Integration Rules Research Articles

MDFEM: Multivariate decomposition finite element method for elliptic PDEs with lognormal diffusion coefficients using higher-order QMC and FEM

We introduce the multivariate decomposition finite element method (MDFEM) for elliptic PDEs with lognormal diffusion coefficients, that is, when the diffusion coefficient has the form a = exp(Z) where Z is a Gaussian random field defined by an infinite series expansion Z(y) = ∑j≥1yjϕj with yj ~ N(0,1) and a given sequence of functions {ϕj}j≥1. We use the MDFEM to approximate the expected value of a linear functional of the solution of the PDE which is an infinite-dimensional integral over the parameter space. The proposed algorithm uses the multivariate decomposition method (MDM) to compute the infinite-dimensional integral by a decomposition into finite-dimensional integrals, which we resolve using quasi-Monte Carlo (QMC) methods, and for which we use the finite element method (FEM) to solve different instances of the PDE. We develop higher-order quasi-Monte Carlo rules for integration over the finite-dimensional Euclidean space with respect to the Gaussian distribution by use of a truncation strategy. By linear transformations of interlaced polynomial lattice rules from the unit cube to a multivariate box of the Euclidean space we achieve higher-order convergence rates for functions belonging to a class of anchored Gaussian Sobolev spaces while taking into account the truncation error. These cubature rules are then used in the MDFEM algorithm. Under appropriate conditions, the MDFEM achieves higher-order convergence rates in terms of error versus cost, i.e., to achieve an accuracy of O(ε) the computational cost is O(ε-1/λ-d′/λ)=O(ε-(p*+d′/τ)/(1-p*)) where ε−1/λ and ε−d′/λ) are respectively the cost of the quasi-Monte Carlo cubature and the finite element approximations, with d′= d (1+δ′) for some δ′ ≥ 0 and d the physical dimension, and 0<p*≤(2+d′/τ)-1 is a parameter representing the sparsity of {ϕj}j≥1.

Targeted Integrity Rule and Its Application

Although Australia introduced the targeted integrity rule to forestall multinationals circumventing the hybrid mismatch rules introduced in 2018, the law including the rule has several areas of interpretation that are new and therefore may result in confusion and conflict.

A cookbook for approximating Euclidean balls and for quadrature rules in finite element methods for nonlocal problems

The implementation of finite element methods (FEMs) for nonlocal models with a finite range of interaction poses challenges not faced in the partial differential equations (PDEs) setting. For example, one has to deal with weak forms involving double integrals which lead to discrete systems having higher assembly and solving costs due to possibly much lower sparsity compared to that of FEMs for PDEs. In addition, one may encounter nonsmooth integrands. In many nonlocal models, nonlocal interactions are limited to bounded neighborhoods that are ubiquitously chosen to be Euclidean balls, resulting in the challenge of dealing with intersections of such balls with the finite elements. We focus on developing recipes for the efficient assembly of FEM stiffness matrices and on the choice of quadrature rules for the double integrals that contribute to the assembly efficiency and also posses sufficient accuracy. A major feature of our recipes is the use of approximate balls, e.g. several polygonal approximations of Euclidean balls, that, among other advantages, mitigate the challenge of dealing with ball-element intersections. We provide numerical illustrations of the relative accuracy and efficiency of the several approaches we develop.

Modeling of an isothermal flow of a magnetohydrodynamic, viscoplastic fluid during forward roll coating process

An isothermal and incompressible flow of a magnetohydrodynamici (MHD) viscoplastic fluid has been examined while passing through the area among two co-rotating rolls. Applying lubrication approximation theory (LAT), the basic flow equations of mass and momentum are derived and subsequently non-dimensionalized. Exact solutions of velocity and pressure gradient distributions are evaluated. To evaluate the complex integrals, Trapezoidal rule for numerical integration is used. The significant engineering parameters like, the roll separation force, maximum pressure and the power transferred to the fluid by the rolls, are also numerically computed. The results demonstrate that the velocity field and pressure gradient are significantly affected due to the presence of the viscoplastic parameter. Magnetic field is found to be the controlling parameter for power input, separating force, and distance among attachment and detachment points, being extremely beneficial for the roll coating process. It has been learnt through this study that MHD can be an effective tool in the coating industry to regulate the engineering quantities in which coating thickness is the most important.

