Uncertainties In Material Properties Research Articles

Probabilistic analysis of geosynthetic-reinforced and pile-supported embankments

This paper presents a probabilistic analysis of a geosynthetics-reinforced and pile-supported (GRPS) embankment by using reliability, sensitivity and uncertainty propagation techniques. Compared to the traditional deterministic evaluation methods, the analyses performed in this work allow designers to rationally consider the uncertainties of material properties and quantify their influences on the embankment performance. Additionally, complementary information/results could be obtained which enables an informed design decision for engineers. This feature is further enhanced in the paper by proposing a new analysis procedure in which four carefully selected probabilistic techniques are combined with an efficient deterministic model. The contribution of the proposed procedure lies in providing a variety of valuable results (e.g., failure probability and sensitivity index) related to the GRPS embankment performance but with a limited computational time. An application of the procedure is presented in the paper. The obtained results revealed that the studied GRPS embankments have relatively low failure probabilities considering a usual traffic loading but could be risky under extreme loading conditions. Based on the sensitivity analysis, all the random parameters are ranked according to their quantified importance. Some discussions are also provided, trying to link the procedure with practical designs and to explain the uncertainty modeling with more details.

Design of a 3D composite structure considering data uncertainties and various failure mechanisms

AbstractThis study has been performed within the research project MuScaBlaDes “Multi‐scale failure analysis with polymorphic uncertainties for optimal design of rotor blades” as part of the DFG Priority Programme (SPP 1886) “Polymorphic Uncertainty Modelling for the Numerical Design of Structures” started in 2016. A three‐dimensional structure consisting of glass fiber reinforced plastic (GFRP) components glued by epoxy resin (EP) is designed considering uncertainties in geometrical and material properties. Different stresses, the vertical deflection, the global stability and the fatigue behavior should fulfill the defined restrictions in the optimization process. A global optimum is searched for four different topologies by minimizing the elastic strain energy density as well as the overall manufacturing costs.

A note on the effect of material uncertainty on acoustic source localization error in anisotropic plates

The uncertainty in material properties of an anisotropic plate may influence the acoustic source localization process undertaken for the plate. To study this effect of material uncertainty, the two moduli of elasticity of an orthotropic plate material are considered in this note as independent random variables and the propagation of this material uncertainty through the wave front shape-based acoustic source localization approach is investigated. Assuming lognormal probability distributions for the two random variables, several design points in lognormal spaces are picked using Latin Hypercube Sampling. Finite element analysis is performed for each design point to simulate the elastic wave propagation due to an acoustic event and wave front shape-based approach is applied to estimate the source location. The time-of-arrivals and source localization errors obtained for each design point are considered as separate response functions at that design point and regression kriging metamodels through the responses at the design points are constructed. Monte Carlo simulations are carried out using these metamodels to obtain the distribution parameters (i.e., ranges, means and standard deviations) of the time-of-arrivals and localization errors. A global sensitivity analysis is performed to estimate the effect of each random variable on the localization errors. It is observed that for lognormally distributed moduli of elasticity with same coefficients of variation, uncertainty in the modulus of elasticity in the major direction affects the source localization accuracy more compared to the uncertainty in the modulus of elasticity in the minor direction, particularly when the ellipse-based technique is used.

Flexural Behavior of Steel Fiber Reinforced Concrete Beams: Probabilistic Numerical Modeling and Sensitivity Analysis

One of the principle issues concerning the practical application of steel fiber reinforced concrete (SFRC) is the uncertainty related to its structural behavior, primarily caused by the partially random distribution and orientation of steel fibers in SFRC structural elements. This paper aims to provide a better understanding of how the variance of material properties of the SFRC affects the flexural behavior of SFRC beams. First, a distributed plasticity fiber finite element model of beam flexural behavior is proposed and validated. Then, probability distributions of selected material properties are defined based on existing probabilistic models and experimental results from the literature. Finally, a variance-based sensitivity analysis is performed using Sobol’ indices to identify uncertainties in material properties that contribute most to the uncertainties related to three characteristic points of a beam’s flexural behavior: first crack, yield, and collapse point. Sensitivity analysis is performed by surrogating the numerical model using polynomial chaos expansion. The variance in residual tensile strength is identified as the main contributor to the variance in the flexural behavior of an SFRC beam used in the case study.

