Ramana V. Grandhi

Structural shape optimization — a survey

Abstract This paper is a survey of structural shape optimization with an emphasis on techniques dealing with shape optimization of the boundaries of two- and three-dimensional bodies. Attention is focused on the special problems of structural shape optimization which are due to a finite element model which must change during the optimization process. These problems include the requirement for sophisticated automated mesh generation techniques and careful choice of design variables. They also include special problems in obtaining sufficiently accurate sensitivity derivatives.

An approximation approach for uncertainty quantification using evidence theory

Abstract Over the last two decades, uncertainty quantification (UQ) in engineering systems has been performed by the popular framework of probability theory. However, many scientific and engineering communities realize that there are limitations in using only one framework for quantifying the uncertainty experienced in engineering applications. Recently evidence theory, also called Dempster–Shafer theory, was proposed to handle limited and imprecise data situations as an alternative to the classical probability theory. Adaptation of this theory for large-scale engineering structures is a challenge due to implicit nature of simulations and excessive computational costs. In this work, an approximation approach is developed to improve the practical utility of evidence theory in UQ analysis. The techniques are demonstrated on composite material structures and airframe wing aeroelastic design problem.

Polynomial Chaos Expansion with Latin Hypercube Sampling for Estimating Response Variability

A computationally efficient procedure for quantifying uncertainty and finding significant parameters of uncertainty models is presented. To deal with the random nature of input parameters of structural models, several efficient probabilistic methods are investigated. Specifically, the polynomial chaos expansion with Latin hypercube sampling is used to represent the response of an uncertain system. Latin hypercube sampling is employed for evaluating the generalized Fourier coefficients of the polynomial chaos expansion. Because the key challenge in uncertainty analysis is to find the most significant components that drive response variability, analysis of variance is employed to find the significant parameters of the approximation model. Several analytical examples and a large finite element model of a joined-wing are used to verify the effectiveness of this procedure.

Structural optimization with frequency constraints

This paper presents a design optimization algorithm for structural weight minimization with multiple frequency constraints. An optimality criterion method based on uniform Lagrangian density for resizing and a scaling procedure to locate the constraint boundary were used in optimization. Multiple frequency constraints of equality and inequality types were addressed. The effectiveness of the algorithm was demonstrated by designing a number of truss structures with as many as 489 design variables. No attempt was made to reduce the number of design variables by such procedures as linking and/or invoking symmetry conditions. The design examples include a 10-bar truss, 200-bar truss, a modified ACOSS-II, and COFS (Control of Flexible Structures) mast truss. All the structures contain nonstructural mass besides their own mass. The algorithm is extremely stable and, in all cases, the optimum designs were obtained in less than 20 iterations regardless of the size of the structure and the number of design variables.

Reliability-based structural optimization using improved two-point adaptive nonlinear approximations

The objective of this paper is to conduct reliability-based structural optimization in a multidisciplinary environment. An efficient reliability analysis is developed by approximating the limit-state functions using two-point adaptive nonlinear approximation. The nonlinear approximation is constructed by using the function values and the first-order gradients at two points of the limit-state function, and its nonlinearity index is automatically changed for different problems. The reduction in computational cost realized in safety index calculation and optimization are demonstrated through two structural problems. This paper presents the safety index computation, analytical sensitivity analysis of the reliability constraints and optimization using frame and turbine blade examples.

Efficient safety index calculation for structural reliability analysis

Abstract This paper proposes an efficient structural reliability analysis algorithm with an adaptive nonlinear approximation of performance function, in which the nonlinearity of the function is controlled by the feedback information from the previous iteration. The proposed analysis procedure includes (1) identifying the most probable point of a limit state, (2) establishing an approximate nonlinear performance function by using intervening variables and (3) applying the HL-RF or the modified HL-RF method. The method is suitable not only for problems that involve explicit performance functions, but also for problems that need computer-intensive structural analysis. Several examples are presented in this paper to demonstrate the performance of the analysis method using large coefficients of variation and nonnormal distribution of random variables.

Safety index calculation using intervening variables for structural reliability analysis

Abstract This paper utilizes the intervening design variables concept for structural reliability analysis. This procedure was traditionally used in structural optimization, whereas this work applies to problems having stochastic information. The present algorithm is on the basis of a safety index algorithm present in Ref. [1] [L. P. Wang and R. V. Grandhi, Efficient safety index calculation for structural reliability analysis. Comput. Struct. 52, 103–111 (1994)], which utilized the nonlinear approximation of performance function in the original space of random variables. The proposed algorithm further develops this procedure by (i) implementing the adaptive nonlinear approximation of performance function in the standard normal space of random variables, (ii) using an improved intervening variable procedure for more accurate approximation, and (iii) adding an additional convergency check on safety index calculation using approximate gradients. The efficiency and robustness of the proposed algorithm are demonstrated by several examples with highly nonlinear, complex, and explicit/implicit performance functions.

Uncertainty Quantification of Structural Response Using Evidence Theory

Over the past decade, classical probabilistic analysis has been a popular approach among the uncertainty quantification methods. As the complexity and performance requirements of a structural system are increased, the quantification of uncertainty becomes more complicated, and various forms of uncertainties should be taken into consideration. Because of the need to characterize the distribution of probability, classical probability theory may not be suitable for a large complex system such as an aircraft, in that our information is never complete because of lack of knowledge and statistical data. Evidence theory, also known as Dempster-Shafer theory, is proposed to handle the epistemic uncertainty that stems from lack of knowledge about a structural system. Evidence theory provides us with a useful tool for aleatory (random) and epistemic (subjective) uncertainties. An intermediate complexity wing example is used to evaluate the relevance of evidence theory to an uncertainty quantification problem for the preliminary design of airframe structures. Also, methods for efficient calculations in large-scale problems are discussed.

Comparison of evidence theory and Bayesian theory for uncertainty modeling

Abstract This paper compares Evidence Theory (ET) and Bayesian Theory (BT) for uncertainty modeling and decision under uncertainty, when the evidence about uncertainty is imprecise. The basic concepts of ET and BT are introduced and the ways these theories model uncertainties, propagate them through systems and assess the safety of these systems are presented. ET and BT approaches are demonstrated and compared on challenge problems involving an algebraic function whose input variables are uncertain. The evidence about the input variables consists of intervals provided by experts. It is recommended that a decision-maker compute both the Bayesian probabilities of the outcomes of alternative actions and their plausibility and belief measures when evidence about uncertainty is imprecise, because this helps assess the importance of imprecision and the value of additional information. Finally, the paper presents and demonstrates a method for testing approaches for decision under uncertainty in terms of their effectiveness in making decisions.

General purpose procedure for reliability based structural optimization under parametric uncertainties

The objective of this paper is to integrate the general purpose structural reliability analysis program, NESSUS (Numerical Evaluation of Stochastic Structures Under Stress), with mathematical optimization capabilities for achieving optimal designs. The field of reliability analysis has matured in the last few years to the level where probabilistic analysis is conducted for a multitude of finite elements by considering the uncertain information on geometry, material properties, loads and boundary conditions. For conducting structural reliability analysis, several disciplines are incorporated in NESSUS to consider different failure modes. This research developed RELOPT (RELiability based structural OPTimization), an automated procedure for design optimization by integrating reliability analysis, sensitivity analysis, function approximations and data base management. Example problems are presented on beam and plate structures with strength and eigenvalue requirements and structural mass minimization.