Robust Optimization With Parameter and Model Uncertainties Using Gaussian Processes

2016 ◽  
Vol 138 (11) ◽  
Author(s):  
Yanjun Zhang ◽  
Mian Li ◽  
Jun Zhang ◽  
Guoshu Li
Keyword(s):  
Robust Optimization ◽  
Gaussian Processes ◽  
Parameter Uncertainty ◽  
Simulation Models ◽  
Optimal Designs ◽  
Optimal Solutions ◽  
Objective Functions ◽  
Design Problems ◽  
Physical Systems ◽  
Design Variables

Uncertainty is unavoidable in engineering design, which may result in variations in the objective functions and/or constraints. The former may degrade the designed performance while the latter can even change the feasibility of the obtained optimal solutions. Taking uncertainty into consideration, robust optimization (RO) algorithms aim to find optimal solutions that are also insensitive to uncertainty. Uncertainty may include variation in parameters and/or design variables, inaccuracy in simulation models used in design problems, and other possible errors. Most existing RO algorithms only consider uncertainty in parameters, but overlook that in simulation models by assuming that the simulation model used can always provide identical outputs to those of the real physical systems. In this paper, we propose a new RO framework using Gaussian processes, considering not only parameter uncertainty but also uncertainty in simulation models. The consideration of model uncertainty in RO could reduce the risk for the obtained robust optimal designs becoming infeasible even if the parameter uncertainty has been considered. Two test examples with different degrees of complexity are utilized to demonstrate the applicability and effectiveness of our proposed algorithm.

Robust Optimization With Parameter and Model Uncertainties Using Gaussian Processes

2016 ◽  
Author(s):  
Yanjun Zhang ◽  
Mian Li ◽  
Jun Zhang ◽  
Guoshu Li
Keyword(s):  
Robust Optimization ◽  
Gaussian Processes ◽  
Parameter Uncertainty ◽  
Simulation Models ◽  
Optimal Designs ◽  
Optimal Solutions ◽  
Objective Functions ◽  
Design Problems ◽  
Physical Systems ◽  
Design Variables

Uncertainty is unavoidable in engineering design, which may result in variations in the objective functions and/or constraints. The former may degrade the designed performance while the latter can even change the feasibility of the obtained optimal solutions. Taking uncertainty into consideration, robust optimization (RO) algorithms aim to find optimal solutions that are also insensitive to uncertainty. Uncertainty may include variation in parameters and/or design variables, inaccuracy in simulation models used in design problems, and other possible errors. Most existing RO algorithms only consider uncertainty in parameters, but overlook that in simulation models by assuming that the simulation model used can always provide identical outputs to those of the real physical systems. In this paper, we propose a new RO framework using Gaussian processes, considering not only parameter uncertainty but also uncertainty in simulation models. The consideration of model uncertainty in RO will significantly reduce the risk for the obtained robust optimal designs becoming infeasible even the parameter uncertainty has been considered. Two test examples with different degrees of complexity are utilized to demonstrate the applicability and effectiveness of our proposed algorithm.

Multi-Objective Design Problem of Tire Wear and Visualization of Its Pareto Solutions2

2006 ◽  
Vol 34 (3) ◽  
pp. 170-194 ◽  
Author(s):  
M. Koishi ◽  
Z. Shida
Keyword(s):  
Inverse Problems ◽  
Conceptual Design ◽  
Dimensional Space ◽  
Design Stage ◽  
Objective Functions ◽  
Pareto Solutions ◽  
Design Problems ◽  
Multi Objective ◽  
Design Engineers ◽  
Design Variables

