Direct dynamic analysis of shells of revolution using high-precision finite elements

For the transient dynamic analysis of structural systems, the direct numerical integration of the equations of motion may be regarded as an alternative to the mode superposition method for linear problems and a necessity for nonlinear problems. When compared to a modal superposition solution, the direct integration approach is attractive in that the eigenvalue problem is avoided. Depending on the amount of information required from the dynamic analysis, e.g. frequencies, frequencies and mode shapes, and/or a complete time history, a direct integration scheme may prove to be more efficient than a modal superposition solution for some linear problems as well.The purpose of this study is to develop and demonstrate a direct integration algorithm which is compatible with an existing high-precision rotational shell finite element. Excellent comparative efficiency for static problems was achieved with this element by the incorporation of the exact geometry, the utilization of high-order interpolation polynomials and, yet, the retention of only a minimum number of nodal variables in the global formulation. Likewise, accurate and efficient results for the free vibration analysis of rotational shells were facilitated by the inclusion of a consistent mass matrix and the utilization of a rationally justified kinematic condensation procedure. The approach to the direct integration stage is strongly tempered by the established characteristics of this element which enable a given shell to be modeled accurately in the spatial domain with a comparatively coarse discretization.The equations of motion for a shell of revolution under conservative loading are derived from Hamilton's variational principle and specialized for the discretization of a rotational shell into curved shell elements. Degrees of freedom in excess of those required to establish minimum (C°) continuity at the nodal circles are eliminated through kinematic condensation. Some guidance as to the proper order of the polynominal approximations for a dynamic analysis is provided by earlier free vibration studies. Whereas the condensation is exact for static problems, it is only approximate for dynamic response and it was found that the accuracy of the eigenvalues obtained for the reduced problem decreases with increasing order of the condensed functions. This tendency is counted by the desirability of using sufficiently high-order interpolations so as to permit accurate stress computations, both at the nodal circles and between nodes since a coarse discretization is necessary to realize maximum efficiency. It was found that cubic polynomials were generally satisfactory from both standpoints, except in localized regions of high stress gradients where quintic polynomials were employed. The finite element discretizations for the direct integration studies were selected on this basis.For the high-precision finite element at hand, the efficiencies achieved in the space domain are demonstrated by the ability to achieve precise solutions with relatively coarse discretization patterns. The resulting comparatively large elements are not subject to accurate representation by diagonal mass matrices so that an implicit, consistent mass approach is followed. Efficiency in the time domain as well rests on the successful modeling of rotational shells subject to dynamic loading using coarse discretitations in space and large increments in time. Computational efficiency and accuracy are demonstrated for various problems documented in the literature, including a shallow spherical cap subject to a step pulse and a hyperboloidal shell under a simulated dynamic wind pressure.     

Two methods for the computation of nonlinear modes of vibrating systems at large amplitudes

The aim of this paper is to present two methods for the calculation of the nonlinear normal modes of vibration for undamped nonlinear mechanical systems: the time integration periodic orbit method and the modal representation method. In the periodic orbit method, the nonlinear normal mode is obtained by making the continuation of branches of periodic orbits of the equation of motion. The terms “periodic orbits” means a closed trajectory in the phase space, which is obtained by time integration. In the modal representation method, the nonlinear normal mode is constructed in terms of amplitude, phase, mode shape, and frequency, with the distinctive feature that the last two quantities are amplitude and total phase dependent. The methods are compared on two DOF strongly nonlinear systems.     

The use of modal superposition in nonlinear dynamics

This paper describes the application of the modal superposition method to the calculation of the nonlinear dynamic response of structures. Although this method has often been used in the analysis of linear response, it was seldom been applied in nonlinear problems. The reasons for this are obvious and seem to be quite valid, but the examples considered herein indicate that modal superposition may be a more useful computational tool than has heretofore been imagined. Four different problems are considered in this paper: a three dimensional cable stayed bridge, a double layered three dimensional cable network, an unstiffened suspension bridge, and a three dimensional elastic-plastic frame. The problems are chosen to exemplify both the strengths and weaknesses of the modal superposition method.     

On the accuracy and efficiency of a finite element tensor product algorithm for fluid dynamics applications

A finite element numerical solution algorithm is derived for application to problems in computational fluid dynamics. Through extension of the basic error extremization constraints offered using the method of weighted residuals, a selective dissipation mechanism is introduced to control phase error and/or non-linearly induced error. Using an implicit integration algorithm, the Jacobian of a Newton iteration procedure is replaced by a tensor matrix product construction for solution economy. Numerical results for multi-dimensional equations, modeling the substantial time derivative within the Navier-Stokes system, are utilized to assess solution accuracy, stability and error control.     

