Probabilistic Finite Element Research Articles

An efficient method for probabilistic finite element analysis with application to contact assessment of CANDU fuel channels

In CANDU reactors, irradiation induced creep and growth of the fuel channel (FC) assembly leads to Pressure Tube (PT) sag and to a reduction in the gap between the PT and the Calandria Tube (CT) (PT–CT gap). Excessive PT sag can lead to contact between the PT and the CT. Uncertainties in the parameters that influence PT–CT contact require probabilistic assessment methods to assess risk of PT-CT contact and demonstrate adherence to CSA Standard N285.8 provisions. Improved models in the prediction of PT–CT contact require computationally expensive 3D finite element analysis (3D FEA), a tool that is prohibitively difficult to use in probabilistic assessments. Current probabilistic analyses rely on 1D FEA models for the prediction of PT-CT contact estimates that are generally less accurate. Therefore, an improved and computationally more efficient approach for conducting probabilistic finite element analyses (PFEA) of CANDU FCs is highly desirable. In this paper, such a PFEA approach is proposed by coupling the multiplicative dimensional reduction method (M-DRM) and the polynomial chaos expansion (PCE) method. The analysis that follows studies the PT–CT gap predictions using deterministic 3D FEA by considering two different PT configurations present within the population of the FCs in a CANDU reactor core. The PFEA is then carried out using 1D and 3D FEA by employing the ABAQUS software suite. The accuracy of the probabilistic analysis results using the proposed approach is initially checked against Monte Carlo Simulation (MCS) results using 1D FEA. Following a good agreement, the proposed approach was applied for PFEA based on 3D FEA. The results indicate that the two different PT orientations significantly influence the probabilistic contact results. They also show that the probabilistic results based on 1D FEA, which are currently used by the nuclear industry, are not reliable. Moreover, it is also shown that a sensitivity analysis can be performed without any additional effort. The low computational cost, predictive capability and the generality of the proposed method are important attributes for adopting it in the future in probabilistic assessments of an entire reactor core.

Spatial variability characteristics of the effective friction angle of Crag deposits and its effects on slope stability

This study investigated the spatial variability characteristics of the effective friction angle of Crag deposits which are granular soils occur in the east of England. Cone Penetration Test data were obtained at 26 locations and interpreted statistically. The distribution characteristic of the effective friction angle of Crag deposits was derived with the mean value, the standard deviation and the correlation length calibrated. Illustrations were also shown on how factors such as ground water pressures and the existence of soft/organic soil zones affect the measurement of the autocovariance function and thus the correlation length. Bayesian inference technique was adopted alongside the method of moments to determine the correlation length. Based on the obtained statistical parameters, both semi-deterministic (based on standard geotechnical design codes) and probabilistic finite element limit analyses were carried out to investigate the stability of slopes in Crag deposits. Slopes of various inclined angles were considered and comparisons between the semi-deterministic and probabilistic results were conducted to improve the understanding of the stability of Crag slopes and to provide insight into the slope stability code used in practice.

Probabilistic Finite Element Modeling of Textile Reinforced SHCC Subjected to Uniaxial Tension.

The paper presents a finite element investigation of the effect of material composition and the constituents’ interaction on the tensile behavior of strain-hardening cement-based composites (SHCC) both with and without textile reinforcement. The input material parameters for the SHCC and continuous reinforcement models, as well for their bond, were adopted from reference experimental investigations. The textile reinforcement was discretized by truss elements in the loaded direction only, with the constitutive relationships simulating a carbon and a polymer textile, respectively. For realistic simulation of macroscopic tensile response and multiple cracking patterns in hybrid fiber-reinforced composites subjected to tension, a multi-scale and probabilistic approach was adopted. SHCC was simulated using the smeared crack model, and the input constitutive law reflected the single-crack opening behavior. The probabilistic definition and spatial fluctuation of matrix strength and tensile strength of the SHCC enabled realistic multiple cracking and fracture localization within the loaded model specimens. Two-dimensional (2D) simulations enabled a detailed material assessment with reasonable computational effort and showed adequate accuracy in predicting the experimental findings in terms of macroscopic stress–strain properties, extent of multiple cracking, and average crack width. Besides material optimization, the model is suitable for assessing the strengthening performance of hybrid fiber-reinforced composites on structural elements.

