An iteration method for predicting static response of nonlinear structural systems with non-deterministic parameters

Confident numerical method is a crucial issue in the field of structural health monitoring. This paper focuses on uncertainty propagation in nonlinear structural systems with non-deterministic parameters. An interval-based iteration method is proposed on the basis of interval analysis and Taylor series expansion. The proposed method aims to improve the bounds of static response calculated by the point-based iteration method. In the proposed method, the iterative interval of static response is updated by revising the lower and upper bounds, respectively, which is the essential difference in comparison with the previous point-based iteration method. In this paper, interval parameters are employed to quantify the non-deterministic parameters instead of random parameters in the case of insufficient sample data. Iterative scheme is established based on the first-order Taylor series expansion. For the implementation of interval-based iteration method, a general procedure is formulated. Moreover, the important source of the limitation of point-based iteration method is revealed profoundly, and the good performance of the proposed method is demonstrated by three numerical comparisons.     

Optimizing radial basis functions by d.c. programming and its use in direct search for global derivative-free optimization

In this paper, we address the global optimization of functions subject to bound and linear constraints without using derivatives of the objective function. We investigate the use of derivative-free models based on radial basis functions (RBFs) in the search step of direct-search methods of directional type. We also study the application of algorithms based on difference of convex (d.c.) functions programming to solve the resulting subproblems which consist of the minimization of the RBF models subject to simple bounds on the variables. Extensive numerical results are reported with a test set of bound and linearly constrained problems.     

An interval uncertainty analysis method for structural response bounds using feedforward neural network differentiation

This paper proposes a new interval uncertainty analysis method for structural response bounds with uncertain‑but-bounded parameters by using feedforward neural network (FNN) differentiation. The information of partial derivative may be unavailable analytically for some complicated engineering problems. To overcome this drawback, the FNNs of real structural responses with respect to structure parameters are first constructed in this work. The first-order and second-order partial derivative formulas of FNN are derived via the backward chain rule of partial differentiation, thus the partial derivatives could be determined directly. Especially, the influences of structures of multilayer FNNs on the accuracy of the first-order and second-order partial derivatives are analyzed. A numerical example shows that an FNN with the appropriate structure parameters is capable of approximating the first-order and second-order partial derivatives of an arbitrary function. Based on the parameter perturbation method using these partial derivatives, the extrema of the FNN can be approximated without requiring much computational time. Moreover, the subinterval method is introduced to obtain more accurate and reliable results of structural response with relatively large interval uncertain parameters. Three specific examples, a cantilever tube, a Belleville spring, and a rigid-flexible coupling dynamic model, are employed to show the effectiveness and feasibility of the proposed interval uncertainty analysis method compared with other methods.     

Adaptive alternating Lipschitz search method for structural analysis with unknown-but-bounded uncertainties

Multisource uncertainties, including property dispersibility of materials and fluctuating service environments, complicate structural design and reliability assessment. In this paper, a novel method named the adaptive alternating Lipschitz search method for structural analysis with unknown-but-bounded uncertainties (or interval uncertainties) is proposed. In contrast to traditional optimization methods that search twice to obtain response bounds, an adaptive alternate iteration strategy is proposed. By sampling step by step, two acquisition functions—named the Lipschitz upper bound and the Lipschitz lower bound—are defined. Structural response bounds can be simultaneously obtained by alternately optimizing the two acquisition functions. The parameter settings do not require adjustments for different types of problems. Additionally, the Bayesian Adaptive Direct Search method is adopted to improve the performance of the strategy. Numerical and experimental cases are presented to demonstrate the validity, accuracy, and efficiency of the proposed methodology. Detailed comparisons indicate that the proposed method is competitive when addressing complicated structural systems with different ranges of uncertainty.     

