Kyung K. Choi

A NEW STUDY ON RELIABILITY-BASED DESIGN OPTIMIZATION

This paper presents a general approach for probabilistic constraint evaluation in the reliability-based design optimization (RBDO). Different perspectives of the general approach are consistent in prescribing the probabilistic constraint, where the conventional reliability index approach (RIA) and the proposed performance measure approach (PMA) are identified as two special cases. PMA is shown to be inherently robust and more efficient in evaluating inactive probabilistic constraints, while RIA is more efficient for violated probabilistic constraints. Moreover, RBDO often yields a higher rate of convergence by using PMA, while RIA yields,singularity in some cases.

Hybrid Analysis Method for Reliability-Based Design Optimization

Reliability-Based Design Optimization (RBDO) involves evaluation of probabilistic constraints, which can be done in two different ways, the Reliability Index Approach (RIA) and the Performance Measure Approach (PMA). It has been reported in the literature that RIA yields instability for some problems but PMA is robust and efficient in identifying a probabilistic failure mode in the RBDO process. However, several examples of numerical tests of PMA have also shown instability and inefficiency in the RBDO process if the Advanced Mean Value (AMV) method, which is a numerical tool for probabilistic constraint evaluation in PMA, is used, since it behaves poorly for a concave performance function, even though it is effective for a convex performance function. To overcome difficulties of the AMV method, the Conjugate Mean Value (CMV) method is proposed in this paper for the concave performance function in PMA. However, since the CMV method exhibits the slow rate of convergence for the convex function, it is selectively used for concave-type constraints. That is, once the type of the performance function is identified, either the AMV method or the CMV method can be adaptively used for PMA during the RBDO iteration to evaluate probabilistic constraints effectively. This is referred to as the Hybrid Mean Value (HMV) method. The enhanced PMA with the HMV method is compared to RIA for effective evaluation of probabilistic constraints in the RBDO process. It is shown that PMA with a spherical equality constraint is easier to solve than RIA with a complicated equality constraint in estimating the probabilistic constraint in the RBDO process. NOMENCLATURE X

Enriched Performance Measure Approach for Reliability-Based Design Optimization.

An enriched performance measure approach is presented for reliability-based design optimization to substantially improve computational efficiency when applied to large-scale applications. In the enriched performance measure approach, four improvements are made over the original performance measure approach: as a way to launch reliability-based design optimization at a deterministic optimum design, as a new enhanced hybrid-mean value method, as an efficient probabilistic feasibility check, and as a fast reliability analysis under the condition of design closeness. It is found that deterministic design optimization helps improve numerical efficiency by reducing some reliability-based design optimization iterations. In reliability-based design optimization, a computational burden on the feasibility check of constraints can be significantly reduced by using a mean value first-order method and by carrying out the refined reliability analysis using the enhanced hybrid-mean value method for e-active and violated constraints. The enhanced hybrid-mean value method is developed to handle nonlinear and/or nonmonotonic constraints in reliability analysis. The fast reliability analysis method is proposed to efficiently evaluate probabilistic constraints under the condition of design closeness. Moreover, two numerical examples are provided to compare the enriched performance measure approach to existing reliability-based design optimization methods from a numerical efficiency and stability point of view.

Shape Design Sensitivity Analysis of Elastic Structures

ABSTRACT ABSTRACT Design problems in which the shape of two- or three-dimensional elastic bodies plays the role of design are studied. Five prototype problems are formulated in a unified variational form, with performance measures involving natural frequency, displacement, and stress in the structure. The material derivative of continuum mechanics and an adjoint variable method of design sensitivity analysis are used to develop an explicit formula for variation of performance measures in terms of normal movement of the boundary of the physical domain. Examples are presented for beams, membranes, shafts, and three-dimensional elastic solids.

An Investigation of Nonlinearity of Reliability-Based Design Optimization Approaches

Because deterministic optimum designs obtained without taking uncertainty into account could lead to unreliable designs, a reliability-based approach to design optimization is preferable using a Reliability-Based Design Optimization (RBDO) method. A typical RBDO process iteratively carries out a design optimization in an original random space (X-space) and a reliability analysis in an independent and standard normal random space (U-space). This process requires numerous nonlinear mappings between X- and U-spaces for various probability distributions. Therefore, the nonlinearity of the RBDO problem will depend on the type of distribution of random parameters, since a transformation between X- and U-spaces introduces additional nonlinearity into the reliability-based performance measures evaluated during the RBDO process. The evaluation of probabilistic constraints in RBDO can be carried out in two ways: using either the Reliability Index Approach (RIA), or the Performance Measure Approach (PMA). Different reliability analysis approaches employed in RIA and PMA result in different behaviors of nonlinearity for RIA and PMA in the RBDO process. In this paper, it is shown that RIA becomes much more difficult to solve for non-normally distributed random parameters because of the highly nonlinear transformations that are involved. However, PMA is rather independent of probability distributions because it only has a small involvement with a nonlinear transformation.

Probabilistic Structural Durability Prediction

Anefe cientreliability analysis method fordurability of structural components subjected to external and inertial loads with time-dependent variable amplitudes is presented. This method is able to support reliability analysis of crack-initiation and crack-propagation lives of practical applications, considering uncertainties such as material properties, manufacturing tolerances, and initial crack size. Three techniques are employed to support the probabilistic durability prediction: 1 ) strain-based approach for multiaxial crack-initiation-life prediction and linear elastic fracture mechanics approach for crack-propagation-life prediction, 2 ) statistics-based approach for reliability analysis, and 3 ) sensitivity analysis and optimization methods for searching the most probable point (MPP) in the random variable space to compute the fatigue failure probability using the e rst-order reliability analysis method. A two-point approximation method is employed to speed up the MPP search. A tracked-vehicle roadarm is presented to demonstrate feasibility of the proposed method.

Shape Design Sensitivity Analysis of Displacement and Stress Constraints

ABSTRACT Structural component shape design sensitivity analysis of functionals that define displacement at a point and mean stress over a small region are considered. Shape design sensitivity formulas for several cases, in which the point and region are fixed of moving, are derived. Functionals that define displacement at a point on the boundary and mean stress over a small region that intersects the boundary are also considered. These are important cases in shape design problems, since maximum displacement and stress are very likely to occur on the boundary at an optimum design.

DESIGN OPTIMIZATION OF DEEPDRAWING PROCESS

The design optimization of deepdrawing process in manufacturing is proposed to control the final shape of the workpiece after elastic springback. The manufac- turing process design problem is formulated to mini- mize the difference between the shape of the desired workpiece geometry and the final analysis result after elastic springback. Nonlinear structural problem that includes elastoplasticity with frictional contact is solved using a meshfree method where the structural domain is discretized by a set of particle points. Continuum- based design sensitivity analysis (DSA) is carried out to efficiently obtain the gradient information for the opti- mization. The shape of the workpiece and the geometry of the rigid die are treated as design variables. The ac- curacy of the sensitivity result is compared with the fi- nite difference result with excellent agreement. The op- timum stamping process improves the quality of the fi- nal product significantly.