Selection Of Collocation Points Research Articles

Stochastic analysis for in-plane dynamic responses of low-speed uniformity of tires due to geometric defects

Tire uniformity is defined as the dynamic mechanical characteristics of pneumatic tires due to the defects originating in tire components during manufacturing processes. These defects lead to deviations in the structural and material parameters of tires and, in turn, induce variations in the dynamic response. In this scenario, this paper proposes a probabilistic model of the low-speed uniformity analysis for predicting the impacts of the uncertainties in the geometric defects and the vertical load on the dynamic responses. The deterministic coupled rigid–flexible ring model, combined with the generalized polynomial chaos expansion theory, is developed to simulate the radial force variation due to radial run-out. In this approach, the non-uniform tire radius and the vertical load are selected as a set of random variables and are approximated by the generalized polynomial chaos expansions. The distributions of these input variables are identified from 200 measured tires by the Pearson model. A non-intrusive probabilistic collocation method is adopted to evaluate the coefficients of the generalized polynomial chaos expansions, and the selection of collocation points is also illustrated. Finally, the distributions of the radial force variation and its first harmonic due to the geometric defects and vertical loads are predicted in comparison with measurements. Moreover, the distributions of the dynamic response caused by each parameter and different variance levels are discussed and verified by the measurement data from 1130 tires.

Physics-informed deep neural network enabled discovery of size-dependent deformation mechanisms in nanostructures

The surface effects based on the Gurtin-Murdoch interface model are incorporated into a physics-informed deep neural network (DNN) to enable the discovery of the size-dependent mechanical response in nanoscale structures for the first time. The DNN directly deals with the minimization of a cost function with contributions from total potential energies and penalties representing the displacement boundary conditions. Following this spirit, the Gurtin-Murdoch interface model is directly implemented into the DNN cost function through additional surface energies associated with the surfaces of the nanostructures without resorting to the implementation of Young-Laplace equations explicitly employed in the analytical approaches. The DNN technique is first validated on both displacement and stress fields vis-à-vis the analytical solution for the modified Kirsch problem with the Gurtin-Murdoch interface under far-field shear loading. Then, the DNN technique is critically and vigorously assessed against the Q4, Q8, and Q9 finite-element results of nanocylinders under diametrical loading. Notably, the accuracy of the DNN is demonstrated to be comparable to the analytical, Q8 and Q9 finite element-based results without the apparent stress discontinuities noticeable in the Q4 element. The numerical example indicates that the selection of collocation points in the DNN approach can be made completely random using the Monte-Carlo simulation without sacrificing the accuracy of the predicted stress field. An inverse load identification problem is also considered to demonstrate the special strength of the DNN approach, namely, both the forward and inverse problems can be handled using the same framework. The present contribution provides a compelling alternative and independent means of identifying surface elasticity effect and hence is a good tool in assessing the finite-element method as well as other approaches’ predictive capability for this class of materials.

Bending analysis of magnetoelectroelastic nanoplates resting on Pasternak elastic foundation based on nonlocal theory

Based on the nonlocal theory and Mindlin plate theory, the governing equations (i.e., a system of partial differential equations (PDEs) for bending problem) of magnetoelectroelastic (MEE) nanoplates resting on the Pasternak elastic foundation are first derived by the variational principle. The polynomial particular solutions corresponding to the established model are then obtained and further employed as basis functions with the method of particular solutions (MPS) to solve the governing equations numerically. It is confirmed that for the present bending model, the new solution strategy possesses more general applicability and superior flexibility in the selection of collocation points. Finally, the effects of different boundary conditions, applied loads, and geometrical shapes on the bending properties of MEE nanoplates are evaluated by using the developed method. Some important conclusions are drawn, which should be helpful for the design and applications of electromagnetic nanoplate structures.

Solving a Poisson Equation with Singularities by the Least-Squares Collocation Method

New h-, p-, and hp-versions of the least-squares collocation method (LSCM) are proposed and implemented for solving the Dirichlet problem for a Poisson equation. Examples are considered of solving problems with singularities such as large gradients, fast growing rate of solution derivatives with increasing order of differentiation, discontinuities of second-order derivatives at angular points of the domain boundary, and solution oscillations with various frequencies for an infinite discontinuity point for derivatives of any order. The new versions of the method are based on a special selection of collocation points in the roots of Chebyshev polynomials of the first kind. The basis functions are defined as products of Chebyshev polynomials. The behavior of numerical solutions on a sequence of grids with an increasing degree of the approximating polynomial is analyzed by using exact analytical solutions. Formulas for the extension operation of transition from a coarse grid to a finer one on a multi-grid complex in the Fedorenko method are obtained.

New collocation method for stochastic response surface reliability analyses

The stochastic response surface method (SRSM) is widely used in engineering reliability analyses due to its efficiency and accuracy. The selection of collocation points in the SRSM has great significance, as it may strongly affect the computed results. This paper investigates the performance of different selection strategies in SRSM, and proposes a new collocation method. First, two commonly used collocation methods—the regression-based collocation method and the linearly independent collocation method—are briefly reviewed; and their limitations in application to reliability analysis are discussed. Then, an improved collocation method that achieves a better tradeoff between efficiency and accuracy is proposed. Four examples are employed to test the performance of the proposed collocation method; and a comparative study is conducted to demonstrate its advantages with respect to some other existing collocation methods.

