Mechanical Quadrature Methods and Extrapolation Algorithms for Boundary Integral Equations with Linear Boundary Conditions in Elasticity

By potential theory, elastic problems with linear boundary conditions are converted into boundary integral equations (BIEs) with logarithmic and Cauchy singularity. In this paper, a mechanical quadrature method (MQMs) is presented to deal with the logarithmic and the Cauchy singularity simultaneously for solving the boundary integral equations. The convergence and stability are proved based on Anselone??s collective compact and asymptotical compact theory. Furthermore, an asymptotic expansion with odd powers of errors is presented, which possesses high accuracy order O(h ³). Using h ³?Richardson extrapolation algorithms (EAs), the accuracy order of the approximation can be greatly improved to O(h ⁵), and an a posteriori error estimate can be obtained for constructing a self-adaptive algorithm. The efficiency of the algorithm is illustrated by examples.     

Mechanical quadrature methods and extrapolation for solving nonlinear boundary Helmholtz integral equations

This paper presents mechanical quadrature methods (MQMs) for solving nonlinear boundary Helmholtz integral equations. The methods have high accuracy of order O(h ³) and low computation complexity. Moreover, the mechanical quadrature methods are simple without computing any singular integration. A nonlinear system is constructed by discretizing the nonlinear boundary integral equations. The stability and convergence of the system are proved based on an asymptotical compact theory and the Stepleman theorem. Using the h ³-Richardson extrapolation algorithms (EAs), the accuracy to the order of O(h ⁵) is improved. To slove the nonlinear system, the Newton iteration is discussed extensively by using the Ostrowski fixed point theorem. The efficiency of the algorithms is illustrated by numerical examples.     

Shape sensitivity analysis of elastic structures by multi-domain boundary element method

Based on the direct differentiation of boundary element equations to design variables, an algorithm of the boundary element method which is suitable for the shape sensitivity analysis of serial multi-domain elastic structures is built. By the algorithm, it is not necessary to form and solve the equations for the structure as a whole. Each domain can be solved separately. Two computational examples are presented to demonstrate the accuracy and versatility of the algorithm.This work has been partially supported by National Natural Science Foundation of China     

Structural–acoustic sensitivity analysis of radiated sound power using a finite element/ discontinuous fast multipole boundary element scheme

The complete interaction between the structural domain and the acoustic domain needs to be considered in many engineering problems, especially for the acoustic analysis concerning thin structures immersed in water. This study employs the finite element method to model the structural parts and the fast multipole boundary element method to model the exterior acoustic domain. Discontinuous higher‐order boundary elements are developed for the acoustic domain to achieve higher accuracy in the coupling analysis. Structural–acoustic design sensitivity analysis can provide insights into the effects of design variables on radiated acoustic performance and thus is important to the structural–acoustic design and optimization processes. This study is the first to formulate equations for sound power sensitivity on structural surfaces based on an adjoint operator approach and equations for sound power sensitivity on arbitrary closed surfaces around the radiator based on the direct differentiation approach. The design variables include fluid density, structural density, Poisson's ratio, Young's modulus, and structural shape/size. A numerical example is presented to demonstrate the accuracy and validity of the proposed algorithm. Different types of coupled continuous and discontinuous boundary elements with finite elements are used for the numerical solution, and the performances of the different types of finite element/continuous and discontinuous boundary element coupling are presented and compared in detail. Copyright © 2016 John Wiley & Sons, Ltd.     

正交各向异性弹性问题的规则化边界元法

本文致力于平面正交各向异性弹性问题的规则化边界元法研究,提出了新的规则化边界元法的理论和方法。对问题的基本解的特性进行了研究,确立基本解的积分恒等式,提出一种基本解的分解技术,在此基础上,结合转化域积分方程为边界积分方程的极限定理,建立了新颖的规则化边界积分方程。和现有方法比,本文不必将问题变换为各向同性的去处理,从而不含反演运算,也有别于Galerkin方法,无需计算重积分,因此所提方法不仅效率高,而且程序设计简单。特别是,所建方程可计算任何边界位移梯度,进而可计算任意边界应力,而不仅限于面力。数值实施时,采用二次单元和椭圆弧精确单元来描述边界几何,使用不连续插值逼近边界函数。数值算例表明,本文算法稳定、效率高,所取得的边界量数值结果与精确解相当接近。     

