An efficient D-vine copula-based coupling uncertainty analysis for variable-stiffness composites

For Variable Stiffness (VS) composites with steered curvilinear tow paths, the fiber orientation angle varies continuously throughout the laminate, and is not required to be straight, parallel and uniform within each ply as in conventional composite laminates. Hence, the thermal properties (conduction), as well as the structural stiffness and strength, vary as functions of location in the laminate, and the associated composite structure is often called a “variable stiffness” composite structure. The steered fibers lead not only to the alteration of mechanical load paths, but also to the alteration of thermal paths that may result in favorable temperature distributions within the laminate and improve the laminate performance. Evaluation of VS laminate performance under thermal loading is the focus of this chapter. Thermal performance evaluations require experimental and numerical analysis of VS laminates under different processing and loading conditions. One of the advantages of using composite materials in many applications is the tailoring capability of the laminate, not only during the design phase but also for manufacturing. Heat transfer through variable conduction and chemical reaction (degree of cure) occurring during manufacturing (curing) plays an important role in the final thermal and mechanical performance, and shape of composite structures. Three case studies are presented in this chapter to evaluate the thermal performance of VS laminates prior to and after manufacturing. The first case study is a numerical analysis that investigates the effect of variable conductivity within the VS laminate on the temperature and degree of cure distribution during its cure cycle. The second case study is a numerical analysis that investigates the transient thermal performance of a rectangular VS composite laminate. Variable thermal conductivity will affect the temperature profile and results are compared to a unidirectional composite. The effect of fiber steering on transient time and steady state solutions is compared to the effect of unidirectional fibers. The third case study is an experimental one that was conducted to evaluate the thermal performance of two variable stiffness panels fabricated using an Advanced Fiber Placement (AFP) Machine.

Design, manufacturing, and testing of a variable stiffness composite cylinder

The design space for high-performance lightweight composite structures has grown considerably since the advent of the variable stiffness concept. In fact, variable stiffness composites have been found to improve buckling performance and dynamic stability, and to tune the structure’s dynamic response by tailoring structural stiffness. Thus, in order to exploit this wider design space, efficient linear analysis tools have an important role in preliminary design of variable stiffness structures, enabling designers to find more effective solutions when considering prestressed dynamically excited aerospace components. Considering this, a multi-domain Ritz method for eigenfrequency, transient and dynamic instability analysis of prestressed variable stiffness laminated doubly-curved shell structures is presented. Working within the first-order shear deformation theory, Sanders–Koiter shell kinematics allow general orthogonal surfaces to be modelled without further assumptions on the shallowness or on the thinness of the structure. The efficiency of the proposed Ritz method is granted by using Legendre orthogonal polynomials as displacement trial functions, while flexibility in modelling and design is given by penalty techniques that allow stiffened variable angle tow shell structures to be modelled as an assembly of shell-like domains. The proposed approach is verified by comparison with published benchmark results and finite element solutions. Original solutions are presented for a prestressed stiffened variable angle tow shell structure, which show great accuracy with an order of magnitude fewer variables when compared to standard finite element procedures, proving the reliability and efficiency of the present method in dealing with the dynamic analysis of multi-part aerospace structures.

Failure analysis of variable stiffness composite plates using continuum damage mechanics models

It is well known that fabrication processes inevitably lead to defects in the manufactured components. However, thanks to the new capabilities of the manufacturing procedures that have emerged during the last decades, the number of imperfections has diminished while numerical models can describe the ground truth designs. Even so, a variety of defects has not been studied yet, let alone the coupling among them. This paper aims to characterise the buckling response of Variable Stiffness Composite (VSC) plates subjected to spatially varying fibre volume content as well as fibre misalignments, yielding a multiscale sensitivity analysis. On the one hand, VSCs have been modelled by means of the Carrera Unified Formulation (CUF) and a layer-wise (LW) approach, with which independent stochastic fields can be assigned to each composite layer. On the other hand, microscale analysis has been performed by employing CUF-based Mechanics of Structure Genome (MSG), which was used to build surrogate models that relate the fibre volume fraction and the material elastic properties. Then, stochastic buckling analyses were carried out following a multiscale Monte Carlo analysis to characterise the buckling load distributions statistically. Eventually, it was demonstrated that this multiscale sensitivity approach can be accelerated by an adequate usage of sampling techniques and surrogate models such as Polynomial Chaos Expansion (PCE). Finally, it has been shown that sensitivity is greatly affected by nominal fibre orientation and the multiscale uncertainty features.

