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Open-source APOST-3D software features a large number of wavefunction analysis tools developed over the past 20 years, aiming at connecting classical chemical concepts with the electronic structure of molecules. APOST-3D relies on the identification of the atom in the molecule (AIM), and several analysis tools are implemented in the most general way so that they can be used in combination with any chosen AIM. Several Hilbert-space and real-space (fuzzy atom) AIM definitions are implemented. In general, global quantities are decomposed into one- and two-center terms, which can also be further grouped into fragment contributions. Real-space AIM methods involve numerical integrations, which are particularly costly for energy decomposition schemes. The current version of APOST-3D features several strategies to minimize numerical error and improve task parallelization. In addition to conventional population analysis of the density and other scalar fields, APOST-3D implements different schemes for oxidation state assignment (effective oxidation state and oxidation states localized orbitals), molecular energy decomposition schemes, and local spin analysis. The APOST-3D platform offers a user-friendly interface and a comprehensive suite of state-of-the-art tools to bridge the gap between theory and experiment, representing a valuable resource for both seasoned computational chemists and researchers with a focus on experimental work. We provide an overview of the code structure and its capabilities, together with illustrative examples. [ABSTRACT FROM AUTHOR]

A new algorithm has been developed to simulate two-dimensional (2D) spectra with correlated anisotropic frequencies faster and more accurately than previous methods. The technique uses finite-element numerical integration on the sphere and an interpolation scheme based on the Alderman–Solum–Grant algorithm. This method is particularly useful for numerical calculations of joint probability distribution functions involving quantities with a parametric orientation dependence. The technique's efficiency also allows for practical least-squares fitting of experimental 2D solid-state nuclear magnetic resonance (NMR) datasets. The simulation method is illustrated for select 2D NMR methods, and a least-squares analysis is demonstrated in the extraction of paramagnetic shift and quadrupolar coupling tensors and their relative orientation from the experimental shifting-d echo 2H NMR spectrum of a NiCl2 · 2D2O salt. [ABSTRACT FROM AUTHOR]

block2 is an open source framework to implement and perform density matrix renormalization group and matrix product state algorithms. Out-of-the-box it supports the eigenstate, time-dependent, response, and finite-temperature algorithms. In addition, it carries special optimizations for ab initio electronic structure Hamiltonians and implements many quantum chemistry extensions to the density matrix renormalization group, such as dynamical correlation theories. The code is designed with an emphasis on flexibility, extensibility, and efficiency and to support integration with external numerical packages. Here, we explain the design principles and currently supported features and present numerical examples in a range of applications. [ABSTRACT FROM AUTHOR]

Typically, the parameters entering a physical simulation model carry some kind of uncertainty, e.g., due to the intrinsic approximations in a higher fidelity theory from which they have been obtained. Global sensitivity analysis (GSA) targets quantifying which parameter uncertainties impact the accuracy of the simulation results, e.g., to identify which parameters need to be determined more accurately. We present a GSA approach based on the Cramérs–von Mises distance. Unlike prevalent approaches, it combines the following properties: (i) it is equally suited for deterministic as well as stochastic model outputs, (ii) it does not require gradients, and (iii) it can be estimated from numerical quadrature without further numerical approximations. Using quasi-Monte Carlo for numerical integration and a first-principles kinetic Monte Carlo model for the CO oxidation on RuO2(110), we examine the performance of the approach. We find that the results agree very well with what is known in the literature about the sensitivity of this model and that the approach converges in a modest number of quadrature points. Furthermore, it appears to be robust against even extreme relative noise. All these properties make the method particularly suited for expensive (kinetic) Monte Carlo models because we can reduce the number of simulations as well as the target variance of each of these. [ABSTRACT FROM AUTHOR]

For a detailed understanding of many processes in nature involving, for example, energy or electron transfer, the theory of open quantum systems is of key importance. For larger systems, an accurate description of the underlying quantum dynamics is still a formidable task, and, hence, approaches employing machine learning techniques have been developed to reduce the computational effort of accurate dissipative quantum dynamics. A downside of many previous machine learning methods is that they require expensive numerical training datasets for systems of the same size as the ones they will be employed on, making them unfeasible to use for larger systems where those calculations are still too expensive. In this work, we will introduce a new method that is implemented as a machine-learned correction term to the so-called Numerical Integration of Schrödinger Equation (NISE) approach. It is shown that this term can be trained on data from small systems where accurate quantum methods are still numerically feasible. Subsequently, the NISE scheme, together with the new machine-learned correction, can be used to determine the dissipative quantum dynamics for larger systems. Furthermore, we show that the newly proposed machine-learned correction outperforms a previously handcrafted one, which, however, improves the results already considerably. [ABSTRACT FROM AUTHOR]

We present a machine learning approach to calculating electronic specific heat capacities for a variety of benchmark molecular systems. Our models are based on data from density matrix quantum Monte Carlo, which is a stochastic method that can calculate the electronic energy at finite temperature. As these energies typically have noise, numerical derivatives of the energy can be challenging to find reliably. In order to circumvent this problem, we use Gaussian process regression to model the energy and use analytical derivatives to produce the specific heat capacity. From there, we also calculate the entropy by numerical integration. We compare our results to cubic splines and finite differences in a variety of molecules in which Hamiltonians can be diagonalized exactly with full configuration interaction. We finally apply this method to look at larger molecules where exact diagonalization is not possible and make comparisons with more approximate ways to calculate the specific heat capacity and entropy. [ABSTRACT FROM AUTHOR]

