Probabilistic Design Optimization Research Articles

Probabilistic shape design optimization of structural components under fatigue

Many failures of structural components are attributed to fatigue due to repeated loading and unloading conditions. The crack growth due to fatigue, represents a critical issue for the integrity and capacity of structural components. Apart from the loading conditions, the shape of the structural components plays an important role in their service life. In this study, extended finite element and level set methods are integrated into a probabilistic shape design optimization framework aiming to improve the service life of structural components under fatigue. In this context, the relation between the geometry of the structural components and their service life is investigated. The effect of uncertain material properties as well as the crack tip initialization, described by random variables is also examined. Comparisons between optimized shapes obtained for various targeted fatigue life values are addressed, while the location of the initial imperfection along with its orientation are found to have a significant effect on the optimal shapes for the components studied.

A Comparative Study on Probabilistic Optimization Methods for Electromagnetic Design

A reliability-based robust design optimization (RBRDO) is developed to ensure the product quality as well as the confidence in product reliability of electromagnetic devices. In the method, the first two statistical moments, mean and variance, of a quality loss function are estimated by the univariate dimension reduction method, while desired probabilistic constraint conditions are assessed by one of the first-order reliability analysis methods, called performance measure approach. For a better understanding between probabilistic design optimization methods, three different formulations of reliability-based design optimization, robust design optimization (RDO), and RBRDO are compared with one another. A simple mathematical model and a brushless direct current (BLDC) motor design problem are tested to demonstrate the features of the three methods and to examine their numerical efficiency.

Uncertainty analysis and probabilistic design optimization of hybrid rocket motors for manned lunar landing

To obtain a conceptual design for a hybrid rocket motor (HRM) to be used as the Ascent Propulsion System in the Apollo lunar module, the deterministic design optimization (DDO) method is applied to the HRM design. Based on the results of an uncertainty analysis of HRMs, an uncertainty-based design optimization (UDO) method is also adopted to improve the design reliability. The HRM design process, which is a multidisciplinary system, is analyzed, and a mathematical model for the system design is established to compute the motor performance from the input parameters, including the input variables and model parameters. The input parameter uncertainties are quantified, and a sensitivity analysis of the model parameter uncertainties is conducted to identify the most important model parameter uncertainties for HRMs. The DDO and probabilistic UDO methods are applied to conceptual designs for an HRM to be used as a substitute for the liquid rocket motor (LRM) of the Ascent Propulsion System. The conceptual design results show that HRMs have several advantages as an alternative to the LRM of the Ascent Propulsion System, including nontoxic propellant combination, small motor volume, and comparable functions, such as restarting and throating. Comparisons of the DDO and UDO results indicate that the UDO method achieves more robust and reliable optimal designs than the DDO method. The probabilistic UDO method can be used to develop better conceptual designs for HRMs.

Advanced probabilistic design and reliability-based design optimization for composite sandwich structure

A composite sandwich structure can improve flexural rigidity and can decrease the weight of a composite laminated plate by 30%. However, it implies significant uncertainty within the construction process of its material properties when compared to other regular metals. Therefore, it is necessary to consider a probabilistic design method based on reliability. This paper calculated the probabilistic margin of safety of a simplified composite sandwich fuselage in order to examine that the classic design method utilizing the safety factor does not ensure the safety of the structure. Reliability-based design optimization (RBDO) was conducted for efficient calculation by using proposed RBDO by moving probability density function (RBDO-MPDF) method in this paper. Crude Monte-Carlo Simulation was used to calculate the probability density function. The results of this paper will be applicable to the improved design methods that ensure the structural reliability and maximized efficiency within the RBDO process.

Multi-Object Multi-Discipline Probabilistic Design of High-Pressure Turbine Blade-Tip Radial Running Clearance

For the probabilistic design of high-pressure turbine (HPT) blade-tip radial running clearance (BTRRC), a distributed collaborative response surface method (DCRSM) was proposed, and the mathematical model of DCRSM was established. From the BTRRC probabilistic design based on DCRSM, the static clearance δ=1.865 mm is demonstrated to be optimal for the BTRRC design considering aeroengine reliability and efficiency. Meanwhile, DCRSM is proved to be of high accuracy and efficiency in the BTRRC probabilistic design. The present study offers an effective way for HPT BTRRC dynamic probabilistic design and provides also a promising method for the further probabilistic optimal design of complex mechanical system.