Abstract Since tires carry out many functions and many of them have tradeoffs, it is important to find the combination of design variables that satisfy well-balanced performance in conceptual design stage. To find a good design of tires is to solve the multi-objective design problems, i.e., inverse problems. However, due to the lack of suitable solution techniques, such problems are converted into a single-objective optimization problem before being solved. Therefore, it is difficult to find the Pareto solutions of multi-objective design problems of tires. Recently, multi-objective evolutionary algorithms have become popular in many fields to find the Pareto solutions. In this paper, we propose a design procedure to solve multi-objective design problems as the comprehensive solver of inverse problems. At first, a multi-objective genetic algorithm (MOGA) is employed to find the Pareto solutions of tire performance, which are in multi-dimensional space of objective functions. Response surface method is also used to evaluate objective functions in the optimization process and can reduce CPU time dramatically. In addition, a self-organizing map (SOM) proposed by Kohonen is used to map Pareto solutions from high-dimensional objective space onto two-dimensional space. Using SOM, design engineers see easily the Pareto solutions of tire performance and can find suitable design plans. The SOM can be considered as an inverse function that defines the relation between Pareto solutions and design variables. To demonstrate the procedure, tire tread design is conducted. The objective of design is to improve uneven wear and wear life for both the front tire and the rear tire of a passenger car. Wear performance is evaluated by finite element analysis (FEA). Response surface is obtained by the design of experiments and FEA. Using both MOGA and SOM, we obtain a map of Pareto solutions. We can find suitable design plans that satisfy well-balanced performance on the map called “multi-performance map.” It helps tire design engineers to make their decision in conceptual design stage.

Multi Objective Design Optimization of Two Bar Truss Using NSGA II and TOPSIS

2014 ◽  
Vol 984-985 ◽  
pp. 419-424
Author(s):  
P. Sabarinath ◽  
M.R. Thansekhar ◽  
R. Saravanan
Keyword(s):  
Design Optimization ◽  
Optimization Problems ◽  
Optimization Methods ◽  
Optimal Solutions ◽  
Objective Functions ◽  
Nsga Ii ◽  
Multi Objective ◽  
Total Displacement ◽  
Design Variables ◽  
Conflicting Objectives

Arriving optimal solutions is one of the important tasks in engineering design. Many real-world design optimization problems involve multiple conflicting objectives. The design variables are of continuous or discrete in nature. In general, for solving Multi Objective Optimization methods weight method is preferred. In this method, all the objective functions are converted into a single objective function by assigning suitable weights to each objective functions. The main drawback lies in the selection of proper weights. Recently, evolutionary algorithms are used to find the nondominated optimal solutions called as Pareto optimal front in a single run. In recent years, Non-dominated Sorting Genetic Algorithm II (NSGA-II) finds increasing applications in solving multi objective problems comprising of conflicting objectives because of low computational requirements, elitism and parameter-less sharing approach. In this work, we propose a methodology which integrates NSGA-II and Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) for solving a two bar truss problem. NSGA-II searches for the Pareto set where two bar truss is evaluated in terms of minimizing the weight of the truss and minimizing the total displacement of the joint under the given load. Subsequently, TOPSIS selects the best compromise solution.

Robust Optimization With Parameter and Model Uncertainties Using Gaussian Processes With Limited Samples

2017 ◽  
Author(s):  
Yanjun Zhang ◽  
Mian Li
Keyword(s):  
Robust Optimization ◽  
Engineering Design ◽  
Gaussian Processes ◽  
Parameter Uncertainty ◽  
Model Uncertainty ◽  
Computational Cost ◽  
Random Errors ◽  
Model Uncertainties ◽  
Uncertainty Model ◽  
Gp Model

Uncertainty is inevitable in engineering design. The existence of uncertainty may change the optimality and/or the feasibility of the obtained optimal solutions. In simulation-based engineering design, uncertainty could have various types of sources, such as parameter uncertainty, model uncertainty, and other random errors. To deal with uncertainty, robust optimization (RO) algorithms are developed to find solutions which are not only optimal but also robust with respect to uncertainty. Parameter uncertainty has been taken care of by various RO approaches. While model uncertainty has been ignored in majority of existing RO algorithms with the hypothesis that the simulation model used could represent the real physical system perfectly. In the authors’ earlier work, a RO framework was proposed to consider both parameter and model uncertainties using the Bayesian approach with Gaussian processes (GP), where metamodeling uncertainty introduced by GP modeling is ignored by assuming the constructed GP model is accurate enough with sufficient training samples. However, infinite samples are impossible for real applications due to prohibitive time and/or computational cost. In this work, a new RO framework is proposed to deal with both parameter and model uncertainties using GP models but only with limited samples. The compound effect of parameter, model, and metamodeling uncertainties is derived with the form of the compound mean and variance to formulate the proposed RO approach. The proposed RO approach will reduce the risk for the obtained robust optimal designs considering parameter and model uncertainties becoming non-optimal and/or infeasible due to insufficiency of samples for GP modeling. Two test examples with different degrees of complexity are utilized to demonstrate the applicability and effectiveness of the proposed approach.