We investigate a class of parametric elliptic semilinear partial differential equations of second order with homogeneous essential boundary conditions, where the coefficients and the right-hand side (and hence the solution) may depend on a parameter. This model can be seen as a reaction-diffusion problem with a polynomial nonlinearity in the reaction term. The efficiency of various numerical approximations across the entire parameter space is closely related to the regularity of the solution with respect to the parameter. We show that if the coefficients and the right-hand side are analytic or Gevrey class regular with respect to the parameter, the same type of parametric regularity is valid for the solution. The key ingredient of the proof is the combination of the alternative-to-factorial technique from our previous work [1] with a novel argument for the treatment of the power-type nonlinearity in the reaction term. As an application of this abstract result, we obtain rigorous convergence estimates for numerical integration of semilinear reaction-diffusion problems with random coefficients using Gaussian and Quasi-Monte Carlo quadrature. Our theoretical findings are confirmed in numerical experiments. [ABSTRACT FROM AUTHOR]

This work reports the magnetic index of aromaticity of cyclophosphazene (c$$ c $$‐P3$$ {}_3 $$N3$$ {}_3 $$H6$$ {}_6 $$) and their halogenated cyclic derivatives: c$$ c $$‐P3$$ {}_3 $$N3$$ {}_3 $$F6$$ {}_6 $$, c$$ c $$‐P3$$ {}_3 $$N3$$ {}_3 $$Cl6$$ {}_6 $$, c$$ c $$‐P3$$ {}_3 $$N3$$ {}_3 $$Br6$$ {}_6 $$, c$$ c $$‐P3$$ {}_3 $$N3$$ {}_3 $$I6$$ {}_6 $$, c$$ c $$‐P3$$ {}_3 $$N3$$ {}_3 $$At6$$ {}_6 $$ and c$$ c $$‐P3$$ {}_3 $$N3$$ {}_3 $$Ts6$$ {}_6 $$. This index, also known as ring‐current strength, is calculated by numerical integration of the magnetically‐induced current density vector field which is generated by a perturbing external magnetic field. Due to the presence of heavy X=Br,I,At$$ \mathrm{X}=\mathrm{Br},\mathrm{I},\mathrm{At} $$ atoms in c$$ c $$‐P3$$ {}_3 $$N3$$ {}_3 $$X6$$ {}_6 $$, important relativistic were expected. Accordingly, all‐electron density functional theory (DFT) calculations were carried out using the four‐component Dirac‐Coulomb (DC) Hamiltonian, including scalar and spin‐orbit relativistic effects. The values were also compared with the corresponding spin‐free (scalar relativistic) ones. [ABSTRACT FROM AUTHOR]

In this work, we attempt to bring together the origins, main results, and recent advances on discrete-event simulation of continuous-time systems. Starting from the early approaches that aimed to represent continuous-time dynamics within the discrete-event system specification (DEVS) formalism framework, the work shows how these ideas gave place to the formalization of the quantized state system (QSS) family of numerical integration algorithms. Then, we describe the QSS algorithms, their properties, their extensions, and the main practical software tools implementing them. We also present a selection of simulation examples illustrating the main features and advantages through comparisons with state-of-the-art continuous-time simulation solvers. [ABSTRACT FROM AUTHOR]

Meshless methods are a type of numerical method used to simulate continuum mechanics problems. These methods have been applied to several types of problems, there are a few works using meshless method focused on dynamic problems, but most works study static loading conditions. The current work aims at using two different meshless methods, the Radial Point Interpolation Method (RPIM) and the Natural Neighbours RPIM (NNRPIM), in a dynamic problem, specifically forced vibrations. This problem requires time integration, therefore three different time integration methods have also been tested, namely: the Central Difference Method (CDM), the Wilson method, and the Newmark method. The CDM is an explicit method, while the other two are implicit. A discretization study was performed to assess the ideal nodal discretization before the numerical and time integration methods are validated. For the implicit methods, different time step lengths were also tested. In the final example damping was introduced. The results prove the validity of the two meshless methods by having similar results to the Finite Element Method (FEM) using the three distinct time integration methods, different loading conditions, and damping. [ABSTRACT FROM AUTHOR]

Integral transformations represent an important mathematical tool for gravitational field modelling. A basic assumption of integral transformations is the global data coverage, but availability of high-resolution and accurate gravitational data may be restricted. Therefore, we decompose the global integration into two parts: (1) the effect of the near zone calculated by the numerical integration of data within a spherical cap and (2) the effect of the far zone due to data beyond the spherical cap synthesised by harmonic expansions. Theoretical and numerical aspects of this decomposition have frequently been studied for isotropic integral transformations on the sphere, such as Hotine's, Poisson's, and Stokes's integral formulas. In this article, we systematically review the mathematical theory of the far-zone effects for the spherical integral formulas, which transform the disturbing gravitational potential or its purely radial derivatives into observable quantities of the gravitational field, i.e. the disturbing gravitational potential and its radial, horizontal, or mixed derivatives of the first, second, or third order. These formulas are implemented in a MATLAB software and validated in a closed-loop simulation. Selected properties of the harmonic expansions are investigated by examining the behaviour of the truncation error coefficients. The mathematical formulations presented here are indispensable for practical solutions of direct or inverse problems in an accurate gravitational field modelling or when studying statistical properties of integral transformations. [ABSTRACT FROM AUTHOR]