Variational boundary integral approach for asymmetric impinging jets of arbitrary two‐dimensional nozzle

We consider asymmetric impinging jets issuing from an arbitrary nozzle. The flow is assumed to be two‐dimensional, inviscid, incompressible, and irrotational. The impinging jet from an arbitrary nozzle has a couple of separated infinite free boundaries, which makes the problem hard to solve. We formulate this problem using the stream function represented with a specific single layer potential. This potential can be extended to the surrounding region of the jet flow, and this extension can be proved to be a bounded function. Using this fact, the formulation yields the boundary integral equations on the entire nozzle and free boundary. In addition, a boundary perturbation produces an extraordinary boundary integral equation for the boundary variation. Based on these variational boundary integral equations, we can provide an efficient algorithm that can treat with the asymmetric impinging jets having arbitrarily shaped nozzles. Particularly, the proposed algorithm uses the infinite computational domain instead of a truncated one. To show the convergence and accuracy of the numerical solution, we compare our solutions with the exact solutions of free jets. Numerical results on diverse impinging jets with nozzles of various shapes are also presented to demonstrate the applicability and reliability of the algorithm.     

Nonlocal elasticity defined by Eringen’s integral model: Introduction of a boundary layer method

In this paper we consider a nonlocal elasticity theory defined by Eringen’s integral model and introduce, for the first time, a boundary layer method by presenting the exponential basis functions (EBFs) for such a class of problems. The EBFs, playing the role of the fundamental solutions, are found so that they satisfy the governing equations on an unbounded domain. Some insight to the theory is given by showing that the EBFs satisfying the Navier equations in the classical elasticity theory also satisfy the governing equations in the nonlocal theory. Some additional EBFs are particularly obtained for the nonlocal theory. In order to use the EBFs on bounded domains, the effects of the boundary conditions are taken into account by truncating the kernel/attenuation function in the constitutive equations. This leads to some residuals in the governing equations which appear near the boundaries. A weighted residual approach is employed to minimize the residuals near the boundaries. The method presented in this paper has much in common with Trefftz methods especially when the influence area of the kernel function is much smaller than the main computational domain. Several one/two dimensional problems are solved to demonstrate the way in which the EBFs can be used through the proposed boundary layer method.     

A Quasilinear Based Procedure for Saturated/Unsaturated Water Movement in Soils

The quasilinear form of Richards equation for one-dimensional unsaturated flow in soils can be readily solved for a wide variety of conditions. However, it cannot explain saturated/unsaturated flow and the constant diffusivity assumption, used to linearise the transient quasilinear equation, can introduce significant error. This paper presents a quasi-analytical solution to transient saturated/unsaturated flow based on the quasilinear equation, with saturated flow explained by a transformed Darcy's equation. The procedure presented is based on the modified finite analytic method. With this approach, the problem domain is divided into elements, with the element equations being solutions to a constant coefficient form of the governing partial differential equation. While the element equations are based on a constant diffusivity assumption, transient diffusivity behaviour is incorporated by time stepping. Profile heterogeneity can be incorporated into the procedure by allowing flow properties to vary from element to element. Two procedures are presented for the temporal solution; a Laplace transform procedure and a finite difference scheme. An advantage of the Laplace transform procedure is the ability to incorporate transient boundary condition behaviour directly into the analytical solutions. The scheme is shown to work well for two different flow problems, for three soil types. The technique presented can yield results of high accuracy if the spatial discretisation is sufficient, or alternatively can produce approximate solutions with low computational overheads by using large sized elements. Error was shown to be stable, linearly related to element size.     

A boundary element method for anisotropic coupled thermoelasticity

Summary A boundary element formulation is presented for the solution of the equations of fully coupled thermoelasticity for materials of arbitrary degree of anisotropy. By employing the fundamental solutions of anisotropic elastostatics and stationary heat conduction, a system of equations with time-independent matrices is obtained. Since the fundamental solutions are uncoupled and time-independent, a domain integral remains in the representation formula which contains the time-dependence as well as the thermoelastic coupling. This domain integral is transformed to the boundary by means of the dual reciprocity method. By taking this approach, the use of dynamic fundamental solutions is avoided, which enables an efficient calculation of system matrices. In addition, the solution of transient processes as well as, free and forced vibration analysis becomes straightforward and can be carried out with standard time-stepping schemes and eigensystem solvers. Another important advantage of the present formulation is its versatility, since it includes a number of simplified thermoelastic theories, viz. the theory of thermal stresses, coupled and uncoupled quasi-static thermoelasticity, and stationary thermoelasticity. The accuracy of the new thermoelastic boundary element method is demonstrated by a number of example problems. Support by the Deutsche Forschungsgemeinschaft (DFG) of the Graduate Collegium Modelling and discretization methods for continua and fluids (GKKS) at the University of Stuttgart is gratefully acknowledged.     

