70,543 research outputs found

### Numerical simulations of a polidisperse sedimentation model by using spectral WENO method with adaptative multiresolution

In this work, we apply adaptive multiresolution (Harten’s approach) characteristic-wise fifth-order Weighted Essentially Non-Oscillatory (WENO) for computing the numerical solution of a polydisperse sedimentation model, namely, the Höfler and Schwarzer model. In comparison to other related works, time discretization is carried out with the ten-stage fourth-order strong stability preserving Runge–Kutta method which is more efficient than the widely used optimal third-order TVD Runge–Kutta method. Numerical results with errors, convergence rates and CPU times are included for four and 11.Departamento Administrativo de Ciencia, Tecnología e Innovación [CO] Colciencias1215-569-33836Métodos adaptativos de multiresolución aplicado a la solución numérica de ciertos modelos descritos matemáticamente por leyes de conservaciónn

### Regular sensitivity calculation and gradient-based optimization of chaotic dynamical systems

A gradient of a quantity-of-interest J with respect to problem parameters can augment the utility of a predictive simulation. By itself, the gradient provides sensitivity information to parameters, which can aid uncertainty quantification. Gradient-based optimization can be used in both scientific and engineering applications, including design optimization, data-assimilated modeling and nonmodal stability analysis.
However, obtaining useful gradients for chaotic systems is challenging. The extreme sensitivity to perturbations that defines chaos amplifies the gradient exponentially in time, which impedes both sensitivity analysis and gradient-based optimization. Fundamentally, any J defined in a chaotic system becomes highly non-convex in time. For such non-convex J, Taylor expansions are useful only in small neighborhoods, which restricts the utility of the gradients, even if computed exactly. Thus they do not indicate a useful parametric sensitivity or guidance toward a useful optimum.
We examine this challenge and investigate routes to circumvent these challenges in two applications. The first is sensitivity computation in particle-in-cell (PIC) simulations involving plasma kinetics. PIC is attractive for representing non-equilibrium plasma distributions in the six-dimensional velocity--position phase-space. To do this, Lagrangian simulation particles represent the position and velocity distribution in a statistical sense. However, computing sensitivity for PIC methods is challenging due to the chaotic dynamics of these particles, and sensitivity techniques remain underdeveloped compared to those for Eulerian discretizations. This challenge is examined from a dual particle--continuum perspective that motivates a new sensitivity discretization. Two routes to sensitivity computation are presented and compared: a direct fully-Lagrangian particle-exact approach provides sensitivities of each particle trajectory, and a new particle-pdf discretization. The new formulation involves a continuum perspective but it is discretized by particles to take the advantages of the same type of Lagrangian particle description leveraged by PIC methods. Since the sensitivity particles in this approach are only indirectly linked to the plasma-PIC particles, they can be positioned and weighted independently for efficiency and accuracy. The corresponding numerical algorithms are presented in mathematical detail. The advantage of the particle-pdf approach in avoiding the spurious chaotic sensitivity of the particle-exact approach is demonstrated for Debye shielding and sheath configurations. In essence, the continuum perspective makes implicit the distinctness of the particles, which is irrelevant to most prediction goals. In this way it circumvents the Lyapunov instability of the N-body PIC system. The cost of the particle-pdf approach is comparable to the baseline PIC simulation.
The other case considered is optimal control of turbulent flow. Evidence supports the possibility of control of turbulence in some applications, in that there seem to be useful, larger-scale components of the flow, which are less chaotic, in the midst of smaller-scale chaotic turbulence fluctuations. While there have been many attempts to extract model descriptions of such components from the full dynamics, such models are often limited in their applicability or accounting for nonlinearity of turbulence. Thus the full dynamics of turbulent flow is needed to be accurately predictive in simulations. However, in this case the sensitivity of the more chaotic turbulent fluctuations masks that of the useful component of the flow control. This challenge is illustrated with a model control problem of the Lorenz system and analyzed in two aspects: the growth of gradients and non-convexity of J. The horseshoe mapping of chaotic dynamical systems is identified as the root-cause mechanism for both aspects of the challenge, and its impact is quantitatively evaluated in various chaotic flow systems, ranging from the Kuramoto--Sivashinsky Equation to a three-dimensional turbulent Kolmogorov flow. A new optimization framework is proposed based on a penalty-based method. In essence, the simulation time is split into multiple intervals and auxiliary states are introduced at intermediate time points, at which the governing equation is not strictly constrained, thus introducing discontinuities in time. These discontinuities allows J to be more convex, thus enlarging search scale in the optimization space. They are exploited in this sense then gradually suppressed with increasingly stronger penalty. This multi-step penalty-based optimization is first demonstrated with a one-dimensional logistic map and the Lorenz example. Then its effectiveness is further demonstrated for more complex chaotic systems and ultimately for turbulent Kolmogorov flow. The proposed method finds a solution that suppresses large-scale pressure fluctuations without laminarization, which suggests its ability to target useful components of the flow in the midst of chaotic turbulence, thereby showing its potential for practical turbulent flow controls. It far outperforms a simple gradient-based search.U of I OnlyAuthor requested U of Illinois access only (OA after 2yrs) in Vireo ETD syste

