一种新型裂尖加强函数的显式动态扩展有限元法

基于扩展有限元方法提出了一种新的裂尖加强函数,与传统三角函数基表征的加强函数相比,该裂尖加强函数通过组合传统的函数基,继承了传统附加函数的特性,同时使得结点的奇异附加自由度减少为2个,减少了总体劲度矩阵的规模,提高了计算效率。通过集中质量矩阵考虑结构的惯性效应,使用显式时间积分方法计算了含裂纹结构的瞬间受载问题,并应用相互作用积分得到裂尖端点处的动态应力强度因子。通过相关算例的对比分析,验证了所提出的裂尖加强函数的有效性,同时表明采用显式时间积分方法进行结构动态响应分析的可行性及准确性。     

Lumped mass finite element implementation of continuum theories with micro‐inertia

Continuum theories can be equipped with additional inertia terms to make them capable of capturing wave dispersion effects observed in micro‐structured materials. Such terms, often called micro‐inertia, are convenient and straightforward extensions of classical continuum theories. Furthermore, the critical time step is usually increased via the inclusion of micro‐inertia. However, the drawback exists that standard finite element discretisation leads to mass matrices that cannot be lumped without losing the micro‐inertia terms. In this paper, we will develop a solution algorithm based on Neumann expansions by which this disadvantage is avoided altogether. The micro‐inertia terms are translated into modifications of the residual force vector, so that the system matrix is the usual lumped mass matrix and all advantages of explicit time integration are maintained. The numerical stability of the algorithm and its effect on the dispersive properties of the model are studied in detail. Numerical examples are used to illustrate the various aspects of the algorithm. Copyright © 2013 John Wiley & Sons, Ltd.     

A fast weakly intrusive multiscale method in explicit dynamics

This paper presents new developments on a weakly intrusive approach for the simplified implementation of space and time multiscale methods within an explicit dynamics software. The ‘substitution’ method proposed in previous works allows to take advantage of a global coarse model, typically used in an industrial context, running separate, refined in space and in time, local analyses only where needed. The proposed technique is iterative, but the explicit character of the method allows to perform the global computation only once per global time step, while a repeated solution is required for the small local problems only. Nevertheless, a desirable goal is to reach convergence with a reduced number of iterations. To this purpose, we propose here a new iterative algorithm based on an improved interface inertia operator. The new operator exploits a combined property of velocity Hermite time interpolation on the interface and of the central difference integration scheme, allowing the consistent upscaling of interface inertia contributions from the lower scale. This property is exploited to construct an improved mass matrix operator for the interface coupling, allowing to significantly enhance the convergence rate. The efficiency and robustness of the procedure are demonstrated through several examples of growing complexity. Copyright © 2014 John Wiley & Sons, Ltd.     

On the numerical stability and mass‐lumping schemes for explicit enriched meshfree methods

Meshfree methods (MMs) such as the element free Galerkin (EFG)method have gained popularity because of some advantages over other numerical methods such as the finite element method (FEM). A group of problems that have attracted a great deal of attention from the EFG method community includes the treatment of large deformations and dealing with strong discontinuities such as cracks. One efficient solution to model cracks is adding special enrichment functions to the standard shape functions such as extended FEM, within the FEM context, and the cracking particles method, based on EFG method. It is well known that explicit time integration in dynamic applications is conditionally stable. Furthermore, in enriched methods, the critical time step may tend to very small values leading to computationally expensive simulations. In this work, we study the stability of enriched MMs and propose two mass‐lumping strategies. Then we show that the critical time step for enriched MMs based on lumped mass matrices is of the same order as the critical time step of MMs without enrichment. Moreover, we show that, in contrast to extended FEM, even with a consistent mass matrix, the critical time step does not vanish even when the crack directly crosses a node. Copyright © 2011 John Wiley & Sons, Ltd.     