Multi-level intelligent design of variable angle tow laminates via image-driven method

Introduction: The greatest part of constructed dams belongs to embankment dams and there are many examples of their failures throughout history. About one-third of the world’s dam failures have been caused by flood overtopping, which indicates that flood overtopping is an important factor affecting reservoir projects’ safety. Moreover, because of a poor understanding of the randomness of floods, reservoir water levels during flood seasons are often lowered artificially in order to avoid overtopping and protect the lives and property of downstream residents. So, estimation of dam overtopping risk with regard to uncertainties is more important than achieving the dam’s safety. This study presents the procedure for risk evaluation of dam overtopping due to various uncertaintiess in inflows and reservoir initial condition. Materials and Methods: This study aims to present a practical approach and compare the different uncertainty analysis methods in the evaluation of dam overtopping risk due to flood. For this purpose, Monte Carlo simulation and Latin hypercube sampling methods were used to calculate the overtopping risk, evaluate the uncertainty, and calculate the highest water level during different flood events. To assess these methods from a practical point of view, the Maroon dam was chosen for the case study. Figure. 1 indicates the work procedure, including three parts: 1) Identification and evaluation of effective factors on flood routing and dam overtopping, 2) Data collection and analysis for reservoir routing and uncertainty analysis, 3) Uncertainty and risk analysis. Figure 1- Diagram of dam overtopping risk evaluation Results and Discussion: Figure 2 shows the results of the computed overtopping risks for the Maroon Dam without considering the wind effect, for the initial water level of 504 m as an example. As it is shown in Figure. 2, the trends of the risk curves computed by the different uncertainty analysis methods are similar. As it can be seen, the risk curves computed by the LHS are slightly higher than those curves computed by the MCS method. Also as it is observed, the differences between risk values of the two methods increase in longer return periods. Variations of overtopping risk with increasing the initial water level and return period related to overtopping risk in the 2-year return period for the initial water level of 470 m are shown in Table1. The results show that elongation of return period plays a more important role in increasing the risk, than the increase of initial water level. T Method 2→2 2→50 2→100 2→1000 2→5000 2→10000 470→470 MCS 1 5 9 23 42.36 58 470→478 2 7 15.6 37 58.34 79 470→485 5.6 13.6 28.6 55.6 85.67 112.6 470→493 10.3 32.6 54 95.6 127.34 152 470→504 40.3 83 117.3 165 200.34 224.3 470→470 LHS 1 5.34 11 25.3 43 60.3 470→478 2.3 8.6 18 39.3 60.67 84 470→485 5.3 17.3 32.6 58.3 89 114.6 470→493 13.3 37.6 57.6 97 133.34 160.3 470→504 41.6 87.3 119.6 173 205 233.3 Figure 2- Overtopping risk in the initial water level of 504 m, without considering the wind effect Conclusions: This study applies MCS and LHS methods to analyze the uncertainty and evaluate the dam overtopping risk consideringthe uncertainties in input variables, such as quintile of flood peak discharge, initial levels of water and spill coefficients. The results show that the uncertainty of water level calculated by MCS is higher than that calculated by LHS. In addition, the overtopping risk calculated by LHS is higher than that calculated by MCS. Furthermore, the increase of inflow rate influences the variations of the overtopping risk more than the increase of the return period. In addition, evaluation of the results indicates that the overtopping risk is an important issue in the Maroon dam. So, a comprehensiverisk analysis procedure in conjunction with uncertainty gives very important information for decision makers to make better judgments in dam operation based on uncertainty in inputs.

The reliability-based design optimization (RBDO) using performance measure approach for problems with correlated input variables requires a transformation from the correlated input random variables into independent standard normal variables. For the transformation with correlated input variables, the two most representative transformations, the Rosenblatt and Nataf transformations, are investigated. The Rosenblatt transformation requires a joint cumulative distribution function (CDF). Thus, the Rosenblatt transformation can be used only if the joint CDF is given or input variables are independent. In the Nataf transformation, the joint CDF is approximated using the Gaussian copula, marginal CDFs, and covariance of the input correlated variables. Using the generated CDF, the correlated input variables are transformed into correlated normal variables and then the correlated normal variables are transformed into independent standard normal variables through a linear transformation. Thus, the Nataf transformation can accurately estimates joint normal and some lognormal CDFs of the input variable that cover broad engineering applications. This paper develops a PMA-based RBDO method for problems with correlated random input variables using the Gaussian copula. Several numerical examples show that the correlated random input variables significantly affect RBDO results.