Robustness Against Large Variations in Multi-Objective Optimization Problems

2013 ◽  
Author(s):  
Weijun Wang ◽  
Stéphane Caro ◽  
Fouad Bennis
Keyword(s):  
Optimization Problems ◽  
Optimal Solutions ◽  
Objective Functions ◽  
Multi Objective Optimization ◽  
Design Environment ◽  
Multi Objective ◽  
Uncertainty Sources ◽  
Design Variables ◽  
Robustness Index ◽  
Environment Parameters

In the presence of multiple optimal solutions in multi-modal optimization problems and in multi-objective optimization problems, the designer may be interested in the robustness of those solutions to make a decision. Here, the robustness is related to the sensitivity of the performance functions to uncertainties. The uncertainty sources include the uncertainties in the design variables, in the design environment parameters, in the model of objective functions and in the designer’s preference. There exist many robustness indices in the literature that deal with small variations in the design variables and design environment parameters, but few robustness indices consider large variations. In this paper, a new robustness index is introduced to deal with large variations in the design environment parameters. The proposed index is bounded between zero and one, and measures the probability of a solution to be optimal with respect to the values of the design environment parameters. The larger the robustness index, the more robust the solution with regard to large variations in the design environment parameters. Finally, two illustrative examples are given to highlight the contributions of this paper.

Multi-objective design optimization of power-cycling hydrodynamic mechanical transmissions

2018 ◽  
Vol 233 (4) ◽  
pp. 1392-1410 ◽  
Author(s):  
Xiaojun Liu ◽  
Dongye Sun ◽  
Datong Qin ◽  
Junlong Liu
Keyword(s):  
Numerical Simulation ◽  
Design Methodology ◽  
Speed Ratio ◽  
Optimal Solutions ◽  
Objective Functions ◽  
Multi Objective ◽  
Design Variables ◽  
Power Cycling ◽  
Automatic Transmissions ◽  
Mechanical Transmissions

The power-cycling hydrodynamic mechanical transmissions have the advantages of continuously adjustable speed ratio and high efficiency compared with the traditional automatic transmissions, so they may be a good substitute for the prior art. For off-highway vehicles which frequently work in low speed and confront great resistance, the zero-speed-ratio torque ratio (TR) of the power-cycling hydrodynamic mechanical transmissions represents the power performance of the vehicles. Furthermore, the complexity of the transmissions is an indispensable consideration for industrial designers. The radial and axial dimensions of the power-cycling hydrodynamic mechanical transmissions are determined by the effective diameter of the torque converter’s circuit and the number of transmissions gears, respectively. In order to optimize the zero-speed-ratio TR and the complexity of the power-cycling hydrodynamic mechanical transmissions, a design methodology is proposed. Considering that there is no explicit mathematical relationship between the design variables and the multi-objective functions, the parametric design and numerical simulation for the torque converter are carried out. The intrinsic mapping between the design variables and the multi-objective functions is fitted by the radial basis function neural network. On this basis, the fast and elitist non-dominated sorting genetic algorithm (NSGA-ΙΙ) is used to solve the multi-objective optimization problem. The numerical simulation for one group of solution selected from the Pareto optimal solutions is conducted. The simulation results indicate that the design methodology proposed in this study is effective. The optimal results show that the zero-speed-ratio TR of the power-cycling hydrodynamic mechanical transmissions is heavily influenced by the radial and axial space of such transmissions. The design optimization helps to find the optimal solutions for the power-cycling HMTs, which are superior to the traditional automatic transmissions and match well the prime mover.