结构体系动力方程求解的显式积分格式的能耗特性

针对作者提出的结构体系动力方程求解的一种显式积分格式,探讨其数值计算能耗特性即算法阻尼特性,导出了其算法阻尼值随体系的物理阻尼和结构体系振动频率值变化的关系,并进一步讨论了该积分格式用于无限介质波动的数值模拟中抑制或消除透射边界引起的计算高频失稳问题,给出了该积分格式与中心差分格式的对比分析算例。     

Linear finite elements with orthogonal discontinuous basis functions for explicit earthquake ground motion modeling

The dynamic explicit finite element method is commonly used in earthquake ground motion modeling. In this method, the element mass matrix is approximately lumped, which may lead to numerical dispersion. On the other hand, the orthogonal finite element method, based on orthogonal polynomial basis functions, naturally derives a lumped diagonal mass matrix and can be applied to dynamic explicit finite element analysis. In this paper, we propose finite elements based on orthogonal discontinuous basis functions, the element mass matrices of which are lumped without approximation. Orthogonal discontinuous basis functions are used to improve the accuracy and reduce the numerical dispersion in earthquake ground motion modeling. We present a detailed formulation of the 4‐node tetrahedral and 8‐node hexahedral elements. The relationship between the proposed finite elements and conventional finite elements is investigated, and the solutions obtained from the conventional explicit finite element method are compared with analytical solutions to verify the numerical dispersion caused by the lumping approximation. Comparison of solutions obtained with the proposed finite elements to analytical solutions demonstrates the usefulness of the technique. Examples are also presented to illustrate the effectiveness of the proposed method in earthquake ground motion modeling in the actual three‐dimensional crust structure. Copyright © 2010 John Wiley & Sons, Ltd.     

Explicit finite element artificial boundary scheme for transient scalar waves in two‐dimensional unbounded waveguide

To simulate the transient scalar wave propagation in a two‐dimensional unbounded waveguide, an explicit finite element artificial boundary scheme is proposed, which couples the standard dynamic finite element method for complex near field and a high‐order accurate artificial boundary condition (ABC) for simple far field. An exact dynamic‐stiffness ABC that is global in space and time is constructed. A temporal localization method is developed, which consists of the rational function approximation in the frequency domain and the auxiliary variable realization into time domain. This method is applied to the dynamic‐stiffness ABC to result in a high‐order accurate ABC that is local in time but global in space. By discretizing the high‐order accurate ABC along artificial boundary and coupling the result with the standard lumped‐mass finite element equation of near field, a coupled dynamic equation is obtained, which is a symmetric system of purely second‐order ordinary differential equations in time with the diagonal mass and non‐diagonal damping matrices. A new explicit time integration algorithm in structural dynamics is used to solve this equation. Numerical examples are given to demonstrate the effectiveness of the proposed scheme. Copyright © 2011 John Wiley & Sons, Ltd.     

Efficient formulation for dynamics of corotational 2D beams

The corotational method is an attractive approach to derive non-linear finite beam elements. In a number of papers, this method was employed to investigate the non-linear dynamic analysis of 2D beams. However, most of the approaches found in the literature adopted either a lumped mass matrix or linear local interpolations to derive the inertia terms (which gives the classical linear and constant Timoshenko mass matrix), although local cubic interpolations were used to derive the elastic force vector and the tangent stiffness matrix. In this paper, a new corotational formulation for dynamic nonlinear analysis is presented. Cubic interpolations are used to derive both the inertia and elastic terms. Numerical examples show that the proposed approach is more efficient than using lumped or Timoshenko mass matrices.     

