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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 report on results from triaxial drained monotonic and cyclic compression tests for four different aggregates (gabbro, limestone, demolished concrete, steel slag), and discuss the applicability of UBCSAND model for simulating the relevant non-linear stress-strain curves. First, UBCSAND is integrated analytically for the elastic and plastic volumetric and deviatoric strains, leading to a pair of novel closed-form solutions expressed in terms of hypergeometric functions. These solutions are inherently free of numerical errors and can be used for the evaluation of results from numerical integration. Second, three calibration techniques for the model parameters are presented and compared: (1) a graphical “chart” solution based on results from forward numerical simulations; (2) a numerical approach based on minimization of residuals; and (3) an informed trial-and-error approach based on the analytical solution. Errors in numerical integration of the model are explored by comparing the analytical solution against results from an Euler integration. Variations in the calibrated parameters are discussed for drained monotonic and cyclic triaxial loading considering different types of aggregates.

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]

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]

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]

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]

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]

Deep learning methods have become attractive recently to accelerate topology optimization (TO) because of their capability to save huge computational costs with negligible sacrifice in the quality of final topologies. However, most current approaches rely on numerical mechanics platforms for training which incur significant computation costs. Further, current approaches are not suited for dynamically changing boundary conditions. This is because these deep learning frameworks once trained using results generated with a specific set of boundary conditions do not readily adapt to others. In this article, we present a novel approach to leverage the abilities of deep learning models for TO. We demonstrate that optimization can be achieved using a neural ordinary differential equation. The evolution of the design variable in each iteration of TO is then achieved by numerical integration of this neural differential equation with respect to the starting design. To improve the quality of the results, two levels of generative adversarial networks are also introduced, at the sequence, and image levels, respectively. The proposed machine learning framework is capable of generating full optimization paths relevant to TO in high resolution within seconds and can address novel unseen boundary conditions. [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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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 complexity of finite element analysis for composite structures can be significantly reduced by representing the connector and adjacent concrete as a macroscopic element. Nevertheless, the prevailing macroscopic models for shear connections predominantly employ nonlinear elastic theory. This approach introduces inaccuracies in estimating structural stiffness and load-bearing capabilities, primarily due to its inability to precisely capture the cumulative effects of plastic damage. In response, this study introduces a novel macroscopic elastoplastic model grounded in plasticity theory, aimed at accurately characterizing the nonlinear behavior of stud connections subjected to concurrent shear and tensile forces. This paper meticulously delineates the implementation of the elastoplastic constitutive model using the backward Euler method for numerical integration. It further articulates the derivation of the consistent tangent stiffness, which aligns with the convergence efficiency of the Newton–Raphson iterative approach. The computation of the element stiffness matrix for a two-node element is executed via the governing equation inherent to the finite element method. An exemplar macroelement test conducted in ABAQUS affirms the implicit backward Euler scheme's stability and consistency across varying tolerances. Validation of the elastoplastic model against empirical test outcomes corroborates its efficacy, demonstrating the model's precision in predicting the load–displacement behavior of stud connections under the influence of shear and tensile forces. [ABSTRACT FROM AUTHOR]

The diffusive viscous (DV) model is a useful tool for interpreting low-frequency seismic attenuation and the influence of fluid saturation on frequency-dependent reflections. Among present methods for the numerical solution of the corresponding DV wave equation, the finite-difference frequency-domain (FDFD) method with complex-valued adaptive coefficients (CVAC) has the advantage of efficiently suppressing both numerical dispersion and numerical attenuation. In this research, the FDFD method with CVAC is first generalized to a 3D DV equation. In addition, the current calculation of CVAC involves the numerical integration of propagation angles, conjugate gradient (CG) iterative optimization, and the sequential selection of initial values, which is difficult and inefficient for implementation. An improved method is developed for calculating CVAC, in which a complex-valued least-squares problem is constructed by substituting the 3D complex-valued plane-wave solutions into the FDFD scheme. The QR-decomposition method is used to efficiently solve the least-squares problem. Numerical dispersion and attenuation analyses reveal that the FDFD method with CVAC requires ∼2.5 spatial points in a wavelength within a dispersion deviation of 1% and an attenuation deviation of 10% for the 3D DV equation. An analytic solution for 3D DV wave equation in homogeneous media is proposed to verify the effectiveness of the proposed method. Numerical examples also demonstrate that the FDFD method with CVAC can obtain accurate wavefield modelling results for 3D DV models with a limited number of spatial points in a wavelength, and the FDFD method with QR-based CVAC requires less computational time than the FDFD method with CG-based CVAC. [ABSTRACT FROM AUTHOR]