The radial basis function-differential quadrature method for elliptic problems in annular domains

Abstract We employ a radial basis function (RBF) - differential quadrature (DQ) method for the numerical solution of elliptic boundary value problems in annular domains. With an appropriate selection of collocation points, for any choice of RBF, both the coefficient and right hand side matrices in the systems appearing in this discretization possess block circulant structures. These linear systems can thus be solved efficiently using matrix decomposition algorithms (MDAs) and fast Fourier transforms (FFTs). In particular, we consider problems governed by the Poisson equation, the inhomogeneous biharmonic equation and the inhomogeneous Cauchy–Navier equations of elasticity. In addition to its simplicity, the proposed method can both achieve high accuracy and solve large-scale problems. The feasibility of the proposed techniques is illustrated by several numerical examples.

The plane waves method for axisymmetric Helmholtz problems

The plane waves method is employed for the solution of Dirichlet and Neumann boundary value problems for the homogeneous Helmholtz equation in two- and three-dimensional domains possessing radial symmetry. The appropriate selection of collocation points and unitary direction vectors in the method leads to circulant and block circulant coefficient matrices in two and three dimensions, respectively. We propose efficient matrix decomposition algorithms which make use of fast Fourier transforms for the solution of the systems resulting from such a discretization. In conjunction with the method of particular solutions, the method is extended to the solution of inhomogeneous axisymmetric Helmholtz problems. Several numerical examples are presented.

A Kansa-Radial Basis Function Method for Elliptic Boundary Value Problems in Annular Domains

We employ a Kansa-radial basis function (RBF) method for the numerical solution of elliptic boundary value problems in annular domains. This discretization leads, with an appropriate selection of collocation points and for any choice of RBF, to linear systems in which the matrices possess block circulant structures. These linear systems can be solved efficiently using matrix decomposition algorithms and fast Fourier transforms. A suitable value for the shape parameter in the various RBFs used is found using the leave-one-out cross validation algorithm. In particular, we consider problems governed by the Poisson equation, the inhomogeneous biharmonic equation and the inhomogeneous Cauchy---Navier equations of elasticity. In addition to its simplicity, the proposed method can both achieve high accuracy and solve large-scale problems. The feasibility of the proposed techniques is illustrated by several numerical examples.

A fast and more universal algorithm for an uncertain acoustic filed in shallow-water

Non-intrusive polynomial chaos expansion (NPCE) method is a fast algorithm with the best performances for an uncertain acoustic filed currently, in which the selection of collocation points is an important factor for the computational accuracy, and some special processing methods such as piecewise probabilistic collocation method, should be adopted when the outputs of acoustic field vary severely with uncertain ocean environmental parameters. A new fast algorithm for uncertain acoustic filed in shallow-water is proposed based on Kriging model. The theoretical description of the new algorithm is given, and numerical simulations are conducted to verify the performances of the proposed algorithm. The physical interpretations are given in detail. The results demonstrate that the proposed algorithm is more accurate than the NPCE method under the same conditions, and any special processing method does not need to be adopted when the outputs of acoustic field vary severely with the uncertain ocean environmental parameters. The weakness of NPCE method can be overcome by the proposed algorithm, which is that the computational cost increases with the stochastic polynomial expanding to a higher order for enhancing the computational accuracy. The selection of sample point of the proposed algorithm is simpler and easier than that of NPCE method, and the calculation errors can be given directly. Thus, the proposed algorithm is more universal than NPCE method.

A Sparse Composite Collocation Finite Element Method for Elliptic SPDEs.

This work presents a stochastic collocation method for solving elliptic PDEs with random coefficients and forcing term which are assumed to depend on a finite number of random variables. The method consists of a hierarchic wavelet discretization in space and a sequence of hierarchic collocation operators in the probability domain to approximate the solution's statistics. The selection of collocation points is based on a Smolyak construction of zeros of orthogonal polynomials with respect to the probability density function of each random input variable. A sparse composition of levels of spatial refinements and stochastic collocation points is then proposed and analyzed, resulting in a substantial reduction of overall degrees of freedom. Like in the Monte Carlo approach, the algorithm results in solving a number of uncoupled, purely deterministic elliptic problems, which allows the integration of existing fast solvers for elliptic PDEs. Numerical examples on two-dimensional domains will then demonstrate the superiority of this sparse composite collocation finite element method compared to the “full composite” collocation finite element method and the Monte Carlo method.

On the numerical solution of some 2-D electromagnetic interface problems by the boundary collocation method

The boundary collocation method is used to solve some 2-D interface problems of microwave heating. The choice of subspace of particular solutions, and the selection of collocation points are discussed. Numerical examples are presented.

Reduced-order models for separation columns—V. Selection of collocation points

In the earlier parts of this paper, a method for developing reduced-order models for separation columns was presented. This article presents some guidelines for determining the degree of order reduction achievable for a given column. An index called ORP (Order Reduction Parameter) is developed to predict a priori, the extent of order reduction achievable and to provide guidelines for choosing a reduced-order model of the appropriate size. This, combined with a comparison with a steady state solution, can provide the engineer with a good estimate of the number and location of collocation points to be used in the reduced-order model. The method is developed using a simple binary absorber and illustrated by application to a more complex, multicomponent distillation column.