Minimum-weight design of elastic sandwich beams with deflection constraints

In this paper, minimum-weight design of an elastic sandwich beam with a prescribed deflection constraint at a given point is investigated. The analysis is based on geometrical considerations using then-dimensional space of discretized specific bending stiffness. Since the present method of analysis is different from the method based on the calculus of variations, the conditions of piecewise continuity and differentiability on specific bending stiffness can be relaxed. Necessary and sufficient conditions for optimality are derived for both statically determinate and statically indeterminate beams. Beams subject to a single loading and beams subject to multiple loadings are analyzed. The degree to which the optimality condition renders the solution unique is discussed. To illustrate the method of solution, two examples are presented for minimum-weight designs under dual loading of a simply supported beam and a beam built in at both ends. The present analysis is also extended to the following problems: (a) optimal design of a beam built in at both ends with piecewise specific stiffness and a prescribed deflection constraint and (b) minimum-cost design of a sandwich beam with prescribed deflection constraints.The results presented in this paper were obtained in the course of research supported partly by the US Army Research Office, Durham, North Carolina, Research Grant No. DA-ARO-31-G1008, and partly by the Office of Naval Research, Contract No. N00014-67-A-0109-0003, Task No. NR 064-496. The authors wish to express their thanks to Professor H. Halkin for pointing out the applicability of optimal control theory to the present problem and to Professor W. Prager for his valuable suggestions.     

Structural weight minimization using necessary and sufficient conditions

This paper considers problems of weight minimization of complex, determinate or indeterminate structures, subject to inequality constraints. Limitations of size of the structural members, allowable stresses, natural frequencies, etc., furnish constraints on the design. We are here principally concerned with the theoretical determination of the necessary and sufficient conditions relating a proposed design to the true constrained minimum-weight design. As an important special case, a study is made of the conditions under which a fully stressed design is a minimum-weight design. Although much attention has been directed toward the fully stressed approach to minimum-weight design, sufficiency conditions and questions relating to global optimality vs local optimality have heretofore not been considered in detail. Example solutions are presented illustrating the application of the present results to design problems. In one such solution, it is demonstrated that, for a broad class of statically determinate structures, sufficiency conditions exist which ensure that the fully stressed design is a globally minimum-weight design.     

Minimum-weight design with piecewise constant specific stiffness

Much of the literature on minimum-weight design is concerned with structures with continuously varying dimensions. Structures of this kind achieve an absolute minimum of weight and thereby provide a useful standard of comparison, but their manufacturing cost is prohibitive in most applications. As a step toward the establishment of optimality conditions for more practical structures, the present paper discusses optimal design of sandwich beams and frames with segmentwise constant cross sections. Optimality is shown to require that the mean square stress in the face sheets has the same value for all segments. The use of the optimality condition is illustrated by examples.This research was supported by the Advanced Research Projects Agency, Project DEFENDER, and was monitored by the US Army Research Office-Durham, Contract No. DA-31-124-ARO-D-257.     

Optimal design of a minimum weight thermal diffuser with constraint on the output thermal power flux

The object of this paper is the development of efficient mathematical and numerical tools to find the optimal shape of a minimum-weight thermal diffuser with a priori specifications on the inward thermal power flux (TPF) and a bound on the outward TPF. The present problem arises in connection with the use of high-power solid state devices in future communications satellites. In a space application the thermal power must ultimately be dissipated to the environment by using heatpipes and correspondingly large radiating areas. However, heatpipes can accept only a limited TPF from a source. Hence we have the requirement of a minimum-weight thermal diffuser with a uniform bound on the outward TPF. Shape optimal design and finite elements methods are used. Complete numerical results are provided.This research was supported by the Natural Sciences and Engineering Research Council of Canada, Strategic Grants G-0573 and G-0654 (Communications) and Operating Grant A-8730.     

Dynamic programming in minimum-weight design of axisymmetric plates

The paper deals with minimum-weight design of axisymmetric plates. The design is subject to restrictions imposed by static equilibrium conditions and various yield criteria governing the failure of the material. Herein, the optimization problem is formulated as a dynamic programming problem and is solved. Such a formulation unifies the design procedure. The paper also discusses the dynamic programming approach.     

