Method Of Feasible Directions Research Articles

Frequency optimization of laminated skew plates

In this study, the frequency optimization of laminated skew plates is investigated. The objective function is to maximize the fundamental frequency of the laminated skew plates and the fiber orientation is considered as design variable. The mathematical formulation is based on the first-order shear deformation theory (FSDT) for the frequency analysis of the laminated skew plates. The modified feasible direction (MFD) method is combined with the finite element solution of laminated skew plates for optimal designs. Therefore, a program based on FORTRAN is used. Finally, the effect of skew angles, boundary conditions, aspect ratios and functionally graded materials on the optimal results is investigated and the results are compared.

Thermal buckling load optimization of laminated skew plates

In this study thermal buckling load optimization of laminated skew plates subjected to uniformly distributed temperature load is investigated. All of the laminates considered are assumed to have fully clamped boundary conditions, though the general boundary conditions may also be studied without any difficulty. The objective function is to maximize the critical temperature capacity of the laminated skew plates and the fibre orientation is considered as design variable. The first-order shear deformation theory is used in the mathematical formulation. Nine-node Lagrangian rectangular plate element is used for the finite element solution of the skew laminates. The temperature field is assumed to be uniform over the plate surface and varies in the thickness direction only. Modified feasible direction method (MFD) is used for the optimization routine. A program based on FORTRAN is used for the optimization problem. Finally, the effect of number of layers, skew angles, aspect ratios, width-to-thickness ratios and antisymmetric lay-up on the optimal designs is investigated and the results are compared.

Multiobjective optimization of laminated composite cylindrical shells for maximum frequency and buckling load

This paper deals with multiobjective optimization of laminated cylindrical shells to maximize a weighted sum of the frequency and buckling load under external load. The layer fibre orientation is used as the design variable and the multiobjective optimization is formulated as the weighted combinations of the frequency and buckling under external load. The first order shear deformation theory is used for the finite element formulation of the laminated shells. Five shell configurations with eight layers are considered as candidate designs. The modified feasible direction method (MFD) is used as optimization routine. For this purpose, a program based on FORTRAN is used for the optimization of the laminated shells. Finally, the effect of different weighting ratios, shell aspect ratio, shell thickness-to-radius ratios and boundary conditions on the optimal designs is investigated and the results are compared.

Generalized Timoshenko modelling of composite beam structures: sensitivity analysis and optimal design

This article describes a new approach to design the cross-section layer orientations of composite laminated beam structures. The beams are modelled with realistic cross-sectional geometry and material properties instead of a simplified model. The VABS (the variational asymptotic beam section analysis) methodology is used to compute the cross-sectional model for a generalized Timoshenko model, which was embedded in the finite element solver FEAP. Optimal design is performed with respect to the layers’ orientation. The design sensitivity analysis is analytically formulated and implemented. The direct differentiation method is used to evaluate the response sensitivities with respect to the design variables. Thus, the design sensitivities of the Timoshenko stiffness computed by VABS methodology are imbedded into the modified VABS program and linked to the beam finite element solver. The modified method of feasible directions and sequential quadratic programming algorithms are used to seek the optimal continuous solution of a set of numerical examples. The buckling load associated with the twist–bend instability of cantilever composite beams, which may have several cross-section geometries, is improved in the optimization procedure.

A comparison of feasible direction methods for the stochastic transportation problem

The feasible direction method of Frank and Wolfe has been claimed to be efficient for solving the stochastic transportation problem. While this is true for very moderate accuracy requirements, substantially more efficient algorithms are otherwise diagonalized Newton and conjugate Frank---Wolfe algorithms, which we describe and evaluate. Like the Frank---Wolfe algorithm, these two algorithms take advantage of the structure of the stochastic transportation problem. We also introduce a Frank---Wolfe type algorithm with multi-dimensional search; this search procedure exploits the Cartesian product structure of the problem. Numerical results for two classic test problem sets are given. The three new methods that are considered are shown to be superior to the Frank---Wolfe method, and also to an earlier suggested heuristic acceleration of the Frank---Wolfe method.