A minimally invasive, efficient method for propagation of full‐field uncertainty in solid dynamics

AbstractWe present a minimally invasive method for forward propagation of material property uncertainty to full‐field quantities of interest in solid dynamics. Full‐field uncertainty quantification enables the design of complex systems where quantities of interest, such as failure points, are not known a priori . The method, motivated by the well‐known probability density function (PDF) propagation method of turbulence modeling, uses an ensemble of solutions to provide the joint PDF of desired quantities at every point in the domain. A small subset of the ensemble is computed exactly, and the remainder of the samples are computed with approximation of the evolution equations based on those exact solutions. Although the proposed method has commonalities with traditional interpolatory stochastic collocation methods applied directly to quantities of interest, it is distinct and exploits the parameter dependence and smoothness of the driving term of the evolution equations. The implementation is model independent, storage and communication efficient, and straightforward. We demonstrate its efficiency, accuracy, scaling with dimension of the parameter space, and convergence in distribution with two problems: a quasi‐one‐dimensional bar impact, and a two material notched plate impact. For the bar impact problem, we provide an analytical solution to PDF of the solution fields for method validation. With the notched plate problem, we also demonstrate good parallel efficiency and scaling of the method.

Portico Farmhouses of the Vistula Delta: Architecture, Current State and Finite Element Modelling of Timber Roof Truss under Material and Cross-section Uncertainty

ABSTRACT The article focuses on the oldest surviving I-type portico houses located in the Vistula Delta, which were built in the 17th and 18th centuries. The study describes the houses’ origins, structure, details of roof structures and technical condition, and it also includes numerical modelling of a roof truss and its mechanical analysis. Numerical models of the roof trusses are defined and analysed by means of the finite element method. Global sensitivity analysis with the use of the polynomial chaos expansion method is used to study the influence of uncertainties in material properties of wood, such as Young’s modulus and the friction coefficient, and also the beam cross-sectional height in the output of the numerical model of the structure. The outcomes show that the uncertainty of beam height exerts the dominating influence on the maximum deflection of the roof truss, hence accurate measurements of geometry are of great importance in the planning of conservation and renovation of such historic structures.

Six sigma robust multi-objective design optimization of flat plate collector system under uncertain design parameters

Flat plate collectors (FPCs) can be considered as an important alternative to reduce the high costs of electrical energy used in heating water. Therefore, the development of FPCs with optimal performance has been much focused upon in recent research initiatives. However, due to the existence of uncertainties in material properties, geometry and environmental conditions, it is imperative to incorporate uncertainty analysis into the design optimization to obtain reliable results. In this work, a six sigma robust multi-objective optimization (SSRMO) of the FPC is developed where uncertain design parameters (UDP) are considered. A combined multi-objective modified imperialist competitive algorithm (MOMICA) and Monte Carlo simulation method (MCSM) is developed and used for the SSRMO. The SSRMO considers simultaneously three objectives functions including the FPC efficiency, the FPC cost and their sensitivities to UDP. The obtained results showed that the FPC design obtained by determinist optimization presents 31% sensitivity in terms of efficiency and 27% sensitivity in terms of total cost. However, the SSRMO provides a robust FPC design with the same optimal performances comparable to those obtained by deterministic method but with much lower sensitivity to UDP. Moreover, the robust FPC performances sensitivities were reduced by 60% compared to literature results.