In this paper, we use a method based on radial basis functions and the collocation method for the numerical solution of a class of Volterra integral equations of the third kind, using zeros of the shifted Legendre polynomial as the collocation points. The principal benefit of this scheme is that it does not require any discretization and so it is independent of the geometry of the domains and can thus be applied to the solution of various kinds of integral equations. The procedure is more flexible for the majority of classes of Volterra integral equations of the third kind. The construction of the suggested technique has been introduced. The convergence analysis of the presented method is investigated. Finally, certain numerical examples are included to show the accuracy and efficiency of the new technique. The numerical results obtained and their comparison with other methods demonstrate the reliability of this method. Our proposed method gives acceptable accuracy with a small use of data, which also reduces the computational costs. [ABSTRACT FROM AUTHOR]

This paper presents the results of modeling the propagation of inharmonic signals from an antenna located on the Earth's surface through the upper layer of the lithosphere and the reflection of these signals from various inhomogeneities in the lithosphere in the case when a video pulse signal of a special form is applied to the antenna. The model used in this study is based on an explicit scheme of numerical integration of Maxwell's equations. In this scheme, the electric and magnetic fields are calculated at the same time points in the same nodes of the spatial grid and splitting in spatial directions and physical processes is also used. The paper studies what information about the nature of the heterogeneity of the lithosphere can be extracted from the shape of the reflected signal. The influence of the parameters of the video pulse signal on the amplitude and shape of the signals reflected from typical inhomogeneities of the lithosphere is also investigated. It is shown that some types of inhomogeneities can be determined by the shape of the reflected signal. [ABSTRACT FROM AUTHOR]

The initial condition for the simulation of a seismically-induced tsunami for a rapid, assumed instantaneous, vertical seafloor displacement is given by the Kajiura low-pass filter integral. This work proposes a new efficient and accurate approach for its numerical evaluation, valid when the sea floor displacement is discretized as a set of rectangular contributions. We compare several truncated quadrature formulae, selecting the optimal one. We verify that we can satisfactorily approximate the initial sea level perturbation as a linear combination of those induced by the elementary sea floor displacements. The methodology is tested on the tsunamigenic Kuril earthquake doublet – a megathrust and an outer-rise – occurred in the Central Kuril Islands in late 2006 and early 2007. We also confirm the importance of the horizontal contribution to the tsunami generation and we consider a simple model of the inelastic deformation of the wedge, on a realistic bathymetry. The proposed approach results accurate and fast enough to be considered relevant for practical applications, and a tool is provided to create tsunami unit source databases for a given region of interest. [ABSTRACT FROM AUTHOR]

Different from the construction process of the traditional finite difference method, we derive a large class of high-accuracy methods for approximating derivatives. Since Taylor expansion is avoided, the requirement for function smoothness in the new methods is greatly reduced. We analyze the approximation errors of the proposed methods and compare their approximation effects. As a typical application, we use the proposed methods to solve Elliptic, Hyperbolic and Parabolic problems numerically. For time-dependent problems, a class of new fully implicit difference schemes and the fourth-order Runge-Kutta method are both used for discretization. When using fully implicit difference schemes, unconditional stability and convergence are proved for hyperbolic and parabolic problems with the periodic boundary conditions. A large number of numerical experiments are provided to demonstrate the theoretical results. [ABSTRACT FROM AUTHOR]

Nonlinear projection equations (NPEs) provide a unified framework for solving various constrained nonlinear optimization and engineering problems. This paper presents a deep learning approach for solving NPEs by incorporating neurodynamic optimization and physics‐informed neural networks (PINNs). First, we model the NPE as a system of ordinary differential equations (ODEs) using neurodynamic optimization, and the objective becomes solving this ODE system. Second, we use a modified PINN to serve as the solution for the ODE system. Third, the neural network is trained using a dedicated algorithm to optimize both the ODE system and the NPE. Unlike conventional numerical integration methods, the proposed approach predicts the end state without computing all the intermediate states, resulting in a more efficient solution. The effectiveness of the proposed framework is demonstrated on a collection of classical problems, such as variational inequalities and complementarity problems. [ABSTRACT FROM AUTHOR]

An analytical approach is presented that allows deriving an exact series-form representation for the flux linkage between two physically large concentric circular coils located on a lossy soil. The expression comes from a three-step analytical procedure. First, the integral expression for the flux linkage is converted into a double integral consisting of a finite and a semi-infinite integral. Next, the semi-infinite integral is recognized to be a well-known tabulated Sommerfeld integral, which may be analytically evaluated straightforwardly. Finally, applying Lommel's expansion allows rewriting the remaining finite integral as a sum of elementary integrals amenable to analytical evaluation. As a result, the flux linkage between the two coils is given as a sum of spherical Hankel functions of the wavenumber in the air and in the ground, multiplied by a coefficient depending on the geometrical dimensions of the coils. The accuracy and robustness of the proposed formulation is tested by comparing its outcomes with those generated by numerical integration of the complete integral representation for the flux linkage and with the results provided by previous analytical approaches to the same problem. It is found that the use of the derived expression for the inductance makes it possible to obtain significant time savings as compared to numerical quadrature schemes. [ABSTRACT FROM AUTHOR]

