# Seminars in 2015

### The linearized problem of Magneto-Photoelasticity

Prof. Vladimir Sharafutdinov

Sobolev Institute of Mathematics

Novosibirsk State University, Russia

Date: Tuesday, January 20, 2015

Abstract: Let a plane polarized electromagnetic wave propagate in a homogeneous isotropic dielectric medium. If a constant exterior magnetic field (gyration vector) is applied to the medium, then the polarization plane rotates along a ray. This phenomenon was discovered by M. Faraday in 1845 and was named the Faraday rotation. In 1970, H. Aben proposed to use the Faraday rotation in photoelasticity and introduced the term magneto-photoelasticity.

We derive the equations of magneto-photoelasticity for a nonhomogeneous background isotropic medium and for a variable gyration vector. They coincide with Aben’s equations in the case of a homogeneous background medium and of a constant gyration vector. We obtain an explicit linearized formula for the fundamental solution under the assumption that variable coefficients of equations are sufficiently small. Then we consider the inverse problem of recovering the variable coefficients from the results of polarization measurements known for several values of the gyration vector. We demonstrate that the data can be easily transformed to a family of Fourier coefficients of the unknown function if the modulus of the gyration vector is agreed with the ray length.

### Killing Tensor Fields on the 2-Torus

Prof. Vladimir Sharafutdinov

Sobolev Institute of Mathematics

Novosibirsk State University, Russia

Date: Thursday, January 22, 2015

Abstract: A symmetric tensor field on a Riemannian manifold is called Killing field if the symmetric part of its covariant derivative is equal to zero. There is a one to one correspondence between Killing tensor fields and first integrals of the geodesic flow which depend polynomially on the velocity. Therefore Killing tensor fields closely relate to the problem of integrability of geodesic flows. In particular, the following question is still open: does there exist a Riemannian metric on the 2-torus which admits an irreducible Killing tensor field of rank ≥ 3? We obtain two necessary conditions on a Riemannian metric on the 2-torus for the existence of Killing tensor fields. The first condition is valid for Killing tensor fields of arbitrary rank and relates to closed geodesics. The second condition is obtained for rank 3 Killing tensor fields and relates to isolines of the Gaussian curvature.

### Axioms of Adaptivity

Prof. Carsten Carstensen

Center for Computational Sciences,

Humboldt-Universität zu Berlin, Germany

Date: 23 Feb 2014

Abstract: Four axioms (A1)--(A4) link estimators and distance functions on a set of admissible refinements together and imply optimality of a standard finite element routine on an abstract level with a loop: solve, estimate, mark, and refine. The presentation provides proofs and examples of the recent review due to C. Carstensen, M. Feischl, M. Page, and D. Praetorius: The axioms of adaptivity, Comput. Math. Appl. 67 (2014)1195–1253 and so discusses the current literature on the mathematics of adaptive finite element methods. The presentation concludes with an overview over several applications of the set of axioms. If time permits, some recent developments are discussed on ongoing joint work with Hella Rabus on separate marking.

### Data-driven Closure Modeling using Functional Inversion and Machine Learning

Prof. Karthik Duraisamy

Department of Aerospace Engineering,

University of Michigan, Ann Arbor.

Date: 19 May, 2015

Abstract: The recent acceleration in computational power and measurement resolution has made possible the availability of extreme scale simulations and data sets. Such data sets have offered powerful physical insights in many application areas of science and engineering. There is significant room for further progress if the highest fidelity computations and high resolution experiments can be utilized to construct truly predictive models of complex, multi-scale systems in regimes of practical interest. While it is intuitively obvious that informing simulations with data can lead to more accurate models, direct infusion of data has not penetrated most applications of computational science, especially those that involve complex and large scale systems. In this talk, a new paradigm for data-driven closure modeling is presented. A key facet of our approach is an inverse modeling strategy that is capable of using data (from high-fidelity computations as well as from experiments) to identify and quantify the functional form of discrepancies within the confines of existing models. A model that accounts for this discrepancy is then reconstructed using Machine Learning techniques and injected into the solution process. Demonstrative applications are shown in closure modeling of transitional and turbulent fluid flows.