Implementation of a novel Six Sigma multi-objective robustness optimization method based on the improved response surface model for bumper system design

In the vehicle industry, a car’s weight and crashworthiness are two conflicting indicators. Owing to uncertainties in loads, geometries, material properties, and operational conditions, deterministic optimization could lead to unreliable or unstable designs and increase the risk of design failure. This study combined the non-dominated sorting genetic algorithm (NSGA-II), Monte Carlo simulation with descriptive sampling (MCSDS), Six Sigma robustness approach, and improved response surface model (IRSM) to develop a general Six Sigma multi-objective reliability-based design optimization method algorithm (6σ-MORO). The proposed method was applied to examine the crashworthiness of a bumper system. In comparison to the response surface model (RSM), the accuracy of the IRSM was improved by only 10 groups of sample points, proving that the IRSM could construct each response surface function efficiently and accurately. Finally, based on the IRSM, a Six Sigma multi-objective robustness optimization was conducted to explore optimal reliability solutions. The results revealed that the proposed method improved the crashworthiness of the bumper system and also ensured its reliability. Note that the optimal design could be fairly promising for engineering design problems with high dimensions.

Evaluation of seismic performance of rotational-friction slip dampers in near-field and far-filed earthquakes

In this study, the performance of rotational-friction slip dampers in steel structures with different heights is investigated by the use of fragility curves. The use of dampers is one of the methods for vibration control of structures by simultaneously increasing both the structural stiffness and damping. Rotational-friction slip dampers are among the passive control devices that dampen the earthquake energy through their stable cyclic behavior. To study the performance of these devices in steel structures, 3, 6 and 9-story steel moment frame structures are designed, and the mentioned dampers are attached to the structure by Chevron braces. To account for the earthquake uncertainty, with the aid of incremental dynamic analysis (IDA), the damper-equipped structure is subjected to both near-field and far-field ground motion records. The acceleration and drift engineering demand parameters are selected as the functions to quantify the damage states, and the design, modeling and material properties uncertainties are considered in accordance with FEMA P-695. Evaluation of statistical results and comparison of the fragility curves, shows that the probability of failure at different damage states decreases when the dampers are added to the structure. This decrease is more remarkable in low-rise structures and near-fault ground motions.

Bayesian optimization for robust design of steel frames with joint and individual probabilistic constraints

This work proposes a Bayesian optimization (BO) method for solving multi-objective robust design optimization (RDO) problems of steel frames under aleatory uncertainty in external loads and material properties. Joint and individual probabilistic constrained RDO problems are formulated to consider two different ways the frame reaches its collapse state. Each problem involves three conflicting objective functions, namely, the total mass of the frame, the mean and variance of the maximum inter-story drift. Since the uncertain objective and probabilistic constraint functions of both problems are implicit within a finite element analysis program and the computation of the probabilistic constraints is an NP-hard problem, BO is used to guide the optimization process toward better solutions after it completes an iteration and offers a set of near Pareto-optimal solutions when it terminates. Specifically, Bayesian regression models called Gaussian processes (GPs) serve as surrogates for the structural responses. Two acquisition functions are then developed for the two RDO problems and a maximization problem of these functions is formulated as a mixed-integer nonlinear programming (MINLP) problem. A new random search coupled with simulated annealing is devised to solve the MINLP problem, thereby locating the most promising point in the input variable space at which the current solutions maximize their chance to be improved and the GP models are refined before the BO starts a new iteration. A test problem and two design examples show that exact or good Pareto-optimal solutions to the RDO problems can be found by the proposed method with 20 iterations.

Influence of input data uncertainties on failure probability of TRISO-coated particle

TRISO coated particle fuel exhibits statistical variations and large uncertainties in physical configurations and material properties from particle to particle. Besides, the dependence of mechanical behavior and failure mechanism of the coating layer on multiple variables (such as density, porosity, anisotropy, fluence and temperature) further increase the complexity of TRISO fuel performance analysis. Consequently, the particle failure probability would change in a relatively wide range. In this paper, we evaluate the effect of input data uncertainties on particle failure probability. First, uncertainty ranges of input data are determined based on literature review and irradiation conditions. Second, a perturbation study on dimensional parameters and material properties is performed. Finally, variation ranges of failure probability upon different input parameters are determined. Results show that the failure rate is sensitive to kernel diameter, SiC thickness and irradiation-induced material properties.

