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In practical engineering applications, there exist two different types of uncertainties: aleatory and epistemic uncertainties. This study attempts to develop a robust design optimization with epistemic uncertainty. For epistemic uncertainties, a possibility- based design optimization improves the failure rate, while a ro- bust design optimization minimizes the product quality loss. In general, product quality loss is described using the first two statistical moments for aleatory uncertainty: mean and standard de- viation. However, there is no metric for product quality loss de- fined when having epistemic uncertainty. This paper first proposes a new metric for product quality loss with epistemic uncertainty, and then a possibility-based robust design optimization. For numerical efficiency and stability, an enriched performance measure approach is employed for possibility-based robust design optimization, and the maximal possibility search is used for a possibility analysis. Three different type of robust objectives are considered for possibility-based robust design optimization: smaller-the- better type (S-Type), larger-the-better type (L-Type), and nominal- the-better type (N-Type). Examples are used to demonstrate the effectiveness of possibility-based robust design optimization using the proposed metric for product quality loss with epistemic uncertainty. [DOI: 10.1115/1.2717232] Keywords: robust design, possibility, epistemic uncertainty, fuzzy, membership function

This paper presents a continuum-based design sensitivity analysis and optimization of high-frequency radiation problems using the energy finite element method and energy boundary element method. The noise radiated from the vibrating structure at a high-frequency range is obtained through a sequential procedure. The structural energy finite element method calculates structural energy distribution, which is then used as the boundary condition for the energy boundary element method to calculate the energy density at a far-field observation point. For design sensitivity analysis, the direct differentiation method calculates the sensitivity of the exterior noise through the sensitivity of the structural energy density obtained from the energy finite element method. The adjoint variable method calculates the adjoint load from an acoustic energy boundary element method reanalysis, and the adjoint response is obtained from a structural energy finite element method reanalysis. The sensitivity information is obtained by carrying out numerical integration only on the structural finite element part. The proposed design sensitivity analysis approach has been applied in the design of automotive and naval structures to search for the best material layout to achieve the lowest noise level at high frequency.

Structural analysis and design optimization have recently been extended to the stochastic approach to consider various uncertainties. However, in areas where it is not possible to produce accurate statistical information for input data, the probabilistic method is not appropriate for stochastic structural analysis and design optimization, because improper modeling of uncertainty could cause a greater degree of statistical uncertainty than those of physical uncertainty. For systems with insufficient information for input data, possibility-based (or fuzzy set) methods have recently been introduced in structural analysis and design optimization. Using possibility methods, the extended fuzzy operations are much simpler than of random variables. Possibility-based design optimization will provide more conservative designs than those from probability methods, and will provide a system level of failure possibility automatically. This paper proposes a new formulation of possibility-based design optimization using the performance measure approach. For the inverse possibility analysis, the maximal possibility search method is proposed to improve numerical efficiency and accuracy comparing with the vertex method and the multilevel-cut method. Two mathematical examples, including a nonmonotonic response and a physical example of vehicle side impact, are used to demonstrate the proposed maximal possibility search method and possibility-based design optimization. DOI: 10.2514/1.16546

The response surface method (RSM) has currently become one of the better-known meta-modeling techniques. However, its approximation errors have placed several restrictions on designers as classical RSM uses the least-squares method (LSM) to find the best-fitting approximation models from the given function data. We discuss how to construct RS models efficiently and accurately using the moving least-squares method (MLSM) combined with sensitivity information. The formulations for incorporating the sensitivity using the MLSM are derived. With this method, several parameters should be determined during the construction of response surfaces. The parametric study and optimization for these parameters are performed. However, because of the discontinuity problem of the optimization, a genetic algorithm is adopted. The correlation coefficient is used for the normalized comparison between the function and gradient errors. Also, the reciprocal condition number is applied to avoid ill-conditioned approximations. Several difficulties and their respective solutions during the approximation processes are described, and the numerical examples are then demonstrated to verify the accuracy and the efficiency of this method. If the sensitivity of each sampling point can be calculated efficiently by utilizing a cheap computation, the proposed method is recognized as very efficient and accurate.

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 [epsilon]-onmonotonic 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.

