Cubature Research Articles

Standardized Telemedicine Software Development Kit with Hybrid Cloud Support

In modern Web development, it is expected that systems operating in the same area can be easily integrated. For common integration points, it is recommended to use a standardized data model and a common interface during the development as this will facilitate further integrations. The use of the cloud infrastructure is increasingly popular in telemedicine, but taking into account the goals, the productivity of the development, the availability of the system and the various regulations, choosing the right solution is not trivial. Inclouded platform consists of numerous currently active telemedical microservices that are working with a common software development kit. This tool provides a standardized data model for document-oriented database systems, has support for public and private clouds by using the classic Data Access Object (DAO) analogy and contains a lot of convenient functions as well. Furthermore, it is found that our solutions can significantly increase development productivity and is confirmed by measurements taken which involved software developers.

Cross section measurements of neutron elastic and inelastic scattering on Fe54

Scattering experiments were performed at the Geel Electron Linear Accelerator (GELINA) to determine the differential cross sections of neutron elastic and inelastic scattering on Fe54, using a highly enriched sample. Neutron energies were calculated via the time-of-flight technique. The scattered neutrons were detected using the ELastic and Inelastic Scattering Array (ELISA), a spectrometer consisting of 32 liquid organic scintillators, able to separate neutron and photon induced events via pulse-shape analysis. The detectors are mounted at eight specific angles with respect to the neutron beam direction to allow the concurrent calculation of both the differential and the integral cross section using the Gauss-Legendre quadrature rule. For elastic scattering, angular distributions and integral cross sections were produced in the energy range from 1 to 8 MeV, whereas for inelastic scattering, partial differential and angle integrated cross sections were estimated from the first excited state of Fe54 in the energy range from 2.5 to 5.5 MeV. The results are compared with the available experimental and evaluated nuclear data, as well as with theoretical calculations performed with the TALYS 1.9 and EMPIRE 3.2 codes. Published by the American Physical Society 2024

Multistage mineralizing episodes of the Proterozoic world-class Volta Grande gold deposit, Amazonian Craton, northern Brazil: Implications for the Bacajá Domain metallogenesis

The easternmost region of the Amazonian Craton in northern Brazil has been the focus of several mining exploration surveys, which led to the identification of the world-class Volta Grande gold deposit (∼6.0 Moz@1.02 g/t). The deposit is inserted in the Bacajá Domain (2.24–2.0 Ga), and part of the mineralization occurs in Proterozoic mylonite, gneiss, and associated metamafic rocks that underwent greenschist to amphibolite-facies metamorphism. Electrum (Au–Ag) occurs as grains in cm-sized quartz high-grade veins and veinlets (up to 34.1 g/t in some rock intervals) that follow the mylonitic foliation and are associated with pervasive carbonate alteration, synchronous with the dynamic metamorphism. Disseminated electrum grains, pyrite, chalcopyrite, arsenopyrite, tellurides, and galena occur associated with potassic, argillic, propylitic, and silicic alteration types mainly in fracture-controlled or pervasive styles. These geological features as a whole suggest an orogenic high-tonnage orogenic gold system. Isotropic core samples along a stratigraphic profile reveal late andesite, dacite, rhyodacite, and rhyolite lava flows and dikes and plutonic rocks such as quartz monzonite, granodiorite, monzodiorite, and subordinate microgranite crosscutting the older metamorphic rocks. These rocks reveal potassic, propylitic, argillic, silicic, and sericitic alteration types. Electrum grains, possibly related to boiling deposition, occur along cm-sized quartz, calcite, and adularia high-grade veins (up to 29.5 g/t in some rock intervals) with chalcopyrite, pyrite, arsenopyrite, tetradymite, and tellurobismuthite. Such mineralogy is compatible with a low-sulfidation epithermal system. Data integration points to at least two distinct mineralizing events leading to the Volta Grande gold deposit – orogenic gold and low-sulfidation epithermal – which explains such large-tonnage and high-grade ore accumulation. It configures a new exploration guide for the Bacajá Domain of the Amazonian Craton, attested by other worldwide regions with similar metamorphic, magmatic, and hydrothermal alteration characteristics.

