Design Sensitivity Method for Sampling-Based RBDO With Fixed COV

2015 ◽  
Author(s):  
Hyunkyoo Cho ◽  
K. K. Choi ◽  
Ikjin Lee ◽  
David Lamb
Keyword(s):  
Random Variables ◽  
Score Function ◽  
Random Variable ◽  
Probability Of Failure ◽  
Design Sensitivity ◽  
Mean Value ◽  
Reliability Based Design ◽  
Design Variables ◽  
Change In The Mean ◽  
The Mean

Conventional reliability-based design optimization (RBDO) uses the means of input random variables as its design variables; and the standard deviations (STDEVs) of the random variables are fixed constants. However, the fixed STDEVs may not correctly represent certain RBDO problems well, especially when a specified tolerance of the input random variable is presented as a percentage of the mean value. For this kind of design problem, the coefficients of variations (COVs) of the input random variables should be fixed, which means STDEVs are not fixed. In this paper, a method to calculate the design sensitivity of probability of failure for RBDO with fixed COV is developed. For sampling-based RBDO, which uses Monte Carlo simulation for reliability analysis, the design sensitivity of the probability of failure is derived using a first-order score function. The score function contains the effect of the change in the STDEV in addition to the change in the mean. As copulas are used for the design sensitivity, correlated input random variables also can be used for RBDO with fixed COV. Moreover, the design sensitivity can be calculated efficiently during the evaluation of the probability of failure. Using a mathematical example, the accuracy and efficiency of the developed method are verified. The RBDO result for mathematical and physical problems indicates that the developed method provides accurate design sensitivity in the optimization process.

Design Sensitivity Method for Sampling-Based RBDO With Varying Standard Deviation

2015 ◽  
Vol 138 (1) ◽  
Author(s):  
Hyunkyoo Cho ◽  
K. K. Choi ◽  
Ikjin Lee ◽  
David Lamb
Keyword(s):  
Standard Deviation ◽  
Design Variable ◽  
Score Function ◽  
Random Variable ◽  
Probability Of Failure ◽  
Design Sensitivity ◽  
Mean Value ◽  
Reliability Based Design ◽  
The Mean ◽  
Sensitivity Method

Conventional reliability-based design optimization (RBDO) uses the mean of input random variable as its design variable; and the standard deviation (STD) of the random variable is a fixed constant. However, the constant STD may not correctly represent certain RBDO problems well, especially when a specified tolerance of the input random variable is present as a percentage of the mean value. For this kind of design problem, the STD of the input random variable should vary as the corresponding design variable changes. In this paper, a method to calculate the design sensitivity of the probability of failure for RBDO with varying STD is developed. For sampling-based RBDO, which uses Monte Carlo simulation (MCS) for reliability analysis, the design sensitivity of the probability of failure is derived using a first-order score function. The score function contains the effect of the change in the STD in addition to the change in the mean. As copulas are used for the design sensitivity, correlated input random variables also can be used for RBDO with varying STD. Moreover, the design sensitivity can be calculated efficiently during the evaluation of the probability of failure. Using a mathematical example, the accuracy and efficiency of the developed design sensitivity method are verified. The RBDO result for mathematical and physical problems indicates that the developed method provides accurate design sensitivity in the optimization process.

scholarly journals Sensitivity Developments for RBDO With Dependent Input Variable and Varying Input Standard Deviation

2017 ◽  
Vol 139 (7) ◽  
Author(s):  
Hyunkyoo Cho ◽  
K. K. Choi ◽  
David Lamb
Keyword(s):  
Standard Deviation ◽  
Random Variable ◽  
Design Sensitivity ◽  
Statistical Correlation ◽  
Mean Value ◽  
Distribution Model ◽  
Target Point ◽  
Probabilistic Constraint ◽  
Reliability Based Design ◽  
Standard Normal

In reliability-based design optimization (RBDO), dependent input random variables and varying standard deviation (STD) should be considered to correctly describe input distribution model. The input dependency and varying STD significantly affect sensitivity for the most probable target point (MPTP) search and design sensitivity of probabilistic constraint in sensitivity-based RBDO. Hence, accurate sensitivities are necessary for efficient and effective process of MPTP search and RBDO. In this paper, it is assumed that dependency of input random variable is limited to the bivariate statistical correlation, and the correlation is considered using bivariate copulas. In addition, the varying STD is considered as a function of input mean value. The transformation between physical X-space and independent standard normal U-space for correlated input variable is presented using bivariate copula and marginal probability distribution. Using the transformation and the varying STD function, the sensitivity for the MPTP search and design sensitivity of probabilistic constraint are derived analytically. Using a mathematical example, the accuracy and efficiency of the developed sensitivities are verified. The RBDO result for the mathematical example indicates that the developed methods provide accurate sensitivities in the optimization process. In addition, a 14D engineering example is tested to verify the practicality and scalability of the developed sensitivity methods.