### A Multilevel higher-order framework for linear wave propagation

Prof. Avijit Chatterjee

Department of Aerospace Engg. <<BR> IIT Bombay

Date: 11 September, 2015

Abstract: Higher-order accurate spatial accuracy in some form is necessary for simulating linear wave propagation in computational aeroacoustics (CAA) and computational electromagnetics (CEM). This stems mainly from the need to accurately model linear wave propagation both in terms of phase and amplitude over long distances and time. Higher-order methods tend to be expensive on a per-grid point basis and lead to large CPU times. Large simulation times, for example, are often encountered in wave scattering problems where the characteristic size of the object is much larger than the wavelength of the incident wave or for objects with re-entrant structures leading to multiple internal reflections. Unlike in boundary value problems, there have not been many instances in literature of successful advances at an algorithm level to offset the disadvantage of long simulation times encountered in higher-order and time-accurate simulation of linear wave propagation. An algebraic multilevel framework is proposed for accelerating higher-order accurate solution of physical systems involving linear wave propagation in an explicit finite- volume framework. As in the mostly finite element based p-multigrid method, the hyperbolic PDE is repeatedly cycled through spatial operators of varying orders of accuracy from highest to lowest on a fixed finite volume grid. The multilevel method is based on higher-order spatially accurate approximations to sufficiently smooth solutions of the linear hyperbolic PDE, being inexpensively and exactly maintained at coarser approximations in the purely advection process which characterizes linear wave propagation. This is unlike the p-multigrid method which relies on efficient smoothing of error components at coarser approximations for an accelerated steady state. Higher-order accuracy at coarser approximation levels in the present multilevel method is achieved exactly via an appropriate correction term based on the relative truncation error τ arising out of the purely advection process of linear wave propagation. Higher-order spatial accuracy can be enforced using the multilevel method at a fraction of the computational cost incurred in a conventional higher-order implementation. Candidate higher-order schemes tested in the framework for solving typical problems in CEM and CAA include ENO, spectral finite volume and ADER.

### Progress in all-Mach number flows and well-balanced methods for the compressible Euler equations

Prof. Christian Klingenberg

University of Würzburg, Germany

Date: 7 October, 2015

Abstract: Many astrophysical systems feature flows which are modeled by the multi-dimensional Euler equations. We discuss two aspects. (1) For the homogeneous Euler equations we look at flow in the low Mach number regime. Here for a conventional finite volume discretization one has excessive dissipation in this regime. We identify inconsistent scaling with low Mach number of numerical flux function as the origin of this problem. We propose a new flux preconditioner that ensures the correct scaling. We demonstrate that our new method is capable of representing flows down to Mach numbers of 10e-10. This is joint work with Wasilij Barsukow, Philipp Edelmann and Fritz Röpke. (2) For the Euler equations with gravity we seek well-balanced methods. We describe a numerical discretization of the compressible Euler equations with a gravitational potential. A pertinent feature of the solutions to these inhomogeneous equations is the special case of stationary solutions with zero velocity, described by a nonlinear PDE, whose solutions are called hydrostatic equilibria. We present a well-balanced method, for which we can ensure robustness, accuracy and stability, since it satisfies discrete entropy inequalities. This is joint work with Christophe Berthon, Praveen Chandrashekar, Vivien Desveaux and Markus Zenk.

### Discontinuous finite volume element methods and its applications

Sarvesh Kumar

Dept. of Mathematics

Indian Institute of Space Science and Technology

Thiruvananthapuram

Date: Oct 20, 2015

Abstract: In this talk, first we would like to address the convergence issues of a standard finite volume element method (FVEM) applied to simple elliptic problems. Then, we discuss discontinuous finite volume element methods (DFVEM) for elliptic problems and its advantages over the standard FVEM. Further, we present a natural extension of DFVEM which was employed for elliptic problems, to the Stokes problems.

