Perturbation and stability analysis of strong form collocation with reproducing kernel approximation

Solving partial differential equations using strong form collocation with nonlocal approximation functions such as orthogonal polynomials and radial basis functions offers an exponential convergence, but with the cost of a dense and ill‐conditioned linear system. In this work, the local approximation functions based on reproducing kernel approximation are introduced for strong form collocation method, called the reproducing kernel collocation method (RKCM). We perform the perturbation and stability analysis of RKCM, and estimate the condition numbers of the discrete equation. Our stability analyses, validated with numerical tests, show that this approach yields a well‐conditioned and stable linear system similar to that in the finite element method. We also introduce an effective condition number where the properties of both matrix and right‐hand side vector of a linear system are taken into consideration in the measure of conditioning. We first derive the effective condition number of the linear systems resulting from RKCM, and show that using the effective condition number offers a tighter estimation of stability of a linear system. The mathematical analysis also suggests that the effective condition number of RKPM does not grow with model refinement. The numerical results are also presented to validate the mathematical analysis. Copyright © 2011 John Wiley & Sons, Ltd.     

A Comparison of Stability and Bifurcation Criteria for a Compressible Elastic Cube

A version of Rivlin’s cube problem is considered for compressible materials. The cube is stretched along one axis by a fixed amount and then subjected to equal tensile loads along the other two axes. A number of general results are found. Because of the homogeneous trivial and non-trivial deformations exact bifurcation results can be found and an exact stability analysis through the second variation of the energy can be performed. This problem is then used to compare results obtained using more general methods. Firstly, results are obtained for a more general bifurcation analysis. Secondly, the exact stability results are compared with stability results obtained via a new method that is applicable to inhomogeneous problems. This new stability method allows a full nonlinear stability analysis of inhomogeneous deformations of arbitrary, compressible or incompressible, hyperelastic materials. The second variation condition expressed as an integral involving two arbitrary perturbations is replaced with an equivalent nonlinear third order system of ordinary differential equations. The positive definiteness condition is thereby reduced to the simple numerical evaluation of zeros of a well behaved function.     

Numerical solution of the Laplacian Cauchy problem by using a better postconditioning collocation Trefftz method

In this paper, the inverse Cauchy problem for Laplace equation defined in an arbitrary plane domain is investigated by using the collocation Trefftz method (CTM) with a better postconditioner. We first introduce a multiple-scale R_k in the T-complete functions as a set of bases to expand the trial solution. Then, the better values of R_k are sought by using the concept of an equilibrated matrix, such that the resulting coefficient matrix of a linear system to solve the expansion coefficients is best-conditioned from a view of postconditioner. As a result, the multiple-scale R_k can be determined exactly in a closed-form in terms of the collocated points used in the collocation to satisfy the boundary conditions. We test the present method for both the direct Dirichlet problem and the inverse Cauchy problem. A significant reduction of the condition number and the effective condition number can be achieved when the present CTM is used, which has a good efficiency and stability against the disturbance from large random noise, and the computational cost is much saving. Some serious cases of the inverse Cauchy problems further reveal that the unknown data can be recovered very well, although the overspecified data are provided only at a 20% of the overall boundary.     

Effective Condition Number for Boundary Knot Method

This study makes the first attempt to apply the effective condition number (ECN) to the stability analysis of the boundary knot method (BKM). We find that the ECN is a superior criterion over the traditional condition number. The main difference between ECN and the traditional condition numbers is in that the ECN takes into account the right hand side vector to estimates system stability. Numerical results show that the ECN is roughly inversely proportional to the numerical accuracy. Meanwhile, using the effective condition number as an indicator, one can fine-tune the user-defined parameters (without the knowledge of exact solution) to ensure high numerical accuracy from the BKM.     

Applicability of the method of fundamental solutions

The condition number of a matrix is commonly used for investigating the stability of solutions to linear algebraic systems. Recent meshless techniques for solving partial differential equations have been known to give rise to ill-conditioned matrices, yet are still able to produce results that are close to machine accuracy. In this work, we consider the method of fundamental solutions (MFS), which is known to solve, with extremely high accuracy, certain partial differential equations, namely those for which a fundamental solution is known. To investigate the applicability of the MFS, either when the boundary is not analytic or when the boundary data are not harmonic, we examine the relationship between its accuracy and the effective condition number.Three numerical examples are presented in which various boundary value problems for the Laplace equation are solved. We show that the effective condition number, which estimates system stability with the right-hand side vector taken into account, is roughly inversely proportional to the maximum error in the numerical approximation. Unlike the proven theories in literature, we focus on cases when the boundary and the data are not analytic. The effective condition number numerically provides an estimate of the quality of the MFS solution without any knowledge of the exact solution and allows the user to decide whether the MFS is, in fact, an appropriate method for a given problem, or what is the appropriate formulation of the given problem.     

