Probabilistic Finite Element Method Research Articles

Fragility and Leakage Risk Assessment of Nuclear Containment Structure under Loss-of-Coolant Accident Conditions Considering Liner Corrosion

The steel liner plays a key role in ensuring the leak-tightness of nuclear containment structures (NCSs). Once the liner is subjected to corrosion, its corrosion effects can severely impact the leak-tight function of the NCS, especially in severe accident scenarios. Therefore, evaluating the corrosion effects of the liner is essential to guaranteeing the safe operation of nuclear power plants. This paper presents a probabilistic safety analysis of the NCS under liner corrosion conditions. Firstly, the corrosion mechanism of the liner is elucidated, and a refined simulation method is developed to investigate the localized corrosion effects of the liner. Utilizing a probabilistic finite element method, the fragility of the NCS under loss-of-coolant accident (LOCA) conditions is evaluated. Finally, the leakage risk of the NCS at different degrees of liner corrosion is discussed. The results indicate that liner corrosion has a significant impact on the median pressure capacity and high-confidence–low-probability pressure capacity of the NCS. With the aggravation of liner corrosion, the safety margin of the NCS decreases, and the total probability of failure increases. Within the scope of this paper, the NCS can fulfill the probabilistic safety requirements.

Generating probability distributions on intervals and spheres: Convex decomposition

This work studies the convexity of a class of positive definite probability measures with density functionsρ=cλ∑n=0∞g(n)Cnλwλ, that are generated through the interval characteristic sequences. We discuss when such measures supported in an interval are convex and demonstrate how convexity is preserved under multiplication by certain multipliers and under the intertwining product. A key message to be delivered from this work is: any integrable function f on the interval with a polynomial expansion that has relatively fast absolute convergence, can be decomposed into a non-unique pair of positive convex interval probabilities (f+,f−)∈Eλ′×Eλ′ up to normalization, in the sense that f=f+−f−, where Eλ′ is a class of convex functions. Consequently not only the study of an interval distribution is reduced to the study of the properties of class Eλ′, but also the discontinuous probabilistic Galerkin schemes can be simplified. As concrete examples the decomposition of polynomials and truncated normal distribution are computed and graphically illustrated. Such interval measures lead to localized spherical probability measures, and as a convex expression of probability measure is a Jensen type inequality for spherical random variables. Based on the spherical information entropy models, one dependent on the domain partition and the other not, we propose three minimization formulations for the probabilistic finite element method, including one that mixes the least-squares and entropy models.

Reliability analysis of pile stabilized earth slopes using weighted uniform simulation method

The finite element method (FEM) is now widely used for stability analysis of earth slopes reinforced with stabilizing piles for the advantages of better simulation of soil–pile interactive behaviors and clearly capturing the complex failure mechanism of pile-reinforced slopes. However, the implementation of probabilistic simulation to model the influence of inherent random variables onto reinforced earth slopes is still challenging due to the extensive computational resources required for the repeated execution of the Finite Element Method (FEM) simulations. This paper proposes a framework for probabilistically analyzing the pile-reinforced earth slope based on a weighted uniform simulation (WUS) method implemented by secondary development of ABAQUS-via-Python codes. Using the proposed method, important information of probabilistic indices can be obtained with less computational cost, and the influence of different pile design parameters on the probabilistic FEM stability analysis of reinforced slopes is systematically studied. The reliability indices β and the most probable failure points (MPPs) for optimal designs of pile-reinforced earth slopes under different pile design parameter combinations (L, Lx/Ls, S, D) are obtained, which can provide useful reference information for conducting slope reinforcement practicing works. In addition, the influence of uncertainty and correlation of soil strength parameters on the reliability of the reinforced slope is also investigated.

Probabilistic finite element analysis of fatigue life of additively manufactured clasp.

The present study was aimed to develop a probabilistic finite element method (FEM) that predicts the variability in the fatigue life of additively manufactured clasp so that it can be used as a virtual test in the design phase before manufacturing. Titanium alloy (Ti-6Al-4V) clasp with integrated chucking part, which was designed for experimental fatigue test to validate the computational method, was investigated. To predict the lower bound, an initial spherical defect was assumed in the region where stress concentration was predicted. The Smith-Watson-Topper (SWT) method, Bäumel & Seeger rule, elasto-plastic FEM, and zooming FEM were used. The influence of assumed initial defect on the fatigue life was significant, and the large variability in the fatigue life was predicted. This study demonstrated that the proposed practical computational method can simulate the large variability in the fatigue life of titanium alloy clasp, which is useful in its design before manufacturing.

