Discontinuous Derivatives Research Articles

Thermal buckling of composite laminates with elastic boundaries

AbstractThe investigation of thermal buckling in composite laminates holds significant theoretical and practical implications. For the commonly encountered rectangular thin plate structures in engineering, this study aims to explore their buckling characteristics under thermal loads in arbitrary elastic boundary conditions. To achieve this, a method based on the principle of minimum potential energy for calculating the buckling load during elastic instability is presented. Initially, transverse constraint springs and rotational constraint springs are set at the four boundaries of the plate structure model, with the stiffness values of these two types of elastic springs designated to simulate arbitrary elastic boundary conditions. The improved Fourier series is employed to address potential discontinuities in the boundary derivatives of displacement functions that may arise with the classical Fourier series form. Subsequently, the potential energy expression for the rectangular plate system is established. By combining this with the principle of minimum potential energy, a system of linear equations is derived through partial differentiation of the unknown Fourier coefficients. Finally, the critical buckling load and other parameters of the rectangular plate are obtained, with reasonable values for the spring stiffness under different boundary conditions provided. The critical buckling temperatures derived from the proposed method are compared with those obtained via finite element analysis. The results indicate that the buckling loads obtained using this method align closely with those from finite element analysis, affirming the validity and convergence of the research methodology.Highlights This study investigates the thermal buckling of composite laminate plates with elastic boundary conditions. Improved Fourier series techniques are employed to analyze thermal buckling in these laminates. The impact of intermediate stiffness on the thermal buckling response of composite materials is examined.

Machine learning-enhanced band gaps prediction for low-symmetry double and layered perovskites.

Density functional theory (DFT) calculations are widely used for material property prediction, but their computational cost can hinder the discovery of novel perovskites. This work explores machine learning (ML) as a faster alternative for predicting band gaps in complex perovskites, focusing on low-symmetry double and layered structures. We employ Support Vector Regression (SVR), Random Forest Regression (RFR), Gradient Boosting Regression (GBR), and Extreme Gradient Boosting (XGBoost) to predict both direct and indirect band gaps. Model performance is evaluated using Mean Absolute Error (MAE), Mean Squared Error (MSE), and R-squared (R²) metrics. Our results reveal SVR as the most effective general model for predicting band gaps in both double and layered perovskites. Interestingly, for double perovskites specifically, XGBoost achieves even higher accuracy when incorporating derivative discontinuity as a feature. Feature importance analysis identifies the standard deviation of valence charges ("Valence (std)") as the most critical factor for band gap prediction across all studied perovskites. This research demonstrates the potential of ML for efficient and accurate band gap prediction in complex perovskites, accelerating material discovery efforts.

Critical Points in the Noiseberg Achievable Region of the Gaussian Z-Interference Channel.

The Gaussian signaling strategy with power control for the Gaussian Z-interference channel with weak interference is reviewed in this paper. In particular, we study the various communication strategies that may arise at various points of the capacity region and identify the locations of the phase transitions between the various strategies. The Gaussian Z-interference channel with weak interference is known to have two critical points in its capacity region, where the slope of the region shows a sudden change. They occur at the points of the unconditional maximum rate for one of the users and the maximum rate that can be accommodated by the other user. In this paper, we discuss additional critical points (locations of phase transitions) in the achievable region of this channel. These turn out to be second-order phase transitions, i.e., we do not observe a discontinuous slope in the achievable rate region, but there is a discontinuity in the second derivative of the rate contour of the achievable region. This review paper is mainly based on some of our ITA (Information Theory and Applications Workshop, UCSD, San Diego, CA, USA) papers since 2011.

Balancing the Contributions to the Gradient Expansion: Accurate Binding and Band Gaps with a Nonempirical Meta-GGA.

The gradient expansion has been a long-standing guide rail in density-functional theory. We here demonstrate that for exchange-correlation approximations that depend on the gradient of the density and the kinetic energy density, i.e., for meta-generalized gradient approximations (meta-GGAs), there is a so far unexploited degree of freedom in the gradient expansion that allows to shift the relative weight of gradient and kinetic energy contributions. As the dependence on the kinetic energy density determines the derivative discontinuity, this allows to construct meta-GGAs that adhere to the known exact constraints, yet have new properties. We demonstrate this with the construction of a meta-GGA that describes both electronic bonds and band gaps with remarkable accuracy.