Comparation of Shell Thickness Integration Rules and Element Types on Single Point Incremental Forming (SPIF) Simulation

Comparation of Shell Thickness Integration Rules and Element Types on Single Point Incremental Forming (SPIF) Simulation

A multi-fidelity integration rule for statistical moments and failure probability evaluations

A multi-fidelity integration rule for statistical moments and failure probability evaluations

STEM Students' Engagement in Horizontal Transfer from Calculus to Physics and their Difficulties

Majority of science students are facing different problems in applying their calculus knowledge to physics courses. Researchers started to develop an integrated approach to address this problem however, many schools are still teaching calculus and physics as two separate subjects. Moreover, there has been no significant research on senior high school students' transfer of learning and difficulties in calculus-based physics subjects. It is crucial because this is when the students first experience applying calculus in a physics context. Hence, the study investigated the engagement of senior high school STEM students to horizontal transfer from Basic Calculus to General Physics subjects and the difficulties they experience in solving calculus-based-physics problems. A correlational study research design was employed to explore the relationship between the students’ physics and calculus performance using a physics worksheet. Both qualitative and quantitative methods were also employed to determine the difficulties of the students in calculus-based-physics problems. The Pearson correlation revealed that there is a significant positive correlation between the students’ physics and calculus performance. Although this could not serve as strong evidence of transfer, this strong correlation implies that senior high school STEM students were able to construct the similarities between the calculus-based physics problems and their calculus schema. As revealed in the questionnaire and the students’ responses in the worksheet and interview, students have difficulty in solving calculus-based physics problems in terms of identifying the variable that needs to be integrated, setting-up the limits of integration, evaluating the limits of integral, and identifying the appropriate rules of integration and applying it in solving the physics problem. These difficulties are rooted in the fact that students have little experience applying calculus in word problems, especially in the physics context.

Short Communication: IPC Salix Cultivar Database Proof-of-Concept

A variety of Salix L. (Willow) tree and shrub cultivars provide resources for significant commercial markets such as bioenergy, environmental applications, basket manufacturing, and ornamental selections. The International Poplar Commission of the Food and Agriculture Organization (IPC FAO) has maintained the Checklist for Cultivars of Salix L. (Willow) since 2015 and now lists 968 epithet records in a Microsoft Excel spreadsheet format. This Proof-of-Concept (POC) investigates using an SQL database to store existing IPC Salix cultivar information and provide users with a format to compare and submit new Salix cultivar entries. The original IPC data were divided into three separate tables: Epithet, Species, and Family. Then, the data were viewed from three different model perspectives: the original Salix IPC spreadsheet data, the Canadian (PWCC), and the Open4st database. Requirements for this process need to balance database integrity rules with the ease of adding new Salix cultivar entries. An integrated approach from all three models proposed three tables: Epithet, Family, and Pedigree. The Epithet and Family tables also included Species data with a reference to a website link for accepted species names and details. The integrated process provides a more robust method to store and report data, but would require dedicated IT personnel to implement and maintain long-term. A potential use case scenario could involve users submitting their Checklist entries to the Salix administrator for review; the entries are then entered into a test environment by IT resources for final review and promotion to a production online environment. Perhaps the most beneficial outcome of this study is the investigation of various strategies and standards for Epithet and Family recording processes, which may benefit the entire Populus and Salix communities.

Two-level model of the grain boundary diffusion under electron beam action

A two-level model of transition zone formation between a polycrystalline substrate and its protective coating under pulsed electron beam exposure with the isolation of individual grain boundaries at the mesolevel has been proposed. Possible chemical reactions and different nature of diffusion in the volume of the grains and along the grain boundaries are taken into account. Physical processes at the mesolevel are reflected in the thermal macro model for the whole sample. Both diffusion coefficients and chemical reaction rates depend on temperature. The different nature of the transition zone formation process between the coating and the substrate is revealed for grains of different sizes. The modification of coatings from tantalum and from silicon on TiNi-surface is studied as examples. The method for effective properties estimation suggested in this work uses the averaging and the integral rule taking into account the form and size of grains and the composition changing during electron beam treatment.

Approximation and application of the Riesz-Caputo fractional derivative of variable order with fixed memory

In this paper, the Riesz-Caputo fractional derivative of variable order with fixed memory is considered. The studied non-integer differential operator is approximated by means of modified basic rules of numerical integration. The three proposed methods are based on polynomial interpolation: piecewise constant, piecewise linear, and piecewise quadratic interpolation. The errors generated by the described methods and the experimental rate of convergence are reported. Finally, an application of the Riesz-Caputo fractional derivative of space-dependent order in continuum mechanics is depicted.

