Zissimos P. Mourelatos

A single-loop method for reliability-based design optimisation

Reliability-based design optimisation (RBDO) can provide optimum designs in the presence of uncertainty. It can, therefore, be a powerful tool for design under uncertainty. The traditional, double-loop RBDO algorithm requires nested optimisation loops, where the design optimisation (outer) loop repeatedly calls a series of reliability (inner) loops. Due to the nested optimisation loops, the computational effort can be prohibitive for practical problems. A single-loop RBDO algorithm is proposed in this paper. Its accuracy is comparable with the double-loop approach and its efficiency is almost equivalent to deterministic optimisation. It collapses the nested optimisation loops into an equivalent single-loop optimisation process by using the Karush–Kuhn–Tucker optimality conditions of the inner reliability loops in the outer design optimisation loop, converting therefore the probabilistic optimisation problem into a deterministic optimisation problem. Two numerical applications, including an automotive vehicle side impact example, demonstrate the accuracy and superior efficiency of the proposed single-loop RBDO algorithm.

A Design Optimization Method Using Evidence Theory

Early in the engineering design cycle, it is difficult to quantify product reliability or compliance to performance targets due to insufficient data or information to model uncertainties. Probability theory cannot be, therefore, used. Design decisions are usually based on fuzzy information that is vague, imprecise qualitative, linguistic or incomplete. Recently, evidence theory has been proposed to handle uncertainty with limited information as an alternative to probability theory. In this paper, a computationally efficient design optimization method is proposed based on evidence theory, which can handle a mixture of epistemic and random uncertainties. It quickly identifies the vicinity of the optimal point and the active constraints by moving a hyperellipse in the original design space, using a reliability-based design optimization (RBDO) algorithm. Subsequently, a derivative-free optimizer calculates the evidence-based optimum, starting from the close-by RBDO optimum, considering only the identified active constraints. The computational cost is kept low by first moving to the vicinity of the optimum quickly and subsequently using local surrogate models of the active constraints only. Two examples demonstrate the proposed evidence-based design optimization method.

A Single-Loop Approach for System Reliability-Based Design Optimization

An efficient single-loop approach for series system reliability-based design optimization problems is presented in this paper. The approach enables the optimizer to apportion the system reliability among the failure modes in an optimal way by increasing the reliability of those failure modes whose reliability can be increased at low cost. Furthermore, it identifies the critical failure modes that contribute the most to the overall system reliability. A previously reported methodology uses a sequential optimization and reliability approach. It also uses a linear extrapolation to determine the coordinates of the most probable points of the failure modes as the design changes. As a result, the approach can be slow and may not converge if the location of the most probable failure point changes significantly. This paper proposes an alternative system RBDO approach that overcomes the above difficulties by using a single-loop approach where the searches for the optimum design and for the most probable failure points proceed simultaneously. An easy to implement active set strategy is used. The maximum allowable failure probabilities of the failure modes are considered as design variables. The efficiency and robustness of the method is demonstrated on three design examples involving a beam, an internal combustion engine and a vehicle side impact. The results are compared with deterministic optimization and the conventional component RBDO formulation.Copyright

DESIGN OPTIMIZATION OF HIERARCHICALLY DECOMPOSED MULTILEVEL SYSTEMS UNDER UNCERTAINTY

This paper presents a methodology for design optimization of hierarchically decomposed systems under uncertainty. We propose an extended, probabilistic version of the deterministic analytical target cascading (ATC) formulation by treating uncertain quantities as random variables and posing probabilistic design constraints. A bottom-to-top coordination strategy is used for the ATC process. Given that first-order approximations may introduce un-acceptably large errors, we use a technique based on the advanced mean value method to estimate uncertainty propagation through the multilevel hierarchy of elements that comprise the decomposed system. A simple yet illustrative hierarchical bilevel engine design problem is used to demonstrate the proposed methodology. The results confirm the applicability of the proposed probabilistic ATC formulation and the accuracy of the uncertainty propagation technique.

A Single-Loop Method for Reliability-Based Design Optimization

Reliability-Based Design Optimization (RBDO) can provide optimum designs in the presence of uncertainty. It can therefore, be a powerful tool for design under uncertainty. The traditional, double-loop RBDO algorithm requires nested optimization loops, where the design optimization (outer) loop, repeatedly calls a series of reliability (inner) loops. Due to the nested optimization loops, the computational effort can be prohibitive for practical problems. A single-loop RBDO algorithm is proposed in this paper for both normal and non-normal random variables. Its accuracy is the same with the double-loop approach and its efficiency is almost equivalent to deterministic optimization. It collapses the nested optimization loops into an equivalent single-loop optimization process by imposing the Karush-Kuhn-Tucker optimality conditions of the reliability loops as equivalent deterministic equality constraints of the design optimization loop. It therefore, converts the probabilistic optimization problem into an equivalent deterministic optimization problem, eliminating the need for calculating the Most Probable Point (MPP) in repeated reliability assessments. Several numerical applications including an automotive vehicle side impact example, demonstrate the accuracy and superior efficiency of the proposed single-loop RBDO algorithm.Copyright

