Optimality criterion for maximum fundamental frequency of free vibrations of frames including axial and bending effects

An optimality criterion for maximum multiple fundamental frequency of free vibrations for structures of prescribed weights is presented. The criterion includes both axial and bending effects and can be used for analysis of truss, beam and frame structures. The error norm based on the criterion is proposed and used to verify trial designs against the optimum. The accompanying iterative procedure reduces this error norm to zero and drives a trial design to the optimum. The modality of the design at the optimum and the corresponding set of Lagrange multipliers are determined automatically.     

Optimality conditions of sliding modes and the maximum principle for control problems with the scalar argument

Various types of optimality criteria and conditions, which define the set of admissible solution, are brought to the canonical form. For the problem set in this form, the optimality conditions of sliding modes are stated. It is shown that the optimality conditions emerge from these conditions in the form of the maximum principle for problems with a scalar argument and an arbitrary combination of the optimality criterion and constraints.     

Regularized trigonometric approximation of optimal periodic control problems

The optimal periodic control problem with the main and the additional optimality criterion for a system described by differential equations is investigated. The energy and raw material consumption are considered as the additional criteria. The problem characterized is approximated by a sequence of discretized optimization problems using trigonometric polynomials and penalty functions. Sufficient conditions are given for the convergence of solutions of discretized problems to the regular optimal solution, which minimizes the additional criterion over the set of solutions minimizing the main criterion. Strong convergence in the case where the energy consumption is the additional criterion is proven.     

Decomposition Method for Solving Three-Index Transportation Problems

A decomposition method based on the sequential modification of the optimality criterion is used for solving the classical three-index transportation problem. The method consists of a sequence of solutions of local problems with three constraints. A monotonic (in the optimality criterion) process that converges to the solution of the original problem is constructed. The solutions of the transportation problem with a linear and quadratic objective function are considered and the numerical results are presented.     

A new optimality criteria method for shape optimization of natural frequency problems

A new shape optimization method for natural frequency problems is presented. The approach is based on an optimality criterion for general continuum solids, which is derived in this paper for the maximization of the first natural frequency with a volume constraint. An efficient redesign rule for frequency problems is developed to achieve the required shape modifications. The optimality criterion is extended to volume minimization problems with multiple frequency constraints. The nonparametric geometry representation creates a complete design space for the optimization problem, which includes all possible solutions for the finite element discretization. The combination with the optimality criteria approach results in a robust and fast convergence, which is independent of the number of design variables. Sensitivity information of objective function and constraints are not required, which allows to solve the structural analysis task using fast and reliable industry standard finite element solvers like ABAQUS, ANSYS, I-DEAS, MARC, NASTRAN, or PERMAS. The new approach is currently being implemented in the optimization system TOSCA.     

Some second-order necessary conditions in optimal control

In optimal control problems any extremal arc which trivially satisfies the Maximum Principle, that is a first-order control variation produces no change in cost, is called singular. Higher-order conditions are then needed to check the optimality of such arcs. Using the Volterra series associated with the variation of the cost functional gives a new context for analyzing singular optimal control problems. A basic optimality criterion for a fixed terminal time Mayer problem is obtained which allows one to derive the necessary conditions for optimality in terms of Lie brackets of vector fields associated with the dynamics of the problem.     

On optimal shapes in materials and structures

In the micromechanics design of materials, as well as in the design of structural connections, the boundary shape plays an important role. The objective may be the stiffest design, the strongest design or just a design of uniform energy density along the shape. In an energy formulation it is proven that these three objectives have the same solution, at least within the limits of geometrical constraints, including the parametrization. Without involving stress/strain fields, the proof holds for 3D-problems, for power-law nonlinear elasticity and for anisotropic elasticity.?To clarify the importance of parametrization, the problem of material/hole design for maximum bulk modulus is analysed. A simple optimality criterion is derived and with a simple superelliptic parametrization, agreement with Hashin-Shtrikman bounds are found. More general examples including nonequal principal strains, nonlinear elasticity and orthotropic elasticity show the versatility of the optimality criterion approach. In spite of this, the mathematical programming approach will be used in the future study of the multiparameter and/or multipurpose problems. Received March 23, 1999     

退化线性规划的一个新的改进的单纯形方法

本文讨论退化线性规划单纯形方法最优解的判定准则和有限主元规则.首先改进简约价值系数向量,提出线性规划单纯形方法最优解的判定准则.并且利用本文的判定准则给出[3]中定理2.3.5(P.84)的一个新的证明.然后提出一种新的混合有限主元规则,在退化情形下通过对单纯形表使用新的混合有限主元规则进行迭代,可以判断当前退化基本可行解或为最优解或给出下次迭代的主元并且跳出循环.最后给出在一组经典的退化线性规划例子下,改进的单纯形方法好的计算表现.     

