Improved numerical integration of thick shell finite elements

A quadratic thick shell element derived from a three-dimensional isoparametric element was first introduced by Ahmad and co-workers in 1968. This element was noted, however, to be relatively inefficient in representing bending deformations in thin shell or thin plate applications. The present paper outlines a selective integration scheme for evaluating the stiffness matrix of the element, in which each component of the strain energy is evaluated separately using a different Gaussian integration grid for each contribution. By this procedure, the excessive bending stiffness of the element, which results from the use of me quadratic interpolation functions, is avoided. The improved performance of this element, as compared with the original thick shell element, is demonstrated by analyses of a variety of thin and thick shell problems.

1 Editors' note: A similar development was outlined by O. C. Zienkiewicz and co-workers in lnt. J. num. Meth. Engng, 3 , 275–290 (1971). Some important details differ between the two papers which are thus complementary.

     

Analysis of static non-linear response by explicit time integration

In general, a system of linear (or non-linear) algebraic equations is solved on an analogue computer by integrating an appropriately defined system of associated first-order differential equations, the steady-state solution of which is the desired solution of algebraic system. This approach usually works very well so long as the associated dynamical system is stable. The idea of using the same approach numerically is also not altogether new. One can, ‘analogously’, construct a dynamical system associated with a given system of non-linear algebraic equations and solve it numerically by any of a number of explicit time integration schemes. In fact, similar ideas provide the basis for certain ‘dynamic relaxation’ methods, incremental loading methods and the widely acclaimed methods of invariant imbedding.     

Transient response of pretwisted delaminated stiffened shell under low velocity impact

The paper presents the study of low velocity impact response of delaminated composite stiffened shell with pretwist employing finite element method for different combination of stiffeners. An eight noded isoparametric shell element along with a three noded isoparametric beam element are employed to model the shell and the stiffener, respectively. The modified Hertzian contact law is considered to compute the contact force, while the Newmark's time integration algorithm is used to solve the time dependent equations of both impactor and shell. The multipoint constraint algorithm is used to model delamination. Finally, the parametric studies are reported.     

Sensitivity of inelastic response to numerical integration of strain energy

The exact solution to the quasi-static, inelastic response of a cantilever beam of rectangular cross-section subjected to a bending moment at the tip is obtained. The material of the beam is assumed to be linearly elastic-linearly strain-hardening. This solution is then compared with three different numerical solutions of the same problem obtained by minimizing the total potential energy using Gaussian quadratures of two different orders and a Newton-Cotes scheme for integrating the strain energy of deformation. Significant differences between the exact dissipative strain energy and its numerical counterpart are emphasized. The consequence of this on the non-linear transient responses of a beam with solid cross-section and that of a thin-walled beam on elastic supports under impulsive loads are examined.     

Degenerated shell element with composite implicit time integration scheme for geometric nonlinear analysis

A degenerated shell element with composite implicit time integration scheme is developed in the present paper to solve the geometric nonlinear large deformation and dynamics problems of shell structures. The degenerated shell element is established based on the eight‐node solid element, where the nodal forces, mass matrices, and stiffness matrices are firstly obtained upon virtual velocity principle and then translated to the shell element. The strain field is modified based on the mixed interpolation of tensorial components method to eliminate the shear locking, and the constitutive relation is modified to satisfy the shell assumptions. A simple and practical computational method for nonlinear dynamic response is developed by embedding the composite implicit time integration scheme into the degenerated shell element, where the composite scheme combines the trapezoidal rule with the three‐point backward Euler method. The developed approach can not only keep the momentum and energy conservation and decay the high frequency modes but also lead to a symmetrical stiffness matrix. Numerical results show that the developed degenerated shell element with the composite implicit time integration scheme is capable of solving the geometric nonlinear large deformation and dynamics problems of the shell structures with momentum and energy conservation and/or decay. Copyright © 2015 John Wiley & Sons, Ltd.     

A-stable two-step time integration methods with controllable numerical dissipation for structural dynamics

A comprehensive study of A-stable linear two-step time integration methods for structural dynamics analysis is presented in this paper. An optimal A-stable linear two-step (OALTS) time integration method is revealed with desirable performance on low-frequency accuracy and high-frequency numerical dissipation properties. The OALTS time integration method is implemented in a direct integration manner for the second-order equations of structural dynamics; is implicit, A-stable, and second-order accurate in displacement, velocity, and acceleration, simultaneously; is easily started; and is numerical dissipation controllable. The OALTS time integration method shows desirable performance on spectral radius distribution, dissipation and dispersion errors, and overshooting behavior, where the results of some typical algorithms in the literature are also compared. Benchmark examples with/without physical damping are performed to validate the accuracy, stability, and efficiency of the OALTS time integration method.     

Multidimensional interpolation and numerical integration by boundary element method

This paper presents a multidimensional interpolation method and a numerical integration for a bounded region using boundary integral equations and a polyharmonic function. In the method using B-spline, points must be assigned in a gridiron layout for two-dimensional cases. In the method presented in this paper, using the polyharmonic function, arbitrary points can be assigned instead of using a gridiron layout, making interpolation an easy process. This method requires the use of boundary geometry of the region and arbitrary internal points. Values at an arbitrary point and the integral value are calculated after solving the discretized boundary integral equations. Numerical integration is performed using this interpolation. In order to investigate the efficiency of this method, several examples are given.     