A noise-vibration-harshness (NVH) design optimization of a complex vehicle structure is presented using finite element and boundary element analyses. The steady-state dynamic behavior of the vehicle is calculated from the frequency response finite element analysis, while the sound pressure level within the acoustic cavity is calculated from the boundary element analysis. A reverse solution process is employed for the design sensitivity calculation using the adjoint variable method. The adjoint load is obtained from the acoustic boundary element re-analysis, while the adjoint solution is calculated from the structural dynamic re-analysis. The evaluation of pressure sensitivity only involves a numerical integration process over the structural part where the design variable is defined. A design optimization problem is formulated and solved, where the structural weight is reduced while the noise level in the passenger compartment is lowered.

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.

During the past decade, numerous endeavors have been made to develop effective reliability-based design optimization (RBDO) methods. Because the evaluation of probabilistic constraints defined in the RBDO formulation is the most difficult part to deal with, a number of different probabilistic design approaches have been proposed to evaluate probabilistic constraints in RBDO. In the first approach, statistical moments are approximated to evaluate the probabilistic constraint. Thus, this is referred to as the approximate moment approach (AMA). The second approach, called the reliability index approach (RIA), describes the probabilistic constraint as a reliability index. Last, the performance measure approach (PMA) was proposed by converting the probability measure to a performance measure. A guide for selecting an appropriate method in RBDO is provided by comparing probabilistic design approaches for RBDO from the perspective of various numerical considerations. It has been found in the literature that PMA is more efficient and stable than RIA in the RBDO process. It is found that PMA is accurate enough and stable at an allowable efficiency, whereas AMA has some difficulties in RBDO process such as a second-order design sensitivity required for design optimization, an inaccuracy to measure a probability of failure, and numerical instability due to its inaccuracy. Consequently, PMA has several major advantages over AMA, in terms of numerical accuracy, simplicity, and stability. Some numerical examples are shown to demonstrate several numerical observations on the three different RBDO approaches.

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 optimization 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.

The design optimization of springback in a deepdrawing process is proposed to control the final shape of the workpiece. The manufacturing process design problem is formulated to minimize the difference between the shape of the desired workpiece geometry and the final analysis result after elastic springback. The rigid die shape and the workpiece thickness are treated as design variables. A nonlinear structural problem that includes finite deformation elastoplasticity with frictional contact is solved using a mesh-free method in which the structural domain is discretized by a set of particles. Continuum-based design sensitivity analysis is carried out to obtain gradient information for the optimization efficiently. The accuracy of the sensitivity result is compared with the finite difference result with excellent agreement. The optimum deepdrawing process significantly improves the quality of the final product.

The article discusses a continuum-based shape design sensitivity formulation for the elasto-plasticity material with a frictional contact condition. Results of analysis of numerical examples indicate that no iteration is required for solving the sensitivity equation which is solved using the mesh-free method.

A continuum-based design sensitivity analysis (DSA) and optimization procedure for hyperelastic structures using reproducing kernel particle method has been developed. Shape DSA and optimization using the meshless methods, which have been tested in an engine mount example, have the potential to eliminate mesh distortion problems that are associated with conventional finite element methodologies. A CAD-based optimization and three-dimensional extension applications are under development.

A continuum-based design sensitivity analysis (DSA) was performed for hyperelastic structures. The technique was performed via a meshless method, known as reproducing the kernel particle method, which eliminates mesh distortion or entanglement that is often encountered when using finite element analysis for large deformation structural analysis and structural shape design optimization. The DSA method confirmed its viability through one- and two-dimensional examples.

The durability of structural components subjected to external and inertial loads with time-dependent variable amplitudes can be determined through an efficient reliability analysis method. Such a method, employed to support a mixed design approach for probabilistic durability, is highly useful for industrial-type applications such as the tracked-vehicle roadarm. It may also be valid for use in low-frequency flexible structures, thermal-induced fatigue and system-level reliability.

A continuum method with material derivative concept enables the second-order shape design sensitivity analysis of three-dimensional linear elastic solids. A shape design velocity field similar to the field for first-order shape design sensitivity helps evaluate the same for second-order. A numerical method that illustrates the validity of second-order shape design sensitivity analysis scheme finds application in several practical problems in structural design. A domain method that allows the expression of design sensitivity data as domain integrals eliminates errors in the finite element analysis data of boundary for structural design.

