On a finite element method for the equations of one-dimensional nonlinear thermoviscoelasticity

A computationally uncoupled numerical scheme for the equations of one-dimensional nonlinear thermoviscoelasticity is proposed. The scheme makes use of the finite element method for the space variable and the different method for the time variable. The existence and uniqueness of the approximate solutions are proved, and bounds of the error are analyzed.     

On the use of the Blatz-Ko constitutive model in nonlinear finite element analysis

The nearly-incompressible material model proposed by Blatz and Ko (Trans. Soc. Rheol., 6, 223–251, 1962) is attractive for its simplicity, and is used currently in several finite element wave propagation codes. A form of the Blatz-Ko model suitable for use within static and implicit dynamic solutions is developed in this paper. Stress point equations for both stress and tangent modulus computations are given, and a typical implementation in FORTRAN is presented. The determination of material properties for the model from laboratory test data is also discussed.     

On finite Newton method for support vector regression

In this paper, we propose a Newton iterative method of solution for solving an ε-insensitive support vector regression formulated as an unconstrained optimization problem. The proposed method has the advantage that the solution is obtained by solving a system of linear equations at a finite number of times rather than solving a quadratic optimization problem. For the case of linear or kernel support vector regression, the finite termination of the Newton method has been proved. Experiments were performed on IBM, Google, Citigroup and Sunspot time series. The proposed method converges in at most six iterations. The results are compared with that of the standard, least squares and smooth support vector regression methods and of the exact solutions clearly demonstrate the effectiveness of the proposed method.     

On a finite dynamic element method for free vibration analysis of structures

This paper explores the concept of finite dynamic elements involving higher order dynamic correction terms in the associated stiffness and mass matrices. Such matrices are then developed for a rectangular prestressed membrane element. Next, efficient analysis techniques for the eigenproblem solution of the resulting quadratic matrix equations are described in detail. These are followed by suitable numerical examples which indicate that employment of such dynamic elements in conjunction with an efficient quadratic matric solution technique will result in a most significant economy in the free vibration analysis of structures.     

Rudimentary considerations for effective line search method in nonlinear finite element analysis

Although the line search method appears theoretically attractive, the computational intensity involved in the search process often prohibits indiscriminate use of this procedure in nonlinear finite element analysis. Effects of the line search method have been scrutinized in order to evaluate its validity in various occurrences. The line search method was implemented and modified to optimize its usefulness for a general class of problems in conjunction with quasi-Newton updates using MSC/NASTRAN. As a result of a considerable amount of numerical experiments, the effective implementation scheme of the line search method is made succinct in this paper. Parametric studies were performed to validate the effectiveness and efficiency of the algorithm. The study shows that the line search method is essential to achieve convergence in some highly nonlinear problems, but it has little impact on the efficiency.     

A modified gradient method for finite element elastoplastic analysis by quadratic programming

The finite analysis problem with piecewise linear constitutive laws is formulated as a linear complementarity and a quadratic programming problem. The solution techniques using the well-known optimization methods of mathematical programming are discussed and a procedure, belonging to the class of gradient methods, is proposed which overcomes the computational difficulties that arise when there is a large number of variables.Through a physical interpretation of the gradient of the objective function, each mathematical step of the proposed optimization technique is translated into a corresponding physical operation on the structure and a mechanical solution procedure with a finite number of steps is derived.Finally, for the incremental analysis problem under non-holonomic constitutive laws the same procedure is adopted combined with the multistage loading technique. Illustrative examples for each of the preceding problems are given.     

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。     

Parallel-vector computations for geometrically nonlinear finite element analysis

Existing procedures for nonlinear finite element analysis are reviewed. Common computational steps among existing methods are identified. Parallel-vector solution strategies for the generation and assembly of element matrices, solution of the resulting system of linear equations, calculations of the unbalanced loads, displacements and stresses are all incorporated into the Newton-Raphson (NR), modified Newton-Raphson (mNR), and BFGS methods. Furthermore, a mixed parallel-vector Choleski-Preconditioned Conjugate Gradient (C-PCG) equation solver is also developed and incorporated into the piecewise linear procedure for nonlinear finite element analysis. Numerical results have indicated that the Newton-Raphson method is the most effective nonlinear procedure and the mixed C-PCG equation solver offers substantial computational advantages in a parallel-vector computer environment.     

On the optimum load step size selection scheme for nonlinear finite element stress analysis

A user-fault-proof algorithm has been proposed for the selection of load increments in a nonlinear finite element stress analysis, which will result in minimum deviations from the given material's stress vs strain curves. The user needs only to specify the starting stress level for the automatic selection and the allowable deviation for the first increment. The selection procedure is fully automatic and hence is particularly suitable for less experienced users.     

