Legendre-Weighted Residual Methods for System of Fractional Order Differential Equations

The numerical approach for finding the solution of fractional order systems of boundary value problems (BPVs) is derived in this paper. The implementation of the weighted residuals such as Galerkin, Least Square, and Collocation methods are included for solving fractional order differential equations, which is broadened to acquire the approximate solutions of fractional order systems with differentiable polynomials, namely Legendre polynomials, as basis functions. The algorithm of the residual formulations of matrix form can be coded efficiently. The interpretation of Caputo fractional derivatives is employed here. We have demonstrated these methods numerically through a few examples of linear and nonlinear BVPs. The results in absolute errors show that the present method efficiently finds the numerical solutions of fractional order systems of differential equations.

Method of Semi Analytic Perturbation Weighted Residuals for Nonlinear Bending of Shallow Shells

The semi? analytic perturbation weighted residuals method was used to solve the nonlinear bending problem of shallow shells, and the fifth order B spline was taken as trial function to seek an efficient method for nonlinear bending problem of shallow shells. The results from the present method are in good agreement with those derived from other methods. The present method is of higher accuracy, lower computing time and wider adaptability. In addition, the design of computer program is simple and it is easy to be programmed.

A WEIGHTED RESIDUAL METHOD FOR ELASTIC-PLASTIC ANALYSIS NEAR A CRACK TIP AND THE CALCULATION OF THE PLASTIC STRESS INTENSITY FACTORS

In this paper, a weighted residual method for the elastic-plastic analysis near a crack tip is systematically given by taking the model of power-law hardening under plane strain condition as a sample. The elastic-plastic solutions of the crack lip field and an approach based on the superposition of the nonlinear finite element method on the complete solution in the whole crack body field, to calculate the plastic stress intensity factors, are also developed. Therefore, a complete analvsis based on the calculation both for the crack tip field and for the whole crack body field is provided.

AN ASSESSMENT OF THE MESHLESS WEIGHTED LEAST-SQUARE METHOD

The meshless weighted least-square (MWLS) method was developed based on the weighted least-square method. The method possesses several advantages, such as high accuracy, high stability and high e?ciency. Moreover, the coe?cient matrix obtained is symmetric and semi- positive de?nite. In this paper, the method is further examined critically. The e?ects of several parameters on the results of MWLS are investigated systematically by using a cantilever beam and an in?nite plate with a central circular hole. The numerical results are compared with those obtained by using the collocation-based meshless method (CBMM) and Galerkin-based meshless method (GBMM). The investigated parameters include the type of approximations, the type of weight functions, the number of neighbors of an evaluation point, as well as the manner in which the neighbors of an evaluation point are determined. This study shows that the displacement accuracy and convergence rate obtained by MWLS is comparable to that of the GBMM while the stress accuracy and convergence rate yielded by MWLS is even higher than that of GBMM. Furthermore, MWLS is much more e?cient than GBMM. This study also shows that the instability of CBMM is mainly due to the neglect of the equi- librium residuals at boundary nodes. In MWLS, the residuals of all the governing equations are minimized in a weighted least-square sense.

Box-Behnken响应面法优化复合益生菌发酵三七药渣工艺及体外抗氧化活性研究

目的以三七药渣为原料,通过益生菌发酵比较不同益生菌以及复合益生菌对各指标成分影响的差异,探究最佳发酵工艺。方法采用单因素、Box-Behnken响应面法优化发酵工艺,并通过体外抗氧化实验评价发酵产物的抗氧化能力。结果最佳发酵工艺为有氧发酵48 h、厌氧发酵36 h,嗜热链球菌、两歧双歧杆菌、嗜酸乳杆菌的比例为2∶3∶1,料液比0.14 g·mL^(-1),接菌量5%,温度33℃。在最佳发酵工艺条件下,三七药渣中性多糖、酸性多糖、总黄酮的含量分别提高了105.64%、96.98%、123.83%,相较于单菌发酵均显著升高。发酵产物清除DPPH、ABTS自由基的IC 50值分别为1.774、3.065 mg·mL^(-1),还原Fe^(3+)的能力为0.138 mmol FeSO_(4)·g^(-1),较未发酵药渣显著增强。结论最佳发酵工艺能显著提高三七药渣中各指标成分含量,显著增强其抗氧化能力,相较单菌发酵各指标成分含量均显著提高,提示上述益生菌之间没有明显的拮抗作用。研究结果为三七药渣的资源化利用提供了科学依据和数据支撑。

