Gaussian Quadrature Rules Research Articles

Two-dimensional frictionless large deformation contact problems using isogeometric analysis and Nitsche’s method

ABSTRACT The simulation of large deformation contact problems has been a tough subject due to the existence of multiple nonlinearities, including geometric nonlinearity and contact interface nonlinearity. In this paper, we develop a novel method to compute the large deformation of 2D frictionless contact by employing Nitsche-based isogeometric analysis. The enforcement of contact constraints as one of the main issues in contact simulation is implemented by using Nitsche’s method, and the node-to-segment scheme is applied to the contact interface discretization. We detailedly derive the discrete formulations for 2D large deformation frictionless contact where NURBS is used for geometrical modeling and the Neo-Hookean hyperelastic materials are applied to describe the deformation of the model. The collocation method with Greville points is employed to integrate the contact interface and the classical Legendre–Gauss quadrature rule is used for contact bodies’ integration. The Lagrange multiplier method and penalty method are also implemented for comparison with Nitsche’s method. Several examples are investigated, and the obtained results are compared with that from commercial software ABAQUS to verify the effectiveness and accuracy of the Nitsche-based isogeometric analysis.

On the computation of Patterson-type quadrature rules

We describe a stable and efficient algorithm for computing positive suboptimal extensions of the Gaussian quadrature rule with one or two degrees less of polynomial exactness than the corresponding Kronrod extension. These rules constitute a particular case of those first considered by Begumisa and Robinson (1991) and then by Patterson (1993) and have been proven to verify asymptotically good properties for a large class of weight functions. In particular, they may exist when the Gauss–Kronrod rule does not. The proposed algorithm is a nontrivial modification of the one introduced by Laurie (1997) for the Gauss–Kronrod quadrature, and it is based on the determination of an associated Jacobi matrix. The nodes and weights of the rule are then given as the eigenvalues and eigenvectors of the matrix, as in the classical Golub–Welsch algorithm (1969).

Finite element-based virtual fields method with pseudo-real deformation fields for identifying constitutive parameters

In this study, a novel virtual fields method (VFM) based on the finite element (FE) scheme, namely FE-VFM, is proposed as an inverse method for identifying the parameters of constitutive models. In the FE-VFM, experimentally measured full-field displacements are mapped onto FE meshes using global and local shape functions, and the internal virtual work is integrated using the Gauss quadrature rule. To validate the new method, a well-designed sensitivity study is conducted using the ideal deformation obtained from FE simulations for anisotropic linear elastic and isotropic plastic materials. In the case of anisotropic elasticity, the residuals of the internal and external virtual work are not significantly affected by the order and size of FE meshes, but the order of the numerical integration has a marginal effect on the quality of the results. Conversely, substantial impacts are obtained for the plastic case, in which the size and order of the FE meshes and the order of the numerical integration are all critical to the accuracy of identification owing to large, localized deformation. Furthermore, the concept of a pseudo-real deformation field is newly proposed as a virtual field, which improves the accuracy of the FE-VFM for optimizing the constitutive parameters of the plastic material. Finally, the inverse identification of the plastic hardening law for press-hardened steel is conducted using the FE-VFM with real experimental data. The results show that the FE-VFM can successfully reproduce the full-field displacements even with relatively low-quality full-field data if an optimum FE mesh is adopted. In addition, the simulated load–displacement curve of notch tension with FE-VFM identified hardening is in good agreement with the experimental results.

Developing the discretization method for fractal‐fractional two‐dimensional Fredholm–Volterra integro‐differential equations

This paper contains the computational approach based on Mott polynomials together with the two‐dimensional Legendre–Gauss (LG) quadrature rule for fractal‐fractional two‐dimensional Fredholm–Volterra (FF‐2D‐FV) integro‐differential equations. Furthermore, we applied the modified operational matrices of integration and fractional derivative in the process of the numerical algorithm to get the approximate solution with high accuracy. The fractal‐fractional derivative is defined in the Atangana–Riemann–Liouville sense. Besides, we estimate the error of the method in Sobolev space. Ultimately, to prove the theoretical claim, we represent the numerical results in the form of tables and diagrams.

Strong asymptotics of Jacobi-type kissing polynomials

We investigate asymptotic behaviour of polynomials satisfying varying non-Hermitian orthogonality relations where and is holomorphic and non-vanishing in a certain neighbourhood in the plane. These polynomials are an extension of so-called kissing polynomials () introduced in Asheim et al. [A Gaussian quadrature rule for oscillatory integrals on a bounded interval. Preprint, 2012 Dec 6. arXiv:1212.1293] in connection with complex Gaussian quadrature rules with uniform good properties in ω. The analysis carried out here is an extension of what was done in Celsus and Silva [Supercritical regime for the kissing polynomials. J Approx Theory. 2020 Mar 18;225:Article ID: 105408]; Deaño [Large degree asymptotics of orthogonal polynomials with respect to an oscillatory weight on a bounded interval. J Approx Theory. 2014 Oct 1;186:33–63], and depends heavily on those works.