Multidisciplinary Design Exploration for Sounding Launch Vehicle using Hybrid Rocket Engine in View of Ballistic Performance

2015 ◽  
Vol 32 (3) ◽  
Author(s):  
Kazuhisa Chiba ◽  
Masahiro Kanazaki ◽  
Atthaphon Ariyarit ◽  
Hideyuki Yoda ◽  
Shoma Ito ◽  
...  
Keyword(s):  
Initial Conditions ◽  
Rocket Engine ◽  
Launch Vehicle ◽  
Test Design ◽  
Objective Functions ◽  
Hybrid Rocket ◽  
Ballistic Performance ◽  
Design Problems ◽  
Design Variables ◽  
Hybrid Rocket Engine

AbstractConceptual design of a launch vehicle with a hybrid rocket engine (HRE) has been implemented using design informatics approach in order to investigate the feasibility of a single-stage hybrid rocket. Two test design problems were formulated by using two objective functions: maximization of downrange and minimization of initial gross weight, seven design variables which describe geometry and initial conditions, and one constraint relative to target altitude. The optimization result reveals the economical performance of hybrid rocket is limited with HRE in terms of the maximum downrange achievable. Moreover, the data-mining result indicates the mechanism of design-variable behavior.

scholarly journals Evaluating Optimization Solvers and Robust Semantics for Simulation-Based Falsification

10.29007/f4vs ◽  
2020 ◽  
Author(s):  
Johan Lidén Eddeland ◽  
Sajed Miremadi ◽  
Knut Åkesson
Keyword(s):  
Objective Function ◽  
Temporal Logic ◽  
Simulation Models ◽  
Cyber Physical Systems ◽  
Objective Functions ◽  
Problem Statement ◽  
Testing Technique ◽  
Physical Systems ◽  
Simulation Based ◽  
Optimization Solvers

Temporal-logic based falsification of Cyber-Physical Systems is a testing technique used to verify certain behaviours in simulation models, however the problem statement typically requires some model-specific tuning of parameters to achieve optimal results. In this experience report, we investigate how different optimization solvers and objective functions affect the falsification outcome for a benchmark set of models and specifications. With data from the four different solvers and three different objective functions for the falsification problem, we see that choice of solver and objective function depends both on the model and the specification that are to be falsified. We also note that using a robust semantics of Signal Temporal Logic typically increases falsification performance compared to using Boolean semantics.

scholarly journals Multilevel Optimization Procedures for Gas Turbine Design: Part II — Solution and Examples

1994 ◽  
Author(s):  
Fabio Ghiglino ◽  
Aristide Massardo
Keyword(s):  
Gas Turbine ◽  
Objective Functions ◽  
Energy Problem ◽  
Multilevel Optimization ◽  
Design Problems ◽  
Analysis And Design ◽  
Design Variables ◽  
Definition Of ◽  
Turbine Design ◽  
Sensitivity Derivative

The mathematical formulations of two procedures for breaking down a large optimization energy problem into a hierarchy of separated but coupled subproblems were presented and discussed in the first part of this work (Ghiglino and Massardo, 1994). Here their applicability is illustrated by formulation of realistic analysis and design problems in the field of gas turbine systems, both for open and closed cycle plants. The definition of objective functions, design variables, constraints and subproblem analyses is discussed; the results obtained with the sensitivity, derivative procedure are compared with the results obtained with the linearization procedure.

Process Optimal Design in Forging by Genetic Algorithm

2002 ◽  
Vol 124 (2) ◽  
pp. 397-408 ◽  
Author(s):  
J. S. Chung ◽  
S. M. Hwang
Keyword(s):  
Genetic Algorithm ◽  
Optimal Design ◽  
Process Design ◽  
Process Model ◽  
Hot Forging ◽  
Objective Functions ◽  
Optimal Design Problem ◽  
Design Problems ◽  
Design Iteration ◽  
Design Variables

A genetic algorithm based approach is presented for process optimal design in forging. In this approach, the optimal design problem is formulated on the basis of the integrated thermo-mechanical finite element process model so as to cover diverse design variables and objective functions, and a genetic algorithm is adopted for conducting design iteration for optimization. The process model, the formulation for process optimal design, and the genetic algorithm are described in detail. The approach is applied to several selected process design problems in cold and hot forging.

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