A GPU‐accelerated nodal discontinuous Galerkin method with high‐order absorbing boundary conditions and corner/edge compatibility

Discontinuous Galerkin finite element schemes exhibit attractive features for accurate large‐scale wave‐propagation simulations on modern parallel architectures. For many applications, these schemes must be coupled with nonreflective boundary treatments to limit the size of the computational domain without losing accuracy or computational efficiency, which remains a challenging task. In this paper, we present a combination of a nodal discontinuous Galerkin method with high‐order absorbing boundary conditions for cuboidal computational domains. Compatibility conditions are derived for high‐order absorbing boundary conditions intersecting at the edges and the corners of a cuboidal domain. We propose a GPU implementation of the computational procedure, which results in a multidimensional solver with equations to be solved on 0D, 1D, 2D, and 3D spatial regions. Numerical results demonstrate both the accuracy and the computational efficiency of our approach.     

基于有限元与边界元耦合法的波浪力分析

利用有限元与边界元耦合法对三维无界区域中直立圆柱所受的波浪力进行进行计算,把整个求解区域分成内域或外域两部分,在内域采用有限元法,对外域采用边界元法,数值计算的结果与理论解吻合良好,表明该方法有效。     

城市轨道车辆车内噪声特性研究

采用有限元法结合边界元法研究按额定速度行驶的城市轨道客车车内噪声特性。车辆行驶工况下车身所受到的结构激励力以车底架上的载荷谱描述,采用有限元计算该激励下车体表面的振动速度,并采用边界元法预报相应的车内声场。最后,分析车身表面不同位置对声场最大声压级点的贡献量及其随频率和空间分布的规律。     

数值模拟在阴极保护中的应用进展

阴极保护是控制金属构件腐蚀的有效方法之一.随着阴极保护以及计算机技术的发展,数值模型在阴极保护体系设计和优化过程中得到了越来越广泛的应用.本文介绍了3种数值模型计算方法及有限元计算过程中边界条件的处理,综述了数值模型计算的发展现状以及面临的问题,阐述了有限元法和边界元法的优缺点.对比分析发现:边界元法可以使整个问题的维度降低一维,减少了计算量,提高精度,简化了建模过程.有限元法要求区域是有界的,可以对阴极保护体系进行三维分析,但计算量较大.边界元法适用于无限域、半无限域问题,有限元法的剖分灵活,更加适用于边界复杂以及求解域内电位场分布变化比较大的体系.在实际应用过程中,如何根据实际情况合理地采用有限元和边界元相结合的方法,发挥两种方法各自优点是阴极保护数值优化技术今后的一个发展方向.     

Weak-form element differential method for solving mechanics and heat conduction problems with abruptly changed boundary conditions

Element differential method (EDM), as a newly proposed numerical method, has been applied to solve many engineering problems because it has higher computational efficiency and it is more stable than other strong-form methods. However, due to the utilization of strong-form equations for all nodes, EDM become not so accurate when solving problems with abruptly changed boundary conditions. To overcome this weakness, in this article, the weak-form formulations are introduced to replace the original formulations of element internal nodes in EDM, which produce a new strong-weak-form method, named as weak-form element differential method (WEDM). WEDM has advantages in both the computational accuracy and the numerical stability when dealing with the abruptly changed boundary conditions. Moreover, it can even achieve higher accuracy than finite element method (FEM) in some cases. In this article, the computational accuracy of EDM, FEM, and WEDM are compared and analyzed. Meanwhile, several examples are performed to verify the robustness and efficiency of the proposed WEDM.     

A physically and geometrically nonlinear scaled‐boundary‐based finite element formulation for fracture in elastomers

This paper is devoted to the formulation of a plane scaled boundary finite element with initially constant thickness for physically and geometrically nonlinear material behavior. Special two‐dimensional element shape functions are derived by using the analytical displacement solution of the standard scaled boundary finite element method, which is originally based on linear material behavior and small strains. These 2D shape functions can be constructed for an arbitrary number of element nodes and allow to capture singularities (e.g., at a plane crack tip) analytically, without extensive mesh refinement. Mapping these proposed 2D shape functions to the 3D case, a formulation that is compatible with standard finite elements is obtained. The resulting physically and geometrically nonlinear scaled boundary finite element formulation is implemented into the framework of the finite element method for bounded plane domains with and without geometrical singularities. The numerical realization is shown in detail. To represent the physically and geometrically nonlinear material and structural behavior of elastomer specimens, the extended tube model and the Yeoh model are used. Numerical studies on the convergence behavior and comparisons with standard Q1P0 finite elements demonstrate the correct implementation and the advantages of the developed scaled boundary finite element. Copyright © 2014 John Wiley & Sons, Ltd.     