考虑地震事件内空间相关性的区域概率地震危险性分析

一次地震事件不同地点的地震动强度参数与结构响应参数之间是空间相关的,相关的地震效应将导致地震损失急剧集中与积累,从而造成灾难性事件,但是现有的区域尺度地震风险评估并没有考虑地震动的空间相关性。该文采用地统计学方法表征地震事件内地震动强度参数的空间相关性,提出了理论半变异函数的几何稳健估计方法,并采用加权最小二乘法实现对短距离空间相关性的准确高效评估。以新潟越冲地震为例,基于指数半变异函数模型,采用加权最小二乘法对几何稳健估计半变异函数值进行拟合,得到了PGA、PGV和0 s~10 s谱加速度的空间相关性函数,提出了变程b值的预测模型,结果表明:相比于短周期谱加速度,长周期谱加速度的空间相关性衰减速率更小,且在T=0.45 s处存在拐点。将所开发的空间相关性模型应用于区域概率地震危险性分析,采用Monte Carlo方法实现了空间相关随机场的区域地震动强度参数的模拟,改进了传统区域地震危险性分析的方法,发现绝大多数情况下考虑空间相关性的年超越概率大于未考虑空间相关性的年超越概率,表明考虑空间相关性对于建筑群落及基础设施风险评估、韧性城市建设等具有重要的指导意义。

DRSTN:深度残差软阈值化网络

在采用深度残差等神经网络模型解决图像分类任务时,特征提取过程损失的一些重要特征会影响模型的分类性能。神经网络“端到端”的学习模式带来的黑盒问题,也会限制其在诸多领域的应用和发展。另外,神经网络模型往往需要较长的训练时间。为了提高深度残差网络模型的分类效果和训练效率,引入了模型迁移方法和软阈值化方法,提出了DRSTN(Deep Residual Soft Thresholding Network)网络,并对此网络结构进行微调,生成了不同版本的DRSTN网络。DRSTN网络的性能得益于3个方面的有机整合:1)通过梯度加权类激活映射(Gradients-weighted Class Activation Mapping,Grad-CAM)方法对网络的特征提取进行可视化,根据可视化结果挑选进一步优化的模型;2)基于模型迁移,研究人员不必全新地搭建模型,可以直接在已有的模型上进行优化,能够节省大量训练时间;3)软阈值化作为非线性变换层嵌入到深度残差网络体系结构中,以消除样本中不相关的特征。实验结果表明,在相同训练条件下,DRSTN_KS(3*3)_RB(2:2:2)网络在CIFAR-10数据集上的分类精度相比SKNet-18,ResNet18和ConvNeXt_tiny网络分别提高了15.5%,8.8%和10.9%;该网络也具有一定的泛化性,在MNIST和Fashion MNIST数据集上能够达到快速的迁移效果,分类精度分别达到99.06%和93.15%。

考虑残余应力的钢桥面板-肋双面焊裂纹应力强度因子计算方法

建立焊接分析有限元模型,对顶板-纵肋双面焊构造的焊接过程进行数值模拟,拟合得到顶板焊趾细节沿板厚方向分布的横向残余应力分布经验公式;建立钢桥面板断裂力学数值模型,结合统一的权函数表达式,推导适用于顶板焊趾处裂纹最深点和表面点应力强度因子的新权函数,并将权函数计算的应力强度因子与有限元计算的应力强度因子进行对比。研究结果表明:顶板-纵肋双面焊顶板焊趾处残余应力沿板厚方向处于拉—压—拉状态,呈正弦函数分布;在二次应力分布下,权函数法与有限元法计算所得顶板焊趾处裂纹最深点应力强度因子最大相对误差为7.4%,表面点应力强度因子最大相对误差为4.1%;在焊接残余应力场下,权函数法与有限元法计算所得顶板焊趾处裂纹最深点应力强度因子最大相对误差为7.6%,表面点应力强度因子最大相对误差为8.6%;权函数法能有效计算钢桥面板-肋双面焊顶板焊趾处疲劳裂纹应力强度因子。

Novel method for random vibration analysis of single-degree-of-freedom vibroimpact systems with bilateral barriers