Approximate response sensitivities for nonlinear problems in explicit dynamic formulation

Explicit and implicit time integration schemes are discussed in the context of sensitivity analysis of dynamic problems. The application of the fully explicit central difference method (CDM) proves to be efficient for many nonlinear problems. In the case of the corresponding dynamic sensitivity problem the CDM is less advantageous both from efficiency and accuracy points of view. Approximate sensitivity expressions are derived in the paper for nonlinear path-dependent problems allowing the application of an unconditionally stable implicit time integration scheme with the time step much larger than the time step of the explicit CDM scheme of the direct problem. The method seems to be particularly suitable for problems of quasi-static nature in which the dynamic terms are artificially introduced to allow explicit CDM solution of highly nonlinear equations. Received January 21, 2000     

无条件稳定的显式降维非线性有限元

针对传统降维非线性有限元计算速度与精确度难以兼顾的问题,提出了一种无条件稳定的显式迭代算法。基于泰勒展开式得到速度、加速度的三阶精度差分表达式从而获得新的有限元显式迭代方程,并分析其单自由度系统下的传递矩阵谱半径。改进迭代方程使谱半径始终小于1从而满足无条件稳定的要求。实验表明,改进后的显式迭代算法在等效阻尼比的精度上优于中心差分法和隐式迭代法;在降维非线性有限元模型计算中的计算耗时优于隐式迭代方法,提高了降维非线性有限元的迭代计算速度。模型在降维后维度数值较高时,仍能维持良好的计算耗时和帧率,保证了模型的精确度。     

Application of the harmonic acceleration method for nonlinear dynamic analysis

The application of the harmonic acceleration method, which shows some advantages in linear transient vibration analysis compared to the other commonly used methods, is extended to nonlinear problems employing the mode superposition technique and an iteration procedure. Stability analysis shows that the method is unconditionally stable. A higher degree of accuracy is achieved. The efficiency of the method is illustrated by some simple numerical examples.     

Some computational aspects of large deformation,rate-dependent plasticity problems

The governing equations for a finite element formulation of boundary value problems for large deformation metal forming processes are derived using a principle of virtual work formulated in a Lagrangian reference system. The updated Lagrangian method is used to simplify the equations. The resulting nonlinear equilibrium equations are solved using Newton's method.A constitutive model for large deformation, rate-dependent plasticity is developed. The model incorporates a one parameter, implicit integration operator for stability and accuracy. The stressrate/strain-rate relation is written in terms of the Jaumann rate of stress.Numerous example problems are solved to demonstrate the effectiveness of the numerical algorithms.     

Modeling and simulation of structural components in recursive closed-loop multibody systems

Passenger cars, transit buses, railroad vehicles, off-highway trucks, earth moving equipment and construction machinery contain structural and light-fabrications (SALF) components that are prone to excessive vibration due to rough terrains and work-cycle loads’ excitations. SALF components are typically modeled as flexible components in the multibody system allowing the analysts to predict elastic deformation and hence the stress levels under different loading conditions. Including SALF component in the multibody system typically generates closed-kinematic loops. This paper presents an approach for integrating SALF modeling capabilities as a flexible body in a general-purpose multibody dynamics solver that is based on joint-coordinates formulation with the ability to handle closed-kinematic loops. The spatial algebra notation is employed in deriving the spatial multibody dynamics equations of motion. The system kinematic topology matrix is used to project the Cartesian quantities into the joint subspace, leading to a condensed set of nonlinear equations with minimum number of generalized coordinates. The proposed flexible body formulation utilizes the component mode synthesis approach to reduce the large number of finite element degrees of freedom to a small set of generalized modal coordinates. The resulting reduced flexible body model has two main characteristics: the stiffness matrix is constant while the mass matrix depends on the elastic modal coordinates. A consistent set of pre-computed inertia shape integrals are identified and used to update the modal mass matrix at each time step. The implementation of the component mode synthesis approach in a closed-loop recursive multibody formulation is presented. The kinematic equations are modified to include the effect of the flexible body modal elastic coordinates. Also, modified constraint equations that include the effect of flexibility at the joint connections and the necessary details of the Jacobian matrix are presented. Baumgarte stabilization approach is used to stabilize the constraint equations without using iterative schemes. A sample results for flexible body impeded in a closed system will be presented to demonstrate the above mentioned approach.     