Mean-square stability of linear stochastic dynamical systems under parametric wide-band excitations

A new simplified condition is developed for determining the exponential meansquare stability margins of linear stochastic dynamical systems. It is well-known that under parametric wide-band noise disturbances, the governing equations of motion of such a system can be approximated by linear Itô stochastic differential equations (SDE). A necessary and sufficient condition for exponential mean square stability of the resulting ltô SDE is that the real parts of all the eigenvalues of the matrix describing the system of second-order moments are negative. Equivalently, the Routh-Hurwitz procedure provides conditions for stability in the form of several inequalities. In this study, it is shown that a necessary condition for the system configuration to correspond to a point on the stability boundary is that the determinant of the matrix describing the system of secondorder moments be zero. This condition is a single algebraic expression allowing for the straightforward calculation of all candidate stability boundaries. In addition, the topological properties of the stability domain are presented and shown to be useful in identifying stability boundaries and stability domains from the developed single stability boundary condition. This simplified condition provides significant advantages in the analytical and numerical estimation of the stability border and stability region of dynamical systems. The usefulness and superiority of the new condition is demonstrated by applications to example dynamical systems, including a long-span bridge model subjected to turbulent wind.     

Exponential stability and stabilization of uncertain T–S fuzzy system with time-varying delay

In this article, the problem of exponential stability and stabilization is investigated for Takagi–Sugeno fuzzy time-varying delay systems with parameter uncertainties. Improved conditions are presented in terms of linear matrix inequalities (LMIs). The desired fuzzy controller is obtained by solving these LMIs. In addition, the new integral inequalities can derive less conservative stability criteria. In the end, some numerical simulations are given to illustrate the validity of the proposed methodology.     

Hybrid and collocation Trefftz methods for traction boundary conditions in linear elastostatics

For linear elastostatics, the particular solutions and the fundamental solutions can be used for the admissible functions, so that only the boundary conditions need to be satisfied, thus called the boundary methods (or Trefftz methods). Hence, how to couple the boundary conditions is a crucial issue. There are basically two types of boundary conditions: (I) the displacement (i.e., Dirichlet) condition and (II) the traction (i.e., Neumann) condition. In this paper, the coupling techniques for the traction (i.e., Neumann condition) are the main theme for boundary methods, because the traction condition may be regarded as the basic condition and the displacement condition as the natural condition. The Lagrange multiplier used for the displacement (i.e., Dirichlet) condition is well known in mathematics community (see Babu?ka, 1973 [1]; Babu?ka et al., 1978 [2]; Li, 1998 [20]; Pitkäranta, 1979 [40]), but the Lagrange multiplier used for the traction (i.e., Neumann) condition is popular for elasticity problems by the Trefftz method in engineering community, which is called the hybrid Trefftz method (HTM) (see de Freitas, 1998 [7]; de Freitas and Ji, 1996 [8], [9]; Jirousek, 1978 [12]; Jirousek and Venkstesh, 1992 [13]; Jirousek and Wroblewski, 1996 [14]; Qin, 2000 [42]). On the other hand, the collocation Trefftz method (CTM) can also be used to directly couple the traction condition without extra-multipliers. In this paper, a brief error analysis for the HTM is given, to provide the optimal convergence rates. Numerical experiments of simple models are carried out to support this analysis. The HTM has the merits: flexibility and robustness, so that it has been widely used in engineering problems. Since the optimal convergence rates are the most important criterion in evaluation of numerical methods, the global performance of the HTM is as good as that of the CTM, although the HTM causes larger condition numbers and requires more CTU time for the computation of the simple models in this paper. More numerical comparisons also show that using the particular solutions is more advantageous than using the fundamental solutions in both HTM and CTM.     

采用粘弹性人工边界单元时显式算法稳定性分析

在土-结构地震反应或近场地震波动问题的分析中,常采用粘弹性人工边界单元将无限域问题转化为近场有限域问题进行计算。由于粘弹性人工边界单元的材料参数和单元尺寸与内部介质单元不同,采用显式时域逐步积分算法时,人工边界区与内部系统的数值稳定条件存在差异,但目前尚未有针对性的分析方法和研究成果,影响了显式数值稳定条件的确定和稳定积分时间步长的正确选取。针对二维粘弹性人工边界单元,该文提出一种分析显式时域逐步积分算法稳定性的方法:建立可代表人工边界区域特征的,包含人工边界单元的若干局部子系统,对各子系统的传递矩阵进行分析,给出采用显式时域逐步积分算法时各子系统的稳定条件解析解。通过对各子系统的稳定条件进行对比分析,获得了采用粘弹性人工边界单元时,显式时域逐步积分算法的统一稳定性条件。当内部介质区也满足该稳定条件时,这一条件成为使整体系统数值计算稳定的充分条件,可用于指导数值分析中离散时间步长的选取。     

