For the finite elements analyses, the stresses and strains and the elements of the deformation-displacement matrices in solid mechanics, the analysts in general are needing actually to obtain the results at the levels of nods of elements, such that the extrapolation functions and especially at the edge nods, don’t allow to offer good precisions. Moreover, the proposed integration formulas for polynomials, which addressed especially to the finite elements developers, such as they could be more favorable for finite elements analysts, and also for the integration of polynomials in general. In addition, such as the same obtained numerical results using the developed formulas and would be otherwise, obtained using Gauss quadrature and exact direct integration ones, thereof the intervention of the first derivatives ordinates herein, mean just that the complete liberty of the choosing points positions, and their contributions in this effect to get
2) The element type: In general, a higher polynomial degree of the shape functions requires a higher integration order to ensure exact numericalintegration regarding the Jacobian determinant. Therefore, the computing effort for the Gaussian quadrature increases. At the same time, the computation effort of the factorization-based method also grows, since a finite element of higher order has a larger number of nodes n. These competing factors must be considered in terms of computational effort and time.
The Meshless Local Petrov-Galerkin (MLPG) method is a numerical framework for solving partial differential equations. This method is unique in that it uses the governing equations in the local symmetric weak form and does not rely on a prescribed node connectivity. MLPG does not specify the numerical tools used for accomplishing the required tasks. In this work, Gaussian quadrature was used for integration over one-dimensional and two-dimensional domains. The Moving Least Squares (MLS) method was used for approximation of data and spatial derivatives of the data. The primitive variable form of the Navier-Stokes equations was used. MLPG results for benchmark fluids applications as well as the Rayleigh Taylor Instability (RTI) are presented. The Shape Function Interpolation Method (SFIM), a novel treatment for gradient type boundary conditions, is introduced. The SFIM method uses the MLS shape functions to identify the nodal boundary value that satisfies the prescribed gradient condition. A capability for computing a inward facing normal vector for a boundary node, based upon the other boundary nodes in the identified region, was also created. Using the standard method for the treatment of Neumann boundary conditions, the quadrature and test function require modification for new geometry types. This feature eliminates the need for modified quadrature and allows the weight function and MLS approximation to use their native Cartesian representations. These two additions minimize the phenomena or geometry specific coding required to apply MLPG to wide range of applications on complex domains.
In what follows, we establish some new error estimates which are sometimes exact and convenient for evaluation of the quadrature error for the Lipschitz con- tinuous functions using a new approach to numericalintegration. The result can be generalized for continuous functions and some types of cubature formulae.
There are a lot of methods that evaluate a deﬁnite integral, numerically. Some of these methods use the end points (closed rules) of integral and some do not (open rules). Some of them are based on using interpolating polynomial. The most popular of such methods are Trapezoidal, Simpson, and mid-point methods which are special cases of Newton-Cots method. There are some other methods that are based on the exact integration of poly- nomials of increasing degree; in which no subdivision of the integration interval are used. Basic properties of these methods can be found in many textbooks such as [1, 18]. The ordinary Taylor’s series has been generalized by many authors. Hardy  introduced a new version of the generalized Taylor’s series that uses Reimann-Liouville fractional integral and Trujillo et al.  obtained a new formula that is based on Reimann-Liouville frac- tional derivatives. For the concept of fractional derivative Odibat  adopted Caputo deﬁnition which is a modiﬁcation of the Reimann-Liouville deﬁnition and introduced a generalized Taylor’s series. Zaid Odibat introduced a generalized method for solving lin- ear partial diﬀerential equations of fractional order [12, 15] and introduced a novel method for nonlinear fractional partial diﬀerential equations . Hashemiparast et al.  intro- duced a method using derivations of function for numericalintegration. There are some good textbooks in this area [11, 16], and some new works have been done on numerical
Numericalintegration methods have witnessed a tremendous development over the last few decades; see, e.g., [2, 3, 14]. In particular, adaptive quadrature rules have nowadays become an integral part of many scientific computing codes. Here, one of the first yet very successful approaches is the application of adaptive Simpson integration or the more accurate Gauss-Kronrod procedures (see, e.g., ). The key points in the design of these methods are, first of all, to keep the number of function evaluations low, and, secondly, to divide the domain of integration in such a way that the features of the integrand function are appropriately and effectively accounted for. The aim of the current article is to propose a complementary adaptive quadra- ture approach that is quite different from previous numericalintegration schemes. In fact, our work is based on exploiting ideas from hp-type adaptive finite element Paul Houston
integration cell. If any of the bounding curves of the integration cell is defined piecewise, the Jacobian determinant in the integrand of the element stiffness matrix (Eq. 7) becomes non-smooth as well. Because the Gaussian quadrature is able to cope with polynomials but not with functions that are defined piecewise, this has to be taken into account by the decomposition algorithm. Having piecewise defined curves as boundary description is a very frequent case, as many geometries in engineering computations use B-Splines or NURBS. The parameter space of these curves is divided into a set of subintervals and the parametric equation of the curve changes at the breakpoint between neighboring subin- tervals. In the context of B-Splines or NURBS, these breakpoints are often referred to as knots.