The functional advantages of tailored stiffness, often seen in nature, are also utilized in composite structures. Advancements in the multiaxis tow placement and automated fiber placement (AFP) machines led to the development of variable angle tow (VAT) composites, also referred further as variable stiffness composites (VSC). These composites are shown to effectively enhance the stress distribution and buckling load capacity of structures with greater flexibility on the design space. This review systematically presents the status of recent research on the topic of VSCs. Various manufacturing techniques of VSC are discussed; constraints and the defects associated with the manufacturing processes are enlisted. The review highlights the optimization studies based on the fiber profile and macro-scale stiffness invariants. Several studies existing in the domain of buckling, vibration, and aeroelastic tailoring of angle tow composites are summarized to connect the important aspects of analysis and present a holistic approach for future studies in this area.

The present dissertation describes the research work carried out towards: I. developing of analytical and numerical models for accurate and efficient analysis of laminates and sandwiches under different loading conditions (static analyses, impulsive loading and low velocity impacts); II. developing of techniques to improve structural performances of composites structures by tuning their energy absorption mechanism; A higher order theory (AD-ZZ) including zig-zag effects, transverse shear continuity, variable transverse displacement and layerwise representation is presented for analysis of laminates and sandwiches. This theory is cast in such a way that the unknowns in the model will involve only five variables like equivalent single layer models: the three displacement components and the two shear rotations of the normals on the reference surface. It is advantageous to formulate the theory in this way, as it provides the opportunity to refine its through-the-thickness representation by adding computational layers in the thickness direction without increasing the number of unknowns. After extensive evaluation of the accuracy obtained by the above mentioned theory for a variety of structures, the AD-ZZ model is enriched by adding a set of continuity functions in order to treat laminates and structures with in-plane discontinuities (e.g. bonded joints, two material wedge) under a unified approach without adding new unknowns. Speaking of techniques to improve structural performances, two approaches are presented: variable stiffness composites and stitching. The former tool works on the strain energy of the structure computing fibre distributions that minimize unwanted energy contributions and maximize the wanted ones. The latter prescribes the insertions of transverse reinforcements in sandwich structures. The homogenized mechanical properties of this structure, required by the AD-ZZ model to perform the analysis, are evaluated through virtual material tests using a 3D FE model. The results obtained applying these techniques demonstrate that both are effective in reducing transverse stresses at critical interfaces and, in certain cases, in improving bending stiffness. New C0 displacement-based and stress-based finite elements for analysis of laminated and sandwich plates based on the AD-ZZ model are developed. To achieve this goal, a new technique that converts derivatives of functional degrees of freedom contained in the AD-ZZ is presented. Both the elements satisfy all the requirements of computationally efficient finite element models for analysis of multilayered structures, namely 1) the number of degrees of freedom is independent from the number of layers; 2) no shear correction factors or penalty number are added in the formulation. Consistent shear fields are obtained for the present finite element formulations. Numerical results demonstrate that these new elements are robust, accurate and computationally efficient for analysis of multilayered structures. In order to study structures subjected to impact loading a new simulation procedure is developed. It is based on the AD-ZZ model and considers the crushing behaviour of soft-like media without performing each time a detailed 3D finite element analysis. Numerical results show that the response of composites undergoing low velocity impact is very accurately predicted with low computational effort. Overall, the present procedure holds great promise for analysis of laminated and sandwich structures undergoing low velocity impact

A new perspective on the simulation of cross-correlated random fields

This paper highlights the shortfalls of Modern Portfolio Theory (MPT). Amongst other flaws, MPT assumes that returns are normally distributed, that correlations are linear and risks are symmetrical. We propose a dynamic and flexible scenario-based approach to portfolio selection that incorporates an investor's economic forecast. Extreme Value Theory (EVT) is used to capture the skewness and kurtosis inherent in asset class returns and account for the volatility clustering and extreme co-movements across asset classes. The estimation consists of using an asymmetric GJR-GARCH model to extract filtered residuals for each asset class return. Subsequently, a marginal cumulative distribution function (CDF) of each asset class is constructed by using a Gaussian-kernel estimation for the interior, together with a generalised Pareto distribution (GPD) for the upper and lower tails. The distribution of exceedance method is applied to find residuals in the tails. A Student's t copula is then fitted to the data to induce correlation between the simulated residuals of each asset class. A Monte Carlo technique is applied to simulate standardised residuals, which represent a univariate stochastic process when viewed in isolation but maintain the correlation induced by the copula. The results are mean-CVaR optimised portfolios, which are derived based on an investor's forward-looking expectation.