A probabilistic finite element method based on random meshes: A posteriori error estimators and Bayesian inverse problems

We present a novel probabilistic finite element method (FEM) for the solution and uncertainty quantification of elliptic partial differential equations based on random meshes, which we call random mesh FEM (RM-FEM). Our methodology allows to introduce a probability measure on classical FEMs to quantify the uncertainty due to numerical errors either in the context of a-posteriori error quantification or for FE based Bayesian inverse problems. The new approach involves only a perturbation of the mesh and an interpolation that are very simple to implement We present a posteriori error estimators and a rigorous a posteriori error analysis based uniquely on probabilistic information for standard piecewise linear FEM. A series of numerical experiments illustrates the potential of the RM-FEM for error estimation and validates our analysis. We furthermore demonstrate how employing the RM-FEM enhances the quality of the solution of Bayesian inverse problems, thus allowing a better quantification of numerical errors in pipelines of computations.

Effect of Aftershocks on Seismic Fragilities of Single-Story Masonry Structures

The effect of aftershocks on the fragility of single-story masonry structures is investigated using probabilistic seismic demand analysis Finite element models of an unreinforced masonry (URM) structure and a confined masonry (CM) structure are established and their seismic response characteristics when subjected to mainshock, aftershock, and the mainshock-aftershock sequence are then comparatively investigated. The effects of aftershocks and the use of confining members on the seismic response are studied. Probabilistic seismic demand models of the structures are built, and fragility curves under various conditions are derived to investigate the effect of aftershocks on structural fragility. The maximum roof displacement and maximum inter-story drift ratio are lower in the confined masonry model than in the unreinforced masonry model; additionally, the probability of exceedance (PE) values of each damage limit state reduced, and those of the mainshock-damaged models subjected to aftershock significantly increase compared to those directly subjected to a same-intensity aftershock. The probability of severe damage or collapse compared with the mainshock-damaged CM model is greater than when each is subjected to a same intensity aftershock. The use of confining members benefits aftershock resistance and reduces the failure probability of the mainshock-damaged structure. The PE values significantly increase with the aftershock scaling factor δ. Therefore, the effect of aftershocks should be considered in the seismic design and analysis of masonry structures.

Modelling and strength prediction of pre-tensioned hybrid bonded joints for structural steel applications

ABSTRACT Strength prediction of bonded joints remains a challenging task made even further complicated for hybrid joints in which adhesive is supplemented by mechanical fasteners. The present paper complements a series of experimental investigations on hybrid joints combining adhesive bonding with pre-tensioned bolts on galvanised and coated steel substrates, and considering two adhesives. The paper starts with modelling the strength of the bonded connection under the simultaneous action of tensile and compressive normal stresses, and shear. It then presents how these data are moulded into appropriate failure criteria and used the latter as input data for a subsequent full probabilistic finite element analysis. Following the paths of Weibull and subsequent researchers, joint capacity is predicted with reasonable accuracy for a dozen different combinations of substrates, adhesive, and pre-tension level – for which, on average, an accuracy slightly below 10% has been achieved. The results deliver meaningful insights into the subtle relationship between adhesive bonding, mechanical fastening, failure criteria, and resulting predictions methods.

Blast Vulnerability of Multi-Storey Masonry Infill Reinforced Concrete Frames

The prototype or experimental study of blast effect on structures is not always possible due to high risk of life and environmental wellbeing. However, advancement in computational science had made it possible to solve such complex and risky problems. Researchers in the past few decades have presented ample computational and numerical studies on blast effect on masonry structure using deterministic approach. The studies have neglected the effect of various uncertainties that may perhaps occur during an actual blast scenario. Therefore, in present study probabilistic Finite Element (FE) analysis was performed on three statistically stable masonry infill RC (Reinforced Concrete) frames having single (G+0), two (G+1) and three (G+2) storey levels respectively to arrive at the blast fragility curves. The probability of failure was computed based on the inelastic tension damage curves developed for the masonry. The effect of scaled distance and different storey levels on blast vulnerability of masonry infill RC frame walls are investigated through these fragility curves. The results indicate a noteworthy effect of scaled distance and storey levels on the blast resistance of the structure. Frame G+0 is found to be comparatively more vulnerable against the blast load effect than other multi-storey frames. The critical value of scaled distance (Z_critical) corresponding to PF 90% in panel P1 was found as 6.67m/kg^1/3, 4.81m/kg^1/3 and 4.66m/kg^1/3 respectively of frames G+0, G+1 and G+2; in panel P2 was found as 4.3m/kg^1/3 and 4.6m/kg^1/3 respectively of frames G+2 and G+3 and; in panel P3 was found as 4.32 m/kg^1/3 of frame G+3. It is also observed that RC frame is negligibly affected with respect to the masonry panel in terms of blast load mitigation.

