Sampling-Based Approach for Design Optimization in the Presence of Interval Variables

2013 ◽  
Author(s):  
David Yoo ◽  
Ikjin Lee
Keyword(s):  
Monte Carlo Simulation ◽  
Design Optimization ◽  
Random Variables ◽  
Delta Function ◽  
Probability Of Failure ◽  
Probabilistic Constraints ◽  
Worst Case ◽  
Performance Function ◽  
Dirac Delta ◽  
Interval Variables

This paper proposes a methodology for sampling-based design optimization in the presence of interval variables. Assuming that an accurate surrogate model is available, the proposed method first searches the worst combination of interval variables for constraints when only interval variables are present or for probabilistic constraints when both interval and random variables are present. Due to the fact that the worst combination of interval variables for probability of failure does not always coincide with that for a performance function, the proposed method directly uses the probability of failure to obtain the worst combination of interval variables when both interval and random variables are present. To calculate sensitivities of constraints and probabilistic constraints with respect to interval variables by the sampling-based method, the behavior of interval variables at the worst case is defined by utilizing the Dirac delta function. Then, Monte Carlo simulation is applied to calculate constraints and probabilistic constraints with the worst combination of interval variables, and their sensitivities. The important merit of the proposed method is that it does not require gradients of performance functions and transformation from X-space to U-space for reliability analysis after the worst combination of interval variables is obtained, thus there is no approximation or restriction in calculating the sensitivities of constraints or probabilistic constraints. Numerical results indicate that the proposed method can search the worst case probability of failure with both efficiency and accuracy and that it can perform design optimization with mixture of random and interval variables by utilizing the worst case probability of failure search.

Optimal Design of Intentional Mistuning for a Bladed Disk With Interval Uncertainty

2015 ◽  
Author(s):  
David Yoo ◽  
Jiong Tang
Keyword(s):  
Optimal Design ◽  
Design Optimization ◽  
Probability Of Failure ◽  
Forced Response ◽  
Response Amplitude ◽  
Interval Uncertainty ◽  
Probabilistic Constraints ◽  
Worst Case ◽  
Bladed Disk ◽  
Gradient Based

This paper presents a methodology for the optimal design of intentional mistuning for a mistuned bladed disk with interval uncertainty. For a bladed disk where blades are weakly coupled, presence of random mistuning can easily induce vibration localization. This phenomenon will lead to great amplification in response amplitude of certain blades. To achieve desired reliability of a bladed disk, amplified response must be reduced to certain level, which requires probabilistic or reliability analysis. In this study, it is considered that blades have random distribution and coupling between blades has interval uncertainty. To treat the interval uncertainty appropriately, the worst-case combination of interval couplings is searched first, then probability of failure is evaluated under the worst-case condition. To increase reliability of a bladed disk, intentional mistuning is used in this study. While applying the intentional mistuning, it is also wanted to minimize the degree of intentional mistuning to minimize the cost of implementation. To find optimal combination of intentional mistuning parameters to achieve dual goals, gradient-based design optimization approach is utilized, which is expected to guarantee efficient convergence. To carry out gradient-based design optimization, sensitivities of objective function and probabilistic constraints with respect to intentional mistuning parameters are derived. During the sensitivity analysis, distribution of forced response amplitude is identified through Gaussian fit and eigenvalue perturbation theory is referred to. Monte Carlo simulation is utilized to accurately calculate probability of failure and its sensitivity. The proposed method is demonstrated with numerical examples of two distinct bladed disks.

Continuous Random Numbers

2019 ◽  
pp. 54-69
Author(s):  
Robert H. Swendsen
Keyword(s):  
Probability Distributions ◽  
Random Variables ◽  
Delta Function ◽  
Ideal Gas ◽  
Dirac Delta Function ◽  
Bayesian Probability ◽  
Random Numbers ◽  
Continuous Variables ◽  
Dirac Delta ◽  
Discrete Random Variables

The theory of probability developed in Chapter 3 for discrete random variables is extended to probability distributions, in order to treat the continuous momentum variables. The Dirac delta function is introduced as a convenient tool to transform continuous random variables, in analogy with the use of the Kronecker delta for discrete random variables. The properties of the Dirac delta function that are needed in statistical mechanics are presented and explained. The addition of two continuous random numbers is given as a simple example. An application of Bayesian probability is given to illustrate its significance. However, the components of the momenta of the particles in an ideal gas are continuous variables.

