Accurate Quadrature Research Articles

Local multiquadric scheme for solving two‐dimensional weakly singular Hammerstein integral equations

AbstractThe main purpose of this article is to present a numerical method for solving two‐dimensional Fredholm‐Hammerstein integral equations of the second kind with weakly singular kernels. The scheme utilizes locally supported (inverse) multiquadric functions constructed on scattered points as a basis in the discrete collocation method. The local (inverse) multiquadrics estimate a function in any dimensions via a small set of data instead of all points in the solution domain. The proposed method uses a special accurate quadrature formula based on the nonuniform Gauss‐Legendre integration rule for approximating singular integrals appeared in the scheme. In comparison with the globally supported (inverse) multiquadric for the numerical solution of integral equations, the proposed method is stable and uses much less computer memory. Moreover, the algorithm of the presented approach is attractive and easy to implement on computers. Since the scheme does not require any mesh generations on the domain, it can be identified as a meshless method. The error analysis of the method is provided. The convergence accuracy of the new technique is examined over several two‐dimensional Hammerstein integral equations and obtained results confirm the theoretical error estimates.

Estimating Orthant Probabilities of High-Dimensional Gaussian Vectors with An Application to Set Estimation

ABSTRACTThe computation of Gaussian orthant probabilities has been extensively studied for low-dimensional vectors. Here, we focus on the high-dimensional case and we present a two-step procedure relying on both deterministic and stochastic techniques. The proposed estimator relies indeed on splitting the probability into a low-dimensional term and a remainder. While the low-dimensional probability can be estimated by fast and accurate quadrature, the remainder requires Monte Carlo sampling. We further refine the estimation by using a novel asymmetric nested Monte Carlo (anMC) algorithm for the remainder and we highlight cases where this approximation brings substantial efficiency gains. The proposed methods are compared against state-of-the-art techniques in a numerical study, which also calls attention to the advantages and drawbacks of the procedure. Finally, the proposed method is applied to derive conservative estimates of excursion sets of expensive to evaluate deterministic functions under a Gaussian random field prior, without requiring a Markov assumption. Supplementary material for this article is available online.

Thin plate spline Galerkin scheme for numerically solving nonlinear weakly singular Fredholm integral equations

ABSTRACTThe present work proposes a numerical method to obtain an approximate solution of non-linear weakly singular Fredholm integral equations. The discrete Galerkin method in addition to thin-plate splines established on scattered points is utilized to estimate the solution of these integral equations. The thin-plate splines can be regarded as a type of free shape parameter radial basis functions which create an efficient and stable technique to approximate a function. The discrete Galerkin method for the approximate solution of integral equations results from the numerical integration of all integrals in the method. We utilize a special accurate quadrature formula via the non-uniform composite Gauss-Legendre integration rule and employ it to compute the singular integrals appeared in the scheme. Since the approach does not need any background meshes, it can be identified as a meshless method. Error analysis is also given for the method. Illustrative examples are shown clearly the reliability and efficiency of the new scheme and confirm the theoretical error estimates.

Application of thin plate splines for solving a class of boundary integral equations arisen from Laplace's equations with nonlinear boundary conditions

This article describes a technique for numerically solving a class of nonlinear boundary integral equations of the second kind with logarithmic singular kernels. These types of integral equations occur as a reformulation of boundary value problems of Laplace's equations with nonlinear Robin boundary conditions. The method uses thin plate splines (TPSs) constructed on scattered points as a basis in the discrete collocation method. The TPSs can be seen as a type of the free shape parameter radial basis functions which establish effective and stable methods to estimate an unknown function. The proposed scheme utilizes a special accurate quadrature formula based on the non-uniform Gauss–Legendre integration rule for approximating logarithm-like singular integrals appeared in the approach. The numerical method developed in the current paper does not require any mesh generations, so it is meshless and independent of the geometry of the domain. The algorithm of the presented scheme is accurate and easy to implement on computers. The error analysis of the method is provided. The convergence validity of the new technique is examined over several boundary integral equations and obtained results confirm the theoretical error estimates.

An Accurate Quadrature for the Numerical Integration of Polynomial Functions

The numerical integration of polynomial functions is one of the most interesting processes for numerical calculus and analyses, and represents thus, a compulsory step especially in finite elements analyses. Via the Gauss quadrature, the users concluded a great inconvenience that is processing at certain points which not required the based in finite element method points for deducting the form polynomials constants. In this paper, the same accuracy and efficiency as the Gauss quadrature extends for the numerical integration of the polynomial functions, but as such at the same points and nods have chosen for the determination of the form polynomials. Not just to profit from the values of the polynomials at those points and nods, but also from their first derivatives, the chosen points positions are arbitrary and the resulted deducted formulas are therefore different, as will be presented bellow and implemented.

