Marcio Gameiro

Three-Dimensional Analysis of Solid Oxide Fuel Cell Ni-YSZ Anode Interconnectivity

A method is described for quantitatively analyzing the level of interconnectivity of solid-oxide fuel cell electrode phases. The method was applied to the three-dimensional microstructure of a Ni-Y2O3-stabilized ZrO2 (Ni-YSZ) anode active layer measured by focused ion beam scanning electron microscopy. Each individual contiguous network of Ni, YSZ, and porosity was identified and labeled according to whether it was contiguous with the rest of the electrode. It was determined that the YSZ phase was 100% connected, whereas at least 86% of the Ni and 96% of the pores were connected. Triple-phase boundary (TPB) segments were identified and evaluated with respect to the contiguity of each of the three phases at their locations. It was found that 11.6% of the TPB length was on one or more isolated phases and hence was not electrochemically active.

Combinatorial-topological framework for the analysis of global dynamics

We discuss an algorithmic framework based on efficient graph algorithms and algebraic-topological computational tools. The framework is aimed at automatic computation of a database of global dynamics of a given m-parameter semidynamical system with discrete time on a bounded subset of the n-dimensional phase space. We introduce the mathematical background, which is based upon Conleys topological approach to dynamics, describe the algorithms for the analysis of the dynamics using rectangular grids both in phase space and parameter space, and show two sample applications.

Validated continuation over large parameter ranges for equilibria of PDEs

Validated continuation was introduced [S. Day, J.-P. Lessard, K. Mischaikow, Validated continuation for equilibria of PDEs, SIAM Journal on Numerical Analysis, in press] as means of checking that the classical continuation method applied to a Galerkin projection of a PDE provides a locally unique equilibrium to the PDE of interest. In this paper, we extend the numerical technique to include a parameter that leads to better bounds on the errors associated with the Galerkin truncation. We test this method on the Swift-Hohenberg and Allen-Cahn equations on one-dimensional domains. For the first equation, we find no numerical obstructions to the validated continuation technique. This is not the case for the Allen-Cahn equation.

Efficient Rigorous Numerics for Higher-Dimensional PDEs via One-Dimensional Estimates

We present an efficient rigorous computational method which is an extension of the work Analytic Estimates and Rigorous Continuation for Equilibria of Higher-Dimensional PDEs (M. Gameiro and J.-P. Lessard, J. Differential Equations, 249 (2010), pp. 2237--2268). The idea is to generate sharp one-dimensional estimates using interval arithmetic which are then used to produce high-dimensional estimates. These estimates are used to construct the radii polynomials which provide an efficient way of determining a domain on which the contraction mapping theorem is applicable. Computing the equilibria using a finite-dimensional projection, the method verifies that the numerically produced equilibrium for the projection can be used to explicitly define a set which contains a unique equilibrium for the PDE. A new construction of the polynomials is presented where the nonlinearities are bounded by products of one-dimensional estimates as opposed to using FFT with large inputs. It is demonstrated that with this approac...

Homology and symmetry breaking in Rayleigh-Bénard convection: Experiments and simulations

Algebraic topology (homology) is used to analyze the state of spiral defect chaos in both laboratory experiments and numerical simulations of Rayleigh-Benard convection. The analysis reveals topological asymmetries that arise when non-Boussinesq effects are present. The asymmetries are found in different flow fields in the simulations and are robust to substantial alterations to flow visualization conditions in the experiment. However, the asymmetries are not observable using conventional statistical measures. These results suggest homology may provide a new and general approach for connecting spatiotemporal observations of chaotic or turbulent patterns to theoretical models.

Applications of robust synchronization to communication systems

In this work, using chaotic systems, we study the role of synchronization on codification and decodification of messages. We first present a general result that is useful to prove uniform dessipativeness for nonautonomous systems of ordinary differential equations. Then some theorems are established to give sufficient conditions to obtain synchronization of coupled systems. The above results are applied to some specfic coupled systems, namely, coupled Lorenz systems, coupled Duffings equations, coupled Chuas systems, etc., showing how to code and decode message using chaotic systems. One of our main results is to obtain the robustness of the synchronization with respect to parameter variation.

Rigorous Numerics for ill-posed PDEs: Periodic Orbits in the Boussinesq Equation

In this paper, we develop computer-assisted techniques for the analysis of periodic orbits of ill-posed partial differential equations. As a case study, our proposed method is applied to the Boussinesq equation, which has been investigated extensively because of its role in the theory of shallow water waves. The idea is to use the symmetry of the solutions and a Newton–Kantorovich type argument (the radii polynomial approach) to obtain rigorous proofs of existence of the periodic orbits in a weighted ℓ1 Banach space of space-time Fourier coefficients with exponential decay. We present several computer-assisted proofs of the existence of periodic orbits at different parameter values.

A Posteriori Verification of Invariant Objects of Evolution Equations: Periodic Orbits in the Kuramoto--Sivashinsky PDE

In this paper, a method for computing periodic orbits of the Kuramoto--Sivashinsky PDE via rigorous numerics is presented. This is an application and an implementation of the theoretical method introduced in [J.-L. Figueras, M. Gameiro, J.-P. Lessard, and R. de la Llave, “A framework for the numerical computation and a posteriori verification of invariant objects of evolution equations,” SIAM J. Appl. Dyn. Syst., to appear]. Using a Newton--Kantorovich-type argument (the radii polynomial approach), existence of solutions is obtained in a weighted

A Framework for the Numerical Computation and A Posteriori Verification of Invariant Objects of Evolution Equations

\ell^\infty

Rigorous numerics for piecewise-smooth systems

Banach space of Fourier coefficients. Once a proof of a periodic orbit is done, an associated eigenvalue problem is solved and Floquet exponents are rigorously computed, yielding proofs that some periodic orbits are unstable. Finally, a predictor-corrector continuation method is introduced to rigorously compute global smooth branches of periodic orbits. An alternative approach and independent implementation of [J.-L. Figueras, M. Gameiro, J.-P. Less...