Determining the initial distribution in space-fractional diffusion by a negative exponential regularization method

In this paper, we consider the backward problem for diffusion equation with space fractional Laplacian, i.e. determining the initial distribution from the final value measurement data. In order to overcome the ill-posedness of the backward problem, we present a so-called negative exponential regularization method to deal with it. Based on the conditional stability estimate and an a posteriori regularization parameter choice rule, the convergence rate estimate are established under a-priori bound assumption for the exact solution. Finally, several numerical examples are proposed to show that the numerical methods are effective.     

Stabilization of a system modeling temperature and porosity fields in a Kelvin�CVoigt-type mixture

In this paper, we investigate the asymptotic behavior of solutions to the initial boundary value problem for the interaction between the temperature field and the porosity fields in a homogeneous and isotropic mixture from the linear theory of porous Kelvin?CVoigt materials. Our main result is to establish conditions which insure the analyticity and the exponential stability of the corresponding semigroup. We show that under certain conditions for the coefficients we obtain a lack of exponential stability. A numerical scheme is given.     

A novel ‘optimal’ exponential‐based integration algorithm for von‐Mises plasticity with linear hardening: Theoretical analysis on yield consistency,accuracy, convergence and numerical investigations

In this communication we propose a new exponential‐based integration algorithm for associative von‐Mises plasticity with linear isotropic and kinematic hardening, which follows the ones presented by the authors in previous papers. In the first part of the work we develop a theoretical analysis on the numerical properties of the developed exponential‐based schemes and, in particular, we address the yield consistency, exactness under proportional loading, accuracy and stability of the methods. In the second part of the contribution, we show a detailed numerical comparison between the new exponential‐based method and two classical radial return map methods, based on backward Euler and midpoint integration rules, respectively. The developed tests include pointwise stress–strain loading histories, iso‐error maps and global boundary value problems. The theoretical and numerical results reveal the optimal properties of the proposed scheme. Copyright © 2006 John Wiley & Sons, Ltd.     

Wave equation model for solving advection–diffusion equation

This paper presents a Wave Equation Model (WEM) to solve advection dominant Advection–Diffusion (A–D) equation. It is known that the operator-splitting approach is one of the effective methods to solve A–D equation. In the advection step the numerical solution of the advection equation is often troubled by numerical dispersion or numerical diffusion. Instead of directly solving the first-order advection equation, the present wave equation model solves a second-order equivalent wave equation whose solution is identical to that of the first-order advection equation. Numerical examples of 1-D and 2-D with constant flow velocities and varying flow velocities are presented. The truncation error and stability condition of 1-D wave equation model is given. The Fourier analysis of WEM is carried out. The numerical solutions are in good agreement with the exact solutions. The wave equation model introduces very little numerical oscillation. The numerical diffusion introduced by WEM is cancelled by inverse numerical diffusion introduced by WEM as well. It is found that the numerical solutions of WEM are not sensitive to Courant number under stability constraint. The computational cost is economical for practical applications.     

异步切换下切换时变时滞系统的保成本控制

切换系统是一种重要的混杂系统,由若干子系统及决定子系统之间切换的切换信号组成。在工程应用中,控制器切换与子系统的切换会存在时延,即异步切换。研究了异步切换下的时变时滞系统的保成本控制问题,利用分段李雅普诺夫函数法和平均驻留时间法,得到保成本控制器存在的充分条件,并从线性矩阵不等式的角度,设计了一个异步切换保成本控制器,使得系统具有鲁棒性能。最后,给出了一个数值例子验证提出的方法的有效性。     

一类具比例时滞细胞神经网络反周期解的指数稳定性

本文针对一类具比例时滞细胞神经网络反周期解的全局指数稳定性进行研究.首先利用非线性变换将一类具比例时滞的细胞神经网络等价变换成一类具常时滞变系数的细胞神经网络.然后通过构造合适的时滞微分不等式和利用不等式技巧,得到了保证该系统反周期解的存在性和全局指数稳定性时滞依赖的充分条件.最后数值算例结果验证所得结论的正确性和与以往文献相比较低的保守性.     