Causal indirect and direct effects provide an interpretable method for decomposing the total effect of an exposure on an outcome into the indirect effect through a mediator and the direct effect through all other pathways. A natural choice for a mediator in a randomized clinical trial is the treatment's targeted biomarker. However, when the mediator is a biomarker, values can be subject to an assay lower limit. The mediator is affected by the treatment and is a putative cause of the outcome, so the assay lower limit presents a compounded problem in mediation analysis. We propose two approaches to estimate indirect and direct effects with a mediator subject to an assay limit: (1) extrapolation and (2) numerical optimization and integration of the observed likelihood. Since these estimation methods solely rely on the so‐called Mediation Formula, they apply to most approaches to causal mediation analysis: natural, separable, and organic indirect, and direct effects. A simulation study compares the two estimation approaches to imputing with half the assay limit. Using HIV interruption study data from the AIDS Clinical Trials Group described in Li et al 2016, AIDS; Lok and Bosch 2021, Epidemiology, we illustrate our methods by estimating the organic/pure indirect effect of a hypothetical HIV curative treatment on viral suppression mediated by two HIV persistence measures: cell‐associated HIV‐RNA and single‐copy plasma HIV‐RNA. [ABSTRACT FROM AUTHOR]

This paper provides a comprehensive and computationally efficient case study for uncertainty quantification (UQ) and global sensitivity analysis (GSA) in a neuron model incorporating ion concentration dynamics. We address how challenges with UQ and GSA in this context can be approached and solved, including challenges related to computational cost, parameters affecting the system's resting state, and the presence of both fast and slow dynamics. Specifically, we analyze the electrodiffusive neuron-extracellular-glia (edNEG) model, which captures electrical potentials, ion concentrations (Na+, K+, Ca2+, and Cl−), and volume changes across six compartments. Our methodology includes a UQ procedure assessing the model's reliability and susceptibility to input uncertainty and a variance-based GSA identifying the most influential input parameters. To mitigate computational costs, we employ surrogate modeling techniques, optimized using efficient numerical integration methods. We propose a strategy for isolating parameters affecting the resting state and analyze the edNEG model dynamics under both physiological and pathological conditions. The influence of uncertain parameters on model outputs, particularly during spiking dynamics, is systematically explored. Rapid dynamics of membrane potentials necessitate a focus on informative spiking features, while slower variations in ion concentrations allow a meaningful study at each time point. Our study offers valuable guidelines for future UQ and GSA investigations on neuron models with ion concentration dynamics, contributing to the broader application of such models in computational neuroscience. [ABSTRACT FROM AUTHOR]

Physics-informed neural networks (PINNs) and their variants have recently emerged as alternatives to traditional partial differential equation (PDE) solvers, but little literature has focused on devising accurate numerical integration methods for neural networks (NNs), which is essential for getting accurate solutions. In this work, we propose adaptive quadratures for the accurate integration of neural networks and apply them to loss functions appearing in low-dimensional PDE discretisations. We show that at opposite ends of the spectrum, continuous piecewise linear (CPWL) activation functions enable one to bound the integration error, while smooth activations ease the convergence of the optimisation problem. We strike a balance by considering a CPWL approximation of a smooth activation function. The CPWL activation is used to obtain an adaptive decomposition of the domain into regions where the network is almost linear, and we derive an adaptive global quadrature from this mesh. The loss function is then obtained by evaluating the smooth network (together with other quantities, e.g., the forcing term) at the quadrature points. We propose a method to approximate a class of smooth activations by CPWL functions and show that it has a quadratic convergence rate. We then derive an upper bound for the overall integration error of our proposed adaptive quadrature. The benefits of our quadrature are evaluated on a strong and weak formulation of the Poisson equation in dimensions one and two. Our numerical experiments suggest that compared to Monte-Carlo integration, our adaptive quadrature makes the convergence of NNs quicker and more robust to parameter initialisation while needing significantly fewer integration points and keeping similar training times. [ABSTRACT FROM AUTHOR]

Context. In the latest Gaia third data release, one can find extremely small proper motion components for the star HD 7977. This, together with the radial velocity measurement lead to the conclusion that this star passed very close to the Sun in the recent past. Aims. Such a very close approach of a one solar mass star must have resulted in noticeable changes in the motion of all Solar System bodies, especially those on less tight orbits, namely long-period comets (LPCs) and transneptunian objects (TNOs). We estimate and discuss these effects. Methods. Our current knowledge on the solar neighbourhood found in the latest Gaia catalogues allowed us to perform numerical integrations and prepare a list of potential stellar perturbers of LPCs. We used this list, made available in the StePPeD database. To study the past motion of LPCs under the simultaneous action of the Galactic potential and passing stars, we used precise original cometary orbits taken from the current CODE catalogue. Results. We examined the reliability of the extremely small proper motion of HD 7977 and conclude that this star can be an unresolved binary; however, according to the astrometry covering more than a century, the current Gaia data cannot be ruled out. We present the parameters of a very close passage of this star near the Sun. We also show examples of the strong influence of this passage on the past motion of some LPCs. We also discuss the possible influence of this perturber on other Solar System bodies. Conclusions. It is possible that 2.47 Myr ago the one solar mass star HD 7977 passed as close as 1000 au from the Sun. Such an event constitutes a kind of dynamical horizon for all studies of the past Solar System bodies' dynamics. [ABSTRACT FROM AUTHOR]

