Nonlinear finite element modeling of the dynamics of unrestrained flexible structures

A finite element modeling and solution technique capable of determining the time response of unrestrained flexible structures which are undergoing large elastic deformations coupled with gross nonsteady translational and rotational motions with respect to an inertial reference frame has been developed. The governing equations of motion are derived using momentum conservation principles and the principle of virtual work. The finite element approximation is applied to the equations of motion and the resulting set of nonlinear second order matrix differential equations is solved timewise by a direct numerical integration scheme based on the trapezoidal rule and Newton-Raphson type iterations. The solution technique is tested on some simple structures consisting of long, slender, uniform beams attached to a rigid mass.     

Fuzzy finite element analysis of imprecisely defined structures with fuzzy nodal force

This paper presents the fuzzy finite element analysis for static displacements of fixed free stepped rectangular beam, truss and simplified bridge structure with fuzzy nodal force. The material and geometric properties of the structures are taken as crisp. Fuzzy finite element analysis of static problem for the above structures converts the problem into fuzzy system of linear equations. As such the coefficient matrix and the right-hand side vector become crisp and fuzzy respectively. A new approach is used here to solve the fuzzy system of linear equations. Numerical results for the three stepped rectangular beam, three-bar truss and simplified bridge with fifteen elements are presented to illustrate the computational aspects of the developed method. The results obtained are depicted in term of plots.     

Dynamic analysis of wind-excited truss tower with friction dampers

This paper presents a rational analytical method for determining dynamic response of wind-excited large truss towers installed with friction dampers and investigates the effectiveness of friction dampers. The analytical procedure involves two models of one truss tower: one is the two-dimensional lumped mass dynamic model and the other is the three-dimensional (3D) finite element static model. These two models are alternately used to fulfill control force transformation, displacement increment transformation, and the numerical integration of equations of motion of the coupled damper-tower system. A three-story truss structure under wind excitation is used to validate the proposed bi-model method through the comparison with a precise 3D dynamic analysis. After a satisfactory comparison, a large space steel television tower is used as an application to examine the feasibility of the bi-model method and the effectiveness of friction dampers. The case study shows that the bi-model method can efficiently compute wind-induced response of the large space television tower with friction dampers and the installation of the friction dampers of proper parameters can significantly reduce wind-induced vibration of the tower.     

Statistical properties of nonlinear mode spectrum for large amplitude forced vibration systems

A method to estimate the statistical properties of a geometrically nonlinear mode spectrum for large amplitude forced vibration systems is presented. It is shown here that this technique is readily applicable to geometrically nonlinear small strain problems in which the equations of motion of structures are derived using the finite element method. A general formulation for problems of this type is presented. The efficiency of the method is illustrated by its application to a nonlinear vibration of an elastic beam.     

基于Patran和MSC Nastran的压电智能桁架结构振动模态分析

用Patran和MSC Nastran分析压电智能桁架结构振动模态,验证基于有限元法建立的智能桁架结构机电耦合动力学模型的正确性和有效性.结果表明：采用Patran和MSC Nastran针对2种典型压电智能桁架结构开展振动模态分析的结果,与采用基于有限元法建立的数学模型计算得到的模态频率及实验测试模态频率近似相等,验证基于有限元法模型的正确性和有效性,为开展主动振动控制器的设计提供模型和技术支持.     

Optimization of nonlinear trusses using a displacement-based approach

A displacement-based optimization strategy is extended to the design of truss structures with geometric and material nonlinear responses. Unlike the traditional optimization approach that uses iterative finite element analyses to determine the structural response as the sizing variables are varied by the optimizer, the proposed method searches for an optimal solution by using the displacement degrees of freedom as design variables. Hence, the method is composed of two levels: an outer level problem where the optimal displacement field is searched using general nonlinear programming algorithms, and an inner problem where a set of optimal cross-sectional dimensions are computed for a given displacement field. For truss structures, the inner problem is a linear programming problem in terms of the sizing variables regardless of the nature of the governing equilibrium equations, which can be linear or nonlinear in displacements. The method has been applied to three test examples, which include material and geometric nonlinearities, for which it appears to be efficient and robust. Received December 4, 2000     

Parallel processing techniques for finite element analysis of nonlinear large truss structures

Methods were developed for parallel processing of finite element solutions of large truss structures. The parallel processing techniques were implemented in two stages, i.e., the repeated forming of the nonlinear global stiffness matrix and the solving of the global system of equations. The Sequent Balance 21000 parallel computer was employed to demonstrate the procedures and the speed-up.     

悬臂梁大变形的向量式有限元分析

为分析悬臂梁的几何非线性行为,用向量式有限元法将结构离散成质点系以及质点间的连接单元.根据牛顿第二定律得到每个质点在内力和外载荷作用下的运动方程以及悬臂梁在每个时刻的变形用该时刻质点系的运动表示.结合刚架元的节点内力和等效质量得出质点位移的迭代计算公式,采用FORTRAN编制计算程序,对悬臂梁分别承受集中载荷和弯矩下的大变形进行算例分析.计算结果与理论解吻合较好,表明该方法能很好地模拟分析悬臂梁的大变形.     