Probabilistic multi-objective optimal design of composite channels using particle swarm optimization

To achieve cost effectiveness and reliability in design, this paper presents a probabilistic multi-objective model for optimal design of composite channels that have a cross-sectional shape of horizontal bottom and parabolic sides. The multiple objectives of channel design include minimizing the cost of channel construction, maximizing the probability of the expected channel capacity being greater than the design flow and minimizing the overtopping probability. In this study, the principles of first-order uncertainty analysis are adopted for handling uncertainty in parameters of the problem. The optimization model is solved using the particle swarm optimization method and Pareto-optimal solutions are generated for various combinations of overtopping probability and exceedance probability of channel capacity. The results show that the presented approach has good potential for exploring different alternative designs of open channels under input parameter uncertainty.

Reliability design optimization of composite structures based on PSO together with FEA

The present work aims to develop a method for reliability-based optimum design of composite structures. A procedure combining particle swarm optimization (PSO) and finite element analysis (FEA) has been proposed. Numerical examples for the reliability design optimization (RDO) of a laminate and a composite cylindrical shell are worked out to demonstrate the effectiveness of the method. Then a design for composite pressure vessels is studied. The advantages and necessity of RDO over the conventional equi-strength design are addressed. Examples show that the proposed method has good stability and is efficient in dealing with the probabilistic optimal design of composite structures. It may serve as an effective tool to optimize other complicated structures with uncertainties.

Probabilistic analysis and optimization of energy absorbing components made of nanofiber enhanced composite materials

An enhanced thermoset polymer matrix with randomly distributed carbon nanofibers (CNFs) is combined with conventional long fibers to form a hybrid composite material for application to impact energy absorbing components. The multi-inclusion method in combination with functionally graded interphase is used for stiffness characterization and shear-lag theory combined with quasi-isotropic lamination approximation is used for strength prediction. Axial crush simulations are performed using MD Nastran with a micromechanics-based progressive failure analysis constitutive model. The stochastic uncertainties in the geometric and material properties of CNF as well as the three-dimensional, non-homogeneous CNF–matrix interphase are represented using probability theory. Through Monte Carlo simulations, these uncertainties are propagated to the homogenized macroscopic properties of the nano-enhanced matrix and subsequently to the stiffness and strength properties of the composite laminate as well as the energy absorbing characteristics of the crush tube. A probabilistic design optimization problem is formulated for minimizing the failure probability associated with the specific energy absorption of the composite tube. A dual surrogate modeling approach is used for approximating the failure probability and solving the optimization problem using sequential quadratic programming. The modeling approach, uncertainty analysis, and probabilistic optimization results are presented and discussed.

Modeling and probabilistic design optimization of a nanofiber-enhanced composite cylinder for buckling

Micromechanical approaches are used in mathematical modeling of randomly distributed carbon nanofibers (CNFs) in a thermoset polymer material. Both CNF waviness and CNF-matrix interphase properties are included in the model. The interphase mechanical properties are considered to vary in a manner similar to functionally graded materials. The effects of stochastic uncertainties on the overall properties of the composite material are represented using the probability theory. The uncertainties are propagated in calculating the axial buckling load probability of a thin-walled composite cylinder, which is then optimized for minimum material volume. The probabilistic design optimization results are presented and discussed.

SIMP type topology optimization procedure considering uncertain load position

In this paper a new type of probabilistic optimal topology design method is elaborated for continuum type of structures where the points of application of the loads are given randomly. In the proposed probabilistic topology optimization method the minimum penalized weight design of the discretized structure is subjected to compliance constraint and side constraints. The compliance expression is probabilistic one. By the use of an appropriate stochastic upperbound theorem, the original stochastic mathematical programming problem is substituted by a deterministic one. The numerical procedure is based on iterative formula which is formed by the use of the first order optimality condition of the Lagrangian function. The application is illustrated by numerical example.

Probabilistic optimal safety with minimum life-cycle cost based on stochastic finite element analysis of steel cable-stayed bridges

This study was intended to efficiently perform the probabilistic safety and optimal design assessment of steel cable-stayed bridges (SCS bridges) using stochastic finite element analysis (SFEA) and expected life-cycle cost (LCC) concept. To that end, advanced probabilistic finite element algorithm (APFEA) which enables to execute the static and dynamic SFEA considering aleatory uncertainties contained in random variable was developed. APFEA is the useful analytical means enabling to conduct the reliability assessment (RA) in a systematic way by considering the result of SFEA based on linearity and nonlinearity of before or after introducing initial tensile force. Appropriateness of APFEA was verified in such a way of comparing the result of SFEA of a simple structure and the result of numerical analysis using Monte Carlo Simulation (MCS) program. The probabilistic method of SCS bridges was set, taking into account of analytical parameters. The dynamic response characteristic by probabilistic method was evaluated using ASFEA, and RA was carried out based on analysis result, thereby quantitatively calculating the probabilistic safety. The optimal design of SCS bridges was determined based on the expected LCC according to the results of SFEA and RA of alternative designs. Moreover, given the potential epistemic uncertainty contained in safety index, failure probability and minimum LCC, the sensitivity analysis was conducted and as a result, a critical distribution phase was illustrated using a cumulative-percentile.