Aims. We study the secular evolution of two planets in mutual deep mean-motion resonance (MMR) in the planar elliptic three-body problem framework for different mass ratios. We do not consider any restriction in the eccentricity of the inner planet e1 or in the eccentricity of the outer planet e2. Methods. The method we used is based on a semi-analytical model that consists of calculating the averaged resonant disturbing function numerically. It is assumed for this that all the orbital elements (except for the mean longitudes) of both planets are constant on the resonant timescale. In order to obtain the secular evolution inside the MMR, we make use of the adiabatic invariance principle, assuming a zero-amplitude resonant libration. We constructed two phase portraits, called the ℋ1 and ℋ2 surfaces, in the three-dimensional spaces (e1, Δϖ, σ) and (e2, Δϖ, σ), where Δϖ is the difference between the planetary longitude of perihelia and σ is the critical angle. These surfaces, which are related through the angular moment conservation, allow us to find the apsidal corotation resonances (ACRs) and to predict the secular evolution of e1, e2, Δϖ, and σ (libration center). Results. While studying the 1:1, 2:1, 3:1, and 3:2 MMR, we found that large excursions in eccentricity can exist in some particular cases. We compared the secular variations of e1, e2, Δϖ, and σ predicted by the model with a numerical integration of the exact equations of motion for different mass ratios. We obtained good matches. Finally, the model was applied to study the secular evolution of the resonant exoplanet systems HD 73526 and HD 31527. They both have a pair of planets and are very close to the deep MMR condition. In the first system, we found that the pair of planets that constitutes the system evolves in a symmetrical ACR, whereas in the second system, we found that planets c and d, which are in an unusual 16:3 MMR, are close to an ACR, but outside its dynamical region, where Δϖ circulates. [ABSTRACT FROM AUTHOR]

We show that independent and uniformly distributed sampling points are asymptotically as good as optimal sampling points for the approximation of functions from Sobolev spaces |$W_p^s(\varOmega)$| on bounded convex domains |$\varOmega \subset{\mathbb{R}}^d$| in the |$L_q$| -norm if |$q

In this paper, based on Yang's fractal theory, the Hermite–Hadamard's inequalities for generalized h -preinvex function are proved. Then, using the local fractional integral identity proposed by Sun [Some local fractional integral inequalities for generalized preinvex functions and applications to numerical quadrature, Fractals 27(5) (2019) 1950071] as auxiliary function, some parameterized local fractional integral inequalities for generalized h -preinvex functions are established. For the special cases of the parameters, some generalized Simpson-type, midpoint-type and trapezoidal inequalities are established. Finally, some applications of these inequalities in numerical integration are proposed. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Atmospheric motion vectors, which can be used to infer wind speed and direction based on the trajectory of cloud movement, are instrumental in enhancing atmospheric wind-field insights, contributing notably to wind-field optimization and forecasting. However, a widespread problem with vector data is their inaccuracy, which, when coupled with the mediocre effectiveness of existing correction methods, limits their practical utility in forecasting, often falling short of expectations. Deep-learning techniques are used to refine atmospheric motion vector data from the FY-4A satellite, notably enhancing data quality. Post-training data undergoes a thorough analysis using a quality evaluation function, followed by its integration into a numerical weather prediction system in order to conduct forecasting experiments. Results indicate a marked improvement in data quality post-error correction by the model, characterized by a significant reduction in root mean square error and a notable increase in correlation coefficients. Furthermore, refined data demonstrate a considerable enhancement in the accuracy of meteorological element forecasts, particularly for Asian and Western Pacific regions. [ABSTRACT FROM AUTHOR]

Fly cutting is widely used in manufacturing of large-scale, high-precision optical components. However, the discontinuity of fly cutting machining leads to significant relative vibrations between the tool and the workpiece. The cutting process generates periodic waves along the cutting direction, which will deteriorate the wavefront characteristics of optical components. Based on the machining dynamics, this paper proposes a direct integration method to predict the waviness error of the machined surface. The cutting force model of fly cutting is established. The multi-mode characteristics of the spindle-tool system are measured by the experimental method. Then, the influence of uncertainties on the calculation results is analyzed by the variance-based sensitivity analysis method. Finally, the plane cutting experiment verifies that the direct integration method effectively predicts the waviness error and its variation trend, and the waviness prediction research is important for optimization of the machining parameters. [ABSTRACT FROM AUTHOR]

A multi-level strategy of numerical integration, incorporating Hammer-type quadrature, adaptive Gauss quadrature, and the improved semi-analytic method, is developed to evaluate the boundary integrals on triangular elements for three dimensional potential problems. The boundary integrals are classified into three categories according to the relative distance between the source point and the integral element. Hammer-type quadrature is used to evaluate the integrals on the element that is far from the source point, while adaptive m -point Gauss quadrature is proposed to evaluate the integrals on the element that is closer to the source point, the number of quadrature points varies with the change of the relative distance. The improved semi-analytic method is proposed for the nearly singular integral on the element that is very close to the source point, where the conformal and sigmoidal transformations are introduced to mitigate the adverse effects caused by the distorted shape of the element. The effective ranges of Hammer-type quadrature and adaptive Gauss quadrature are investigated and a reasonable scheme for their activation is provided. The present multi-level strategy exhibits high accuracy, robustness, and efficiency for the integrals with 1/ r , 1/ r 3 , and 1/ r 5. It makes the fast multipole boundary element method more competitive and applicable in the analysis of large complex structures. [ABSTRACT FROM AUTHOR]