scholarly journals Evaluation of Bearing Capacity of Strip Footing Using Random Layers Concept

2015 ◽  
Vol 37 (3) ◽  
pp. 31-39 ◽  
Author(s):  
Marek Kawa ◽  
Dariusz Łydżba
Keyword(s):  
Monte Carlo ◽  
Bearing Capacity ◽  
Failure Mechanism ◽  
Design Theory ◽  
Cohesive Soil ◽  
Probability Of Failure ◽  
Mean Value ◽  
Strip Footing ◽  
Random Field Theory ◽  
The Mean

Abstract The paper deals with evaluation of bearing capacity of strip foundation on random purely cohesive soil. The approach proposed combines random field theory in the form of random layers with classical limit analysis and Monte Carlo simulation. For given realization of random the bearing capacity of strip footing is evaluated by employing the kinematic approach of yield design theory. The results in the form of histograms for both bearing capacity of footing as well as optimal depth of failure mechanism are obtained for different thickness of random layers. For zero and infinite thickness of random layer the values of depth of failure mechanism as well as bearing capacity assessment are derived in a closed form. Finally based on a sequence of Monte Carlo simulations the bearing capacity of strip footing corresponding to a certain probability of failure is estimated. While the mean value of the foundation bearing capacity increases with the thickness of the random layers, the ultimate load corresponding to a certain probability of failure appears to be a decreasing function of random layers thickness.

Uniform saddlepoint approximations

1988 ◽  
Vol 20 (3) ◽  
pp. 622-634 ◽  
Author(s):  
J. L. Jensen
Keyword(s):  
Random Variables ◽  
Random Variable ◽  
Tail Probability ◽  
Poisson Random Variable ◽  
Compound Poisson ◽  
Saddlepoint Approximations ◽  
The Mean ◽  
General Conditions

The validity of the saddlepoint expansion evaluated at the point y is considered in the limit y tending to ∞. This is done for the expansions of the density and of the tail probability of the mean of n i.i.d. random variables and also for the expansion of the tail probability of a compound Poisson sum , where N is a Poisson random variable. We consider both general conditions that ensure the validity of the expansions and study the four classes of densities for X1 introduced in Daniels (1954).

The Largest Missing Value in a Sample of Geometric Random Variables

2014 ◽  
Vol 23 (5) ◽  
pp. 670-685 ◽  
Author(s):  
MARGARET ARCHIBALD ◽  
ARNOLD KNOPFMACHER
Keyword(s):  
Random Variables ◽  
Random Variable ◽  
Missing Value ◽  
Asymptotic Expressions ◽  
Mean And Variance ◽  
The Mean ◽  
Special Case

We consider samples of n geometric random variables with parameter 0 < p < 1, and study the largest missing value, that is, the highest value of such a random variable, less than the maximum, that does not appear in the sample. Asymptotic expressions for the mean and variance for this quantity are presented. We also consider samples with the property that the largest missing value and the largest value which does appear differ by exactly one, and call this the LMV property. We find the probability that a sample of n variables has the LMV property, as well as the mean for the average largest value in samples with this property. The simpler special case of p = 1/2 has previously been studied, and verifying that the results of the present paper coincide with those previously found for p = 1/2 leads to some interesting identities.

Reliability-Based Topology Optimization Using Mean-Value Second-Order Saddlepoint Approximation

2018 ◽  
Vol 140 (3) ◽  
Author(s):  
Dimitrios I. Papadimitriou ◽  
Zissimos P. Mourelatos
Keyword(s):  
Topology Optimization ◽  
Limit State ◽  
Random Variables ◽  
Saddlepoint Approximation ◽  
Probability Of Failure ◽  
Mean Value ◽  
Second Order ◽  
State Function ◽  
Sensitivity Derivatives ◽  
Reliability Method

A reliability-based topology optimization (RBTO) approach is presented using a new mean-value second-order saddlepoint approximation (MVSOSA) method to calculate the probability of failure. The topology optimizer uses a discrete adjoint formulation. MVSOSA is based on a second-order Taylor expansion of the limit state function at the mean values of the random variables. The first- and second-order sensitivity derivatives of the limit state cumulant generating function (CGF), with respect to the random variables in MVSOSA, are computed using direct-differentiation of the structural equations. Third-order sensitivity derivatives, including the sensitivities of the saddlepoint, are calculated using the adjoint approach. The accuracy of the proposed MVSOSA reliability method is demonstrated using a nonlinear mathematical example. Comparison with Monte Carlo simulation (MCS) shows that MVSOSA is more accurate than mean-value first-order saddlepoint approximation (MVFOSA) and more accurate than mean-value second-order second-moment (MVSOSM) method. Finally, the proposed RBTO-MVSOSA method for minimizing a compliance-based probability of failure is demonstrated using two two-dimensional beam structures under random loading. The density-based topology optimization based on the solid isotropic material with penalization (SIMP) method is utilized.

