Method Of Feasible Directions Research Articles

Optimal design of rib structures using the topology optimization technique

The objective of this work is to search the optimal shapes and locations of ribs in order to increase the stiffness of structures using the topology optimization technique. In this approach, an initial rib structure consisting of many discrete shell elements is added to the existing structure and the optimization procedure is applied to determine the density of each element for the minimal mean compliance of the whole system. The remaining shell elements determined to have higher densities by an optimization procedure will remain attached to the structure and become the rib structure. This is different from the existing approach which tries to derive the rib design from the optimized thickness distribution of the structure. A penalty term is also introduced in the objective function in addition to the mean compliance to prohibit the intermediate densities. The mean compliance is computed by the finite element method in which each element has a variable density. Young's modulus of each element is derived by assuming a quadratic relation between the density and Young's modulus. The optimization is performed by the feasible direction method using the densities of elements as the design variables. The total material usage is used as a constraint in the optimization to provide lightweight structures.

Self-Tuning Norm-Relaxed Method of Feasible Directions

This paper presents three updating techniques for the scaling matrix or the scalar weight used in the norm-relaxed method of feasible directions, a generalization of the popular Pironneau–Polak algorithm. These techniques include variable metric updates and tuning of a scalar weight in a way characteristic of trust-region methods, and also techniques based on the idea of multiple directions, where the update decision is made by comparing results of searching along several directions determined by distinct values of weights. Numerical results obtained on a standard test set are provided. These results indicate that the updating techniques allow considerable computational savings when compared with the original Pironneau-Polak method.

Multi-objective Optimization of a Flexible Rotor in Magnetic Bearings With Critical Speeds and Control Current Constraints

This paper presents the single objective optimization and the multi-objective optimization for a flexible rotor system with magnetic bearings. The weight of rotor shaft and the transmitted forces at the magnetic bearings are minimized either individually or simultaneously under the constraints on the critical speeds and the control currents of magnetic bearings. The design variables are the cross-sectional area of the shaft, the bias currents of magnetic bearings, and the positions of the disk and the magnetic bearings. The dynamic characteristics are analyzed using the generalized polynomial expansion method and the sensitivity analysis is also studied. For single objective optimization, the method of feasible directions (MFD) is applied. For multi-objective optimization, the weighting method (WM), the goal programming method (GPM), and the fuzzy method (FM) are employed. It is found that the system design can be significantly affected by the choices of the bias currents of magnetic bearings, the position of the disk with unbalance, and the magnetic bearings. The results also show that a better compromised design can always be obtained for multi-objective optimization.

A combined method for smooth equilibrium problems with nonlinear constraints

An iterative method for finding equilibrium points in the case of a smooth convex-concave function and a convex feasible set is proposed. This set is defined by smooth concave functions. The method is based on combining and extending ideas contained in feasible direction methods and relaxation methods for optimization problems and convex inequalities

Convergence analysis of norm-relaxed method of feasible directions

This paper gives a complete treatment of the asymptotic rate of convergence for a class of feasible directions methods, including those studied by Pironneau and Polak and by Cawood and Kostreva. Rate estimates of Pironneau and Polak are sharpened in an analysis which shows the dependence on certain parameters of the direction-finding subproblem and the problem functions. Special cases of interior optimal solution, linear constraints, and fixed matrix norm are analyzed in detail. Numerical verification is provided.

Minimum weight design of structures under frequency and frequency response constraints

A technique for the determination of the least weight design which satisfies the desired frequency and frequency response constraints, plus upper and lower bounds on the design variables, is introduced. The design algorithm is based on Zoutendijk's method of feasible directions [1]. This method requires the analytical gradients of the objective function and the constraint functions which are active at a given stage in the design process. A considerable improvement in convergence has been achieved by considering each pushoff factor as a linear function of the corresponding active constraint. An initial step length based on a pre-selected decrement of the objective function is used. The results, obtained using an IBM-compatible microcomputer, assert that the method leads to good results even when a very limited number of modes is used in calculating the sensitivities of the frequency response function. This makes the method very efficient with regard to both accuracy, computation time and suitability for use on microcomputers.