The vibroimpact systems with bilateral barriers are often encountered in practice.However,the dynamics of the vibroimpact system with bilateral barriers is full of challenges.Few closed-form solutions were obtained.In this paper,we propose a novel method for random vibration analysis of single-degree-of-freedom(SDOF)vibroim-pact systems with bilateral barriers under Gaussian white noise excitations.A periodic approximate transformation is employed to convert the equations of the motion to a con-tinuous form.The probabilistic description of the system is subsequently defined through the corresponding Fokker-Planck-Kolmogorov(FPK)equation.The closed-form station-ary probability density function(PDF)of the response is obtained by solving the reduced FPK equation and using the proposed iterative method of weighted residue together with the concepts of the circulatory probability flow and the potential probability flow.Finally,the versatility of the proposed approach is demonstrated by its application to two typical examples.Note that the solution obtained by using the proposed method can be used as the benchmark to examine the accuracy of approximate solutions obtained by other methods.

Anti-plane deformations around arbitrary-shaped canyons on a wedge-shape half-space:moment method solutions

The wave propagation behavior in an elastic wedge-shaped medium with an arbitrary shaped cylindrical canyon at its vertex has been studied.Numerical computation of the wave displacement field is carried out on and near the canyon surfaces using weighted-residuals(moment method).The wave displacement fields are computed by the residual method for the cases of elliptic,circular,rounded-rectangular and flat-elliptic canyons,The analysis demonstrates that the resulting surface displacement depends,as in similar previous analyses,on several factors including,but not limited,to the angle of the wedge,the geometry of the vertex,the frequencies of the incident waves,the angles of incidence,and the material properties of the media.The analysis provides intriguing results that help to explain geophysical observations regarding the amplification of seismic energy as a function of site conditions.

AN EFFICIENT PREDICTION METHOD FOR THE TWO-DIMENSIONAL COUPLED STRUCTURAL-ACOUSTIC ANALYSIS

A wave number method （WNM） is proposed to deal with the two-dimensional coupled structural-acoustic problem. Based on an indirect Trefftz approach, the displacement and the pressure response are approximated respectively by a set of wave functions, which exactly satisfy the governing equations and are independent of the size of the coupled system. The wave functions comprise the exact solutions of the homogeneous part of the governing equations and some particular solution functions, which arise from the external excitation. The weighting coefficients of the wave functions can be obtained by enforcing the pressure approximation to satisfy the boundary conditions and it is performed by applying the weighted residual formulation. The example is computed by the WNM and the BEM. The results show that, the WNM can attain the same accuracy and convergence as the BEM with less degrees of freedom.

An analysis of the incompressible viscous flows problem by meshless method

Generally the incompressible viscous flow problem is described by the Navier-Stokes equation. Based on the weighted residual method the discrete formulation of element-free Galerkin is inferred in this paper. By the step-bystep computation in the field of time, and adopting the least-square estimation of the-same-order shift, this paper has calculated both velocity and pressure from the decoupling independent equations. Each time fraction Newton-Raphson iterative method is applied for the velocity and pressure. Finally, this paper puts the method into practice of the shear-drive cavity flow, verifying the validity, high accuracy and stability.

AN ITERATIVE PARALLEL ALGORITHM OF FINITE ELEMENT METHOD

In this paper, a parallel algorithm with iterative form for solving finite element equation is presented. Based on the iterative solution of linear algebra equations, the parallel computational steps are introduced in this method. Also by using the weighted residual method and choosing the appropriate weighting functions, the finite element basic form of parallel algorithm is deduced. The program of this algorithm has been realized on the ELXSI-6400 parallel computer of Xi'an Jiaotong University. The computational results show the operational speed will be raised and the CPU time will be cut down effectively. So this method is one kind of effective parallel algorithm for solving the finite element equations of large-scale structures.

AN APPROXIMATE METHOD FOR DETERMING THE VELOCITY PROFILE IN A LAMINAR BOUNDARY-LAYER ON FLAT PLATE

In this paper, using the integration method, it is sought to solve the problem for the laminar boundary_layer on a flat plate. At first, a trial function of the velocity profile which satisfies the basical boundary conditions is selected. The coefficients in the trial function awaiting decision are decided by using some numerical results of the boundary_layer differential equations. It is similar to the method proposed by Peng Yichuan, but the former is simpler. According to the method proposed by Peng, when the awaiting decision coefficients of the trial function are decided, it is sought to solve a third power algebraic equation. On the other hand, in this paper, there is only need for solving a linear algebraic equation. Moreover, the accuracy of the results of this paper is higher than that of Peng.