A unified approach to structural weight minimization

It is shown that the two classical approaches to structural optimization have now reached a stage where they employ the same basic principles. Indeed, the well-known optimality criteria approach can be viewed as transforming the initial problem in a sequence of simple explicit problems in which the constraints are approximated from virtual work considerations. On the other hand, the mathematical programming approaches have progressively evoluated to a linearization method using the reciprocals of the design variables — this powerful method is proven here to be identical to a generalized optimality criteria approach. Finally, new efficient methods are proposed: (a) a hybrid optimality criterion based on first-order approximations of the most critical stress constraints and zeroth-order approximations of the others and (b) a mixed method which lies between a strict primal mathematical programming method and a pure optimality criteria (or linearization) approach. Simple numerical problems illustrate the concepts established in the paper.     

Optimal shakedown loading for circular plates

Optimization of shakedown loading under constrained residual displacement is considered for elastic and perfectly plastic circular plates. The load variation bounds, which satisfy the optimality criterion in concert with plate-strength and stiffness requirements, are identified. The actual strain fields of the plate depend on the loading history. Thus, the load optimization problem at shakedown is stated as a pair of problems that are executed in parallel: the main load optimization and the verification of the prescribed magnitudes of the bounds on the residual deflections. The problem must be solved by iteration. The Rozen projected gradient method is applied. A mechanical interpretation of the Rozen optimality criterion is given, which permits the simplification of the mathematical model for load optimization and the formulation of the solution algorithm. Numerical examples include circular annular plates with and without a rigid inclusion. The results are valid under the assumption of small displacements.     

Certainty equivalence control with forcing: revisited

Certainty equivalence control with forcing has been shown to be optimal for several stochastic adaptive control problems with the average cost per unit time criterion. Recently researchers have started looking at stochastic adaptive control problems with a view to minimizing the rate of increase of the learning loss. This criterion is stronger than the average cost per unit time criterion. Certainty equivalence control with forcing does not usually suffice for the learning loss criterion and one has to develop fairly complicated schemes in order to achieve optimality. The objective of this paper is to see how well one might be able to do with a certainty-equivalence-control-with-forcing type of scheme. In particular we construct a class of such schemes whose learning loss is O((log n)^1+δ) for δ > 0, whereas optimal schemes typically have a O(log n)learning loss.     

A constraint-based approach for the shift design personnel task scheduling problem with equity

This paper presents an industrial problem which arises in a company specialized in drug evaluation and pharmacology research. The aim is to build employee timetables covering the demand given by a set of fixed tasks. The optimality criterion concerns the equity of the workload sharing. A solution to this problem is the assignment of all tasks whose resulting working shifts respect tasks requirements as well as legal and organizational constraints. Scheduling problems usually consider a fixed set of shifts which have to be assigned to a given number of employees whereas in our problem shifts are not fixed and are deduced from the task assignment. In the following, we refer to this problem as the shift-design personnel task scheduling problem with an equity criterion (SDPTSP-E), in reference to the shift minimization personnel task scheduling problem (SMPTSP). Even if the SDPTSP-E is related to several problems, none of them allow to grasp its full complexity. Consequently, we propose a dedicated method based on constraint programming. Several branching and exploration strategies are proposed and tested.     

Finite strain topology optimization based on phase-field regularization

In this paper the topology optimization problem is solved in a finite strain setting using a polyconvex hyperelastic material. Since finite strains is considered the definition of the stiffness is not unique. In the present contribution, the objective of the optimization is minimization of the end-displacement for a given amount of material. The problem is regularized using the phase-field approach which leads to that the optimality criterion is defined by a second order partial differential equation. Both the elastic boundary value problem and the optimality criterion is solved using the finite element method. To approach the optimal state a steepest descent approach is utilized. The interfaces between void and full material are resolved using an adaptive finite element scheme. The paper is closed by numerical examples that clearly illustrates that the presented method is able to find optimal solutions for finite strain topology optimization problems.     