二维非局部线弹性平面问题的辛分析

将二维非局部线弹性理论引入到Hamilton体系下,基于变分原理推导得出了二维线弹性理论的对偶方程和相应的边界条件.在分析验证对偶方程的准确性的基础上,该套方法被应用于二维弹性平面波问题的求解.将精细积分与扩展的W-W算法相结合在Hamilton体系下建立了求解平面Rayleigh波的数值算法.从推导到计算的保辛性确保了辛体系非局部理论与算法的准确性.通过对不同算例的数值计算,分析和对比了非局部理论方法与传统局部理论方法的差别,并进一步指出了该套算法的适用性和优势所在.     

各向异性位势平面问题的规则化边界元法

基于转化域方程为边界积分方程的极限定理及一个新颖的基本解分解技术, 建立间接变量规则化边界积分方程, 它有效地避免了奇异积分的直接计算. 与已有方法比,该方法不将问题变换为各向同性的问题去处理, 因而无需反演运算, 也有别于Galerkin方法, 无需计算重积分. 可计算任意边界位势梯度, 而不仅限于法向通量. 针对椭圆边界的边值问题, 提交一种精确单元来描述边界几何. 数值算例表明, 所提算法稳定且效率高, 所得数值结果与精确解吻合较好.      

Quasi-Green＇s function method for free vibration of simply-supported trapezoidal shallow spherical shell

The idea of quasi-Green＇s function method is clarified by considering a free vibration problem of the simply-supported trapezoidal shallow spherical shell. A quasi- Green＇s function is established by using the fundamental solution and boundary equation of the problem. This function satisfies the homogeneous boundary condition of the prob- lem. The mode shape differential equations of the free vibration problem of a simply- supported trapezoidal shallow spherical shell are reduced to two simultaneous Fredholm integral equations of the second kind by the Green formula. There are multiple choices for the normalized boundary equation. Based on a chosen normalized boundary equa- tion, a new normalized boundary equation can be established such that the irregularity of the kernel of integral equations is overcome. Finally, natural frequency is obtained by the condition that there exists a nontrivial solution to the numerically discrete algebraic equations derived from the integral equations. Numerical results show high accuracy of the quasi-Green＇s function method.     

模拟相变问题的精细积分边界元法

将精细积分边界元法和界面追踪法相结合求解相变问题。因为边界元法只需要将待求解空间域的边界离散,方便连续追踪移动界面位置和重构网格,所以边界元法适合应用于移动边界问题的模拟。首先,利用精细积分边界元法在固相区域和液相区域分别求解相应的瞬态热传导控制方程,从而求得温度场和边界热流密度。然后,根据固-液相变界面上的能量平衡方程,利用热流密度求得相变界面的移动速度,再采用界面追踪法预测移动相变界面的位置变化。最后,给出了几个数值算例,并通过与参考解的对比验证本文方法的准确性。     

General hyperbolic difference formulas for linear and quasilinear hyperbolic equations

The method of non-standard finite elements was used to develop multilevel difference schemes for linear and quasilinear hyperbolic equations with Dirichlet boundary conditions. A closed form equation of kth-order accuracy in space and time (O(Δt^k, Δx^k)) was developed for one-dimensional systems of linear hyperbolic equations with Dirichlet boundary conditions. This same equation is also applied to quasilinear systems. For the quasilinear systems a simple iteration technique was used to maintain the kth-order accuracy. Numerical results are presented for the linear and non-linear inviscid Burger's equation and a system of shallow water equations with Dirichlet boundary conditions.     