Correlation propagation for uncertainty analysis of structures based on a non-probabilistic ellipsoidal model

Traditional non-probabilistic methods for uncertainty propagation problems evaluate only the lower and upper bounds of structural responses, lacking any analysis of the correlations among the structural multi-responses. In this paper, a new non-probabilistic correlation propagation method is proposed to effectively evaluate the intervals and non-probabilistic correlation matrix of the structural responses. The uncertainty propagation process with correlated parameters is first decomposed into an interval propagation problem and a correlation propagation problem. The ellipsoidal model is then utilized to describe the uncertainty domain of the correlated parameters. For the interval propagation problem, a subinterval decomposition analysis method is developed based on the ellipsoidal model to efficiently evaluate the intervals of responses with a low computational cost. More importantly, the non-probabilistic correlation propagation equations are newly derived for theoretically predicting the correlations among the uncertain responses. Finally, the multi-dimensional ellipsoidal model is adopted again to represent both uncertainties and correlations of multi-responses. Three examples are presented to examine the accuracy and effectiveness of the proposed method both numerically and experimentally.     

An enhanced subinterval analysis method for uncertain structural problems

This paper proposes an enhanced subinterval analysis method to predict the bounds of structural response with interval parameters, which could deal with problems with relatively large uncertainties of the parameters. The intervals are first divided into several subintervals, and two expansion routes are then constructed based on the sensitivity analysis. Two subinterval sets are selected according to the expansion points on the routes, and the first order Taylor expansion method is then adopted to complete the subinterval analysis. Based on the selected subinterval sets, the upper and lower bounds of the structural response are further obtained by employing the interval union operation. An adaptive convergence approach is presented to determine the appropriate number of subintervals. Four numerical examples are investigated to demonstrate the validity of the proposed method.     

Interval static analysis of multi-cracked beams with uncertain size and position of cracks

This paper deals with beams under static loads, in presence of multiple cracks with uncertain parameters. The crack is modelled as a linearly-elastic rotational spring and, following a non-probabilistic approach, both stiffness and position of the spring are taken as uncertain-but-bounded parameters.A novel approach is proposed to compute the bounds of the response. The key idea is a preliminary monotonicity test, which evaluates sensitivity functions of the beam response with respect to the separate variation of every uncertain parameter within the pertinent interval. Next, two alternative procedures calculate lower and upper bounds of the response. If the response is monotonic with respect to all the uncertain parameters, the bounds are calculated by a straightforward sensitivity-based method making use of the sensitivity functions built in the monotonicity test. In contrast, if the response is not monotonic with respect to even one parameter only, the bounds are evaluated via a global optimization technique. The presented approach applies for every response function and the implementation takes advantage of closed analytical forms for all response variables and related sensitivity functions.Numerical results prove efficiency and robustness of the approach, which provides very accurate bounds even for large uncertainties, avoiding the computational effort required by the vertex method and Monte Carlo simulation.     

Meshless method of lines for the numerical solution of generalized Kuramoto-Sivashinsky equation

In this paper, the numerical solution of the generalized Kuramoto-Sivashinsky equation is presented by meshless method of lines (MOL). In this method the spatial derivatives are approximated by radial basis functions (RBFs) giving an edge over finite difference method (FDM) and finite element method (FEM) because no mesh is required for discretization of the problem domain. Only a set of scattered nodes is required to approximate the solution. The numerical results in comparison with exact solution using different radial basis functions (RBFs) prove the efficiency and accuracy of the method.     

Robust topology optimization for multi-material structures under interval uncertainty

In this paper, we propose an efficient method to design robust multi-material structures under interval loading uncertainty. The objective of this study is to minimize the structural compliance of linear elastic structures. First, the loading uncertainty can be decomposed into two unit forces in the horizontal and vertical directions based on the orthogonal decomposition, which separates the uncertainty into the calculation coefficients of structural compliance that are not related to the finite element analysis. In this manner, the time-consuming procedure, namely, the nested double-loop optimization, can be avoided. Second, the uncertainty problem can be transformed into an augmented deterministic problem by means of uniform sampling, which exploits the coefficients related to interval variables. Finally, an efficient sensitivity analysis method is explicitly developed. Thus, the robust topology optimization (RTO) problem considering interval uncertainty can be solved by combining orthogonal decomposition with uniform sampling (ODUS). In order to eliminate the influence of numerical units when comparing the optimal results to deterministic and RTO solutions, the relative uncertainty related to interval objective function is employed to characterize the structural robustness. Several multi-material structure optimization cases are provided to demonstrate the feasibility and efficiency of the proposed method, where the magnitude uncertainty, directional uncertainty, and combined uncertainty are investigated.     

