Pascal Polynomial Research Articles

Cross-sectional analysis of initially curved composite beams with rectangular and circular sections using an asymptotic meshless method

An analytical meshless cross-sectional dimensional reduction method is developed to perform structural analysis of initially curved composite beams. By introducing the Pascal polynomials in the Cartesian coordinates system for beams with rectangular cross-sections and in the polar coordinates system for the beams with circular cross-sections to the warping field, a three-dimensional beam problem can be split into a two-dimensional cross-sectional analysis and a one-dimensional along the reference line of the beam. The obtained cross-sectional stiffness matrix based on higher-order of the strain energy can be incorporated into a one-dimensional finite element solution for initially curved composite beams, for the purpose of strain and stress analysis. Finally, the three-dimensional strain through the thickness of the beam can be achieved by the recovery analysis. The present method takes advantage of Pascal polynomials so that the solution procedure will be more simplified and computationally more efficient compared to three-dimensional finite element (3D FE) method. The proposed method for different isotropic and anisotropic beams is examined and compared with the literature, 3D FE, and Variational Asymptotic Beam Sectional (VABS) package which is a finite element based cross-sectional analysis tool for composite beams.

A local meshfree radial point interpolation method for Berger equation arising in modelling of thin plates

In this study, two-dimensional (2D) Berger equation arising in deflection of thin plates is solved numerically. A local meshfree collocation technique based on radial point interpolation is developed for the Berger equation. As radial basis function thin plate splines are used to avoid selecting optimal shape parameter. The method is used to solve both linear and non-linear Berger equation on regular and irregular geometries. Obtained results during numerical simulations, are compared with different methods exist in literatrue such as method of fundamental solutions, local Kansa’s method and Pascal polynomial based meshless method. The comparisons affirm the accuracy and reliability of the suggested method.

A meshless collocation method based on Pascal polynomial approximation and implicit closest point method for solving reaction–diffusion systems on surfaces

A meshless collocation method based on Pascal polynomial approximation and implicit closest point method for solving reaction–diffusion systems on surfaces

A strong-form meshfree computational method for plane elastostatic equations of anisotropic functionally graded materials via multiple-scale Pascal polynomials

A strong-form meshfree method is proposed for solving plane elastostatic equations of anisotropic functionally graded materials. Any general function may be the grading function and it is changing smoothly from location to location in the material. The proposed method is based on Pascal polynomial basis and multiple-scale technique and it is a genuinely meshfree method since no numerical integrations over domains and meshing processes are required for considered problems. Implementation of the proposed method is straightforward and the method gives very accurate results. Stability of the solutions are examined numerically in occurrence of random noise. Some certain test problems with known exact solutions are solved both on regular and irregular geometries. Acquired solutions by the suggested method are compared with the exact solutions as well as with solutions of some existing numerical techniques in literature, such as boundary element, meshless local Petrov–Galerkin and radial basis function based meshless methods, to show accuracy of the proposed method.

The local meshless method based on Pascal polynomial basis functions for solving fourth-order PDEs

In this paper, we present a localized method based on Pascal polynomial basis functions to solve fourth-order partial differential equations (PDEs) even with variable coefficients. The proposed algorithm is simple and effective, since applying Pascal polynomial basis functions can avoid the derivation of the closed-form particular solutions for higher order PDEs. Also, the localized formulation can alleviate the ill-conditioned problem of the resulting coefficient matrix. Five numerical examples are presented to demonstrate the accuracy and effectiveness of the proposed method in both regular and irregular domains.

Proper orthogonal decomposition Pascal polynomial-based method for solving Sobolev equation

PurposeThis study aims to use the polynomial approximation method based on the Pascal polynomial basis for obtaining the numerical solutions of partial differential equations. Moreover, this method does not require establishing grids in the computational domain.Design/methodology/approachIn this study, the authors present a meshfree method based on Pascal polynomial expansion for the numerical solution of the Sobolev equation. In general, Sobolev-type equations have several applications in physics and mechanical engineering.FindingsThe authors use the Crank-Nicolson scheme to discrete the time variable and the Pascal polynomial-based (PPB) method for discretizing the spatial variables. But it is clear that increasing the value of the final time or number of time steps, will bear a lot of costs during numerical simulations. An important purpose of this paper is to reduce the execution time for applying the PPB method. To reach this aim, the proper orthogonal decomposition technique has been combined with the PPB method.Originality/valueThe developed procedure is tested on various examples of one-dimensional, two-dimensional and three-dimensional versions of the governed equation on the rectangular and irregular domains to check its accuracy and validity.