We introduce a predictor-corrector discretization scheme for the numerical integration of a class of stochastic differential equations and prove that it converges with weak order 1.0. The key feature of the new scheme is that it builds up sequentially (and recursively) in the dimension of the state space of the solution, hence making it suitable for approximations of high-dimensional state space models. We show, using the stochastic Lorenz 96 system as a test model, that the proposed method can operate with larger time steps than the standard Euler--Maruyama scheme and, therefore, generate valid approximations with a smaller computational cost. We also introduce the theoretical analysis of the error incurred by the new predictor-corrector scheme when used as a building block for discrete-time Bayesian filters for continuous-time systems. Finally, we assess the performance of several ensemble Kalman filters that incorporate the proposed sequential predictor-corrector Euler scheme and the standard Euler--Maruyama method. The numerical experiments show that the filters employing the new sequential scheme can operate with larger time steps, smaller Monte Carlo ensembles, and noisier systems. [ABSTRACT FROM AUTHOR]

Partial differential equations (PDEs) with uncertain or random inputs have been considered in many studies of uncertainty quantification. In forward uncertainty quantification, one is interested in analyzing the stochastic response of the PDE subject to input uncertainty, which usually involves solving high-dimensional integrals of the PDE output over a sequence of stochastic variables. In practical computations, one typically needs to discretize the problem in several ways: approximating an infinite-dimensional input random field with a finite-dimensional random field, spatial discretization of the PDE using, e.g., finite elements, and approximating high-dimensional integrals using cubatures such as quasi--Monte Carlo methods. In this paper, we focus on the error resulting from dimension truncation of an input random field. We show how Taylor series can be used to derive theoretical dimension truncation rates for a wide class of problems and we provide a simple checklist of conditions that a parametric mathematical model needs to satisfy in order for our dimension truncation error bound to hold. Some of the novel features of our approach include that our results are applicable to nonaffine parametric operator equations, dimensionally truncated conforming finite element discretized solutions of parametric PDEs, and even compositions of PDE solutions with smooth nonlinear quantities of interest. As a specific application of our method, we derive an improved dimension truncation error bound for elliptic PDEs with lognormally parameterized diffusion coefficients. Numerical examples support our theoretical findings. [ABSTRACT FROM AUTHOR]

The author considers the problem of calculating integrals of rapidly oscillating functions from some classes of differential functions, in particular, in the case of the interpolation class of functions, where a fixed table of its values specifies the information operator. Quadrature formulas for calculating integrals of rapidly oscillating functions that are optimal in terms of accuracy and in the order of accuracy are constructed. The optimal estimates for the error of the method are obtained. [ABSTRACT FROM AUTHOR]

We investigate the numerical approximation of integrals over |$\mathbb{R}^{d}$| equipped with the standard Gaussian measure |$\gamma $| for integrands belonging to the Gaussian-weighted Sobolev spaces |$W^{\alpha }_{p}(\mathbb{R}^{d}, \gamma)$| of mixed smoothness |$\alpha \in \mathbb{N}$| for |$1 < p < \infty $|⁠. We prove the asymptotic order of the convergence of optimal quadratures based on |$n$| integration nodes and propose a novel method for constructing asymptotically optimal quadratures. As for related problems, we establish by a similar technique the asymptotic order of the linear, Kolmogorov and sampling |$n$| -widths in the Gaussian-weighted space |$L_{q}(\mathbb{R}^{d}, \gamma)$| of the unit ball of |$W^{\alpha }_{p}(\mathbb{R}^{d}, \gamma)$| for |$1 \leq q < p < \infty $| and |$q=p=2$|⁠. [ABSTRACT FROM AUTHOR]

A seminumerical approach proposed many years ago for describing gravitational collapse in the post-quasi-static approximation is modified in order to avoid the numerical integration of the basic differential equations the approach is based upon. For doing that we have to impose some restrictions on the fluid distribution. More specifically, we shall assume the vanishing complexity factor condition, which allows for analytical integration of the pertinent differential equations and leads to physically interesting models. Instead, we show that neither the homologous nor the quasi-homologous evolution are acceptable since they lead to geodesic fluids, which are unsuitable for being described in the post-quasi-static approximation. Also, we prove that, within this approximation, adiabatic evolution also leads to geodesic fluids, and therefore, we shall consider exclusively dissipative systems. Besides the vanishing complexity factor condition, additional information is required for a full description of models. We shall propose different strategies for obtaining such an information, which are based on observables quantities (e.g., luminosity and redshift), and/or heuristic mathematical ansatz. To illustrate the method, we present two models. One model is inspired in the well-known Schwarzschild interior solution, and another one is inspired in Tolman VI solution. [ABSTRACT FROM AUTHOR]