Exchange-correlation potential built on the derivative discontinuity of electron density.

Electronic structures are fully determined by the exchange-correlation (XC) potential. In this work, we develop a new method to construct reliable XC potentials by properly mixing the exact exchange and the local density approximation potentials in real space. The spatially dependent mixing parameter is derived based on the derivative discontinuity of electron density and is first-principle. We derived the equations for solving the mixing parameter and proposed an approximation to simplify these equations. Based on this approximation, this new method gives reasonable predictions for the ionization energies, fundamental gaps, and singlet-triplet energy differences for various molecular systems. The impact of the approximation on the constructed XC potentials is examined, and it is found that the quality of the XC potentials can be further improved by removing the approximation. This work demonstrates that the derivative discontinuity of electron density is a promising constraint for constructing high-quality XC potentials.

Deep Interface Alternation Method (DIAM) based on domain decomposition for solving elliptic interface problems

The interface problem is highly challenging due to its non-smoothness, discontinuity, and interface complexity. In this paper, a new and simple Deep Interface Alternation Method (DIAM) is developed to solve elliptic interface problems to avoid dealing with interfaces. It combines the ideas of domain decomposition methods and deep learning methods. Specifically, we first transform the interface problem with discontinuous derivatives into multiple continuous subproblems based on the Dirichlet–Dirichlet algorithm of domain decomposition. Then, we establish different fully connected neural networks for each subproblem to approximate parallelly the continuous solutions in the subdomain. The interface information is especially exchanged among the different loss functions of each subdomain neural network while minimizing the loss functions of each subdomain separately to obtain solutions to the entire interface problem. Numerical experiments were conducted on two-dimensional and three-dimensional elliptical interface problems with different coefficient contrasts and interface complexity. The results indicate that the Deep Interface Alternation Method has effectiveness and accuracy.

Full counting statistics of 1d short range Riesz gases in confinement

We investigate the full counting statistics of a harmonically confined 1d short range Riesz gas consisting of N particles in equilibrium at finite temperature. The particles interact with each other through a repulsive power-law interaction with an exponent k > 1 which includes the Calogero–Moser model for k = 2. We examine the probability distribution of the number of particles in a finite domain [−W,W] called number distribution, denoted by N(W,N) . We analyze the probability distribution of N(W,N) and show that it exhibits a large deviation form for large N characterized by a speed N3k+2k+2 and by a large deviation function (LDF) of the fraction c=N(W,N)/N of the particles inside the domain and W. We show that the density profiles that create the large deviations display interesting shape transitions as one varies c and W. This is manifested by a third-order phase transition exhibited by the LDF that has discontinuous third derivatives. Monte–Carlo simulations based on Metropolis–Hashtings (MH) algorithm show good agreement with our analytical expressions for the corresponding density profiles. We find that the typical fluctuations of N(W,N) , obtained from our field theoretic calculations are Gaussian distributed with a variance that scales as Nνk , with νk=(2−k)/(2+k) . We also present some numerical findings on the mean and the variance. Furthermore, we adapt our formalism to study the index distribution (where the domain is semi-infinite (−∞,W]) , linear statistics (the variance), thermodynamic pressure and bulk modulus.

Asymptotics for viscous Burgers equation under impulsive forcing

An initial value problem to the viscous Burgers equation under the influence of an external forcing of distribution type is considered. The existence and uniqueness of weak solutions for it are investigated. To do this, we linearize the viscous Burgers equation and show the existence and uniqueness of common boundary function for the associated initial-boundary problems. Jump discontinuity of the spatial derivative of the solution makes a key role in determining large time asymptotics to the solutions. The explicit representation of the common boundary function and its asymptotic behavior are discussed for a specific case of the jump discontinuity. Here, the common boundary is obtained in terms of Associated Legendre functions of the first and second kind. For the general case, asymptotic behavior of the common boundary of the associated initial-boundary problems is used to determine the large time asymptotics for the Cauchy problem to the viscous Burgers equation.

Tilted-Plane Structure of the Energy of Finite Quantum Systems.