Solving a nonlinear heat equation with nonlocal boundary conditions by a method of nonlocal boundary shape functions

When a nonlinear heat equation is subjected to nonlocal boundary conditions, the difficulty might arise because time-dependent integral conditions present in the problem. To overcome this difficulty, we derive a nonlocal boundary shape function (NLBSF) to satisfy initial condition and two nonlocal boundary conditions. Then, letting the free function in the NLBSF be the Pascal polynomials, the generated new bases automatically satisfy all the specified conditions. The solution is thus expanded in terms of these bases. After collocating points inside the space-time domain to satisfy the nonlinear heat equation and employing a novel splitting and linearizing technique to solve the resulting linear system, we can quickly find accurate solution of the nonlocal boundary conditions problem of nonlinear heat equation. Examples confirm the high accuracy and efficiency of the proposed iterative method.

Structural analysis of composite tubes using a meshless analytical dimensional reduction method

AbstractA polynomial‐based model in conjunction with dimensional reduction method are presented to perform cross‐sectional analysis and to determine strain distribution in composite tubes under bending loading. For beam structures with tubular cross‐section, the Variational Asymptotic Method (VAM) has been employed to decompose a three‐dimensional (3D) elasticity problem into a two‐dimensional cross‐sectional analysis and a one‐dimensional analysis along the length. This greatly reduces computational time as compared to 3D Finite Element Method (FEM). There also exists publically available Variational Asymptotic Beam Sectional Analysis (VABS), a FEM cross‐sectional analysis tool based on VAM. For VABS, FEM mesh for the beam section needs to be generated. For the case of composite tubes with many layers, the mesh generation consumes efforts and is unnecessary. We introduce a new meshless dimensional reduction method for the analysis of composite tubes. This method utilizes Pascal polynomials in polar coordinates to model the warping functions. Using this, one can obtain stiffness constants and 3D strains of the composite tube. This method is straightforward, meshless, with similar computation time as VABS, and is much more efficient than conventional 3D FEM. The accuracy of the proposed method is examined by comparing the obtained results with 3D FE (ANSYS), VABS, literature, and experiment.

An efficient meshfree method based on Pascal polynomials and multiple-scale approach for numerical solution of 2-D and 3-D second order elliptic interface problems

In this work, we propose an efficient meshfree method based on Pascal polynomials and multiple-scale approach for numerical solutions of two-dimensional (2-D) and three-dimensional (3-D) elliptic interface problems which may have discontinuous coefficients and curved interfaces with or without sharp corners. The proposed method uses Pascal polynomials as basis functions and utilizes multiple-scale approach for stabilizing numerical solutions. It is a well known fact that using polynomial basis without any modifications to obtain numerical solutions of partial differential equations may be peculiar owing to ill conditioned resultant coefficient matrix which is formed after process of discretization. Hence, as a remedy to the highly ill-conditioned coefficient matrix we employ multiple-scale approach. The main idea behind the multiple-scale approach is to make norm of all columns of resultant coefficient matrix equal to each other. The proposed method is a truly strong-form meshfree method since we do not need any mesh or integration process in problem domain, these features makes the implementation of the method very simple in computer environment. The efficiency of the proposed method is tested by some test problems which may have smooth interface or interface with sharp corners. Stability of the proposed method is investigated numerically in the presence of noise effect. Further, to show accuracy of the proposed method we present some comparisons with available numerical methods in literature, such as direct meshless local Petrov-Galerkin method, matched interface and boundary method, spectral element method and some meshless methods based on radial basis functions. The obtained numerical results and their comparisons confirm applicability of the proposed method for 2-D and 3-D steady state elliptic interface problems.

A homogenization function technique to solve the 3D inverse Cauchy problem of elliptic type equations in a closed walled shell

We solve the inverse Cauchy problem of elliptic type partial differential equations in an arbitrary 3D closed walled shell for recovering unknown data on an inner surface, with the over-specified Cauchy boundary conditions given on an outer surface. We first derive a homogenization function in the 3D domain to annihilate the Dirichlet as well as the Neumann data on the outer surface. Then, we can transform the inverse Cauchy problem to solve a direct problem inside the closed walled shell, using the homogenization technique and the domain type collocation method. The boundary functions are constructed from the 3D Pascal polynomials multiplied by an elementary boundary function, which are adopted as the bases to expand the numerical solution of the transformed elliptic equation. A simple scaling regularization is employed to reduce the condition number of the linear system to determine the expansion coefficients. Several numerical examples are presented to show that the novel method can overcome the highly ill-posed property of the inverse Cauchy problem in the 3D closed walled shell. The proposed algorithm is robust against large noise up to and very time saving to obtain accurate solution.