Improving the biomechanical performance of screws fixation in a customized mandibular reconstruction prosthesis based on reliability measure

Background: The customized prosthesis is a new method for the reconstruction of large mandibular defects. The ability of dental rehabilitation to improve masticatory functions while maintaining the aesthetics of the main anatomy of the patient's jaw. But the most important problem with all custom prosthesis is the poor performance of screw fixation strength the connections at the bone-plate interface. Materials and Methods: This study was performed to investigate the effect of the number and layout of screws to improve the strength of the bone–prosthesis interface. Due to the inherent variability of input parameters, Analysis of the biomechanical performance of screw fixation strength, a probabilistic finite element method approach has been used. Random input parameters include mechanical properties of the cortical bone, cancellous bone, titanium alloy (Ti6Al4V), and bite force. The layout of the screws was designed in 6 models. Criteria for evaluating the biomechanical performance of screw fixation strength include maximum stress and strain of von Mises cortical bone around the screws. The Monte-Carlo method was used for finite element simulation. Results: The most critical screw in all models is screw No.1, which by increasing the number of screws and correcting the layout shape, the values of maximum stress and strain in the bone around screw No.1 has decreased by 26.7% and 46.3%, respectively, and increased the reliability of the screw connection performance by 25% and 28%, respectively. Conclusion: Finally, in the reconstruction of a large lateral mandibular defect by the customized prosthesis, the strength of the prosthesis to connect to the remaining mandible bone can be improved by increasing the number and modifying the layout of the screws.

Mandibular reconstruction system reliability analysis using probabilistic finite element method

The aim of this study was to design for mandibular reconstruction of large lateral defect with minimum target reliability with designated confidence interval under bite force range of 300 ± 102 N. The performance of the models has been evaluated by numerical analysis considering the uncertainty of input parameters. Computer-Aided design was used to develop the models of three designs according to the patient's anatomy and to achieve to near symmetry of the mandible. Stress-strength modeling was utilized for the probabilistic physics of failure analysis under assumption of a quasi-static load. Monte-Carlo simulation was also applied for probabilistic finite element analysis and reliability assessment. The sensitivity analysis of the models was developed to reflect the significance of the variables in the models. The deterministic stress analysis shows that the highest stress and the second maximum stress are 110 MPa and 85 MPa for cortical bone around the screws, respectively. Also, it is determined that the maximum plate stress of the titanium conventional plate model is 580 MPa. The reconstruction system success rate was improved in all models by observing the anatomy of the patient's mandible in the plate designs by computer-aided design and additive manufacturing techniques. Based on the results, the reliability of plate strength and pull-out screws strength are 99.99% and 96.71% for the fibula free flap model, respectively, and 99.99% and 94.17%, respectively, for the customized prosthesis model. Probability sensitivity factors showed that uncertainty in the elastic modulus of the cortical bone has the greatest effect on the probability of screws loosening.

Condition Assessment of Joints in Steel Truss Bridges Using a Probabilistic Neural Network and Finite Element Model Updating

The condition of joints in steel truss bridges is critical to railway operational safety. The available methods for the quantitative assessment of different types of joint damage are, however, very limited. This paper numerically investigates the feasibility of using a probabilistic neural network (PNN) and a finite element (FE) model updating technique to assess the condition of joints in steel truss bridges. A two-step identification procedure is developed to achieve damage localization and severity assessment. A series of FE models with single or multiple damages are simulated to generate the training and testing data samples and validate the effectiveness of the proposed approach. The influence of noise on the identification accuracy is also evaluated. The results show that the change rate of modal curvature (CRMC) can be used as a damage-sensitive input of the PNN and the accuracy of preliminary damage localization can exceed 90% when suitable training patterns are utilized. Damaged members can be localized in the correct substructure even with noise contamination. The FE model updating method used can effectively quantify the joint deterioration severity and is robust to noise.

