New series expansions for fundamental solutions of linear elastostatics in 2D

Series expansions of fundamental solutions are essential to algorithms and analysis of the null field method (NFM) and to analysis of the method of fundamental solutions (MFS). For linear elastostatics, new Fourier series expansions of FS are derived, directly from integration. The new expansions of the FS are simpler than those in Chen et?al. (J Mech 26(3):393?C401, 2010), thus facile to application in NFM and MFS. The new series expansions of FS in this paper are important to both theory and computation of linear elastostatics. Some computation of the MFS for linear elastostatics is provided, where the expansions of fundamental solutions are a basis tool in analysis. Numerical results of a simple example are reported, accompanied with error analysis.     

Knowledge-based method for the validation of complex simulation models

In traditional research, the validation of simulation models is mainly based on statistical analysis and simulation error evaluating. As simulation models become more and more complex, simulation behavior is more complicated and heavily dependent on simulation conditions, traditional method cannot be applied directly. On the other hand, measured data needed for traditional methods cannot always be available. What’s more, the validation of complex models is usually costly and time-consuming.This research makes effort to settle the problems above. It provides three aspects of advantage for model validation. First, complicated simulation behavior is abstracted and classified as five categories, among which behavior relationship and aggregative behavior is unique for complex simulation models. And the analysis and validation method of each kind of behavior is proposed. Second, besides measured data of real system, it proposed to utilize experience of expert and other kind of domain knowledge for validation task. And third, the simulation analysis and validation method proposed can be implemented in knowledge system and accomplish validation task automatically.Simulation output analysis is the most important step in model validation. Besides classical continuous dynamic fitting and statistical consistency analysis methods, we propose a domain knowledge-based method for the validation of relationship among behavior segments and aggregative behavior which is unique for complex simulation models. Domain knowledge takes the role of reference in model validation, and it varies from measured data of real system to qualitative experience of experts. Knowledge-based system is implemented based on domain knowledge and validation techniques proposed.Validation of electromagnetic rail gun simulation models are introduced as an example. It has been found that this method can provides an automatic validation way for complex simulation models, and validation tasks can be accomplished efficiently.     

Comparisons of fundamental solutions and particular solutions for Trefftz methods

In the Trefftz method (TM), the admissible functions satisfying the governing equation are chosen, then only the boundary conditions are dealt with. Both fundamental solutions (FS) and particular solutions (PS) satisfy the equation. The TM using FS leads to the method of fundamental solutions (MFS), and the TM using PS to the method of particular solutions (MPS). Since the MFS is one of TM, we may follow our recent book [20], [21] to provide the algorithms and analysis. Since the MFS and the MPS are meshless, they have attracted a great attention of researchers. In this paper numerical experiments are provided to support the error analysis of MFS in Li [15] for Laplace's equation in annular shaped domains. More importantly, comparisons are made in analysis and computation for MFS and MPS. From accuracy and stability, the MPS is superior to the MFS, the same conclusion as given in Schaback [24]. The uniform FS is simpler and the algorithms of MFS are easier to carry out, so that the computational efforts using MFS are much saved. Since today, the manpower saving is the most important criterion for choosing numerical methods, the MFS is also beneficial to engineering applications. Hence, both MFS and MPS may serve as modern numerical methods for PDE.     

Stability analysis via condition number and effective condition number for the first kind boundary integral equations by advanced quadrature methods,a comparison

In our previous study [Huang et al., 2008, 2009, 2010 [21], [24], [20]; Huang and Lu, 2004 [22], [23]; Lu and Huang, 2000 [38]], we have proposed advanced (i.e., mechanical) quadrature methods (AQMs) for solving the boundary integral equations (BIEs) of the first kind. These methods have high accuracy O(h³), where $h={\max }_{1?m?d}{h}_m$ and h_m (m=1,…,d) are the mesh widths of the curved edge ${\Gamma }_m$. The algorithms are simple and easy to carry out, because the entries of discrete matrix are explicit without any singular integrals. Although the algorithms and error analysis of AQMs are discussed in Huang et al. (2008, 2009, 2010) [21], [24], [20], Huang and Lu (2004) [22], [23], Lu and Huang (2000) [38], there is a lack of systematic stability analysis. The first aim of this paper is to explore a new and systematic stability analysis of AQMs based on the condition number (Cond) and the effective condition number (Cond_eff) for the discrete matrix K_h. The challenging and difficult lower bound of the minimal eigenvalue is derived in detail for the discrete matrix of AQMs for a typical BIE of the first kind. We obtain Cond=O(h_min^?1) and Cond_eff=O(h_min^?1), where $h_{\min }={\min }_{1?m?d}{h}_m$, to display excellent stability. Note that Cond_eff = O(Cond) is greatly distinct to the case of numerical partial differential equations (PDEs) in Li et al. (2007, 2008, 2009, 2010) [26], [31], [32], [33], [34], [35], [36], [37], Li and Huang (2008) [27], [28], [29], [30], Huang and Li (2006) [19] where Cond_eff is much smaller than Cond. The second aim of this paper is to explore intrinsic characteristics of Cond_eff, and to make a comparison with numerical PDEs. Numerical experiments are carried out for three models with smooth and singularity solutions, to support the analysis made.     

