Investigation of the fiber packing density in a unidirectional composite

Recommendations are made for designing a unidirectional composite with an optimal arrangement of the fibers. The effect of the fiber diameter distribution law on the packing density is investigated.All-Union Scientific-Research Institute of Glass-Reinforced Plastics and Glass Fiber, Moscow. Moscow Institute of Electronic Machine-Building. Translated from Mekhanika Polimerov, No. 3, pp. 538–541, May–June, 1976.     

Optimization of the axial stability of equal-strength reinforced cylindrical pressure vessels

The class of reinforced cylindrical shells obtained by spiral winding with reinforcing ring or longitudinal layers is considered. The structure of the shells is optimal with respect to internal pressure. The reinforcement angle is selected so as to maximize the critical axial load.Translated from Mekhanika Polimerov, No. 6, pp. 1123–1126, November–December, 1973.     

Optimal winding of filament-wound reinforced plastic shells designed to resist buckling under external pressure or axial compression

The optimal method of winding the reinforcement of thin cylindrical shells subjected to external pressure or axial loading is investigated. Various types of winding — normal, simple oblique, crossed, and isotropic — are considered. An approximate formula is obtained for the critical external pressure. This formula is used to analyze the possibility of raising the critical load by adopting oblique winding. In the case of axial compression different buckling modes are examined and the optimal winding determined.Central Scientific-Research Institute of Machine Building, Moscow. Translated from Mekhanika Polimerov, Vol. 4, No. 5, pp. 864–875, September–October, 1968.     

Optimization of a cylindrical composite shell with a viscoelastic core in axial compression

The problem of the optimal design of a composite shell in creep is formulated. The progressive buckling of a cross-wound reinforced cylindrical shell supported on a viscoelastic core is considered as a particular case. The reinforcement structure and shell thickness corresponding to minimum weight for a given load and service life are found.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 3, pp. 442–446, May–June, 1975.     

Gauss–Man i n Conne ction s for Arrangements, I E igenvalues

We construct a formal connection on the Aomoto complex of an arrangement of hyperplanes, and use it to study the Gauss–Manin connection for the moduli space of the arrangement in the cohomology of a complex rank one local system. We prove that the eigenvalues of the Gauss–Manin connection are integral linear combinations of the weights which define the local system.     

Dynamical optimization of ribbed cylindrical shells under the action of periodic nonharmonic loads

Ribbed cylindrical shells that are under the action of a nonharmonic cylindrical load are considered. The problem of optimal (with respect to the criterion of minimum mass) projection is formulated in terms of nonlinear programming. Some results of a numerical experiment about the choice of optimal parameters are shown, which was carried out on an IBM with the help of the method of random search.Translated from Dinamicheskie Sistemy, No. 9, pp. 22–27, 1990.     

Dynamical stability and optimization of parameters of cylindrical shells under short-time influence of loads that increase in a power-wise way

Some criteria of dynamical nonstability are derived for smooth cylindrical shells under actions of axial contracting loads that increase fast in a power-wise way. In terms of nonlinear programming the optimization problem is formulated that determines parameters of shells of minimal mass under some restrictions with respect to local and global loss of stability and also with respect to strength. With the help of the Kuhn-Tucker theorem some variants of the optimal solution are obtained. As an example, we consider two characteristic forms of load: linearly increasing load, and a load that increases in the quadratic way. We study the influence of basic parameters of these loads on the mass, and geometric dimensions of the optimal shells. Some numerical examples are given.Translated from Dinamicheskie Sistemy, No. 8, pp. 47–53, 1989.     

Propagation of acoustic waves in a fibrous fluid-filled material

The homogenization method is used for the analysis of acoustic wave propagation in a unidirectional fibrous material for the acoustic control of the fiber arrangement in manufacturing composites. Acoustic equations for a rigid periodic structure filled by a nonviscous fluid are obtained by two-dimensional asymptotic expansions. A regular square and triangular arrangement of the fibers with a round cross-section are considered. The analysis reveals that the velocity of acoustic waves is significantly affected by both the volume content and the fiber arrangement.Moscow State Academy of Chemical Engineering, Russia. Translated from Mekhanika Kompozitnykh Materialov, Vol. 33, No. 5, pp. 651–655 September–October, 1997.     

