Ashok D. Belegundu

A shape optimization approach based on natural design variables and shape functions

Abstract The general problem of concern is to find the optimum shape of an elastic body, which requires minimizing an objective function subject to stress, displacement, frequency, and manufacturing constraints. The basic approach so far has been to choose a set of geometric design variables that define the shape of the structure. Typically the design variables have been chosen as coefficients of splines and polynomials, coordinates of ‘control’ nodes, and other geometric parameters. An automatic finite element discretization scheme that uses geometric entities such as lines, arcs, splines, and blending functions, is then used to relate changes in position of interior grid points in the finite element mesh to changes in the design variables. In this paper, a set of natural design variables is chosen as the design variables defining the shape. Specifically, the design variables are the magnitudes of a set of fictitious loads applied on the structure. The displacements produced by these fictitious loads, or natural shape functions, are added onto the initial mesh to obtain the new shape. Consequently, a linear relationship is established between changes in grid point locations and design variables through a finite element analysis. Plane elasticity problems are solved using the new approach. The quality of the finite element meshes produced and other salient features of the shape optimal design problem are discussed.

Robustness of Design Through Minimum Sensitivity

The problem of designing mechanical systems or components under uncertainty is considered. The basic idea is to ensure quality control at the design stage by minimizing sensitivity of the response to uncertain variables by proper selection of design variables . The formulation does not involve probability distributions. It is proved, however, that when the response is linear in the uncertain variable, reduction in sensitivity implies lesser probability of failure. The proof is generalized to the non-linear case under certain restrictions. In one example, the design of a three-bar truss is considered. The length of one of the bars is considered to be the uncertain variable while cross-sectional areas are the design variables. The sensitivity of the x-displacement is minimized. The constrained optimization problem is solved using a nonlinear programming code. A criterion which can help identify some of the problems where robustness in design is critical is discussed.

Material tailoring of structures to achieve a minimum radiation condition

A strategy is developed for designing structures that radiate sound inefficiently in light fluids. The problem is broken into two steps. First, given a frequency and overall geometry of the structure, a surface velocity distribution is found that produces a minimum radiation condition. This particular velocity distribution is referred to as the ‘‘weak radiator’’ velocity profile. Second, a distribution of Young’s modulus and density distribution is found for the structure such that it exhibits the weak radiator velocity profile as one of its mode shapes. In the first step, a finite element adaptation of the integral wave equation is combined with the Lagrange multiplier theorem to obtain a surface velocity distribution that minimizes the radiated sound power. In the second step, extensive use of structural finite element modeling as well as linear programming techniques is made. The result is a weak radiator structure. When compared to a structure with uniform material properties, the weak radiator struct...

A general optimization strategy for sound power minimization

A general approach for minimizing radiated acoustic power of a baffled plate excited by broad band harmonic excitation is given. The steps involve a finite element discretization for expressing acoustic power and vibration analysis, analytical design sensitivity analysis, and the use of gradient-based optimization algorithms. Acoustic power expressions are derived from the Rayleigh integral for plates. A general methodology is developed for computing design sensitivities using analytical expressions. Results show that analytical sensitivity analysis is important from both computational time and accuracy considerations. Applications of the optimization strategy to rectangular plates and an engine cover plate are presented. Thicknesses are chosen as design variables.

A Computational Study of Transformation Methods for Optimal Design

In this paper computational aspects of transformatio n methods are studied. Transformation methods, which include sequential unconstrained minimization techniques (SUMTs) and multiplier methods, are based on solving a sequence of unconstrained minimization problems. An efficient technique is given to compute gradients of the transformation function. An operations count is given to demonstrate savings of the suggested technique over two other techniques used in the literature. Computer programs implementing the use of this technique in SUMTs and multiplier algorithms are developed. Applications of these programs on a set of structural design problems are given. Multiplier methods are found to be very stable and reliable, even on some relatively difficult problems, and they perform better than SUMTs. Some ways of improving the efficiency of transformation methods are given, together with possible extensions of using the suggested approach to compute gradients of implicit functions.

Analytical Sensitivity of Acoustic Power Radiated from Plates

A general formulation to obtain the analytical expressions for the sensitivity of the acoustic power radiated by a vibrating structure to one of its design variables is described. The formulation, which is based on finite elements, is applied to both single frequency and broad band harmonic excitation of plates. The sensitivity coefficients indicate the effect of changing various design or modeling parameters on the acoustic power and can be used to optimize the structure for minimum sound radiation. Analytical sensitivity estimates are compared with finite difference values. Results show that analytical sensitivity analysis is important from both computational time and accuracy points of view

Design Approach for Minimizing Sound Power from Vibrating Shell Structures

A design approach for minimizing sound power radiation from vibrating thin shell structures is presented. The method couples finite element analysis for determining structural modes and vibrations with a boundary element/wave superposition code for determining sound power radiation. Noise reduction is accomplished herein by optimal placement and sizing of small point masses on the structure. These masses alter the critical mode shapes so as to reduce sound power. The simulated annealing technique is used to determine the location and/or magnitude of the point masses. A computer program has been developed for design. Design examples are presented with the use of the computer program.

A systematic approach for generating velocity fields in shape optimization

Design velocity fields affect every stage of the shape optimization process. The progress of the optimization process, distortion of the finite element mesh, and final shape are sensitive to the quality of velocity fields. It is important to identify and generate effective velocity fields at the beginning of the process. This paper provides several criteria to determine the effectiveness of velocity fields. A systematic approach for generating these velocity fields using deformation fields is developed. The use of interactive procedures is shown to be indispensable for ensuring the effectiveness and quality of design velocity fields. General strategies and guidelines for generating velocity fields are given. Concepts of weight-reducing, stress-reducing, form-preserving, and smooth basis shapes are presented. Normalization of velocity fields is discussed. A method for controlling mesh distortion during the shape optimization process is given based on an explicit limit on the design change to prevent the Jacobian from vanishing. Two- and three-dimensional design problems are solved.

An optimization algorithm based on the method of feasible directions

The theory and implementation of an optimization algorithm code based on the method of feasible directions are presented. Although the method of feasible directions was developed during the 1960s, the present implementation of the algorithm includes several modifications to improve its robustness. In particular, the search direction is generated by solving a quadratic program which uses an interior method based on a variation of Karmarkars algorithm. The constraint thickness parameter is dynamically adjusted to yield usable-feasible directions. The theory is discussed with emphasis on the important and often overlooked role played by the various parameters guiding the iterations within the program. Also discussed is a robust approach for handling infeasible starting points. The code was validated by solving a variety of structural optimization test problems that have known solutions (obtained by other optimization codes). A variety of problems from different infeasible starting points has been solved successfully. It is observed that this code is robust and accurate. Further research is required to improve its numerical efficiency while retaining its robustness.

A sensitivity interpretation of adjoint variables in optimal design

Abstract The adjoint method of computing derivatives of cost and constraint functions with respect to design variables requires the calculation of certain adjoint variables. Until now, the adjoint variables have been looked upon only as some intermediate vectors needed to calculate design derivatives. In this paper, they are shown to have an important significance. They represent the sensitivity of the cost and constraint functions with respect to the loading or forcing function in the design problem. A sensitivity theorem for the adjoint variables is presented for structural, mechanical dynamic, and distributed parameter systems. These results offer some immediate practical advantages, such as a method for computing influence coefficients for structural systems, and a method for verifying (debugging) the analytical calculation of adjoint variables in development of a computer code.