Strength Optimization of Laminated Composite Plates

A procedure to select the optimum fiber orientations and determine the maximum load-bearing capacity of the simply supported symmetrically laminated plates combined the bidirectional tension loads and bending moments is described. The fiber orientation is considered a design variable. The optimization problem consists of two stages. The objective of the first stage is to maximize the strength of the laminated plates by determining the fiber orientations optimally while the objective of the second stage is to maximize each of the bidirectional tension loads and bending moments subject to the Tsai—Wu failure criterion for the optimum fiber orientations obtained from the first stage. The modified feasible direction method (MFD) is used as an optimization procedure. Therefore, a FORTRAN program is used for the finite element analysis and optimization routine. Also, the optimum fiber orientations are obtained using the golden section (GS) method to compare the results. Finally, the effect of the different plate aspect ratio, width-to thickness ratio, the uncertainties in the material properties and the uncertainties in the fiber orientations on the optimum results is investigated and the results are compared.

A superlinearly convergent strongly sub-feasible SSLE-type algorithm with working set for nonlinearly constrained optimization

In this paper, by means of a new efficient identification technique of active constraints and the method of strongly sub-feasible direction, we propose a new sequential system of linear equations (SSLE) algorithm for solving inequality constrained optimization problems, in which the initial point is arbitrary. At each iteration, we first yield the working set by a pivoting operation and a generalized projection; then, three or four reduced linear equations with a same coefficient are solved to obtain the search direction. After a finite number of iterations, the algorithm can produced a feasible iteration point, and it becomes the method of feasible directions. Moreover, after finitely many iterations, the working set becomes independent of the iterates and is essentially the same as the active set of the KKT point. Under some mild conditions, the proposed algorithm is proved to be globally, strongly and superlinearly convergent. Finally, some preliminary numerical experiments are reported to show that the algorithm is practicable and effective.

Material properties of the ovine mitral valve anterior leaflet in vivo from inverse finite element analysis

We measured leaflet displacements and used inverse finite-element analysis to define, for the first time, the material properties of mitral valve (MV) leaflets in vivo. Sixteen miniature radiopaque markers were sewn to the MV annulus, 16 to the anterior MV leaflet, and 1 on each papillary muscle tip in 17 sheep. Four-dimensional coordinates were obtained from biplane videofluoroscopic marker images (60 frames/s) during three complete cardiac cycles. A finite-element model of the anterior MV leaflet was developed using marker coordinates at the end of isovolumic relaxation (IVR; when the pressure difference across the valve is approximately 0), as the minimum stress reference state. Leaflet displacements were simulated during IVR using measured left ventricular and atrial pressures. The leaflet shear modulus (G(circ-rad)) and elastic moduli in both the commisure-commisure (E(circ)) and radial (E(rad)) directions were obtained using the method of feasible directions to minimize the difference between simulated and measured displacements. Group mean (+/-SD) values (17 animals, 3 heartbeats each, i.e., 51 cardiac cycles) were as follows: G(circ-rad) = 121 +/- 22 N/mm2, E(circ) = 43 +/- 18 N/mm2, and E(rad) = 11 +/- 3 N/mm2 (E(circ) > E(rad), P < 0.01). These values, much greater than those previously reported from in vitro studies, may result from activated neurally controlled contractile tissue within the leaflet that is inactive in excised tissues. This could have important implications, not only to our understanding of mitral valve physiology in the beating heart but for providing additional information to aid the development of more durable tissue-engineered bioprosthetic valves.

Covering a compact polygonal set by identical circles

The paper considers the problem of covering compact polygonal set by identical circles of minimal radius. A mathematical model of the problem based on Voronoi polygons is constructed and its characteristics are investigated. On the ground of the characteristics a modification of the Zoutendijk feasible directions method is developed to search local minima. A special approach is suggested to choose starting points. Many computational examples are given.

Frequency optimization of laminated folded composite plates

In this study, frequency optimization of symmetrically laminated folded composite plates is presented. The folded plates are modelled as a composite structure that consists of flat plates. The fundamental frequencies of one- and two-fold folded plates are maximized with respect to fibre orientations. The first-order shear deformation theory (FSDT) is used for vibration analysis of laminated folded plates. The modified feasible direction (MFD) method is used for the optimization routine. Therefore a program based on FORTRAN is used. Finally, the significant effects of length-to-thickness ratios, crank angles, different plate lengths and boundary conditions on the optimum results are demonstrated.