Secure pseudorandom bit generators and point sets with low star-discrepancy

The star-discrepancy is a quantitative measure for the irregularity of distribution of a point set in the unit cube that is intimately linked to the integration error of quasi-Monte Carlo algorithms. These popular integration rules are nowadays also applied to very high-dimensional integration problems. Hence multi-dimensional point sets of reasonable size with low discrepancy are badly needed. A seminal result from Heinrich, Novak, Wasilkowski and Woźniakowski shows the existence of a positive number C such that for every dimension d there exists an N-element point set in [0,1)d with star-discrepancy of at most Cd∕N. This is a pure existence result and explicit constructions of such point sets would be very desirable. The proofs are based on random samples of N-element point sets which are difficult to realize for practical applications.In this paper we propose to use secure pseudorandom bit generators for the generation of point sets with star-discrepancy of order O(d∕N). This proposal is supported theoretically and by means of numerical experiments.

How are expectancies and values cognitively combined to determine behavioral intentions? The role of expectancy‐value theory, information integration and behavioral outcomes in dietary intentions

SummaryThe expectancy‐value (EV) framework is among the most prominent psychological approaches to predicting behavioral intentions. However, how expectancies and values are cognitively integrated (e.g., multiplicatively, additively or averaging) to produce intentions has yet to be tested. This research combined EV and Information Integration Theory methodologies to identify how EV integration determined dietary intentions among 80 participants (Meanage = 19.4 years, 61 females). Participants were presented with scenarios depicting three potential outcomes of junk‐food consumption (weight gain, increased disease risk, and time savings). Expectancies (no information/low/medium/high) were varied within‐subjects, and between‐subject values were trichotomized for each outcome. Participants indicated their dietary intentions in this 3×4 mixed design for all three outcomes. Expectancies and values were integrated additively to produce dietary intentions in the context of weight gain and disease‐risk, but the integration rule for time savings could not be determined. Theoretical implications and practical applications of these results are also discussed.

An exponentially fitted integration scheme for a class of quasilinear singular perturbation problems

A class of quasilinear singularly perturbed two point boundary value problems in ordinary differential equations exhibiting boundary layer at the left end of the underlying interval is considered and a new exponentially fitted integration scheme on a uniform mesh is devised for its numerical solution. The devised scheme is obtained by introducing a suitable constant fitting factor in a new three term recurrence relationship derived by the application of outer region solution obtained by asymptotic expansion procedure and a combination of the exact and approximate rules of integration with finite difference approximations of the first derivative. Value of the fitting factor is determined using the theory of singular perturbations and used to take care of rapid changes in the solution. Resulting tridiagonal algebraic system of equations is solved by the Thomas algorithm. Convergence of the scheme is analyzed. Three numerical example problems are solved and computational results are tabulated to show the accuracy and efficiency of the method. It is easily observed that the derived scheme is able to produce accurate results with minimal computational effort for all the values of the mesh size h when the perturbation parameter e tends to zero. Both the theoretical and numerical analysis of the method reveals that the method is able to produce uniformly convergent results with quadratic convergence rate.

Efficient equilibrium-based stress recovery for isogeometric laminated curved structures

This work extends the stress recovery for laminated composite solid plates, proposed in [1,2], to curved structures. Based on 3D Isogeometric Analysis (IGA) computations and equilibrium, this procedure uses a single element through the thickness in combination with a calibrated layer-by-layer integration rule or a homogenized approach, allowing for an inexpensive and accurate approximation in terms of in-plane stresses (and their derivatives), while through-the-thickness stress components are poorly approximated. Relying on the high-order continuity properties of IGA shape functions, an accurate out-of-plane stress state can also be recovered by means of direct integration of the equilibrium equations in strong form. The a posteriori step application, which is straightforward in the context of solid plates, is not trivial in the case of curved geometries. In fact, the notion of in-plane and out-of-plane directions is not clear when modeling this kind of structures in the global reference system, while adopting curvilinear coordinates to express the equilibrium gives rise to additional coupled terms that require an iterative process to resolve the balance of momentum equation. Therefore, we propose to apply the recovery locally, which, despite leading to more elaborated stress derivative terms because of the increasing geometry complexity, still allows for a direct reconstruction as the resolvent system is uncoupled. Several numerical results show the good performance of this approach particularly for composite stacks with significant radius-to-thickness ratio and number of plies.