An h‐hierarchical adaptive procedure for the scaled boundary finite‐element method

The scaled boundary finite‐element method (a novel semi‐analytical method for solving linear partial differential equations) involves the solution of a quadratic eigenproblem, the computational expense of which rises rapidly as the number of degrees of freedom increases. Consequently, it is desirable to use the minimum number of degrees of freedom necessary to achieve the accuracy desired. Stress recovery and error estimation techniques for the method have recently been developed. This paper describes an h‐hierarchical adaptive procedure for the scaled boundary finite‐element method. To allow full advantage to be taken of the ability of the scaled boundary finite‐element method to model stress singularities at the scaling centre, and to avoid discretization of certain adjacent segments of the boundary, a sub‐structuring technique is used. The effectiveness of the procedure is demonstrated through a set of examples. The procedure is compared with a similar h‐hierarchical finite element procedure. Since the error estimators in both cases evaluate the energy norm of the stress error, the computational cost of solutions of similar overall accuracy can be compared directly. The examples include the first reported direct comparison of the computational efficiency of the scaled boundary finite‐element method and the finite element method. The scaled boundary finite‐element method is found to reduce the computational effort considerably. Copyright © 2002 John Wiley & Sons, Ltd.     

Use of higher‐order shape functions in the scaled boundary finite element method

The scaled boundary finite element method is a novel semi‐analytical technique, whose versatility, accuracy and efficiency are not only equal to, but potentially better than the finite element method and the boundary element method for certain problems. This paper investigates the possibility of using higher‐order polynomial functions for the shape functions. Two techniques for generating the higher‐order shape functions are investigated. In the first, the spectral element approach is used with Lagrange interpolation functions. In the second, hierarchical polynomial shape functions are employed to add new degrees of freedom into the domain without changing the existing ones, as in the p‐version of the finite element method. To check the accuracy of the proposed procedures, a plane strain problem for which an exact solution is available is employed. A more complex example involving three scaled boundary subdomains is also addressed. The rates of convergence of these examples under p‐refinement are compared with the corresponding rates of convergence achieved when uniform h‐refinement is used, allowing direct comparison of the computational cost of the two approaches. The results show that it is advantageous to use higher‐order elements, and that higher rates of convergence can be obtained using p‐refinement instead of h‐refinement. Copyright © 2005 John Wiley & Sons, Ltd.     

Forming tubular hexahedral screws—Process development by means of a combined finite element-boundary element approach

The aim of the present paper is twofold: first, to propose a new forming process that is capable of producing hexahedral heads with washers in tubular screws and, second, to present a numerical approach to solve the plastic deformation of the tubular preforms and the elastic deformation of the dies.The methodology draws from independent determination of mechanical properties and fabrication of prototypes with a forming tool designed and fabricated by the authors to the development of a direct boundary integral element formulation for solving the elastic deformation of the dies inside an existing in-house finite element computer program.The performance of the proposed forming process is assessed by experimentation and results and observations are explained in the light of the proposed numerical approach based on the combination of the finite element (FE) with the boundary element (BE) methods.Because project and design of tooling is often only dependent on boundary solutions for the elastic deformation of dies and for the contact between the workpiece and the dies, the proposed FE-BE approach seems as a cost-effective methodology that avoids discretization of dies by finite elements, reduces the overall size and improves the performance of the resulting computer models.     

Quasi-static analysis of viscoplastic plates by the boundary element method

A direct boundary element method (BEM) is developed for the determination of the time-dependent inelastic deflection of plates of arbitrary planform and under arbitrary boundary conditions to general lateral loading history. The governing differential equation is the nonhomogeneous biharmonic equation for the rate of small transverse deflection. The boundary integral formulation is derived by using a combination of the BEM and finite element methodology. The plate material is modelled as elastic-viscoplastic. Numerical examples for sample problems are presented to illustrate the method and to demonstrate its merits.     