Numerical recipes for elastodynamic virtual element methods with explicit time integration

We present a general framework to solve elastodynamic problems by means of the virtual element method (VEM) with explicit time integration. In particular, the VEM is extended to analyze nearly incompressible solids using the B-bar method. We show that, to establish a B-bar formulation in the VEM setting, one simply needs to modify the stability term to stabilize only the deviatoric part of the stiffness matrix, which requires no additional computational effort. Convergence of the numerical solution is addressed in relation to stability, mass lumping scheme, element size, and distortion of arbitrary elements, either convex or nonconvex. For the estimation of the critical time step, two approaches are presented, ie, the maximum eigenvalue of a system of mass and stiffness matrices and an effective element length. Computational results demonstrate that small edges on convex polygonal elements do not significantly affect the critical time step, whereas convergence of the VEM solution is observed regardless of the stability term and the element shape in both two and three dimensions. This extensive investigation provides numerical recipes for elastodynamic VEMs with explicit time integration and related problems.     

采用粘弹性人工边界时显式算法稳定性的改善研究

在显式动力计算中引入粘弹性人工边界时,受人工边界刚度和阻尼等因素影响,整体模型的数值积分稳定性将变得更为严格,这在一定程度上限制了粘弹性人工边界在大规模显式动力计算中的应用。该文基于对采用粘弹性人工边界的显式时域逐步积分算法稳定性条件的分析及其影响因素的研究,提出通过对人工边界附加集中质量来改善其数值积分稳定性的方法,发展了稳定性更优的改进粘弹性人工边界。为确定合理的人工边界质量值,利用基于局部子系统的稳定性分析方法推导得到改进粘弹性人工边界的稳定性条件,通过比较分析给出人工边界质量参数的建议值。采用该建议值后,粘弹性人工边界区的数值积分稳定性条件优于内部计算域的稳定性条件,整体计算模型的稳定性由内部计算域控制,此时可以用常规的稳定性判别准则来确定临界时间积分步长。数值算例表明,该文提出的粘弹性人工边界数值积分稳定性改善方法在提高计算效率的同时保持原人工边界的计算精度,具有较强的实用性。     

Improvement of PUFEM for the numerical solution of high‐frequency elastic wave scattering on unstructured triangular mesh grids

The purpose of this paper, which builds on previous work (Int. J. Numer. Meth. Engng 2009; 77 :1646–1669), is to improve a numerical scheme based on the partition of unity finite element method (PUFEM) for the solution of the time harmonic elastic wave equations. The approach consists to approximate the displacement field by the standard finite element shape functions, enriched locally by superimposing pressure (P) and shear (S) plane waves. The aim is to accurately model two‐dimensional elastic wave problems on relatively coarse mesh grids, capable of containing many wavelengths per nodal spacing, for wide ranges of frequencies. This allows us to relax the traditional requirement of about 10 nodal points per S wavelength. In this work, an exact integration scheme for the linear triangular finite element is developed to evaluate the oscillatory integrals arising from the use of the PUFEM. The main contribution here consists in developing an explicit closed‐form solution for two‐dimensional wave‐based integrals, when the phase variation is linear in the local coordinate element system. The evaluation of the element mass matrix is performed from appropriate edge integrals. All other element matrices, obtained by adequate splitting of the element stress tensor matrix, are simply deduced from the element mass matrix entries. The results show clearly that the proposed integration scheme evaluates accurately the entries of the global matrix with drastic reduction of the computational time. Numerical tests dealing with the scattering of S elastic plane waves by a circular rigid body show that, for the same discretization level, it is possible to improve the accuracy by using large elements associated with high numbers of approximating plane waves rather than using small elements with less plane waves. However, this increases the conditioning and the fill‐in of the global matrix. At high frequency, it is even possible to push the number of degrees of freedom per S wavelength under 2 and still achieve good accuracy. Finally, some remarks on the choice of the numbers of P and S plane waves leading to better accuracy and conditioning are discussed. Copyright © 2010 John Wiley & Sons, Ltd.     

