结合横向振荡和空间正交的向量血流速度测量

传统的彩色多普勒成像只能测量与超声波束平行的血流速度分量,且依赖于超声波束与血管之间的夹角。超声向量血流成像是一种更加先进的血流成像技术,该方法可以直接获得血流速度的实际大小和方向,因此不依赖于超声波束与血管之间的夹角。本文从向量血流测量方法之一的横向声场法入手,简要概括了横向振荡(Transverse Oscillation, TO)法和空间正交(Spatial Quadrature, SQ)法两种方法的基本原理、成像过程及各自的优缺点,并提出了一种互相结合的方法,即奇偶振荡法(Odd Even Oscillation, OEO),该方法利用SQ法快速进行波束合成,利用TO法计算最终的速度矢量,克服了TO法和SQ法各自的缺点,能够有效解决TO法成像计算量大以及SQ法出现混叠和对噪声灵敏度高的问题。     

On the equivalence of the time domain differential quadrature method and the dissipative Runge–Kutta collocation method

Numerical solutions for initial value problems can be evaluated accurately and efficiently by the differential quadrature method. Unconditionally stable higher order accurate time step integration algorithms can be constructed systematically from this framework. It has been observed that highly accurate numerical results can also be obtained for non‐linear problems. In this paper, it is shown that the algorithms are in fact related to the well‐established implicit Runge–Kutta methods. Through this relation, new implicit Runge–Kutta methods with controllable numerical dissipation are derived. Among them, the non‐dissipative and asymptotically annihilating algorithms correspond to the Gauss methods and the Radau IIA methods, respectively. Other dissipative algorithms between these two extreme cases are shown to be B‐stable (or algebraically stable) as well and the order of accuracy is the same as the corresponding Radau IIA method. Through the equivalence, it can be inferred that the differential quadrature method also enjoys the same stability and accuracy properties. Copyright © 2001 John Wiley & Sons, Ltd.     

Method for numerical integration of rapidly oscillating functions in diffraction theory

The quadrature of general, highly oscillatory integrals is a relatively complicated computational problem that occurs in a wide range of practical applications, e.g. in physics, chemistry, and image analysis. It is often necessary to use a high number of nodal points with classical quadrature formulas in order to achieve a required accuracy of numerical integration of rapidly oscillating functions. However, an increase in integration points leads to a larger computational time. Our work describes and analyses a method for numerical integration of rapidly oscillating functions, which enables to obtain the required accuracy with a smaller number of nodal points than classical integration rules and with a relatively simple integration formula. The proposed method is generally suitable for integration of rapidly oscillating functions in various applications. The method is demonstrated in examples of calculation of the diffraction integral in optics, where the integrand is often a rapidly oscillatory function. The results can be readily adapted to similar problems from other fields. Copyright © 2009 John Wiley & Sons, Ltd.     

An energy-conserving and decaying time integration method for general nonlinear dynamics

An energy-conserving and decaying time integration method is derived from the discretized energy balance equation in this paper. Based on the Gauss-Legendre quadrature rule, the new method evaluates the mean internal force by collecting the internal forces of several quadrature points in the time interval. Compared with previous conserving methods, the new method avoids the reconstruction of finite element models as used in the energy-momentum method, and the computation of energy functions as used in the constraint energy method, so it is simple and straightforward, and has no difficulty in implementation for dynamics with general nonlinear resilience including nonlinear damping force and internal force. By employing enough quadrature nodes, this method can preserve system energy accurately for conservative system, and then the unconditional stability is achieved automatically. In addition, the energy-decaying method is developed by introducing the energy dissipation term to the energy balance equation, and two free parameters controlling the numerical dissipation are determined based on linear spectral analysis. Finally, the performance of the proposed method is checked on several examples and the results are compared with that of the trapezoidal rule.     

A study on time schemes for DRBEM analysis of elastic impact wave

 The precise integration and differential quadrature methods are two new unconditionally stable numerical schemes to approximate time derivative with more than the second order accuracy. Recent studies showed that compared with the Houbolt and Newmark methods, they produced more accurate solutions with large time step for the problems where response is primarily dominated by low and intermediate frequency modes. This paper aims to investigate these time schemes in the context of the dual reciprocity BEM (DRBEM) formulation of various shock-excited scalar elastic wave problems, where high modes have important affect on traction response. The Houbolt method was widely recommended in many literatures for such DRBEM dynamic formulations. However, this study found that the damped Newmark algorithm was the most efficient and accurate for impact traction analysis in conjunction with the DRBEM. The precise integration and differential quadrature methods are shown inapplicable for such shock-excited problems due to the absence of numerical damping. On the other hand, we also found that to achieve the same order of accuracy, the differential quadrature method required much less computing effort than the precise integration method due to the use of the Bartels–Stewart algorithm solving the resulting Lyapunov matrix analogization equation. Received 6 November 2000     