Estimating failure probability of IBBC connection using direct coupling of reliability approach and finite element method

This paper aims to produce estimates of probabilities of exceeding specified structural performance thresholds for two types of I beam to box column (IBBC) connections used in steel moment resisting frames (SMRF), when subjected to earthquake ground motions, and to conduct sensitivity analysis of affecting parameters. This is done through coupling the finite element and reliability theory approaches, which allows the analyst to define all uncertainties associated with model parameters. In order to take into account, the record to record variability, incremental dynamic analysis was carried out and statistical properties of the random variable of seismic loading were obtained for the second story of a sample 12-story SMRF at the global collapse state. Variability in geometry, material strength, as well as seismic loading, were then taken into account and considered as inputs of a probabilistic finite element model of IBBC connection. Two damage states were identified based on experimental observations for the two types of connections considered. Consequently, two damage indices and two limit state functions corresponding to each damage state were defined. Finally, the probability of each damage state and union of them were calculated using finite element analysis coupled with reliability theory approaches. In addition, through sensitivity analysis, important random variables, which have more influence on connection performance were addressed. According to the results obtained, the conclusion was that finite element reliability method can allow an advanced safety assessment of IBBC connection.

Impact of structural uncertainties on the buckling strength of cylindrical GLARE panels subjected to uniform compression

The objective of this article is the investigation of the elastic buckling strength of cylindrical simply supported GLARE (GLAss REinforced) panels subjected to axial compression using probabilistic analysis methods, so that the effect of uncertainties associated with material properties and dimensions of the panels on their elastic buckling load can be evaluated. The mechanical properties of aluminum along with the dimensions of aluminum and unidirectional (UD) glass-epoxy layers are considered to be random input variables whereas the critical buckling load is defined as a random output parameter. The employed eigenvalue buckling analysis and the probabilistic finite element analysis were carried out with ANSYS software. The Probabilistic Design System (PDS), along with the Monte Carlo Simulation and the Latin Hypercube Sampling method were used for the calculations. It is found that the thickness of aluminum layers has the strongest effect on the buckling strength, among the considered random input variables. It is also demonstrated that there is a considerable probability for the buckling load of GLARE panels to be overestimated when a deterministic analysis is conducted.

Probabilistic Finite Element Analysis of Cooled High-Pressure Turbine Blades—Part A: Holistic Description of Manufacturing Variability

Abstract Modern high-pressure turbine (HPT) blade design stands out due to its high complexity comprising three-dimensional blade features, multipassage cooling system (MPCS), and film cooling to allow for progressive thermodynamic process parameters. During the last decade, probabilistic design approaches have become increasingly important in turbomachinery to incorporate uncertainties such as geometric variations caused by manufacturing scatter and deterioration. Within this scope, the first part of this two-part article introduces parametric models for cooled turbine blades that enable probabilistic finite element (FE) analysis taking geometric variability into account to aim at sensitivity and robustness evaluation. The statistical database is represented by a population of more than 400 blades whose external geometry is captured by optical measurement techniques and 34 blades that are digitized by computed tomography (CT) to record the internal geometry and the associated variability, respectively. Based on these data, parametric models for airfoil, profiled endwall (PEW), wedge surface (WSF), and MPCS are presented. The parametric airfoil model that is based on the traditional profile theory is briefly described. In this regard, a methodology is presented that enables to adapt this airfoil model to a given population of blades by means of Monte Carlo-based optimization. The endwall variability of hub and shroud are parametrized by radial offsets that are applied to the respective median endwall geometry. WSFs are analytically represented by planes. Variations of the MPCS are quantified based on the radial distribution of cooling passage centroids. Thus, an individual MPCS can be replicated by applying adapted displacement functions to the core passage centroids. For each feature that is considered within this study, the accuracy of the parametric model is discussed with respect to the variability that is present in the investigated blade population and the measurement uncertainty. Within the scope of the second part of this article, the parametric models are used for a comprehensive statistical analysis to reveal the parameter correlation structure and probability density functions (PDFs). This is required for the subsequent probabilistic finite element analysis involving real geometry effects.