Deterministic and probabilistic buckling response of fiber–metal laminate panels under uniaxial compression

PurposeThe purpose of this paper is to study the deterministic elastic buckling behavior of cylindrical fiber–metal laminate panels subjected to uniaxial compressive loading and the investigation of GLAss fiber-REinforced aluminum laminate (GLARE) panels using probabilistic finite element method (FEM) analysis.Design/methodology/approachThe FEM in combination with the eigenvalue buckling analysis is used for the construction of buckling coefficient–curvature parameter diagrams of seven fiber–metal laminate grades, three glass-fiber composites and monolithic 2024-T3 aluminum. The influences of uncertainties concerning material properties and laminate dimensions on the buckling load are studied with sensitivity analyses.FindingsIt is found that aluminum has a stronger impact on the buckling behavior of the fiber–metal laminate panels than their constituent uni-directional or woven composites. For the classical simply supported boundary conditions, it is found that there is an approximately linear relation between the buckling coefficient and the curvature parameter when the diagrams are plotted in double logarithmic scale. The probabilistic calculations demonstrate that there is a considerable probability to overestimate the buckling load of GLARE panels with deterministic calculations.Originality/valueIn this study, the deterministic and probabilistic buckling response of fiber metal laminate panels is investigated. It is shown that realistic structural uncertainties could substantially affect the buckling strength of aerospace components.

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.

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.

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.

Three-dimensional soft tissue prediction in orthognathic surgery: a clinical comparison of Dolphin, ProPlan CMF, and probabilistic finite element modelling

Three-dimensional surgical planning is used widely in orthognathic surgery. Although numerous computer programs exist, the accuracy of soft tissue prediction remains uncertain. The purpose of this study was to compare the prediction accuracy of Dolphin, ProPlan CMF, and a probabilistic finite element method (PFEM). Seven patients (mean age 18years; five female) who had undergone Le Fort I osteotomy with preoperative and 1-year postoperative cone beam computed tomography (CBCT) were included. The three programs were used for soft tissue prediction using planned and postoperative maxillary position, and these were compared to postoperative CBCT. Accurate predictions were obtained with each program, indicated by root mean square distances: RMSDolphin=1.8±0.8mm, RMSProPlan=1.2±0.4mm, and RMSPFEM=1.3±0.4mm. Dolphin utilizes a landmark-based algorithm allowing for patient-specific bone-to-soft tissue ratios, which works well for cephalometric radiographs but has limited three-dimensional accuracy, whilst ProPlan and PFEM provide better three-dimensional predictions with continuous displacements. Patient or population-specific material properties can be defined in PFEM, while no soft tissue parameters are adjustable in ProPlan. Important clinical considerations are the topological differences between predictions due to the three algorithms, the non-negligible influence of the mismatch between planned and postoperative maxillary position, and the learning curve associated with sophisticated programs like PFEM.

Microstructural topology optimization for minimizing the sound pressure level of structural–acoustic coupled systems with multi-scale random parameters

ABSTRACTThe main aim of this article is to present a robust microstructural topology optimization methodology for structural–acoustic coupled systems with multi-scale random parameters. During the microstructural topology optimization, both the uncertainty at the macro-scale, which comes from the physical parameters of the acoustic medium or the external load, and the uncertainty existing in the constituent material properties of the microstructure at the micro-scale are considered as random parameters. A homogenization-based probabilistic finite element method (HPFEM) is first developed for quantifying the structural–acoustic system with multi-scale random parameters. The use of the HPFEM transforms the problem of microstructural topology optimization with multi-scale random parameters to an augmented deterministic microstructural topology optimization problem. This provides a computationally cheap alternative to Monte Carlo-based optimization algorithms. A numerical example of a hexahedral box is given to demonstrate the efficiency of the proposed method.

