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.     

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.     

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.     

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.     

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.     

Multilevel Monte Carlo method with applications to stochastic partial differential equations

In this work, the approximation of Hilbert-space-valued random variables is combined with the approximation of the expectation by a multilevel Monte Carlo (MLMC) method. The number of samples on the different levels of the multilevel approximation are chosen such that the errors are balanced. The overall work then decreases in the optimal case to O(h ^?2) if h is the error of the approximation. The MLMC method is applied to functions of solutions of parabolic and hyperbolic stochastic partial differential equations as needed, for example, for option pricing. Simulations complete the paper.     

Free vibrational analysis of axially loaded bending-torsion coupled beams: a dynamic finite element

In this article, a dynamic finite element formulation for the free vibration analysis of axially loaded bending-torsion coupled beams is presented. Based on the Euler–Bernoulli and St. Venant beam theories, the exact solutions of the differential equations governing the uncoupled vibrations of an axially loaded uniform beam are found. Then, employing these solutions as basis functions, the analytical expressions for uncoupled bending and torsional dynamic shape functions are derived. Exploiting the principle of virtual work, together with the variable approximations based on the resulting shape functions, leads to a single frequency dependent element matrix which has both mass and stiffness properties. The application of the theory is demonstrated by an illustrative example of a bending-torsion coupled beam with cantilever end conditions, for which the influence of axial force on the natural frequencies is studied. The correctness of the theory is confirmed by the published results and numerical checks.     

Interval optimization of dynamic response for structures with interval parameters

This paper presents an interval optimization method for the dynamic response of structures with interval parameters. The matrices of structures with interval parameters are given. Combining the interval extension of function with the perturbation theory of dynamic response, the method for interval dynamic response analysis is derived. The interval optimization problem is transformed into a corresponding deterministic one. Because the mean values and the uncertainties of the interval parameters can be elected as the design variables, more information of the optimization results can be obtained by the present method than that obtained by the deterministic one. The present method is implemented for a truss structure and a frame structure. The numerical results show that the method is effective.     

Application of boundary-domain element method to the free vibration problem of plate structures

The boundary-domain element method is applied to the free vibration problem of thin-walled plate structures. The static fundamental solutions are used for the derivation of the integral equations for both in-plane and out-of-plane motions. All the integral equations to be implemented are regularized up to an integrable order and then discretized by means of the boundary-domain element method. The entire system of equations for the plate structures composed of thin elastic plates is obtained by assembling the equations for each plate component satisfying the equilibrium and compatibility conditions on the connected edge as well as the boundary conditions. The algebraic eigenvalue equation is derived from this system of equations and is able to be solved by using the standard solver to obtain eigenfrequencies and eigenmodes. Numerical analysis is carried out for a few example problems and the computational aspects are discussed.