A finite element method for the survival probability of nonlinear oscillators

The calculation of the survival probability of a nonlinear structure which is subjected to random excitation presents major difficulties. Here the concern is with a nonlinear single degree of freedom system subjected to either stationary or nonstationary broad band Gaussian excitation. The survival probability of such a system can be found as a solution to the backward Kolmogorov equation, which is solved here using the Finite Element method. Application of the method to a number of examples shows good agreement with published results and demonstrates that the computer time required by the method is low.     

Formulation of a nonlinear compatible finite element for the analysis of laminated composites

Increased use of laminated composites has pointed out the need for better analytic tools. These tools must be able to correctly account for normal shear in the laminates and should be able to solve nonlinear problems. The finite element method is applied in this paper to analyze laminated composites using an approach which includes both nonlinear and normal shear effects. A finite element is formulated from basic elasticity equations. The most unique characteristic of the element is the manner in which normal shear is handled. The normal to a reference surface is allowed to not only rotate but also to change shape. Not only is displacement continuity imposed at lamina interfaces, but also slope continuity is guaranteed. After a complete general formulation of the finite element is presented, the method is specialized to plates so that the convergence characteristics, the accuracy and the applicability of the element can be studied. The finite element does an excellent job of analyzing laminated composites, and gives the analyst the ability to do nonlinear analysis of laminated composites under a variety of loading and boundary conditions.     

A combined Rayleigh-Ritz/finite element method for the nonlinear analysis of stiffened plated structures

The paper describes a hybrid method for the non-linear analysis of steel plates and stiffened plating in which local (finite element) displacement functions are supplemented or replaced by global functions. The method is applied to the collapse analysis of box-girder bridges.     

Identification of nonlinear elastic solids by a finite element method

The problem of utilizing experimental data to characterize the stress constitutive function for a nonlinear elastic solid is formulated as an inverse boundary value problem. The use of finite element discretization is extended by introducing a technique of material parameterization that utilizes finite elements defined over the domain of the stress constitutive function. The discretized identification problem is then reduced to a system of nonlinear algebraic equations that couples the data set and the discretized boundary value problem. The effect of errors in measured data is minimized by employing a weighted least squares error norm to generate the equations from which the unknown material parameters are obtained. An illustrative numerical experimental is included.     

On nonlinear finite element analysis in single-, multi- and parallel-processors

Numerical solution of nonlinear equilibrium problems of structures by means of Newton-Raphson type iterations is reviewed. Each step of the iteration is shown to correspond to the solution of a linear problem, therefore the feasibility of the finite element method for nonlinear analysis is established. Organization and flow of data for various types of digital computers, such as single-processor/single-level memory, single-processor/two-level-memory, vector-processor/two-level-memory, and parallel-processors, with and without sub-structuring (i.e. partitioning) are given. The effect of the relative costs of computation, memory and data transfer on substructuring is shown. The idea of assigning comparable size substructures to parallel processors is exploited. Under Cholesky type factorization schemes, the efficiency of parallel processing is shown to decrease due to the occasional shared data, just as that due to the shared facilities.     

On the automatic solution of nonlinear finite element equations

An algorithm for the automatic incremental solution of nonlinear finite element equations in static analysis is presented. The procedure is designed to calculate the pre- and post-buckling/collapse response of general structures. Also, eigensolutions for calculating the linearized buckling response are discussed. The algorithms have been implemented and various experiences with the techniques are given.     

Coupling of a nonlinear finite element structural method with a Navier-Stokes solver

A new three-dimensional viscous aeroelastic solver is developed in the present work. A well validated full Navier-Stokes code is coupled with a nonlinear finite element plate model. Implicit coupling between the computational fluid dynamics and structural solvers is achieved using a subiteration approach. Computations of several benchmark static and dynamic plate problems are used to validate the finite element portion of the code. This coupled aeroelastic scheme is then applied to the problem of three-dimensional panel flutter. Inviscid and viscous supersonic results match previous computations using the same aerodynamic method coupled with a finite difference structural solver. For the case of subsonic flow, multiple solutions consisting of static, upward and downward deflections of the panel are discussed. The particular solution obtained is shown to be sensitive to the cavity pressure specified underneath the panel.     

The theorems of geometric variation for nonlinear finite element analysis

This paper illustrates the application of the theorems of geometric variation to the reanalysis of nonlinear finite element structures. The formulations presented here are compared for efficiency with the Newton-Raphson methods for a number of problems. This comparison indicates the efficiency of the proposed techniques and the type of problems where the techniques may be used profitably.