70,543 research outputs found

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

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    In this work, we apply adaptive multiresolution (Harten鈥檚 approach) characteristic-wise fifth-order Weighted Essentially Non-Oscillatory (WENO) for computing the numerical solution of a polydisperse sedimentation model, namely, the H枚fler and Schwarzer model. In comparison to other related works, time discretization is carried out with the ten-stage fourth-order strong stability preserving Runge鈥揔utta method which is more efficient than the widely used optimal third-order TVD Runge鈥揔utta method. Numerical results with errors, convergence rates and CPU times are included for four and 11.Departamento Administrativo de Ciencia, Tecnolog铆a e Innovaci贸n [CO] Colciencias1215-569-33836M茅todos adaptativos de multiresoluci贸n aplicado a la soluci贸n num茅rica de ciertos modelos descritos matem谩ticamente por leyes de conservaci贸nn

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

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    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

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    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

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    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

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    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

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    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

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    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, s1,,sKs_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 s1,,sKs_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

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    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

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    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

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    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 NN points, we can use NN 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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