On stochastic finite element method for linear elastostatics by the Taylor expansion

The main aim of the paper is to present an application of the Taylor expansion in formulation and computational implementation of the perturbation-based stochastic finite element method. Random-input parameters as well as all-state functions included in static equilibrium equations are expanded in this approach around their expectations via Taylor series up the order given a priori. It further enables a dual computational approach for determination of probabilistic moments of the state functions—a formation and the solution of increasing order equilibrium equations and, on the other hand, polynomial approximation of deterministic state functions with respect to a given input random parameter. Theoretical and technical details of such methodology are explained also; some elementary engineering application with analytical solution is available to derive explicitly fundamental probabilistic moments of the resulting state function.     

Monte Carlo simulation-compatible stochastic field for application to expansion-based stochastic finite element method

In order to endow the expansion-based stochastic formulation with the capability of representing the characteristic behavior of stochastic systems, i.e., the non-linear dependence of the response variability on the coefficient of variation of the stochastic field, a Monte Carlo simulation-compatible stochastic field is suggested. Through a theoretical comparison of displacement vectors in the Monte Carlo method and an expansion-based scheme, it is found that the stochastic field adopted in the expansion-based scheme is not compatible with that appearing in the Monte Carlo simulation. The Monte Carlo simulation-compatible stochastic field is established by means of enforcing the compatibility between the stochastic fields in the expansion-based scheme and the Monte Carlo simulation. Employing the stochastic field suggested in this study, the response variability is reproduced with high precision even for uncertain fields with a moderately large coefficient of variation. Furthermore, the formulation proposed here can be used as an indirect Monte Carlo scheme by directly substituting the numerically simulated random fields into the covariance formula. This yields a pronounced reduction in the computation cost while resulting in virtually the same response variability as the Monte Carlo technique.     

Generalized perturbation-based stochastic finite element method in elastostatics

Generalised nth order stochastic perturbation technique that can be applied to solve some boundary value or boundary initial problems in computational physics and/or engineering with random coefficients is presented here. This technique is implemented in conjunction with the finite element method (FEM) to model 1D linear elastostatics problem with a single random variable. Main motivation of this work is to improve essentially the accuracy of the stochastic perturbation technique, which in its second order realization was ineffective for large variations of the input random fields. The nth order approach makes it possible to specify the accuracy of the computations a priori for the expected values and variances separately. The symbolic computer program is employed to perform computational studies on convergence of the first two probabilistic moments for simple unidirectional tension of the bar. These numerical studies verify the influence of coefficient of variation of the random input and, in the same time, of the perturbation parameter on the first four probabilistic moments of the final solution vector.     

Perturbation based stochastic finite element method for homogenization of two-phase elastic composites

The main idea of the paper is to apply the second order perturbation and stochastic second central moment technique to solve the homogenization problem. In order to determine the effective elasticity tensor, the prevailing computational methodology discussed in the literature so far was the Monte-Carlo simulation providing appropriate expected values and higher order probabilistic moments of the effective tensor components. The technique applied in this paper aims at significantly reducing the computational cost of the simulation without sacrificing the solution accuracy. The numerical example substantiates this claim in the case of a periodic fiber-reinforced plane strain composite with random fiber and matrix Young’s moduli.     

Uncertainty-based loads analysis for spacecraft: Finite element model validation and dynamic responses

The stochastic correlation and validation of spacecraft structural dynamic models and the stochastic launcher-satellite coupled loads analysis present several points of interest and involve novel aspects. This paper describes the main objectives of two studies in progress under the technical management of the European Space Agency and performed by consortia of European industries and university. The paper presents an overview of the major aspects related to the implementation of the approach, the identified methods and tools and the future developments.     

Computational aspects of the stochastic finite element method

We present an overview of the stochastic finite element method with an emphasis on the computational tasks involved in its implementation.      

