrate the potential advantages of the proposed numerical approach.
Yukun Li: Discontinuous Galerkin methods for the Allen-Cahn equation and the mean curvature flow
This talk is concerned with some new convergence results for interior penalty discontinuous Galerkin (IPDG) approximations of the Allen-Cahn and its sharp interface limit known as the mean curvature flow. The main result to be presented is the convergence of the numerical interfaces to the sharp interface of the mean curvature flow as both the numerical mesh parameters and the phase field parameter (called the interaction length) tend to zero. The crux for establishing this result is to derive, by a nonstandard technique, error estimates for the IPDG solutions which blows up only polynomially (instead of exponentially) in the reciprocal of the phase field parameter. Numerical experiments will also be presented to gauge the performance of the proposed IPDG methods.
|End of Spring 2014 semester||Thank you for your attendance, Happy Summmer!|
|Computational/Applied Math Seminar - FALL 2013 - Wed 3:35 Ayres 122|
|Aug 28||Yi Zhang, Math, UTK||Finite Element Methods for Fourth Order Variational Inequalities|
|Sep 4||Abner Salgado , Math, UTK||Electrowetting on Dielectric: Modelling, Analysis and Computation|
|Sep 11||Yi Zhang, Math, UTK||Finite Element Methods for Fourth Order Variational Inequalities, Part 2|
|Sep 18||- none - SEARCDE on Sep21-22||—|
|Sep 25||Shawn Walker , Louisiana State||A New Mixed Formulation For a Sharp Interface Model of Stokes Flow and Moving Contact Lines|
|Oct 2||Vasilios Alexiades UTK||Comparison of time steppers for laser ablation|
|Oct 9||Lukas Vlcek, ORNL||Development of Realistic Molecular and Mesoscale Models of Fluids|
|Oct 16||- none - Fall break eve||—|
|Oct 23||Miroslav Stoyanov , ORNL||Algorithm Based Solver Resilience with Respect to Silent Hardware Faults|
|Oct 30||Jun Jia , ORNL||Accelerating time integration using spectral deferred correction|
|Nov 6||Jean-Luc Guermond , Texas A&M||Revisiting first-order viscosity for continuous finite element approximation of nonlinear conservation equations|
|Nov 13||Timo Heister , Clemson U||Grad-Div based preconditioning for incompressible flow problems|
|Nov 20||Harbir Antil , George Mason U||Convergence of reduced basis method|
|Dec 11||Rich Lehoucq , Sandia National Lab||A Nonlocal Vector Calculus and Nonlocal Balance Laws|