Showing papers on "Minimum weight published in 1994"

The Complexity of Multiterminal Cuts

Abstract: In the multiterminal cut problem one is given an edge-weighted graph and a subset of the vertices called terminals, and is asked for a minimum weight set of edges that separates each terminal from all the others. When the number $k$ of terminals is two, this is simply the mincut, max-flow problem, and can be solved in polynomial time. It is shown that the problem becomes NP-hard as soon as $k=3$, but can be solved in polynomial time for planar graphs for any fixed $k$. The planar problem is NP-hard, however, if $k$ is not fixed. A simple approximation algorithm for arbitrary graphs that is guaranteed to come within a factor of $2-2/k$ of the optimal cut weight is also described.

Spanning trees short or small

Abstract: We study the problem of finding small trees. Classical network design problems are considered with the additional constraint that only a specified number $k$ of nodes are required to be connected in the solution. A prototypical example is the $k$MST problem in which we require a tree of minimum weight spanning at least $k$ nodes in an edge-weighted graph. We show that the $k$MST problem is NP-hard even for points in the Euclidean plane. We provide approximation algorithms with performance ratio $2\sqrt{k}$ for the general edge-weighted case and $O(k^{1/4})$ for the case of points in the plane. Polynomial-time exact solutions are also presented for the class of decomposable graphs which includes trees, series-parallel graphs, and bounded bandwidth graphs, and for points on the boundary of a convex region in the Euclidean plane. We also investigate the problem of finding short trees, and more generally, that of finding networks with minimum diameter. A simple technique is used to provide a polynomial-time solution for finding $k$-trees of minimum diameter. We identify easy and hard problems arising in finding short networks using a framework due to T. C. Hu.

A Lagrangian relax-and-cut approach for the sequential ordering problem with precedence relationships

Abstract: The sequential ordering problem with precedence relationships was introduced in Escudero [7]. It has a broad range of applications, mainly in production planning for manufacturing systems. The problem consists of finding a minimum weight Hamiltonian path on a directed graph with weights on the arcs, subject to precedence relationships among nodes. Nodes represent jobs (to be processed on a single machine), arcs represent sequencing of the jobs, and the weights are sums of processing and setup times. We introduce a formulation for the constrained minimum weight Hamiltonian path problem. We also define Lagrangian relaxation for obtaining strong lower bounds on the makespan, and valid cuts for further tightening of the lower bounds. Computational experience is given for real-life cases already reported in the literature.

Computational methods for the diameter restricted minimum weight spanning tree problem.

Abstract: Let G be a simple undirected graph with non-negative edge weights. In this paper we consider the following combinatorial optimization problem : Find, in G, a minimum weight spanning tree having diameter at most D. This problem is trivial for D :S 3 and NP-complete for D :: 4. In this paper we develop and implement a number of Branch and Bound algorithms for this problem. Computational results, based on simulated problems, are discussed.

Spanning trees short or small

Abstract: We study the problem of finding small trees. Classical network design problems are considered with the additional constraint that only a specified number k of nodes are required to be connected in the solution. A prototypical example is the kMST problem in which we require a tree of minimum weight spanning at least k nodes in an edge-weighted graph. We show that the kMST problem is NP-hard even for points in the Euclidean plane. We provide approximation algorithms with performance ratio 2v/ for the general edge-weighted case and O(k1/4) for the case of points in the plane. Polynomial-time exact solutions are also presented for the class of treewidth-bounded graphs, which includes trees, series-parallel graphs, and bounded bandwidth graphs, and for points on the boundary of a convex region in the Euclidean plane. We also investigate the problem of finding short trees and, more generally, that of finding networks with minimum diameter. A simple technique is used to provide a polynomiM-time solution for finding k-trees of minimum diameter. We identify easy and hard problems arising in finding short networks using a framework due to T. C. Hu.

Truss Topology Optimization with Simultaneous Analysis and Design

Abstract: Strategies for topology optimization of trusses for minimum weight subject to stress and displacement constraints by simultaneous analysis and design (SAND) are considered. The ground structure approach is used. A penalty function formulation of SAND is compared with an augmented Lagrangian formulation. The efficiency of SAND in handling combinations of general constraints is tested. A strategy for obtaining an optimal topology by minimizing the compliance of the truss is compared with a direct weight minimization solution to satisfy stress and displacement constraints. It is shown that for some problems, starting from the ground structure and using SAND is better than starting from a minimum compliance topology design and optimizing only the cross sections for minimum weight under stress and displacement constraints. A member elimination strategy to save CPU time is discussed.

Supersonic transport wing minimum weight design integrating aerodynamics and structures

Abstract: An approach is presented for determining the minimum weight design of aircraft wing models which takes into consideration aerodynamics-structure coupling when calculating both zeroth-order information needed for analysis and first-order information needed for optimization. When performing sensitivity analysis, coupling is accounted for by using a generalized sensitivity formulation. The results presented show that the aeroelastic effects are calculated properly and noticeably reduce constraint approximation errors. However, for the particular example selected, the error introduced by ignoring aeroelastic effects are not sufficient to significantly affect the convergence of the optimization process. Trade studies are reported that consider different structural materials, internal spar layouts, and panel buckling lengths. For the formulation, model, and materials used in this study, an advanced aluminum material produced the lightest design while satisfying the problem constraints. Also, shorter panel buckling lengths resulted in lower weights by permitting smaller panel thicknesses and generally, unloading the wing skins and loading the spar caps. Finally, straight spars required slightly lower wing weights than angled spars.