Hierarchical parallelisation for the solution of stochastic finite element equations

As an example application the elliptic partial differential equation for steady groundwater flow is considered. Uncertainties in the conductivity may be quantified with a stochastic model. A discretisation by a Galerkin ansatz with tensor products of finite element functions in space and stochastic ansatz functions leads to a certain type of stochastic finite element system (SFEM). This yields a large system of equations with a particular structure. They can be efficiently solved by Krylov subspace methods, as here the main ingredient is the multiplication with the system matrix and the application of the preconditioner. We have implemented a “hierarchical parallel solver” on a distributed memory architecture for this. The multiplication and the preconditioning uses a—possibly parallel—deterministic solver for the spatial discretisation as a building block in a black-box fashion. This paper is concerned with a coarser grained level of parallelism resulting from the stochastic formulation. These coarser levels are implemented by running different instances of the deterministic solver in parallel. Different possibilities for the distribution of data are investigated, and the efficiencies determined. On up to 128 processors, systems with more than 5 × 10⁷ unknowns are solved.     

Probabilistic fiber element modeling of reinforced concrete structures

A computational model based on a stochastic fiber element model is developed in this study. This model can be utilized for probabilistic evaluation of reinforced concrete (RC) members. The stochastic fiber element model is developed by combining the conventional fiber element formulation and the midpoint method for random field representation, to account for spatial variability of material and geometrical properties within a structural member. Three verification examples show the capability of the developed model in estimating the nonlinear structural behavior including softening. As an application of the developed computational model, a probabilistic strength analysis of a RC column is conducted in terms of the axial load-bending moment interaction. An approach of evaluating RC structural systems using the developed probabilistic computational model is also presented.     

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.     

平面薄膜自由振动的有限元分析

构造6节点三角形单元,适合于平面薄膜自由振动的有限元分析.文中采用面积坐标,给出单元的形函数,根据哈密顿原理建立薄膜自由振动方程,推导其单元刚度矩阵和单元质量矩阵.3个典型算例表明,6节点三角形单元的计算结果比ANSYS三角形单元更接近理论解,具有更高的精度.     

Non-stationary response of large, non-linear finite element systems under stochastic loading

A very efficient and straightforward numerical procedure for the computation of statistical second moment characteristics of large, non-linear finite element systems under stochastic loading is presented. For the modeling of both the loading and the response of the system an orthogonal series expansion of the corresponding covariance matrix, the so-called Karhunen-Loève expansion is applied, allowing to incorporate potentially available statistical data of an excitation process directly into the analysis. The non-linear equation of motion is linearized by the method of equivalent statistical linearization. According to the present capabilities of this linearization technique, one-dimensional hysteretic elements are used for modeling the non-linear system behavior. The mode acceleration method is applied in order to reduce significantly the size of the system equation and thus increasing the computational efficiency of the proposed procedure. Contrary to methodologies based on state space formulations, this procedure relies on deterministic step by step integration, implying that the dimension of the system equation is the same as in a purely deterministic analysis.     

The use of finite element techniques for calculating the dynamic response of structures to moving loads

This paper presents a technique for using standard finite element packages for analysing the dynamic response of structures to time-variant moving loads. To illustrate the method and for validation purposes, the technique is first applied to a simply supported beam subject to a single load moving along the beam. Finally, it is applied to the problem that initiated the work: calculation of the effects of two-dimensional motion of the trolley on the response of the base structure of a mobile gantry crane model.     

Nonlinear tracking in a diffusion process with a Bayesian filter and the finite element method

A new approach to nonlinear state estimation and object tracking from indirect observations of a continuous time process is examined. Stochastic differential equations (SDEs) are employed to model the dynamics of the unobservable state. Tracking problems in the plane subject to boundaries on the state-space do not in general provide analytical solutions. A widely used numerical approach is the sequential Monte Carlo (SMC) method which relies on stochastic simulations to approximate state densities. For off-line analysis, however, accurate smoothed state density and parameter estimation can become complicated using SMC because Monte Carlo randomness is introduced. The finite element (FE) method solves the Kolmogorov equations of the SDE numerically on a triangular unstructured mesh for which boundary conditions to the state-space are simple to incorporate. The FE approach to nonlinear state estimation is suited for off-line data analysis because the computed smoothed state densities, maximum a posteriori parameter estimates and state sequence are deterministic conditional on the finite element mesh and the observations. The proposed method is conceptually similar to existing point-mass filtering methods, but is computationally more advanced and generally applicable. The performance of the FE estimators in relation to SMC and to the resolution of the spatial discretization is examined empirically through simulation. A real-data case study involving fish tracking is also analysed.     

