An O(k²+kh²+h²) Accurate Two-level Implicit Cubic Spline Method for One Space Dimensional Quasi-linear Parabolic Equations

In this piece of work, using three spatial grid points, we discuss a new two-level implicit cubic spline method of O(k2 + kh2 + h4) for the solution of quasi-linear parabolic equation , 0 0 subject to appropriate initial and Dirichlet boundary conditions, where h > 0, k > 0 are grid sizes in space and time-directions, respectively. The cubic spline approximation produces at each time level a spline function which may be used to obtain the solution at any point in the range of the space variable. The proposed cubic spline method is applicable to parabolic equations having singularity. The stability analysis for diffusion- convection equation shows the unconditionally stable character of the cubic spline method. The numerical tests are performed and comparative results are provided to illustrate the usefulness of the proposed method.

二阶非线性抛物方程的B样条有限元法

首先,用二次B样条有限元法求解Fisher-Kolmogorov(FK)方程,证明半离散格式与全离散格式解的稳定性与收敛性;其次,用Crank-Nicolson方法离散时间变量,得到近似解的收敛阶为O((Δt)^(2)+h^(3));最后,用数值算例验证了理论分析结果及B样条有限元法的有效性.

基于B样条与鲸鱼优化算法的机械臂轨迹规划

为了提高机械臂的工作效率,构建一种基于B样条与鲸鱼优化算法(whale optimization algorithm,WOA)的机械臂时间最优轨迹规划方法.用蒙特卡罗法描绘机械臂的工作空间,用B样条对给出的路径点进行插值处理,根据机械臂各个关节的性能,引入角速度与角加速度约束,同时加入边界条件.在构建时间最优的目标函数后,利用引入惯性权重值的WOA对机械臂运行时间进行优化.用Matlab进行仿真验证,结果表明构建的算法在时间优化方面效果好于传统的5次多项式方法,并且角速度与角加速度曲线连续平滑,验证了算法的有效性和可行性.

基于B-Spline的位移监测数据动态优化拟合

为更好地进行工程位移的监测,对基坑施工过程钢支撑水平位移激光监测数据特征、误差类型及其成因进行了分析,研究了B-Spline在海量监测数据去噪中的适用性,并通过引入待估参数的B样条基函数修正,建立最小二乘控制下位移-时间关系的高精度拟合优化。结果表明:待估参数ζ_(i)的修正重构,把监测数据序列转化为参数估计问题,使位移时变关系的最优化拟合成为可能;移动时间坐标方法,可实现等时距监测数据的动态处理,优化计算时间;B-Spline融合最小二乘法,在最小方差控制下实现监测数据-时间关系最优拟合,形成光滑、连续的高精度拟合曲线,有利于提高监测反馈控制的精度和可靠性。实例应用验证了成果的可靠性和适用性。

ANALYSIS OF BENDING, VIBRATION AND STABILITY FOR THIN PLATE ON ELASTIC FOUNDATION BY THE MULTIVARIABLE SPLINE ELEMENT METHOD

In this paper, the bicubic splines in product form are used to construct the multi-field functions for bending moments, twisting moment and transverse displacement of the plate on elastic foundation. The multivariable spline element equations are derived, based on the mixed variational principle. The analysis and calculations of bending, vibration and stability of the plates on elastic foundation are presented in the paper. Because the field functions of plate on elastic foundation are assumed independently, the precision of the field variables of bending moments and displacement is high.

2D numerical manifold method based on quartic uniform B-spline interpolation and its application in thin plate bending

A new numerical manifold （NMM） method is derived on the basis of quartic uniform B-spline interpolation. The analysis shows that the new interpolation function possesses higher-order continuity and polynomial consistency compared with the conven- tional NMM. The stiffness matrix of the new element is well-conditioned. The proposed method is applied for the numerical example of thin plate bending. Based on the prin- ciple of minimum potential energy, the manifold matrices and equilibrium equation are deduced. Numerical results reveal that the NMM has high interpolation accuracy and rapid convergence for the global cover function and its higher-order partial derivatives.

两条B样条曲线求交的高效计算方法

曲线曲面间求交计算在CG和CAD中有着广泛的应用.牛顿法等迭代法计算效率高但需要良好的初始值;裁剪法具有良好的鲁棒性但计算效率不理想,尤其是对于相切情况的求交问题.为此,提出一种计算2条B样条曲线交点的混合方法.首先提出一种高效的线性复杂度裁剪方法,用于获得良好的初始值;然后提出一种与导数无关且效率更高的改进的割线法,用于验证贯穿性相交情况;最后提出一个相切情况下收敛阶为2的迭代公式,其性能远优于现有的牛顿法和裁剪法.理论上,混合方法若与根隔离法相结合,可以应用于更多类型曲线间的求交问题.数值实验结果表明,与现有的同类方法相比,在贯穿情况下,所提方法的计算效率提高约10%,在相切情况下则提高约100%~300%.