In this paper, according to the element discretization characteristic of the geometric model, a complex function theory based transformation boundary element method is proposed to perform the steady-state thermal analysis of plate structures. The numerical implementation of the proposed method can be attributed to a numerical integration problem in element analysis, which is mainly divided into the following two steps: firstly, a bidirectional transformation method based on distance function and complex function is proposed to remove the weak singularities of the fundamental solution in integral equation, which is suitable for the weakly singular integrals of narrow strip discrete elements; secondly, a dual sinh method based on complex poles is developed to remove the influence of the nearly singular integrals in the analysis of the plate structures. With this method, the numerical integration in the integral equation can be accurately evaluated, which ensures the accurate calculation of the thermal physical variables of the plate structures. Several numerical examples are presented and discussed to prove the accuracy and versatility of the presented method. The results show that the presented method has the superiority of dealing with plate structure problems. [ABSTRACT FROM AUTHOR]

This paper presents a new accurate method to compute the mechanical response of pavement structures using an Artificial Neural Network (ANN) model coupled with Multi-Layer Elastic Analysis (MLEA). The ANN model is used to improve the numerical integration of the response function used in the MLEA method. It requires four inputs: total pavement thickness, the diameter of the contact area, radial distance, and depth of the response point; and it was trained on one million hypothetical pavement structures. The developed method has been validated by a comparative analysis against boundary conditions, finite element analysis, and available MLEA solutions using various hypothetical pavement structures. The results demonstrate that the developed solution gives excellent response in the vicinity of the pavement surface together with a significant improvement in computational efficiency. [ABSTRACT FROM AUTHOR]

The stability of axially functionally graded (AFG) heavy columns was analyzed. Consideration in stability analysis of the column is given to the free vibration and bucking problems. The mass density and Young's modulus of the AFG heavy column vary along the column axis through a power-law function. Unified modeling of the differential equations with the associated boundary conditions governing the deformed shape of the free vibrations and buckling of the column was developed. Using a combination of direct numerical integration method and numerical solution method of nonlinear equation, differential equations were solved to calculate the natural frequency and the critical buckling load. Calculation results for the natural frequency and buckling load compare well with the FEM results. As a result of numerical experiments, the effects of material and geometric properties on the natural frequency and the buckling load were reported and discussed in detail. [ABSTRACT FROM AUTHOR]

This paper develops a thermodynamic entropy-based life prediction model to estimate the low-cycle fatigue (LCF) life of the nickel-based superalloy GH4169 at elevated temperature (650 °C). The gauge section of the specimen was chosen as the thermodynamic system for modeling entropy generation within the framework of the Chaboche viscoplasticity constitutive theory. Furthermore, an explicitly numerical integration algorithm was compiled to calculate the cyclic stress–strain responses and thermodynamic entropy generation for establishing the framework for fatigue life assessment. A thermodynamic entropy-based life prediction model is proposed with a damage parameter based on entropy generation considering the influence of loading ratio. Fatigue lives for GH4169 at 650 °C under various loading conditions were estimated utilizing the proposed model, and the results showed good consistency with the experimental results. Finally, compared to the existing classical models, such as Manson–Coffin, Ostergren, Walker strain, and SWT, the thermodynamic entropy-based life prediction model provided significantly better life prediction results. [ABSTRACT FROM AUTHOR]

In this work, we lay the foundations for computing the behavior of a quantum heat engine whose working medium consists of an ensemble of non-harmonic quantum oscillators. In order to enable this analysis, we develop a method based on the Schrödinger picture. We investigate different possible choices on the basis of expanding the density operator, as it is crucial to select a basis that will expedite the numerical integration of the time-evolution equation without compromising the accuracy of the computed results. For this purpose, we developed an estimation technique that allows us to quantify the error that is unavoidably introduced when time-evolving the density matrix expansion over a finite-dimensional basis. Using this and other ways of evaluating a specific choice of basis, we arrive at the conclusion that the basis of eigenstates of a harmonic Hamiltonian leads to the best computational performance. Additionally, we present a method to quantify and reduce the error that is introduced when extracting relevant physical information about the ensemble of oscillators. The techniques presented here are specific to quantum heat cycles; the coexistence within a cycle of time-dependent Hamiltonian and coupling with a thermal reservoir are particularly complex to handle for the non-harmonic case. The present investigation is paving the way for numerical analysis of non-harmonic quantum heat machines. [ABSTRACT FROM AUTHOR]

The aim of current investigation is to study the dynamic behavior of shells made of unsymmetric laminated composites. Based on the higher-order shell model with twelve parameters, a shell element with four nodes is designed and equipped with the ANS method to circumvent the transverse shear locking. Next, by using the Hamilton's principle, the mass and the stiffness matrices as well as the load vector are extracted. Additionally, the standard Newmark approach for the numerical time integration is employed. Some problems involving plates and cylindrical and spherical shells are solved to verify the performance of the proposed finite element formulation. [ABSTRACT FROM AUTHOR]

