Optimal Modification of Shape for Two-Dimensional Elastic Bodies

A variational formulation is presented for shape optimal design of two-dimensional elastic bodies. Optimal modification or remodel of a specified shape is determined for reinforcing (addition of material), lightening (removal of material), or redistribution of material by a specified amount. Necessary conditions are obtained in analytical form to provide the basis for an algorithm to solve the optimal remodel design problem computationally. The method is demonstrated for examples in both torsion and plane stress problems, including cases in which the optimal modification calls for introduction of a hole into an original shape.     

A Study of Two-Dimensional Plane Elasticity Finite Elements for Optimal Design

ABSTRACT

Design of structures using a variety of two-dimensional finite elements is considered in this paper. An efficient technique of computing first-order derivatives of pointwise stress constraints for simple and higher-order two-dimensional (membrane) finite elements is presented. Computational aspects of design sensitivity vector calculation, using a semi-analytical method versus traditional methods are presented. Implementation of a fully-stressed design approach to find a suitable initial estimate leads to increased computational efficiency. These aspects of the design procedure are illustrated through analysis of numerical examples. Experience indicates that a suitable mix of low and higher-order elements yields the most efficient and accurate design model.     

Viscosity Solutions of Dynamic-Programming Equations for the Optimal Control of the Two-Dimensional Navier-Stokes Equations

In this paper we study the infinite-dimensional Hamilton-Jacobi equation associated with the optimal feedback control of viscous hydrodynamics. We resolve the global unique solvability problem of this equation by showing that the value function is the unique viscosity solution.     

Optimal Design in Two-Dimensional Conductivity for a General Cost Depending on the Field

We explore an optimal design problem in two-dimensional conductivity for a rather general cost depending on the underlying field. Through a typical variational reformulation that has been explored recently, we provide a simplified relaxed version which is amenable to numerical simulation, and prove that it is a true relaxation under a main structural hypothesis. Several important cases are covered including a linear cost in the gradient and a convex, isotropic functional (in particular, the pth power of the field for any ). For the isotropic, non-quadratic case, our computations do not require an explicit form of the constrained quasiconvexification of the equivalent vector variational problem. That structural assumption ties together the underlying state equation and the integral cost.     

Discrete Models of Problems in the Elastic Theory of Composites in Cylindrical Coordinates. 2. Two-Dimensional Problems

The base factors and global mesh equations corresponding to classes of basic two-dimensional problems of the elastic theory of composites are constructed in circular cylindrical coordinates (axisymmetric problems in polar coordinates). The domain occupied by the composite may be of arbitrary connectivity and configuration. The difference scheme or the global system of linear algebraic equations that corresponds to any differential problem from the given class of problems may be written explicitly     

A Method for Solving Two-Dimensional Nonlinear Boundary-Value Problems of Magnetoelasticity for Thin Shells

A method is proposed to solve two-dimensional nonlinear problems of magnetoelasticity for thin shells. A problem is formulated and two-dimensional nonlinear equations of magnetoelasticity for current-carrying shells are derived__________Translated from Prikladnaya Mekhanika, Vol. 41, No. 5, pp. 32–39, May 2005.     

Approximation of the Base Pressure and Determination of the Optimal Height of the Rear Face of a Two-Dimensional Afterbody

On the basis of the available experimental and calculated data, approximate relations for determining the base pressure behind the rear face of a two-dimensional body in Mach 0 to 4 flow are derived, the relative thickness of the turbulent boundary layer on the body ranging from 0 to . Using these relations, the optimum afterbody contours giving a two-dimensional body maximum thrust are determined. The rear face heights of these contours are determined for arbitrary afterbody lengths and boundary layer thicknesses at M = 1–4.     

Duality among Optimal Design Problems for Torsional Vibration

ABSTRACT

Four types of mass and frequency optimization problems are stated for free torsional vibration of thin-walled cylinders subject to constraints on wall thickness and frequencies of vibration. It is shown, using Pontryagin's method, that the mathematical structure of all four problems is similar and leads to identical classes of optimal thickness distributions. These duality relations are used in an example to construct an optimal frequency solution from the solutions for both maximum and minimum mass problems. General relations among the governing parameters for the four problems are stated. The results of Grinev and Filippov and of Thermann for the abnormal optimization problems are verfied as a specific limiting example of the general results.     

A General Formulation and Solution Scheme for Fractional Optimal Control Problems

Accurate modeling of many dynamic systems leads to a set of Fractional Differential Equations (FDEs). This paper presents a general formulation and a solution scheme for a class of Fractional Optimal Control Problems (FOCPs) for those systems. The fractional derivative is described in the Riemann–Liouville sense. The performance index of a FOCP is considered as a function of both the state and the control variables, and the dynamic constraints are expressed by a set of FDEs. The Calculus of Variations, the Lagrange multiplier, and the formula for fractional integration by parts are used to obtain Euler–Lagrange equations for the FOCP. The formulation presented and the resulting equations are very similar to those that appear in the classical optimal control theory. Thus, the present formulation essentially extends the classical control theory to fractional dynamic system. The formulation is used to derive the control equations for a quadratic linear fractional control problem. An approach similar to a variational virtual work coupled with the Lagrange multiplier technique is presented to find the approximate numerical solution of the resulting equations. Numerical solutions for two fractional systems, a time-invariant and a time-varying, are presented to demonstrate the feasibility of the method. It is shown that (1) the solutions converge as the number of approximating terms increase, and (2) the solutions approach to classical solutions as the order of the fractional derivatives approach to 1. The formulation presented is simple and can be extended to other FOCPs. It is hoped that the simplicity of this formulation will initiate a new interest in the area of optimal control of fractional systems.     

