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At least 19 records

A piecewise linear approximation scheme for hereditary optimal control problems

An approximation scheme based on 'piecewise linear' approximations of L2 spaces is employed to formulate a numerical method for solving quadratic optimal control problems governed by linear retarded functional differential equations. This piecewise linear method is an extension of the so called averaging technique. It is shown that the Riccati equation for the linear approximation is solved by simple transformation of the averaging solution. Thus, the computational requirements are essentially the same. Numerical results are given.

Cliff, E. M.↗

Piecewise linear approximation for hereditary control problems

Finite dimensional approximations are presented for linear retarded functional differential equations by use of discontinuous piecewise linear functions. The approximation scheme is applied to optimal control problems when a quadratic cost integral has to be minimized subject to the controlled retarded system. It is shown that the approximate optimal feedback operators converge to the true ones both in case the cost integral ranges over a finite time interval as well as in the case it ranges over an infinite time interval. The arguments in the latter case rely on the fact that the piecewise linear approximations to stable systems are stable in a uniform sense. This feature is established using a vector-component stability criterion in the state space R(n) x L(2) and the favorable eigenvalue behavior of the piecewise linear approximations.

Propst, Georg↗

Piecewise linear approximation for hereditary control problems

This paper presents finite-dimensional approximations for linear retarded functional differential equations by use of discontinuous piecewise linear functions. The approximation scheme is applied to optimal control problems, when a quadratic cost integral must be minimized subject to the controlled retarded system. It is shown that the approximate optimal feedback operators converge to the true ones both in the case where the cost integral ranges over a finite time interval, as well as in the case where it ranges over an infinite time interval. The arguments in the last case rely on the fact that the piecewise linear approximations to stable systems are stable in a uniform sense.

Propst, Georg↗

Finite element mesh configurations using isoenergetics and equalized energy levels

The concept of equalizing energy levels was shown to be a viable additional criterion in laying out finite element grids according to the isoenergetic discretization technique. Similar problems specifically with respect to mesh refinement in piecewise approximation theory are being researched. Common criteria in both areas are developed in an effort to cope with the question of discretization for improved piecewise approximations.

Turcke, D. J.↗

Approximate Analysis of Semiconductor Laser Arrays

Simplified equation yields useful information on gains and output patterns. Theoretical method based on approximate waveguide equation enables prediction of lateral modes of gain-guided planar array of parallel semiconductor lasers. Equation for entire array solved directly using piecewise approximation of index of refraction by simple functions without customary approximation based on coupled waveguid modes of individual lasers. Improved results yield better understanding of laser-array modes and help in development of well-behaved high-power semiconductor laser arrays.

Marshall, William K.↗

Evaluation of digitally corrected ERTS imagery

Utilizing all digital processing techniques, precision rectified ERTS multispectral imagery have been produced. Precision geometric correction is accomplished by: (1) utilizing a low order piecewise approximation to the image distortions; (2) incorporating spacecraft attitude refinement derived from ground control points utilized in a Kalman filter; (3) utilizing interpolation techniques which preserve value and slope continuity in the image data. Imagery produced is represented in a UTM projection, and has been evaluated for precision by statistical and analytical methods. Extrapolations to modest size computers indicate good throughput with no sacrifice in precision.

Rifman, S. S.↗

A new approach in cascade flow analysis using the finite element method

A new approach in analyzing the potential flow past cascades and single airfoils using the finite element method is developed. In this analysis the circulation around the airfoil is not externally imposed but is directly computed in the numerical solution. Different finite element discretization patterns, orders of piecewise approximation, and grid sizes are used in the solution. The results obtained are compared with existing experimental measurements and exact solutions in cascades and single airfoils.

Baskharone, E.↗

Optimization of Car Body under Constraints of Noise, Vibration, and Harshness (NVH), and Crash

To be competitive on the today's market, cars have to be as light as possible while meeting the Noise, Vibration, and Harshness (NVH) requirements and conforming to Government-man dated crash survival regulations. The latter are difficult to meet because they involve very compute-intensive, nonlinear analysis, e.g., the code RADIOSS capable of simulation of the dynamics, and the geometrical and material nonlinearities of a thin-walled car structure in crash, would require over 12 days of elapsed time for a single design of a 390K elastic degrees of freedom model, if executed on a single processor of the state-of-the-art SGI Origin2000 computer. Of course, in optimization that crash analysis would have to be invoked many times. Needless to say, that has rendered such optimization intractable until now. The car finite element model is shown. The advent of computers that comprise large numbers of concurrently operating processors has created a new environment wherein the above optimization, and other engineering problems heretofore regarded as intractable may be solved. The procedure, shown, is a piecewise approximation based method and involves using a sensitivity based Taylor series approximation model for NVH and a polynomial response surface model for Crash. In that method the NVH constraints are evaluated using a finite element code (MSC/NASTRAN) that yields the constraint values and their derivatives with respect to design variables. The crash constraints are evaluated using the explicit code RADIOSS on the Origin 2000 operating on 256 processors simultaneously to generate data for a polynomial response surface in the design variable domain. The NVH constraints and their derivatives combined with the response surface for the crash constraints form an approximation to the system analysis (surrogate analysis) that enables a cycle of multidisciplinary optimization within move limits. In the inner loop, the NVH sensitivities are recomputed to update the NVH approximation model while keeping the Crash response surface constant. In every outer loop, the Crash response surface approximation is updated, including a gradual increase in the order of the response surface and the response surface extension in the direction of the search. In this optimization task, the NVH discipline has 30 design variables while the crash discipline has 20 design variables. A subset of these design variables (10) are common to both the NVH and crash disciplines. In order to construct a linear response surface for the Crash discipline constraints, a minimum of 21 design points would have to be analyzed using the RADIOSS code. On a single processor in Origin 2000 that amount of computing would require over 9 months! In this work, these runs were carried out concurrently on the Origin 2000 using multiple processors, ranging from 8 to 16, for each crash (RADIOSS) analysis. Another figure shows the wall time required for a single RADIOSS analysis using varying number of processors, as well as provides a comparison of 2 different common data placement procedures within the allotted memories for each analysis. The initial design is an infeasible design with NVH discipline Static Torsion constraint violations of over 10%. The final optimized design is a feasible design with a weight reduction of 15 kg compared to the initial design. This work demonstrates how advanced methodology for optimization combined with the technology of concurrent processing enables applications that until now were out of reach because of very long time-to-solution.