Developing chaotic systems-on-a-chip is gaining much attention due to its great potential in securing communication, encrypting data, generating random numbers, and more. The digital implementation of chaotic systems strives to achieve high performance in terms of time, speed, complexity, and precision. In this paper, the focus is on developing high-speed Field Programmable Gate Array (FPGA) cores for chaotic systems, exemplified by the Lorenz system. The developed cores correspond to numerical integration techniques that can extend to the equations of the sixth order and at high precision. The investigation comprises a thorough analysis and evaluation of the developed cores according to the algorithm complexity and the achieved precision, hardware area, throughput, power consumption, and maximum operational frequency. Validations are done through simulations and careful comparisons with outstanding closely related work from the recent literature. The results affirm the successful creation of highly efficient sixth-order Lorenz discretizations, achieving a high throughput of 3.39 Gbps with a precision of 16 bits. Additionally, an outstanding throughput of 21.17 Gbps was achieved for the first-order implementation coupled with a high precision of 64 bits. These outcomes set our work as a benchmark for high-performance characteristics, surpassing similar investigations reported in the literature. [ABSTRACT FROM AUTHOR]

Computer simulation of continuous chaotic systems is usually performed using numerical methods. The discretization may introduce new properties into finite-difference models compared to their continuous prototypes and can therefore lead to new types of dynamical behavior exhibited by discrete chaotic systems. It is known that one can control the dynamics of a discrete system using a special class of integration methods. One of the applications of such a phenomenon is chaos-based communication systems, which have recently attracted attention due to their high covertness and broadband transmission capability. Proper modulation of chaotic carrier signals is one of the key problems in chaos-based communication system design. It is challenging to modulate and demodulate a chaotic signal in the same way as a conventional signal due to its noise-like shape and broadband characteristics. Therefore, the development of new modulation–demodulation techniques is of great interest in the field. One possible approach here is to use adaptive numerical integration, which allows control of the properties of the finite-difference chaotic model. In this study, we describe a novel modulation technique for chaos-based communication systems based on generalized explicit second-order Runge–Kutta methods. We use a specially designed test bench to evaluate the efficiency of the proposed modulation method and compare it with state-of-the-art solutions. Experimental results show that the proposed modulation technique outperforms the conventional parametric modulation method in both coverage and noise immunity. The obtained results can be efficiently applied to the design of advanced chaos-based communication systems as well as being used to improve existing architectures. [ABSTRACT FROM AUTHOR]

This paper explores two sample-based methods for analysing multistable systems: basin stability and basin entropy. Both methods rely on many numerical integration trials conducted with diverse initial conditions. The collected data is categorised and used to compute metrics that characterise solution stability, phase space structure, and system dynamics predictability. Basin stability assesses the overall likelihood of reaching specific solutions, while the basin entropy measure aims to capture the structure of attraction basins and the complexity of their boundaries. Although these two metrics complement each other effectively, their original procedures for computation differ significantly. This paper introduces a universal approach and algorithm for calculating basin stability and entropy measures. The suitability of these procedures is demonstrated through the analysis of two non-linear systems. [ABSTRACT FROM AUTHOR]

The spherical ensemble is a well-known ensemble of N repulsive points on the two-dimensional sphere, which can realized in various ways (as a random matrix ensemble, a determinantal point process, a Coulomb gas, a Quantum Hall state...). Here we show that the spherical ensemble enjoys remarkable convergence properties from the point of view of numerical integration. More precisely, it is shown that the numerical integration rule corresponding to N nodes on the two-dimensional sphere sampled in the spherical ensemble is, with overwhelming probability, nearly a quasi-Monte-Carlo design in the sense of Brauchart-Saff-Sloan-Womersley for any smoothness parameter s ≤ 2. The key ingredient is a new explicit sub-Gaussian concentration of measure inequality for the spherical ensemble. [ABSTRACT FROM AUTHOR]

This paper introduces a novel dynamical model, building upon the existing dynamical model for Deimos in the current numerical ephemerides, which only encompasses the simple libration effects of Deimos. The study comprehensively incorporates the rotational dynamics of Deimos influenced by the torque exerted by the major celestial bodies (Mars, the Sun) in the solar system within the inertial space. Consequently, a full dynamical model is formulated to account for the complete coupling between the rotation and orbit of Deimos. Simultaneously, employing precision orbit determination methods used for artificial satellites, we develop an adjustment model for fitting data to the complete model. The 12-order Adams–Bashforth–Moulton (ABM) integration algorithm is employed to synchronously integrate the 12 state variables of the full model to obtain the orbit of Deimos.The difference in the orbits obtained by integrating the full model over a period of 10 years and those obtained by the simplified model is at the order of 10 km. After precise orbit determination, this difference decreases to below 100 m, so numerical simulation results indicate that the full dynamical model and adjustment model are stable and reliable. Simultaneously, the integration of the Deimos third-order gravity field in the full model over a 10-year period induces only meter-level positional changes. This suggests that when constructing the complete model, the utilization of a second-order gravity field alone is sufficient. Compared to the simple model, the polar axis of Deimos in the inertial space exhibits a more complex oscillation in the full model. Additionally, the full model calculates that the minimum moment of inertia principal axis of Phobos has an amplitude of approximately 0.5 degrees in the longitude direction and does not exceed 2 degrees in the latitude direction. This work further advances the current dynamical model for Deimos and establishes the foundational model for the generation of a new set of precise numerical ephemerides for Deimos. [ABSTRACT FROM AUTHOR]

