Quadrature Integration Research Articles

A finite element approach for nonlinear, transient heat conduction problems with convection, radiation or contact boundary conditions

Heat conduction problems with convection and/or radiation boundary conditions arise in nuclear and other engineering fields. This paper presents a finite element approach to solve transient, three-dimensional (3-D), nonlinear heat conduction problems using quadratic tetrahedral elements with area/volume coordinates and quadrature integrations. Boundary conditions modeled include surface convection, to-ambient radiation, and surface-to-surface radiation (using the radiosity matrix method, RMM). Instead of the commonly used hemi-cube method, L′Huilier’s theorem is used here to calculate view factors. Contact constraints are also imposed. Interface condition is set up for the interfaces of different material. The one- and two-step backward differentiation are used for time integration. Nonlinearities introduced by temperature-dependent material thermal properties and the Stefan–Boltzmann law for radiation are handled using the Picard and/or Newton–Raphson methods. The convergence rate of the numerical scheme is verified using an analytical solution obtained using the method of manufactured solutions (MMS). Several practical cases are simulated, including that of two spheres in contact with each other, and results are compared to those obtained using commercial codes.

Energy harvesting and dynamic response of SMA nano conical panels with nanocomposite piezoelectric patch under moving load

Nowadays, the application of advanced and intelligent materials such as piezoelectric, shape memory alloys (SMA), and multi-phase materials, are highly demanded, especially in self-powered energy structures. Therefore, to obtain maximum efficiency, it is crucial to study and analysis the mechanical behavior of the energy production rate of the employed materials. This research theoretically evaluated the impacts of moving load and the use of a piezoelectric patch on the level of energy harvesting and dynamic behavior of a Nano Conical Panel (NCP) made from SMA located on a frictional substrate using the First-order Shear Deformation Theory (FSDT). Based on mixture rule, boron nitride nanotubes (BNNTs) with smart properties are used to reinforce the piezoelectric patch. A two-phase nonlocal method was adopted to take nano-scale effects into account. Mechanical behaviors such as friction and torsional viscoelastic were considered for the foundation layer. To reproduce the pseudo-elastic characteristics of SMA, the Hernandez-Lagoudas model was employed. In addition to these, Integral Quadrature (IQ), Differential Quadrature (DQ), and Newmark methods were coupled together to solve the equations regarding the output voltage, electric power and dynamic deflection. The computational outcomes showed that by applying a positive voltage to a piezoelectric patch, the maximum strain, as well as the energy harvesting capacity of NCP, were increased. Placing the NCP on a frictional viscoelastic-torsional layer led to deteriorated peak values of dimensionless dynamic displacement and dimensionless voltage by about 50.3% and 44.5%, respectively, compared with those of the system without the medium. At load resistance of 400 nano ohm, increasing the moving load velocity to 6.5 nm/ns, piezoelectric patch length to NCP length ratio up to 0.5, and applying CFCF boundary conditions led to enhancement of maximum output power by 30%, 88.3%, and 377%, respectively. Also, increasing the weight fraction of Boron Nitride Nanotubes (BNNT) up to 0.4% in the piezoelectric layer was found to be a suitable strategy to simultaneously decrease the dimensionless dynamic displacement and enhance dimensionless voltage by 23.4% and 21%, respectively.

A Mixed-Effects Model in Which the Parameters of the Autocorrelated Error Structure Can Differ between Individuals

Research in psychology has seen a rapid increase in the usage of experience sampling methods and daily diary methods. The data that result from using these methods are typically analyzed with a mixed-effects or a multilevel model because it allows testing hypotheses about the time course of the longitudinally assessed variable or the influence of time-varying predictors in a simple way. Here, we describe an extension of this model that does not only allow to include random effects for the mean structure but also for the residual variance, for the parameter of an autoregressive process of order 1 and/or the parameter of a moving average process of order 1. After we have introduced this extension, we show how to estimate the parameters with maximum likelihood. Because the likelihood function contains complex integrals, we suggest using adaptive Gauss-Hermite quadrature and Quasi-Monte Carlo integration to approximate it. We illustrate the models using a real data example and also report the results of a small simulation study in which the two integral approximation methods are compared.