A method for computing the sensitivity of the critical load of nonlinear structural systems taking into account the size and the shape is given. The effects of strain, displacement and material nonlinearities are analyzed in the kinematic equations derived. The use of the method produces precise sensitivity of the critical load and the numerical computation process is faster than the estimated critical load procedure.

Structural dynamic and design sensitivity analyses are undergoing substantial research. This is necessary to find functional methods for transient dynamic analysis and optimization and an efficient unified continuum-based sizing design sensitivity analysis method is proposed. This approach is appropriate for built-up structures with nonproportional damping because it integrates mode superposition and direct differentiation methods. The exclusion of derivatives of stiffness and mass matrices makes the method reliable.

A continuum design sensitivity analysis theory for dynamic frequency response of structural systems is developed. This must allow implementation using existing finite element codes. The Cosmic/Nastran finite element codes were chosen for the study. Results show that when perturbation ratio is large, design sensitivity does not agree with finite difference results. Design sensitivity results are more accurate when evaluated with better finite element analysis results.

To represent input variability accurately, an input distribution model for random variables should be constructed using many data. However, for certain input variables, engineers may have only their intervals, which represent input uncertainty. In practical engineering applications, both random and interval variables could exist at the same time. To consider both input variability and uncertainty, inverse reliability analysis should be carried out considering both random and interval variables--mixed variables--and their mathematical correlation in a performance measure. In this paper, an iterative most probable point (MPP) search method has been developed for the mixed-variable problem. The update procedures for MPP search are developed considering the features of mixed variables in the inverse reliability analysis. MPP search for random and interval variables proceed simultaneously to consider the mathematical correlation. An interpolation method is introduced to find a better candidate MPP without additional function evaluations. Mixed-variable design optimization (MVDO) has been formulated to obtain cost-effective and reliable design in the presence of mixed variables. In addition, the design sensitivity of a probabilistic constraint has been developed for an effective and efficient MVDO procedure. Using numerical examples, it is found that the developed MPP search method finds an accurate MPP more efficiently than the generic optimization method does. In addition, it is verified that the developed method enables the MVDO process with a small number of function evaluations. [ABSTRACT FROM AUTHOR]

Stiffness and mass matrices need not be distinguished to accomplish a unifield continuum-based sizing design sensitivity analysis of eigenvectors. Sensitivity results are possible with the addition of a pair of load-dependent Ritz vectors (LDRVs) to the eigenvector basis.

Byline: Byeng D. Youn, Kyung K. Choi, Jun Tang Mechanical fatigue subject to external and inertia transient loads in the service life of mechanical systems often leads to structural failure due to accumulated damage. A structural durability analysis that predicts the fatigue life of mechanical components subject to dynamic stresses and strains is a computer-intensive, multidisciplinary simulation process, since it requires the integration of several computer-aided engineering tools and large amounts of data communication and computation. Uncertainties in geometric dimensions result in the indeterministic fatigue life of a mechanical component. Uncertainty propagation to structural fatigue under transient dynamic loading is not only numerically complicated but also extremely expensive. It is, therefore, a challenge to develop a durability-based design-optimisation process and reliability analysis to ascertain whether the optimal design is reliable. The objective of this paper is to develop an integrated CAD-based computer-aided engineering process to effectively carry out design optimisation for structural durability, yielding a manufacturable, durable, and cost-effective product. In addition, a reliability analysis is executed to assess the reliability of the deterministic optimal design.

A novel design potential method that integrates the probabilistic constraint evaluation closely into the design optimization process is presented for the robust system parameter design. From a broader perspective, it is shown that the probalistic constraints can be evaluated using either the conventional reliability index approach or the proposed performance measure approach. The performance measure approach is inherently robust and is more effective when the probabilistic constraint is inactive. The reliability index approach is more effective for for the violated probabilistic constraint, but it could yield singularity when the probabilistic constraint is inactive. Moreover, the close coupling of performance probability analysis and design optimization illustrated in a proposed unified system space. The design potential method, which is developed to take full advantage of the important design information obtained from the previous probabilistic constraint evaluation, can significantly accelerate the convergence of the reliability-based design optimization process.