A new system of singular integral equations for a curvilinear crack in bonded materials

The modified complex potentials (MCPs) functions are used to develop a new system of singular integral equations (SIEs) for a curvilinear crack in the upper part of bonded materials subjected to shear mode stress with the help of continuity conditions for resultant force and displacement functions. The unknown dislocation distribution function is mapped into a square root singularity function by using curved length coordinate method and the traction along the crack as the right hand term. The Gaussian quadrature rules were used to obtain the numerical solution for a new system of SIEs in order to compute the nondimensional stress intensity factors (SIFs) for these problems. Our results agree with those of the previous works. The findings have revealed that the nondimensional SIFs depend on the elastic constant ratio, crack geometries and the position of the cracks.

Internality of generalized averaged Gauss quadrature rules and truncated variants for modified Chebyshev measures of the first kind

It is desirable that a quadrature rule be internal, i.e., that all nodes of the rule live in the convex hull of the support of the measure. Then the rule can be applied to approximate integrals of functions that have a singularity close to the convex hull of the support of the measure. This paper investigates whether generalized averaged Gauss quadrature formulas for modified Chebyshev measures of the first kind are internal. These rules are applied to estimate the error in Gauss quadrature rules associated with modified Chebyshev measures of the first kind. It is of considerable interest to be able to assess the error in quadrature rules in order to be able to choose a rule that gives an approximation of the desired integral of sufficient accuracy without having to evaluate the integrand at unnecessarily many nodes. Some of the generalized averaged Gauss quadrature formulas considered are found not to be internal. We will show that some truncated variants of these rules are internal, and therefore can be applied to estimate the error in Gauss quadrature rules also when the integrand has singularities on the real axis close to the interval of integration.

Global‐phase portrait and large‐degree asymptotics for the Kissing polynomials

AbstractWe study a family of monic orthogonal polynomials that are orthogonal with respect to the varying, complex‐valued weight function, , over the interval , where is arbitrary. This family of polynomials originally appeared in the literature when the parameter was purely imaginary, that is, , due to its connection with complex Gaussian quadrature rules for highly oscillatory integrals. The asymptotics for these polynomials as have recently been studied for , and our main goal is to extend these results to all in the complex plane. We first use the technique of continuation in parameter space, developed in the context of the theory of integrable systems, to extend previous results on the so‐called modified external field from the imaginary axis to the complex plane minus a set of critical curves, called breaking curves. We then apply the powerful method of nonlinear steepest descent for oscillatory Riemann–Hilbert problems developed by Deift and Zhou in the 1990s to obtain asymptotics of the recurrence coefficients of these polynomials when the parameter is away from the breaking curves. We then provide the analysis of the recurrence coefficients when the parameter approaches a breaking curve, by considering double scaling limits as approaches these points. We see a qualitative difference in the behavior of the recurrence coefficients, depending on whether or not we are approaching the points or some other points on the breaking curve.

High-order Combined Multi-step Scheme for Solving Forward Backward Stochastic Differential Equations

In this work, in order to obtain higher-order schemes for solving forward backward stochastic differential equations, we propose a new multi-step scheme by adopting the high-order multi-step method in Zhao et al. (SIAM J. Sci. Comput., 36(4): A1731-A1751, 2014) with the combination technique. Two reference ordinary differential equations containing the conditional expectations and their derivatives are derived from the backward component. These derivatives are approximated by using the finite difference methods with multi-step combinations. The resulting scheme is a semi-discretization in the temporal direction involving conditional expectations, which are solved by using the Gaussian quadrature rules and polynomial interpolations on the spatial grids. Our new proposed multi-step scheme allows for higher convergence rate up to ninth order, and are more efficient. Finally, we provide a numerical illustration of the convergence of the proposed method.

A Gaussian quadrature rule for Fourier-type highly oscillatory integrals in the presence of stationary points

In this article, a Gaussian quadrature rule was developed for the approximation of Fourier-type highly oscillatory integrals (FHOIs) with the phase functions of form g(x)=xr,r=2s,s≥1,x∈[−1,1]. The presence of these rules was demonstrated numerically through an analytical-numerical process, and questions were also addressed regarding the number of the nodes of the quadrature rule tend to the endpoints ±1 and the number of those tend to the so-called stationary points (where the integrand does not oscillate locally) as ω→∞. With proper assumptions, an asymptotic order was obtained for the numerical method, which indicated that the quadrature error decreased rapidly with the increase in oscillation parameter ω or the quadrature points. Furthermore, the behavior of the quadrature nodes as ω→∞ and ω→0 was analyzed to demonstrate the optimal asymptotic order and optimal polynomial order of the proposed method. To illustrate the efficiency and accuracy of the proposed method, some numerical examples were considered as well. In addition, the proposed method was compared to other numerical methods.