Triangulations intersect nicely

We show that there is a matching between the edges of any two triangulations of a planar point set such that an edge of one triangulation is matched either to the identical edge in the other triangulation or to an edge that crosses it. This theorem also holds for the triangles of the triangulations and in general independence systems. As an application, we give some lower bounds for the minimum-weight triangulation which can be computed in polynomial time by matching and network-flow techniques. We exhibit an easy-to-recognize class of point sets for which the minimum-weight triangulation coincides with the greedy triangulation.     

Dynamic programming and the optimum design of rotating disks

The minimum-weight design of elastic rotating disks under various design criteria has been studied using ideas of dynamic programming and invariant imbedding. It is shown that the criterion of minimum potential energy furnishes a remarkably simple solution in terms of a linear partial differential equation subject to initial conditions. The treatment of more general design constraints leads to the development of a number of stable, two-sweep iterative procedures. A series of numerical examples incorporating various design constraints is presented to show the accuracy and feasibility of the method. The sensitivity of the radial stresses to small changes in the thickness of the disk is noticed. On the other hand, the high stability of the volume with respect to the design criteria is stressed.This work was partially supported by a University of California grant. The author wishes to thank Mr. R. Todeschini, who carried out the numerical computations.     

Minimum-weight design of continuous beams under displacement and stress constraints

Explicit optimality conditions for minimum-weight design of elastic sandwich beams with segmentwise constant structural stiffness, subject to displacement and mean-square stress constraints, are obtained. An iterative procedure that combines the use of the optimality conditions with finite-element analysis is proposed and is illustrated by numerical examples. These examples suggest that very few iterations are necessary to obtain a good approximation to the optimal design. It is shown that, for practical purposes, the optimization problem may be simplified by using the optimality conditions derived for statically determinate beams instead of those valid for statically indeterminate beams.This research was sponsored by the U.S. Army Research Office, Durham, North Carolina. The authors are grateful to Professor W. Prager for helpful comments.     

On the duals of geometric Goppa codes from norm-trace curves

In this paper we study the dual codes of a wide family of evaluation codes on norm-trace curves. We explicitly find out their minimum distance and give a lower bound for the number of their minimum-weight codewords. A general geometric approach is performed and applied to study in particular the dual codes of one-point and two-point codes arising from norm-trace curves through Goppaʼs construction, providing in many cases their minimum distance and some bounds on the number of their minimum-weight codewords. The results are obtained by showing that the supports of the minimum-weight codewords of the studied codes obey some precise geometric laws as zero-dimensional subschemes of the projective plane. Finally, the dimension of some classical two-point Goppa codes on norm-trace curves is explicitely computed.     

Optimal design of rotating disk for given radial displacement of edge

Minimum-weight design of axially-symmetric, rotating, elastic disks is usually discussed in terms of uniform strength. In this paper, an alternative constraint is used: under the combined influence of centrifugal forces and a uniform radial traction along the circular edge, the radial displacement at this edge is to have a prescribed value. A necessary and sufficient optimality condition is derived, and its use in the determination of optimal disk profiles is illustrated by examples. It is shown that the resulting disks are far from satisfying the condition of uniform strength although their weights are only very slightly smaller than those of the corresponding disks of uniform strength.This research was sponsored by the Air Force Flight Dynamics Laboratory, Wright-Patterson Air Force Base, Ohio, under Contract No. F33615-C-1826.     

Minimum-weight design of structures with natural-frequency constraints

The problem of minimum-weight design of structures with several natural-frequency constraints is considered in this paper. The problem is solved by using a combined finite element method and sequential linear programming (FEM-SLP) formulation. The unique features of the present approach include the use of the assumed mode reanalysis formulation for the repeated eigensolution and the associated sensitivity analysis. The present approach has been implemented using the general-purpose finite element program . Two examples are included to illustrate the effectiveness of the present approach.     

Enumeration of Pareto optimal multi-criteria spanning trees – a proof of the incorrectness of Zhou and Gen's proposed algorithm

The minimum spanning tree (MST) problem is a well-known optimization problem of major significance in operational research. In the multi-criteria MST (mc-MST) problem, the scalar edge weights of the MST problem are replaced by vectors, and the aim is to find the complete set of Pareto optimal minimum-weight spanning trees. This problem is NP-hard and so approximate methods must be used if one is to tackle it efficiently. In an article previously published in this journal, a genetic algorithm (GA) was put forward for the mc-MST. To evaluate the GA, the solution sets generated by it were compared with solution sets from a proposed (exponential time) algorithm for enumerating all Pareto optimal spanning trees. However, the proposed enumeration algorithm that was used is not correct for two reasons: (1) It does not guarantee that all Pareto optimal minimum-weight spanning trees are returned. (2) It does not guarantee that those trees that are returned are Pareto optimal. In this short paper we prove these two theorems.     