Optimum design of bending-torsion coupled beams with frequency or aeroelastic constraints

The optimum (minimum mass) design of vibrating beams with noncolinear mass axes and elastic axes is investigated. These beams, which exhibit coupling between bending and torsional displacements, are analysed using a dynamic stiffness matrix method which ensures that all natural frequencies can be found with certainty and to any desired accuracy. Optimization, which is carried out using the method of feasible directions, allows cross-sectional thicknesses to vary independently for a number of sections (elements) along the beam length. Results are presented to illustrate the optimum design of such non-uniform beams, constrained by either a minimum value of fundamental frequency or a minimum separation between the first bending and first torsional frequencies. The latter designs are compared with the optimum design of similar beams, constrained instead by a minimum value of flutter speed. Comparison shows that, as expected, increasing separation between first bending and first torsional frequencies increases flutter speed. However, the designs produced by each approach have a significantly different thickness distribution.

Norm-relaxed method of feasible directions: application in structural optimization

This paper examines the norm-relaxed method of feasible directions applied to solving structural optimization problems. The novelty of the method lies in elliptical bounding of the length of the direction computed by a subproblem characteristic of a popular algorithm of Pironneau and Polak. It is shown that updating the scaling matrix generating the elliptical norm results in computational savings over the original method of Pironneau and Polak. The new method is evaluated on several classical structural optimization problems and benchmarked against other feasible directions codes, i.e. CONMIN and DOT.

Prediction of optimal preform thickness distribution in blow molding

AbstractA finite element optimization method is presented that determines the optimal thickness profile of a preform for a blow‐molded part having the required wall thickness distribution. The optimization method is based on a method of feasible directions and the design variables are thicknesses of finite elements. The step size is determined by using a Brent 1‐dimensional minimization method. Two methods for determining feasible directions are compared. A finite element model is formulated based on the thin membrane approximation, which relates the preform wall thickness distribution to the wall thickness distribution in the blow‐molded part. Triangular or conical frustra elements are used to describe deformed shapes of the preform for 2‐dimensional axisymmetric deformations. An incompressible nonlinear elastic constitutive model describes the material response. The accuracy of the finite element model used is evaluated by comparing the model predictions to published experimental data and computer simulations.

Thickness optimization for maximum buckling loads in composite laminated plates

Isotropic plates and composite laminated plates are modeled using variable thickness, shear deformable, finite plate elements. The method of feasible directions is used to determine optimal thickness distributions over the plate that yield maximum uniaxial and biaxial buckling loads. Plates that have thicknesses tailored uniaxially and biaxially are considered. Optimum isotropic plate designs indicate a complex interaction occurs between inplane load redistribution and bending/twisting stiffness redistribution during the optimization process. Optimization produces increases in uniaxial buckling loads of over 200% for isotropic plates compared to uniform plates. Much smaller improvements are possible when biaxial compressive loads are present. Composite materials also allow for increased efficiency in the redistribution of loads and stiffnesses. Improvements in both uniaxial and biaxial buckling loads of over 160% are shown possible in composite plates compared to uniform quasiisotropic plates. The effects of transverse shear stiffness on the optimum design and corresponding buckling load for both isotropic and composite laminated plates are studied by analyzing plates with different width/thickness ratios. The improvement in the buckling loads through thickness optimization is shown to decrease as the effects of transverse shear increase.