Probabilistic Finite Element Analysis of Cooled High-Pressure Turbine Blades—Part B: Probabilistic Analysis

Abstract Modern high-pressure turbine (HPT) blade design stands out due to high complexity comprising three-dimensional blade features, multipassage cooling system (MPCS), and film cooling to allow for progressive thermodynamic process parameters. During the last decade, probabilistic design approaches have become increasingly important in turbomachinery to incorporate uncertainties such as geometric variations caused by manufacturing scatter. In Part B of this two-part article, real geometry effects are considered within a probabilistic finite element (FE) analysis that aims at sensitivity evaluation. The knowledge about the geometric variability is derived based on a blade population of more than 400 individuals by means of parametric models that are introduced in Part A. The HPT blade population is statistically assessed, which allows for reliable sensitivity analysis and robustness evaluation taking the variability of the airfoil, profiled endwalls (PEWs) at hub and shroud, wedge surfaces (WSFs), and the MPCS into account. The probabilistic method—Monte Carlo simulation (MCS) using an extended Latin hypercube sampling (eLHS) technique—is presented subsequently. Afterward, the FE model that involves thermal, linear-elastic stress, and creep analysis is described briefly. Based on this, the fully automated process chain involving computer-aided design (CAD) model creation, FE mesh morphing, FE analysis, and postprocessing is executed. Here, the mesh morphing process is presented involving a discussion of the mesh quality. The process robustness is assessed and quantified referring to the impact on input parameter correlation. Finally, the result quantities of the probabilistic FE simulation are evaluated in terms of sensitivities. For this purpose, regions of interest are determined, wherein the statistical analysis is conducted to achieve the sensitivity ranking. A significant influence of the considered geometric uncertainties onto mechanical output quantities is observed, which motivates to incorporate these in modern design strategies or robust optimization.

Computation-Effective Structural Performance Assessment Using Gaussian Process-Based Finite Element Model Updating and Reliability Analysis

Structural health monitoring data has been widely acknowledged as a significant source for evaluating the performance and health conditions of structures. However, a holistic framework that efficiently incorporates monitored data into structural identification and, in turn, provides a realistic life-cycle performance assessment of structures is yet to be established. There are different sources of uncertainty, such as structural parameters, computer model bias and measurement errors. Neglecting to account for these factors results in unreliable structural identifications, consequent financial losses, and a threat to the safety of structures and human lives. This paper proposes a new framework for structural performance assessment that integrates a comprehensive probabilistic finite element model updating approach, which deals with various structural identification uncertainties and structural reliability analysis. In this framework, Gaussian process surrogate models are replaced with a finite element model and its associate discrepancy function to provide a computationally efficient and all-round uncertainty quantification. Herein, the structural parameters that are most sensitive to measured structural dynamic characteristics are investigated and used to update the numerical model. Sequentially, the updated model is applied to compute the structural capacity with respect to loading demand to evaluate its as-is performance. The proposed framework’s feasibility is investigated and validated on a large lab-scale box girder bridge in two different health states, undamaged and damaged, with the latter state representing changes in structural parameters resulted from overloading actions. The results from the box girder bridge indicate a reduced structural performance evidenced by a significant drop in the structural reliability index and an increased probability of failure in the damaged state. The results also demonstrate that the proposed methodology contributes to more reliable judgment about structural safety, which in turn enables more informed maintenance decisions to be made.

Development and calibration of a probabilistic finite element hip capsule representation

The objective of this study was to develop a probabilistic representation of the hip capsule, which is calibrated to experimental capsular torque-rotation behavior and captures the observed variability for use in population-based studies. A finite element model of the hip capsule was developed with structures composed of a fiber-reinforced membrane, represented by 2D quadrilateral elements embedded with tension-only non-linear spring. An average capsule representation was developed by calibrating ligament properties (linear stiffness, reference strain) so that torque-rotation behavior matched mean cadaveric data. A probabilistic capsule was produced by determining the ligament property variability which represented ±2 SD measured in the experiment. Differences between experimental and model kinematics across all positions had RMS error of 4.7°. Output bounds from the optimized probabilistic capsule representation were consistent with ±2 SD of experimental data; the overall RMS error was 5.1°. This model can be employed in population-based finite element studies of THA to assess mechanics in realistic scenarios considering implant design, as well as surgical and patient factors.