Cross element integration for superconvergent frequency computation with cubic isogeometric formulation

A superconvergent cross element integration technique is presented for the cubic isogeometric formulation referring to the frequency computation of wave equations. More specifically, a four-element integration cell with 11-point quadrature and an intermediate two-element integration cell with 6-point quadrature are developed in accordance with the optimization of discrete isogeometric frequency error. These cross-element quadrature rules stand in sharp contrast to the conventional 4-point element based integration for the cubic isogeometric formulation, which needs 16 quadrature points within four elements per spatial dimension, especially for multi-dimensional scenarios. Meanwhile, a superconvergence with two added accuracy orders upon the 6th order accurate standard cubic isogeometric approach is naturally embedded in the proposed cross element integration technique by construction. Consequently, both efficiency and accuracy advantages are simultaneously realized in the proposed cross element integration method for cubic isogeometric formulation. The efficacy of the proposed superconvergent methodology is consistently validated by several representative numerical examples.

Vibration frequency and mode localization characteristics of strain gradient variable-thickness microplates

Based on the modified strain gradient theory and Kirchhoff–Love assumptions, a free vibration model is established for variable-thickness microplates with three material length scale parameters, and the corresponding Euler–Lagrange equation is derived. Keeping the microplate volume constant, three kinds of thickness variations along the X-direction are considered: linear, parabolic, and cubic variations. Further, a C2-type four-node 36-degree-of-freedom variable-thickness differential quadrature finite element is developed to solve the resulting variable-coefficient boundary value problem by combining the Gauss–Lobatto and differential quadrature rules. Galerkin approximate solution for fully simply supported microplates is provided as a benchmark for numerical method verification. The convergence and accuracy of the model are verified through several examples. Finally, the vibration frequencies of the microplates under different thickness variation types, taper ratios, thickness ratios, material length scale parameters, and boundary conditions are examined through parametric analysis. Also, the influence of each factor on the mode localization of the microplates is quantitatively characterized by the modal assurance criterion (MAC) for the first time. The numerical results show that the thickness variation has a minor effect on the vibration frequency but a significant influence on the mode shape when the volume of the microplate is constant. The strain gradient, taper ratio, and discontinuous boundary have a remarkable effect on the distribution of higher-order mode shape contour lines and the regions where vibration localization occurs.

Computation of pairs of related Gauss-type quadrature rules

The evaluation of Gauss-type quadrature rules is an important topic in scientific computing. To determine estimates or bounds for the quadrature error of a Gauss rule often another related quadrature rule is evaluated, such as an associated Gauss-Radau or Gauss-Lobatto rule, an anti-Gauss rule, an averaged rule, an optimal averaged rule, or a Gauss-Kronrod rule when the latter exists. We discuss how pairs of a Gauss rule and a related Gauss-type quadrature rule can be computed efficiently by a divide-and-conquer method.

Multipoint stress mixed finite element methods for linear viscoelasticity with weak symmetry

In this paper, we propose two Multipoint Stress Mixed Finite Element (MSMFE) methods for linear viscoelasticity with weak symmetry on quadrilateral grids. The methods are constructed based on the lowest order Brezzi–Douglas–Marini mixed finite element spaces for elastic and viscous stress, piecewise constant velocity and piecewise constant (linear) vorticity. A special quadrature rule is applied for local stress and vorticity elimination. This results in a positive definite cell centered velocity–vorticity or only velocity system at each time step. Unconditional energy-dissipation of the MSMFE methods is proved rigorously. The accuracy of all the numerical solutions in their nature norms are established to be first order space convergence, both for the semi-discrete and fully-discrete formulations. Numerical results validate the theory results of the proposed methods.

Prediction of Laser Welding Distortion in Cylindrical Shell Structures Using Direct Application of Inherent Strain

In this study, a simplified analysis method by directly applying inherent strain (DIS) at integration points to predict welding distortion of cylindrical shell structures was suggested. The 3D thermo-elastoplastic method demands substantial computational time. The strain as direct boundary (SDB) simplified method using virtual temperature encounters challenges in consecutive thermal structural analysis including welding distortion. The suggested method (that is, DIS) offers the advantage of enabling continuous thermal structural analysis after welding distortion analysis, with computational efficiency. To calculate inherent strain, lap laser welding with 2 mm and 2.6 mm SUS304 plates was performed, measuring fusion and heat-affected zone sizes via cross-sectional observations. Using the calculated inherent strain and our suggested direct input method, we conducted welding analysis and welding test on cylindrical shell structures. Comparison of welding distortion under same conditions revealed radial distortion difference within approximately 10%, corresponding with the experimental result and confirming a computational time reduction by over 2,000 times than those of thermo-elastoplastic analysis. Therefore, an efficient and accurate prediction of welding distortion was achievable through the suggested method.