Optimization of the stressed state by a distribution of elastic parameters in elastic bodies

For designing an isotropic inhomogeneous body having a variable elastic modulus and a prescribed shape under the influence of an external force load we give a formulation and solution of the problem of determining the design that is optimal with respect to stress. The problem of optimal design reduces to a certain problem in the theory of elasticity for a non-linearelastic material. As an example we consider the problem of optimal design of an inhomogeneous cylinder. Four figures. Bibliography: 9 titles.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 30, 1989, pp. 78–82.     

Optimal investment and reinsurance of an insurer with model uncertainty

We introduce a novel approach to optimal investment–reinsurance problems of an insurance company facing model uncertainty via a game theoretic approach. The insurance company invests in a capital market index whose dynamics follow a geometric Brownian motion. The risk process of the company is governed by either a compound Poisson process or its diffusion approximation. The company can also transfer a certain proportion of the insurance risk to a reinsurance company by purchasing reinsurance. The optimal investment–reinsurance problems with model uncertainty are formulated as two-player, zero-sum, stochastic differential games between the insurance company and the market. We provide verification theorems for the Hamilton–Jacobi–Bellman–Isaacs (HJBI) solutions to the optimal investment–reinsurance problems and derive closed-form solutions to the problems.     

On the optimal structural design for a nonconservative,elastic stability problem

The design of a cantilever column under a follower load is considered with the aim of maximizing the critical value of the load. The optimality condition is derived, and a modified Ritz method is used to determine an approximate solution for the bending stiffness. Results are obtained numerically for the case of a sandwich column with constant bending stiffness in each of two segments. It is found that, for the same structural weight, the optimal design yields a critical load significantly higher than that for a uniform column.This research was supported in part by the US Army Research Office–Durham and in part by the United States Navy under Grant No. NONR N00014-67-A-0191-0009.     

On formulations of optimal design problems with respect to reliability criteria for reinforced shells

We discuss problems of the application of various reliability theories in models of optimal design of structures. Using the example of the problem of optimizing (with respect to the minimal mass criterion) the parameters of an inclined spherical shell reinforced by a network of ribs and subject to a random load stated as a nonlinear programming problem we give a comparative analysis of the optimal designs obtained using the theory of Bolotin and the theory of Kapur-Lamberson. One figure. One table. Bibliography: 5 titles.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 30, 1989, pp. 82–87.     

Statistical analysis of the bearing capacity of rim-spoke flywheels made of reinforced plastics

Conclusion The suggested numerical method makes it possible to plot the distribution functions of the bearing capacity of RSF at the structural level of their design elements (rim, spokes) with a view to the scatter of the strength and strain properties of the used materials and the geometry of arrangement of the structure. This involves taking into account the random nature of the radial distribution in the rim material of both strain and strength properties, which makes it possible to investigate more fully the processes of failure really occurring in RSF. The suggested method is the basis for stipulating a scientifically substantiated safety factor (the theoretical bearing capacity) in dependence on the required reliability; this makes it possible to use a differentiated approach to problems of devising RSF for actual purposes. Our investigations may serve as the basis for solving problems of optimal planning of RSF according to the conditions of the required reliability.Translated from Mekhanika Kompozitnykh Materialov, No. 6, pp. 1041–1047, November–December, 1987.     

Methods of design of extremal experiments in problems of vector optimization of composite shells

Using the methods of the theory of design of extremal experiments and vector optimization we solve a problem of optimal design of cylindrical shells of composite materials with respect to the criteria of minimal weight and maximum critical load. The use of the three-dimensional scheme of reinforcement of material leads to improvements in the weight of the shell in comparison with the two-dimensional reinforcement scheme.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 28, 1988, pp. 70–74.     