Probabilistic finite element analysis of the deflection on a beam

The reliability of complex systems can be quantified using probabilistic finite element analysis. A widely accepted approach is interlacing of probabilistic methods with commercial finite element solvers for evaluating the fatigue life. In this paper, the deflection of a beam is calculated using finite element method (ABAQUS® commercial software), then the results of the simulated model are employed in MATLAB®, which is used to determine the probability, reliability and the reliability index. It was found that the reliability index is around 1.2428, while the reliability value is calculated to be around 0.8930. From the calculations the values of the probability of failure (POF) was found to be around 0.1070. Finally, the results of the calculated maximum deflection have been compared and a good agreement is found.

Comparative Approaches to Probabilistic Finite Element Methods for Slope Stability Analysis

Probabilistic slope stability analyses are often preferable to deterministic methods when soils are inherently heterogeneous, or the reliability of geotechnical parameters is largely unknown. These methods are suitable for evaluating the risk of slope failure by producing a range of potential scenarios for the slope stability factor of safety. Several probabilistic methods including the Point Estimate Method, Monte Carlo Method and Random Finite Element Method, can be combined with the Finite Element technique. In this study, various shear strength distributions are considered for three different probabilistic Finite Element Methods to determine Factor of Safety and Probability of Failure distributions, based on the associated method of slope stability analysis. Results are presented for a case study of an Australian open-cut coal mine, with a range of shear strength parameter distributions for coal and interseam cohesive materials considered. Coal and interseam shear strength parameters are varied independently, to determine the effects of each material on the slope Factor of Safety.

Minimising tibial fracture after unicompartmental knee replacement: A probabilistic finite element study

BackgroundPeriprosthetic tibial fracture after unicompartmental knee replacement is a challenging post-operative complication. Patients have an increased risk of mortality after fracture, the majority undergo further surgery, and the revision operations are less successful. Inappropriate surgical technique increases the risk of fracture, but it is unclear which technical aspects of the surgery are most problematic and no research has been performed on how surgical factors interact. MethodsFirstly, this study quantified the typical variance in surgical cuts made during unicompartmental knee replacement (determined from bones prepared by surgeons during an instructional course). Secondly, these measured distributions were used to create a probabilistic finite element model of the tibia after replacement. A thousand finite element models were created using the Monte Carlo method, representing 1000 virtual operations, and the risk of tibial fracture was assessed. FindingsMultivariate linear regression of the results showed that excessive resection depth and making the vertical cut too deep posteriorly increased the risk of fracture. These two parameters also had high variability in the prepared synthetic bones. The regression equation calculated the risk of fracture from three cut parameters (resection depth, vertical and horizonal posterior cuts) and fit the model results with 90% correlation. InterpretationThis study introduces for the first time the application of a probabilistic approach to predict the aetiology of fracture after unicompartmental knee replacement, providing unique insight into the relative importance of surgical saw cut variations. Targeted changes to operative technique can now be considered to seek to reduce the risk of periprosthetic fracture.

A Bayesian model updating with incomplete complex modal data

Abstract Finite element (FE) model updating using measured dynamic data has gained much attention in view of its contribution towards important areas like model-correction, structural health monitoring. Usually model updating is observed to be performed using measured modal data like modal frequencies and mode shapes – indeed very few works are observed to provide scope for using damping measurements in FE model updating. The present work proposes a Bayesian probabilistic FE model updating using multiple sets of complex modal data (viz. complex modal frequencies and complex mode shapes) and thereby enabling the scope for incorporating damping measurements in updating. Moreover there remains the opportunity for using incomplete complex modal data and avoiding the requirement of mode matching. In this paper, a computationally efficient model updating approach based on maximum a posteriori (MAP) is formulated where all the mass, stiffness and damping parameters are updated in a sequential iterative manner. Detailed formulations are obtained considering damping to be contributed by classical damping (Rayleigh damping) and non-classical viscous damping. Besides, formulations for uncertainty estimation and probabilistic damage detection are also developed. The proposed approach is validated using two numerical examples while utilizing incomplete complex modal data. Performance in updating is evaluated for both undamaged and damaged cases. The proposed approach is observed to be successful for updating of FE model as well as for detection of changes/damages in probabilistic manner, while being computationally efficient as well.