Soil-structure interaction and wave propagation problems in 2D by a Duhamel integral based approach and the convolution quadrature method

In this paper, a new methodology for analyzing wave propagation problems, originally presented and checked by the authors for one-dimensional problems [18], is extended to plane strain elastodynamics. It is based on a Laplace domain boundary element formulation and Duhamel integrals in combination with the convolution quadrature method (CQM) [13], [14]. The CQM is a technique which approximates convolution integrals, in this case the Duhamel integrals, by a quadrature rule whose weights are determined by Laplace transformed fundamental solutions and a multi-step method. In order to investigate the accuracy and the stability of the proposed algorithm, some plane wave propagation and interaction problems are solved and the results are compared to analytical solutions and results from finite element calculations. Very good agreement is obtained. The results are very stable with respect to time step size. In the present work only multi-region boundary element analysis is discussed, but the presented technique can easily be extended to boundary element – finite element coupling as will be shown in subsequent publications.     

Uncertainty analysis of elastostatic problems incorporating a new hybrid stochastic-spectral finite element method

This article aims to present a combination of stochastic finite element and spectral finite element methods as a new numerical tool for uncertainty quantification. One of the well-established numerical methods for reliability analysis of engineering systems is the stochastic finite element method. In this article, a commonly used version of the stochastic finite element method is combined with the spectral finite element method. Furthermore, the spectral finite element method is a numerical method employing special orthogonal polynomials (e.g., Lobatto) and quadrature schemes (e.g., Gauss-Lobatto-Legendre), leading to suitable accuracy, and much less domain discretization with excellent convergence as well. The proposed method of this article is a hybrid method utilizing efficiencies of both methods for analysis of stochastically linear elastostatic problems. Moreover, a spectral finite element method is proposed for numerical solution of a Fredholm integral equation followed by the present method, to provide further efficiencies to accelerate stochastic computations. Numerical examples indicate the efficiency and accuracy of the proposed method.     

地震作用下水-轴对称柱体相互作用的子结构分析方法

针对水-轴对称柱体动力相互作用问题,提出了一种地震作用下水-结构相互作用的时域子结构分析方法。基于三维不可压缩水体的波动方程和边界条件,利用分离变量法将其转换为环向解析、竖向和径向数值的二维模型;基于比例边界有限元推导了截断边界处无限域水体的动力刚度方程,并将水体内域有限元方程和人工边界处的动水压力进行耦合,从而得到结构表面的动水压力方程;将轴对称柱体结构的有限元方程与动水压力方程耦合,从而得到水-轴对称柱体结构系统的时域有限元方程;数值算例验证该文提出的水-轴对称动力相互作用的子结构方法,结果表明:该文方法具有很高的精度和计算效率。通过对水中轴对称结构地震响应和自振频率的分析表明:地震动水压力对结构自振频率和动力响应的影响随水深的增加而增大。     

Stress recovery and error estimation for the scaled boundary finite‐element method

The scaled boundary finite‐element method is a novel semi‐analytical technique, combining the advantages of the finite element and the boundary element methods with unique properties of its own. This paper develops a stress recovery procedure based on a modal interpretation of the scaled boundary finite‐element method solution process, using the superconvergent patch recovery technique. The recovered stresses are superconvergent, and are used to calculate a recovery‐type error estimator. A key feature of the procedure is the compatibility of the error estimator with the standard recovery‐type finite element estimator, allowing the scaled boundary finite‐element method to be compared directly with the finite element method for the first time. A plane strain problem for which an exact solution is available is presented, both to establish the accuracy of the proposed procedures, and to demonstrate the effectiveness of the scaled boundary finite‐element method. The scaled boundary finite‐element estimator is shown to predict the true error more closely than the equivalent finite element error estimator. Unlike their finite element counterparts, the stress recovery and error estimation techniques work well with unbounded domains and stress singularities. Copyright © 2002 John Wiley & Sons, Ltd.     

Combination of Newmark method and natural element method for elastodynamics

The natural element method (NEM) is a meshless method. The trial and test functions of the NEM are constructed using natural neighbor interpolations which are based on the Voronoi tessellation of a set of nodes. The NEM interpolation is linear between adjacent nodes on the boundary of the convex hull, which makes imposition of essential boundary conditions easy to implement. We investigate the performance of the NEM combined with the Newmark method for problems of elastodynamics in this article. Applications are considered for a cantilever beam with different initial load conditions. The NEM numerical results are compared with the finite element method. NEM shows promise for these applications.     