Optimal Geometric Multigrid Preconditioners for HDG-P0 Schemes for the reaction-diffusion equation and the Generalized Stokes equations

We present the lowest-order hybridizable discontinuous Galerkin schemes with numerical integration (quadrature), denoted as HDG-P0 for the reaction-diffusion equation and the generalized Stokes equations on conforming simplicial meshes in two- and three-dimensions. Here by lowest order, we mean that the (hybrid) finite element space for the global HDG facet degrees of freedom (DOFs) is the space of piecewise constants on the mesh skeleton. A discontinuous piecewise linear space is used for the approximation of the local primal unknowns. We give the optimal a priori error analysis of the proposed HDG-P0 schemes, which hasn’t appeared in the literature yet for HDG discretizations as far as numerical integration is concerned. Moreover, we propose optimal geometric multigrid preconditioners for the statically condensed HDG-P0 linear systems on conforming simplicial meshes. In both cases, we first establish the equivalence of the statically condensed HDG system with a (slightly modified) nonconforming Crouzeix–Raviart (CR) discretization, where the global (piecewise-constant) HDG finite element space on the mesh skeleton has a natural one-to-one correspondence to the nonconforming CR (piecewise-linear) finite element space that live on the whole mesh. This equivalence then allows us to use the well-established nonconforming geometry multigrid theory to precondition the condensed HDG system. Numerical results in two- and three-dimensions are presented to verify our theoretical findings.

Moho Geometry Estimation Beneath the Gulf of Guinea From Satellite Gravity Data Based on a Regularized Non‐Linear Gravity Inversion

AbstractIn this work, we present a new gravimetric Moho model over the Gulf of Guinea computed from the inversion of the global gravity EIGEN‐6C4 model. The computation procedure contains a regularized non‐linear inversion algorithm. Before applying the inversion algorithm, the gravity data from the EIGEN‐6C4 model have been corrected from gravimetric effects of topography, bathymetry, and sediments. To calculate these corrections, a forward modeling has been applied using tesseroids to take into account the earth curvature, and via the Legendre Quadrature integration technique. The inversion procedure of the Bouguer disturbances integrates the use of hyperparameters in particular the regularization parameter, the anomalous Moho density contrast and the reference Moho depth. This regularized non‐linear inversion method has been tested on synthetic data contaminated by a normally distributed noise. Through this test, the inversion method showed its effectiveness in estimating a Moho model. The application of this method on the Gulf of Guinea has produced a high‐resolution gravimetric model, whose depths vary between 7.75 and 44.50 km. The greatest depths (greater than 40 km) are located along the eastern part of the study area. The comparison of our gravimetric model with the isostatic Airy, GEMMA (GOCE Exploitation for Moho Modeling and Applications) and CRUST1.0 Moho models, globally shows good similarities. The comparison of our gravimetric Moho model to the GEMMA gravimetric model exhibits large residuals around the continental margin, thus reflecting the limit of the inversion method for areas with great Moho depth gradients. The differences observed between our gravimetric model and the CRUST1.0 model in certain regions such as southern Nigeria show the impact of the neglect of anomalous crustal and mantle sources in the accuracy of the solutions obtained after inversion. However, our gravimetric model yields an improved and high‐resolution representation of the Moho undulations over the Gulf of Guinea compared to previous gravimetric Moho models.

On projection method for numerical solution of hypersingular integral equations of the first kind combined with quadrature methods

The Fredholm integral equations of the first kind with two regular and hypersingular kernels are considered on [−1, 1]. The hypersingular kernel is considered smooth enough with no additional conditions. A projection method based on second kind Chebyshev polynomials approximation, combined with quadrature integration method, is developed to obtain high accurate approximations. The proposed method reduces the underlying integral equation to a system of algebraic equations. Several illustrative examples are provided to show the efficiency of the method.