### MAS: A versatile Landau-fluid eigenvalue code for plasma stability analysis in general geometry

We have developed a new global eigenvalue code, Multiscale Analysis for
plasma Stabilities (MAS), for studying plasma problems with wave toroidal mode
number n and frequency omega in a broad range of interest in general tokamak
geometry, based on a five-field Landau-fluid description of thermal plasmas.
Beyond keeping the necessary plasma fluid response, we further retain the
important kinetic effects including diamagnetic drift, ion finite Larmor
radius, finite parallel electric field, ion and electron Landau resonances in a
self-consistent and non-perturbative manner without sacrificing the attractive
efficiency in computation. The physical capabilities of the code are evaluated
and examined in the aspects of both theory and simulation. In theory, the
comprehensive Landau-fluid model implemented in MAS can be reduced to the
well-known ideal MHD model, electrostatic ion-fluid model, and drift-kinetic
model in various limits, which clearly delineates the physics validity regime.
In simulation, MAS has been well benchmarked with theory and other gyrokinetic
and kinetic-MHD hybrid codes in a manner of adopting the unified physical and
numerical framework, which covers the kinetic Alfven wave, ion sound wave,
low-n kink, high-n ion temperature gradient mode and kinetic ballooning mode.
Moreover, MAS is successfully applied to model the Alfven eigenmode (AE)
activities in DIII-D discharge #159243, which faithfully captures the frequency
sweeping of RSAE, the tunneling damping of TAE, as well as the polarization
characteristics of KBAE and BAAE being consistent with former gyrokinetic
theory and simulation. With respect to the key progress contributed to the
community, MAS has the advantage of combining rich physics ingredients,
realistic global geometry and high computation efficiency together for plasma
stability analysis in linear regime.Comment: 40 pages, 21 figure

### Implicit high-order gas-kinetic schemes for compressible flows on three-dimensional unstructured meshes

In the previous studies, the high-order gas-kinetic schemes (HGKS) have
achieved successes for unsteady flows on three-dimensional unstructured meshes.
In this paper, to accelerate the rate of convergence for steady flows, the
implicit non-compact and compact HGKSs are developed. For non-compact scheme,
the simple weighted essentially non-oscillatory (WENO) reconstruction is used
to achieve the spatial accuracy, where the stencils for reconstruction contain
two levels of neighboring cells. Incorporate with the nonlinear generalized
minimal residual (GMRES) method, the implicit non-compact HGKS is developed. In
order to improve the resolution and parallelism of non-compact HGKS, the
implicit compact HGKS is developed with Hermite WENO (HWENO) reconstruction, in
which the reconstruction stencils only contain one level of neighboring cells.
The cell averaged conservative variable is also updated with GMRES method.
Simultaneously, a simple strategy is used to update the cell averaged gradient
by the time evolution of spatial-temporal coupled gas distribution function. To
accelerate the computation, the implicit non-compact and compact HGKSs are
implemented with the graphics processing unit (GPU) using compute unified
device architecture (CUDA). A variety of numerical examples, from the subsonic
to supersonic flows, are presented to validate the accuracy, robustness and
efficiency of both inviscid and viscous flows.Comment: arXiv admin note: text overlap with arXiv:2203.0904

### Quantum Mechanics Lecture Notes. Selected Chapters

These are extended lecture notes of the quantum mechanics course which I am
teaching in the Weizmann Institute of Science graduate physics program. They
cover the topics listed below. The first four chapter are posted here. Their
content is detailed on the next page. The other chapters are planned to be
added in the coming months.
1. Motion in External Electromagnetic Field. Gauge Fields in Quantum
Mechanics.
2. Quantum Mechanics of Electromagnetic Field
3. Photon-Matter Interactions
4. Quantization of the Schr\"odinger Field (The Second Quantization)
5. Open Systems. Density Matrix
6. Adiabatic Theory. The Berry Phase. The Born-Oppenheimer Approximation
7. Mean Field Approaches for Many Body Systems -- Fermions and Boson

### A family of total Lagrangian Petrov-Galerkin Cosserat rod finite element formulations

The standard in rod finite element formulations is the Bubnov-Galerkin
projection method, where the test functions arise from a consistent variation
of the ansatz functions. This approach becomes increasingly complex when highly
nonlinear ansatz functions are chosen to approximate the rod's centerline and
cross-section orientations. Using a Petrov-Galerkin projection method, we
propose a whole family of rod finite element formulations where the nodal
generalized virtual displacements and generalized velocities are interpolated
instead of using the consistent variations and time derivatives of the ansatz
functions. This approach leads to a significant simplification of the
expressions in the discrete virtual work functionals. In addition, independent
strategies can be chosen for interpolating the nodal centerline points and
cross-section orientations. We discuss three objective interpolation strategies
and give an in-depth analysis concerning locking and convergence behavior for
the whole family of rod finite element formulations.Comment: arXiv admin note: text overlap with arXiv:2301.0559