Computation of bounds for eigenvalues of structures with interval parameters

In this paper, the computation of eigenvalue bounds for generalized interval eigenvalue problem is considered. Two algorithms based on the properties of continuous functions are developed for evaluating upper and lower eigenvalue bounds of structures with interval parameters. The method can provide the tightest bounds within a given precision. Numerical examples illustrate the effectiveness of the proposed method.     

Lower bounds of error estimates for singular perturbation problems with Jacobi-Spectral approximations

With weighted orthogonal Jacobi polynomials, we study spectral approximations for singular perturbation problems on an interval. The singular parameters of the model are included in the basis functions, and then its stiff matrix is diagonal. Considering the estimations for weighted orthogonal coefficients, a special technique is proposed to investigate the a posteriori error estimates. In view of the difficulty of a posteriori error estimates for spectral approximations, we employ a truncation projection to study lower bounds for the models. Specially, we present the lower bounds of a posteriori error estimates with two different weighted norms in details.     

A modified method of approximate particular solutions for solving linear and nonlinear PDEs

The method of approximate particular solutions (MAPS) was first proposed by Chen et al. in Chen, Fan, and Wen, Numer Methods Partial Differential Equations, 28 (2012), 506–522. using multiquadric (MQ) and inverse multiquadric radial basis functions (RBFs). Since then, the closed form particular solutions for many commonly used RBFs and differential operators have been derived. As a result, MAPS was extended to Matérn and Gaussian RBFs. Polyharmonic splines (PS) has rarely been used in MAPS due to its conditional positive definiteness and low accuracy. One advantage of PS is that there is no shape parameter to be taken care of. In this article, MAPS is modified so PS can be used more effectively. In the original MAPS, integrated RBFs, so called particular solutions, are used. An additional integrated polynomial basis is added when PS is used. In the modified MAPS, an additional polynomial basis is directly added to the integrated RBFs without integration. The results from the modified MAPS with PS can be improved by increasing the order of PS to a certain degree or by increasing the number of collocation points. A polynomial of degree 15 or less appeared to be working well in most of our examples. Other RBFs such as MQ can be utilized in the modified MAPS as well. The performance of the proposed method is tested on a number of examples including linear and nonlinear problems in 2D and 3D. We demonstrate that the modified MAPS with PS is, in general, more accurate than other RBFs for solving general elliptic equations.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1839–1858, 2017     

Sobolev bounds on functions with scattered zeros, with applications to radial basis function surface fitting

In this paper we discuss Sobolev bounds on functions that vanish at scattered points in a bounded, Lipschitz domain that satisfies a uniform interior cone condition. The Sobolev spaces involved may have fractional as well as integer order. We then apply these results to obtain estimates for continuous and discrete least squares surface fits via radial basis functions (RBFs). These estimates include situations in which the target function does not belong to the native space of the RBF.

     

Exact bounds for the sensitivity analysis of structures with uncertain-but-bounded parameters

Based on interval mathematical theory, the interval analysis method for the sensitivity analysis of the structure is advanced in this paper. The interval analysis method deals with the upper and lower bounds on eigenvalues of structures with uncertain-but-bounded (or interval) parameters. The stiffness matrix and the mass matrix of the structure, whose elements have the initial errors, are unknown except for the fact that they belong to given bounded matrix sets. The set of possible matrices can be described by the interval matrix. In terms of structural parameters, the stiffness matrix and the mass matrix take the non-negative decomposition. By means of interval extension, the generalized interval eigenvalue problem of structures with uncertain-but-bounded parameters can be divided into two generalized eigenvalue problems of a pair of real symmetric matrix pair by the real analysis method. Unlike normal sensitivity analysis method, the interval analysis method obtains informations on the response of structures with structural parameters (or design variables) changing and without any partial differential operation. Low computational effort and wide application rang are the characteristic of the proposed method. Two illustrative numerical examples illustrate the efficiency of the interval analysis.     