scholarly journals Game Techniques to Solve Students' Adjustment Problems: Evidance From Indonesia

2021 ◽  
Vol 15 ◽  
pp. 191-201
Author(s):  
Edy Irawan ◽  
Syarifuddin Dahlan ◽  
Een Y. Haenilah ◽  
Tubagus Ali Rachman Puja Kesuma ◽  
Albet Maydiantoro ◽  
...  
Keyword(s):  
Mean Value ◽  
Control Group ◽  
Adjustment Problems ◽  
Control Group Design ◽  
Guidance Services ◽  
Post Test ◽  
Change In The Mean ◽  
The Mean ◽  
Experimental Group ◽  
Group Guidance

This research is motivated by the problems of students who fail to achieve happiness in their lives; this is caused by the inability of students to make adjustments to all forms of change in maintaining survival. Efforts that can be made to overcome this problem are to provide group guidance services with game techniques. This study aims to test whether the game technique in group guidance services is effective for improving students' self-adjustment. The method used in this research is experimental research using Pretest-Posttest Control Group Design. The results showed that there was a change in the mean value at the pre-test of 93.43 for the experimental class and 92.57 for the control class and the mean value at the post-test of 151.64 for the experimental class and 98.71 for the control class. So that the mean value at the time of post-test increased by 58.21 for the experimental group and 06.14 for the control group. These changes mean that group guidance with game techniques is empirically proven to be effective in increasing student adaptation

scholarly journals On the Accuracy of Error Propagation Calculations by Analytic Formulas Obtained for the Inverse Transformation

2019 ◽  
Vol 64 (3) ◽  
pp. 217
Author(s):  
V. I. Romanenko ◽  
N. V. Kornilovska
Keyword(s):  
Error Propagation ◽  
Random Variable ◽  
Mean Value ◽  
Inverse Transformation ◽  
First Order ◽  
Order Of Magnitude ◽  
The Mean ◽  
Normally Distributed

The accuracy of error propagation calculations is estimated for the transformation x → y = f(x) of the normally distributed random variable x. The estimation is based on the formulas for the error propagation obtained for the inverse transformation y → x of the normally distributed random variable y. In the general case, the calculation accuracy for the mean value and the variance of the random variable y is shown to be of the first order of magnitude in the variance of the random variable x.

Uniform saddlepoint approximations

1988 ◽  
Vol 20 (03) ◽  
pp. 622-634 ◽  
Author(s):  
J. L. Jensen
Keyword(s):  
Random Variables ◽  
Random Variable ◽  
Tail Probability ◽  
Poisson Random Variable ◽  
Compound Poisson ◽  
Saddlepoint Approximations ◽  
The Mean ◽  
General Conditions

The validity of the saddlepoint expansion evaluated at the point y is considered in the limit y tending to ∞. This is done for the expansions of the density and of the tail probability of the mean of n i.i.d. random variables and also for the expansion of the tail probability of a compound Poisson sum , where N is a Poisson random variable. We consider both general conditions that ensure the validity of the expansions and study the four classes of densities for X 1 introduced in Daniels (1954).

Reliability-Based Design Optimization on Qualitative Objective With Limited Information

2018 ◽  
Vol 140 (12) ◽  
Author(s):  
Khaldon T. Meselhy ◽  
G. Gary Wang
Keyword(s):  
Objective Function ◽  
Design Optimization ◽  
Probability Distributions ◽  
Random Variables ◽  
Limited Information ◽  
Reliability Based Design Optimization ◽  
Mathematical Formula ◽  
Reliability Based Design ◽  
Design Structure ◽  
Design Variables

Reliability-based design optimization (RBDO) algorithms typically assume a designer's prior knowledge of the objective function along with its explicit mathematical formula and the probability distributions of random design variables. These assumptions may not be valid in many industrial cases where there is limited information on variable variability and the objective function is subjective without mathematical formula. A new methodology is developed in this research to model and solve problems with qualitative objective functions and limited information about random variables. Causal graphs and design structure matrix are used to capture designer's qualitative knowledge of the effects of design variables on the objective. Maximum entropy theory and Monte Carlo simulation are used to model random variables' variability and derive reliability constraint functions. A new optimization problem based on a meta-objective function and transformed deterministic constraints is formulated, which leads close to the optimum of the original mathematical RBDO problem. The developed algorithm is tested and validated with the Golinski speed reducer design case. The results show that the algorithm finds a near-optimal reliable design with less initial information and less computation effort as compared to other RBDO algorithms that assume full knowledge of the problem.

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