Statistical correlation between elastic properties of plain-weave composite and its influence on structure reliability

Statistical correlation between in-plane elastic properties of plain-weave composite (E-glass fabric/epoxy and carbon fabric/epoxy) is studied by probabilistic finite element method. The in-plane elastic properties including Young’s moduli, shear moduli and Poisson’s ratios are derived from finite element model on a composite unit-cell subjected periodic boundary constraint. It is shown that significant statistical correlation exists between the in-plane elastic properties and it depends on the variations of both the constituent properties and the unit-cell geometry. Two numerical examples are present on the reliability evaluation of unidirectional and multi-layer laminated plates bearing multi-axial in-plane loads, and this is conducted at two scenarios: considering and neglecting the statistical correlation. Monte-carlo (MC) method is employed to conduct the reliability calculation and Gibbs sampling is used for drawing samples where there is statistical correlation between random variables. For the multi-layer laminated plates, it is observed that neglecting the statistical correlation would result an 15–20% overestimation on the laminate failure probability, which indicates the importance of consideration of the statistical correlation in the process of reliability evaluation or reliability based design of plain-weave composite structures.

Evaluation of Seismic Risk for Bridges using F-R-M Method: A Combination of Finite Element Method and Neural Network Model

Earthquake action in the design of the bridge belongs to the accidental action. Earthquake is difficult to predict, so the frequency of its occurrence is very small when the structure is in use. Once it occurs, the seismic value is very large and destructive, therefore the risk assessment of the damage caused by the earthquake to the bridge structure becomes the main problem of bridge engineering design. Our country is an earthquake-prone area, and bridges built there have relatively large adjacent spans, of which the quality and stiffness distribution is uneven, due to the limitation of terrain and construction conditions. The existing seismic codes are not applicable to the seismic risk assessment method of the bridge; therefore, it is very important and meaningful to study the seismic risk assessment method. In this paper, a new method for estimating seismic risk probability of bridge seismic risk in earthquake-prone area, which is a combination of neural network simulation and probabilistic finite element method (F-R-M) is established. In the method, firstly, the multiple seismic wave of bridge history analysis and get corresponding seismic response to the structure of the bridge, then RBF neural network is established, with numerical field type, seismic wave energy, seismic wave peak number, the pier moment as input parameters, to train and test the network of RBF. After the success of the training, the Monte Carlo method can be adopted to generate a large number of random numbers with the simulation of neural network to get bridge structures data subjected to random seismic response, and the structure failure probability can be calculated.

Optimum Design of the Random Vibration Structural System Using Two Steps Approach with Monte Carlo Simulation

Structural dynamic system involves random variables conditions such as material property, geometric parameters and applied loads. This uncertainties result from the structural parameter are carefully considered the dynamic structural response in displacement, stress, and natural frequencies. The random vibrational system must be designed to withstand a certain amount of the fluctuation with respect to the uncertainties. Harmonic response of a spring-mass system is mathematically modelled with the probabilistic finite element method using the Monte Carlo simulation. The aim of this paper is to find the optimal lowest frequency for the spring-mass system with random input variables and response parameters to the displacements. The probabilistic design is carried out using ANSYS probabilistic design module in a commercial application software and then the optimal design is sequentially solved. An efficient and practical optimal design evaluation method is proposed for the design of the harmonic system. The numerical results are obtained where the next highest frequency of the system and displacements treated as constraints.

A New Model to Study Fatigue in Dental Implants Based on Probabilistic Finite Elements and Cumulative Damage Model.

The aim of this study was to predict the fatigue life of two different connections of a dental implant as in load transfer to bone. Two three-dimensional models were created and assembled. All models were subjected to a natural masticatory force of 118 N in the angle of 75° to the occlusal plane. All degrees of freedom in the inferior border of the cortical bone were restrained, and the mesial and distal borders of the end of the bone section were constrained. Fatigue material data and loads were assumed as random variables. Maximum principal stresses on bone were evaluated. Then, the probability of failure was obtained by the probabilistic approach. The maximum principal stress distribution predicted in the cortical and trabecular bone is 32 MPa for external connection and 39 MPa for internal connection. A mean life of 103 and 210 million cycles were obtained for external and internal connection, respectively. Probability cumulative function was also evaluated for both connection types. This stochastic model employs a cumulative damage model and probabilistic finite element method. This methodology allows the possibility of measured uncertainties and has a good precision on the results.

Safety Assessment of Composite Armor under Ballistic Impact

The failure predictions are made for composite targets provided with different ply arrangements under ballistic impact. The material properties and initial projectile velocity are considered as random parameters. 3D probabilistic finite element method is employed to obtain the dynamic response using explicit time domain solver. The stochastic stresses obtained at critical locations and strength parameters of random nature; contribute to damage initiation. Fiber failure initiation models adopted to establish the performance functions. The probability of failure (Pf) of anti-symmetric cross ply arrangement is found to be minimum in comparison to other ply lay-ups namely, symmetric cross ply, symmetric angle ply and anti-symmetric angle ply. Sensitivity analysis shows that the Pf is significantly influenced by the mean values and scatters of the shear strength (T13) and initial velocity. This information may be used effectively for design optimization to achieve better strength and lighter weight of composite for body armors.