Structural optimization with approximate sensitivities

Computational efficiency in structural optimization can be enhanced if the intensive computations associated with the calculation of the sensitivities, that is, gradients of the behavior constraints, are reduced. Approximation to gradients of the behavior constraints that can be generated with a small amount of numerical calculations is proposed. Structural optimization with these approximate sensitivities produced correct optimum solution. Approximate gradients performed well for different nonlinear programming methods, such as the sequence of unconstrained minimization technique, method of feasible directions, sequence of quadratic programming and sequence of linear programming. Structural optimization with approximate gradients can reduce by one third the CPU time that would otherwise be required to solve the problem with explicit closed-form gradients. The proposed gradient approximation shows potential to reduce intensive computation that has been associated with traditional structural optimization.

Vibration and optimum design of rotating laminated blades

The vibration and optimum design of a rotating laminated blade subject to constraints on the dynamic behavior are investigated. Restrictions on multiple natural frequencies as well as the maximum dynamic deflections of rotating laminated blades are considered as constraints on the dynamic behavior of the system. Aerodynamic forces acting on the blade are simulated as harmonic excitations. The optimality criterion method and the modified method of feasible directions have been successfully developed for optimizing the weight of the rotating laminated blade. Effects of radius of the disk, aspect ratio, and rotating speed on the system dynamic behaviors and/or the optimum design are also studied. The vibration analysis shows that most of the bending modes can be significantly affected by the rotating speed and the radius of the disk. Results also show that the optimum weight with constraints on the dynamic response is higher than that with frequency constraints. Moreover, results show that the weight of the rotating laminated blade can be greatly reduced at the optimum design stage.

Algorithmic modeling of TES processes

TES (transform-expand-sample) is a versatile class of stationary stochastic processes which can model arbitrary marginals, a wide variety of autocorrelation functions, and a broad range of sample path behaviors. TES parameters are of two kinds: the first kind is used for the exact fitting of the empirical distribution (histogram), while the second kind is used for approximating the empirical autocorrelation function. Parameters of the first kind are easy to determine algorithmically, but those of the second kind require a hard heuristic search on a large parametric function space. This paper describes an algorithmic procedure which can replace the heuristic search, thereby largely automating TES modeling. The algorithm is cast in nonlinear programming setting with the objective of minimizing a weighted sum of squared differences between the empirical autocorrelations and their candidate TES model counterparts. It combines a brute-forte search with steepest-descent nonlinear programming using Zoutendijk's feasible direction method. Finally, we illustrate the efficacy of our approach via three examples: two from the domain of VBR (variable bit rate) compressed video and one representing results from a laser intensity experiment. >

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 1960's, 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 Karmarkar's 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.

Design of L-Q regulators for state constrained continuous-time systems

A new methodology to the design of LQ regulators for continuous-time systems subject to linear state constraints is proposed, consisting of two parts. In the first one, the positive invariance of the polyhedron defined by the constraints in the state space is imposed, guaranteeing thereby that the constraints will not be violated. Furthermore, the admissible constrained controllers are parameterized via the determination of the elements which are fixed in the state feedback matrix. This parameterization enables in a second part the L-Q regulator to be obtained from the solution of a parameter optimization problem subject to linear constraints, for which it is proposed a specialized feasible directions method.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

On the robustness and efficiency of the SAM algorithm for structural optimization

AbstractA parameter study involving a feasibility factor, adaptive tolerance refinement and active set and move limits is conducted on the interior feasible direction method with successive approximations (SAM). The study involves the sizing design of five structural problems of which one is a large power transmission tower to be designed in adherence with a standard design code. It is concluded that a move limit is dispensable for this application while adaptive tolerance refinement makes the algorithm more efficient. Optimal ranges for the feasibility factor and active set limits are suggested. This study reaffirms and quantifies the exceptional stability and robustness of the SAM algorithm.

A DESIGN OPTIMIZATION PROCEDURE FOR MINIMIZING DRIVE SYSTEM WEIGHT OF HIGH SPEED PROP-ROTORS

Abstract An optimization procedure is developed to address the problem of minimizing the drive system weight of high speed prop-rotor aircraft which are required to demonstrate fixed-wing-like efficiencies in high speed forward flight and maintain acceptable hover figure of merit similar to helicopters. The optimization is performed using the method of feasible directions. A hybrid approximate analysis procedure is also used to reduce the computational effort of using exact analysis for every function evaluation necessary within the optimizer. The results compared to a reference rotor show significant weight reductions. The aerodynamic performance of the optimized rotor, analyzed at “off-design” points to judge the strength of the optimization problem formulation and the validity of the resulting design, shows considerable improvements. The results are compared to the reference values and significant reduction in the weight is achieved.