A general mass lumping scheme for the variants of the extended finite element method

A new strategy for the mass matrix lumping of enriched elements for explicit transient analysis is presented. It is shown that to satisfy the kinetic energy conservation, the use of zero or negative masses for enriched degrees of freedom of lumped mass matrix may be necessary. For a completely cracked element, by lumping the mass of each side of the interface into the finite element nodes located at the same side and assigning zero masses to the enriched degrees of freedom, the kinetic energy for rigid body translations is conserved without transferring spurious energy across the interface. The time integration is performed by adopting an explicit-implicit technique, where the regular and enriched degrees of freedom are treated explicitly and implicitly, respectively. The proposed method can be viewed as a general mass lumping scheme for the variants of the extended finite element methods because it can be used irrespective of the enrichment method. It also preserves the optimal critical time step of an intact finite element by treating the enriched degrees of freedom implicitly. The accuracy and efficiency of the proposed mass matrix are validated with several benchmark examples.     

Finite element transient dynamic analysis of isotropic and fibre reinforced composite plates using a higher-order theory

A higher-order shear deformable C° continuous finite element is developed and employed to investigate the transient response of isotropic, orthotropic and layered anisotropic composite plates. The governing ordinary linear differential equations are integrated using the central difference explicit time integration scheme. A special mass matrix diagonalization scheme is adopted which conserves the total mass of the element and includes the effects due to rotary inertia terms. Numerical results for deflections and stresses are presented for rectangular plates under various boundary conditions and loadings. The parametric effects of the time step, finite element mesh, lamination scheme and orthotropy on the transient response are investigated. The numerical results are compared with those available in the literature, and with the results obtained by solving the same problems using the Mindlin plate element.     

Temporal‐spatial dispersion and stability analysis of finite element method in explicit elastodynamics

This work presents the temporal‐spatial (full) dispersion and stability analysis of plane square linear and biquadratic serendipity finite elements in explicit numerical solution of transient elastodynamic problems. Here, the central difference method, as an explicit time integrator, is exploited. The paper complements and extends the previous work on spatial/grid dispersion analysis of plane square biquadratic serendipity finite elements. We report on a computational strategy for temporal‐spatial dispersion relationships, where eigenfrequencies from grid/spatial dispersion analysis are adjusted to comply with the time integration method. Besides that, an ‘optimal’ lumped mass matrix for the studied finite element types is proposed and investigated. Based on the temporal‐spatial dispersion and stability analysis, relationships suggesting the ‘proper’ choice of mesh size and time step size from knowledge of the loading spectrum are presented. Copyright © 2015 John Wiley & Sons, Ltd.     

A faster and accurate explicit algorithm for quasi‐harmonic dynamic problems

It is known that the explicit time integration is conditionally stable. The very small time step leads to increase of computational time dramatically. In this paper, a mass‐redistributed method is formulated in different numerical schemes to simulate transient quasi‐harmonic problems. The essential idea of the mass‐redistributed method is to shift the integration points away from the Gauss locations in the computation of mass matrix for achieving a much larger stable time increment in the explicit method. For the first time, it is found that the stability of explicit method in transient quasi‐harmonic problems is proportional to the softened effect of discretized model with mass‐redistributed method. With adjustment of integration points in the mass matrix, the stability of transient models is improved significantly. Numerical experiments including 1D, 2D and 3D problems with regular and irregular mesh have demonstrated the superior performance of the proposed mass‐redistributed method with the combination of smoothed finite element method in terms of accuracy as well as stability. Copyright © 2016 John Wiley & Sons, Ltd.     

基于有限质点法的多尺度精细化分析

多尺度作为一种精细化分析的建模手段,能有效平衡结构分析中的计算效率和精度。有限质点法是一种能准确分析结构非线性行为的方法,在多种复杂行为分析中得到应用。该文利用有限质点法以质点为基本元素和显式积分的特点,基于梁、壳等低维单元的平截面假定,将多尺度连接处的质点分为主质点和从质点,集成从质点的质量、质量惯性矩阵、力、力矩等物理量至主质点,求解主质点运动方程后,由位移约束条件求得从质点位移,从而实现不同维度单元的多尺度连接。算例表明该多尺度方法对梁-壳、梁-固体及壳-固体的连接是有效的,在几何非线性及动力问题中具有良好的精度及稳定性,适合于结构复杂行为分析。     