Iteration Algorithm for Propagation of Mixed Uncertainties in Reliability Analysis

2011 ◽  
Vol 199-200 ◽  
pp. 569-574
Author(s):  
Chang Cong Zhou ◽  
Zhen Zhou Lu ◽  
Qi Wang
Keyword(s):  
Structural Reliability ◽  
Random Variables ◽  
Point Estimate ◽  
Iteration Algorithm ◽  
Fourth Moment ◽  
Performance Function ◽  
Fuzzy Variables ◽  
Point Estimate Method ◽  
Interval Variables ◽  
Reliability Method

For structural reliability problems simultaneously involving random variables, interval variables and fuzzy variables, an iteration algorithm is proposed to deal with the propagation of uncertainties. Corresponding to assumed membership value in the membership level interval [0,1], the membership interval of fuzzy variables can be obtained. After the fuzzy variables and the interval variables’ effects on the extreme value of performance function are alternately and iteratively analyzed with the random variables’ effects on statistical properties of the performance function, converged design point can be calculated, and the reliability can be as well calculated by the fourth moment algorithm based on the point estimate method. Finally the membership function of the reliability can be solved. Owing to the faster convergence of the iteration algorithm, the efficiency of the proposed algorithm is highly improved compared to the conventional numerical simulation method. And the adoption of fourth moment method proves much better in accuracy than only applying first order reliability method in the iteration algorithm. After the basic concept and process of the proposed algorithm is detailed, several numerical and engineering examples are studied to demonstrate its advantage both in efficiency and accuracy.

General Robust Design by Semi-Second-Order Taylor Expansion

2008 ◽  
Author(s):  
Xiaoping Du
Keyword(s):  
Design Optimization ◽  
Robust Design ◽  
Taylor Expansion ◽  
Random Variables ◽  
Second Order ◽  
Robust Design Optimization ◽  
Integration Strategy ◽  
Interval Variables ◽  
Standard Deviations ◽  
Robustness Assessment

The purpose of robust design optimization is to minimize variations in design performances and therefore to make the design insensitive to uncertainties. Current robust design methods fall into two types — probabilistic robust design and worst-case (interval) robust design. The former method is used when random variables are involved. In this method, robustness is measure by standard deviations of design performances. The later method is used when uncertainties are represented by intervals. The widths of design performances are then used to measure robustness. In many engineering application, both random variables and interval variables may exist simultaneously. In this paper, a general approach to robust design optimization is proposed. The generality comes from the ability to handle both random and interval variables. To alleviate the computational burden, we employ a previously developed general robustness assessment method — semi-second-order Taylor expansion method, to evaluate the maximum and minimum standard deviations of a design performance. An efficient integration strategy of the general robustness assessment and optimization is proposed. With the integration strategy, the number of function calls can be reduced while good accuracy is maintained. A robust shaft design problem is given for demonstration.

Hybrid Analysis Method for Reliability-Based Design Optimization

2003 ◽  
Vol 125 (2) ◽  
pp. 221-232 ◽  
Author(s):  
Byeng D. Youn ◽  
Kyung K. Choi ◽  
Young H. Park
Keyword(s):  
Design Optimization ◽  
Mean Value ◽  
Performance Measure ◽  
Equality Constraint ◽  
Probabilistic Constraints ◽  
Probabilistic Constraint ◽  
Reliability Based Design Optimization ◽  
Performance Function ◽  
Reliability Based Design ◽  
Advanced Mean Value

Reliability-based design optimization (RBDO) involves evaluation of probabilistic constraints, which can be done in two different ways, the reliability index approach (RIA) and the performance measure approach (PMA). It has been reported in the literature that RIA yields instability for some problems but PMA is robust and efficient in identifying a probabilistic failure mode in the optimization process. However, several examples of numerical tests of PMA have also shown instability and inefficiency in the RBDO process if the advanced mean value (AMV) method, which is a numerical tool for probabilistic constraint evaluation in PMA, is used, since it behaves poorly for a concave performance function, even though it is effective for a convex performance function. To overcome difficulties of the AMV method, the conjugate mean value (CMV) method is proposed in this paper for the concave performance function in PMA. However, since the CMV method exhibits the slow rate of convergence for the convex function, it is selectively used for concave-type constraints. That is, once the type of the performance function is identified, either the AMV method or the CMV method can be adaptively used for PMA during the RBDO iteration to evaluate probabilistic constraints effectively. This is referred to as the hybrid mean value (HMV) method. The enhanced PMA with the HMV method is compared to RIA for effective evaluation of probabilistic constraints in the RBDO process. It is shown that PMA with a spherical equality constraint is easier to solve than RIA with a complicated equality constraint in estimating the probabilistic constraint in the RBDO process.

scholarly journals Reliability Based Design Optimization of Reinforced Concrete Frames Using Genetic Algorithm

2021 ◽  
Author(s):  
Mohammadreza Seify Asghshahr
Keyword(s):  
Genetic Algorithm ◽  
Reinforced Concrete ◽  
Design Optimization ◽  
Reliability Index ◽  
Random Variables ◽  
Probabilistic Constraints ◽  
Rc Frame ◽  
Reliability Based Design Optimization ◽  
Form Analysis ◽  
Reliability Based Design

