PoCET: a Polynomial Chaos Expansion Toolbox for Matlab

We investigate subcritical resonant heat transfer in a heated periodic grooved channel by modulating the flow with an oscillation of random amplitude. This excitation effectively destabilizes the flow at relatively low Reynolds number and establishes strong communication between the grooved flow and the Tollmien–Schlichting channel waves, as revealed by various statistical quantities we analysed. Both single-frequency and multi-frequency responses are considered, and an optimal frequency value is obtained in agreement with previous deterministic studies. In particular, we employ a new approach, the multi-element generalized polynomial chaos (ME-gPC) method, to model the stochastic velocity and temperature fields for uniform and Beta probability density functions (PDFs) of the random amplitude. We present results for the heat transfer enhancement parameter $E$ for which we obtain mean values, lower and upper bounds as well as PDFs. We first study the dependence of the mean value of $E$ on the magnitude of the random amplitude for different Reynolds numbers, and we demonstrate that the deterministic results are embedded in the stochastic simulation results. Of particular interest are the PDFs of $E$, which are skewed with their peaks increasing towards larger values of $E$ as the Reynolds number increases. We then study the effect of multiple frequencies described by a periodically correlated random process. We find that the mean value of $E$ is increased slightly while the variance decreases substantially in this case, an indication of the robustness of this excitation approach. The stochastic modelling approach offers the possibility of designing ‘smart’ PDFs of the stochastic input that can result in improved heat transfer enhancement rates.

We address the problem of approximating the moments of the solution, , of an Itô stochastic differential equation (SDE) with drift and diffusion terms over a time grid . In particular, we assume an explicit numerical scheme for the generation of sample paths and then obtain recursive equations that yield any desired noncentral moment of as a function of the initial condition . The core of the methodology is the decomposition of the numerical solution into a “central part” and an “effective noise” term. The central term is computed deterministically from the ordinary differential equation (ODE) that results from eliminating the diffusion term in the SDE, while the effective noise accounts for the stochastic deviation from the numerical solution of the ODE. For simplicity, we describe the proposed methodology based on an Euler–Maruyama integrator, but other explicit numerical schemes can be exploited in the same way. We also apply the moment approximations to construct estimates of the 1-dimensional marginal probability density functions of based on a Gram–Charlier expansion. Both for the approximation of moments and 1-dimensional densities, we describe how to handle the cases in which the initial condition is fixed (i.e., for some deterministic and known ) or random. In the latter case, we resort to polynomial chaos expansion (PCE) schemes in order to approximate the target moments. The methodology has been inspired by the PCE and differential algebra methods used for uncertainty propagation in astrodynamics problems. Hence, we illustrate its application for the quantification of uncertainty in a 2-dimensional Keplerian orbit perturbed by a Wiener noise process.

Sparse polynomial chaos expansions of frequency response functions using stochastic frequency transformation

We consider the semilinear heat equation ut = Δu + up both in ℝN and in a bounded domain with homogeneous Dirichlet boundary conditions, with 1 < p < ps where ps is the Sobolev exponent. This problem has solutions with finite‐time blowup; that is, for large enough initial data there exists T < ∞ such that u is a classical solution for 0 < t < T, while it becomes unbounded as t ↗ T. In order to understand the situation for t > T, we consider a natural approximation by reaction problems with global solution and pass to the limit. As is well‐known, the limit solution undergoes complete blowup: after it blows up at t = T, the continuation is identically infinite for all t > T.We perform here a deeper analysis of how complete blowup occurs. Actually, the singularity set of a solution that blows up as t ↗ T propagates instantaneously at time t = T to cover the whole space, producing a catastrophic discontinuity between t = T− and t = T+. This is called the “avalanche.” We describe its formation as a layer appearing in the limit of the natural approximate problems. After a suitable scaling, the initial structure of the layer is given by the solution of a limit problem, described by a simple ordinary differential equation. As t proceeds past T, the solutions of the approximate problems have a traveling wave behavior with a speed that we compute. The situation in the inner region depends on the type of approximation: a conical pattern is formed with time when we approximate the power up by a flat truncation at level n, while for tangent truncations we get an exponential increase in time and a diffusive spatial pattern. © 2003 Wiley Periodicals, Inc.

The kinetics of slag metal reactions are complex and often transient, in the sense that interfacial area, the equilibrium driving force, temperature gradients, and fluid properties are changing with time. This highly transient behavior is challenging to model using simple ordinary differential equations, and new theoretical approaches must be developed to deal with the complexity associated with these systems. Three examples from recent studies are described to illustrate methods of analyzing transient behavior. The first is desulfurization of steel in ladle metallurgy where the equilibrium driving force is changing with time, and the second is the case of reacting droplets in oxygen steelmaking where “bloating” of the droplet has a dramatic effect on the kinetics. The third is in the case of reactions between an alloy droplet and slag that result in large changes in interfacial area due to surface tension driven flows.