Probabilistic finite element method for large tumor radiofrequency ablation simulation and planning

A challenging problem of radiofrequency ablation (RFA) in liver surgery is to accurately estimate the shapes and sizes of RFA lesions whose formation depends on intrinsic variations of the thermal–electrical properties of soft tissue. Large tumors, which can be as long as 10 cm or more, further complicate the problem. In this paper, a probabilistic bio-heating finite element (FE) model is proposed and developed to predict RFA lesions. Uncertainties of RFA lesions are caused by the probabilistic nature of five thermal–electrical liver properties: thermal conductivity, liver tissue density, specific heat, blood perfusion rate and electrical conductivity. Confidence levels of shapes and sizes of lesions are generated by the FE model incorporated with the mean-value first-order second-moment (MVFOSM) method. Based on the probabilistic FE method, a workflow of RFA planning is introduced to enable clinicians to preoperatively view the predicted RFA lesions in three-dimension (3D) within a hepatic environment. Accurate planning of the RFA needle placements can then be achieved based on the interactive simulation and confidence level selection.

Shape-instability life scatter prediction of 40Cr steel: Damage-coupled crystal plastic probabilistic finite element method

Disposable mechanical elements, which are widely used in spacecrafts, undergo fracture primarily via shape-instability failure. A cyclic crystal plasticity model based on the probabilistic finite element method is developed with the aim of predicting the shape-instability lifespan. In the model, a two-dimensional (2D) finite element model is established, based on the Voronoi tessellation, to simulate the morphology of the microstructure measured during electron backscatter diffraction of 40Cr steel. A damage-coupled cyclic crystal plasticity model is used to describe the mechanical behavior and material degradation. In addition, the material properties described by a stochastic distribution are randomly generated and then assigned to the elements in the 2D model in order to simulate the heterogeneity of the material. The validity of the prediction model is revealed through a comparison of the simulation and experimental results. Furthermore, the sensitivities of the control parameters, such as the element size, variance of the stochastic distribution, and morphology of the microstructure, are determined. The presented model allows the determination of the shape-instability lifespan of disposable mechanical elements. This model accurately predicts, and hence negates the need for many measurements of, the lifespan and therefore has significant potential for use in an appropriate field.

A Study of Probabilistic FEMs for a Slope Reliability Analysis Using the Stress Fields

In this paper, the applicability and the effectiveness of the probabilistic finite element methods (FEMs) such as the perturbation method, and the Spectral Stochastic Finite Element Method (SSFEM) applied to the reliability analysis of the slope stability have been studied. The results were checked by the Monte Carlo simulation and a direct coupling ap-proach combining the deterministic finite elements code and First Order Reliability Method (FORM) algorithm. These methods are presented considering the spatial variation of soil strength parameters and Young modulus. The random field is used to describe the spatial variation. Also, the reliability analysis is conducted using a performance function formulat-ed in terms of the stochastic stress mobilized along the sliding surface. The present study shows that the perturbation method and SSFEM can be considered as practical methods to conduct a second moment analysis of the slope stability taking into account the spatial variability of soil properties since good results are obtained with acceptable estimated rela-tive errors. Finally, the perturbation method is performed to delimit the location of the critical probabilistic sliding surfac-es and to evaluate the effect of the correlation length of soil strength parameters on the safety factor. In addition, the two methods are used to estimate the probability density and the cumulative distribution function of the factor of safety.

Random Response Analysis of Vibration Transfer Path Systems with Translational and Rotational Motions <sup></sup>

This paper based on the generalized probabilistic perturbation finite element method solves the random response analysis problem of vibration transfer path systems with translational and rotational motions. The effective random response analysis approaches are achieved using Kronecker algebra, matrix calculus, generalized second moment technique of vector-valued functions and matrix-valued functions. For the vibration transfer path system with multi-dimensional paths, the random response is described correctly and expressly in time domain as uncertain factors, which include mass, damping, stiffness and position, are considered. The mathematical expressions of the first order and second order moments for the random vibration response of vibration transfer path are obtained. According to the corresponding numerical example, the results of calculation are consistent with the results of Monte-Carlo simulation, which shows the method is feasible theoretically.