This paper introduces a new framework for reliability based design optimization (RBDO) of the reinforced concrete (RC) frames. This framework is constructed based on the genetic algorithm (GA) and finite element reliability analysis (FERA) to optimize the frame weight by selecting appropriate sections for structural elements under deterministic and probabilistic constraints. Modulus of elasticity of the concrete and steel bar, dead load, live load, and earthquake equivalent load are considered as random variables. Deterministic constraints include the code design requirements that must be satisfied for all the frame elements according to the nominal values of the aforementioned random variables. On the other hand, this framework provides the minimum required reliability index as the probabilistic constraint. The first-order reliability method (FORM) using the Newton-type recursive relationship will be used to compute the reliability index. The maximum inter-story drift is considered as an engineering demand parameter to define the limit-state function in FORM analysis. To implement the proposed framework, a mid-rise five-story RC frame is selected as an example. Based on the analysis results, increasing the minimum reliability index from 6 to 7 causes an 11 % increase in the weight of the selected RC frame as an objective function. So, we can obtain a trade-off between the optimized frame weight and the required reliability index utilizing the developed framework. Furthermore, the high values of the reliability index for the frame demonstrate the conservative nature of code requirements for interstory drift limitations based on the linear static analysis method.

Hybrid Analysis Method for Reliability-Based Design Optimization

2001 ◽  
Author(s):  
Kyung K. Choi ◽  
Byeng D. Youn
Keyword(s):  
Design Optimization ◽  
Mean Value ◽  
Performance Measure ◽  
Equality Constraint ◽  
Probabilistic Constraints ◽  
Probabilistic Constraint ◽  
Reliability Based Design Optimization ◽  
Performance Function ◽  
Reliability Based Design ◽  
Advanced Mean Value

Abstract Reliability-Based Design Optimization (RBDO) involves evaluation of probabilistic constraints, which can be done in two different ways, the Reliability Index Approach (RIA) and the Performance Measure Approach (PMA). It has been reported in the literature that RIA yields instability for some problems but PMA is robust and efficient in identifying a probabilistic failure mode in the RBDO process. However, several examples of numerical tests of PMA have also shown instability and inefficiency in the RBDO process if the Advanced Mean Value (AMV) method, which is a numerical tool for probabilistic constraint evaluation in PMA, is used, since it behaves poorly for a concave performance function, even though it is effective for a convex performance function. To overcome difficulties of the AMV method, the Conjugate Mean Value (CMV) method is proposed in this paper for the concave performance function in PMA. However, since the CMV method exhibits the slow rate of convergence for the convex function, it is selectively used for concave-type constraints. That is, once the type of the performance function is identified, either the AMV method or the CMV method can be adaptively used for PMA during the RBDO iteration to evaluate probabilistic constraints effectively. This is referred to as the Hybrid Mean Value (HMV) method. The enhanced PMA with the HMV method is compared to RIA for effective evaluation of probabilistic constraints in the RBDO process. It is shown that PMA with a spherical equality constraint is easier to solve than RIA with a complicated equality constraint in estimating the probabilistic constraint in the RBDO process.

Augmented Single Loop Single Vector Algorithm Using Nonlinear Approximations of Constraints in Reliability-Based Design Optimization

2019 ◽  
Vol 141 (10) ◽  
Author(s):  
Petter N. Lind ◽  
Mårten Olsson
Keyword(s):  
Design Optimization ◽  
Order Approximation ◽  
Nonlinear Problems ◽  
Probability Of Failure ◽  
Reliability Based Design Optimization ◽  
Performance Function ◽  
Single Loop ◽  
Nonlinear Approximations ◽  
Reliability Based Design ◽  
Probe Point

Reliability-based design optimization (RBDO) aims at minimizing a function of probabilistic design variables, given a maximum allowed probability of failure. The most efficient methods available for solving moderately nonlinear problems are single loop single vector (SLSV) algorithms that use a first-order approximation of the probability of failure in order to rewrite the inherently nested structure of the loop into a more efficient single loop algorithm. The research presented in this paper takes off from the fundamental idea of this algorithm. An augmented SLSV algorithm is proposed that increases the rate of convergence by making nonlinear approximations of the constraints. The nonlinear approximations are constructed in the following way: first, the SLSV experiments are performed. The gradient of the performance function is known, as well as an estimate of the most probable failure point (MPP). Then, one extra experiment, a probe point, per performance function is conducted at the first estimate of the MPP. The gradient of each performance function is not updated but the probe point facilitates the use of a natural cubic spline as an approximation of an augmented MPP estimate. The SLSV algorithm using probing (SLSVP) also incorporates a simple and effective move limit (ML) strategy that also minimizes the heuristics needed for initiating the optimization algorithm. The size of the forward finite difference design of experiment (DOE) is scaled proportionally with the change of the ML and so is the relative position of the MPP estimate at the current iteration. Benchmark comparisons against results taken from the literature show that the SLSVP algorithm is more efficient than other established RBDO algorithms and converge in situations where the SLSV algorithm fails.

Dirac delta regularization

2020 ◽  
Author(s):  
Matheus Pereira Lobo
Keyword(s):  
Delta Function ◽  
Dirac Delta Function ◽  
Dirac Delta ◽  
Finite Result

I present a finite result for the Dirac delta "function."

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