移动质量-梁振动的时空有限元法数值分析

为了获得移动质量沿梁匀速运动的系统动态响应,建立了时空有限元数值求解模型.考虑移动质量惯性项,得到移动质量-梁时变系统的动力学方程.应用时空有限元法.得到了移动集中质量作用下Bernoulli-Euler梁离散单元的质量矩阵、刚度矩阵.与Newmark-β法、Wilson-θ法计算结果进行比较,时空有限无法计算梁的动态响应的精度更高.     

随机振动和谐波载荷下基于相对位移分析的结构间隙变化评估

为正确评估紧凑结构在外载荷作用下的安全性,推导随机振动和谐波载荷下相对位移的理论计算方法,考虑响应点之间相关性,并以某工程结构有限元分析为例,对随机振动和谐波载荷下结构中的间隙变化进行评估。将谐波载荷与随机振动载荷进行等效转换,数值模拟结果表明2种方法的计算结果完全吻合,相互验证分析方法的有效性和正确性。     

谱元法在求解刚架结构动力学问题中的应用

把谱元法应用于刚架结构的动力学响应计算和分析中.建立了杆和梁的谱单元动力学刚度阵,针对刚架结构组装了整体动力学刚度阵,建立了整体结构的运动方程,计算了结构的固有频率和时域响应,并与采用有限元方法得到的结果进行了对比.从结果中可以看出谱元法在数值模拟中的独特优势.     

A simulation-and-regression approach for stochastic dynamic programs with endogenous state variables

We investigate the optimum control of a stochastic system, in the presence of both exogenous (control-independent) stochastic state variables and endogenous (control-dependent) state variables. Our solution approach relies on simulations and regressions with respect to the state variables, but also grafts the endogenous state variable into the simulation paths. That is, unlike most other simulation approaches found in the literature, no discretization of the endogenous variable is required. The approach is meant to handle several stochastic variables, offers a high level of flexibility in their modeling, and should be at its best in non time-homogenous cases, when the optimal policy structure changes with time. We provide numerical results for a dam-based hydropower application, where the exogenous variable is the stochastic spot price of power, and the endogenous variable is the water level in the reservoir.     

Biomechanical study of lumbar spine with dynamic stabilization device using finite element method

A finite element model of the human lumbar spine was developed for the parametric study of the stiffness of a dynamic stabilization device. Spinal segments (L2-L5) were used to investigate the effect of dynamic stabilization and the influence on the mobility of adjacent intervertebral segments. Three spines were analysed and compared: (1) a lumbar spine with intact discs, used as a reference; (2) a fused spine with a fixation device after the the facetectomy and the total laminectomy; and (3) a spine stabilized with a dynamic stabilization device after the facetectomy and the total laminectomy.The Range of Motion (ROM) and the intervertebral disc pressure of L3-L4 and the ROM and disc pressure of the adjacent segments were examined to determine the influence of the implant on the adjacent segments.In the case of the dynamically stabilized spine, the total ROM was greater than that of the fused spine but similar to that of the intact spine. Furthermore, the disc pressure on the adjacent segments in the fused spine was greater than that of the intact spine, but the disc pressure of the dynamically stabilized spine was similar to the intact spine. In particular, the dynamic stabilization device having a stiffness of 10-15 N/mm made the destabilized spine more similar to the intact spine.Results indicated that the use of dynamic stabilization devices restored functionality closer to that of the intact spine as compared to the fused spine. The stiffness value utilized in the device was determined to be an important design parameter in manufacturing the dynamic stabilization device.     

用有限元法计算特征值问题的一种新的动态凝聚方法

有限元法是常用的建模方法,由于所建模型具有较大的自由度,通常需要进行降阶处理.一般来讲,模型前几阶特征值和特征向量可以较精确地得到,利用所得到的特征值和主振型分量(在特征向量中与所给定的主自由度对应的振型分量),本文提出了一种新的动态凝聚方法,该方法是通过迭代方式,利用所得到的特征值和主振型分量对Guyan降阶法所得到的降阶模型进行修正.与同类方法相比,本文方法具有较高的计算精度和很小的计算量,且迭代收敛的稳定性很好.最后本文给出了一个计算实例.