Optimal shape design and unilateral boundary value problems: Part II

The shape optimization of an elastic body in contact with a rigid surface is considered. An existence result for optimal shapes as well as a numerical realization are stated. From several numerical results it can be seen that minimizing the total potential energy of the system leads to an even distribution of contact forces on the contact boundary, even in the cases when the contact involves friction.     

Optimal shape of an elastic structure

This paper treats an optimal design problem of an elastic upright structure which can be regarded as a column. The optimal shape of the structure, which makes the lowest natural frequency as large as possible, is determined under several equality and inequality constraints. A necessary condition for optimality is derived, and a numerical procedure for obtaining the optimal shape is discussed. The piecewise Hermite interpolation technique and the Rayleigh-Ritz method are employed for computing the smallest eigenvalue and the corresponding eigenfunction. Several numerical results are presented.     

A Boundary value technique for solving singularly perturbed,fixed end-point optimal control problems

A method is proposed to solve fixed end-point, linear optimal control problems with quadratic cost and singularly perturbed state. After translating the problem into a two-point boundary value problem, we choose two points t₁, t₂ ? [t₀, t_f] and let τ = (t-t₀)/? and σ = (t_f-t)/?. The τ-scaled, original and σ-scaled boundary value problems are then solved on the intervals [t₀, t₁], [t₁, t₂] and [t₂, t_f] respectively. A test example is solved to illustrate the method.     

Sequential conjugate gradient-restoration algorithm for optimal control problems with non-differential constraints and general boundary conditions,part I

In this paper, a new member of the family of sequential gradient-restoration algorithms for the solution of optimal control problems is presented. This is an algorithm of the conjugate gradient type, which is designed to solve two classes of optimal control problems, called Problem P1 and Problem P2 for easy indentification. Problem P1 involves minimizing a functional I subject to differential constraints and general boundary conditions. It consists of finding the state x (t), the control u (t), and the parameter pi so that the functional I is minimized, while the constraints and the boundary conditions are satisfied to a predetermined accuracy. Problem P2 extends Problem P1 to include non-differential constraints to be satisfied everywhere along the interval of integration. The approach taken is a sequence of two-phase cycles, composed of a conjugate gradient phase and a restoration phase. The conjugate gradient phase involves one iteration and is designed to decrease the value of the functional, while the constraints are satisfied to first order. The restoration phase involves one or more iterations; each restorative iteration is designed to force constraint satisfaction to first order, while the norm squared of the variations of the control, the parameter, and the missing components of the initial state is minimized. The resulting algorithm has several properties: (i) it produces a sequence of feasible solutions; (ii) each feasible solution is characterized by a value of the functional I which is smaller than that associated with any previous feasible solution; and (iii) for the special case of a quadratic functional subject to linear constraints, the variations of the state, control, and parameter produced by the sequence of conjugate gradient phases satisfy various orthogonality and conjugacy conditions. The algorithm presented here differs from those of References 1-4, in that it is not required that the state vector be given at the initial point. Instead, the initial conditions can be absolutely general. In analogy with References 1-4, the present algorithm is capable of handling general final conditions; therefore, it is suitable for the solution of optimal control problems with general boundary conditions. The importance of the present algorithm lies in that many optimal control problems either arise naturally in the present format or can be brought to such a format by means of suitable transformations.⁵ Therefore, a great variety of optimal control problems can be handled. This includes: (i) problems with control equality constraints, (ii) problems with state equality constraints, (iii) problems with state-derivative equality constraints, (iv) problems with-control inequality constraints, (v) problems with state inequality constraints, (vi) problems with state-derivative inequality constraints, and (vii) Chebyshev minimax problems. Several numerical examples are presented in Part 2 (Reference 6) in order to illustrate the performance of the algorithm associated with Problem P1 and Problem P2. The numerical results show the feasibility as well as the convergence characteristics of the present algorithm.     

Sequential conjugate gradient-restoration algorithm for optimal control problems with non-differential constraints and general boundary conditions,part 2

In Reference 1, Wu and Miele developed the sequential conjugate gradient-restoration algorithm for minimizing a functional subject to differential constraints, with or without non-differential constraints, and general boundary conditions. In this paper, several numerical examples are presented. Some of these pertain to a quadratic functional subject to linear constraints, and some pertain to a non-quadratic functional subject to non-linear constraints. These examples demonstrate the feasibility as well as the convergence characteristics of the sequential conjugate gradient-restoration algorithm.     

Optimal boundary impedance for the minimization of reflection: (I) Asymptotic solutions by the geometrical optics method

In the design of surfaces which absorb waves, the impedance boundary condition is used as an effective means of diminishing the reflection. In this paper, we use the geometrical optics method to approximate the optimal impedance value which minimizes the reflected field for the scalar wave equation with a monochromatic source. Our treatment yields good results for optimal impedance in the asymptoticity region of the geometrical optics solution.     

Renewable hydrogen and ammonia for combined heat and power systems in remote locations: Optimal design and scheduling

Using hydrogen (H) and ammonia (NH) for renewable energy storage has the potential to enable economical power and heat supply with high renewable penetrations, especially in remote locations which are characterized by high energy costs. In this work we assess the economic competitiveness of renewable combined heat and power (CHP) systems in Mahaka HI, Nantucket MA, and Northwest Arctic Borough (NWAB) AK by optimally designing these systems for scenarios in which power and heat can be purchased over a range of historical energy prices as well as when 100% renewable supply is required. We use a combined optimal design and scheduling model which minimizes annualized net present cost by determining optimal technology selection and size simultaneously with optimal schedules for each period of a system operating horizon aggregated from full year hourly resolution data via a consecutive temporal clustering algorithm. We find that renewable generation meets at least 85% of power demands and 75% of heat demands under the lowest energy prices investigated. Higher conventional energy prices lead to increased renewable penetration which is facilitated by renewable NH as a seasonal energy storage medium, as are 100% renewable CHP systems. NH is used for power generation with heat cogeneration in all three locations, as well as directly for heating in NWAB. On an annual cost basis, NH-enabled 100% renewable CHP is only 3% more expensive in Mahaka and NWAB than systems which can purchase energy at the lowest prices, while it is 15% more expensive in Nantucket.     

Optimal control problems with delays in state and control variables subject to mixed control–state constraints

Optimal control problems with delays in state and control variables are studied. Constraints are imposed as mixed control–state inequality constraints. Necessary optimality conditions in the form of Pontryagin's minimum principle are established. The proof proceeds by augmenting the delayed control problem to a nondelayed problem with mixed terminal boundary conditions to which Pontryagin's minimum principle is applicable. Discretization methods are discussed by which the delayed optimal control problem is transformed into a large‐scale nonlinear programming problem. It is shown that the Lagrange multipliers associated with the programming problem provide a consistent discretization of the advanced adjoint equation for the delayed control problem. An analytical example and numerical examples from chemical engineering and economics illustrate the results. Copyright © 2008 John Wiley & Sons, Ltd.