New variable transformations for evaluating nearly singular integrals in 2D boundary element method

This work presents a further development of the distance transformation technique for accurate evaluation of the nearly singular integrals arising in the 2D boundary element method (BEM). The traditional technique separates the nearly hypersingular integral into two parts: a near strong singular part and a nearly hypersingular part. The near strong singular part with the one-ordered distance transformation is evaluated by the standard Gaussian quadrature and the nearly hypersingular part still needs to be transformed into an analytical form. In this paper, the distance transformation is performed by four steps in case the source point coincides with the projection point or five steps otherwise. For each step, new transformation is proposed based on the approximate distance function, so that all steps can finally be unified into a uniform formation. With the new formulation, the nearly hypersingular integral can be dealt with directly and the near singularity separation and the cumbersome analytical deductions related to a specific fundamental solution are avoided. Numerical examples and comparisons with the existing methods on straight line elements and curved elements demonstrate that our method is accurate and effective.     

Numerical solution of two-dimensional mixed problems with variable coefficients by the boundary-domain integral and integro-differential equation methods

This paper presents a formulation of the boundary-domain integral equation (BDIE) and the boundary-domain integro-differential equation (BDIDE) methods for the numerical solution of two-dimensional mixed boundary-value problems (BVP) for a second-order linear elliptic partial differential equation (PDE) with variable coefficients. The methods use a specially constructed parametrix (Levi function) to reduce the BVP to a BDIE or BDIDE. The numerical formulation of the BDIDE employs an approximation for the boundary fluxes in terms of the potential function within the domain cells; therefore, the solution is fully described in terms of the variation of the potential function along the boundary and domain. Linear basis functions localised on triangular elements and standard quadrature rules are used for the calculation of boundary integrals. For the domain integrals, we have implemented Gaussian quadrature rules for two dimensions with Duffy transformation, by mapping the triangles into squares and eliminating the weak singularity in the process. Numerical examples are presented for several simple problems with square and circular domains, for which exact solutions are available. It is shown that the present method produces accurate results even with coarse meshes. The numerical results also show that high rates of convergence are obtained with mesh refinement.     

A multi-scale modeling of surface effect via the modified boundary Cauchy–Born model

In this paper, a new multi-scale approach is presented based on the modified boundary Cauchy–Born (MBCB) technique to model the surface effects of nano-structures. The salient point of the MBCB model is the definition of radial quadrature used in the surface elements which is an indicator of material behavior. The characteristics of quadrature are derived by interpolating data from atoms laid in a circular support around the quadrature, in a least-square scene. The total-Lagrangian formulation is derived for the equivalent continua by employing the Cauchy–Born hypothesis for calculating the strain energy density function of the continua. The numerical results of the proposed method are compared with direct atomistic and finite element simulation results to indicate that the proposed technique provides promising results for modeling surface effects of nano-structures.     

Some benchmark problems and basic ideas on the accuracy of boundary element analysis

Taken the linear elasticity problems as examples, some benchmark problems are presented to investigate the influence of calculation error and discretization error on the accuracy of boundary element analysis. For the conventional boundary element analysis based on singular kernel function of fundamental solution and using Gaussian elimination method, the main calculation error comes from the integration of kernel and shape function product on each element. Based on some benchmark problems of “simple problem” without discretization error, it can be observed that sometimes a large number of integration points in Gaussian quadrature should be adopted. To guarantee the integration accuracy reliably, an improved adaptive Gaussian quadrature approach is presented and verified. The main error of boundary element analysis is the discretization error, provided the calculation error has been reduced effectively. Based on some benchmark problems, it can be observed that for the bending problems both the constant and linear element are not efficient, the quadratic element with a reasonable refined mesh is required to guarantee the accuracy of boundary element analysis. An error indicator to guide the mesh refinement in the convergence test towards the converged accurate results based on the distribution of boundary effective stress is presented and verified.     