Positivity-preserving and entropy-bounded discontinuous Galerkin method for the chemically reacting, compressible Euler equations. Part II: The multidimensional case

In this second part of our two-part paper, we extend to multiple spatial dimensions the one-dimensional, fully conservative, positivity-preserving, and entropy-bounded discontinuous Galerkin scheme developed in the first part for the chemically reacting Euler equations. Our primary objective is to enable robust and accurate solutions to complex reacting-flow problems using the high-order discontinuous Galerkin method without requiring extremely high resolution. Variable thermodynamics and detailed chemistry are considered. Our multidimensional framework can be regarded as a further generalization of similar positivity-preserving and/or entropy-bounded discontinuous Galerkin schemes in the literature. In particular, the proposed formulation is compatible with curved elements of arbitrary shape, a variety of numerical flux functions, general quadrature rules with positive weights, and mixtures of thermally perfect gases. Preservation of pressure equilibrium between adjacent elements, especially crucial in simulations of multicomponent flows, is discussed. Complex detonation waves in two and three dimensions are accurately computed using high-order polynomials. Enforcement of an entropy bound, as opposed to solely the positivity property, is found to significantly improve stability. Mass, total energy, and atomic elements are shown to be discretely conserved.

A spectral finite element Reissner–Mindlin shell formulation with NURBS-based geometry definition

A curved non-isoparametric Reissner–Mindlin shell element is developed for analyzing thin-walled structures. The standard kinematic description of the element requires the calculation of the director vector. To address this demand accurately, similar to isogeometric analysis (IGA), the geometry is defined by utilization of the non-uniform rational B-splines (NURBS) imported directly from computer-aided design (CAD) files. Then, shape functions of the Legendre spectral element method (SEM) are used to interpolate the displacements. Consequently, the shell director vector and Jacobian of the transformation are calculated properly according to the presented formulation. On the other hand, in Legendre SEM combined with Gauss–Lobatto–Legendre quadrature, the integration points and the element nodes coincide. Thus, the easily computable local coordinate systems at the integration points can be used directly as nodal basis systems. A separate calculation of nodal basis systems at control points, which is the source of either complexity or error in IGA shells, is not required. Given the condition number of the stiffness matrix in the developed method, super high-order elements can also be used. Very high order p-refined elements are used in addition to h-refinement of the mesh to show the capability of higher order elements to analyze problems without mesh refinement. The validity and convergence rate of the method are investigated and verified through various cases of h- and p-refinement in challenging obstacle course problems.

A numerical technique for solving nonlinear singularly perturbed Fredholm integro-differential equations

This study deals with two numerical schemes for solving a class of singularly perturbed nonlinear Fredholm integro-differential equations. The nonlinear terms are linearized using the quasi-linearization technique. On the layer adapted Shishkin mesh, the numerical solution is initially calculated using the finite difference scheme for the differential part and quadrature rule for the integral part. The method proves to be first-order convergent in the discrete maximum norm. Then, using a post-processing technique we significantly enhance the accuracy from first order to second order. Further, a hybrid scheme on the nonuniform mesh is also constructed and analyzed whose solution converges uniformly, independent of the perturbation parameter and directly gives second order accuracy. Parameter uniform error estimates are demonstrated and the theoretical results are validated through some numerical tests.

Numerical Integration of Some Arbitrary Functions over an Ellipsoid by Discretizing into Hexahedral Elements for Biomaterial Studies

This study mathematically examines chemical and biomaterial models by employing the finite element method. Unshaped biomaterials’ complex structures have been numerically analyzed using Gaussian quadrature rules. It has been analyzed for commercial benefits of chemical engineering and biomaterials as well as biorefinery fields. For the computational work, the ellipsoid has been taken as a model, and it has been transformed by subdividing it into six tetrahedral elements with one curved face. Each curved tetrahedral element is considered a quadratic and cubic tetrahedral element and transformed into standard tetrahedral elements with straight faces. Each standard tetrahedral element is further decomposed into four hexahedral elements. Numerical tests are presented that verify the derived transformations and the quadrature rules. Convergence studies are performed for the integration of rational, weakly singular, and trigonometric test functions over an ellipsoid by using Gaussian quadrature rules and compared with the generalized Gaussian quadrature rules. The new transformations are derived to compute numerical integration over curved tetrahedral elements for all tests, and it has been observed that the integral outcomes converge to accurate values with lower computation duration.