Reliability-based topology optimization for heterogeneous composite structures under interval and convex mixed uncertainties

With the rapid development of intelligent manufacturing and smart design, the optimal design conception for composite laminates is arising more attentions in both academic and engineering fields. Note that most of the current researches focus on the deterministic laminate design whereas the uncertainty-oriented topological design scheme for heterogeneous composites are rarely considered. In view of this, this study will present a new layout optimization process which combines the hybrid uncertainty quantification analysis and non-probabilistic reliability-based topology optimization (NRBTO). Owing to the sample limitation of multi-source uncertainties in material properties and external loads, the non-statistical convex-theoretical set theory is first explored to determine displacement response bounds of multi-layer structures under anisotropic constitutive relationship. As the safety judgment, a novel reliability index corresponding to the local stiffness features is defined via interval and convex mixed uncertainties, and its explicit expression is further deduced by the set-interference principle. To ensure the robustness and convergence of the large-scale iteration in NRBTO framework, the adjoint equations are also analyzed for the design sensitivity updating and the gradient algorithm is involved as well. Several typical optimization cases are presented to demonstrate the validity and effectiveness of the developed methodology.

Statistical Analysis for Transmission Error of Gear System with Mechanical and Thermal Deformation Uncertainties

We establish a robust algorithm to analyze the influence of system uncertainties on the transmission error of a spur gear pair under 2D simplification. The algorithm provides a way of generating smooth cutter profiles with machining uncertainties and measuring the thermal deformation through the uncertainties in material properties. Then, it produces realizations of gear tooth profiles based on the analytical method for accuracy and computational efficiency. Numerical investigations show the statistical analysis on the tooth contact analysis by comparing steel and plastic gears. It is worthwhile remarking that the plastic gear is susceptible to the geometric error caused by thermal deformation. Moreover, although the impact of thermal deformation on steel gear may seem slim, it can have a noticeable influence when it exists with mechanical uncertainties together.

The stochastic aeroelastic response analysis of helicopter rotors using deep and shallow machine learning

This paper addresses the influence of manufacturing variability of a helicopter rotor blade on its aeroelastic responses. An aeroelastic analysis using finite elements in spatial and temporal domains is used to compute the helicopter rotor frequencies, vibratory hub loads, power required and stability in forward flight. The novelty of the work lies in the application of advanced data-driven machine learning (ML) techniques, such as convolution neural networks (CNN), multi-layer perceptron (MLP), random forests, support vector machines and adaptive Gaussian process (GP) for capturing the nonlinear responses of these complex spatio-temporal models to develop an efficient physics-informed ML framework for stochastic rotor analysis. Thus, the work is of practical significance as (i) it accounts for manufacturing uncertainties, (ii) accurately quantifies their effects on nonlinear response of rotor blade and (iii) makes the computationally expensive simulations viable by the use of ML. A rigorous performance assessment of the aforementioned approaches is presented by demonstrating validation on the training dataset and prediction on the test dataset. The contribution of the study lies in the following findings: (i) The uncertainty in composite material and geometric properties can lead to significant variations in the rotor aeroelastic responses and thereby highlighting that the consideration of manufacturing variability in analyzing helicopter rotors is crucial for assessing their behaviour in real-life scenarios. (ii) Precisely, the substantial effect of uncertainty has been observed on the six vibratory hub loads and the damping with the highest impact on the yawing hub moment. Therefore, sufficient factor of safety should be considered in the design to alleviate the effects of perturbation in the simulation results. (iii) Although advanced ML techniques are harder to train, the optimal model configuration is capable of approximating the nonlinear response trends accurately. GP and CNN followed by MLP achieved satisfactory performance. Excellent accuracy achieved by the above ML techniques demonstrates their potential for application in the optimization of rotors under uncertainty.