A parallel mixed time integration algorithm for nonlinear dynamic analysis

This paper presents a parallel mixed time integration algorithm formulated by synthesising the implicit and explicit time integration techniques. The proposed algorithm is an extension of the mixed time integration algorithms [Comput. Meth. Appl. Mech. Engng 17/18 (1979) 259; Int. J. Numer. Meth. Engng 12 (1978) 1575] being successfully employed for solving media-structure interaction problems. The parallel algorithm for nonlinear dynamic response of structures employing mixed time integration technique has been devised within the broad framework of domain decomposition. Concurrency is introduced into this algorithm, by integrating interface nodes with explicit time integration technique and later solving the local submeshes with implicit algorithm. A flexible parallel data structure has been devised to implement the parallel mixed time integration algorithm. Parallel finite element code has been developed using portable Message Passing Interface software development environment. Numerical studies have been conducted on PARAM-10000 (Indian parallel supercomputer) to test the accuracy and also the performance of the proposed algorithm. Numerical studies indicate that the proposed algorithm is highly adaptive for parallel processing.     

On nonlinear dynamic analysis using substructuring and mode superposition

The solution of nonlinear dynamic equilibrium equations using mode superposition and substructuring is studied. The objective is to design schemes that in some analyses can significantly decrease the computational effort involved when compared to a complete direct integration solution. Specific schemes for mode superposition analysis and substructuring are proposed. These techniques have been implemented in ADINA. The results of a few sample analyses are presented and recommendations are given on the use of these procedures in practical analysis.     

A feasible direction algorithm for general nonlinear semidefinite programming

This paper deals with nonlinear smooth optimization problems with equality and inequality constraints, as well as semidefinite constraints on nonlinear symmetric matrix-valued functions. A new semidefinite programming algorithm that takes advantage of the structure of the matrix constraints is presented. This one is relevant in applications where the matrices have a favorable structure, as in the case when finite element models are employed. FDIPA_GSDP is then obtained by integration of this new method with the well known Feasible Direction Interior Point Algorithm for nonlinear smooth optimization, FDIPA. FDIPA_GSDP makes iterations in the primal and dual variables to solve the first order optimality conditions. Given an initial feasible point with respect to the inequality constraints, FDIPA_GSDP generates a feasible descent sequence, converging to a local solution of the problem. At each iteration a feasible descent direction is computed by merely solving two linear systems with the same matrix. A line search along this direction looks for a new feasible point with a lower objective. Global convergence to stationary points is proved. Some structural optimization test problems were solved very efficiently, without need of parameters tuning.     

A Hybrid Implicit-Explicit Adaptive Multirate Numerical Scheme for Time-Dependent Equations

We develop a hybrid implicit and explicit adaptive multirate time integration method to solve systems of time-dependent equations that present two significantly different scales. We adopt an iteration scheme to decouple the equations with different time scales. At each iteration, we use an implicit Galerkin method with a fast time-step to solve for the fast scale variables and an explicit method with a slow time-step to solve for the slow variables. We derive an error estimator using a posteriori analysis which controls both the iteration number and the adaptive time-step selection. We present several numerical examples demonstrating the efficiency of our scheme and conclude with a stability analysis for a model problem.     

On data dependence of stability domains,exponential stability and stability radii for implicit linear dynamic equations

We shall deal with some problems concerning the stability domains, the spectrum of matrix pairs, the exponential stability and its robustness measure for linear implicit dynamic equations of arbitrary index. First, some characterizations of the stability domains corresponding to a convergent sequence of time scales are derived. Then, we investigate how the spectrum of matrix pairs, the exponential stability and the stability radii for implicit dynamic equations depend on the equation data when the structured perturbations act on both the coefficient of derivative and the right-hand side.     

Nonlinear transient responses of initially stressed composite plates

The nonlinear transient response of initially stressed composite plates is investigated using the finite element method. A nine-node isoparametric quadrilateral element is developed to model laminated plates under initial deformation and initial stress according to the Mindlin plate theory and von Karman large deflection assumptions. In the time integration, the Newmark constant acceleration method in conjunction with an efficient and accurate iteration scheme is used. Numerical results for deflections and bending moments for isotropic and laminated plates are obtained.     