New Numerical Methods for Solving the Initial Value Problem Based on a Symmetrical Quadrature Integration Formula Using Hybrid Functions

In this study, we construct new numerical methods for solving the initial value problem (IVP) in ordinary differential equations based on a symmetrical quadrature integration formula using hybrid functions. The proposed methods are designed to provide an efficient and accurate solution to IVP and are more suitable for problems with non-smooth solutions. The key idea behind the proposed methods is to combine the advantages of traditional numerical methods, such as Runge–Kutta and Taylor’s series methods, with the strengths of modern hybrid functions. Furthermore, we discuss the accuracy and stability analysis of these methods. The resulting methods can handle a wide range of problems, including those with singularities, discontinuities, and other non-smooth features. Finally, to demonstrate the validity of the proposed methods, we provide several numerical examples to illustrate the efficiency and accuracy of these methods.

Static Analysis of Anisotropic Doubly-Curved Shell Subjected to Concentrated Loads Employing Higher Order Layer-Wise Theories

In the present manuscript, a Layer-Wise (LW) generalized model is proposed for the linear static analysis of doublycurved shells constrained with general boundary conditions under the influence of concentrated and surface loads. The unknown field variable is modelled employing polynomials of various orders, each of them defined within each layer of the structure. As a particular case of the LW model, an Equivalent Single Layer (ESL) formulation is derived too. Different approaches are outlined for the assessment of external forces, as well as for non-conventional constraints. The doubly-curved shell is composed by superimposed generally anisotropic laminae, each of them characterized by an arbitrary orientation. The fundamental governing equations are derived starting from an orthogonal set of principal coordinates. Furthermore, generalized blending functions account for the distortion of the physical domain. The implementation of the fundamental governing equations is performed by means of the Generalized Differential Quadrature (GDQ) method, whereas the numerical integrations are computed employing the Generalized Integral Quadrature (GIQ) method. In the post-processing phase, an effective procedure is adopted for the reconstruction of stress and strain through-the-thickness distributions based on the exact fulfillment of three-dimensional equilibrium equations. A series of systematic investigations are performed in which the static response of structures with various curvatures and lamination schemes, calculated by the present methodology, have been successfully compared to those ones obtained from refined finite element three-dimensional simulations. Even though the present LW approach accounts for a two-dimensional assessment of the structural problem, it is capable of well predicting the three-dimensional response of structures with different characteristics, taking into account a reduced computational cost and pretending to be a valid alternative to widespread numerical implementations.

Free Vibration Characteristics of Bidirectional Graded Porous Plates with Elastic Foundations Using 2D-DQM

This manuscript develops for the first time a mathematical formulation of the dynamical behavior of bi-directional functionally graded porous plates (BDFGPP) resting on a Winkler–Pasternak foundation using unified higher-order plate theories (UHOPT). The kinematic displacement fields are exploited to fulfill the null shear strain/stress at the bottom and top surfaces of the plate without needing a shear factor correction. The bi-directional gradation of materials is proposed in the axial (x-axis) and transverse (z-axis) directions according to the power-law distribution function. The cosine function is employed to define the distribution of porosity through the transverse z-direction. Equations of motion in terms of displacements and associated boundary conditions are derived in detail using Hamilton’s principle. The two-dimensional differential integral quadrature method (2D-DIQM) is employed to transform partial differential equations of motion into a system of algebraic equations. Parametric analysis is performed to illustrate the effect of kinematic shear relations, gradation indices, porosity type, elastic foundations, geometrical dimensions, and boundary conditions (BCs) on natural frequencies and mode shapes of BDFGPP. The effect of the porosity coefficient on the natural frequency is dependent on the porosity type. The natural frequency is dependent on the coupling of gradation indices, boundary conditions, and shear distribution functions. The proposed model can be used in designing BDFGPP used in nuclear, marine, aerospace, and civil structures based on their topology and natural frequency constraints.