Using Cross Validation to Design Conservative Surrogates

this work we use safety margins to conservatively compensate for fitting errors associated with surrogates. We propose the use of cross validation for estimating the required safety margin for a desired level of conservativeness (percentage of safe predictions). The approach was tested on three algebraic examples for two basic surrogates: namely, kriging and polynomial response surface. For these examples we found that cross validation is effective for selecting the safety margin. We also applied the approach to the probabilistic design optimization of a composite laminate. This design under uncertainty example showed that the approach can be successfully used in engineering applications.

Study on Probabilistic Optimal Design of Diaphragm Spring

The mathematical model of probabilistic optimization is established for diaphragm spring. The process and the method of design with an example are presented. The result shows the model and the method of design are feasible, and perfect optimal result is attained. Vehicle has mass production and its conditions of use are changeable, which determine design parameters and operating parameters of its parts have complex randomness. Diaphragm spring clutch is now widely used by many kinds of vehicles. Research of diaphragm spring parameters’ probabilistic and statistics model and the study of its optimal design have great significance for the design, R & D, manufacturing and use of automotive products and parts. Studies in the references[1,2,3] have shown that the root cause of diaphragm spring’s fatigue fracture is the original crackles near the concave inner edge point, so there is a need to do probabilistic optimal design for diaphragm spring’s structural parameters to improve the fatigue life.

Strength Properties of Poled Lead Zirconate Titanate Subjected to Biaxial Flexural Loading in High Electric Field

The mechanical strength of poled lead zirconate titanate (PZT) has been studied using ball‐on‐ring (BoR) biaxial flexure tests with a high electric field applied concurrently. Both the as‐received and the aged PZT specimens were tested. The Weibull plot and a 95% confidence ratio ring were used to characterize the responses of mechanical strength under various electric loading conditions. A fractographical study has been conducted at the same time, and the fracture origins or strength‐limiting flaws of tested PZT specimens have been identified and characterized accordingly. The fracture toughness was further estimated to correlate with the obtained fracture stresses and flaws. It has been observed that electric field affects the mechanical strength of poled PZT, and the degree of the effect depends on the sign and magnitude of the applied electric field. Within the examined electric field range of −3 to +3 times the coercive field, an increasing electric field resulted in a rapid strength decrease and a sharp increase with the turning point around the negative coercive field. Surface‐located volume‐distributed flaws were identified to be strength limiting for this PZT material. Variations of the mechanical strength with the electric field were believed to be related to the domain switching and amount of switchable domains. An aging effect on the mechanical strength of poled PZT could be significant, especially in the OC condition. These results and observations have the potential to serve probabilistic reliability analysis and design optimization of multilayer PZT piezo actuators.

Convergence control of the iterative procedure for performance-measure-based probabilistic structural design optimization

The advanced mean value (AMV) iterative scheme is commonly used to evaluate probabilistic constraints in the performance measure approach (PMA) for probabilistic structural design optimization (PSDO). However, the iterative procedure of PSDO may fail to converge. In this article, the chaotic dynamics theory is suggested to investigate and attack the non-convergence difficulties of PMA-based PSDO. Essentially, the AMV iterative formula forms a discrete dynamical system with control parameters. If the AMV iterative sequences present the numerical instabilities of periodic oscillation, bifurcation, and even chaos in some control parameter interval, then the outer optimization loop in PSDO cannot converge and acquire the correct optimal design. Furthermore, the stability transformation method (STM) of chaos feedback control is applied to perform the convergence control of AMV, in order to capture the desired fixed points in the whole control parameter interval. Meanwhile, PSDO is solved by the approaches of PMA two-level and PMA with the sequential approximate programming (SAP)—PMA with SAP. Numerical results of several examples illustrate that STM can smoothly overcome the convergence failure of PSDO resulting from the periodic oscillation, bifurcation, and chaotic solutions of AMV iterative procedure for evaluating the probabilistic constraints. Moreover, the probabilistic optimization with uniform random variables, which is widely recognized as a highly nonlinear and fairly difficult problem, can be attacked through introducing the strategy of chaos control. In addition, the approach of PMA with SAP combining with STM is quite effective and efficient.