Electrodynamic tethers produce low thrust through interaction of the electric current in the tether with the Earth's magnetic field. The thrust is comparable with that of ion rockets and Hall thrusters, and they have the added advantage that they are propellantless, allowing them to produce an order of magnitude greater velocity changes than ion rockets. However, the long conductors of such electrodynamic thrusters typically exhibit unstable behaviors with higher currents. Instability affects both libration and bending modes of tether motion and significantly limits the performance characteristics of electrodynamic tether thrusters. Previous concepts for electrodynamic tethers have proposed stabilizing them by vertically under the gravity gradient, but this passive gravity-gradient stabilization severely limits the current in the conductor, and therefore limits the thrust. Two methods have been developed to stabilize electrodynamic tethers and improve their performance. First, the system spins with an average spin rate significantly higher than the orbital rate, increasing tether tension for a more robust and controllable tether system, and providing a better orientation of the tether with respect to the magnetic field for orbital maneuvering. Second, electric current variation is used to control both the tether spin parameters and the tether bending modes. It is shown that a controlled, spinning electrodynamic tether can consistently deliver a much higher thrust compared with the traditional hanging tether configuration. Minimum-time orbit transfers with spinning tethers can be described approximately by a set of relatively simple ordinary differential equations using Pontryagin's Principle. These techniques were developed to control the dynamics of the Spinning Electrodynamic Tether (SET) system. This uses a conductor two to ten kilometers long as an electrodynamic thruster for a low-thrust orbit transfer vehicle. The SET was simulated with a PC-based computer program to evaluate its orbit transfer capabilities. This vehicle is capable of repeated large orbit changes in low earth orbit, totaling >50 km/sec each year for several years.

The solute drag effect of phosphorus (P) in the single-phase austenite (γ-Fe) region was studied under isothermal annealing conditions at temperatures of 1050 °C, 1150 °C, 1250 °C and 1350 °C. High-temperature laser scanning confocal microscopy (HT-LSCM) was used to observe and quantify in situ grain growth on the surface of three different samples containing 0.026 wt.-% P, 0.044 wt.-% P and 0.102 wt.-% P. The dependence of the arithmetic mean grain sizes on time and temperature were modeled mathematically using a simple ordinary differential equation (ODE) according to normal grain growth (NGG) theory. As no other major effects, i.e., Zener-pinning by precipitates, occurred under the selected experimental conditions, grain growth interference was only considered by grain boundary (GB) segregation of P. Thus, the total grain boundary mobility (M) was directly determined depending on the P content and isothermal annealing temperature. The fitted GB mobility values enabled the determination of an average binding energy value between impurity P atoms and grain boundaries (E0 = -1.1 - -0.6 eV) in the system. Finally, the GB segregation of P in γ-Fe was derived from the observed grain growth kinetics. The results showed reasonable agreement with calculations using segregation enthalpies from the literature.

A simple model of the tracer flux from the Mururoa lagoon to the Pacific

In this paper, a hardware-efficient auditory neuron model, whose nonlinear dynamics is described by an ergodic cellular automaton, is designed. It is shown that the model can reproduce first pitch-shift effect induced by three tones, which is one of the nonlinear sound processing functions observed in the biological auditory system. The model and a simple ordinary differential equation (ODE) auditory neuron model are implemented on field-programmable gate arrays (FPGAs) and it is demonstrated that the model designed in this paper employs fewer circuit elements for FPGA implementation and consumes lower power than the simnle ODE model.

The aim of this paper is to conduct a statistical analysis of the effects of the fillet radii on the dynamics of the falling plate using the nonintrusive spectral projection (NISP) method. The free fall of two-dimensional cards immersed in a fluid was studied using a deterministic and stochastic numerical approach. The motion is characterized by the fluid-body interaction described by coupling the Navier–Stokes and rigid body dynamic equations. The model’s predictions have been validated using both experimental and numerical data available in the literature. In the stochastic simulations, the fillet radius of the plate was considered a random variable characterized by a uniform probability density function introducing, in this way, some uncertainties in the plate’s trajectory. To take into account the uncertainties, we employed the NISP method based on polynomial chaos expansion. The analysis was focused on finding the ensemble mean trajectory and error bar for a confidence interval of 95 % for both tumbling and fluttering regimes.

Quantifying Uncertainty in CFD

Numerical analysis of the Burgers’ equation in the presence of uncertainty

In reality, the behavior and nature of nonlinear dynamical systems are ubiquitous in many practical engineering problems. The mathematical models of such problems are often governed by a set of coupled second-order differential equations to form multi-degree-of-freedom (MDOF) nonlinear dynamical systems. It is extremely difficult to find the exact and analytical solutions in general. In this paper, the homotopy analysis method is presented to derive the analytical approximation solutions for MDOF dynamical systems. Four illustrative examples are used to show the validity and accuracy of the homotopy analysis and modified homotopy analysis methods in solving MDOF dynamical systems. Comparisons are conducted between the analytical approximation and exact solutions. The results demonstrate that the HAM is an effective and robust technique for linear and nonlinear MDOF dynamical systems. The proof of convergence theorems for the present method is elucidated as well.

Robustness of stochastic expansions for the stability of uncertain nonlinear dynamical systems – Application to brake squeal

In model‐based optimization in the presence of model‐plant mismatch, the set of model parameter estimates which satisfy an identification objective may not result in an accurate prediction of the gradients of the cost‐function and constraints. To ensure convergence to the optimum, the predicted gradients can be forced to match the measured gradients by adapting the model parameters. Since updating all available parameters is impractical due to estimability problems and overfitting, there is a motivation for adapting a subset of parameters for updating the predicted outputs and gradients. This article presents an approach to select a subset of parameters based on the sensitivities of the model outputs and of the cost function and constraint gradients. Furthermore, robustness to uncertainties in initial batch conditions is introduced using a robust formulation based on polynomial chaos expansions. The improvements in convergence to the process optimum and robustness are illustrated using a fed‐batch bioprocess. © 2017 American Institute of Chemical Engineers AIChE J, 63: 2660–2670, 2017