Localized meshless methods based on polynomial basis functions for solving axisymmetric equations

In this paper, two localized meshless methods based on polynomial basis functions are utilized to solve axisymmetric problems. In the first approach, we applied the localized method of particular solutions (LMPS) and the closed-form particular solution to simplify the two-stage approach using Chebyshev polynomial as the basis functions for solving axisymmetric problems. We also propose the modified local Pascal polynomial method (MLPM) to compare the results with LMPS. Since only the low order polynomial basis functions are used, no preconditioning treatment is required and the solution is quite stable. Four numerical examples are given to demonstrate the effectiveness of the proposed methods.

A Meshfree Computational Approach Based on Multiple-Scale Pascal Polynomials for Numerical Solution of a 2D Elliptic Problem with Nonlocal Boundary Conditions

A two-dimensional (2D) elliptic problem with nonlocal boundary conditions on both regular and irregular domains is solved numerically by Pascal polynomial basis unified with multiple-scale technique. Very accurate numerical solutions and quite reasonable condition numbers are obtained with the proposed method which is also a truly meshfree method since difficult meshing processes or numerical integrations over domains are not needed for considered problems. Four test problems are solved to show the accuracy and efficiency of the proposed method. Also stability of the method is studied against large noise effect.

Numerical solution to the deflection of thin plates using the two-dimensional Berger equation with a meshless method based on multiple-scale Pascal polynomials

• Two-dimensional linear and nonlinear Berger equation investigated. • Meshless multiple-scale Pascal polynomials method applied to fourth order equations. • Accurate results on complexly shaped domains with low number of collocation points . • Efficiency of proposed method demonstrated based on six problems. In this study, we employ Pascal polynomial basis in the two-dimensional Berger equation, which is a fourth order partial differential equation with applications to thin elastic plates. The polynomial approximation method based on Pascal polynomial basis can be readily adapted to obtain the numerical solutions of partial differential equations. However, a drawback with the polynomial basis is that the resulting coefficient matrix for the problem considered may be ill-conditioned. Due to this ill-conditioned behavior, we use a multiple-scale Pascal polynomial method for the Berger equation. The ill-conditioned numbers can be mitigated using this approach. Multiple scales are established automatically by selecting the collocation points in the multiple-scale Pascal polynomial method. This method is also a meshless method because there is no requirement to establish complex grids or for numerical integration. We present the solutions of six linear and nonlinear benchmark problems obtained with the proposed method on complexly shaped domains. The results obtained demonstrate the accuracy and effectiveness of the proposed method, as well showing its stability against large noise effects.

Recovering a general space-time-dependent heat source by the coupled boundary integral equation method

ABSTRACTAs to recover a time-dependent heat source under an extra temperature measured at an interior point, we can reformulate it to be a three-point boundary value problem. We can develop a coupled boundary integral equation method, wherein by selecting two sets of adjoint test eigenfunctions in two sub domains and using polynomials as the trial functions of unknown heat source, the time-dependent heat source is recovered very well and quickly. Four numerical examples, including a discontinuous one, demonstrate the efficiency for the ill-posed inverse heat source problem in a large time duration and under a large noise up to 10–30%. Then, selecting three sets of adjoint test eigenfunctions in three domains: problem domain and two sub domains, and using the Pascal polynomials as trial functions, the unknown space-time-dependent heat source is recovered very fast and accurately from the solution of three coupled boundary integral equations.

A fast multiple-scale polynomial solution for the inverse Cauchy problem of elasticity in an arbitrary plane domain

The polynomial expansion method together with the collocation technique is a cheap yet simple method to solve the Navier equations of elasticity, which is easily arranged to satisfy the governing equations and boundary conditions pointwise. In this paper we propose a novel numerical algorithm for the solution of an overspecified/underspecified Cauchy problem of linear elasticity in an arbitrary plane domain, by using the multiple-scale Pascal polynomial expansion method (MSPEM), of which the scales are determined a priori by the collocation points, according to the idea of equilibrated matrix. In the numerical tests of a direct problem as well as the overspecified/underspecified Cauchy problems, the MSPEM is very accurate and stable against large relative noise up to 20% for the unknown displacements recovery problem, and up to 100% absolute noise for the recovery of unknown loading force. The present method is convergent very fast for most cases within 100 iteration steps.