A Methodology for Trading-Off Performance and Robustness Under Uncertainty

Mathematical optimization plays an important role in engineering design, leading to greatly improved performance. Deterministic optimization, however, may result in undesired choices because it neglects uncertainty. Reliability-based design optimization (RBDO) and robust design can improve optimization by considering uncertainty. This paper proposes an efficient design optimization method under uncertainty, which simultaneously considers reliability and robustness. A mean performance is traded-off against robustness for a given reliability level of all performance targets. This results in a probabilistic multiobjective optimization problem. Variation is expressed in terms of a percentile difference, which is efficiently computed using the advanced mean value method. A preference aggregation method converts the multiobjective problem to a single-objective problem, which is then solved using an RBDO approach. Indifference points are used to select the best solution without calculating the entire Pareto frontier. Examples illustrate the concepts and demonstrate their applicability.

Time-Dependent Reliability Estimation for Dynamic Problems Using a Niching Genetic Algorithm

A time-dependent reliability analysis method is presented for dynamic systems under uncertainty using a niching genetic algorithm (GA). The system response is modeled as a parametric random process. A double-loop optimization algorithm is used. The inner loop calculates the maximum response in time, using a hybrid (global-local) optimization algorithm. A global GA quickly locates the vicinity of the global maximum, and a gradient-based optimizer subsequently refines its location. A time-dependent problem is, therefore, transformed into a time-independent one. The outer loop calculates multiple most probable points (MPPs), which are commonly encountered in vibration problems. The dominant MPPs with the highest contribution to the probability of failure are identified. A niching GA is used because of its ability to simultaneously identify multiple solutions. All potential MPPs are initially identified approximately, and their location is efficiently refined using a gradient-based optimizer with local metamodels. For computational efficiency, the local metamodels are built using mostly available sample points from the niching GA. Among all MPPs, the significant and independent ones are identified using a correlation analysis. Approximate limit states are built at the identified MPPs, and the system failure probability is estimated using bimodal bounds. The vibration response of a cantilever plate under a random oscillating pressure load and a point load is used to illustrate the present method and demonstrate its robustness and efficiency. A finite-element model is used to calculate the plate response.

VALIDATING DESIGNS THROUGH SEQUENTIAL SIMULATION-BASED OPTIMIZATION

Computational simulation models support a rapid design process. Given model approximation and operating conditions uncertainty, designers must have confidence that the designs obtained using simulations will perform as expected. This paper presents a methodology for validating designs as they are generated during a simulation-based optimization process. Current practice focuses on validation of simulation models throughout the entire design space. In contrast, the proposed methodology requires validation only at design points generated during optimization. The goal of such validation is confidence in the resulting design rather than in the underlying simulation model. The proposed methodology is illustrated on a simple cantilever beam design subject to vibration.Copyright

An Efficient Re-Analysis Methodology for Probabilistic Vibration of Large-Scale Structures

Probabilistic analysis and design of large-scale structures requires repeated finite-element analyses of large models, and each analysis is expensive. This paper presents a methodology for probabilistic analysis and reliability-based design optimization of large-scale structures that consists of two re-analysis methods, one for estimating the deterministic vibratory response and another for estimating the probability of the response exceeding a certain level. The deterministic re-analysis method can analyze efficiently large-scale finite-element models consisting of tens or hundreds of thousand degrees of freedom and design variables that vary in a wide range. The probabilistic re-analysis method calculates very efficiently the system reliability for different probability distributions of the random variables by performing a single Monte Carlo simulation of one design. The methodology is demonstrated on probabilistic vibration analysis and reliability-based design optimization of a realistic vehicle model. It is shown that the computational cost of the proposed re-analysis method for a single reliability analysis is about 1/20 of the cost of the same analysis using MSC/NASTRAN. Moreover, the probabilistic re-analysis approach enables a designer to perform reliability-based design optimization of the vehicle at a cost almost equal to that of a single reliability analysis. Without using the probabilistic re-analysis approach, it would be impractical to perform reliability-based design optimization of the vehicle.

Calculation of Journal Bearing Dynamic Characteristics Including Journal Misalignment and Bearing Structural Deformation

A detailed journal bearing analysis for accurate evaluation of film dynamic characteristics is presented. The new formulation is based on a local perturbation of the oil film at each computational node that captures the important effects of journal misalignment and bearing structural deformation in rotor dynamics and engine NVH applications. The new algorithm is an extension to the classical approach of evaluating film dynamic characteristics based on journal eccentricity perturbation. The governing equations for the oil film pressure, stiffness, and damping are solved using a finite difference approach and their output is validated with numerical results from the literature.