APPLICATION OF SUBREGION FUNCTION METHOD FOR SOLVING BEAM-BOARD STRUCTURE

In this paper, the solution to the structure consisting of a bead and a board is given as a result of the application of the subregion function method which was suggested in ref. [1]. The same problem is also computed with finite element method. The comparison between the two results shows that the application of the subregion function in the method of weighted residuals is practical and effective, especially for solving compound structures.

Wave Number Method for Three-Dimensional Steady-State Acoustic Problems

Based on the indirect Trefftz approach, a wave number method （WNM） is proposed to deal with three-dimensional steady-state acoustic problems. In the WNM, the dynamic pressure response variable is approximated by a set of wave functions, which exactly satisfy the Helmholtz equation. The set of wave functions comprise the exact solutions of the homogeneous part of the governing equations and some particular solution functions. The unknown coefficients of the wave functions can be obtained by enforcing the pressure approximation to satisfy the boundary conditions. Compared with the boundary element method （BEM）, the WNM have a smaller system matrix, and is applicable to the radiation problems since the wave functions are independent of the domain size. A 3D acoustic cavity is exemplified to show the properties of the method. The results show that the wave number method is more efficient than the BEM, and it is fairly accurate.

Solution of Electrohydrodynamic Flow Equation Using an Orthogonal Collocation Method Based on Mixed Interpolation

In this Paper, we have proposed a new weighted residual method known as orthogonal collocation-based on mixed interpolation (OCMI). Mixed interpolation uses the classical polynomial approximation with two correction terms given in the form of sine and cosine function. By these correction terms, we can control the error in the solution. We have applied this approach to a non-linear boundary value problem (BVP) in ODE which governs the electrohydrodynamic flow in a cylindrical conduit. The solution profiles shown in the figures are in good agreement with the work of Paullet (1999) and Ghasemi et al. (2014). Our solution is monotonic decreasing and satisfies , where, α governs the strength of non-linearity and for large values of α solutions are . The residual errors are given in Table 1 and Table 2 which are significantly small. Comparison of residual errors between our proposed method, Least square method and Homotopy analysis method is also given and shown via the Table 3 where as the profiles of the residual error are depicted in Figures 4-8. Table and graphs show that efficiency of the proposed method. The error bound and its L2-norm with relevant theorems for mixed interpolation are also given.

Vortex-Source Method for the 3D Incompressible Irrotational Flow

The weighted residuals method was used for obtaining the boundary integral representation of the velocity of the three-dimensional inviscid irrotational flow. It is shown that velocity in an arbitrary point of domain can be expressed through its values on the boundary. Boundary integral equations of the second kind for solving boundary-valued problems of the first and second kinds are developed. The result has been also generalised to the case of solenoidal vector fields with potential vorticity. It is shown that the resulting integral equations are Fredholm integral equations of the second kind and allow effective numerical solving of corresponding boundary-valued problems. Examples of numerical solutions for a sphere and an ellipsoid are given for demonstration of efficiency of the offered method.

基于ARIMA-OSELM的火电厂SCR入口NOx浓度预测建模研究

针对火力发电厂燃烧系统运行工况复杂、迟延较大,导致选择性催化还原烟气脱硝系统(SCR)中入口NO x质量浓度难以准确测量的问题,提出了一种基于ARIMA-OSELM神经网络组合模型的火电厂SCR入口NO x浓度预测方法,分别从最优权重和残差优化2个组合角度进行对比研究。将该方法应用于某火力发电厂SCR入口浓度预测中,结果表明:基于ARIMA-OSELM残差优化的组合模型预测精度最高,其效果优于ARIMA-OSELM最优权重的组合预测模型以及单一ARIMA和OSELM神经网络预测模型,评价指标F MAPE、M RMSE和R 2分别为0.190、1.364和0.978。

弹性地基梁板的径向基函数逼近求解方法

基于Winkler模型对弹性地基梁和弹性地基板的挠度求解提出了一种新方法。利用加权余量法中的配点法结合径向基函数逼近得到挠度近似解,该方法无需划分网格且无需积分是未知数与方程数目一一对应的强形式解法。数值实验表明:该近似解结果与理论解基本吻合,计算效果与杂交边界法相当,优于滑动最小二乘法;此外,根据边界条件可赋予附加未知量物理意义,将部分特征矩阵构造成单位阵从而降低了待求方程组的求解难度,减小误差,为特征矩阵附加行的编写提供了新思路。