A finite element large displacement analysis of stiffened plates

A finite element analysis of the large deflection behaviour of stiffened plates using the isoparametric quadratic stiffened plate bending element is presented. The evaluation of fundamental equations of the stiffened plates is based on Mindlin's hypothesis. The large deflection equations are based on von Kármán's theory. The solution algmrithm for the assembled nonlinear equilibrium equations is based on the Newton-Raphson iteration technique. Numerical solutions are presented for rectangular plates and skew stiffened plates.     

Dynamics of flexible-link mechanisms

Dynamic response of flexible-link mechanism (structure) when subjected to external dynamic load is analyzed. The non-linear equations of motion are discretized using the finite element method (FEM) and Newmark time integration method combined with Newton-Raphson iterative technique is used for solution. From the dynamic response of the structure, the required kinematic characteristics are derived. Typical examples are worked out for illustration.     

The problem of optimal control of the orientation of an orbit of a spacecraft as a deformable figure

The problem of optimal control of the orbit orientation of a spacecraft regarded as a deformable figure is studied. The problem of optimal re-orientation of an orbit is formulated as a problem of optimal control of the motion of the center of mass of a spacecraft with a movable right end of the trajectory and is solved based on the Pontryagin maximum principle. To describe the orientation of an instantaneous orbit, a new quaternion osculating element that replaces three classical angular elements of the orbit is applied. Necessary optimality conditions are obtained; several first integrals of the system of equations of the boundary-value problem of the maximum principle are found; transformations that reduce the dimension of the system of differential equations of the boundary-value problem (without their complication) are proposed; the proposed approach is analyzed, and an example of numerical solution of the problem is presented     

Eigenvalue inverse formulation for optimising vibratory behaviour of truss and continuous structures

This paper presents formulations for inverse optimisation of vibration behaviour of finite element models of both truss and continuous structures. The proposed algorithms determine the required modifications on truss and continuous structures to achieve specified natural frequencies. The modification can be carried out globally or locally on the structures stiffness and matrices and the formulation can also be used to add new structural members to achieve the desired response. Numerical examples and finite element implementation of the developed method are provided to demonstrate the feasibility of the formulations.     

有限元GPU加速计算的实现方法

研究基于GPU的有限元求解中的总刚矩阵生成和线性方程组求解问题.通过对单元着色和分组完成总刚矩阵的生成,并以行压缩存储(Compressed Sparse Row,CSR)格式存储,用预处理共轭梯度法求解所生成的大规模线性稀疏方程组.在CUDA(Compute Unified Device Architecture)平台上完成程序设计,并用GT430 GPU对弹性力学的平面问题和空间问题进行试验.结果表明,总刚矩阵生成和方程组求解分别得到最高11.7和8的计算加速比.     

Tracing post-limit-point paths with reduced basis technique

A reduced basis technique and a problem-adaptive computational algorithm are presented for predicting the post-limit-point paths of structures. In the proposed approach the structure is discretized by using displacement finite element models. The nodal displacement vector is expressed as a linear combination of a small number of vectors and a Rayleigh-Ritz technique is used to approximate the finite element equations by a small system of nonlinear algebraic equations.To circumvent the difficulties associated with the singularity of the stiffness matrix at limit points, a constraint equation, defining a generalized arc-length in the solution space, is added to the system of nonlinear algebraic equations and the Rayleigh-Ritz approximation functions (or basis vectors) are chosen to consist of a nonlinear solution of the discretized structure and its various order derivatives with respect to the generalized arc-length. The potential of the proposed approach and its advantages over the reduced basis-load control technique are outlined. The effectiveness of the proposed approach is demonstrated by means of numerical examples of structural problems with snap-through and snap-back phenomena.     

Substructuring and equation system solutions in finite element analysis

Dealing with large structural systems it is convenient and often almost necessary to employ a substructure technique in order not to exceed the available computer capacity. Efficient solution of systems of equations is very essential in finite element analysis. In this paper, first, a procedure is described as to how to systematically reduce the number of assembled equations on different substructure levels. The size of a system of equations to be solved at a time is reduced by condensation of different types of unknowns at the element level. Second, a generalized Gaussian elimination scheme is presented for the solution of systems of equations in substructuring. The same scheme may be very useful in solutions of mixed finite element models, equation systems with Lagrangian multipliers and moderately ill-conditioned problems. The system properties and the initially neatly banded forms of system equations are preserved during the solution process. Lastly, three examples illustrate solutions of some complex structural problems. The method presented is applicable to a wide range of physical problems and can achieve a substantial computer economy.     

