An unsteady adaptive stochastic finite elements formulation for rigid-body fluid–structure interaction

Fluid–structure interactions provide design constraints in many fields, yet methods available for their analysis normally assume that the structural properties are exactly known. In this contribution, these properties are more realistically modeled using random fields. Stochastic finite element methods are applied to perform uncertainty and reliability analysis on fluid–structure interaction problems with random input parameters. As an example we consider panel divergence and panel flutter. Numerical simulations demonstrate the appropriateness of sensitivitybased methods for characterization of the statistical moments of the critical points as well as for the determination of the probability of occurrence of undesired phenomena

Structural stability and reliability of the underground steel tanks with the Stochastic Finite Element Method

The Unsteady Adaptive Stochastic Finite Elements (UASFE) method resolves the effect of randomness in numerical simulations of single-mode aeroelastic responses with a constant accuracy in time for a constant number of samples. In this paper, the UASFE framework is extended to multi-frequency responses and continuous structures by employing a wavelet decomposition pre-processing step to decompose the sampled multi-frequency signals into single-frequency components. The effect of the randomness on the multifrequency response is then obtained by summing the results of the UASFE interpolation at constant phase for the different frequency components. Results are presented for the three-dimensional transonic AGARD 445.6 wing aeroelastic benchmark.

The lifespan of a reinforced concrete (RC) structure can be greatly influenced by the spatial variability of its material characteristics which, in particular, explains the observed or measured reduction of the tensile strength at first crack when the volume under tension increases. This paper discusses the ability of accounting for the spatial variability of the tensile strength of concrete in RC structures using a stochastic finite element (SFE) method based on random field simulations. In this work, the generation of random fields on the concrete tensile strength aims at computing the force corresponding to the first crack occurrence, and the reduced tensile strength of the structure. The method can be applied in particular to large‐sized structures, which show a pronounced size effect, for different types of loading. The method consists of, first, estimating the mean of the random field, using the analytical approach of the weakest link and localization method (WL ). Then, the discretized random field is defined on a particular 2D or 3D grid, and it is finally projected on the finite element mesh of the studied structure. The study of the parameters that influence the prediction of the cumulative density functions (CDFs) of the rupture force or the tensile strength is highlighted using experimental series of concrete beams having different volumes and subjected to 4‐point bending loading. Moreover, the SFE method is applied to a RC tie‐beam under tensile loading, characterized by a weak stress gradient, which complicates the prediction of crack positions.

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.

Until recently, stochastic structural mechanics has addressed the issue of deterministic structures subjected to random loading. With the availability of more accurate analysis and design tools, however, quantifying the sensitivity of model predictions to uncertainty in the mechanical properties of structures has become possible. It has been observed, with the help of these tools, that this type of uncertainty can be more significant to the overall predictions of a particular structural model than the more traditional fluctuations attributed to external loads. In view of that, recent procedures have been developed for representing uncertainties in the parameters of a structural model, as well as, for propagating this uncertainty to obtain the associated uncertainty in the predicted response. The stochastic finite element method is a procedure for performing such an analysis, whereby the spatial extent of the structure has been represented within the context of the finite element method. This chapter describes a recent implementation of the stochastic finite element method that combines theoretical rigor with generality and efficiency of implementation. Specifically, the Spectral Stochastic Finite Element Method (SSFEM) presented in this chapter addresses the situation where the uncertain material properties are realizations of a spatially fluctuating random field. The formulation relies on discretizing the random processes using spectral expansions, thus eliminating the correlation between the requisite mesh size to meet energy-based convergence criteria, and the scales of fluctuation of the random material properties involved.

We introduce novel adaptive methods to approximate moments of solutions of partial differential Equations (PDEs) with uncertain parametric inputs. A typical problem in Uncertainty Quantification is the approximation of the expected values of quantities of interest of the solution, which requires the efficient numerical approximation of high-dimensional integrals. We perform this task by a class of deterministic quasi-Monte Carlo integration rules derived from Polynomial lattices, that allows to control a-posteriori the integration error without querying the governing PDE and does not incur the curse of dimensionality. Based on an abstract formulation of adaptive finite element methods (AFEM) for deterministic problems, we infer convergence of the combined adaptive algorithms in the parameter and physical space. We propose a selection of examples of PDEs admissible for these algorithms. Finally, we present numerical evidence of convergence for a model diffusion PDE.