Structural reliability analysis by univariate decomposition and numerical integration

This paper presents a new and alternative univariate method for predicting component reliability of mechanical systems subject to random loads, material properties, and geometry. The method involves novel function decomposition at a most probable point that facilitates the univariate approximation of a general multivariate function in the rotated Gaussian space and one-dimensional integrations for calculating the failure probability. Based on linear and quadratic approximations of the univariate component function in the direction of the most probable point, two mathematical expressions of the failure probability have been derived. In both expressions, the proposed effort in evaluating the failure probability involves calculating conditional responses at a selected input determined by sample points and Gauss–Hermite integration points. Numerical results indicate that the proposed method provides accurate and computationally efficient estimates of the probability of failure.     

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.     

Generalized numerical integration of moments

Sampling points and weighting coefficients of the Gaussian type are presented for integrands typically encountered in axisymmetric finite elements. The proposed method is a generalization of Gaussian Integration of Moments for non-zero limits of integration. The method achieves one extra order of accuracy in the integration of polynomials as compared with the Gauss-Legendre method with the same number of sampling points. Although the locations of sampling points require the solution of non-linear equations, analytical solutions are presented for the cases of one and two sampling points. Special cases of these general expressions are shown to include both Gauss-Legendre integration corresponding to an integration range at a considerable distance from the axis of symmetry, and Fishman integration corresponding to an integration range whose lower limit lies on the axis of symmetry.     

A numerical integration approach by weight factors and its accuracy

A new parabolic approach for numerical integration using weight coefficients is proposed. Several basic approaches for integration have been analysed: linear (trapezoidal), parabolic (new form), Simpson's, Newton–Cotes and Gaussian. The weight coefficients are transformed into the same form for all methods. Simpson's and Newton's formulas (for n = 3) are obtained as special cases of the general parabolic formula. A simple procedure to estimate the accuracy of the integral for final intervals is given.     

Multidimensional numerical integration for meshless BEM

The conventional boundary element method (BEM) uses internal cells for the domain integrals, when nonlinear problems or problems with domain effects are solved. In the conventional BEM, however, the merit of the BEM, which is easy preparation of data, is lost. This paper presents numerical integration for a meshless BEM, which does not require internal cells. This method uses arbitrary internal points instead of internal cells. First, a multidimensional interpolation method for distribution in an arbitrary domain is shown using boundary integral equations. This method requires values on a boundary of a region and values at arbitrary internal points. In this paper, multidimensional numerical integration is proposed using the above multidimensional interpolation method. This integration is useful for inelastic problems and thermal stress analysis with arbitrary internal heat generation. This method is based on an improved multiple-reciprocity BEM (triple-reciprocity BEM) for heat conduction analysis with heat generation. In order to investigate the efficiency of this method, several numerical examples are given.     

非稳态雷暴冲击风场的瞬态数值模拟

冲击射流是雷暴冲击风场研究中最常用的模拟方法。大部分物理试验和数值模拟都是假定射流速度入口风速不随时间变化,而实际雷暴冲击风的下沉气流速度都是随时间连续变化的。在雷暴冲击风的全生命周期内一般会经历逐渐增大到最大值后再逐渐减小的过程。对于重要的工程结构抗风设计而言,得到雷暴冲击风全生命周期的风速时程信息十分有必要。基于冲击射流三维足尺模型模拟雷暴冲击风风场,在入口处引入一个更符合雷暴冲击风实际演化过程的衰减函数,采用大涡模拟数值分析获得了雷暴冲击风的非稳态风场,并得到其随时间衰减的瞬态演化过程。结果表明,该模拟方法可以较好地再现雷暴冲击风场的非稳态过程,为进一步讨论非稳态雷暴冲击风场中的结构风荷载特性奠定了基础。     

Transient propagation of anti-plane cracks opened by time varying loads

The problem of an anti-plane crack opened by a time varying concentrated load at the origin, and propagating self-similarly in an elastic body was treated. The analysis is based on Chaplygin's transformation and the solution is effected by using the conformal mapping technique. The problem was reduced to a mixed boundary value problem for an analytic function in the half-plane, which is solvable by the Keldysh-Sedov formula.

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Résumé On traite du problème de l'ouverture, puis de la propagation, d'une fissure antiplanaire dans un corps élastique, sous l'effet d'une charge concentrée à son origine, et caractérisée par des variations dans le temps. L'analyse est basée sur une transformation de Chaplygin, solutionnée en recourant à une technique de représentation conforme. Le problème a été réduit à un problème mixte de valeurs aux limites d'une fonction analytique dans le demi-plan, laquelle peut être résolue par une fonction de Keldish-Sedov.

     

A new family of time integration methods for heat conduction problems using numerical green’s functions

This paper is concerned with the formulation and numerical implementation of a new class of time integration schemes applied to linear heat conduction problems. The temperature field at any time level is calculated in terms of the numerical Green’s function matrix of the model problem by considering an analytical time integral equation. After spatial discretization by the finite element method, the Green’s function matrix which transfers solution from t to t + Δt is explicitly computed in nodal coordinates using efficient implicit and explicit Runge-Kutta methods. It is shown that the stability and the accuracy of the proposed method are highly improved when a sub-step procedure is used to calculate recursively the Green’s function matrix at the end of the first time step. As a result, with a suitable choice of the number of sub-steps, large time steps can be used without degenerating the numerical solution. Finally, the effectiveness of the present methodology is demonstrated by analyzing two numerical examples.     

多维非平稳随机地震响应分析的混合型精细时程积分

在一维虚拟激励法的基础上,通过激励功率谱的分解,建立多维非平稳地震激励作用下结构随机响应的虚拟激励法。根据演变随机激励的调制函数取不同形式,推导不同的特解精细积分格式,并混合应用这些格式于结构响应时变功率谱分析,可获得高效精确的数值结果。最后,通过算例讨论地震动各分量相关性对非对称结构地震响应的影响,得到一些有意义的结论。