A class of reducible quadrature rules for the second-kind Volterra integral equations using barycentric rational interpolation

Reducible quadrature rules constitute a well-established class of direct quadrature methods for approximating solutions to Volterra integral equations. Unlike interpolatory quadrature formulae, the reducible quadrature rule can be constructed without the need for additional calculations of moments. This paper investigates the reducible quadrature rule by employing barycentric rational interpolation to solve the underlying initial value problem and analyzes its convergence and stability. It is found that these quadrature rules exhibit high-order convergence rates accompanied by extensive stability regions. Several numerical illustrations are provided to verify the theoretical results.

Legendre quadrature for the discretization of 1D radiating panels

AbstractIn [A. Capozzoli, C. Curcio, A. Liseno, MMS, Pizzo Calabro, Italy, 2022], the problem of modeling a source/scatterer using an equivalent radiator has been addressed and an approach has been given and numerically assessed.Once dimensioned the radiating panel, a practical implementation can be provided by a non-uniform array. The element positions should be chosen so that the array is capable to approximate, with an adequate accuracy, the fields radiated by the equivalent radiator. Here, the array element positioning is performed by exploiting a quadrature rule which takes into account that the singular functions supported on the region of interest associated to the most significant singular values of the radiation operator are related to those supported on the equivalent panel by a radiation integral. The quadrature rule enables also to choose a set of weights which are essential in the definition of the element excitation coefficients from the knowledge of the source distribution on the equivalent panel. For simplicity, a one-dimensional problem with a Legendre quadrature rule is considered. The approach is numerically assessed by checking the capability of the array to radiate, with a satisfactory degree of accuracy, the singular functions associated to the region of interest.

A second-order phase field-lattice Boltzmann model with equation of state inputting for two-phase flow containing soluble surfactants

In this paper, we propose a phase field-lattice Boltzmann (LB) model with an equation of state (EOS) inputting for two-phase flow containing soluble surfactants. In this model, both the order parameter for the phase field and the surfactant concentration are described by second-order partial differential equations, along with Navier–Stokes equations for the flow field. Changes in surfactant concentration do not affect the order parameter distribution; hence, an unwanted sharpening effect cannot arise. Most importantly, in the existing models, the EOS of surface tension is determined by posterior simulation tests instead of being directly set as an input parameter before the simulations. Hence, it is difficult to determine the model parameters in practical applications. To address this issue, we systematically develop a fully analytical EOS for surface tension based on the Gibbs–Duhem equation. Subsequently, an approximate explicit form for EOS is provided by utilizing the Jacobi–Gauss quadrature rule. Furthermore, a multiple-relaxation-time LB scheme is utilized to numerically solve the governing equations of three physical fields. Two benchmark examples are simulated to validate the accuracy of the present model. The consistency between the numerical results and the analytical EOS is verified. Moreover, the dynamics of droplets with surfactant in simple shear flow is investigated, unveiling the profound impact of various factors, such as surfactant bulk concentration, capillary number, and viscosity ratio, on single droplet deformation and two equal-sized droplets interaction. A detailed exploration of the fluid mechanism involved in two-phase flow with soluble surfactants is presented.

On the one-point quadrature discretization in peridynamics: A novel perspective from Monte Carlo integration

One-point quadrature discretization is one of the most popular discretizations for peridynamic simulation, but it suffers from poor accuracy. Employing smoothly decaying influence functions or precise volume correction can address this, but the former applies only to specific materials, and the latter may be computationally expensive and limited to square or cubic meshes. This paper discusses one-point quadrature discretization in the context of Monte Carlo integration and presents the quadrature rule, considering the smoothed acceptance-rejection method. In this context, the volume correction factor is explained as the probability of two particles interacting with each other. We summarize several volume correction algorithms from the literature and then present a new one based on the proposed quadrature rule. Numerical tests were conducted in both 2D and 3D. According to the comparison, the proposed algorithm only costs approximately 0.2% to 40% of the time that precise volume correction needs to compute the quadrature weights, while its integration accuracy is still at the same level. The results suggest that the continuity of the acceptance-rejection rule and the precision of the expectation of the horizon volume primarily matter in the accuracy of quadrature rules.