Writer identification using redundant writing patterns and dual-factor analysis of variance

Writer identification (WI) is a typical pattern recognition problem with the goal of recognizing the writer of a text from images of his or her handwriting. For handwriting-based applications, a new approach is required that can confirm the writer based on a very small amount of available text. However, due to the limited number of characters and the diverse contents of the available samples, it is still challenging to achieve a high recognition rate on large-scale databases. To solve this problem, this paper proposes an efficient method for offline text-independent WI using redundant writing patterns and dual-factor analysis of variance (DF-ANOVA). In this method, a search for writing patterns in a limited number of handwritten texts is first conducted to generate meaningful codebooks. Then, these writing patterns are described by means of the simplified Wigner distribution function (WDF) and improved directional index histogram (DIH) descriptors to extract both structural and texture features, and the two corresponding feature distances are integrated by means of a fuzzy integral rule to make the final decision. The Fisher discriminant ratio (FDR) is used for feature selection, and DF-ANOVA is used to eliminate the influence of factors associated with the labels of samples on the feature distance. The results of evaluations on the IAM (96.92%), Firemaker (96.4%), and Uyghur2016 (100%) databases illustrate that this system shows strong robustness to an increasing number of writers and a decreasing number of words in the sample texts.

Evaluation of weighted fusion for scalar images in multi-sensor network

The regular image fusion method based on scalar has the problem how to prioritize and proportionally enrich image details in multi-sensor network. Based on multiple sensors to fuse and manipulate patterns of computer vision is practical. A fusion (integration) rule, bit-depth conversion, and truncation (due to conflict of size) on the image information are studied. Through multi-sensor images, the fusion rule based on weighted priority is employed to restructure prescriptive details of a fused image. Investigational results confirm that the associated details between multiple images are possibly fused, the prescription is executed and finally, features are improved. Visualization for both spatial and frequency domains to support the image analysis is also presented.

Judgments of happiness during trail running: Pleasure, engagement, and meaning

ObjectivesIt has been theorized that happiness is derived from three major, unique sets of life experience: pleasure, engagement, and meaning. The present study examined the mental processes by which individuals combined five information cues (relatedness, autonomy, competence, mental vitality and physical vitality) when judging the degree of happiness felt by a trail runner during a run. Design/methodThe participants (104 adult male athletes; Mage = 32.70; SD = 10.86) rated pleasure, engagement, and meaning in 32 scenarios built from combinations of these cues. ResultsThe results of multivariate and univariate analyses of variance indicated that all five cues had a positive effect on judgments of pleasure, engagement, and meaning. The participants used three different information integration rules, depending on the pathway to happiness being probed. ConclusionsThe information integration and the integration rules highlighted the different contributions of pleasure, engagement, and meaning in cognitive building of happiness.

Two-dimensional reward evaluation in mice

When choosing among multi-attribute options, integrating the full information may be computationally costly and time-consuming. So-called non-compensatory decision rules only rely on partial information, for example when a difference on a single attribute overrides all others. Such rules may be ecologically more advantageous, despite being economically suboptimal. Here, we present a study that investigates to what extent animals rely on integrative rules (using the full information) versus non-compensatory rules when choosing where to forage. Groups of mice were trained to obtain water from dispensers varying along two reward dimensions: volume and probability. The mice’s choices over the course of the experiment suggested an initial reliance on integrative rules, later displaced by a sequential rule, in which volume was evaluated before probability. Our results also demonstrate that while the evaluation of probability differences may depend on the reward volumes, the evaluation of volume differences is seemingly unaffected by the reward probabilities.

Further Assessment of Three Bathe Algorithms and Implementations for Wave Propagation Problems

This paper further analyzes three Bathe algorithms ([Formula: see text]-Bathe, [Formula: see text]-Bathe and [Formula: see text]-Bathe) with their unknown properties revealed. The analysis shows firstly that three Bathe algorithms can cover two common integration schemes, trapezoidal rule and backward Euler formula, and that the second-order [Formula: see text]-Bathe algorithm is algebraically identical to the [Formula: see text]-Bathe algorithm. Via formulation of the generalized two-sub-step Newmark algorithm, it is shown that the common Newmark method cannot be considered as a special case of the [Formula: see text]-Bathe algorithm. For wave propagation problems, optimal Courant–Friedrichs–Lewy (CFL) numbers for reducing dispersion errors are found for the three Bathe algorithms by considering spatial and temporal discretizations simultaneously, while the modified integration rules are used for the element mass and stiffness matrices to reduce the anisotropy in wave propagating directions. The recommended optimal algorithmic parameters are given for the three Bathe algorithms to help users effectively solve various dynamic and wave propagation problems.

On some new double dynamic inequalities associated with Leibniz integral rule on time scales

In 2020, El-Deeb et al. proved several dynamic inequalities. It is our aim in this paper to give the retarded time scales case of these inequalities. We also give a new proof and formula of Leibniz integral rule on time scales. Beside that, we also apply our inequalities to discrete and continuous calculus to obtain some new inequalities as special cases. Furthermore, we study boundedness of some delay initial value problems by applying our results as application.