Post by Steve Yenisch on May 13, 2009 23:28:44 GMT -5
Paper: www.mae.ufl.edu/nkim/ISSMO/LuisCelorrioPaper.pdf
Presentation: www.mae.ufl.edu/nkim/ISSMO/LuisCelorrioPresentation.pdf
ABSTRACT
A deterministic optimization does not account for the uncertainties in the design variables and parameters. Modern competitive market demands have required the designers to introduce techniques for obtaining optimized designs that are also reliable. In the past twenty-five years, researchers have proposed a variety of methods to obtain optimum and reliable designs. These methods are addressed in Reliability-Based Design Optimization (RBDO). There are various types of RBDO approaches: Double-Loop methods, Decoupled methods and Single-Loop methods. This paper studies the efficiency of the various RBDO approaches applied to problems with dependent nonnormal random input variables. Usually, the joint cumulative distribution function of this random vector is seldom available. In practice, it is further recognized that a general random vector could only be characterized reliably up to the marginal distributions and a measure of dependence between such as the popular linear correlation matrix. However, such limited information could not be enough to defined uniquely a general random vector.
First Order Reliability Method (FORM) is the most widely used method for reliability analysis. This method usually requires a iso-probabilistic transformation from the dependent non normal input random variables in the original space to standard normal random variables in the standard space. Since only marginal distributions and the linear correlation matrix are available to define the input random vector, the Nataf transformation is often the first choice. Recently, Nataf transformation has been considered from the copula viewpoint because it is the composition of two functions: the Gaussian copula and a linear transformation. The Gaussian copula accurately constructs a joint cumulative distribution function for several types of dependent random vectors when the marginal distributions and the linear correlation matrix are available. Since this information could be obtained from real data, Nataf transformation could be used in many practical RBDO applications. However, using a Gaussian copula and, therefore, a linear correlation matrix to model a general random vector might generate several pitfalls and difficulties.
A structural design example with dependent loads is provided in this paper to show the practical applicability of the Nataf transformation in RBDO. Several RBDO approaches are considered. The numerical efficiency and convergence are checked. The computational cost is assessed by the amount of optimization iterations and the total of performance functions evaluations.
Keywords: Structural Reliability, Reliability Based Design Optimization, Nataf Transformation, Copulas, Correlated random variables
Presentation: www.mae.ufl.edu/nkim/ISSMO/LuisCelorrioPresentation.pdf
Efficiency Analysis of Reliability Based Design Optimization approaches for dependent non-normal random vectors
Luis Celorrio Barragué1, Eduardo Martínez de Pisón2, Carmen Bao3
1 Mechanical Engineering Department, University of La Rioja, Logroño, Spain, luis.celorrio@unirioja.es
2 Mechanical Engineering Department, University of La Rioja, Logroño, Spain, eduardo.mtnezdepison@unirioja.es
3 Mechanical Engineering Department, University of La Rioja, Logroño, Spain, carmen.bao@unirioja.es
Luis Celorrio Barragué1, Eduardo Martínez de Pisón2, Carmen Bao3
1 Mechanical Engineering Department, University of La Rioja, Logroño, Spain, luis.celorrio@unirioja.es
2 Mechanical Engineering Department, University of La Rioja, Logroño, Spain, eduardo.mtnezdepison@unirioja.es
3 Mechanical Engineering Department, University of La Rioja, Logroño, Spain, carmen.bao@unirioja.es
ABSTRACT
A deterministic optimization does not account for the uncertainties in the design variables and parameters. Modern competitive market demands have required the designers to introduce techniques for obtaining optimized designs that are also reliable. In the past twenty-five years, researchers have proposed a variety of methods to obtain optimum and reliable designs. These methods are addressed in Reliability-Based Design Optimization (RBDO). There are various types of RBDO approaches: Double-Loop methods, Decoupled methods and Single-Loop methods. This paper studies the efficiency of the various RBDO approaches applied to problems with dependent nonnormal random input variables. Usually, the joint cumulative distribution function of this random vector is seldom available. In practice, it is further recognized that a general random vector could only be characterized reliably up to the marginal distributions and a measure of dependence between such as the popular linear correlation matrix. However, such limited information could not be enough to defined uniquely a general random vector.
First Order Reliability Method (FORM) is the most widely used method for reliability analysis. This method usually requires a iso-probabilistic transformation from the dependent non normal input random variables in the original space to standard normal random variables in the standard space. Since only marginal distributions and the linear correlation matrix are available to define the input random vector, the Nataf transformation is often the first choice. Recently, Nataf transformation has been considered from the copula viewpoint because it is the composition of two functions: the Gaussian copula and a linear transformation. The Gaussian copula accurately constructs a joint cumulative distribution function for several types of dependent random vectors when the marginal distributions and the linear correlation matrix are available. Since this information could be obtained from real data, Nataf transformation could be used in many practical RBDO applications. However, using a Gaussian copula and, therefore, a linear correlation matrix to model a general random vector might generate several pitfalls and difficulties.
A structural design example with dependent loads is provided in this paper to show the practical applicability of the Nataf transformation in RBDO. Several RBDO approaches are considered. The numerical efficiency and convergence are checked. The computational cost is assessed by the amount of optimization iterations and the total of performance functions evaluations.
Keywords: Structural Reliability, Reliability Based Design Optimization, Nataf Transformation, Copulas, Correlated random variables