Discriminantal Arrangements, Fiber Polytopes and Formality

Manin and Schechtman defined the discriminantal arrangement of a generic hyperplane arrangement as a generalization of the braid arrangement. This paper shows their construction is dual to the fiber zonotope construction of Billera and Sturmfels, and thus makes sense even when the base arrangement is not generic. The hyperplanes, face lattices and intersection lattices of discriminantal arrangements are studied. The discriminantal arrangement over a generic arrangement is shown to be formal (and in some cases 3–formal), though it is in general not free. An example of a free discriminantal arrangement over a generic arrangement is given.     

Control of damped vibrations of a rod by means of a concentrated force or moment loading

The optimal local loading regimes of a vibrating rod (one endpoint of the rod is fixed, while the second is stress-free), which ensure the damping of certain modes of elastic vibration over a minimal time interval, for a minimal norm of the control function and also under the simultaneous minimization of the time and the norm of the load function are given.Translated from Matematicheskie Metody i Fiziko-Mekhanicheskie Polya, No. 25, pp. 71–75, 1987.     

Space-filling curves for 2-simplicial meshes created with bisections and reflections

Numerical experiments in J. Maubach: Local bisection refinement and optimal order algebraic multilevel preconditioners, PRISM-97 conference Proceedings, 1977, 121–136 indicated that the refinement with the use of local bisections presented in J. Maubach: Local bisection refinement for n-simplicial grids generated by reflections, SIAM J. Sci. Comput. 16 (1995), 210–227 leads to highly locally refined computational 2-meshes which can be very efficiently load-balanced with the use of a space-filling curve. This paper introduces the construction of this curve which can be produced at almost no costs, proofs that all its properties are invariant under local bisection, and comments on the 3-dimensional case.With the use of a space-filling curve (which passes through all triangular elements), load balancing over several processors is trivial: The load can be distributed over N processors by cutting the curve into N almost equilength parts. Each processor then operates on the triangles which are passed by its part of the curve.     

Little-Preemptive Scheduling on Unrelated Processors

We consider the preemptive scheduling of n independent jobs on m unrelated machines to minimize the makespan. Preemptive schedules with at most 2m–3 preemptions are built, which are optimal when the maximal job processing time is no more than the optimal schedule makespan. We further restrict the maximal job processing time and obtain optimal schedules with at most m–1 preemptions. This is better than the earlier known best bound of 4m ²–5m+2 on the total number of preemptions. Without the restriction on the maximal job processing time, our (2m–3)-preemptive schedules have a makespan which is no more than either of the following two magnitudes: (a) the maximum between the longest job processing time and the optimal preemptive makespan, and (b) the optimal nonpreemptive makespan. Our (m–1)-preemptive schedules might be at most twice worse than an optimal one.     

Concerning the critical load for an orthotropic cylindrical shell with a viscoelastic core

The problem of determining the critical load for a cylindrical shell with a viscoelastic core is considered. An expression for the critical load under quasistationary loading is obtained on the basis of a generalization of the Winkler hypothesis.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 6, pp. 1049–1053, November–December, 1973.     

Optimal dividend strategies for a risk process under force of interest

In the classical Cramér–Lundberg model in risk theory the problem of maximizing the expected cumulated discounted dividend payments until ruin is a widely discussed topic. In the most general case within that framework it is proved [Gerber, H.U., 1968. Entscheidungskriterien fuer den zusammengesetzten Poisson-prozess. Schweiz. Aktuarver. Mitt. 1, 185–227; Azcue, P., Muler, N., 2005. Optimal reinsurance and dividend distribution policies in the Cramér–Lundberg model. Math. Finance 15 (2) 261–308; Schmidli, H., 2008. Stochastic Control in Insurance. Springer] that the optimal dividend strategy is of band type. In the present paper we discuss this maximization problem in a generalized setting including a constant force of interest in the risk model. The value function is identified in the set of viscosity solutions of the associated Hamilton–Jacobi–Bellman equation and the optimal dividend strategy in this risk model with interest is derived, which in the general case is again of band type and for exponential claim sizes collapses to a barrier strategy. Finally, an example is constructed for Erlang(2)-claim sizes, in which the bands for the optimal strategy are explicitly calculated.