A differential quadrature hierarchical finite element method and its applications to vibration and bending of Mindlin plates with curvilinear domains

A differential quadrature hierarchical finite element method (DQHFEM) is proposed by expressing the hierarchical finite element method matrices in similar form as in the differential quadrature finite element method and introducing interpolation basis on the boundary of hierarchical finite element method elements. The DQHFEM is similar as the fixed interface mode synthesis method but the DQHFEM does not need modal analysis. The DQHFEM with non‐uniform rational B‐splines elements were shown to accomplish similar destination as the isogeometric analysis. Three key points that determine the accuracy, efficiency and convergence of DQHFEM were addressed, namely, (1) the Gauss–Lobatto–Legendre points should be used as nodes, (2) the recursion formula should be used to compute high‐order orthogonal polynomials, and (3) the separation variable feature of the basis should be used to save computational cost. Numerical comparison and convergence studies of the DQHFEM were carried out by comparing the DQHFEM results for vibration and bending of Mindlin plates with available exact or highly accurate approximate results in literatures. The DQHFEM can present highly accurate results using only a few sampling points. Meanwhile, the order of the DQHFEM can be as high as needed for high‐frequency vibration analysis. Copyright © 2016 John Wiley & Sons, Ltd.     

正交各向异性平面问题边界元——边境元素法研究

在文献[1]中,本文作者研究了正交各向异性平面问题边界元素法的有关基本理论和计算公式,在上述工作的基础上,本文进一步研究各向异性平面问题边界邻域的应力分析。当采用边界元素法分析应力时,由于边界积分的奇异性,边界邻域应力的计算结果往往存在一定误差。为解决此问题,本文提出一个基于修正余能原理的所谓边境元素,包括四节点边境元素、八节点边境元素和三节点边境元素等。在边界元素法求解的基础上,进一步利用本文所述边境元素法,得到了非常满意的计算结果。      

BEM and FEM analysis of Signorini contact problems with friction

In this paper, we present a comparative study of the boundary element method (BEM) and the finite element method (FEM) for analysis of Signorini contact problems in elastostatics with Coulomb's friction law. Particularities of each method and comparison with the penalty method are discussed. Numerical examples are included to demonstrate the present formulations and to highlight its performance.     

矩形薄板弯曲问题的U变换-有限元法

该文扩展了U变换-有限元法分析弹性矩形薄板的范围。通过构造一个与简支、固支或二种边界条件组合的矩形板的等效系统,使刚度矩阵成为循环矩阵,采用U变换,成功解耦了有限元矩阵方程,使得有限元计算只须在一个单元上进行。给出了承受板中集中载荷和对边均布弯矩两种载荷形式下的板中挠度解析表达式。所得到的级数解不仅计算效率高,还能给出误差估计的显式表达式,能够直接掌控计算精度。算例中考察了几种不同边界条件下的计算结果,与已有理论结果的对比说明,该方法提高了计算的精度和效率。     

Parallel computing of high‐speed compressible flows using a node‐based finite‐element method

An efficient parallel computing method for high‐speed compressible flows is presented. The numerical analysis of flows with shocks requires very fine computational grids and grid generation requires a great deal of time. In the proposed method, all computational procedures, from the mesh generation to the solution of a system of equations, can be performed seamlessly in parallel in terms of nodes. Local finite‐element mesh is generated robustly around each node, even for severe boundary shapes such as cracks. The algorithm and the data structure of finite‐element calculation are based on nodes, and parallel computing is realized by dividing a system of equations by the row of the global coefficient matrix. The inter‐processor communication is minimized by renumbering the nodal identification number using ParMETIS. The numerical scheme for high‐speed compressible flows is based on the two‐step Taylor–Galerkin method. The proposed method is implemented on distributed memory systems, such as an Alpha PC cluster, and a parallel supercomputer, Hitachi SR8000. The performance of the method is illustrated by the computation of supersonic flows over a forward facing step. The numerical examples show that crisp shocks are effectively computed on multiprocessors at high efficiency. Copyright © 2003 John Wiley & Sons, Ltd.