变截面舵面结构全场瞬态变形测量及振动特性分析

搭建了一套全场非接触应变测量系统,对变截面舵面结构受随机载荷激励作用下的变形及振动特性进行实验研究。实验以2A12铝合金材料制成的变截面舵面结构为试验对象,采用高分辨率的相机记录舵面结构全场清晰的散斑图像,然后利用三维数字图像相关方法对其结构表面的连续变形进行直接测量,同时利用傅里叶变换对采集到的散斑图像进行振动特性分析,获得变截面舵面结构的模态频率和模态振型。为验证非接触测量得到的振动特性结果的准确性,用小质量加速度传感器得到的振动响应信号进行对比。试验结果表明,在0~500Hz的测试频段内,三维数字图像相关方法辨识变截面舵面结构前三阶模态参数与加速度传感器测量结果非常吻合,前三阶固有频率误差在2%以内,阻尼比误差在6%以内,前二阶模态振型均为弯曲模式,第三阶模态振型为弯曲扭转耦合模式,验证了三维数字图像相关的全场振动测量技术的有效性,为高温环境及非线性变截面舵面结构非接触振动测试提供依据。     

Improved solution methods for inelastic rate problems

The objective of the paper is an assessment of the incremental solution methods for the analysis of inelastic rate problems. In particular, the possibilities of the initial load method are explored to improve the accuracy and stability of the traditional explicit operators by higher-order time expansions and implicit weighting schemes.The convergence limitations are examined for different classes of inelastic growth laws (viscous flow, viscoelasticity, viscoplasticity) which restrict the time step because of the iterative solution of the implicit algorithm. The range and rate of convergence of the initial load method (constant stiffness predictor-corrector iteration) is enlarged by tangential gradient techniques which account for the inelastic response in the structural stiffness matrix. In this way the time step restriction disappears although at a considerable increase of computational expense because of the costly computation and decomposition of structural gradients within each iteration cycle (Newton-Raphson methods).As compared to the linear single-step methods, the cubic Hermitian time expansions furnish far better accuracy than the traditional linear expansions for very little increase of computational cost. Stability and convergence limits correspond to those of the lower-order operators, whereby the implicit midstep of backward weighting schemes are most advantageous. In this context it is worth noting that aging or strain-hardening effects in the inelastic growth law reduce dramatically the time step restrictions of the iterative initial load solution methods (predictor-corrector schemes), as compared to the simplest creep model in which the inelastic growth law depends only on stress, e.g. for viscous flow and viscoplasticity.     

A reduction method for nonlinear structural dynamic analysis

A computational algorithm for predicting the nonlinear dynamic response of a structure is presented. The nonlinear system of ordinary differential equations resulting from the finite element discretization is highly reduced by means of a Rayleigh-Ritz analysis. The basis vectors are chosen to be the current tangent eigenmodes together with some modal derivatives that indicate the way in which the spectrum is changing. Only a few basis updatings are required during the whole time integration.The truncation error introduced at every change of basis is pointed out as the cause for a divergence-type behaviour, and some means for eliminating it are discussed.Results for examples involving large displacements are shown and compared to the results obtained by integrating the complete system of equations.     

An iterative matrix-free method in implicit immersed boundary/continuum methods

The objective of this paper is to present an iterative solution strategy for implicit immersed boundary/continuum methods. An overview of the newly proposed immersed continuum method in conjunction with the traditional immersed boundary method will also be presented. As a key ingredient of the fully implicit time integration, a matrix-free combination of Newton–Raphson iteration and GMRES iterative linear solver is proposed.     

Incremental Deformation Subspace Reconstruction

Recalculating the subspace basis of a deformable body is a mandatory procedure for subspace simulation, after the body gets modified by interactive applications. However, using linear modal analysis to calculate the basis from scratch is known to be computationally expensive. In the paper, we show that the subspace of a modified body can be efficiently obtained from the subspace of its original version, if mesh changes are small. Our basic idea is to approximate the stiffness matrix by its low‐frequency component, so we can calculate new linear deformation modes by solving an incremental eigenvalue decomposition problem. To further handle nonlinear deformations in the subspace, we present a hybrid approach to calculate modal derivatives from both new and original linear modes. Finally, we demonstrate that the cubature samples trained for the original mesh can be reused in fast reduced force and stiffness matrix evaluation, and we explore the use of our techniques in various simulation problems. Our experiment shows that the updated subspace basis still allows a simulator to generate visual plausible deformation effects. The whole system is efficient and it is compatible with other subspace construction approaches.