A meshless method for numerical solution of the one-dimensional wave equation with an integral condition using radial basis functions

The hyperbolic partial differential equation with an integral condition arises in many physical phenomena. In this paper, we propose a numerical scheme to solve the one-dimensional hyperbolic equation that combines classical and integral boundary conditions using collocation points and approximating the solution using radial basis functions (RBFs). The results of numerical experiments are presented, and are compared with analytical solution and finite difference method to confirm the validity and applicability of the presented scheme.     

Uncertainty analysis for structures with hybrid random and interval parameters using mathematical programming approach

A novel computational method, namely the unified perturbation mathematical programming (UPMP) approach, for hybrid uncertainty analysis of engineering structures is proposed in this paper. The presented study considers a mixture of random and interval system parameters which are frequently encountered in engineering applications. Within the UPMP approach, matrix perturbation theory is adopted in combination with the mathematical programming approach. The proposed computational method provides a non-simulative hybrid uncertainty analysis framework, which is competent to offer the extreme bounds of the statistical characteristics (i.e., mean and variance) of any concerned structural responses in computationally tractable fashion. In order to thoroughly explore various intricate aspects of the engineering system involving hybrid uncertainties, systematic numerical experiments have also been conducted. Diverse statistical analyses are implemented to identify the bounded probability profile of the uncertain structural responses. Both academic and practical engineering structures are investigated to justify the applicability, accuracy and efficiency of the proposed UPMP approach.     

Unsteady seepage analysis using local radial basis function-based differential quadrature method

Numerical simulation of two-dimensional transient seepage is developed using radial basis function-based differential quadrature method (RBF-DQ). To the best of the authors’ knowledge, this is the first application of this method to seepage analysis. For the general case of irregular geometry and unstructured node distribution, the local form of RBF-DQ was used. The multiquadric type of radial basis functions was selected for the computations, and the results were compared with analytical, finite element method, and existing numerical solutions from the literature. Results of this study show that localized RBF-DQ can produce accurate results for the analysis of seepage. The method is meshfree and easy to program, but as with previous applications of RBFs, requires careful selection of suitable shape parameters. A practical method for estimating suitable shape parameters is discussed. For time integration, Crank–Nicolson, Galerkin and finite difference methods were applied, leading to stable results.     

A computational modeling of the behavior of the two-dimensional reaction–diffusion Brusselator system

This paper studies a meshfree technique for the numerical solution of the two-dimensional reaction–diffusion Brusselator system along with Dirichlet and Neumann boundary conditions. Combination of collocation method using the radial basis functions (RBFs) with first order accurate forward difference approximation is employed for obtaining meshfree solution of the problem. Different types of RBFs are used for this purpose. The method is shown to converge to the only equilibrium point of the system. Performance of the proposed method is successfully tested in terms of various error norms. In the case of non-availability of exact solution, performance of the new method is compared with the results obtained from the existing methods [7] and [8]. The elementary stability analysis is established theoretically and is also supported by numerical results.     

Global sensitivity analysis using support vector regression

Global sensitivity analysis (GSA) plays an important role in exploring the respective effects of input variables on response variables. In this paper, a new kernel function derived from orthogonal polynomials is proposed for support vector regression (SVR). Based on this new kernel function, the Sobol’ global sensitivity indices can be computed analytically by the coefficients of the surrogate model built by SVR. In order to improve the performance of the SVR model, a kernel function iteration scheme is introduced further. Due to the excellent generalization performance and structural risk minimization principle, the SVR possesses the advantages of solving non-linear prediction problems with small samples. Thus, the proposed method is capable of computing the Sobol’ indices with a relatively limited number of model evaluations. The proposed method is examined by several examples, and the sensitivity analysis results are compared with the sparse polynomial chaos expansion (PCE), high dimensional model representation (HDMR) and Gaussian radial basis (RBF) SVR model. The examined examples show that the proposed method is an efficient approach for GSA of complex models.