Homogenized functions to recover [formula omitted]/[formula omitted] by solving a small scale linear system of differencing equations

In order to recover the unknown heat source H(t)/H(x) in the heat conduction equation, we introduce homogenized functions and differencing equations, which can significantly simplify the solution procedures for the inverse heat source recovery problem. We only need to solve a few linear equations in the problem domain, since the initial condition/boundary conditions and a supplementary condition are satisfied automatically, and the differencing technique is employed to eliminate the heat source terms in the linear equations. The Pascal polynomials and eigenfunctions are adopted as trial functions to expand the trial solutions, and then some small scale linear systems are solved to determine the expansion coefficients. Because the ill-posedness of the inverse heat source recovery problem is greatly reduced, the present methods are accurate and stable against large noise up to 20%, which are confirmed by the numerical tests.

A multiple-scale Pascal polynomial for 2D Stokes and inverse Cauchy–Stokes problems

Abstract The polynomial expansion method is a useful tool for solving both the direct and inverse Stokes problems, which together with the pointwise collocation technique is easy to derive the algebraic equations for satisfying the Stokes differential equations and the specified boundary conditions. In this paper we propose two novel numerical algorithms, based on a third–first order system and a third–third order system, to solve the direct and the inverse Cauchy problems in Stokes flows by developing a multiple-scale Pascal polynomial method, of which the scales are determined a priori by the collocation points. To assess the performance through numerical experiments, we find that the multiple-scale Pascal polynomial expansion method (MSPEM) is accurate and stable against large noise.

A multiple-scale Pascal polynomial triangle solving elliptic equations and inverse Cauchy problems

The polynomial expansion method is a useful tool to solve partial differential equations (PDEs). However, the researchers seldom use it as a major medium to solve PDEs due to its highly ill-conditioned behavior. We propose a single-scale and a multiple-scale Pascal triangle formulations to solve the linear elliptic PDEs in a simply connected domain equipped with complex boundary shape. For the former method a constant parameter R0 is required, while in the latter one all introduced scales are automatically determined by the collocation points. Then we use the multiple-scale method to solve the inverse Cauchy problems, which is very accurate and very stable against large noise to 20%. Numerical results confirm the validity of the present multiple-scale Pascal polynomial expansion method.

Pascal-Interpolation-Based Noninteger Delay Filter and Low-Complexity Realization

This paper proposes a new method for designing the polynomial-interpolation-type noninteger-delay filter with a new structure formulation. Since the design formu- lation and the new realization structure are based on the discrete Pascal transform (DPT) and Pascal interpolation, we call the resulting filter Pascal noninteger-delay filter. The kth-order Pascal polynomial is used to pass through the given (k + 1) data points in achieving the kth-order Pascal filter. The Pascal noninteger-delay filter is a real- time filter that consists of two sections, which can be real- ized into the front-section and the back-section. The front- section contains multiplication-free digital filters, and the number of multiplications in the back-section just linearly increases as order becomes high. Since the new Pascal filter has low complexity and structure can adjust non- integer delay online, it is more suited for fast delay tuning. Consequently, the polynomial-interpolation-type delay filter can be achieved by using the Pascal approach with high efficiency and low-complexity structure.

The modified Pascal polynomial approximation and filter design method

SUMMARYPolynomial approximations are extensively used in analog and IIR digital filter design. In this paper, a comprehensive filter design and an optimization procedure are presented explicitly using a filter‐appropriate modified Pascal polynomial. The so‐designed all‐pole Pascal filters exhibit non‐equiripple passband and monotonic transition and stopband responses. The order of the new Pascal filters is calculated from the order inequality which, although it cannot be analytically solved, leads to a nomograph that has been created and is presented here. Inevitably, the mathematical complexity introduced by the nature of the Pascal polynomials makes the analytical expression of the poles of the transfer function unfeasible and for that reason poles are given by means of appropriate tables. The design method is demonstrated in several detailed examples and Pascal filters are compared with their all‐pole counterparts, Butterworth and Chebyshev, over which they reveal certain advantages and disadvantages. Copyright © 2010 John Wiley & Sons, Ltd.