The piecewise linearity condition on the total energy with respect to the total magnetization of finite quantum systems is derived using the infinite-separation-limit technique. This generalizes the well-known constancy condition, related to static correlation error, in approximate density functional theory. The magnetic analog of Koopmans' theorem in density functional theory is also derived. Moving to fractional electron count, the tilted-plane condition is derived, lifting certain assumptions in previous works. This generalization of the flat-plane condition characterizes the total energy surface of a finite system for all values of electron count N and magnetization M. This result is used in combination with tabulated spectroscopic data to show the flat-plane structure of the oxygen atom, among others. We find that derivative discontinuities with respect to electron count sometimes occur at noninteger values. A diverse set of tilted-plane structures is shown to occur in d-orbital subspaces, depending on chemical coordination. General occupancy-based total-energy expressions are demonstrated thereby to be necessarily dependent on the symmetry-imposed degeneracies.

Calculation of Plates of Different Shapes Using Generalized Equations of the Finite Difference Method

Statement of the problem. The article reviewed nonaxisymmetric tasks for calculating slabs in shape of a circle sector, wedge and a slab formed by a connection of rectangular and circular areas. The solution is based on generalized equations of the finite differences method. The algorithm allows us to take into account finite discontinuities of the desired function, the right side of the original differential equations, as well as discontinuities of derivatives of these functions without accumulation of the grid and using points outside the contour. It resolves oneself into a system of differential algebraic equations generated for field and contour points. In this paper the initial differential equations and their numerical approximation are provided. Results. Calculations of a semicircle and a wedge under different boundary conditions are presented, as well as an uncut slab in the form of a semicircle with an annular support and a composite slab are consi-dered. The solution of the above problems has been carried out on the minimum possible computational grid, comparison with the available solutions has been provided, and static checks of the calculation have been carried out. Conclusions. Along with the most common numerical method for calculating buildings and structures, it is possible and quite effective to use these equations. It is proved by means of some examples that such an approximation has a number of advantages: the results allow us to talk about the sufficient accuracy of the solution with a small number of partitions; fundamental simplicity and universality of the method.

Extended N-centered ensemble density functional theory of double electronic excitations.

A recent work (arXiv:2401.04685) has merged N-centered ensembles of neutral and charged electronic ground states with ensembles of neutral ground and excited states, thus providing a general and in-principle exact (so-called extended N-centered) ensemble density functional theory of neutral and charged electronic excitations. This formalism made it possible to revisit the concept of density-functional derivative discontinuity, in the particular case of single excitations from the highest occupied Kohn-Sham (KS) molecular orbital, without invoking the usual "asymptotic behavior of the density" argument. In this work, we address a broader class of excitations, with a particular focus on double excitations. An exact implementation of the theory is presented for the two-electron Hubbard dimer model. A thorough comparison of the true physical ground- and excited-state electronic structures with that of the fictitious ensemble density-functional KS system is also presented. Depending on the choice of the density-functional ensemble as well as the asymmetry of the dimer and the correlation strength, an inversion of states can be observed. In some other cases, the strong mixture of KS states within the true physical system makes the assignment "single excitation" or "double excitation" irrelevant.

Harmonic balance for differential constitutive models under oscillatory shear

Harmonic balance (HB) is a popular Fourier–Galerkin method used in the analysis of nonlinear vibration problems where dynamical systems are subjected to periodic forcing. We adapt HB to find the periodic steady-state response of nonlinear differential constitutive models subjected to large-amplitude oscillatory shear flow. By incorporating the alternating-frequency-time scheme into HB, we develop a computer program called FLASH (acronym for Fast Large Amplitude Simulation using Harmonic balance), which makes it convenient to apply HB to any differential constitutive model. We validate FLASH by considering two representative constitutive models, viz., the exponential Phan-Thien–Tanner model and a nonlinear temporary network model. In terms of accuracy and speed, FLASH typically outperforms the conventional approach of solving initial value problems by numerical integration via time-stepping methods often by several orders of magnitude. Exceptions to this rule are sometimes encountered for materials that are strongly shear thinning or described by constitutive models with discontinuous derivatives. We discuss how FLASH can be conveniently extended for other nonlinear constitutive models, which opens up potential applications in model calibration and selection, and stability analysis.