Computation of error bounds via generalized Gauss–Radau and Gauss–Lobatto rules

Many functionals of a large symmetric matrix of interest in science and engineering can be expressed as a Stieltjes integral with a measure supported on the real axis. These functionals can be approximated by quadrature rules. Golub and Meurant proposed a technique for computing upper and lower error bounds for Stieltjes integrals with integrands whose derivatives do not change sign on the convex hull of the support of the measure. This technique is based on evaluating pairs of a Gauss quadrature rule and a suitably chosen Gauss–Radau or Gauss–Lobatto quadrature rule. However, when derivatives of the integrand change sign on the convex hull of the support of the measure, this technique is not guaranteed to give upper and lower error bounds for the functional. We describe an extension of the technique by Golub and Meurant that yields upper and lower error bounds for the functional in situations when only some derivatives of the integrand do not change sign on the convex hull of the support of the measure. This extension is based on the use of pairs of Gauss, and suitable generalized Gauss–Radau or Gauss–Lobatto rules. New methods to evaluate generalized Gauss–Radau and Gauss–Lobatto rules also are described.

Hierarchic models for the free vibration analysis of functionally gradient plates

Hierarchic models for the modal analyses of ceramic–metal functional gradient (FG) plates are introduced. This study was motivated by the fact that the free vibration analyses of those structures are usually conducted by the first- or higher-order shear deformation theories (SDT). Meanwhile, a hierarchic model does not only include these classical theories, but their model level varies from the first order to the 3D full elasticity. In fact, in the hierarchic models, the part of displacement field through the thickness is assumed a priori using polynomials with the desired highest order. Hence, they can be numerically implemented using 2D finite element or mesh-free method, by discretizing only the mid-surface of 3D structures. The global mass and global stiffness matrices of FG plates are computed by adopting the Gaussian quadrature rules for the mid-surface integral and the trapezoidal rule for the thickness-wise integral. The proposed hierarchic models are numerically illustrated and their characteristics are investigated. In addition, the modal characteristics of FG plates are parametrically examined to the key factors of gradient layer. The hierarchic models approach the same limit and show a sequence of model accuracy. Meanwhile, the modal responses of metal-ceramic FG plate structures are dependent of the volume fraction pattern and the relative thickness ratio of gradient layer.

Efficient quadrature rules for the singularly oscillatory Bessel transforms and their error analysis

In this paper, we focus on the computation and analysis of the highly oscillatory Bessel transforms with endpoint singularities of algebraic and logarithmic type. Based on the modification of the numerical steepest descent method, we present a new and efficient quadrature rule. Firstly, we divide the considered integrals into two parts by $J_{m}(z)=\frac {1}{2}\left [H_{m}^{(1)}(z)+H_{m}^{(2)}(z)\right ]$ , where each part can be transformed into the Fourier-type integrals. Then, we use the Cauchy’s residue theorem to convert these Fourier-type integrals into the infinite integrals on $[0,+\infty )$ . Next, the resulting infinite integrals can be efficiently calculated by constructing some appropriate Gaussian quadrature rules. In addition, we conduct error analysis in inverse powers of the frequency parameter. Finally, several numerical examples are provided to show the efficiency and accuracy of the proposed method.

Detecting optimality and extracting solutions in polynomial optimization with the truncated GNS construction

A basic closed semialgebraic subset of {mathbb {R}}^{n} is defined by simultaneous polynomial inequalities p_{1}ge 0,ldots ,p_{m}ge 0. We consider Lasserre’s relaxation hierarchy to solve the problem of minimizing a polynomial over such a set. These relaxations give an increasing sequence of lower bounds of the infimum. In this paper we provide a new certificate for the optimal value of a Lasserre relaxation to be the optimal value of the polynomial optimization problem. This certificate is to check if a certain matrix has a generalized Hankel form. This certificate is more general than the already known certificate of an optimal solution being flat. In case we have detected optimality we will extract the potential minimizers with a truncated version of the Gelfand–Naimark–Segal construction on the optimal solution of the Lasserre relaxation. We prove also that the operators of this truncated construction commute if and only if the matrix of this modified optimal solution is a generalized Hankel matrix. This generalization of flatness will enable us to prove, with the use of the GNS truncated construction, a result of Curto and Fialkow on the existence of quadrature rule if the optimal solution is flat and a result of Xu and Mysovskikh on the existence of a Gaussian quadrature rule if the modified optimal solution is a generalized Hankel matrix . At the end, we provide a numerical linear algebraic algorithm for detecting optimality and extracting solutions of a polynomial optimization problem.