In this paper, we apply a numerical integration strategy recently developed for determining the deformation shapes of structures constituted by Cosserat rods, to predict the behavior of panto-cylinders. Panto-cylinders have, as microstructure, a set of two families of helicoidal beams interconnected by perfect or elastic joints. The pivot's free rotation axis is, in the reference configuration, orthogonal to the cylindrical surfaces spanned by the beams. We perform a series of numerical simulations looking for the mechanical parameters which exalt the chirality effects in the structure. For the performed compression, extensions, shear, and torsion tests, we find chiral deformation patterns with a dependence on the type of joint and its length. [ABSTRACT FROM AUTHOR]

Many studies have shown that the brain can process subliminal numerals, i.e., participants can categorize a subliminal number into two categories: greater than 5 or less than 5. In the context of many studies on the unconscious integration of multiple subliminal stimuli, the issue of whether multiple subliminal numbers can be integrated is contentious. The same-different task is regarded as a perfect tool to explore unconscious integration. In the two experiments reported, we used a same-different task in which a pair of masked prime numbers was followed by a pair of target numbers, and participants were asked to decide whether the two target numbers were on the same (both smaller or larger than 5) or different sides (one smaller, the other larger than 5) of 5 in magnitude. The results indicated that the prime numbers could be categorized unconsciously, which was reflected by the category priming effect, and that the unconscious category relationship of the two prime numbers could affect the judgment on the category relationship of the two target numbers, as reflected by the response priming effect. The duration of the prime-to-target interstimulus interval (ISI) was also manipulated, showing a positive compatibility effect (PCE) of category priming and a negative compatibility effect (NCE) of response priming no matter whether the ISI was short (50 ms) or long (150 ms). The NCE, which occurred when the prime-to-target ISI was relatively short in this study, contradicted the conventional view but was consistent with previous results of unconscious integration based on an attention modulation mechanism. Importantly, this study provided evidence for the still-under-debate issue of numerical information integration. [ABSTRACT FROM AUTHOR]

Loose coupling exists in the full ceramic bearing-pedestal-foundation system within wide temperature ranges, which brings difficulties to the status monitoring and fault identification. In response to this problem, a dynamic model including the coupling effects in the system is established considering thermal-related fit clearance and pedestal looseness clearance within wide temperature ranges. The numerical integration method is applied in the simulation of system dynamic responses with the loose faults, and the relationship between the loosening impact force and peak values in vibrations is set up. The coupling mechanism between the two faults is thereby revealed, and the results are verified through experiments. Results show that the occurrences of outer ring looseness and pedestal looseness lead to new frequency components in dynamic responses, and the amplitudes of corresponding frequencies increase with the looseness clearances. The real-time running status of the system can be obtained by identifying the amplitude of the fault feature frequencies. This study reveals the fault coupling mechanism in full ceramic bearing-pedestal-foundation system within wide temperature ranges and provides theoretical basis and technical support for the safe and reliable operations for related equipment. [ABSTRACT FROM AUTHOR]

General expressions for the vectorial formulation of the equations of motion under geopotential perturbations are provided. These expressions are based on the apsidal frame formulation, and ease the identification of the fast and slow components of the motion, as well as the consequent averaging of the potential over the mean anomaly. The mean variation equations of the eccentricity and angular momentum vector, while redundant, comprise a non-singular differential system that is efficiently integrated numerically, and naturally complements analogous vectorial formulations of the third-body and solar radiation pressure disturbing effects. Still, classical approaches in orbital elements may show better efficiency in the integration of moderate and high eccentricity orbits due to the almost linear character of the variations. • Geopotential perturbations are reformulated in vectorial form. • The vectorial formulation is free from singularities. • The long-term behavior is obtained by averaging techniques. • The numerical integration of the vectorial mean elements is highly accurate and stable. [ABSTRACT FROM AUTHOR]

The optimization of scenarios and design of real-time-control in tokamaks, especially for machines still in design phase, requires a comprehensive exploration of solutions to the Grad–Shafranov (GS) equation over a high-dimensional space of plasma and coil parameters. Emulators can bypass the numerical issues in the GS equation, if a large enough library of equilibria is available. We train an ensemble of neural networks to emulate the typical shape-control targets (separatrix at midplane, X-points, divertor strike point, flux expansion, and poloidal beta) as a function of plasma parameters and active coil currents for the range of plasma configurations relevant to spherical tokamaks with a super-X divertor, with percent-level accuracy. This allows a quick calculation of the classical-control shape matrices, potentially allowing real-time calculation at any point in a shot with submillisecond latency. We devise a hyperparameter sampler to select the optimal network architectures and quantify uncertainties on the model predictions. To generate the relevant training set, we devise a Markov-chain Monte Carlo algorithm to produce large libraries of forward Grad–Shafranov solutions without the need for user intervention. The algorithm promotes equilibria with desirable properties, while avoiding parameter combinations resulting in problematic profiles or numerical issues in the integration of the GS equation. [ABSTRACT FROM AUTHOR]