Shifted Gegenbauer-Gauss Collocation Method for Solving Fractional Neutral Functional-Differential Equations with Proportional Delays

In this paper, the shifted Gegenbauer-Gauss collocation (SGGC) method is applied to fractional neutral functional-differential equations with proportional delays. The technique we have used is based on shifted Gegenbauer polynomials and Gauss quadrature integration. The shifted Gegenbauer-Gauss method reduces solving the generalized fractional pantograph equation fractional neutral functional-differential equations to a system of algebraic equations. Reasonable numerical results are obtained by selecting few shifted Gegenbauer-Gauss collocation points. Numerical results demonstrate its accuracy, and versatility of the proposed techniques.

Improved Technique for Autonomous Vehicle Motion Planning Based on Integral Constraints and Sequential Optimization

The study is dedicated to elaborating and analyzing a technique for autonomous vehicle (AV) motion planning based on sequential trajectory and kinematics optimization. The proposed approach combines the finite element method (FEM) basics and nonlinear optimization with nonlinear constraints. There were five main innovative aspects introduced in the study. First, a 7-degree polynomial was used to improve the continuity of piecewise functions representing the motion curves, providing 4 degrees of freedom (DOF) in a node. This approach allows using the irregular grid for roadway segments, increasing spans where the curvature changes slightly, and reducing steps in the vicinity of the significant inflections of motion boundaries. Therefore, the segment length depends on such factors as static and moving obstacles, average road section curvature, camera sight distance, and road conditions (adhesion). Second, since the method implies splitting the optimization stages, a strategy for bypassing the moving obstacles out of direct time dependency was developed. Thus, the permissible area for maneuvering was determined using criteria of safety distance between vehicles and physical limitation of tire–road adhesion. Third, the nodal inequality constraints were replaced by the nonlinear integral equality constraints. In contrast to the generally distributed approach of restricting the planning parameters in nodes, the technique of integral equality constraints ensures the disposition of motion parameters’ curves strictly within the preset boundaries, which is especially important for quite long segments. In this way, the reliability and stability of predicted parameters are improved. Fourth, the seamless continuity of both the sought parameters and their derivatives is ensured in transitional nodes between the planning phases and adjacent global coordinate systems. Finally, the problem of optimization rapidity to match real-time operation requirements was addressed. For this, the quadrature integration approach was implemented to represent and keep all the parameters in numerical form. The study considered cost functions, limitations stipulated by the vehicle kinematics and dynamics, as well as initial and transient conditions between the planning stages. Simulation examples of the predicted trajectories and curves of kinematic parameters are demonstrated. The advantages and limitations of the proposed approach are highlighted.

Performance of novel multipath ultrasonic phased array flowmeter using Gaussian quadrature integration

In this paper, phased array transducer is used in multipath ultrasonic flow meters to measure the velocities in discrete lines from a fixed position without requiring mechanical movement. Gaussian quadrature integration formulas are applied to evaluate the measurement performance of multipath ultrasonic phased array flowmeter downstream single bend based on different velocity profile models. The flow field downstream single bend is modeled using CFD simulation. Multipath ultrasonic phased array flowmeter is examined at different distances from the elbow. The results reveal that at 20D downstream elbow, the standard deviation of relative errors using Gaussian quadrature based on single bend model is about 1% but based on symmetric models is about 3.5%. Far from the elbow, the standard deviation of relative errors based on symmetric models are about 0.5%, and based on the single bend model is about 3.5%.