The finite element method has become a powerful tool for the numerical solution of a wide range of engineering problem, particularly when analytical solutions are not available or very difficult to arriving the results. Numerical methods for integration approximate a definite integral of a given function by a weighted sum of function values at specified points. There are many quadrature methods available for approximating integrals. Surface integrals are used in multiple areas of physics and engineering. In particular, they are used for Problems involving calculations of mass of a shell, center of mass and moments of inertia of a shell, fluid flow and mass flow across a surface, electric charge distributed over a surface, plate bending, plane strain, heat conduction over a plate, and similar problems in other areas of engineering which are very difficult to analyse using analytical techniques, These problems can be solved using the finite element method. The method proposed here is termed as Generalized Gaussian rules, since the Generalized Gaussian quadrature nodes and weights for products of polynomial and logarithmic function given in  by Ma et al. are used in this paper
While testing functions are deﬁned analytically, they are often used numerically by employing a number of integration points on testing domains. Hence, at the implementation level, testing functions are reduced to a set of testing vectors with proper weights that are combinations of integration weights and the values of the used testing functions. Considering that the accuracy of MFIE heavily depends on the choice of the testing functions, it may be possible to select (and even optimize) the testing vectors numerically such that the accuracy of MFIE is improved, without deriving any analytical expression for the testing functions. In this study, we show that this approach is feasible and such numerical testing functions can be designed and constructed based on the compatibility of MFIE systems with the corresponding EFIE solutions. The designed testing functions provide more accurate results for both MFIE and the combined-ﬁeld integral equation (CFIE) that is more popular for closed conductors. The proposed numerical approach and its advantages for multi-frequency applications are demonstrated on two diﬀerent scattering problems.
The numericalintegration of signals given in tabular form is usually conducted using quadrature formulas, and experimental errors are not taken into consideration. In fact, quadrature formulas yield unpredictable results for various reasons. The first reason lies in the impossibility of establishing a priori smoothness of input data, and the second is that there is no way to evaluate the result of integration. Theoretical approximations of the error of quadrature formulas (3) may not be useful for actual values due to the inability to accurately calculate the residual term. The traditional approach consists of applying various smoothing filters. In this case, algorithms are used that do not depend on the task of integrating itself, which leads to excessive smoothing. The authors propose a new method for solving the problem of numerically integrating inaccurate signals that minimizes the residual term of the quadrature formula (4) for the set of unknown values of the signal by using ill-posed problem algorithms .
Abstract. It is well-known that the trapezoidal rule, while being only second-order accurate in general, improves to spectral accuracy if applied to the integration of a smooth periodic function over an entire period on a uniform grid. More precisely, for the function that has a square integrable derivative of order r the convergence rate is o N −(r−1/2)
C o cal integration based on interpolating poly- nstruction of quadrature rules for numeri- nomials is done by many authors that these poly- nomials are used to find weights corresponding to nodes. To see some quadrature rules based on polynomials, one can refer to - and the references therein. As we know, wavelets anal- ysis plays an important role in different areas of mathematics -. So, many authors ap- plied wavelets to approximate the solution of in- tegral equations, ordinary differential equations and partial differential equations. Recently, in , the authors applied Haar wavelets and hybrid functions to find numerical solution of definite in- tegrals with constant limits. In , the authors extended the scope of applicability of the method
The method of moments (MoM) is an efficient way to analyze the electromagnetic scattering problems formulated in terms of integral equations [1–3]. The conventional straightforward application of MoM involves low-order basis functions with the mesh cells’ size in the order of one tenth of a wavelength, leading to a dense system of linear equations. With N being the number of unknowns, the memory requirement is O(N 2 ), and the solution complexity is O(N 3 ) for a direct solver and O(N 2 ) for an iterative one. Several techniques have been proposed to reduce the memory demands as well as the solution complexity of the conventional MoM. Fast integral equation solvers, such as the multilevel fast multiple method (MLFMM) [4, 5], adaptive integral method (AIM) [6, 7] and its close counterpart, the precorrected FFT (PC-FFT) [8, 9], reach the solution complexity and memory
differential Equation (1) by certain finite difference. Extrapolation principles in  are applied to improve the performance of the method presented in . Kia Dithelm in  studied that a fast algorithm for the numerical solution of initial value problems of the form (1) in the sense of Caputo identifies and discusses potential prob- lems in the development of generally applicable schemes. More recently, Lagrange multiplier method and the homotopy perturbation method are used to solve numerically multi-order fractional differential equation see . Micul  considered the problem
mentioned in  and  of perturbed trapezoid type for k.k ∞ - norm. The inequality is then functional for a Divider of the interval [ρ,  to acquire many composite quadrature rules. The inequality is also functional to special means by Appropriately choosing selecting the function involved to get Certain direct association between different means.
Runge-Kutta methods -, are used for resolving this type of differential equations linear systems. These methods don’t need the calculus of partial de- rivatives of the perturbation function, but they have the problem of a lot of in- termediate calculations with a computational cost increase. It is preferred that the numerical methods used to resolve this type of systems verify the property of integrating exactly the homogeneous problem. The methods based on the Schei- fele - functions show this good property     . Runge-Kutta methods also have this property but it is necessary to do a transformation in the differential equations. This process is complicated because we need to change the integration order, furthermore the characteristic equation roots have to be known. The - function series methods are based on a refinement of Taylor ex- pansion   and they are applied to problems where the solution has a near sinusoidal behaviour which is not possible to be effectively resolved by Fourier series and Taylor’s expansions.
A critical point in the construction of the quadrature matrices is the inversion of special Vandermonde matrices. In the present implementation, only the selection of non-uniform Chebyshev abscissas was performed with excellent results. In practice, using Chebyshev polynomial bases, it can be expected that quadratures of much higher order could be used to solve problems demanding such sophistication.