Two computational strategies for the evaluation of the spherical harmonic coefficients of the gravitational potential due to a generally shaped homogeneous polyhedral source are examined in detail. The techniques are implemented numerically for the known asteroid shape models of Eros and Didymos. The aim of the investigation is to quantify specific numerical aspects of the two algorithms, such as the accuracy of the techniques compared to a closed analytical solution for varying distance between source and computation point, the band-limited spectral analysis of the obtained spherical harmonic models and the convergence behavior of the corresponding series expansion in the vicinity of the characteristic Brillouin sphere. From a computational point of view, the line integral approach demands approximately three times the CPU time of Werner's method. The two sets of spherical harmonic coefficients are 100% correlated up to degree 45 for Eros and up to degree 49 for Didymos. Approaching degree 100, the correlation by degree decreases by 0.0004% for Eros and by 0.004% for Didymos, the corresponding values for the correlation by order being 0.0002% and 0.304%. Inside the Brillouin sphere and approaching its boundary, the numerical agreement of the gravitational potential between the line integral method and the analytical solution is at the 1E-4 level, while with Werner's approach at the 1E-7 level. At a distance of 33.5 km outside the Brillouin sphere for Eros and 2.2 km for Didymos, both methods are identical, reaching an agreement level with the analytical solution of 1E-11 level for Eros and 1E-14 for Didymos. In terms of spherical harmonic representation, the series defined by the line integral approach converges faster to the analytical value for the gravitational potential by 4 degrees. [ABSTRACT FROM AUTHOR]

This paper presents a method to simulate fracture propagation in a damage zone in 2D, whereby Cohesive Zone (CZ) elements are assigned the same Continuum Damage Mechanics (CDM) model as their adjacent finite elements. Material hardening is a result of damage‐induced stiffness reduction in the finite and cohesive elements, while softening is modeled by cohesive debonding only. The damage components calculated at the numerical integration points of each finite element are averaged and that average is assigned to the nodes of the adjacent CZ elements. CZ element debonding triggers when a critical damage threshold is reached, beyond which, a linear softening law is used in both mode I and mode II. The proposed model is calibrated against published triaxial compression tests conducted on shale. Single‐CZ element simulations confirm that, below the critical damage threshold, elastic deformation energy is stored and energy is dissipated by damage propagation and irreversible deformation. Passed the critical damage threshold, stored elastic energy is released in the form of cohesive debonding energy. Borehole breakout simulations indicate that the proposed CDM‐based CZ model can predict the formation of spiral‐shaped fractures around a free circular cavity under biaxial stress. Simulations of fracture propagation in textured materials suggest that cohesive debonding within and between grains is triggered by the coalescence of localized zones of continuum damage, and that the grain size distribution affects both the fracture pattern and the softening behavior of the specimen considered. [ABSTRACT FROM AUTHOR]

In this work, we adopt a problem-based learning approach to develop an integrative bachelor project that relies on competences acquired from basic courses in mathematics, mechanics and computation. In this sense, the proposed project tries to pave the way from "apparently disconnected" concepts gained through previous studies toward the field of computational mechanics. On this basis, we study the motion of a particle along an arbitrary curve in space subject only to the gravitational field, the so-called sliding bead. Although this is a classic problem in mechanics, it has a substantial richness from a theoretical and practical point of view where the student reinforcing abstract skills and exploring different aspects of its numerical solution are allowed. Along this path, we start with the modeling process. This is followed by a stability analysis of the governing differential equations. Later on, we present a moderate introduction to classical numerical time integration by considering several numerical schemes and the well-known Matlab add-on Simulink. Finally, we present a brief discussion about how the proposed project allows articulating concepts of mathematics, mechanics, and computation in engineering programs of studies at horizontal and vertical levels. [ABSTRACT FROM AUTHOR]

Algorithms for generating uniform random points on a triaxial ellipsoid are non-trivial to verify because of the non-analytical form of the surface area. In this paper, a formula for the surface area of an ellipsoidal patch is derived in the form of a one-dimensional numerical integration problem, where the integrand is expressed using elliptic integrals. In addition, analytical formulae were obtained for the special case of a spheroid. The triaxial ellipsoid formula was used to calculate patch areas to investigate a set of surface sampling algorithms. Particular attention was paid to the efficiency of these methods. The results of this investigation show that the most efficient algorithm depends on the required coordinate system. For Cartesian coordinates, the gradient rejection sampling algorithm of Chen and Glotzer is best suited to this task, when paired with Marsaglia's method for generating points on a unit sphere. For outputs in polar coordinates, it was found that a surface area rejection sampler is preferable. [ABSTRACT FROM AUTHOR]

This article deals with the numerical analysis of a class of singularly perturbed delay Volterra integro-differential equations exhibiting multiple boundary layers. The discretization of the considered problem is done using an implicit difference scheme for the differential term and a composite numerical integration rule for the integral term. The analysis of the discrete scheme consists of two parts. First, we establish an a priori error estimate that is used to prove robust convergence of the discrete scheme on Shishkin and Bakhvalov type meshes. Next, we establish the maximum norm a posteriori error estimate that involves difference derivatives of the approximate solution. The derived a posteriori error estimate gives the computable and guaranteed upper bound on the error. Numerical experiments confirm the theory. [ABSTRACT FROM AUTHOR]