Analytic method for finding stationary solutions to generalized nonlinear Schrödinger equations

We present a method to generate analytic, stationary solutions to generalized nonlinear Schrödinger equations with complicated dispersion profiles. The experimental observation of such solutions in a fiber laser was recently reported. Our method proceeds in a systematic fashion and does not require initial guesses. The solutions take the form of an exponentially converging sum of functions, each with an infinite number of discontinuous derivatives at the origin, which disappear upon summation. They can be found conveniently using a symbolic manipulation program. Our method can not only find stable solitons, but also higher order solutions that tend to be unstable and that are difficult to find numerically.

Exploiting a derivative discontinuity estimate for accurate G0W0 ionization potentials and electron affinities

The GW approximation has become an important tool for predicting charged excitations of isolated molecules and condensed systems. Its popularity can be attributed to many factors, including a favorable scaling and relatively good accuracy. In practical applications, the GW is often performed as a one-shot perturbation known as G0W0 . Unfortunately, G0W0 suffers from a strong starting point dependence and is often not as accurate as one would need. Self-consistent GW methodologies alleviate these problems but come with a marked increase in computational cost. In this manuscript, we propose the use of an estimate of the exchange-correlation derivative discontinuity to provide a remarkably good starting point for G0W0 calculations, yielding ionization potentials and electron affinities with eigenvalue self-consistent GW quality at no additional cost. We assess the quality of the resulting methodology with the GW100 benchmark set and compare its advantages over other similar methods.

Ensemble Ground State of a Many-Electron System with Fractional Electron Number and Spin: Piecewise-Linearity and Flat-Plane Condition Generalized.

Description of many-electron systems with a fractional electron number (Ntot) and fractional spin (Mtot) is of great importance in physical chemistry, solid-state physics, and materials science. In this Letter, we provide an exact description of the zero-temperature ensemble ground state of a general, finite, many-electron system and characterize the dependence of the energy and the spin-densities on both Ntot and Mtot, when the total spin is at its equilibrium value. We generalize the piecewise-linearity principle and the flat-plane condition and determine which pure states contribute to the ground-state ensemble. We find a new derivative discontinuity, which manifests for spin variation at a constant Ntot, as a jump in the Kohn-Sham potential. We identify a previously unknown degeneracy of the ground state, such that the total energy and density are unique, but the spin-densities are not. Our findings serve as a basis for development of advanced approximations in density functional theory and other many-electron methods.

Optimization of Topological Reconfiguration in Electric Power Systems Using Genetic Algorithm and Nonlinear Programming with Discontinuous Derivatives

This article addresses a comprehensive analysis of power electrical systems, employing a combined approach of genetic algorithms and mathematical optimization through nonlinear programming with discontinuous derivatives (DNLP) in GAMS. The primary objective is to minimize economic losses and associated costs faced by the network operator following disruptive events. The analysis is divided into two fundamental aspects. Firstly, it addresses the topological reconfiguration of the network, involving the addition of lines and distributed energy resources such as distributed generation. To determine the optimal topological reconfiguration, a genetic algorithm was developed and implemented. This approach aims to restore electrical service to the maximum load within the system. Secondly, an optimal energy dispatch was performed for each generator, considering the variation in load throughout the day. The system’s load curve is taken into account to determine the optimal energy distribution. Thus, the problem of economic losses is approached from two perspectives: the minimization of costs due to nonsupplied electrical energy and the determination of efficient energy dispatch for each generator after network reconfiguration. For the analysis and case studies, simulations were conducted on the IEEE 9- and 30-node test systems. The results demonstrate the effectiveness of the proposed solution, evaluated in terms of reduced load shedding and economic losses.