In uncertainty analysis and reliability-based multidisciplinary design and optimization (RBMDO) of engineering structures, the saddlepoint approximation (SA) method can be utilized to enhance the accuracy and efficiency of reliability evaluation. However, the random variables involved in SA should be easy to handle. Additionally, the corresponding saddlepoint equation should not be complicated. Both of them limit the application of SA for engineering problems. The moment method can construct an approximate cumulative distribution function of the performance function based on the first few statistical moments. However, the traditional moment matching method is not very accurate generally. In order to take advantage of the SA method and the moment matching method to enhance the efficiency of design and optimization, a fourth-moment saddlepoint approximation (FMSA) method is introduced into RBMDO. In FMSA, the approximate cumulative generating functions are constructed based on the first four moments of the limit state function. The probability density function and cumulative distribution function are estimated based on this approximate cumulative generating function. Furthermore, the FMSA method is introduced and combined into RBMDO within the framework of sequence optimization and reliability assessment, which is based on the performance measure approach strategy. Two engineering examples are introduced to verify the effectiveness of proposed method.

Decline rate-time equations, including uncertainty characteristics of production data, were developed to estimate the probabilistic reserves and production life of hydrocarbon reservoirs. The uncertainty characteristic, or residual function, is a statistical distribution of the difference between field data and the deterministic production decline curve. The reserves may be estimated directly by summing the production rates, including the residual function, which is referred to as the summation method. Alternatively, other equations for estimating the probabilistic reserves and production life are also derived, based on the integration of the decline time-rate equations that include uncertainty. This is referred to as the integration method. In both methods, Monte-Carlo simulation is used. The uncertainty analysis is applied to the production data from two gas fields located in Taiwan: the Chinshui and the Tiechenshan gas fields. In the case studies, production data for the Chinshui gas field shows an exponential decline, whereas the Tiechenshan gas field exhibits hyperbolic decline. The residual function for the Chinshui gas field is a normal distribution with a mean of zero. In the Tiechenshan gas field, the residual function approximates a normal distribution. Thus, the determined probabilistic reserves for both fields also have a normal distribution. The range of reserves, which is defined as the difference between proven and mean reserves, is estimated using both summation and integration methods. The Chinshui gas field is 10.6% of mean reserve value, whereas for the Tiechenshan range of probabilisitc reserves of value is 8.9%. The methods derived in this study are more accurate than those determined by the conventional probability method where the range of reserves depends on the assumed ranges of the parameters used. A set of new decline curves equations in residual form, instead of conventional equations of decline curves, are derived. In addition, the equations of cumulative production in residual form are derived. A new method based on the equations of decline curves in residual form and those of cumulative production in residual form are developed (uncertainty analysis method). Case studies with conventional decline curve analysis, probabilistic decline curve analysis, and a new method of uncertainty analysis are presented. The results of the three methods are compared and discussed.

Uncertainty analysis is a key element of sound techno-economic analysis (TEA) of CO2 Capture and Storage (CCS) technologies and systems, and in the communication of TEA results. Many CCS technologies are relatively novel, with only few large-scale projects constructed and in operation to date. Therefore, uncertainties in technology performance and costs are often substantial, making it imperative that they be characterized and reported. Although uncertainty analysis itself is not novel, with some methods already frequently used by the CCS TEA community, a document that provides a comprehensive overview of methods and approaches, as well as guidance on their selection and use, is still lacking. Given its importance, we seek to fill this gap by providing a critical review of uncertainty analysis methods along with guidance on the selection and use of these methods for CCS TEAs, highlighting good practice and examples from the CCS literature. The paper starts by identifying the different audiences for CCS TEAs, the different modelling approaches available for CCS technology performance and cost analysis, and the different roles that uncertainty analysis may play. It then continues to discuss established, as well as emerging, uncertainty analysis methods and addresses how and when each method is best used, as well as common pitfalls. We argue that the most commonly used method of one-parameter-at-a-time ‘local’ sensitivity analysis may often be a suboptimal choice, and that other approaches may be more suitable or lead to more insight, especially since uncertainty analysis software is becoming more widespread and easier to use. Finally, the paper discusses the benefits of advanced uses of uncertainty analysis in, for instance, the design of CCS experiments or in the design and planning of CCS infrastructure. Sound uncertainty analysis has an important role to play in TEAs of CCS technologies and systems, and there are many opportunities to bring the use of uncertainty analysis to a higher level than currently practiced. This review of and guidance on available methods is intended to help accelerate continued methods development and their application to more robust and meaningful CCS performance and costing studies.