Reliability analysis of slope Stability using stochastic finite element method

Shear lag analysis by the adaptive finite element method: 2. Analysis of complex plated structures

A procedure is presented that allows monitoring of the dynamic regime of pelagic ecosystems in the North Atlantic Ocean. It uses a diversity index (mean number of species per con- tinuous plankton recorder (CPR) sample) and species assemblage indicators calculated from all calanoid copepods (108 species) identified by the CPR survey since 1958. In any region of the North Atlantic covered by the CPR survey, the procedure calculates the values of these ecosystem indica- tors using a constant number of samples, selected randomly, and checks any bias associated with the spatial and temporal heterogeneity of the sampling. The procedure is described and illustrated in this study using CPR data in the central North Sea. The usefulness of this tool is demonstrated by a study of the sensitivity of the indicators to the number of CPR samples employed in the estimation of monthly or annual means. Firstly, it is shown that on an annual scale the procedure improves the quality of estimations. However, when long-term changes of a plankton indicator are examined for a particular month, it is crucial to use an appropriate number of CPR samples to detect a significant link between the biological and hydro-climatic environment. Secondly, the importance of using species assemblage indicators is demonstrated. They allowed the detection and the ecological interpretation of both episodic events and the regime shift in the central North Sea. It is shown that the regime shift in the North Sea had a strong impact on plankton community structure and was marked by an increase in diversity related to a progressive increase in warm-water species and a decrease in cold-water species. This study also has important implications for researchers using the CPR data. It demonstrates the robustness of the CPR data, but on the other hand also shows that the spatial and temporal heterogeneity should be assessed with care before interpreting the long-term changes of a CPR-derived plankton indicator.

Posets are easily testable

In this paper, we develop an adaptive high-order surface finite element method (FEM) incorporating spectral deferred correction method for chain contour discretization to solve polymeric self-consistent field equations on general curved surfaces. The high-order surface FEM is obtained by the high-order surface geometrical approximation and the high-order function space approximation. Numerical results demonstrate that the precision order of these methods is consistent with theoretical prediction. In order to describe the sharp interface in the strongly segregated system more accurately, an adaptive FEM equipped with a new $Log$ marking strategy is proposed. The $Log$ marking strategy can not only label the elements that need to be refined or coarsened, but also give the refined or coarsened times, which can make full use of the information of a posterior error estimator and improve the efficiency of the adaptive algorithm. To demonstrate the power of our approach, we investigate the self-assembled patterns of diblock copolymers on several distinct curved surfaces. Numerical results illustrate the efficiency of the proposed method, especially for strongly segregated systems with economical discretization nodes.

To analysis the reliability of Dongjiang Bridge, a steel truss bridge stiffened with rigid cables, a new analysis method that combined with each advantage of some common reliability computing method was put forward, which could get the probability distribution and numeric attributes of complex bridge structure's response expediently, and then get the reliability index and failure probability. A stochastic finite element model was established to analyze this bridge, in which some parameters such as material, geometric dimensioning, loads, and so on, were simulated as stochastic variables. The analysis result show that the lowest reliability index of main member bar's stress is 5.10, more than the allowable value 4.7, and the corresponding failure probability is 1.71×10-7. The reliability index of mid-span deflection in serviceability limit states reach up to 6.99 and its failure probability is 1.53×10-7. All results indicate that the strength and stiffness of Dongjiang Bridge has higher reliability.

A Critical Study on the Haldar and Mahadevan's Reliability Analysis

Chapter 3 presents the fundamentals of the Stochastic Finite Element Method in the framework of the stochastic formulation of the virtual work principle. The resulting stochastic partial differential equations are solved with either non-intrusive Monte Carlo simulation methods, or intrusive approaches such as the versatile spectral stochastic finite element method. Additional approximate methodologies such as the Neumann and Taylor series expansion methods are also presented together with some exact analytic solutions that are available for statically determinate stochastic structures. The concept of the variability response function is then developed and generalized for general stochastic finite element systems.