Reliability-Based Optimization für Multiple Constraints with Evolutionary Algorithms

Diplomarbeit von David Daum

Introduction:

In handling real-world optimization problems, it is often the case that the underlying decision variables and parameters cannot be controlled exactly as specified. For example, if a deterministic consideration of an optimization problem results in an optimal dimension of a cylindrical member to have a 50 mm diameter, there exists no manufacturing process which will guarantee the production of a cylinder having exactly a 50 mm diameter. Every manufacturing process has a finite machine precision and the dimensions are expected to vary around the specified value. Similarly, the strength of a material often does not remain fixed for the entire length of the material and is expected to vary from point to point. When such variations in decision variables and parameters are expected in practice, an obvious question arises: How reliable is the optimized design against failure when the suggested parameters cannot be adhered to? This question is important because in most optimization problems the deterministic optimum lies at the intersection of a number of constraint boundaries. Thus, if no uncertainties in parameters and variables are expected, the optimized solution is the best choice, but if uncertainties are expected, in most occasions, the optimized solution will be found to be infeasible, violating one or more constraints. These uncertainties, which are either controllable (e.g.imensions) or uncontrollable (e.g. material properties), are present and need to be accounted for in the design process.

Assuming that the variables follow a probability distribution in practice, reliability-based design optimization (RBDO) methods find a reliable solution which is feasible with a pre-specified probability. In most RBDO problems, failure probability and costs are violating objectives, which means that when one is lowered, the other may rise. Therefore, it is important to identify the uncertain variables which have an impact on the problem and describe them with different probability distributions based on statistical calculations. Then, the ordinary deterministic constraint is replaced by a stochastic constraint which is only restricting the probability of failure for a solution, not the failure itself. This can be done for each constraint or for the complete set of constraints, for the complete structure.

Different methods for evaluating the reliability of a solution exist. If the cumulative density function (CDF) with its borders is integrable, the reliability can be calculated analytically and serve directly as an input for the optimization.Unfortunately, most problems include complex distributions with complex constraints, which makes it impossible to calculate the exact value. One straightforward method is to use Monte-Carlo simulation; however, this gets computationally expensive when the desired reliability is very high. As engineering technology advances, many real-world design problems include complex and expensive calculations like simulation processes as finite element (FEA) or computational fluid dynamics (CFD). Since the constraint functions for every sample have to be evaluated, even a small sample size becomes impractical due to the computational burden.

A more common and faster approach is the evaluation of reliability with first- or secondorder reliability methods (FORM/SORM), which are based on linear and quadratic approximations of the constraint functions.

In this work, we use an approach based on the first-order approximation FORM of the constraint function. For the reliability analysis, we include all constraints for reaching high level of accuracy. Furthermore, we propose a method for identifying inactive and active constraints in terms of reliability which increases the computational efficiency.

This work is organized as follows: In Chapter 2, we give an overview of the related work and the basic literature. Chapter 3 provides a short introduction to single and multiobjective optimization together with the foundations of genetic algorithms. Chapter 4 gives an overview of reliability-based optimization and introduces some basic concepts for reliability analysis via first and second order approximation. Based on the evaluation of more than one constraint the concept of structural reliability is introduced in Chapter 5, including our approach to identify active and inactive constraints regarding the reliability. Chapter 6 reports about the combination of reliability-based optimization and multi-objective optimization. In Chapter 7, we show the results of our test cases and also of a real-world engineering problem. The summary and an outlook on future research are given in Chapter 8.

Table of Contents:

	Kurzzusammenfassung	2
	Abstract	3
	Contents	4
	List of figures	7
	List of tables	9
1.	Introduction	1
2.	Related Work	4
3.	Optimization	14
3.1	Single Objective Optimization	15
3.2	Multi-Objective Optimization	16
3.2.1	Multi-Objective Optimization Problem	17
3.2.2	Classical Methods	18
3.2.3	Pareto Dominance	19
3.3	Evolutionary Optimization	21
3.3.1	Biological Evolution	22
3.3.2	Genetic and Evolutionary Algorithms	24
4.	Contents
3.3.3	Evolutionary Optimization	25
3.3.4	NSGA2: A Multi Objective EA	31
4.	Reliability-based Design Optimization	34
4.1	Most Probable Point	36
4.1.1	Performance Measure Approach (PMA)	40
4.1.2	Reliability Index Approach RIA	41
4.2	Search Algorithm for the MPP	42
4.2.1	A Fast Approximation based on the RIA	42
4.3	Reliability Analysis	44
4.3.1	Simulation Methods	45
4.3.2	Single-loop Methods	46
4.3.3	Double-loop Methods	47
4.3.4	Decoupled Methods	47
5.	Structural Reliability	50
5.1	Foundations	50
5.2	Proposed Active Constraint Approach	52
5.3	Implementation	55
6.	Multi-Objective Reliability-Based Optimization	57
6.1	Fixed Reliability with Multiple Objectives	57
6.2	Reliability as an Objective	58
7.	Simulation Results	60
7.1	Two-Variable Test Problem	60
7.2	Two-Variable Multi-Modal Test Problem	65
7.3	A Car Side-Impact Problem	67
7.4	Multi-Objective Optimization for a Specified Reliability	72
7.4.1	Two-Objective Car Side-Impact Problem	73
8.	Conclusions	77
8.1	Summary	77
8.2	Outlook	79
	Bibliography	81

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