Trigonometric approximation of optimal periodic control problemsfor systems with inertial controllers

A constrained optimal periodic control (OPC) problem for nonlinear systems with inertial controllers is considered. A sequence of approximate problems containing trigonometric polynomials for approximation of the state, control, and functions in the state equations and in the optimality criterion is formulated. Sufficient conditions for a sequence of nearly optimal solutions of approximate problems to be norm-convergent to the basic problem optimal solution are derived. It is pointed out that the direct approximation approach in the space of state and control combined with the finite-dimensional optimization methods such as the space covering and gradient-type methods makes probable the finding of the global optimum for OPC problems     

A note on risk-sensitive control of invariant models

According to Assaf, a dynamic programming problem is called invariant if its transition mechanism depends only on the chosen action. This paper studies properties of risk-sensitive invariant problems with a general state space. The main result establishes the optimality equation for the risk-sensitive average cost criterion without any restrictions on the risk factor. Moreover, a practical algorithm is provided for solving the optimality equation in case of a finite action space.     

Optimality criterion approach using tapered finite elements with nodal averaging technique for two-dimensional problems

Use of tapered finite elements with three design variables per element and nodal averaging technique in the optimality criterion approach is studied in this paper. Minimum volume design of a uniformly heated square plate with a single temperature constraint is obtained using the proposed method. The present results with a lower order finite element mesh compare very well with those obtained with optimality criterion approach using constant thickness elements and with mathematical programming techniques using a higher order finite element mesh.     

Performance Properties of Large Scale Parallel Systems

There are several metrics that characterize the performance of a parallel system, such as parallel execution time, speedup, and efficiency. A number of properties of these metrics have been studied. For example, it is a well known fact that given a parallel architecture and a problem of a fixed size, the speedup of a parallel algorithm does not continue to increase with increasing number of processors. It usually tends to saturate or peak at a certain limit. Thus, it may not be useful to employ more than an optimal number of processors for solving a problem on a parallel computer. This optimal number of processors depends on the problem size, the parallel algorithm, and the parallel architecture. In this paper we study the impact of parallel processing overheads and the degree of concurrency of a parallel algorithm on the optimal number of processors to be used when the criterion for optimality is minimization of the parallel execution time. We then study a more general criterion of optimality and show how operating at the optimal point is equivalent to operating at a unique value of efficiency that is characteristic of the criterion of optimality and the properties of the parallel system under study. We put the technical results derived in this paper in perspective with similar results that have appeared in the literature before and show how this paper generalizes and/or extends these earlier results.     

Optimal shakedown design of beam structures

The optimal design of plane beam structures made of elastic perfectly plastic material is studied according to the shakedown criterion. The design problem is formulated by means of a statical approach on the grounds of the shakedown lower bound theorem, and by means of a kinematical approach on the grounds of the shakedown upper bound theorem. In both cases two different types of design problems are formulated: one searches for the minimum volume design whose shakedown limit load is assigned; the other searches for the design of the assigned volume whose shakedown limit load is maximum. The optimality conditions of the four problems above are found by the use of a variational approach; such conditions prove the equivalence of the two types of design problems, provide useful information on the structural behaviour in optimality conditions, and constitute a fifth possible way to determine the optimal design. Whatever approach is used, the strong non-linearity of the corresponding problem does not allow the finding of the analytical solution. Consequently, in the application stage suitable numerical procedures must be employed. Two numerical examples are given.     

The FEM optimization of active vibration control in structures by solving the corresponding two-point-boundary-value problem

An algorithm for solving optimal active vibration control problems by the finite element method (FEM) is presented. The optimality equations for the problem are derived from Pontryagin’s principle in the form of a set of the fourth order ordinary differential equations that, together with the initial and final boundary conditions, constitute the boundary value problem in the time domain, which in control is referred to as a two-point-boundary-value problem. These equations decouple in the modal space and can be solved by the FEM technique. An analogy between the optimality equations and the governing equations for a set of certain static beams permits obtaining numerical solutions to the optimal control problem with the help of standard ‘structural’ FEM software. The optimal action of actuators is automatically calculated by applying the independent modal space control concept. The structure’s response to actuation forces is also determined and can independently be verified for spillover effects. As an illustration, the algorithm is used for the analysis of optimal action of actuators to attenuate vibrations of an elastic fin.