We also talk about the applications of DFVEM for the approximation of miscible displacement problems. The mathematical model which describe the miscible displacement of one incompressible fluid by another in a porous medium is modeled by two coupled nonlinear partial differential equations; one is pressure-velocity equation and other is concentration equation. In this talk, we discuss a mixed FVEM for the approximation of the pressure-velocity equation and a DFVEM for the concentration equation. Also, a priori error estimates for velocity, pressure and concentration will be shown. Some Numerical experiments will be presented to substantiate the validity of the theoretical results.

### Automatic differentiation for the stochastic ode of finance

Olivier Pironneau

LJLL-UPMC, Paris VI

Date: 16 Nov, 2015

Abstract: In finance modeling uses almost always one or several stochastic differential or integro-differential equations (SODE). Agents in the financial world are computing daily thousands of sensitivities of expected value of solutions of SODE. We will answer the following questions:

1. Can these be computed by using automatic differentiation of computer programs

2. How does it compare with Mike Giles' Vibrato and Malliavin calculus ?

3. Furthermore how does it compare with the PDE approach ?

### Strategies for active flow control of high-speed jets

Aniruddha Sinha

Department of Aerospace Engineering

IIT Bombay

Date: 17 Nov, 2015

Abstract: The global challenges of improving performance, reducing wastage, and minimizing environmental impact for all flow systems require understanding and active control. The particular application chosen here is a high-speed turbulent jet controlled for either noise mitigation or mixing enhancement. Plasma actuators had previously demonstrated promise in achieving these goals in an open-loop control configuration. This talk will mainly focus on two strategies for active flow control. The first is a simple and robust technique called extremum-seeking control that doesn't require any model. However, for the same reason, it has a slow response speed. The second is a reduced-order modeling approach geared towards conventional model-based feedback control – it is designed to have a faster response. The model is developed using a combination of detailed experimentation and theoretical analysis, and it reproduces the overall effects of such actuation on the jet. The three main ingredients of the model are stochastic estimation, proper orthogonal decomposition, and Galerkin projection. Extensions of both the approaches to other flows are discussed.

### Wall-resolved LES of supersonic channel flows: Subgrid-scale closures for internal energy equation formulation

Sriram Raghunath

Mechanical Engineering

Michigan State University, United States

Date: 11 December 2015, Friday

Abstract: Subgrid-scale closures for wall-resolved explicit filtered LES of Supersonic Channel Flow with different thermal-wall boundary conditions are studied at bulk flow Mach number 1.5 and reference Reynolds number in wall units Re_tau = 190 and 340. Internal energy formulation is used to account for temperature in the system of equations, and three flow conditions : 1) two walls symmetrically isothermal, 2) symmetrically isothermal walls at different temperatures and 3) uniform property flow with viscous work subtracted from the energy equation, are taken up for study. In explicit filtered LES, the fields carry additional implicit filtering due to numerical error from truncation and the error is magnified due to nonlinearity of the viscous dissipation term containing velocity gradients. Mixed tensor-diffusivity model coupled with a dynamic Smagorinsky term is used for modeling SGS stress and SGS heat flux. The principle of partially reconstructing the resolved but unrepresented scales of the tensor-diffusivity model is extended to SGS viscous dissipation using a semi-empirical mixed model. The relative significance of the SGS terms are studied from a set of DNS data and a priori tests of the models are presented. Explicitly filtered LES is performed with three configurations of SGS viscous dissipation models on two levels of grid refinement. Arguments in favor of using a dynamically computed thermal diffusivity in place of extended mixing length analogy with a subgrid-scale Prandtl number, are made and validated. A new pair of two-term mixed models for viscous dissipation that accounts for the sub-filter scale stress work and global dissipation of the large-scales are proposed. From the large-eddy simulations of uniform property flows, the grid resolution was found to be a dominant factor influencing the accuracy of the simulations. In the variable property flows, the two term models are found to yield better prediction of dissipation and hence the mean temperature profiles. Comparing the LES on both the coarse and fine grid, the variable property simulation at Re_{\tau} = 340 yielded better results in terms of Reynolds stress correlation and {\it a posteriori} estimates of the SGS terms, indicating some favorable performance of the SGS models at a higher Reynolds number. The hot-wall half of the asymmetric channel simulations show a different behavior compared to the other flow cases in terms of SGS heat flux and viscous dissipation.