有阻尼体系动力分析的一种显式差分法

本文利用拉格朗日型的二次插值函数近似位移反应，建立了速度、加速度的差商近似公式，进而推得集中质量阻尼体系动力分析的一种显式差分法。该显式差分格式具有二阶计算精度，与目前常用的有阻尼体系动力求解的几种格式相比，具有明显减少计算工作量的优点。     

Free vibration analysis of arches using curved beam elements

The natural frequencies and mode shapes for the radial (in‐plane) bending vibrations of the uniform circular arches were investigated by means of the finite arch (curved beam) elements. Instead of the complicated explicit shape functions of the arch element given by the existing literature, the simple implicit shape functions associated with the tangential, radial (or normal) and rotational displacements of the arch element were derived and presented in matrix form. Based on the relationship between the nodal forces and the nodal displacements of a two‐node six‐degree‐of‐freedom arch element, the elemental stiffness matrix was derived, and based on the equation of kinetic energy and the implicit shape functions of an arch element the elemental consistent mass matrix with rotary inertia effect considered was obtained. Assembly of the foregoing elemental property matrices yields the overall stiffness and mass matrices of the complete curved beam. The standard techniques were used to determine the natural frequencies and mode shapes for the curved beam with various boundary conditions and subtended angles. In addition to the typical circular arches with constant curvatures, a hybrid beam constructed by using an arch segment connected with a straight beam segment at each of its two ends was also studied. For simplicity, a lumped mass model for the arch element was also presented. All numerical results were compared with the existing literature or those obtained from the finite element method based on the conventional straight beam element and good agreements were achieved. Copyright © 2003 John Wiley & Sons, Ltd.     

Effects of time integration schemes on discontinuous deformation simulations using the numerical manifold method

Numerical stability by using certain time integration scheme is a critical issue for accurate simulation of discontinuous deformations of solids. To investigate the effects of the time integration schemes on the numerical stability of the numerical manifold method, the implicit time integration schemes, ie, the Newmark, the HHT‐α, and the WBZ‐α methods, and the explicit time integration algorithms, ie, the central difference, the Zhai's, and Chung‐Lee methods, are implemented. Their performance is examined by conducting transient response analysis of an elastic strip subjected to constant loading, impact analysis of an elastic rod with an initial velocity, and excavation analysis of jointed rock masses, respectively. Parametric studies using different time steps are conducted for different time integration algorithms, and the convergence efficiency of the open‐close iterations for the contact problems is also investigated. It is proved that the Hilber‐Hughes‐Taylor‐α (HHT‐α), Wood‐Bossak‐Zienkiewicz‐α (WBZ‐α), Zhai's, and Chung‐Lee methods are more attractive in solving discontinuous deformation problems involving nonlinear contacts. It is also found that the examined explicit algorithms showed higher computational efficiency compared to those implicit algorithms within acceptable computational accuracy.     

Inelastic transient dynamic analysis of Reissner–Mindlin plates by the D/BEM

A direct domain/boundary element method (D/BEM) for dynamic analysis of elastoplastic Reissner–Mindlin plates in bending is developed. Thus, effects of shear deformation and rotatory inertia are included in the formulation. The method employs the elastostatic fundamental solution of the problem resulting in both boundary and domain integrals due to inertia and inelasticity. Thus, a boundary as well as a domain space discretization by means of quadratic boundary and interior elements is utilized. By using an explicit time‐integration scheme employed on the incremental form of the matrix equation of motion, the history of the plate dynamic response can be obtained. Numerical results for the forced vibration of elastoplastic Reissner–Mindlin plates with smooth boundaries subjected to impulsive loading are presented for illustrating the proposed method and demonstrating its merits. Copyright © 2000 John Wiley & Sons, Ltd.