Stability analysis of method of fundamental solutions for mixed boundary value problems of Laplace’s equation

Since the stability of the method of fundamental solutions (MFS) is a severe issue, the estimation on the bounds of condition number Cond is important to real application. In this paper, we propose the new approaches for deriving the asymptotes of Cond, and apply them for the Dirichlet problem of Laplace’s equation, to provide the sharp bound of Cond for disk domains. Then the new bound of Cond is derived for bounded simply connected domains with mixed types of boundary conditions. Numerical results are reported for Motz’s problem by adding singular functions. The values of Cond grow exponentially with respect to the number of fundamental solutions used. Note that there seems to exist no stability analysis for the MFS on non-disk (or non-elliptic) domains. Moreover, the expansion coefficients obtained by the MFS are oscillatingly large, to cause the other kind of instability: subtraction cancelation errors in the final harmonic solutions.     

Stability analysis of Trefftz methods for the stick-slip problem

The stick-slip problem is a two-dimensional Stokes flow problem, and is classified into biharmonic equation with crack singularities. The collocation Trefftz method (CTM) is used to provide the very accurate solutions and leading coefficients. In this paper, the error analysis is made, to show the exponential convergence rates, and the new stability analysis is explored more in detail. We derive the bounds of effective and traditional condition numbers, to have the polynomial and the exponential growth rates, respectively. The moderate effective condition number is a suitable criterion of stability for the CTM solution of the stick-slip problem, while the huge condition number is misleading. Besides, numerical experiments are carried out to support the stability analysis. Hence the effective condition number becomes a new trend of stability for numerical partial differential equations.     

Cauchy problems of Laplace's equation by the methods of fundamental solutions and particular solutions

The Cauchy problems of Laplace's equation are ill-posed with severe instability. In this paper, numerical solutions are solicited by the method of fundamental solutions (MFS) and the method of particular solutions (MPS). We focus on the analysis of the MFS, and derive the bounds of errors and condition numbers. The analysis for the MPS can also be obtained similarly. Numerical experiments and comparisons are reported for the Cauchy and Dirichlet problems by the MPS and the MFS. The Cauchy noise data and the regularization are also adopted in numerical experiments. Both the MFS and the MPS are effective to Cauchy problems. The MPS is superior in accuracy and stability; but the MFS owns simplicity of algorithms, and earns flexibility for a wide range of applications, such as Cauchy problems. These conclusions also coincide with [37]. The basic analysis of error and stability is explored in this paper, and applied to the Cauchy data. There are many reports on numerical Cauchy problems, see the survey paper in [12]; most of them are of computational aspects. The strict analysis of this paper may, to a certain degree, fill up the existing gap between theory and computation of Cauchy problems by the MFS and the MPS. Moreover, comprehensive analysis and compatible computation are two major characteristics of this paper, which may enhance the study of numerical Cauchy problems forward to a higher and advanced level.     

Numerical methods for calculating pressure distribution in gas bearings

In gas bearings, the pressure distribution is governed by a non-linear Reynolds equation. In order to solve this equation two numerical methods, the conservative difference scheme and the finite element method, are provided in this paper. They are superior to the finite difference method of Colemman [2]. Use of the finite element method is advocated because of its flexibility in solving the Reynolds equation.     

高效液相色谱-蒸发光散射法分析海参脑苷脂和神经酰胺的含量

目的建立了一种高效液相色谱-蒸发光散射检测方法(high performance liquid chromatography coupled with evaporative light-scattering detection,HPLC-ELSD)同时分析海参中脑苷脂和神经酰胺含量的方法。方法以TSKgel CN-80Ts柱为分离柱,正己烷-异丙醇-二氯甲烷-甲醇为流动相,采用梯度洗脱方式,流速1.0m L/min,漂移管温度50℃,氮气流速2.4 L/min。结果在优化条件下,18 min内完成分离,脑苷脂的线性范围为2.5~250μg/m L,检出限为0.5μg/m L(S/N=3),相关系数R0.99,神经酰胺的线性范围为5~150μg/m L,检出限为0.05μg/m L(S/N=3),相关系数R0.99,方法稳定性良好;不同种海参中脑苷脂和神经酰胺含量存在较为明显的差异,含量范围分别为0~4.44 mg/g和0.24~2.21 mg/g。其中,猪参的脑苷脂含量最高为4.44 mg/g,而白底靴参的脑苷脂未能检出。阿拉斯加红参的神经酰胺含量最高为2.21 mg/g,而海地瓜神经酰胺含量最少。结论所建立的方法灵敏、准确、快速,可应用于食品中脑苷脂和神经酰胺的同时定量分析。