In this paper, the numerical algorithms for calculating the values of the left- and right-sided Riemann–Liouville fractional integrals and the Riesz fractional integral using spline interpolation techniques are derived. The linear, quadratic and three variants of cubic splines are taken into account. The estimation of errors using analytical methods are derived. We show four examples of numerical evaluation of the mentioned fractional integrals and determine the experimental rate of convergence for each derived algorithm. The high-precision calculations are executed using the 128-bit floating-point numbers and arithmetic routines. [ABSTRACT FROM AUTHOR]

In this paper, we compare the performances of two Butcher-based block hybrid methods for the numerical integration of initial value problems. We compare the condition numbers of the linear system of equations arising from both methods and the absolute errors of the solution obtained. The results of the numerical experiments illustrate that the better conditioned method outperformed its less conditioned counterpart based on the absolute errors. In addition, after applying our method on some examples, it was discovered that the absolute errors in this work were better than those of a recent study in the literature. Hence, we recommend this method for the numerical solution of stiff and non-stiff initial value problems. [ABSTRACT FROM AUTHOR]

Progress in better understanding and modeling Earth surface systems requires an ongoing integration of data and numerical models. Advances are currently hampered by technical barriers that inhibit finding, accessing, and executing modeling software with related datasets. We propose a design framework for Data Components, which are software packages that provide access to particular research datasets or types of data. Because they use a standard interface based on the Basic Model Interface (BMI), Data Components can function as plug-and-play components within modeling frameworks to facilitate seamless data–model integration. To illustrate the design and potential applications of Data Components and their advantages, we present several case studies in Earth surface processes analysis and modeling. The results demonstrate that the Data Component design provides a consistent and efficient way to access heterogeneous datasets from multiple sources and to seamlessly integrate them with various models. This design supports the creation of open data–model integration workflows that can be discovered, accessed, and reproduced through online data sharing platforms, which promotes data reuse and improves research transparency and reproducibility. [ABSTRACT FROM AUTHOR]

With a treatment policy strategy, therapies are evaluated regardless of the disturbance caused by intercurrent events (ICEs). Implementing this estimand is challenging if subjects are not followed up after the ICE. This circumstance can be dealt with using delta adjustment (DA) or reference‐based (RB) imputation. In the survival field, DA and RB imputation have been researched so far using multiple imputation (MI). Here, we present a fully analytical solution. We use the illness–death multistate model with the following transitions: (a) from the initial state to the event of interest, (b) from the initial state to the ICE, and (c) from the ICE to the event. We estimate the intensity function of transitions (a) and (b) using flexible parametric survival models. Transition (c) is assumed unobserved but identifiable using DA or RB imputation assumptions. Various rules have been considered: no ICE effect, DA under proportional hazards (PH) or additive hazards (AH), jump to reference (J2R), and (either PH or AH) copy increment from reference. We obtain the marginal survival curve of interest by calculating, via numerical integration, the probability of transitioning from the initial state to the event of interest regardless of having passed or not by the ICE state. We use the delta method to obtain standard errors (SEs). Finally, we quantify the performance of the proposed estimator through simulations and compare it against MI. Our analytical solution is more efficient than MI and avoids SE misestimation—a known phenomenon associated with Rubin's variance equation. [ABSTRACT FROM AUTHOR]

Space debris near the synchronous orbit poses significant collision risks to spacecraft in this area. Therefore, it is necessary to analyze the orbital dynamics of space debris in order to predict its orbital evolution and prevent it from colliding with the spacecraft. Due to the combined effect of the dissipative forces and Earth's 1:1 tesseral resonance, the space debris with high area-to-mass ratio near the synchronous orbit exhibits two special dynamical behaviors: resonance capture and resonance jump. By using a simplified model, this paper analyzes in detail the orbital evolution of the debris of the resonance capture and resonance jump, and explores the conditions for their occurrence. It is found that the dynamical behavior of space debris around the resonance capture or resonance jump is highly sensitive to its initial conditions. In addition, the long-term direct numerical integrations with the full dynamical model for the orbital evolution of the debris are performed. The difference between the phase space parameters of the resonance capture and resonance jump for space debris is discussed, and the pendulum-like oscillation of the resonant angle is also studied. • Resonance capture and resonance jump for space debris around the synchronous orbit. • Initial conditions of resonance capture and resonance jump of space debris are found. • Resonance capture and resonance jump of debris are sensitive to initial conditions. • Resonant angle of near-synchronous debris exhibits pendulum-like oscillation. [ABSTRACT FROM AUTHOR]