A meshless Galerkin scheme for the approximate solution of nonlinear logarithmic boundary integral equations utilizing radial basis functions

This paper presents a computational scheme to solve nonlinear logarithmic singular boundary integral equations. These types of integral equations arise from boundary value problems of Laplace’s equations with nonlinear Robin boundary conditions. The discrete Galerkin method together with the (inverse) multiquadric radial basis functions established on scattered points is utilized to approximate the solution. The discrete Galerkin method for solving boundary integral equations results from the numerical integration of all integrals in the method. The proposed scheme uses a special accurate quadrature formula via the nonuniform Gauss–Legendre integration rule to compute logarithm-like singular integrals appeared in the scheme. Since the numerical method developed in the current paper does not require any mesh generations on the boundary of the domain, it is meshless and does not depend to the domain form. We also investigate the error analysis of the proposed method. Illustrative examples show the reliability and efficiency of the new scheme and confirm the theoretical error estimates.

Influence of Grain Inclination Angle on Shear Buckling of Laminated Timber Sheathing Products

Recent advances in timber production industries have enabled production of new innovative laminated timber products having layers with grain inclination angle. This paper is aimed to study influence of grain inclination angle in the laminated veneer lumber (LVL) and plywood sheathings on their shear buckling loads. Two extreme edge conditions of simply supported and clamped edges are considered. First, an accurate differential quadrature (DQ) computational code is developed using MAPLE programming software to obtain eigen buckling values and their corresponding eigen mode shapes. Next, for convenience of engineering calculations, approximate algebraic formulae are presented to predict critical shear buckling loads and mode shapes of LVL and plywood panels having layers with grain inclination angle, with adequate accuracy. Furthermore, finite element (FE) modelling is conducted for several cases using ANSYS software to show validity and accuracy of the predicted results for the problem. It is shown that the highest shear buckling loads of LVL sheathings is achievable when the inclination angle of about 30° with respect to the shorter edges is considered for production of LVL panels, whereas the same angle with respect to the long edges of the LVL sheathings results in a relatively lower buckling load. Considering similar inclination angle with respect to any edges of a plywood sheathings will also results in its highest pre-buckling capacity. It is also demonstrated that, under optimal design and certain loading circumstances, LVL shows a higher shear buckling capacity compared to a similar plywood sheathing.

Formula omitted]adaptation for Shallow Water flows: conservation, well balancedness, efficiency

We investigate the potential of the so-called “relocation” mesh adaptation in terms of resolution and efficiency for the simulation of free surface flows in the near shore region. Our work is developed in three main steps. First, we consider several Arbitrary Lagrangian Eulerian (ALE) formulations of the shallow water equations on moving grids, and provide discrete analogue in the Finite Volume and Residual Distribution framework. The compliance to all the physical constraints, often in competition, is taken into account. We consider different formulations allowing to combine volume conservation (DGCL) and equilibrium (Well-Balancedness), and we clarify the relations between the so-called pre-balanced form of the equations [7]), and the classical upwinding of the bathymetry term gradients [1]. Moreover, we propose a simple remap of the bathymetry based on high accurate quadrature on the moving mesh which, while preserving an accurate representation of the initial data, also allows to retain mass conservation within an arbitrary accuracy. Second, the coupling of the resulting schemes with a mesh partial differential equation is studied. Since the flow solver is based on genuinely explicit time stepping, we investigate the efficiency of three coupling strategies in terms of cost overhead w.r.t. the flow solver. We analyze the role of the solution remap necessary to evaluate the error monitor controlling the adaptation, and propose simplified formulations allowing a reduction in computational cost. The resulting ALE algorithm is compared with the rezoning Eulerian approach with interpolation proposed e.g. in Tang and Tang [17]. An alternative cost effective Eulerian approach, still allowing a full decoupling between adaptation and flow evolution steps is also proposed. Finally, a thorough numerical evaluation of the methods discussed is performed. Numerical results on propagation, and inundation problems shows that the best compromise between accuracy and CPU time is provided by a full ALE formulation. If a loose coupling with the mesh adaptation is sought, then the cheaper Eulerian approach proposed is shown to provide results quite close to the ALE.