A Chain Decomposition Algorithm for the Proof of a Property on Minimum Weight Triangulations

Abstract: In this paper, a chain decomposition algorithm is proposed and studied. Using this algorithm, we prove a property on minimum weight triangulations of points in the Euclidean plane, which shows that a special kind of line segments must be in the minimum weight triangulations and in the greedy triangulation of a given point set.

Minimum weight wheel rim

Abstract: A design arrangement is given, by means of which, the weight of a composite wheel rim may approach minimum. This is accomplished by locating high strength fiber bundles at extremes of the section. At other, less critical locations in the section less dense reinforcing fibers are used to save weight. Variations of this arrangement are applicable to a wide variety of wheels (i.e., bicycle, automobile, motorcycle, airplane, etc.).

Three-dimensional shape optimization using the boundary element method

Abstract: A practical design sensitivity calculation technique of displacements and stresses for three-dimensional bodies based on the direct differentiation method of discrete boundary integral equations is formulated in detail. Then the sensitivity calculation technique is applied to determine optimum shapes of minimum weight subjected to stress constraints, where an approximated subproblem is constructed repeatedly and solved sequentially by the mathematical programming method. The shape optimization technique suggested here is applied to determine optimum shapes of a cavity in a cube and a connecting rod

The preparata codes and a class of 4‐designs

Abstract: An extension theorem for t-designs is proved. As an application, a class of 4-(4m + 1,5,2) designs is constructed by extending designs related to the 3-designs formed by the minimum weight vectors in the Preparata code of length n = 4m, m ≥ 2. © 1994 John Wiley & Sons, Inc.

Application of optimization techniques to weight reduction of automobile bodies

Abstract: In order to find the optimum design which ensures a high quality and a light weight at the same time, it is necessary ot have effective prediction methods at the design stage. Described here is an application of structural optimization using FEM (Finite Element Method) to obtain design information for minimum weight. This paper also proposes the utilization of Design of Experiment in structural optimization for more efficient analysis.

Optimization for minimum sensitivity to uncertain parameters

Abstract: A procedure to design a structure for minimum sensitivity to uncertainties in problem parameters is described. The approach is to minimize directly the sensitivity derivatives of the optimum design with respect to fixed design parameters using a nested optimization procedure. The procedure is demonstrated for the design of a bimetallic beam for minimum weight with insensitivity to uncertainties in structural properties. The beam is modeled with finite elements based on two dimensional beam analysis. A sequential quadratic programming procedure used as the optimizer supplies the Lagrange multipliers that are used to calculate the optimum sensitivity derivatives. The method was perceived to be successful from comparisons of the optimization results with parametric studies.

Global convergence of the stress ratio method for truss sizing

Abstract: It is proved that if the well-known stress ratio method is applied to a “perturbed” stress-constrained minimum weight problem for truss structures, then the generated sequence of iteration points always converges to a global optimum. The most interesting step in the proof is a transformation of the stress-constrained problem to an equivalent unconstrained problem in which a combination of weight and compliance should be minimized. After the transformation, the stress ratio method becomes in fact a “successive linearization” method for solving this unconstrained problem.

Integration of structural optimization in the general design process for aircraft

Abstract: Formal mathematical optimization methods have been developed during the past 10-15 yr for the structural design of aircraft. Together with a reliable analysis program like finite element methods (FEM), they provide powerful tools for the structural design. They are efficient in at least two ways: 1) by producing designs that meet all specified requirements at minimum weight in one step and 2) relieving the engineer from a timeconsuming search for modifications that give better results, they allow more creative design modifications. MBB has developed an optimization code called MBB-LAGRANGE which uses mathematical programming and gradients to fulfill different constraints simultaneously. A method for solving large linear equation systems by an iterative method is described to show the effort which went into the program, formulating the physical problem in a very efficient mathematical way. Some examples depicting the successful application of the MBBLAGRANGE code are presented. This article closes with an outlook on how the optimization problem could be enlarged to include also the shape and size of airplanes.

Minimum Weight Design of Reticular Space Structures: A Computer Aided System

Abstract: In this work, by the definition of a computer aided system, we describe an optimisation procedure of reticular space structures based on the search of the minimum volume of the bars. The design system is based on classical methods of solution of minimum problems, the Feasible Regulation Method and the Optimality Criteria Method, adapted to this problem of defining again the stability and convergence parameters. The research of the optimal solution is carried out with reference to popular commercial sections (tubes) taking into account dimensional constrains imposed by the industrial production and design codes. The results of some numerical examples show the versatility and the efficiency of the design system with regard to reticular structures of any shape and dimension.