基于微分求积法及V-变换的大规模动力系统快速数值计算方法

针对大规模动力系统动态响应的数值计算,传统的微分求积法通常在时间域上逐步离散、整体求解,存在“维数灾”问题。在多级高阶时域微分求积法的基础上,提出了基于V-变换的大规模动力系统动态响应的快速数值计算方法。利用微分求积法的加权系数矩阵满足V-变换这一重要特性,将离散后的雅可比矩阵方程进行解耦分块,推导形成了多级分块递推计算方法。数值算例表明,即使采用相当于Newmark方法2s倍的步长,微分求积法的计算精度仍比Newmark方法要高出2~3个数量级。进一步对3个不同规模的算例系统进行了测试,结果表明：相对于传统的数值计算方法,多级分块递推计算方法可以获得较大的加速比,能够显著提高大规模动力系统动态响应的计算效率。     

微分求积模拟二维流体中流函数约束的施加方法研究

采用微分求积法数值求解流函数-涡度方程来模拟二维流体时会遇到流函数的超约束问题,即虽然流函数方程为二阶偏微分方程,但在每个固体边界上都存在两个约束条件:一个Dirichlet条件和一个Neumann条件。以二维驱动方腔流动为例,对该问题进行深入分析,进而提出一种新的超约束处理方法,即在边界涡度的计算中考虑Neumann条件,而仅将Dirichlet条件施加于流函数方程。数值结果显示该方法可行,且计算效率较高。同时给出前人提出的单层法和双层法进行比较。试算表明单层法对于网格数的奇偶性很敏感,不适于处理该问题。与双层法对比后发现:该方法计算精度较高,且由于回避了超约束问题而更加方便于使用。     

Equilibrium of an elastic finite cylinder: Filon's problem revisited

This paper deals with an analytical solution of the axisymmetric boundary-value problem of the theory of elasticity for a finite circular cylinder with free ends and arbitrary loaded curved surface. The object of this paper is to employ the method of superposition to obtain accurate values of the stress field near the boundaries. The classical Filon (1902) problem of uniformly distributed tangential load applied along two rings at the curved surface is addressed in full detail. The distribution of stresses along some typical sections of the cylinder are shown graphically.     

Meshless and wavelets based complex quadrature of highly oscillatory integrals and the integrals with stationary points

In this paper, we formulate and employ efficient and accurate methods for numerical evaluation of highly oscillatory integrals and integrals having stationary points. Two new approaches using radial basis function (RBF) and wavelets are discussed. The first approach is related to meshless method (MM) which is based on multiquadric (MQ) RBF, and is specially designed for integrands having oscillatory character. This approach stems from the Levin's method. In this procedure, the solution is obtained by solving the corresponding ODE or PDE instead of finding a numerical solution of the integration problem. In situations when the integrand has stationary points, MM fails to deliver. We opt for quadrature rules based on Haar wavelets and hybrid functions. The proposed methods are tested on a number of benchmark tests considered in available literature. The performance of the new methods is compared with the existing methods. Better accuracy of the proposed methods is reported for a variety of problems.     

A generalized dimension‐reduction method for multidimensional integration in stochastic mechanics

A new, generalized, multivariate dimension‐reduction method is presented for calculating statistical moments of the response of mechanical systems subject to uncertainties in loads, material properties, and geometry. The method involves an additive decomposition of an N‐dimensional response function into at most S‐dimensional functions, where S?N; an approximation of response moments by moments of input random variables; and a moment‐based quadrature rule for numerical integration. A new theorem is presented, which provides a convenient means to represent the Taylor series up to a specific dimension without involving any partial derivatives. A complete proof of the theorem is given using two lemmas, also proved in this paper. The proposed method requires neither the calculation of partial derivatives of response, as in commonly used Taylor expansion/perturbation methods, nor the inversion of random matrices, as in the Neumann expansion method. Eight numerical examples involving elementary mathematical functions and solid‐mechanics problems illustrate the proposed method. Results indicate that the multivariate dimension‐reduction method generates convergent solutions and provides more accurate estimates of statistical moments or multidimensional integration than existing methods, such as first‐ and second‐order Taylor expansion methods, statistically equivalent solutions, quasi‐Monte Carlo simulation, and the fully symmetric interpolatory rule. While the accuracy of the dimension‐reduction method is comparable to that of the fourth‐order Neumann expansion method, a comparison of CPU time suggests that the former is computationally far more efficient than the latter. Copyright © 2004 John Wiley & Sons, Ltd.     