The Impact of Radiative Transfer at Reduced Spectral Resolution in Large‐Eddy Simulations of Convective Clouds

AbstractMany radiative transfer schemes approximate the spectral integration over ∼105 to ∼106 wavelengths with correlated k‐distributions methods that typically require only 101–102 spectral integration points (g‐points). The exact number of g‐points is then chosen as an optimal balance between computational costs and accuracy, normally assessed in terms of a number of radiative quantities. How this radiative accuracy propagates to simulation accuracy, however, is not straightforward. In this study, we therefore explore the sensitivity of cloud properties in large‐eddy simulations (LES) to the accuracy of radiative fluxes and heating rates. We first generate smaller sets of g‐points from existing k‐distributions by repeatedly combining adjacent g‐points while maintaining the highest possible accuracy on a chosen set of radiative metrics. Next, we perform three sets of LES with varying cloud—radiation coupling pathways, and therefore different requirements for the accuracy of the radiative transfer computations, to investigate how these smaller and thus less accurate k‐distributions affect simulation characteristics. The decrease in radiative accuracy with 3–4 times smaller k‐distributions results in biases in cloud properties that are relative small compared to their temporal fluctuations. These results show potential for speeding up radiative transfer computations in cloud‐resolving models by reducing the resolved spectral detail. However, more statistically converged simulations and a wider set of case studies is required to fully assess the robustness of our results.

A Collocation Numerical Method for Highly Oscillatory Algebraic Singular Volterra Integral Equations

The highly oscillatory algebraic singular Volterra integral equations cannot be solved directly. A collocation numerical method is proposed to overcome the difficulty created by the highly oscillatory algebraic singular kernel. This paper is composed primarily of two methods—the piecewise constant collocation method and the piecewise linear collocation method—in which uniformly distributed nodes serve as collocation points. For the efficient computation of highly oscillatory and algebraic singular integrals, the steepest descent method as well as the Gauss–Laguerre and generalized Gauss–Laguerre quadrature rules are employed. Consequently, the resulting linear system is solved for the unknown function approximated by the Lagrange interpolation polynomial. Detailed theoretical analysis is carried out and numerical experiments showing high accuracy are also presented to confirm our analysis.

An Efficient Quadrature Rule for Highly Oscillatory Integrals with Airy Function

In this work, our primary focus is on the numerical computation of highly oscillatory integrals involving the Airy function. Specifically, we address integrals of the form ∫0bxαf(x)Ai(−ωx)dx over a finite or semi-infinite interval, where the integrand exhibits rapid oscillations when ω≫1. The inherent high oscillation and algebraic singularity of the integrand make traditional quadrature rules impractical. In view of this, we strategically partition the interval into two segments: [0,1] and [1,b]. For integrals over the interval [0,1], we introduce a Filon-type method based on a two-point Taylor expansion. In contrast, for integrals over [1,b], we transform the Airy function into the first kind of Bessel function. By applying Cauchy’s integration theorem, the integral is then reformulated into several non-oscillatory and exponentially decaying integrals over [0,+∞), which can be accurately approximated by the generalized Gaussian quadrature rule. The proposed methods are accompanied by rigorous error analyses to establish their reliability. Finally, we present a series of numerical examples that not only validate the theoretical results but also showcase the accuracy and efficacy of the proposed method.

Numerical evaluation of oscillatory integrals via automated steepest descent contour deformation

Steepest descent methods combining complex contour deformation with numerical quadrature provide an efficient and accurate approach for the evaluation of highly oscillatory integrals. However, unless the phase function governing the oscillation is particularly simple, their application requires a significant amount of a priori analysis and expert user input, to determine the appropriate contour deformation, and to deal with the non-uniformity in the accuracy of standard quadrature techniques associated with the coalescence of stationary points (saddle points) with each other, or with the endpoints of the original integration contour. In this paper we present a novel algorithm for the numerical evaluation of oscillatory integrals with general polynomial phase functions, which automates the contour deformation process and avoids the difficulties typically encountered with coalescing stationary points and endpoints. The inputs to the algorithm are simply the phase and amplitude functions, the endpoints and orientation of the original integration contour, and a small number of numerical parameters. By a series of numerical experiments we demonstrate that the algorithm is accurate and efficient over a large range of frequencies, even for examples with a large number of coalescing stationary points and with endpoints at infinity. As a particular application, we use our algorithm to evaluate cuspoid canonical integrals from scattering theory. A Matlab implementation of the algorithm is made available and is called PathFinder.