Influence of structural and material property uncertainties on biomechanics of intervertebral discs - Implications for disc tissue engineering

This study investigated how variations of structural and material properties of human intervertebral discs (IVDs) affect the biomechanical responses of the IVDs under simulated physiological loading conditions using a stochastic finite element (SFE) model. An SFE method, which combined an anatomic FE model of human lumbar L3-4 segment and probabilistic analysis of its structural and material properties, was used to generate a dataset of 500 random disc samples with varying structural and material properties. The sensitivity of the biomechanical responses, including intervertebral displacements/rotations, intradiscal pressures (IDP), fiber stresses and matrix strains of annulus fibrosus (AF), were systematically quantified under various physiological loading conditions, including a 500N compression and 7.5Nm moments in the 3 primary rotations. Significant variations of the IDPs, IVD displacements/rotations, and stress/strain distributions were found using the dataset of 500 ramdom disc samples. Under all the loading conditions, the IDPs were positively correlated with the Poisson's ratio of the NP (r=0.46 to 0.75, p=0.004-0.001) and negatively with the Young's modulus of the annulus matrix (r=-0.48 to -0.65, p=0.003-0.001). The primary intervertebral rotations were significantly affected by the Young's modulus of the annulus matrix (r=-0.44 to -0.71, p=0.001-0.032) and the orientations of the annular fibers (r=-0.45 to -0.69, p=0.001-0.029). The heterogeneity of structures and material properties of the IVD had distinct effects on the biomechanical performances of the IVD. These data could help improve our understanding of the intrinsic biomechanics of the IVD and provide references for optimal design of tissue engineered discs by controlling structural and material properties of the disc components.

GPR-inferred fracture aperture widening in response to a high-pressure tracer injection test at the Äspö Hard Rock Laboratory, Sweden

We assess the performance of the Ground Penetrating Radar (GPR) method in fractured rock formations of very low transmissivity (e.g. T ≈ 10−9–10−10 m2/s for sub-mm apertures) and, more specifically, to image fracture widening induced by high-pressure injections. A field-scale experiment was conducted at the Äspö Hard Rock Laboratory (Sweden) in a tunnel situated at 410 m depth. The tracer test was performed within the most transmissive sections of two boreholes separated by 4.2 m. The electrically resistive tracer solution composed of deionized water and Uranine was expected to lead to decreasing GPR reflections with respect to the saline in situ formation water. The injection pressure was 5000 kPa leading to an injection rate of 8.6 mL/min (at steady state) that was maintained during 25 h, which resulted in a total injected volume of 13 L. To evaluate the fracture pathways between the boreholes, we conducted 3-D surface-based GPR surveys before and at the end of the tracer tests, using 160 MHz and 450 MHz antennas. Difference GPR data between the two acquisitions highlight an increasing fracture reflectivity in-between the boreholes at depths corresponding to the injection interval. GPR-based modeling suggests that the observed increasing reflectivity is not due to the tracer solution, but rather to a 50% widening of the fracture. Considering prevailing uncertainties in material properties, a hydromechanical analysis suggests that such a degree of widening is feasible. This research demonstrates that field-scale in situ GPR experiments may provide constraints on fracture widening by high-pressure injection and could help to constrain field-scale elastic parameters in fractured rock.

Reliability-based Design Optimization of Laminated Composite Structures under Delamination and Material Property Uncertainties

Delamination is a major type of defects from manufacturing process in laminated composite structures, exhibiting uncertain characteristics which are represented by mixed random variables of continuous in-plane position and delamination size, as well as discrete through-the-thickness position, which should be considered in the design phase of composite structures. Thus, this paper presents a reliability-based design framework for optimal design of composite stacking sequence, for the first time to consider both delamination and material property uncertainties from manufacturing process. Mixed continuous-discrete random variables are involved, and a reliability analysis method is newly proposed to tackle this mixed-variable problem, maintaining high levels of both accuracy and efficiency. The formula of total probability is first employed to formulate the reliability constraint so that discrete and continuous random variables are decoupled; Monte Carlo simulation is then used for reliability analysis with respect to continuous random variables, and a surrogate modelling approach based on Gaussian process regression model is adopted to reduce computation costs. As design variables of ply angles are limited to a discrete set, a genetic algorithm with some proposed improvements is used to handle discrete design variables. Consequently, an optimization framework is devised for composite laminate design under uncertainties from manufacturing process, which is verified via case studies of a cantilever composite plate and can also be extended as a general solution for other composite laminates. With integrated consideration of manufacturing limitations and imperfections as well as structural performances, the presented framework provides a valuable tool in composite structure design.