A univariate dimension-reduction method for multi-dimensional integration in stochastic mechanics

This paper presents a new, univariate dimension-reduction method for calculating statistical moments of response of mechanical systems subject to uncertainties in loads, material properties, and geometry. The method involves an additive decomposition of a multi-dimensional response function into multiple one-dimensional functions, an approximation of response moments by moments of single random variables, and a moment-based quadrature rule for numerical integration. The resultant moment equations entail evaluating N number of one-dimensional integrals, which is substantially simpler and more efficient than performing one N-dimensional integration. The proposed method neither requires the calculation of partial derivatives of response, nor the inversion of random matrices, as compared with commonly used Taylor expansion/perturbation methods and Neumann expansion methods, respectively. Nine numerical examples involving elementary mathematical functions and solid-mechanics problems illustrate the proposed method. Results indicate that the univariate dimension-reduction method provides more accurate estimates of statistical moments or multidimensional integration than first- and second-order Taylor expansion methods, the second-order polynomial chaos expansion method, the second-order Neumann expansion method, statistically equivalent solutions, the quasi-Monte Carlo simulation, and the point estimate method. While the accuracy of the univariate dimension-reduction method is comparable to that of the fourth-order Neumann expansion, a comparison of CPU time suggests that the former is computationally far more efficient than the latter.     

Analytical integration formulae for linear isoparametric finite elements

Exact and approximate analytical expressions can be derived for integrals arising in finite element methods, employing isoparametric linear quadrilaterals in two space dimensions with bilinear basis functions. The formulae associated with rectangular elements, arbitrarily oriented in space, can be shown to be a special case. The proposed method provides considerable savings in computational effort, in comparison with a numerical method that employs Gaussian quadrature procedures. In addition, the method, when applied to a quadrilateral inscribable in a circle, can be shown to produce better accuracy than the associated (2 × 2) Gaussian quadrature formulae.     

Wave characteristics in gratings by linear superposition of retarded fields

We present a formalism for the wave characteristics in gratings and periodic dielectrics based on the linear superposition of retarded fields. The idea is based on the physical picture that an incident field affects the charges in the material forming the gratings and hence leads to oscillating current and charge densities, which in turn generate more fields via the retarded potential. A set of self-consistent equations for the electric field and current and charge densities is derived. Expressions for the electric field everywhere, including the reflected and transmitted fields, are derived. The formalism is then applied to the calculation of diffraction efficiency so as to illustrate its application and to establish its validity by comparing results with the rigorous coupled-wave method. We further generalize the formalism to include possible anisotropy and nonlinearity in the response of the material forming the grating.     

An XFEM method for modeling geometrically elaborate crack propagation in brittle materials

We present a method for simulating quasistatic crack propagation in 2‐D which combines the extended finite element method (XFEM) with a general algorithm for cutting triangulated domains, and introduce a simple yet general and flexible quadrature rule based on the same geometric algorithm. The combination of these methods gives several advantages. First, the cutting algorithm provides a flexible and systematic way of determining material connectivity, which is required by the XFEM enrichment functions. Also, our integration scheme is straightforward to implement and accurate, without requiring a triangulation that incorporates the new crack edges or the addition of new degrees of freedom to the system. The use of this cutting algorithm and integration rule allows for geometrically complicated domains and complex crack patterns. Copyright © 2011 John Wiley & Sons, Ltd.     

The evaluation of improper integrals encountered in the use of conformal transformation

Methods are described and reviewed for the accurate numerical evaluation of improper integrals encountered in conformal transformation solutions involving boundaries of relatively complicated shape. Four methods are reviewed for the solution of integrals containing end-point singularities. Two new methods are discussed for the solution of integrals containing both end point singularities and simple poles within the range of integration. One method uses a combination of a simple recursive formula and the coefficients of a Chebyshev series and a second method involves subtracting out the singularity and the use of Gauss–Jacobi quadrature. Both methods can give results of high accuracy and an upper limit of the error is readily found. A numerical example is taken which is typical of the application to practical problems and this brings out a comparison of the two methods.     

Three-dimensional tomographic reconstruction of an absorptive perturbation with diffuse photon density waves

A three-dimensional tomographic reconstruction algorithm for an absorptive perturbation in tissue is derived. The input consists of multiple two-dimensional projected views of tissue that is backilluminated with diffuse photon density waves. The algorithm is based on a generalization of the projection-slice theorem and consists of depth estimation, image deconvolution, filtering, and backprojection. The formalism provides estimates of the number of views necessary to achieve a given spatial resolution in the reconstruction. The algorithm is demonstrated with data simulated to mimic the absorption of a contrast agent in human tissue. The effects of noise and uncertainties in the depth estimate are explored.