Thermal buckling load optimization of angle-ply laminated cylindrical shells

In this study, thermal buckling load optimization of symmetrically laminated cylindrical shells subjected to uniformly distributed temperature load is investigated. The objective function is to maximize the critical temperature capacity of laminated shells and the fibre orientation is considered as design variable. The first-order shear deformation theory is used to study thermal buckling response of the laminates. The modified feasible direction method is used as optimization routine. For this purpose, a program based on FORTRAN is used for the optimization of shells. Finally, the effects of number of layers, length-to-radius ratio and boundary conditions on the optimum results are investigated.

Effects of nonuniform boundary conditions on the buckling load optimization of laminated composite plates

This paper deals with buckling load optimization of symmetrically laminated composite thin plates under the uniaxial and biaxial compressive loads with discontinuities along the boundaries. The objective of the design problem is to maximize the buckling load of the laminated plates and the fibre orientation is taken as design variable. The classical laminate plate theory is used for the mathematical formulations of the laminates. The modified feasible direction method is used as optimization routine. Therefore a program based on FORTRAN is used. Finally, the effect of nonuniform boundary conditions, aspect ratios and load ratios on the optimum results is illustrated.

A norm-relaxed method of feasible directions for finely discretized problems from semi-infinite programming

In this paper, a class of finely discretized Semi-Infinite Programming (SIP) problems is discussed. Combining the idea of the norm-relaxed Method of Feasible Directions (MFD) and the technique of updating discretization index set, we present a new algorithm for solving the Discretized Semi-Infinite (DSI) problems from SIP. At each iteration, the iteration point is feasible for the discretized problem and an improved search direction is computed by solving only one direction finding subproblem, i.e., a quadratic program, and some appropriate constraints are chosen to reduce the computational cost. A high-order correction direction can be obtained by solving another quadratic programming subproblem with only equality constraints. Under weak conditions such as Mangasarian–Fromovitz Constraint Qualification (MFCQ), the proposed algorithm possesses weak global convergence. Moreover, the superlinear convergence is obtained under Linearly Independent Constraint Qualification (LICQ) and other assumptions. In the end, some elementary numerical experiments are reported.

Integrated feasible direction method and genetic algorithm for optimal planning of harmonic filters with uncertainty conditions

This paper presents an integrated approach of feasible direction method and genetic algorithm (FDM + GA) to optimize the planning of large-scale passive harmonic filters (PHF). The objective is to minimize the total demand distortion of harmonic currents, total harmonic distortion of load bus voltages, filters loss and the cost of total installation of LC tuned filters. Reactive power compensation and constraints of individual harmonics have been considered in the design process, and the constraints of harmonics with orders lower than the filter tuned-points have been set stricter to avoid amplifying non-characteristic harmonics as well. In order to determine a set of weights of objective function representing the relative importance of each term, the simplest and most efficient form of triangular membership functions have been employed. The designed FDM + GA heuristic is applied to the harmonic problems in a chemical plant with three 6-pulse rectifiers. Three design schemes are compared to demonstrate the performance of the FDM + GA heuristic. Finally, expectations and standard deviations of objective values are used to present the effects of filter parameter detuning, loading uncertainty, and changing of system impedance.

On the Convergence of the Pshenichnyi-Pironneau-Polak Minimax Algorithm with an Active Set Strategy

The purpose of this technical note is to present a proof of convergence of the Pshenichnyi-Pironneau-Polak (PPP) minimax algorithm (see Algorithm 2.4.1 in Polak, Optimization: Algorithms and Consistent Approximations, Springer, [1997]), modified to use an active set strategy. This active set strategy was formally derived in Polak (Optimization: Algorithms and Consistent Approximations, Springer, [1997]) from those used in the methods of feasible directions developed by Zoutendijk (Methods of Feasible Directions, Elsevier, [1960]) and Polak (Computational Methods in Optimization: A Unified Approach, Academic, [1971]). The resulting e-Active PPP algorithm was presented as Algorithm 2.4.34, in Polak (Optimization: Algorithms and Consistent Approximations, Springer, [1997]), without any proofs.