DSC‐Ritz method for high‐mode frequency analysis of thick shallow shells

This paper addresses a challenging problem in computational mechanics—the analysis of thick shallow shells vibrating at high modes. Existing methods encounter significant difficulties for such a problem due to numerical instability. A new numerical approach, DSC‐Ritz method, is developed by taking the advantages of both the discrete singular convolution (DSC) wavelet kernels of the Dirichlet type and the Ritz method for the numerical solution of thick shells with all possible combinations of commonly occurred boundary conditions. As wavelets are localized in both frequency and co‐ordinate domains, they give rise to numerical schemes with optimal accurate, stability and flexibility. Numerical examples are considered for Mindlin plates and shells with various edge supports. Benchmark solutions are obtained and analyzed in detail. Experimental results validate the convergence, stability, accuracy and reliability of the proposed approach. In particular, with a reasonable number of grid points, the new DSC‐Ritz method is capable of producing highly accurate numerical results for high‐mode vibration frequencies, which are hitherto unavailable to engineers. Moreover, the capability of predicting high modes endows us the privilege to reveal a discrepancy between natural higher‐order vibration modes of a Mindlin plate and those calculated via an analytical relationship linking Kirchhoff and Mindlin plates. Copyright © 2004 John Wiley & Sons, Ltd.     

大型数控车床进给伺服系统的建模与分析

针对某大型数控车床的横向进给伺服系统,建立了考虑摩擦和传动刚度的综合力学模型和数学模型。通过数值仿真分析了低速进给下静动摩擦力差值和传动刚度变化对工作台输出的影响,得出了工作台产生爬行运动的可能性条件。该型数控车床横向进给系统现场实验的结果验证了理论分析的合理性。所得结论为数控机床进给伺服系统的稳定性和加工精度的提高提供了研究基础。     

波动谱元模拟中透射边界稳定性分析

无限域波动数值模拟中,人工边界的稳定性是获得可靠模拟结果的前提。具有高阶精度的谱元法和透射边界两者结合的数值模拟方案显示出较好的模拟精度和数值稳定性,然而,仍然存在数值失稳现象,其失稳机理和稳定条件尚不明确,相应的理论分析极为欠缺。该文针对透射边界在高阶谱元法中的稳定性,依据高阶谱单元中非等间距节点的周期延拓特点,通过构建内域和边界数值格式的向量形式来分析人工边界反射系数。进而保证边界对谱元法中存在的真实模态和虚假模态的反射系数均小于等于1,从而得到透射边界的稳定条件,其表现为无量纲边界参数和谱元参数之间的关系,其含义为透射边界人工波速与介质物理波速的比值限定在一定范围内。同时揭示了透射边界引发高频失稳的机理,即边界对谱元法中虚假模态的反复反射放大所致。最后采用数值实验验证了透射边界稳定条件。     

An efficient and unconditionally stable numerical algorithm for nonlinear structural dynamics

This article proposes an algorithm for express solutions in nonlinear structural dynamics. Our strategy is to adopt a typical time integrator and accept the solution after a constant number of iterations using a constant Jacobian matrix. Its success may not be initially obvious, but we demonstrate that the proposed algorithm not only is fully operational but also inherits the advantages of the host time integrators such as the unconditional stability, the order of accuracy, and the numerical dissipation that helps suppress the spurious higher mode oscillation. The use of a constant Jacobian matrix plays the key role in minimizing the computational expense associated with matrix operations. We first study the optimization of the number of iterations, then present the consistency and stability analysis followed by some examples verifying these features, and conclude by showing the exponential efficiency improvement in a response history analysis of a high-rise building fully equipped with nonlinearities.     

Convex multilevel decomposition algorithms for non-monotone problems

A convex, multilevel decomposition algorithm is proposed in this paper for the solution of static analysis problems involving non-monotone, possibly multivalued laws. The theory is developed here for a model structure with non-monotone interface or boundary conditions. First the non-monotone laws are written in the form of a difference of two monotone functions. Under this decomposition, the non-linear elastostatic analysis problem is equivalent to a system of convex variational inequalities and to non-convex min-min problems for appropriately defined Lagrangian functions. The solution(s) of each one of the aforementioned problems describe the position(s) of static equilibrium of the considered structure. In this paper a multilevel optimization scheme, due to Auchmuty,¹ is used for the numerical solution of the problem. The most interesting feature of this method, from the computational mechanics' standpoint, is the fact that each one of the subproblems involved in the multilevel algorithm is a convex optimization problem, or, in terms of mechanics, an appropriately modified monotone ‘unilateral’ problem. Thus, existing algorithms and software can be used for the numerical solution with minor modifications. Numerical results concerning the calculation of elastic and rigid stamp problems and of material inclusion problems with delamination and non-monotone stick-slip frictional effects illustrate the theory.