Two-stream interaction problem and its application to mass transport

Classical boundary layer theory is effectively extended to study two-stream interaction problems. In particular, by proper scaling we invent a universal curve which relates the second-derivative initial condition of the normalized stream function to the first-derivative free-stream condition. This relationship is independent of any fluid property, and the Blasius flow profiles in respective streams are connected by matching at the interface. Accurate approximations to the universal curve are constructed by piecewise third-order polynomials. This technique finds an excellent application to mass transport problem of solvents from one immiscible fluid to another, namely, extraction of fluid as an interfacial problem. Analytical formula is obtained to show explicit dependence of mass flux on the nonlinear Blasius profiles, partition coefficient, Schmidt numbers, ratio of mass transfer coefficients, ratio of kinematic viscosities as well as the contact length of the two immiscible fluids. In other words, given these physical parameters, we directly obtain the mass flux simply by numerical quadrature of integrals involving the Blasius profiles.

Vibration of 2D FG Higher Order Plates using Differential Integral Quadrature Method

Vibration of 2D FG Higher Order Plates using Differential Integral Quadrature Method

Time domain boundary integral equations and convolution quadrature for scattering by composite media

We consider acoustic scattering in heterogeneous media with piecewise constant wave number. The discretization is carried out using a Galerkin boundary element method in space and Runge-Kutta convolution quadrature in time. We prove well-posedness of the scheme and providea prioriestimates for the convergence in space and time.

Seismic analysis and optimization of concrete bridge under the moving train utilizing numerical methods and adaptive improved harmony search algorithm

Abstract In this paper, the simultaneous actions of moving force and seismic load on the dynamic displacement and optimization of the concrete bridge are studied. The sinusoidal shear deformation beam theory is employed for the modelling of the concrete bridge mathematically. The structural damping of the concrete bridge is assumed by the Kelvin–Voigt theory. Utilizing the method of energy and Hamilton’s law, the equations of motion are obtained. Three mixed numerical methods, including the integral quadrature, harmonic differential quadrature method, and Newmark technique, are presented for the numerical outcomes of the differential equations. Utilizing adaptive improved harmony search, improved harmony search, harmony search, and global harmony search algorithms, the optimization process of the concrete bridge is examined. The mentioned algorithm is improved adaptively by utilizing dynamic deflection. The harmony memory is corrected at first and second adjustments, respectively, based on emotional bandwidth and step size randomly. The optimum conditions of the concrete bridge are evaluated with various harmony existing search methods. The role of multiple parameters, including the velocity and acceleration of moving load, length and thickness of bridge, boundary conditions, and the amplitude of carrying load, in the dynamic displacement of the bridge is studied. The numerical results indicate that with increasing the velocity and acceleration of the moving train, the dynamic displacement of the concrete bridge increases. In addition, with increasing the length of the bridge, the time of maximum deflection (i.e. when the train is in the middle of the bridge) is increased. It is concluded for the concrete bridge under the seismic load that the optimum values of the bridge’s length and thickness are decreased (about 24%) and increased (about 21%), respectively. The optimum values of amplitude, velocity, and acceleration of moving train are decreased, respectively, about 34%, 33%, and 29% in the case of the concrete bridge under the earthquake load. In addition, the optimum length of the concrete bridge is decreased significantly, with increasing the moving load amplitude, velocity, and acceleration.

Mathematical modelling, numerical analysis and damage of dams subjected to hydrodynamic pressure

Mathematical modelling and numerical analysis of a concrete dam with combined effects of water-sediment-foundation-earthquake are the main and novel contributions of this study. For this purpose, a concrete dam is modeled by exponential shear deformation beam theory (ESDBT). The hydro-dynamic pressure of water due to earthquake load is obtained by Helmholtz equation assuming various boundary conditions of reservoir and wave reflection coefficient. In addition, the forces of foundation and sediment are assumed by vertical and axial springs. On the basis of Hamilton's principle, the final motion equations are derived and solved by Harmonic differential quadrature (HDQ), Integral Quadrature (IQ) and with Newmark methods for calculating the dynamic deflection of the concrete dam. The Drucker–Prager criterion is applied for damage analysis of the concrete dam. The influences of various important parameters such as wave reflection coefficient, sediment type and its depth as well as foundation are investigated on the dynamic deflection, hydrodynamic pressure as a function of water depth, time history of hydrodynamic pressure and Drucker–Prager damage index of the concrete dam. The results indicated that with enhancing the sediment depth, the dynamic deflection and hydrodynamic pressure are decreased. In addition, the hydrodynamic pressure increases the dynamic deflection of the concrete dam about 35% with respect to hydrostatics pressure. With increasing the wave reflection constant, the absolute of Drucker–Prager damage index and the safety of concrete dam are reduced at the crest, middle and heel of the concrete dam.