Corotational mixed finite element formulation for thin-walled beams with generic cross-section

The corotational technique is adopted here for the analysis of three-dimensional beams. The technique exploits the technology that applies to a two-noded element, a coordinate system which continuously translates and rotates with the element. In this way, the rigid body motion is separated out from the deformational motion. In this paper, a mixed formulation are adopted for the derivation of the local element tangent stiffness matrix and nodal forces. The mixed finite element formulation is based on an incremental form of the two-field Hellinger–Reissner variational principle to permit elasto-plastic material behavior. The local beam kinematics is based on a low-order nonlinear strain expression using Bernoulli assumption. The present formulation captures both the Saint–Venant and warping torsional effects of thin-walled open cross-sections. Shape functions that satisfy the nonlinear local equilibrium equations are selected for the interpolation of the stress resultants. In particular, for the torsional forces and the twist rotation degree of freedom, a family of hyperbolic interpolation functions is adopted in lieu of conventional polynomials. Governing equations are expressed in a weak form, and the constitutive equations are enforced at each integration cross-section along the element length. A consistent state determination algorithm is proposed. This local element, together with the corotational framework, can be used to analyze the nonlinear buckling and postbuckling of thin-walled beams with generic cross-section. The present corotational mixed element solution is compared against the results obtained from a corotational displacement-based model having the same beam kinematics and corotational framework. The superiority of the mixed formulation is clearly demonstrated.     

Methods of Estimating the Envelope of Elastic Oscillations of the Flexible Spacecraft

A problem concerned with studying of dynamics of angular motion of large space structures and flexible spacecraft was considered. It stems from the need to take into account multiple elastic modes when analyzing and designing their control systems. It was proposed to seek a solution to this problem, which is especially noticeable when modeling processes with high-frequency oscillations, on the way to replacing the precise solution of the equations of these motions by an approximate solution, that is, to determine a relatively low-frequency envelope of the precise solution, which allows one to extend the step of integration and, therefore, reduce the time of modeling. Two approaches to estimating the envelop were proposed, and their efficiencies were compared by the example of modeling angular stabilization of the flexible satellite with five elastic modes.     

Dynamic inelastic structural analysis by boundary element methods

Summary Boundary element methodologies for the determination of the response of inelastic two-and three-dimensional solids and structures as well as beams and flexural plates to dynamic loads are briefly presented and critically discussed. Elastoplastic and viscoplastic material behaviour in the framework of small deformation theories are considered. These methodologies can be separated into four main categories: those which employ the elastodynamic fundamental solution in their formulation, those which employ the elastostatic fundamental solution in their formulation, those which combine boundary and finite elements for the creation of an efficient hybrid scheme and those representing special boundary element techniques. The first category, in addition to the boundary discretization, requires a discretization of those parts of the interior domain expected to become inelastic, while the second category a discretization of the whole interior domain, unless the inertial domain integrals are transformed by the dual reciprocity technique into boundary ones, in which case only the inelastic parts of the domain have to be discretized. The third category employs finite elements for one part of the structure and boundary elements for its remaining part in an effort to combine the advantages of both methods. Finally, the fourth category includes special boundary element techniques for inelastic beams and plates and symmetric boundary element formulations. The discretized equations of motion in all the above methodologies are solved by efficient step-by-step time integration algorithms. Numerical examples involving two-and three-dimensional solids and structures and flexural plates are presented to illustrate all these methodologies and demonstrate their advantages. Finally, directions for future research in the area are suggested.     

Linear Systems with Large Uncertainties,with Applications to Truss Structures

Linear systems whose coefficients have large uncertainties arise routinely in finite element calculations for structures with uncertain geometry, material properties, or loads. However, a true worst case analysis of the influence of such uncertainties was previously possible only for very small systems and uncertainties, or in special cases where the coefficients do not exhibit dependence. This paper presents a method for computing rigorous bounds on the solution of such systems, with a computable overestimation factor that is frequently quite small. The merits of the new approach are demonstrated by computing realistic bounds for some large, uncertain truss structures, some leading to linear systems with over 5000 variables and over 10000 interval parameters, with excellent bounds for up to about 10% input uncertainty. Also discussed are some counterexamples for the performance of traditional approximate methods for worst case uncertainty analysis.     

Parallel numerical simulation with domain decomposition for seismic response analysis of shield tunnel

The research presented in this paper deals with parallel explicit finite element simulation with domain decomposition for seismic response analysis of shield tunnel, which is the non-linear and time dependent behaviour of complex structures in engineering. Such simulations have to provide high accuracy in the prediction of deformations and stability, by taking into account the long-term influences of the non-linear behaviour of the material as well as the large deformation and contact conditions. The limiting factors of the computer simulation are the computer run time and the memory requirement during solution of large-scale problems. To overcome these problems, a domain decomposition method and dynamic-explicit time integration procedure are used, and the latter is used for the solution of the semi-discrete equations of motion, which is very suited for parallel processing. Using the high performance computer ShenWei-I, the seismic response of shield tunnel in Shanghai is processed, and the number of nodes and elements in final finite element model exceeded 4.0 million and 3.8 million, respectively. The results reveal the weak position in the shield tunnel under Shanghai seismic wave. The paper provides references for the antiseismic design of the shield tunnel.