一种采用B样条插值的改进能量算子在电梯电机轴承故障诊断中的应用

针对传统Teager能量解调算子方法对电梯运行系统中存在的强背景噪声较为敏感的不足,提出了一种改进的能量解调算子方法;采用了B样条技术与传统Teager能量算子方法进行结合,其中建立的B样条曲线对信号进行插值起到滤波作用;然后再利用Teager能量算子对滤波信号进行转换;最后利用傅里叶变换得到转换信号的频谱图从而揭示故障特征;所提出的基于B样条插值的能量解调方法不仅保留了传统能量解调算法的优点,如较高的解调精度和优秀的时间分辨率等,并且可以在强噪声背景下提取出微弱轴承故障特征;经实验验证实现了提高强背景噪声下的轴承故障检测的性能,能够在故障退化的早期检测故障,满足了实际工况下故障诊断上的应用。

Two 8-node quadrilateral spline elements by B-net method

Isoparametric quadrilateral elements are widely used in the finite element method, but the accuracy of the isoparametric quadrilateral elements will drop obviously deteriorate due to mesh distortions. Spline functions have some properties of simplicity and conformality. Two 8-node quadrilateral elements have been developed using the trian- gular area coordinates and the B-net method, which can ex- actly model the quadratic field for both convex and concave quadrangles. Some appropriate examples are employed to evaluate the performance of the proposed elements. The nu- merical results show that the two spline elements can obtain solutions which are highly accurate and insensitive to mesh distortions.

B样条函数法的任意荷载近似计算方法

为优化B样条函数法数值计算流程,提升解决复杂荷载问题的能力。本文基于B样条函数法,通过与相关文献数值算例结果进行比较分析,研究任意荷载的新型计算方法。论文以解圆柱壳结构模型为数值算例,基于文中提出的计算方法,构建出三组不同的单方向荷载逼近函数,分别求解得到特殊点的位移、弯矩、轴力,以及位移和内力沿坐标轴方向的分布,与精确解(扁壳解法)进行比较研究,分析近似计算方法的精确性与实用性。结果显示,在特殊点处二次插值方法的相对误差均低于1.5%,三次、四次插值方法的相对误差均低于0.6%,沿坐标轴方向的分布与精确解均一致,表明本方法具有精度高、处理复杂荷载简便、易于程序编写、构造函数灵活等特点。本方法用逼近函数替代原函数便于数值积分的想法,为未来B样条函数法处理复杂荷载提供新思路。

Performance of global look-up table strategy in digital image correlation with cubic B-spline interpolation and bicubic interpolation

Global look-up table strategy proposed recently has been proven to be an efficient method to accelerate the interpolation, which is the most time-consuming part in the iterative sub-pixel digital image correlation （DIC） algorithms. In this paper, a global look-up table strategy with cubic B-spline interpolation is developed for the DIC method based on the inverse compositional Gauss-Newton （IC-GN） algorithm. The performance of this strategy, including accuracy, precision, and computation efficiency, is evaluated through a theoretical and experimental study, using the one with widely employed bicubic interpolation as a benchmark. The global look-up table strategy with cubic B-spline interpolation improves significantly the accuracy of the IC-GN algorithm-based DIC method compared with the one using the bicubic interpolation, at a trivial price of computation efficiency.

Numerical solution of Poisson equation with wavelet bases of Hermite cubic splines on the interval

A new wavelet-based finite element method is proposed for solving the Poisson equation. The wavelet bases of Hermite cubic splines on the interval are employed as the multi-scale interpolation basis in the finite element analysis. The lifting scheme of the wavelet-based finite element method is discussed in detail. For the orthogonal characteristics of the wavelet bases with respect to the given inner product, the corresponding multi-scale finite element equation can be decoupled across scales, totally or partially, and suited for nesting approximation. Numerical examples indicate that the proposed method has the higher efficiency and precision in solving the Poisson equation.

Development of quadrilateral spline thin plate elements using the B-net method

The quadrilateral discrete Kirchhoff thin plate bending element DKQ is based on the isoparametric element Q8, however, the accuracy of the isoparametric quadrilateral elements will drop significantly due to mesh distortions. In a previous work, we constructed an 8-node quadrilateral spline element L8 using the triangular area coordinates and the B- net method, which can be insensitive to mesh distortions and possess the second order completeness in the Cartesian co- ordinates. In this paper, a thin plate spline element is devel- oped based on the spline element L8 and the refined tech- nique. Numerical examples show that the present element indeed possesses higher accuracy than the DKQ element for distorted meshes.