Net saving improvement of capacitor banks in power distribution systems by increasing daily size switching number: A comparative result analysis by artificial intelligence

AbstractThis paper studies the effect of the number of switching (NOS) per day of capacitor banks on loss reduction in radial distribution systems. To this aim, the daytime (more precisely, 24 h) is divided into different numbers of time segments (equal to the same NOS) for capacitors’ size switching. The resulting non‐linear programming with discontinuous derivatives (called DNLP) model is solved subject to related constraints. The results reveal the impact of hourly switching of capacitor banks on further loss reduction (namely 118.4435, 83.7856, and 101.738 MWh for three IEEE systems) and higher net savings (i.e. k$5.6067, k$4.2772, and k$5.3542 for the same systems) of radial distribution systems compared to daily switching. Then, the hyper‐tuned Random Forest model is trained based on the IEEE 69‐bus network, fine‐tuned by the IEEE 10‐bus network, and fitted by the IEEE 33‐bus network to have an intelligent multi‐classification task with the highest accuracy. Numerical simulation, in both classic and intelligent parts, is presented to demonstrate the performance of DeepOptaCap. For the final step, DeepOptaCast is compared to other intelligent models of Light Gradient Boosting Method (LGBM), Decision Tree, and XGBoost, regarding KPIs of mean absolute percentage error, root mean squared percentage error, mean absolute error, root mean squared error, and coefficient of determination to demonstrate the model's superiority.

Two classes of third-order weighted compact nonlinear schemes for Hamilton-Jacobi equations

The solutions of Hamilton-Jacobi (HJ) equations may contain discontinuous derivatives, which brings numerical difficulties in capturing sharply these derivatives. This paper presents a third-order weighted compact nonlinear scheme (WCNS) and a third-order perturbed WCNS (WCNS-P) for solving HJ equations. A linear combination of hybrid cell-edge and cell-node values of the function is applied to approximate the spatial derivatives of the function. For the WCNS, the upwind-biased nonlinear weighted interpolation is used for the unknown cell-edge values, while for the WCNS-P perturbations with a free parameter is introduced to the upwind-biased linear interpolation. The free parameter can be optimized to reduce numerical errors in smooth regions. To enhance the numerical stability, a monotone polynomial interpolation method is designed to switch the WCNS-P to the WCNS. Several numerical experiments are performed to test the order of accuracy and the discontinuity-capturing ability of the two schemes.

Density functional theory of material design: fundamentals and applications—II

Abstract This is the second and the final part of the review on density functional theory (DFT), referred to as DFT-II. In the first review, DFT-I, we have discussed wavefunction-based methods, their complexity, and basics of density functional theory. In DFT-II, we focus on fundamentals of DFT and their implications for the betterment of the theory. We start our presentation with the exact DFT results followed by the concept of exchange-correlation (xc) or Fermi-Coulomb hole and its relationship with xc energy functional. We also provide the exact conditions for the xc-hole, xc-energy and xc-potential along with their physical interpretation. Next, we describe the extension of DFT for non-integer number of electrons, the piecewise linearity of total energy and discontinuity of chemical potential at integer particle numbers, and derivative discontinuity of the xc potential, which has consequences on fundamental gap of solids. After that, we present how one obtains more accurate xc energy functionals by going beyond the LDA. We discuss the gradient expansion approximation (GEA), generalized gradient approximation (GGA), and hybrid functional approaches to designing better xc energy functionals that give accurate total energies. However, these functionals fail to predict properties like the ionization potential and the band gap. Thus, we next describe different methods of modelling these potentials and results of their application for calculation of the band gaps of different solids to highlight accuracy of different xc potentials. Finally, we conclude with a glimpse on orbital-free density functional theory and the machine learning approach.

A regularization–correction approach for adapting subdivision schemes to the presence of discontinuities

Linear approximation methods suffer from Gibbs oscillations when approximating functions with jumps. Essentially non oscillatory subcell-resolution (ENO-SR) is a local technique avoiding oscillations and with a full order of accuracy, but a loss of regularity of the approximant appears. The goal of this paper is to introduce a new approach having both properties of full accuracy and regularity. In order to obtain it, we propose a three-stage algorithm: first, the data is smoothed by subtracting an appropriate non-smooth data sequence; then a chosen high order linear approximation operator is applied to the smoothed data and finally, an approximation with the proper jump or corner (jump in the first order derivative) discontinuity structure is reinstated by correcting the smooth approximation with the non-smooth element used in the first stage. This new procedure can be applied as subdivision scheme to design curves and surfaces both in point-value and in cell-average contexts. Using the proposed algorithm, we are able to construct approximations with high precision, with high piecewise regularity, and without smearing nor oscillations in the presence of discontinuities. These are desired properties in real applications as computer aided design or car design, among others.