NOVEL REGULARIZED BOUNDARY INTEGRAL EQUATIONS FOR POTENTIAL PLANE PROBLEMS

The universal practices have been centralizing on the research of regulariza-tion to the direct boundary integal equations (DBIEs). The character is elimination of singularities by using the simple solutions. However, up to now the research of regular ization to the first kind integral equations for plane potential problems has never been found in previous literatures. The presentation is mainly devoted to the research on the regularization of the singular boundary integral equations with indirect unknowns. A novel view and idea is presented herein, in which the regularized boundary integral equations with indirect unknowns without including the Cauchy principal value (CPV) and Hadamard-finite-part (HFP) integrals are established for the plane potential problems. With some numerical results, it is shown that the better accuracy and higher efficiency, especially on the boundary, can be achieved by the present system.     

POD-identification reduced order model of linear transport equations for control purposes

Intrusive reduced order modeling techniques require access to the solver's discretization and solution algorithm, which are not available for most computational fluid dynamics codes. Therefore, a nonintrusive reduction method that identifies the system matrix of linear fluid dynamical problems with a least-squares technique is presented. The methodology is applied to the linear scalar transport convection-diffusion equation for a 2D square cavity problem with a heated lid. The (time-dependent) boundary conditions are enforced in the obtained reduced order model (ROM) with a penalty method. The results are compared and the accuracy of the ROMs is assessed against the full order solutions and it is shown that the ROM can be used for sensitivity analysis by controlling the nonhomogeneous Dirichlet boundary conditions.     

AN IMPLICIT SERIES PRECISE INTEGRATION ALGORITHM FOR STRUCTURAL NONLINEAR DYNAMIC EQUATIONS

Nonlinear dynamic equations can be solved accurately using a precise integration method. Some algorithms exist, but the inversion of a matrix must be calculated for these algorithms. If the inversion of the matrix doesn‘t exist or isn‘t stable, the precision and stability of the algorithms will be affected. An explicit series solution of the state equation has been presented. The solution avoids calculating the inversion of a matrix and its precision can be easily controlled. In this paper, an implicit series solution of nonlinear dynamic equations is presented.The algorithm is more precise and stable than the explicit series solution and isn‘t sensitive to the time-step. Finally, a numerical example is presented to demonstrate the effectiveness of the algorithm.     

功能梯度材料动态断裂力学的径向积分边界元法

采用径向积分边界元法分析功能梯度材料动态断裂力学问题. 该方法使用与弹性模量无关的弹性静力学开尔文基本解作为问题的基本解,在导出的边界-域积分方程中含有由材料的非均质性和惯性项引起的域积分,通过径向积分法将域积分转化为等效的边界积分,得到只含边界积分的纯边界积分方程;从而建立只需边界离散的无内部网格边界元算法. 采用候博特方法求解关于时间二阶导数的系统离散的常微分方程组. 最后通过数值算例验证本文方法的精度和有效性.      

THE MESHLESS VIRTUAL BOUNDARY METHOD AND ITS APPLICATIONS TO 2D ELASTICITY PROBLEMS

A novel numerical method for eliminating the singular integral and boundary effect is processed. In the proposed method, the virtual boundaries corresponding to the numbers of the true boundary arguments are chosen to be as simple as possible. An indirect radial basis function network (IRBFN) constructed by functions resulting from the indeterminate integral is used to construct the approaching virtual source functions distributed along the virtual boundaries. By using the linear superposition method, the governing equations presented in the boundaries integral equations (BIE) can be established while the fundamental solutions to the problems are introduced. The singular value decomposition (SVD) method is used to solve the governing equations since an optimal solution in the least squares sense to the system equations is available. In addition, no elements are required, and the boundary conditions can be imposed easily because of the Kronecker delta function properties of the approaching functions. Three classical 2D elasticity problems have been examined to verify the performance of the method proposed. The results show that this method has faster convergence and higher accuracy than the conventional boundary type numerical methods.     

一个求解浅水波方程的时域分段展开算法

发展了一种时域分段展开自适应方法求解一维非线性浅水波方程。通过时域分段展开,将一个非线性的时空耦合初边值问题转化为一系列的线性空间边值问题,并采用有限元方法递推求解;通过展开阶数的递进,实现了分段时域的自适应计算,当不同步长时可保持稳定的计算精度。研究结果表明,当步长较大而Heun’s法、四阶Runge-Kutta法不能得到合理结果时,本文算法仍能保证足够的计算精度。