6th World Congresses of Structural and Multidisciplinary Optimization Rio de Janeiro, 30 May - 03 June 2005, Brazil
Reliability Based Topology Optimization using a Hybrid Cellular Automaton Algorithm Neal M. Patel1a , Harish Agarwal2 , Andr´ es Tovar3 , John E. Renaud1b (1) Department of Aerospace and Mechanical Engineering. University of Notre Dame, Notre Dame, Indiana 46556, USA. (a) Graduate Research Assistant,
[email protected] (b) Professor,
[email protected]. (2) General Electric Global Research, Niskayuna, New York, 12309, USA,
[email protected] (3) Department of Mechanical and Mechatronic Engineering. Universidad Nacional de Colombia. Cr. 30 45-03, Bogota, Colombia. Assistant Professor,
[email protected]
1. Abstract In this research, a reliability based topology optimization (RBTO) for structural design methodology using the Hybrid Cellular Automata (HCA) method is proposed. More specifically, a decoupled reliability based design optimization (RBDO) approach is utilized, so that the topology optimization is separate from the reliability analysis. In this paper, a maximum allowable displacement failure mode is considered. In this methodology, starting from a continuum design space of uniform material distribution and initial uncertain variable values, a deterministic topology optimization is followed by a reliability assessment of the resulting structure to determine the most probable point of failure (MPP) for the current structure. The MPP is determined with respect to the maximum allowable deflection of the structure when loaded. This is generally a computationally expensive process using traditional techniques due to the large number of design variables associated with topology optimization problem. However, combining the efficient methods of the non-gradient HCA algorithm with the decoupled approach for RBDO aims to reduce this burden. The topology optimization was without constraint in previous applications of the HCA method. To accommodate RTBO, a mechanism for a global constraint for maximum allowable displacement is developed. This paper details the methodology for the six-sigma design of structures using topology optimization. 2. Keywords: Hybrid Cellular Automaton, Reliability Based Topology Optimization, Decoupled Approach. 3. Introduction The objective of reliability based design optimization (RBDO) is to mediate between cost and safety. RBDO is a probabilistic optimization method that aims to minimize variations caused by design uncertainties. In deterministic optimization, designs are often driven to the limits of the design constraints, neglecting tolerances in modeling and simulation uncertainties. Therefore, resulting optimized designs can be unreliable with a high probability of failure. The probabilistic RBDO approach allows for the design at specific risk and target reliability level accounting for a the various sources of uncertainties. In probabilistic optimization methods, these variational uncertainties are modeled as random variables. In this respect, the deterministic analysis can be viewed as an extension of the probabilistic analysis, where the deterministic quantities are a trivial instance of the random variables. Reliability based topology optimization (RBTO) extends this reliability notion to the area of structural topology optimization. In this paper, we consider a discretized continuum design domain, where each density element is a design variable. Traditional topology optimization methods drive the topology of a structure to an optimum design based on a single constraint on mass. However, nothing can be said about the reliability of the resulting topology since it does not account for uncertainties and modes of failure that the structure realistically would require, such as variabilities in loading and maximum levels of performance constraints, e.g. those on displacement and stress. Therefore, this RBTO formulation incorporates additional structural performance constraints account for uncertainties and modes of failure. However, because of the large number of design variables associated with continuum topology optimization problems, RBTO methods are inherently computationally time prohibitive because of additional required system analysis associate with RBDO. Karmanda et al. proposed a reliability-based
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methodology for structural topology optimization using a heuristic strategy that aims to reduce mass while improving the reliability level of the structure without greatly increasing its weight [6]. In this approach, the structural mass this driven a minimum that satisfies the performance constraint. In previous work, the Hybrid Cellular Automaton (HCA) method was developed for structural synthesis of continuum material where the state of each cell is defined by both density and strain energy [7, 10]. The change in density is evaluated using a local CA rule, while the strain energy is evaluated using a global structural analysis via the finite element method (FEM). In this research, RBTO has the same objective as the deterministic topology optimization; minimize strain energy and mass. Here we consider the mode of failure to be the maximum deflection of the structure when loaded. Therefore, a constraint on maximum displacement of the structure is implement as well as a similar displacement constraint formulation for the limit-state function. Agarwal presents a decoupled RBDO approach is employed such that the topology optimization is separate from the reliability analysis [1]. In this methodology, starting from an initial design domain of full material and uncertain parameters, such as loads, a complete topology optimization is followed by a reliability assessment of the structure; because the main optimization and the reliability assessment phases are detached, we refer to this a decoupled approach. The reliability formulation used in this investigation is known as the performance measure approach (PMA) where the reliability index β is included as a constraint in this subproblem and the random variables are driven to the Most Probably Point (MMP) of failure for the current structural design with respect to the displacement constraint [5]. The MPP must satisfy the specified reliability index β. A new topology optimization is executed using these uncertain values determined in the reliability subproblem and the process is repeated until convergence. For the problems presented in this paper, the design variables for the problem are the densities of the CA elements that make up the design domain. A First Order Reliability Method (FORM) is used to perform reliability analyses each structural topology where uncertain input parameters are characterized by statistically independent, normally distributed random variables. The elastic modulus and applied load(s) are considered as the random uncertain variables for the problems presented in this paper. 4. Reliability Based Design Optimization Optimized designs based on a deterministic formulation are usually associated with a high probability of failure because of the violation of certain probabilistic constraints. This is particularly true if the probabilistic constraints are active at the deterministic optima. In today’s competitive marketplace, it is very important that the resulting designs are optimum and at the same time reliable. Optimized designs without considering the variability of design variables and parameters can be subjected to failure in service. So to achieve the objective of obtaining reliable optimum designs, a designer has to replace a deterministic optimization with a Reliability-based Design Optimization (RBDO), where the critical probabilistic constraints are replaced with reliability constraints, as shown below, min s.t.
f (x, v) gR (x, v) ≥ 0 gD j = 1, .., Ndet j (x, v) ≥ 0 xl ≤ x ≤ xu ,
where x represents the elemental design variables from the topology optimization and gR represents reliability constraints. They are either constraints on probabilities of failure corresponding to each probabilistic constraint, or a single constraint on the overall system probability of failure. The reliability constraints gR can be formulated as follows, giR g
R
= Pallowi − Pi
i = 1, . . . , Nprob
= Pallowsys − Psys ,
(1) (2)
where Pi is the failure probability of the probabilistic constraint giR at a given design and Pallowi is the allowable probability of failure for this failure mode, Psys is the system failure probability at a given design and Pallowsys is the allowable system probability of failure. These probabilities of failure are
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usually estimated by employing standard reliability techniques. Reliability analysis is a tool to compute the reliability index β or the probability of failure corresponding to a given failure mode or for the entire system [3]. The uncertainties are modeled as continuous random variables V = (V1 , V2 , ..., Vn )T , with known (or assumed) continuously differentiable distribution functions, FV (v). In this paper, the random variables are denoted as V to distinguish themselves from the deterministic design variables, x, and the ith random probabilistic constraint can be denoted as giR (V). In the following, v denotes a realization of the set of random variables V. Letting giR (V) ≤ 0 represent the failure domain and giR (V) = 0 be the so-called limit state function. It is almost impossible to find an analytical solution to the above integral. In standard reliability techniques, a probability distribution transformation T :