Approximation Techniques for Broad-Band Acoustic Radiated Noise Design Optimization Problems

This study investigates the applicability of various approximation methods to broadband radiated noise design optimization problems. Low-order series approximations of dynamic response may be used to replace full numerical system solutions to effect significant computer cost savings during design iterations. Also, the ease of evaluating the approximate functions may be further exploited by using global optimization search methods, such as simulated annealing, at individual design iterations. The combination of approximating radiated noise spectra and evaluating the approximate spectra for all possible design alternatives greatly increases the possibility of finding a truly optimal design. The effectiveness of the approximations is measured by considering optimization accuracy, evaluated by the algorithm’s ability to find a global or near-global minimum independent of the initial design; computational efficiency, based on the number of numerical design analyses required for convergence; and generality, where the method should be relatively independent of the problem type. Finite element models of three test cases with varying performance goals and design parameters were used to evaluate the optimization methods. Shell thicknesses, shell loss factors, and rib stiffener locations were varied to minimize structural weight and manufacturing costs while lowering broad-band radiated noise levels below a specified goal. First-order Taylor and half-quadratic series approximation optimization approaches were compared to traditional local minimization methods (Modified Method of Feasible Directions and Broydon-Fletcher-Goldfarb-Shanno). For all test cases, the approximation approaches found the global optimum design more frequently than the local minimization methods. Also, the half-quadratic method converged using fewer design evaluations than the first-order Taylor method for most test cases.

Functionally graded metal matrix composite tubes

Current advanced manufacturing techniques allow the continuous variation of fiber or inclusion volume fraction in metal matrix composites. With this technology, it is now possible to tailor a composite to the expected loads by using the constituent materials to redistribute the stress and strain states through the material. Lighter and more structurally efficient components will be obtained through this grading process. The focus of this paper is the evaluation of the effects of material property and fiber volume grading on the overall mechanical response of metal matrix composite tubes subjected to mechanical loadings. This is accomplished through the development of a fully elastic-plastic axisymmetric generalized plane strain tube model. This analytical model incorporates a micromechanics algorithm in order to determine the elastic-plastic response of a heterogeneous fiber-reinforced composite cylinder. An arbitrary number of heterogeneous concentric cylinders can be included in the model, each with independent material properties. The inelastic analysis is performed through the method of successive elastic solutions. The optimization algorithm used in conjunction with this solution procedure utilizes the method of feasible directions and accepts any combination of design variables, constraints, and objective functions. As an example of the effectiveness of this grading, it is possible to vary the fiber volume fraction in an SiC/Ti-24Al-11Nb tube in such a way that the effective stress at the critical inner surface of an internally pressurized tube is reduced. For a 50.8 mm (2in) thick tube with an internal radius of 25.4 mm (1 in) and an internal pressure of 206.8 MPa (30 ksi), a uniform 40% fiber volume fraction distribution results in a tube that begins to plastically yield at the inner radius. By grading the fiber volume fraction, the tube now behaves elastically under the same pressure loading, allowing the tube to have a wall thickness of 25.4 mm (1 in) before plastic yielding begins. This grading results in a 60% weight saving.

Norm-relaxed method of feasible directions for solving nonlinear programming problems

A variation of the Polak method of feasible directions for solving nonlinear programming problems is shown to be related to the Topkis and Veinott method of feasible directions. This new method is proven to converge to a Fritz John point under rather weak assumptions. Finally, numerical results show that the method converges with fewer iterations than that of Polak with a proper choice of parameters.