The method of feasible directions for mathematical programming problems with preconvex constraints

The convergence of the method of feasible directions is proved for the case of the smooth objective function and a constraint in the form of the difference of convex sets (the so-called preconvex set). It is shown that the method converges to the set of stationary points, which generally is narrower than the corresponding set in the case of a smooth function and smooth constraints. The scheme of the proof is similar to that proposed earlier by Karmanov.

Thermal buckling load optimization of laminated composite plates

In this study, the applicability of the Modified Feasible Direction (MFD) method on the thermal buckling optimization of laminated plates subjected to uniformly distributed temperature load is investigated. The objective function is to maximize the critical temperature capacity of laminated plates and the fiber orientation is considered as design variable. The first-order shear deformation theory is used in the mathematical formulation. For this purpose, a program based on FORTRAN is used for the optimization of laminated plates. Finally, the effect of aspect ratio, antisymmetric lay-up, boundary condition, material anisotropy, ratio of coefficients of thermal expansion, and hybrid laminates on the results is investigated and the results are compared.

Distributed algorithm for optimal sequence and power allocation in uplink code division multiple access systems

An alternative approach is presented for centralised algorithms that design optimal sequences and powers under quality of service constraints in the uplink of a code division multiple access systems. The authors propose a distributed algorithm, where each user designs its optimal codeword in such a way to transmit minimum power, based on certain feedback information sent from the base station. The authors define the user cost function, which is the user power written as a convex function of user codewords for a defined signal-to-interference plus noise ratio target. For the constrained optimisation problem, optimal codewords are designed based on feasible direction method and the optimal user powers are the minimum values of the user cost functions. For the optimal configuration, the matched filter employed at receiver will have the same performance as the minimum mean squared error filter. Even if the optimal user powers are unique, the optimal codewords do not correspond to a unique allocation, but rather to an entire class of codewords that can be related by unitary transformations. The algorithm works properly in the presence of multiple access interference with white or coloured additive noise and requires no ordering. The proposed algorithm is analysed and illustrated with numerical examples obtained from simulations.

Frequency optimization of laminated composite angle-ply plates with circular hole

The fundamental frequencies of simply supported symmetrically laminated composite angle-ply plates with central circular holes with a given material system are maximized with respect to fibre orientations. The first-order shear deformation (FSDT) theory is used for finite element analysis. Modified feasible direction (MFD) method is used for the optimization routine. Also, the optimum fibre orientations are obtained using golden section (GS) method to compare with MFD method. Finally, the significant effects of different number of layers, boundary conditions, plate aspect ratios, hole diameter-to-width ratio, material anisotropy and antisymmetric lay-up on the results are demonstrated.

Determination of Minimum Thrust Requirement for a Passenger Aircraft

This paper describes a methodology aimed at a rapid assessment of the minimum thrust requirement for a given aircraftconfiguration.Thislatterinvolvesatightcouplingbetween flightmissionanalyses,engineperformance,and optimization techniques. The flight performance analysis is used to assess the relatively significant constraints, thus rapidly identifying the feasible design space to meet specific requirements, considering a tradeoff between operation constraints and objectives often of conflicting nature. With conventional procedures, it is rather difficult to find the optimumdesignpoint(matchpoint),becauseofmanyinvolvedobjectivesandconstraints.However,byadoptingthe actual methodology using a subroutine called KSOPT, which solves constraints optimization problem using Kreisselmeier–Steinhauser envelope function formulation, associated with a deterministic optimization algorithm, either sequential quadratic programming or modified method of feasible direction, the overall procedure becomes simpler, because there is only one objective and a constraint, thus avoiding separate optimizations. This present methodologyisthereforemore effectivefrom thepointof viewofrapidityinsearchingforthe optimum matchpoint, and may provide guidance in identifying a potential propulsion system and the necessary data serving to develop a derivative (growth) engine. Nomenclature CD = drag coefficient CL = lift coefficient D = drag force F = vector of objective functions g = constraints of inequality L = lift force M = Mach number S = lifting surface T = total thrust V = velocity W = weight X = vector of variables Subscripts CA = climb in altitude CR = cruise ICS = initial climb segment LA = landing TO = takeoff KS = envelope function