Solving weakly singular integral equations utilizing the meshless local discrete collocation technique

The current work presents a computational scheme to solve weakly singular integral equations of the second kind. The discrete collocation method in addition to the moving least squares (MLS) technique established on scattered points is utilized to estimate the solution of integral equations. The MLS scheme approximates a function without mesh refinement over the domain that includes a locally weighted least squares polynomial fitting. The discrete collocation technique for the approximate solution of integral equations results from the numerical integration of all integrals in the method. We utilize an accurate quadrature formula based on the use of non-uniform composite Gauss-Legendre integration rule and employ it to compute the singular integrals appeared in the approach. The proposed scheme does not require any meshes, so it can be called as the meshless local discrete collocation (MLDC) method. Error analysis is also given for the method. Illustrative examples are shown clearly the reliability and efficiency of the new scheme and confirm the theoretical error estimates.

The free vibration analysis of point supported rectangular plates using quadrature element method

In this study, solutions of point supported isotropic plates are presented by using hybrid approach of Quadrature Element Method (QEM). The Hybrid QEM technique consists of a collocation method in conjunction with Galerkin finite element technique, to combine the high accurate and rapid converging of Differential Quadrature Method (DQM) for efficient solution of differential equation. To show the validity of the solutions, obtained results are compared with other known solutions methods for point supported rectangular plates. In addition different solutions are carried out for different type boundary conditions, different locations and number of point supports. Obtained results for first vibration modes on plates were also tested using a commercial finite element code and it was seen that they are in good agreement.

A Time Finite Element Method Based on the Differential Quadrature Rule and Hamilton’s Variational Principle

An accurate and efficient Differential Quadrature Time Finite Element Method (DQTFEM) was proposed in this paper to solve structural dynamic ordinary differential equations. This DQTFEM was developed based on the differential quadrature rule, the Gauss–Lobatto quadrature rule, and the Hamilton variational principle. The proposed DQTFEM has significant benefits including the high accuracy of differential quadrature method and the generality of standard finite element formulation, and it is also a highly accurate symplectic method. Theoretical studies demonstrate the DQTFEM has higher-order accuracy, adequate stability, and symplectic characteristics. Moreover, the initial conditions in DQTFEM can be readily imposed by a method similar to the standard finite element method. Numerical comparisons for accuracy and efficiency among the explicit Runge–Kutta method, the Newmark method, and the proposed DQTFEM show that the results from DQTFEM, even with a small number of sampling points, agree better with the exact solutions and validate the theoretical conclusions.

Geometrically accurate, efficient, and flexible quadrature techniques for the tetrahedral finite cell method

We illustrate the importance of geometrically accurate volume quadrature for obtaining optimal accuracy with non-boundary-fitted finite element discretizations, when the problem domain is defined by sharp boundaries. We consider the tetrahedral finite cell method (TetFCM) and replace its recursive subdivision based integration approach with geometrically accurate quadrature rules that emanate from higher-order geometric parametrizations of cut tetrahedral elements. The element-wise parametrization procedure relies on the identification of the intersection topology and a series of higher-order mappings based on Lagrange polynomials. We demonstrate with several 3D examples that geometrically faithful local parametrization ensures optimal accuracy, while significantly reducing the number of quadrature points with respect to recursive subdivision. On the other hand, we highlight the strength of subdivision quadrature in the context of a patient-specific workflow for the simulation-based performance analysis of coupled bone/implant configurations. In particular, we show that accuracy, flexibility and computational efficiency of the TetFCM critically depend on flexibly applying the two different quadrature variants for fuzzy imaging data and sharp boundary representations, respectively.

High order accurate conservative remapping scheme on polygonal meshes using a posteriori MOOD limiting

In this article we present a high order accurate 2D conservative remapping method for a general polygonal mesh. This method conservatively projects piecewise constant data from an old mesh onto a possibly uncorrelated new one. First an arbitrary (high) accuracy polynomial reconstruction operator is built. Then, the exact intersection between the old and new mesh is constructed, leading to a submesh of old subcells paving any new cell. Last, a high order accurate quadrature rule is designed to integrate the arbitrary high order accurate polynomials on the subcells to get a final new piecewise constant cell average on any new cell. The technique is limited a posteriori in such a way that effective high accuracy is maintained on smooth solutions while an essentially non-oscillatory behavior is observed on an irregular solution. Moreover, intrinsic physical properties of the system of variables such as positivity can be ensured. Numerical results assess that such a method is effective on problems for scalar remapping (smooth and discontinuous). We have also considered the remapping of multiple coupled quantities, such as mass, momentum and energy. The results confirm that this remap method is efficient in situations containing irregular/discontinuous profiles and smooth parts, for which some physical admissible constraints such as positivity must be ensured.