### An iterative warping and clustering algorithm to estimate multiple wave-shape functions from a nonstationary oscillatory signal

Nonsinusoidal oscillatory signals are everywhere. In practice, the
nonsinusoidal oscillatory pattern, modeled as a 1-periodic wave-shape function
(WSF), might vary from cycle to cycle. When there are finite different WSFs,
$s_1,\ldots,s_K$, so that the WSF jumps from one to another suddenly, the
different WSFs and jumps encode useful information. We present an iterative
warping and clustering algorithm to estimate $s_1,\ldots,s_K$ from a
nonstationary oscillatory signal with time-varying amplitude and frequency, and
hence the change points of the WSFs. The algorithm is a novel combination of
time-frequency analysis, singular value decomposition entropy and vector
spectral clustering. We demonstrate the efficiency of the proposed algorithm
with simulated and real signals, including the voice signal, arterial blood
pressure, electrocardiogram and accelerometer signal. Moreover, we provide a
mathematical justification of the algorithm under the assumption that the
amplitude and frequency of the signal are slowly time-varying and there are
finite change points that model sudden changes from one wave-shape function to
another one.Comment: 39 pages, 11 figure

### Heat kernel-based p-energy norms on metric measure spaces

We focus on heat kernel-based p-energy norms (1<p<\infty) on bounded and
unbounded metric measure spaces, in particular, weak-monotonicity properties
for different types of energies. Such properties are key to related studies,
under which we generalise the convergence result of Bourgain-Brezis-Mironescu
(BBM) for p\neq2. We establish the equivalence of various p-energy norms and
weak-monotonicity properties when there admits a heat kernel satisfying the
two-sided estimates. Using these equivalences, we verify various
weak-monotonicity properties on nested fractals and their blowups. Immediate
consequences are that, many classical results on p-energy norms hold for such
bounded and unbounded fractals, including the BBM convergence and
Gagliardo-Nirenberg inequality.Comment: 39 pages with 1 figur

### Aero-thermal analysis of a laminar separation bubble subjected to varying free-stream turbulence: Large Eddy Simulation

A quantitative analysis illustrating salient features of a Laminar Separation
Bubble (LSB), its transition forming coherent structures, and associated heat
transfer has been performed on a flat plate for varying free stream turbulence
(fst) between 1.2% to 10.3%. A well-resolved Large Eddy Simulation (LES)
developed in-house is used for the purpose. Flow separation has been induced by
imposing an adverse pressure gradient on the upper boundary of a Cartesian
domain. Isotropic perturbations are introduced at the inlet to mimic grid
turbulence. With an increase of fst, an upstream shift in the mean reattachment
point has been observed while the onset of separation remains almost invariant,
shrinking the bubble length significantly. The transition of the shear layer is
triggered by the Kelvin-Helmholtz (K-H) instability for fst of less than 3.3%,
while Klebanoff modes (Kmodes) dictate the flow transition at fst greater than
6.5%. Further, a mixed mode, i.e., both K-H and K-modes, contribute to the flow
transition at a moderate level of fst, lying between 3.3% and 6.5%. Thus, the
roll-up of the shear layer appears in the second half of the bubble shedding
large-scale vortices that keep their identity far downstream at low fst levels.
On the contrary, the streamwise streaks via K-modes prior to the separation are
found to interact with the LSB, resulting in an earlier breakdown of the shear
layer with abundant small-scale vortices downstream at the moderate to high fst
levels. However, higher surface-normal heat flux is associated with large-scale
energetic coherent vortices

### A hybrid quantum algorithm to detect conical intersections

Conical intersections are topologically protected crossings between the
potential energy surfaces of a molecular Hamiltonian, known to play an
important role in chemical processes such as photoisomerization and
non-radiative relaxation. They are characterized by a non-zero Berry phase,
which is a topological invariant defined on a closed path in atomic coordinate
space, taking the value $\pi$ when the path encircles the intersection
manifold. In this work, we show that for real molecular Hamiltonians, the Berry
phase can be obtained by tracing a local optimum of a variational ansatz along
the chosen path and estimating the overlap between the initial and final state
with a control-free Hadamard test. Moreover, by discretizing the path into $N$
points, we can use $N$ single Newton-Raphson steps to update our state
non-variationally. Finally, since the Berry phase can only take two discrete
values (0 or $\pi$), our procedure succeeds even for a cumulative error bounded
by a constant; this allows us to bound the total sampling cost and to readily
verify the success of the procedure. We demonstrate numerically the application
of our algorithm on small toy models of the formaldimine molecule
(\ce{H2C=NH}).Comment: 15 + 10 pages, 4 figure

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