Algebraic characterization of minimum weight codewords of cyclic codes

Abstract: We consider primitive cyclic codes of length n over GF(q), where n=q/sup m/-1, and for any such code with defining set I(C), we define a system of algebraic equations, S/sub I(C/)(w), constructed with the Newton identities for the weight w. We prove that algebraic solutions of this system are in correspondence with all codewords of C of weight lower than w. To deal effectively with the system S/sub I(C/)(w), we compute a Groebner basis of this system, which gives pertinent information on minimum weight codewords. A few examples are given. >

Solving a Class of Stochastic Minimization Problems

Abstract: This work gives a methodology for analyzing a class of discrete minimization problems with random element weights. The minimum weight solution is shown to be an absorbing state in a Markov chain, while the distribution of weight of the minimum weight element is shown to be of phase type. We then present two-sided bounds for matroids with NBUE distributed weights, as well as for weights with bounded positive hazard rates. We illustrate our method using a realistic military communications problem.

Study of structurally efficient graphite-thermoplastic trapezoidal-corrugation sandwich and semisandwich panels

Abstract: The structural efficiency of compression-loaded trapezoidal-corrugation sandwich and semi-sandwich composite panels is studied to determine their weight savings potential. Sandwich panels with two identical face sheets and a trapezoidal corrugated core between them, and semi-sandwich panels with a corrugation attached to a single skin are considered. An optimization code is used to find the minimum weight designs for critical compressive load levels ranging from 3,000 to 24,000 lb/in. Graphite-thermoplastic panels based on the optimal minimum weight designs were fabricated and tested. A finite-element analysis of several test specimens was also conducted. The results of the optimization study, the finite-element analysis, and the experiments are presented.

Section fully utilized in prestressed concrete

Abstract: A unified procedure is presented for the flexural design of non-standard prestressed concrete members of minimum weight, with direct section dimensioning. The design procedure is based on assumed section efficiency, effectiveness of prestress, and covers the whole domain of load, beam depth, and span, above and below critical. Standard sections can also be used. A numerical design example is worked out to illustrate the proposed method.

Investigation of Methods for the Structural Weight Analysis of a Mach 2.4 Axisymmetric Inlet

Abstract: Structural design and analysis tools appropriate for estimating the structural weight of an axisymmetric inlet designed for Mach 2.4 cruise were evaluated. Little information regarding the inlet mechanical design is available in the preliminary design phase, so it is necessary to first develop a reasonable structural design before estimating the inlet weight. The Internally Pressurized Structure Synthesis and Optimization (IPSSO) program, employing an analytical approach, was chosen for evaluation due to its combined design and analysis capabilities. The inlet design produced by IPSSO was then analyzed using the NASTRAN finite element program. The finite element analysis was performed to help identify the limitations of the analytically based code as well as to evaluate NASTRAN for this application. Comparison between the IPSSO inlet weight and that of a similar inlet developed by the Boeing Commercial Airplane Group was also made. Program evaluation concluded that the combined use of IPSSO to create an initial design and NASTRAN to perform a numerical analysis would provide the capability to evaluate a limited number of inlet design The development of a new tool for the minimum weight design and analysis of inlet structures would be required for greater flexibility in evaluating inlet conceptual designs.

Analysis and design optimization of laminated composite structures using symbolic computation.

Abstract: The present study involves the analysis and design optimization of thin and thick laminated composite structures using symholic computation. The fibre angle and wall thickness of balanced and unbalanced thin composite pressure vessels are optimized subject to a strength criterion in order to maximise internal pressure or minimise weight , and the effects of axial and torsional forces on the optimum design are investigated. Special purpose symbolic computation routines are developed in the C programming language for the transformation of coordinate axes, failure analysis and the calculation of design sensitivities. In the study of thin-walled laminated structures, the analytical expression for the thickness of a laminate under in-plane loading and its sensitivity with respect to the fibre orientation are determined in terms of the fibre orientation using symbolic computation. In the design optimization of thin composite pressure vessels, the computational efficiency of the optimization algorithm is improved via symbolic computation. A new higher-order theory which includes the effects of transverse shear and normal deformation is developed for the analysis of laminated composite plates and shells with transversely isotropic layers. The Mathematica symbolic computation package is employed for obtaining analytical and numerical results on the basis of the higher-order theory. It is observed that these numerical results are in excellent agreement with exact three-dimensional elasticity solutions. The computational efficiency of optimization algorithms is important and therefore special purpose symbolic computation routines are developed in the C programming language for the design optimization of thick laminated structures based on the higher-order theory. Three optimal design problems for thick laminated sandwich plates are considered, namely, the minimum weight, minimum deflection and minimum stress design. In the minimum weight problem, the core thickness and the fibre content of the surface layers are optimally determined by using equations of micromechanics to express the elastic constants. In the minimum deflection problem, the thicknesses of the surface layers are chosen as the design variables. In the minimum stress problem, the relative thicknesses of the layers are computed such that the maximum normal stress will be minimized. It is shown that this design analysis cannot be performed using a classical or shear-deformable theory for the thick panels under consideration due to the substantial effect of normal deformation on the design variables.