A DC-Stable, Well-Balanced, Calderón Preconditioned Time Domain Electric Field Integral Equation

The marching-on-in-time (MOT) solution of the time domain electric field integral equation (TD-EFIE) has traditionally suffered from a number of problems, including: 1) instability; 2) spurious static contributions plaguing the solution; 3) low-frequency breakdown; and 4) dense discretization breakdown. The first issue can be resolved by employing proper space-time Galerkin discretization schemes and accurate quadrature methods. The second and the third issue have been resolved by the quasi-Helmholtz Projected TD-EFIE (qHP-TDEFIE). This contribution introduces a multiplicative preconditioner which can be applied to the qHP-TDEFIE, without further modifying the original scheme. This preconditioner is based on Calderon techniques and guarantees that the MOT system can be solved efficiently using iterative methods, not only for large time step sizes but also for dense spatial discretizations, and for both simply and multiply connected geometries.

Unstructured finite element-based digital image correlation with enhanced management of quadrature and lens distortions

Like subset-based methods, the very first finite element versions of digital image correlation were closely related to the regular structure of images, as they were based on regular quadrilateral elements corresponding to an integer number of pixels. The use of unstructured meshes, to exploit the full potential of FE-DIC in structural mechanics, is now widespread. Most of the time, the formulation, the quadrature and the definition of the region of interest still rely on the pixels grid. In this paper, a formulation in the physical coordinate system and not in the image frame is proposed for 2D digital image correlation. In addition to a more precise definition of the region of interest, it allows the use of a more accurate quadrature rule. It is also shown that lens distortions can be successfully taken into account directly with such a formalism.

Efficient and accurate numerical quadrature for immersed boundary methods

One question in the context of immersed boundary or fictitious domain methods is how to compute discontinuous integrands in cut elements accurately. A frequently used method is to apply a composed Gaussian quadrature based on a spacetree subdivision. Although this approach works robustly on any geometry, the resulting integration mesh yields a low order representation of the boundary. If high order shape functions are employed to approximate the solution, this lack of geometric approximation power prevents exponential convergence in the asymptotic range. In this paper we present an algorithmic subdivision approach that aims to be as robust as the spacetree decomposition even for close-to-degenerate cases—but remains geometrically accurate at the same time. Based on 2D numerical examples, we will show that optimal convergence rates can be obtained with a nearly optimal number of integration points.

An accurate quadrature technique for the contact boundary in 3D finite element computations

This paper presents a new numerical integration technique for 3D contact finite element implementations, focusing on a remedy for the inaccurate integration due to discontinuities at the boundary of contact surfaces. The method is based on the adaptive refinement of the integration domain along the boundary of the contact surface, and is accordingly denoted RBQ for refined boundary quadrature. It can be used for common element types of any order, e.g. Lagrange, NURBS, or T-Spline elements. In terms of both computational speed and accuracy, RBQ exhibits great advantages over a naive increase of the number of quadrature points. Also, the RBQ method is shown to remain accurate for large deformations. Furthermore, since the sharp boundary of the contact surface is determined, it can be used for various purposes like the accurate post-processing of the contact pressure. Several examples are presented to illustrate the new technique.

Modification of the Euler quadrature formula for functions with a boundary-layer component

The Euler quadrature formula for the numerical integration of functions with a boundary-layer component on a uniform grid is investigated. If the function under study has a rapidly growing component, the error can be significant. A uniformly accurate quadrature formula is constructed by modifying the Hermite interpolation formula so that the resulting one is exact for the boundary-layer component. An analogue of the Euler formula that is exact for the boundary-layer component is constructed. It is proved that the resulting composite quadrature formula is third-order accurate in space uniformly with respect to the boundary-layer component and its derivatives.

A modified Nyström–Clenshaw–Curtis quadrature for integral equations with piecewise smooth kernels

The Nyström–Clenshaw–Curtis (NCC) quadrature is a highly accurate quadrature which is suitable for integral equations with semi-smooth kernels. In this paper, we first introduce the NCC quadrature and point out that the NCC quadrature is not suitable for certain integral equation with well-behaved kernel functions such as e−|t−s|. We then modify the NCC quadrature to obtain a new quadrature which is suitable for integral equations with piecewise smooth kernel functions. Applications of the modified NCC quadrature to Wiener–Hopf equations and a nonlinear integral equation are presented.

A self-correcting quadrature voltage controlled oscillator

A self-correcting quadrature voltage controlled oscillator (QVCO) with phase correcting loop is proposed. It comprises the QVCO core and phase correcting loop, which corrects the quadrature phase error. Two LC VCOs, buffers, and phase shifters are coupled in circular configuration to achieve IQ symmetry. This paper introduces the idea of realizing QVCO with low phase noise and accurate quadrature phase by using the phase correcting loop. The simulation results based on the 65 nm CMOS process show that the self-correcting QVCO has a phase error less than 0.5° and 1 MHz offset phase noise of −120 dBc/Hz at 3.7 GHz with 49% tuning range.