Higher-order particle representation for particle-in-cell simulations

In this paper we present an alternative approach to the representation of simulation particles for unstructured electrostatic and electromagnetic PIC simulations. In our modified PIC algorithm we represent particles as having a smooth shape function limited by some specified finite radius, r0. A unique feature of our approach is the representation of this shape by surrounding simulation particles with a set of virtual particles with delta shape, with fixed offsets and weights derived from Gaussian quadrature rules and the value of r0. As the virtual particles are purely computational, they provide the additional benefit of increasing the arithmetic intensity of traditionally memory bound particle kernels. The modified algorithm is implemented within Sandia National Laboratories' unstructured EMPIRE-PIC code, for electrostatic and electromagnetic simulations, using periodic boundary conditions. We show results for a representative set of benchmark problems, including electron orbit, a transverse electromagnetic wave propagating through a plasma, numerical heating, and a plasma slab expansion. Good error reduction across all of the chosen problems is achieved as the particles are made progressively smoother, with the optimal particle radius appearing to be problem-dependent.

A New Application of Gauss Quadrature Method for Solving Systems of Nonlinear Equations

In this paper, we introduce a new three-step Newton method for solving a system of nonlinear equations. This new method based on Gauss quadrature rule has sixth order of convergence (with n=3). The proposed method solves nonlinear boundary-value problems and integral equations in few iterations with good accuracy. Numerical comparison shows that the new method is remarkably effective for solving systems of nonlinear equations.

Model reduction for fractional elliptic problems using Kato's formula

<p style='text-indent:20px;'>We propose a novel numerical algorithm utilizing model reduction for computing solutions to stationary partial differential equations involving the spectral fractional Laplacian. Our approach utilizes a known characterization of the solution in terms of an integral of solutions to local (classical) elliptic problems. We reformulate this integral into an expression whose continuous and discrete formulations are stable; the discrete formulations are stable independent of all discretization parameters. We subsequently apply the reduced basis method to accomplish model order reduction for the integrand. Our choice of quadrature in discretization of the integral is a global Gaussian quadrature rule that we observe is more efficient than previously proposed quadrature rules. Finally, the model reduction approach enables one to compute solutions to multi-query fractional Laplace problems with orders of magnitude less cost than a traditional solver.</p>

Numerical solution of nonlinear fractional optimal control problems using generalized Bernoulli polynomials

AbstractThis article introduces a new class of basis functions, namely, the generalized Bernoulli polynomials (GBP). The GBP are adopted for solving nonlinear fractional optimal control problems (NFOCP) generated by nonlinear fractional dynamical systems (NFDS) and boundary conditions (BC). The corresponding operational matrices (OM) of fractional derivatives (FD) expand the solution of the problem in terms of the GBP. The method transforms the NFOCP into systems of nonlinear algebraic equations. First, the state and control variables are approximated by the GBP with unknown coefficients and parameters and substituted in the objective function, NFDS and BC. Then, the Gaussian quadrature rule and the OM of FD allow the formulation of a constrained problem, which is solved using Lagrange multipliers. The accuracy of the method is tested by means of several examples and the results confirm its good performance.

A novel quadrilateral element for analysis of functionally graded porous plates/shells reinforced by graphene platelets

This paper firstly presents numerical analyses of functionally graded porous plates/shells with graphene platelets (GPLs) reinforcement using a novel four-node quadrilateral element with five degrees of freedom per node, namely SQ4P, based on the first-order shear deformation theory and Chebyshev polynomials. The novelty of the present element is to use the high-order shape functions which satisfy the interpolation condition at the points based on Chebyshev polynomials to build the new flat four-node element for analysis of plate/shell structures. The Chebyshev polynomials are a sequence of orthogonal polynomials that are described recursively and the values of these polynomials belong to the interval [−1,1] as well as vanish at the Gauss points. Full Gauss quadrature rule is used to establish the stiffness matrix, geometric stiffness matrix, mass matrix and load vector. Various dispersions of GPLs and internal pores into the metal matrix through the thickness of structure are considered with the rule of a mixture and the Halpin–Tsai model for evaluating effective material properties across the thickness. The influence of weight fraction, porosity coefficient and dimensions of GPLs, distribution of GPLs and porosity into metal matrix are fully studied via several numerical examples from static bending to free vibration and buckling response. Numerical results and comparison with other solutions from available references suggest that the present element has enough reliability and validity to use in structural analysis. With regular and irregular meshes, these results are in close agreement with the exact solutions by using the suitable value for the order of the shape functions.

Explicit Gaussian Quadrature Rules for $$C^1$$ Cubic Splines with Non-uniform Knot Sequences

Explicit Gaussian Quadrature Rules for $$C^1$$ Cubic Splines with Non-uniform Knot Sequences