# Worst Case Optimization (WCO)

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## Robust Design Optimization (RDO)

It is evident that deterministic optimization without considering the uncertainties of input parameters will find a global optimum that lies on one or several constraints boundaries in most cases. With a small deviation on the solution, this one could easily violate one or more constraints and fall into the failure domain. Moreover, if the global optimum lies on a very narrow valley of objective function, even a slight variation in the variables could result in a significant change for the performance. In order to reduce the impact of disturbance and improve the robustness, RDO method is proposed.
Within the RDO process, the statistical variability of the design parameter is considered and the original deterministic input parameter is replaced by a random input parameter . For the sake of simplicity and without loss of generality, in this manuscript all the input variables with uncertainty follow a Gaussian law where the mean value is denoted and the standard deviation is denoted which is considered constant.
where represents the initial function objective, has the same dimension with , is the joint probability density function of . The mean and standard deviation of constraints can be calculated in the same way. However, the analytical evaluation of the integrals in (1.15) seems impossible to compute in most practical problems, for that reason, some numerical methods are proposed to obtain a quite good approximate result of these moments.
The techniques can be separated roughly into three groups, the first is the simulation methods, and the most general method in this group is the well-established Monte Carlo simulation with a huge quantity of samples. The second group is the perturbation methods and the most typical one is Taylor based Method of Moments (Padulo, Forth, & Guenov, 2008). The last one is approximation methods for instance by using chaos polynomials. The former two type of methods will be used in this manuscript and the following sections will show more details of the mentioned typical methods: Monte Carlo simulation and Taylor based Method of Moments.

### Monte Carlo simulation

Monte Carlo method is a simulation method commonly used in particle physics, or it can also introduce a statistical approach to assess risk in a finance. The purpose of the Monte Carlo simulation is to estimate quantities like mean, PDF… of the output parameters when the input parameters of the model are random variables. It is a numerical method for statistical simulation that uses sequences of random numbers to perform the simulation.
The aim of RDO is to find the mean value of design variables within a feasible space leading to lower sensitivity of the objective and constraints to uncertainty on design variables. The process for all these formulations are almost the same: Firstly, the input parameters are modelled by their distributions. Secondly, for the reason of simplicity and less time consuming, the Taylor Based Method of Moments is used to propagate the uncertainty from input to output parameters and calculate the statistical moments of the objective function and constraints. At last, robust formulations are used with optimization algorithms to solve the new robust problem.

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#### Reliability-Based Design Optimization (RBDO)

The method RBDO aims to find the optimal design which satisfies a given targeted reliability level represented by probability of failure. In sampling method (Hit or Miss), the probability of failure is related to the number of points in a sampling around the mean value that fall outside of the reliable domain. This means that RBDO attempts to find the optimal design in order to have the probability of failure smaller than a given targeted value. The formulation of RBDO

General Introduction
Chapter 1: Methods adapted to fast models
1 State of the art
1.1 Worst Case Optimization (WCO)
1.1.1 Worst-vertex-based WCO
1.2 Robust Design Optimization (RDO)
1.2.1 Monte Carlo simulation
1.2.2 Taylor Based Method of Moments
1.2.3 Robust formulations
1.3 Reliability-Based Design Optimization (RBDO)
1.3.1 Probability of failure
1.3.2 Double-loop method
1.3.3 Single-loop method
1.3.4 Sequential decoupled method
1.4 Reliability-Based Robust Design Optimization (RBRDO)
1.4.1 Nominal-the-best type
1.4.2 Smaller-the-better type
1.4.3 Larger-the-better type
2 Numerical investigations on methods
2.1 Comparison of WCO methods
2.2 Comparison of RDO methods
2.3 Comparison of RBDO methods
2.4 Comparison of RBRDO methods
3 Conclusion
Chapter 2: Methods adapted to heavy models
1 State of the art
1.1 Surrogate models
1.2 Kriging
1.2.1 Gaussian process
1.2.2 Simple Kriging (SK)
1.2.3 Ordinary Kriging (OK)
1.2.4 Universal Kriging (UK)
1.3 Meta-model based design optimization
1.4 Efficient Global Optimization (EGO)
1.5 Infill Searching Criteria (ISC)
2 Adaptive methods for optimization with uncertainty
2.1.1 New infill strategy for WCO
2.1.2 Mathematical example
2.2.1 New infill strategy for RDO
2.2.2 Mathematical example
2.3.1 Infill Strategies for Double-loop method
2.3.2 Infill Strategies for Single-loop method
2.3.3 Infill Strategies for Sequential method
2.3.4 Mathematical Example
2.4.1 New infill strategy for RBRDO
2.4.2 Mathematical example
3 Conclusion
Chapter 3: Transformer
1 Models
1.1 Analytic model
1.2 Finite element model
1.3 Comparison AM and FEM
2 Optimization problem
3 Comparison of methods for fast models
3.1 WCO methods
3.2 RDO methods
3.3 RBDO methods
3.4 RBRDO methods
4 Comparison of methods for heavy models
4.1 WCO methods
4.2 RDO methods
4.3 RBDO methods
4.4 RBRDO methods
5 Conclusion
Conclusion and perspectives
Reference

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