Minimum—weight perfect matching for nonintrinsic distances on the line

We consider a minimum-weight perfect matching problem on the line and establish a “bottom-up” recursion relation for weights of partial minimum-weight matchings. Bibliography: 11 titles.     

Min-weight double-tree shortcutting for Metric TSP: Bounding the approximation ratio

The Metric Traveling Salesman Problem (TSP) is a classical NP-hard optimization problem. The double-tree shortcutting method for Metric TSP yields an exponentially-sized space of TSP tours, each of which approximates the optimal solution within at most a factor of 2. We consider the problem of finding among these tours the one that gives the closest approximation, i.e. the minimum-weight double-tree shortcutting. Previously, we gave an efficient algorithm for this problem, and carried out its experimental analysis. In this paper, we address the related question of the worst-case approximation ratio for the minimum-weight double-tree shortcutting method. In particular, we give lower bounds on the approximation ratio in some specific metric spaces: the ratio of 2 in the discrete shortest path metric, 1.622 in the planar Euclidean metric, and 1.666 in the planar Minkowski metric. The first of these lower bounds is tight; we conjecture that the other two bounds are also tight, and in particular that the minimum-weight double-tree method provides a 1.622-approximation for planar Euclidean TSP.     

Subharmonics near an equilibrium point for hamitonian systems

This work is devoted to the study of subharmonic solutions near an equilibrium for certain Hamiltonian systems. We impose a weak condition on the Floquet exponents of the linearized system, and a superquadratic condition on the higher order term. This last condition is reduced to the center manifold. This research was supported in part by the Air Force Office of Scientific Research under Grant AFOSR-87-0202.     

On the existence of equivalent finite invariant measures

Summary Necessary and sufficient conditions are given for the existence of a finite measure which is equivalent to a given measure and invariant with respect to each transformation in a given commutative semigroup of measurable null-invariant point transformations. This result was already known for denumerably generated semigroups. A complementary result is proved which states that if one such equivalent measure exists, then there exists a unique equivalent measure which agrees with the original measure on the invariant sets.Research sponsored by the Air Force Office of Scientific Research, Office of Aerospace Research, United States Air Force, under Grant No. AFOSR-68-1394.     

On the optimal design of plates for a given deflection

The optimum design problem of an elastic plate for a given deflection is considered. The design variable is chosen to be the thickness of the plate. Using the principle of stationary mutual potential energy first introduced by Shield and Prager, a sufficient optimality condition (which makes the volume stationary) is derived for plates under bending caused by general loading conditions. As an example, the optimum profile of a simply supported circular plate under a given rotationally symmetric loading is obtained.     

Dynamics of three coupled limit cycle oscillators with application to artificial intelligence

We study a system of three limit cycle oscillators which exhibits two stable steady states. The system is modeled by both phase-only oscillators and by van der Pol oscillators. We obtain and compare the existence, stability and bifurcation of the steady states in these two models. This work is motivated by application to the design of a machine which can make decisions by identifying a given initial condition with its associated steady state.     

Thrust nozzle optimization including boundary-layer effects

An optimization analysis employing the calculus of variations and a numerical relaxation scheme for applying the results are presented for the design of axisymmetric, maximumthrust nozzle contours including boundary-layer effects. The method is illustrated by comparison with a known solution without boundary-layer effects. A parametric study of boundary-layer effects for a range of parameters typical of scramjet thrust nozzles is presented.This work was sponsored by the Air Force Aero-Propulsion Laboratory, Wright-Patterson Air Force Base, Ohio, under Contract No. F33615-67-C-1068.     

Two-Connected Augmentation Problems in Planar Graphs

Given a weighted undirected graph G and a subgraph S of G, we consider the problem of adding a minimum-weight set of edges of G to S so that the resulting subgraph satisfies specified (edge or vertex) connectivity requirements between pairs of nodes of S. This has important applications in upgrading telecommunication networks to be invulnerable to link or node failures. We give a polynomial algorithm for this problem when S is connected, nodes are required to be at most 2-connected, and G is planar. Applications to network design and multicommodity cut problems are also discussed.