A linear outer approximation of line losses for DC-based optimal power flow problems
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In this work, two quaternary diamond-like semiconductors (DLSs) in the Cu-Zn-Ge-Se system, the known Cu 2 ZnGeSe 4 (also referred to as CZGSe) and the new Cu 4 ZnGe 2 Se 7 , are compared in terms of their crystal structures, electronic structures and physicochemical properties. Both compounds were prepared by high-temperature, solid-state synthesis at 800 degrees C. Single crystal X-ray diffraction was used to determine the structure of Cu 4 ZnGe 2 Se 7 . The structures of both Cu 2 ZnGeSe 4 and Cu 4 ZnGe 2 Se 7 can be considered as derivatives of cubic diamond. Cu 4 ZnGe 2 Se 7 , with a reduced symmetry due to the cation ordering pattern, adopts the Cu 4 NiSi 2 S 7 structure type, with space group C2. The corner-sharing tetrahedra in Cu 4 ZnGe 2 Se 7 are slightly distorted because the charge for some of the S 2 - anions is not compensated by the first-nearest-neighbor cations. Rietveld refinements using synchrotron X-ray powder diffraction data were used to assess the phase purity of the samples and confirm the bulk structural behavior. Diffuse reflectance UV/Vis/NIR spectroscopy shows that Cu 2 ZnGeSe 4 and Cu 4 ZnGe 2 Se 7 have direct optical bandgaps of 1.38 and 0.91 eV, respectively. Electronic structure calculations implementing density functional theory confirm the direct bandgap for Cu 4 ZnGe 2 Se 7 , with a calculated value of 0.62 eV. Both compounds are air stable, thermally stable up to relatively high temperatures, undergo phase transitions and have wide windows of optical clarity. The Kurtz-Perry powder technique was used to determine the second harmonic generation (SHG) responses using a commercial AgGaSe 2 standard. While Cu 4 ZnGe 2 Se 7 displays a weak SHG response, Cu 2 ZnGeSe 4 exhibits a response that is greater than several benchmark materials, with a large SHG coefficient, χ (2) , of 43 ± 6 pm/V at λ = 2900 nm.
We present an integrated experimental and theoretical study of the dynamics and rheology of self-crosslinked, slightly charged, temperature responsive soft poly(N-isopropylacrylamide) (pNIPAM) microgels over a wide range of concentration and temperature spanning the sharp change in particle size and intermolecular interactions across the lower critical solution temperature (LCST). Dramatic, non-monotonic changes in viscoelasticity are observed as a function of temperature, with distinct concentration dependence in the dense fluid, glassy, and soft-jammed regimes. Motivated by our experimental observations, we formulate a minimalistic model for the size dependence of a single microgel particle and the change of the interparticle interaction from purely repulsive to attractive upon heating. Using microscopic equilibrium and time-dependent statistical mechanical theories, theoretical predictions are quantitatively compared with experimental measurements of the shear modulus. Good agreement is found for the nonmonotonic temperature behavior that originates as a consequence of the competition between reduced microgel packing fraction and increasing interparticle attractions. Furthermore, testable predictions are made for nonlinear rheological properties such as the yield stress and strain. To our knowledge, this is the first attempt to quantitatively understand in a unified manner the viscoelasticity of dense, temperature-responsive microgel suspensions spanning a wide range of temperatures and concentrations.
Uncertainty quantification techniques such as the time-dependent generalized polynomial chaos (TD-gPC) use an adaptive orthogonal basis to better represent the stochastic part of the solution space (aka random function space) in time. However, because the random function space is constructed using tensor products, TD-gPC-based methods are known to suffer from the curse of dimensionality. Here, we introduce a new numerical method called the flow-driven spectral chaos (FSC) which overcomes this curse of dimensionality at the random-function-space level. The proposed method is not only computationally more efficient than existing TD-gPC-based methods but is also far more accurate. The FSC method uses the concept of enriched stochastic flow maps to track the evolution of a finite-dimensional random function space efficiently in time. To transfer the probability information from one random function space to another, two approaches are developed and studied herein. In the first approach, the probability information is transferred in the mean-square sense, whereas in the second approach the transfer is done exactly using a new theorem that was developed for this purpose. The FSC method can quantify uncertainties with high fidelity, especially for the long-time response of stochastic dynamical systems governed by ODEs of arbitrary order. Six representative numerical examples, including a nonlinear problem (the Van-der-Pol oscillator), are presented to demonstrate the performance of the FSC method and corroborate the claims of its superior numerical properties. Finally, a parametric, high-dimensional stochastic problem is used to demonstrate that when the FSC method is used in conjunction with Monte Carlo integration, the curse of dimensionality can be overcome altogether.
Here we present a hybrid method for time-dependent particle transport problems that combines Monte Carlo (MC) estimation with deterministic solutions based on discrete ordinates. For spatial discretizations, the MC algorithm computes a piecewise constant solution and the discrete ordinates use bilinear discontinuous finite elements. From the hybridization of the problem, the resulting problem solved by Monte Carlo is scattering free, resulting in a simple, efficient solution procedure. Between time steps, we use a projection approach to “relabel” collided particles as uncollided particles. In conclusion, from a series of standard 2-D Cartesian test problems we observe that our hybrid method has improved accuracy and reduction in computational complexity of approximately an order of magnitude relative to standard discrete ordinates solutions.
The main challenge of large-scale numerical simulation of radiation transport is the high memory and computation time requirements of discretization methods for kinetic equations. In this work, we derive and investigate a neural network-based approximation to the entropy-based closure method to accurately compute the solution of the multi-dimensional moment system with a low memory footprint and competitive computational time. We extend methods developed for the standard entropy-based closure to the regularized entropy-based closures. The main idea is to interpret structure-preserving neural network approximations of the regularized entropy-based closure as a two-stage approximation to the original entropy-based closure. We conduct a numerical analysis of this approximation and investigate optimal parameter choices. Our numerical experiments demonstrate that the method has a much lower memory footprint than traditional methods with competitive computation times and simulation accuracy. The code and all trained networks are provided on GitHub.
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