Online condition monitoring of floating wind turbines drivetrain by means of digital twin

This paper presents a digital twin (DT) condition monitoring approach for drivetrains on floating offshore wind turbines. Digital twin in this context consists of torsional dynamic model, online measurements and fatigue damage estimation which is used for remaining useful life (RUL) estimation. At first, methods for system parameter estimation are presented. The digital twin model provides sufficient inputs for the load observers designed in specific points of the drivetrain to estimate the online load and subsequently stress in the different components. The estimated real-time stress values feed the degradation model of the components. The stochastic degradation model proposed for estimation of real-time fatigue damage in the components is based on a proven model-based approach which is tested under different drivetrain operations, namely normal, faulty and overload conditions. The uncertainties in model, measurements and material properties are addressed, and confidence interval for the estimations is provided by a detailed analysis on the signal behavior and using Monte Carlo simulations. A test case, using 10 MW drivetrain, has been demonstrated.

A probabilistic creep model incorporating test condition, initial damage, and material property uncertainty

Uncertainty is prevalent in the creep resistance of alloys, where at elevated temperature and low pressure, rupture can range across logarithmic decades. In this study, a probabilistic continuum-damage-mechanics (CDM)-based model is derived to capture the uncertainty of creep resistance. To meet this objective, creep data for alloy 304 Stainless Steel is gathered. A constitutive model, “Sinh”, is calibrated deterministically to determine the statistical variability of the material properties. Three sources of uncertainty are injected into the model: test condition (stress and temperature), initial damage, and material properties. Probabilistic simulations are carried out by (a) calibrating probability distribution functions (pdfs) for each source of uncertainty (b) randomly sampling the pdfs using Monte Carlo methods and (c) executing simulations to replicate the uncertain creep behavior. A sensitivity analysis is performed to evaluate the relative effect of each source of uncertainty. In full probabilistic simulations, the cumulative uncertainty of creep behavior is evaluated. The probabilistic model accurately predicts the creep deformation and rupture of the available experiments. The probabilistic model is validated for interpolation but lacks extrapolation ability. Several future works are proposed to further improve the model.

Biomechanical Effects of a Cross Connector in Sacral Fractures - A Finite Element Analysis.

Background: Spinopelvic fractures and approaches of operative stabilization have been a source of controversial discussion. Biomechanical data support the benefit of a spinopelvic stabilization and minimally invasive procedures help to reduce the dissatisfying complication rate. The role of a cross connector within spinopelvic devices remains inconclusive. We aimed to analyze the effect of a cross connector in a finite element model (FE model).Study Design: A FE model of the L1-L5 spine segment with pelvis and a spinopelvic stabilization was reconstructed from patient-specific CT images. The biomechanical relevance of a cross connector in a Denis zone I (AO: 61-B2) sacrum fracture was assessed in the FE model by applying bending and twisting forces with and without a cross connector. Biomechanical outcomes from the numerical model were investigated also considering uncertainties in material properties and levels of osseointegration.Results: The designed FE model showed comparable values in range-of-motion (ROM) and stresses with reference to the literature. The superiority of the spinopelvic stabilization (L5/Os ilium) ± cross connector compared to a non-operative procedure was confirmed in all analyzed loading conditions by reduced ROM and principal stresses in the disk L5/S1, vertebral body L5 and the fracture area. By considering the combination of all loading cases, the presence of a cross connector reduced the maximum stresses in the fracture area of around 10%. This difference has been statistically validated (p < 0.0001).Conclusion: The implementation of a spinopelvic stabilization (L5/Os ilium) in sacrum fractures sustained the fracture and led to enhanced biomechanical properties compared to a non-reductive procedure. While the additional cross connector did not alter the resulting ROM in L4/L5 or L5/sacrum, the reduction of the maximum stresses in the fracture area was significant.