A General Formulation and Solution Scheme for Fractional Optimal Control Problems

Accurate modeling of many dynamic systems leads to a set of Fractional Differential Equations (FDEs). This paper presents a general formulation and a solution scheme for a class of Fractional Optimal Control Problems (FOCPs) for those systems. The fractional derivative is described in the Riemann–Liouville sense. The performance index of a FOCP is considered as a function of both the state and the control variables, and the dynamic constraints are expressed by a set of FDEs. The Calculus of Variations, the Lagrange multiplier, and the formula for fractional integration by parts are used to obtain Euler–Lagrange equations for the FOCP. The formulation presented and the resulting equations are very similar to those that appear in the classical optimal control theory. Thus, the present formulation essentially extends the classical control theory to fractional dynamic system. The formulation is used to derive the control equations for a quadratic linear fractional control problem. An approach similar to a variational virtual work coupled with the Lagrange multiplier technique is presented to find the approximate numerical solution of the resulting equations. Numerical solutions for two fractional systems, a time-invariant and a time-varying, are presented to demonstrate the feasibility of the method. It is shown that (1) the solutions converge as the number of approximating terms increase, and (2) the solutions approach to classical solutions as the order of the fractional derivatives approach to 1. The formulation presented is simple and can be extended to other FOCPs. It is hoped that the simplicity of this formulation will initiate a new interest in the area of optimal control of fractional systems.     

The Jacobi-Rosochatius Problem on an Ellipsoid: the Lax Representations and Billiards

In this article, we construct the Lax representations of the geodesic flow, the Jacobi-Rosochatius problem and its perturbations by means of separable polynomial potentials on an ellipsoid. We prove complete integrability in the case of a generic symmetric ellipsoid and describe analogous systems on complex projective spaces. Also, we consider billiards within an ellipsoid under the influence of the Hook and Rosochatius potentials between the impacts. A geometric interpretation of the integrability analogous to the classical Chasles and Poncelet theorems is given.     

The Geometry of the Solution Set of Nonlinear Optimal Control Problems

In an optimal control problem one seeks a time-varying input to a dynamical systems in order to stabilize a given target trajectory, such that a particular cost function is minimized. That is, for any initial condition, one tries to find a control that drives the point to this target trajectory in the cheapest way. We consider the inverted pendulum on a moving cart as an ideal example to investigate the solution structure of a nonlinear optimal control problem. Since the dimension of the pendulum system is small, it is possible to use illustrations that enhance the understanding of the geometry of the solution set. We are interested in the value function, that is, the optimal cost associated with each initial condition, as well as the control input that achieves this optimum. We consider different representations of the value function by including both globally and locally optimal solutions. Via Pontryagin’s maximum principle, we can relate the optimal control inputs to trajectories on the smooth stable manifold of a Hamiltonian system. By combining the results we can make some firm statements regarding the existence and smoothness of the solution set.     

Penalty Methods for Numerical Approximations of Optimal Boundary Flow Control Problems

Abstract

This article is concerned with penalty methods for solving optimal Dirichlet control problems governed by the steady-state and time-dependent Navier-Stokes equations. We present, in two different versions, the penalized methods for solving the steady-slate Dirichlet control problems. These approaches are implemented and compared numerically. We also generalize the penalty methods to the time-dependent case. Scmidiscrete and fully discrete approximations of time-dependent Dirichlet control problems are discussed and implemented. Numerical results for solving both the steady-state and the time dependent Dirichlet control problems are reported.     

An Optimal Condition of Compactness for Elasticity Problems Involving one Directional Reinforcement

This paper deals with the homogenization of a homogeneous elastic medium reinforced by very stiff strips in dimension two. We give a general condition linked to the distribution and the stiffness of the strips, under which the nature of the elasticity problem is preserved in the homogenization process. This condition is sharper than the one used in Briane and Camar-Eddine (J. Math. Pures Appl. 88:483–505, 2007) and is shown to be optimal in the case where the strips are periodically arranged. Indeed, a fourth-order derivative term appears in the limit equation as soon as the condition is no more satisfied. In the periodic case the influence of oscillations in the medium surrounding the strips is also considered. The homogenization method is based both on a two-scale convergence for the strips and the use of suitable oscillating test functions. This allows us to obtain a distributional convergence of two of the three entries of the stress tensor contrary to the Γ-convergence approach of Briane and Camar-Eddine (J. Math. Pures Appl. 88:483–505, 2007).     