A Blend of Analytical and Numerical Methods to Compute Orthogonal Image Moments over a Unit Disk

Accurate image moment computation is critical because they are used in a variety of fields, including image reconstruction and object recognition. Orthogonal polynomials are frequently used to compute moments due to their numerous intersecting and important theoretical properties. In polar coordinate systems, orthogonal polynomials such as the Polar Harmonic Fourier Transform (PHFT) and Polar Harmonic Transformation (PHT) are defined. However, the images are defined by a Cartesian coordinate system. To obtain image moments, a double integration over a unit circle over the product of the PHFT or PHT and the image function must be performed. The choice of double integration techniques and domain for the double integral has a significant effect on the precision of the computed moments. We have proposed a method for using the entire unit circle as an integration domain in this study, for image reconstruction. We used the PHFT and PHT to apply this technique to computing moments for the reconstruction. The proposed method outperforms other state-of-the-art methods, including Gaussian quadrature numerical integration method (GQM) and zeroth order approximation (ZOA) on variety of scenarios using three benchmarked image sets. We have demonstrated experimentally that this technique significantly improves the accuracy of image moments in higher order moments, as measured by reduced mean squared error (MSE). Additionally, the proposed method significantly improved the moment’s rotational (an improvement of 1-3% as compared to GQM and an improvement of 37-93% as compared to ZOA) and scaling invariance (an improvement of 0-500% as compared to GQM and an improvement of 3-7000% as compared to ZOA).

Nonlinear Static Stability of Imperfect Bio-Inspired Helicoidal Composite Beams

The objective of this manuscript is to develop, for the first time, a mathematical model for the prediction of buckling, postbuckling, and nonlinear bending of imperfect bio-inspired helicoidal composite beams with nonlinear rotation angle. The equilibrium nonlinear integrodifferential equations of imperfect (curved) helicoidal composite beams are derived from the Euler–Bernoulli kinematic assumption. The differential integral quadrature method (DIQM) and Newton-iterative method are employed to evaluate the response of imperfect helicoidal composite beams. Following the validation of the proposed model, numerical studies are performed to quantify the effect of rotation angle, imperfection amplitude, and foundation stiffness on postbuckling and bending behaviors of helicoidal composite beams. The perfect beam buckles through a pitchfork bifurcation. However, the imperfect beam snaps through the buckling type. The critical buckling load increases with the increasing value of elastic foundation constants. However, the nonlinear foundation constant has no effect in the case of perfect beams. The present model can be exploited in the analysis of bio-inspired structure, which has a failure similar to a metal and low interlaminar shear stress, and is used extensively in numerous engineering applications.

New, Simple and Accurate Approximation for the Gaussian Q Function With Applications

In this letter, we propose novel, accurate approximation for the Gaussian <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$Q$ </tex-math></inline-formula> function which is expressed as the sum of simple exponentials. To do so, we use the composite Gauss quadrature numerical integration method incorporating a special mid point rule. The nuances on the accuracy as well as an insight on the tractability of the proposed approximation is exhaustively presented in this letter. We show that this approximation facilitates the symbol error probability of square quadrature amplitude modulation technique over the versatile Fluctuating Beckmann fading model and the practical Fisher-Snedecor <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathcal {F}$ </tex-math></inline-formula> distribution. Lastly, the analysis is justified with the help of Monte-Carlo simulations.