融合改进斜四邻域A^(*)与三次B样条曲线的路径规划算法

针对传统A^(*)算法八邻域搜索路径节点多导致搜索及规划路径速度慢的问题,提出1种融合改进斜四邻域A^(*)与三次B样条曲线的路径规划算法。在传统八邻域搜索启发下,删除原本上下左右4个方向上的搜索路径,保留右上、右下、左上及左下4个方向的搜索路径,使算法在搜索时速度更快,搜索路径节点数更少;引入三次B样条曲线,优化路径上的尖锐拐点,使路径更平滑。采用提出的算法与其他2种算法在3种不同分辨率及不同障碍率的地图上进行实验验证。结果表明:相比于传统A^(*)算法,改进斜四邻域A^(*)算法的搜索时间缩短83.4%,路径节点数减少89.7%;引入的三次B样条曲线使斜四邻域A^(*)算法规划出的路径更平滑,提出的算法可极大提高路径搜索及规划的有效性。

Simulations of solitary waves of RLW equation by exponential B-spline Galerkin method

In this paper, an approximate function for the Galerkin method is composed using the combination of the exponential B-spline functions. Regularized long wave equation （RLW） is integrated fully by using an exponential B-spline Galerkin method in space together with Crank-Nicolson method in time. Three numerical examples related to propagation of sin- gle solitary wave, interaction of two solitary waves and wave generation are employed to illustrate the accuracy and the efficiency of the method. Obtained results are compared with some early studies.

A cubic quadrilateral spline element with concave shapes

Basic requirement for applying isoparametric element is that the element has to be convex and no violent distortion is allowed. In this paper, a cubic quadrilateral spline element with 12 nodes has been developed using the triangular area coordinates and the B-net method, which can exactly model the cubic field for quadrilateral element with both convex and concave shapes. Neither mapping nor coordinate transformation is required and the spline element can obtain high accuracy solutions and insensitive to mesh distortions.

基于B样条的(保边界的)高精度符号距离函数重建

针对隐式符号距离函数重建这一问题,提出一种高精度的符号距离函数重建方法.首先,基于快速行进法获得离散的符号距离场;其次,针对离散符号距离场进行B样条自适应拟合;最后,为了提高重建的边界拟合精度,在拟合的目标函数中考虑了边界误差,得到了在边界上具有较高精度的离散符号距离函数重建结果.实验结果表明,通过上述方法获得的隐式符号距离函数重建结果,在保证符号距离函数拟合良好的情况下其零水平集能很好的贴合原始边界.

Estimation of the water–oil–gas relative permeability curve from immiscible WAG coreflood experiments using the cubic B-spline model

Immiscible water-alternating-gas（WAG） flooding is an EOR technique that has proven successful for water drive reservoirs due to its ability to improve displacement and sweep efficiency.Nevertheless,considering the complicated phase behavior and various multiphase flow characteristics,gas tends to break through early in production wells in heterogeneous formations because of overriding,fingering,and channeling,which may result in unfavorable recovery performance.On the basis of phase behavior studies,minimum miscibility pressure measurements,and immiscible WAG coreflood experiments,the cubic B-spline model（CBM） was employed to describe the three-phase relative permeability curve.Using the Levenberg-Marquardt algorithm to adjust the vector of unknown model parameters of the CBM sequentially,optimization of production performance including pressure drop,water cut,and the cumulative gas-oil ratio was performed.A novel numerical inversion method was established for estimation of the water-oil-gas relative permeability curve during the immiscible WAG process.Based on the quantitative characterization of major recovery mechanisms,the proposed method was validated by interpreting coreflood data of the immiscible WAG experiment.The proposed method is reliable and can meet engineering requirements.It provides a basic calculation theory for implicit estimation of oil-water-gas relative permeability curve.

Numerical Solution of the Seventh Order Boundary Value Problems Using B-Spline Method

We develop a numerical method for solving the boundary value problem of The Linear Seventh Ordinary Boundary Value Problem by using the seventh-degree B-Spline function. Formulation is based on particular terms of order of seventh order boundary value problem. We obtain Septic B-Spline formulation and the Collocation B-spline method is formulated as an approximation solution. We apply the presented method to solve an example of seventh order boundary value problem in which the result shows that there is an agreement between approximate solutions and exact solutions. Resulting in low absolute errors shows that the presented numerical method is effective for solving high order boundary value problems. Finally, a general conclusion has been included.

B-Spline Collocation Method for Solving Singularly Perturbed Boundary Value Problems

We use fifth order B-spline functions to construct the numerical method for solving singularly perturbed boundary value problems. We use B-spline collocation method, which leads to a tri-diagonal linear system. The accuracy of the proposed method is demonstrated by test problems. The numerical results are found in good agreement with exact solutions.