Optimal Design Problems for Elastic Bodies by Use of the Maximum Principle

A class of optimal design problems for elastic bodies requires minimization of a nonconvex energy functional with respect to both kinematical variables and a design variable. Such functionals for rod, beam, sphere, and cylinder problems are special cases of a single functional. Minimization of this functional using Pontryagin's Maximum Principle to handle the nonconvexity and the inequality constraints leads to the choice of the design variable as taking on only its upper and lower bound values, agreeing with some previous results found for specific problems. A generalization to extensible beam-column problems is discussed.     

Solution of Fixed Final State Optimal Control Problems via Simple Cell Mapping

A strategy is proposed to solve the fixed final state optimalcontrol problem using the simple cell mapping method. A non-uniform timestep simple cell mapping is developed to create a general database fromwhich solutions of various optimal control problems can be obtained. Atwo-stage backward search algorithm is proposed to eliminate degeneratedpaths often associated with the simple cell mapping. The proposed methodcan accurately delineate the switching curves and eliminate false limitcycles in the solution. The method is applied to two optimal controlproblems with bang-bang control. The well-known minimum time controlproblem of moving a point mass from any initial condition to the originof the phase plane is studied first. This example has exact solutionsavailable which provide a yardstick to examine the accuracy of themethod. The cell size dependence of the solution accuracy is studiednumerically. The second example is a variable stiffness feedback controlproblem with tuning range saturation. The strategy proposed is able toprovide the switching curves in the phase plane. This result has notbeen obtained before.     

On the Optimal Location of Singularities Arising in Variational Problems of Nonlinear Elasticity

Experiments on elastomers have shown that triaxial tension can induce a material to exhibit holes that were not previously evident. Analytic work in nonlinear elasticity has established that such cavity formation may indeed be an elastic phenomenon: sufficiently large prescribed boundary deformations yield a hole-creating deformation as the energy minimizer whenever the elastic energy is of slow growth. One of the many unanswered problems is where such holes will form. In this paper we suggest a new method, which is based upon asymptotics and linear elasticity, that can be used to determine the optimal location for hole creation. Using this method we show that, under reasonable hypotheses, the center is (locally) the best position for a solitary hole to form in an elastic ball. This revised version was published online in August 2006 with corrections to the Cover Date.     

Homogenization of Stiff Plates and Two-Dimensional High-Viscosity Stokes Equations

The paper deals with the homogenization of stiff heterogeneous plates. Assuming that the coefficients are equi-bounded in L ¹, we prove that the limit of a sequence of plate equations remains a plate equation which involves a strongly local linear operator acting on the second gradients. This compactness result is based on a div-curl lemma for fourth-order equations. On the other hand, using an intermediate stream function we deduce from the plates case a similar result for high-viscosity Stokes equations in dimension two, so that the nature of the Stokes equation is preserved in the homogenization process. Finally, we show that the L ¹-boundedness assumption cannot be relaxed. Indeed, in the case of the Stokes equation the concentration of one very rigid strip on a line induces the appearance of second gradient terms in the limit problem, which violates the compactness result obtained under the L ¹-boundedness condition.     

Simulations of Spatially Developing Two-Dimensional Shear Layers and Jets

A computational study of spatially evolving two-dimensional free shear flows has been performed using direct numerical simulation of the Navier–Stokes equations in order to investigate the ability of these two-dimensional simulations to predict the overall flow-field quantities of the corresponding three-dimensional “real” turbulent flows. The effects of inflow forcing on these two-dimensional flows has also been studied. Simulations were performed of shear layers, as well as weak (large co-flow and relatively weak shear) and strong (small co-flow and relatively strong shear) jets. Several combinations of discrete forcing with and without a broadband background spectrum were used. Although spatially evolving direct simulations of shear layers have been performed in the past, no such simulations of the plane jet have been performed to the best of our knowledge. It was found that, in the two-dimensional shear layers, external forcing led to a strong increase in the initial growth of the shear-layer thickness, followed by a region of decreased growth as in physical experiments. The final downstream growth rate was essentially unaffected by forcing. The mean velocity profile and the naturally evolving growth rate of the shear layer in the case of broadband forcing compare well with experimental data. However, the total and transverse fluctuation intensities are larger in the two-dimensional simulations with respect to experimental data. In the weak-jet simulations it was found that symmetric forcing completely overwhelms the natural tendency to transition to the asymmetric jet column mode downstream. It was observed that two-dimensional simulations of “strong” jets with a low speed co-flow led to a fundamentally different flow with large differences even in mean velocity profiles with respect to experimental data for planar jets. This was a result of the dominance of the two-dimensional mechanism of vortex dipole ejection in the flow due to the lack of spanwise instabilities. Experimental studies of planar jets do not show vortex dipole formation and ejection. A three-dimensional “strong”-jet simulation showed the rapid evolution of three-dimensionality effectively preventing this two-dimensional mechanism, as expected from experimental results. Received: 25 November 1996 and accepted 17 April 1997     

On Stationary Solutions of Two-Dimensional Euler Equation

We study the geometry of streamlines and stability properties for steady state solutions of the Euler equations for an ideal fluid.