An Efficient Re-Analysis Methodology for Probabilistic Vibration of Large-Scale Structures

Probabilistic analysis and design of large-scale structures requires repeated finite-element analyses of large models, and each analysis is expensive. This paper presents a methodology for probabilistic analysis and reliability-based design optimization of large-scale structures that consists of two re-analysis methods, one for estimating the deterministic vibratory response and another for estimating the probability of the response exceeding a certain level. The deterministic re-analysis method can analyze efficiently large-scale finite-element models consisting of tens or hundreds of thousand degrees of freedom and design variables that vary in a wide range. The probabilistic re-analysis method calculates very efficiently the system reliability for different probability distributions of the random variables by performing a single Monte Carlo simulation of one design. The methodology is demonstrated on probabilistic vibration analysis and reliability-based design optimization of a realistic vehicle model. It is shown that the computational cost of the proposed re-analysis method for a single reliability analysis is about 1/20 of the cost of the same analysis using MSC/NASTRAN. Moreover, the probabilistic re-analysis approach enables a designer to perform reliability-based design optimization of the vehicle at a cost almost equal to that of a single reliability analysis. Without using the probabilistic re-analysis approach, it would be impractical to perform reliability-based design optimization of the vehicle.

Effects of electric field and biaxial flexure on the failure of poled lead zirconate titanate

Reliable design of lead zirconate titanate (PZT) piezo stack actuators demands that several issues, including electromechanical coupling and ceramic strength-size scaling, be scrutinized. This study addresses those through the use of ball-on-ring (BoR) biaxial flexure strength tests of a PZT piezoelectric material that is concurrently subjected to an electric field. The Weibull strength distributions and fracture surfaces were examined. The mechanical failures were further analyzed in terms of internal stress, energy release rate, and domain-switching toughening. Both the sign and the magnitude of an electric field had a significant effect on the strength of poled PZT within the tested range. A surface flaw type with an average depth of around 18 microm was identified to be the strength limiter and responsible for the failure of the tested PZT under both mechanical and electromechanical loadings. With a value of 0.74 MPa.m(1/2) in the absence of electric field, the fracture toughness of the poled PZT was affected by an applied electric field just as the strength was affected. These results and observations have the potential to serve probabilistic reliability analysis and design optimization of multilayer PZT piezo actuators.

Probabilistic optimal design of laminates using improved particle swarm optimization

A new approach to the particle swarm optimization (PSO) is proposed for the solution of non-linear optimization problems with constraints, and is applied to the reliability-based optimum design of laminated composites. Special mutation-interference operators are introduced to increase swarm variety and improve the convergence performance of the algorithm. The reliability-based optimum design of laminated composites is modelled and solved using the improved PSO. The maximization of structural reliability and the minimization of total weight of laminates are analysed. The stacking sequence optimization is implemented in the improved PSO by using a special coding technique. Examples show that the improved PSO has high convergence and good stability and is efficient in dealing with the probabilistic optimal design of composite structures.

A probabilistic design optimization for MEMS tunable capacitors

This paper presents a design optimization method for MEMS parallel-plate capacitors under fabrication uncertainties. The objective of the optimization problem is to maximize the production yield considering the fabrication tolerances. The method utilizes aspects of the advanced first-order second-moment (AFOSM) reliability method in probabilistic design to find a linearized feasible region for performance functions and uses an analytical double-bounded-probability distribution function (DB-PDF) to approximate the distribution of random variables. Then, it attempts to place the tolerance box in such a way that the portions of the box with higher yield lies in the feasible region. The yield is directly estimated using the joint cumulative distribution function (CDF) over the tolerance box requiring no numerical integration and saving considerable computational complexity for multidimensional problems. For this reason, any arbitrary distribution can be considered for random parameters and the problem is not restricted to normality assumptions. Numerical examples, verified by Monte-Carlo simulations, demonstrate that optimal designs significantly increase the yield. The advantage of the proposed design optimization method is that the yield can be maximized in early design stages without tightening tolerances or increasing the fabrication cost and complexity. The application of the presented method is not limited to tunable capacitors and can be extended to other MEMS devices.

Spacecraft Reliability-Based Design Optimization Under Uncertainty Including Discrete Variables

computationally expensive sampling techniques. The computational cost of optimization approaches becomes prohibitivewhenconsideringdiscretetechnology andredundancychoicesasvariables.Thisworkpresentsagenetic algorithm with Monte Carlo sampling for probabilistic reliability-based design optimization of satellite systems. In thisapproach,confidence-levelconstraintsensurethatsystemreliabilityrequirementsaremetwithhighprobability. Thegeneticalgorithm–MonteCarlosamplingapproachiscomparedtoadeterministic margin-basedapproachthat enforcesmarginsorsafetyfactorsonthereliabilityofindividualcomponents.Thecomparisonshowsthatthegenetic algorithm–Monte Carlo sampling approach produces satellite designs that have low launch mass (a surrogate for cost) while achieving reliability requirements at specified high confidence levels, while the genetic algorithm– deterministic margin-based approach produces heavy satellite designs with excessive redundancy. Based on this work, extensions of a genetic algorithm-based approach for discrete optimization under uncertainty that may require less computational effort appear possible.