151 research outputs found
Hermite-Gauss functions in the analysis of a category of semiconductor optical devices
Available from British Library Document Supply Centre-DSC:DXN017041 / BLDSC - British Library Document Supply CentreSIGLEGBUnited Kingdo
Perturbed, Entropy-Based Closure for Radiative Transfer
We derive a hierarchy of closures based on perturbations of well-known
entropy-based closures; we therefore refer to them as perturbed entropy-based
models. Our derivation reveals final equations containing an additional
convective and diffusive term which are added to the flux term of the standard
closure. We present numerical simulations for the simplest member of the
hierarchy, the perturbed M1 or PM1 model, in one spatial dimension. Simulations
are performed using a Runge-Kutta discontinuous Galerkin method with special
limiters that guarantee the realizability of the moment variables and the
positivity of the material temperature. Improvements to the standard M1 model
are observed in cases where unphysical shocks develop in the M1 model.Comment: 35 pages, 8 figure
Numerical optimal control with applications in aerospace
This thesis explores various computational aspects of solving nonlinear, continuous-time dynamic optimization problems (DOPs) numerically. Firstly, a direct transcription method for solving DOPs is proposed, named the integrated residual method (IRM). Instead of forcing the dynamic constraints to be satisfied only at a selected number of points as in direct collocation, this new approach alternates between minimizing and constraining the squared norm of the dynamic constraint residuals integrated along the whole solution trajectories. The method is capable of obtaining solutions of higher accuracy for the same mesh compared to direct collocation methods, enabling a flexible trade-off between solution accuracy and optimality, and providing reliable solutions for challenging problems, including those with singular arcs and high-index differential-algebraic equations.
A number of techniques have also been proposed in this work for efficient numerical solution of large scale and challenging DOPs. A general approach for direct implementation of rate constraints on the discretization mesh is proposed. Unlike conventional approaches that may lead to singular control arcs, the solution of this on-mesh implementation has better numerical properties, while achieving computational speedups. Another development is related to the handling of inactive constraints, which do not contribute to the solution of DOPs, but increase the problem size and burden the numerical computations. A strategy to systematically remove the inactive and redundant constraints under a mesh refinement framework is proposed.
The last part of this work focuses on the use of DOPs in aerospace applications, with a number of topics studied. Using example scenarios of intercontinental flights, the benefits of formulating DOPs directly according to problem specifications are demonstrated, with notable savings in fuel usage. The numerical challenges with direct collocation are also identified, with the IRM obtaining solutions of higher accuracy, and at the same time suppressing the singular arc fluctuations.Open Acces
Rigorous data-driven computation of spectral properties of Koopman operators for dynamical systems
Koopman operators are infinite-dimensional operators that globally linearize
nonlinear dynamical systems, making their spectral information useful for
understanding dynamics. However, Koopman operators can have continuous spectra
and infinite-dimensional invariant subspaces, making computing their spectral
information a considerable challenge. This paper describes data-driven
algorithms with rigorous convergence guarantees for computing spectral
information of Koopman operators from trajectory data. We introduce residual
dynamic mode decomposition (ResDMD), which provides the first scheme for
computing the spectra and pseudospectra of general Koopman operators from
snapshot data without spectral pollution. Using the resolvent operator and
ResDMD, we also compute smoothed approximations of spectral measures associated
with measure-preserving dynamical systems. We prove explicit convergence
theorems for our algorithms, which can achieve high-order convergence even for
chaotic systems, when computing the density of the continuous spectrum and the
discrete spectrum. We demonstrate our algorithms on the tent map, Gauss
iterated map, nonlinear pendulum, double pendulum, Lorenz system, and an
-dimensional extended Lorenz system. Finally, we provide kernelized
variants of our algorithms for dynamical systems with a high-dimensional
state-space. This allows us to compute the spectral measure associated with the
dynamics of a protein molecule that has a 20,046-dimensional state-space, and
compute nonlinear Koopman modes with error bounds for turbulent flow past
aerofoils with Reynolds number that has a 295,122-dimensional
state-space
Probabilistic Framework for Sensor Management
A probabilistic sensor management framework is introduced, which maximizes the utility of sensor systems with many different sensing modalities by dynamically configuring the sensor system in the most beneficial way. For this purpose, techniques from stochastic control and Bayesian estimation are combined such that long-term effects of possible sensor configurations and stochastic uncertainties resulting from noisy measurements can be incorporated into the sensor management decisions
Real-Space Mesh Techniques in Density Functional Theory
This review discusses progress in efficient solvers which have as their
foundation a representation in real space, either through finite-difference or
finite-element formulations. The relationship of real-space approaches to
linear-scaling electrostatics and electronic structure methods is first
discussed. Then the basic aspects of real-space representations are presented.
Multigrid techniques for solving the discretized problems are covered; these
numerical schemes allow for highly efficient solution of the grid-based
equations. Applications to problems in electrostatics are discussed, in
particular numerical solutions of Poisson and Poisson-Boltzmann equations.
Next, methods for solving self-consistent eigenvalue problems in real space are
presented; these techniques have been extensively applied to solutions of the
Hartree-Fock and Kohn-Sham equations of electronic structure, and to eigenvalue
problems arising in semiconductor and polymer physics. Finally, real-space
methods have found recent application in computations of optical response and
excited states in time-dependent density functional theory, and these
computational developments are summarized. Multiscale solvers are competitive
with the most efficient available plane-wave techniques in terms of the number
of self-consistency steps required to reach the ground state, and they require
less work in each self-consistency update on a uniform grid. Besides excellent
efficiencies, the decided advantages of the real-space multiscale approach are
1) the near-locality of each function update, 2) the ability to handle global
eigenfunction constraints and potential updates on coarse levels, and 3) the
ability to incorporate adaptive local mesh refinements without loss of optimal
multigrid efficiencies.Comment: 70 pages, 11 figures. To be published in Reviews of Modern Physic
A Review of Computational Stochastic Elastoplasticity
Heterogeneous materials at the micro-structural level are usually subjected to several uncertainties. These materials behave according to an elastoplastic model, but with uncertain parameters. The present review discusses recent developments in numerical approaches to these kinds of uncertainties, which are modelled as random elds like Young's modulus, yield stress etc. To give full description of random phenomena of elastoplastic materials one needs adequate mathematical framework. The probability theory and theory of random elds fully cover that need. Therefore, they are together with the theory of stochastic nite element approach a subject of this review. The whole group of di erent numerical stochastic methods for the elastoplastic problem has roots in the classical theory of these materials. Therefore, we give here the classical formulation of plasticity in very concise form as well as some of often used methods for solving this kind of problems. The main issues of stochastic elastoplasticity as well as stochastic problems in general are stochastic partial di erential equations. In order to solve them we must discretise them. Methods of solving and discretisation are called stochastic methods. These methods like Monte Carlo, Perturbation method, Neumann series method, stochastic Galerkin method as well as some other very known methods are reviewed and discussed here
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