Kodiyalam, Srinivas↗

Offset truss hex solar concentrator

A solar energy concentrator system comprises an offset reflector structure made up of a plurality of solar energy reflector panel sections interconnected with one another to form a piecewise approximation of a portion of a (parabolic) surface of revolution rotated about a prescribed focal axis. Each panel section is comprised of a plurality of reflector facets whose reflective surfaces effectively focus reflected light to preselected surface portions of the interior sidewall of a cylindrically shaped solar energy receiver. The longitudinal axis of the receiver is tilted at an acute angle with respect to the optical axis such that the distribution of focussed solar energy over the interior surface of the solar engine is optimized for dynamic solar energy conversion. Each reflector panel section comprises a flat, hexagonally shaped truss support framework and a plurality of beam members interconnecting diametrically opposed corners of the hexagonal framework recessed within which a plurality of (spherically) contoured reflector facets is disposed. The depth of the framework and the beam members is greater than the thickness of a reflector facet such that a reflector facet may be tilted (for controlling the effective focus of its reflected light through the receiver aperture) without protruding from the panel section.

White, John E.↗

On High-Order Upwind Methods for Advection

In the fourth installment of the celebrated series of five papers entitled "Towards the ultimate conservative difference scheme", Van Leer (1977) introduced five schemes for advection, the first three are piecewise linear, and the last two, piecewise parabolic. Among the five, scheme I, which is the least accurate, extends with relative ease to systems of equations in multiple dimensions. As a result, it became the most popular and is widely known as the MUSCL scheme (monotone upstream-centered schemes for conservation laws). Schemes III and V have the same accuracy, are the most accurate, and are closely related to current high-order methods. Scheme III uses a piecewise linear approximation that is discontinuous across cells, and can be considered as a precursor of the discontinuous Galerkin methods. Scheme V employs a piecewise quadratic approximation that is, as opposed to the case of scheme III, continuous across cells. This method is the basis for the on-going "active flux scheme" developed by Roe and collaborators. Here, schemes III and V are shown to be equivalent in the sense that they yield identical (reconstructed) solutions, provided the initial condition for scheme III is defined from that of scheme V in a manner dependent on the CFL number. This equivalence is counter intuitive since it is generally believed that piecewise linear and piecewise parabolic methods cannot produce the same solutions due to their different degrees of approximation. The finding also shows a key connection between the approaches of discontinuous and continuous polynomial approximations. In addition to the discussed equivalence, a framework using both projection and interpolation that extends schemes III and V into a single family of high-order schemes is introduced. For these high-order extensions, it is demonstrated via Fourier analysis that schemes with the same number of degrees of freedom 𝐾 per cell, in spite of the different piecewise polynomial degrees, share the same sets of eigenvalues and thus, have the same stability and accuracy. Moreover, these schemes are accurate to order 2𝐾−1, which is higher than the expected order of 𝐾.

high-order methods↗

Global Optimization via Quadratic Disjunctive Programming for Water Networks Design with Energy Recovery

Generalized disjunctive programming (GDP) models with bilinear and concave constraints, often seen in water network design, are challenging optimization problems. This work proposes quadratic and piecewise linear approximations for nonlinear terms to reformulate GDP models into quadratic GDP (QGDP) models that suitable solvers may solve more efficiently. We illustrate the benefits of the quadratic reformulation with a water treatment network design problem in which nonconvexities arise from bilinear terms in the mixers’ mass balances and concave investment cost functions of treatment units. Given the similarities with water network design problems, we suggest quadratic approximation for the GDP model for the optimal design of a large-scale reverse electrodialysis (RED) process. This power technology can recover energy from salinity differences between by-product streams of the water sector, such as desalination brine mixed with regenerated wastewater effluents. The solver Gurobi excels in handling QGDP problems, but weighing the problem’s precision and tractability balance is crucial. The piecewise linear approximation yields more accurate, yet larger QGDP models that may require longer optimization times in large-scale process synthesis problems.

Water Networks↗

On approximating hereditary dynamics by systems of ordinary differential equations

The paper deals with methods of obtaining approximate solutions to linear retarded functional differential equations (hereditary systems). The basic notion is to project the infinite dimensional space of initial functions for the hereditary system onto a finite dimensional subspace. Within this framework, two particular schemes are discussed. The first uses well-known piecewise constant approximations, while the second is a new method based on piecewise linear approximating functions. Numerical results are given.

Cliff, E. M.↗

A Theoretical and Computational Revisit of Conversions Between Whitham’s F-Function and Equivalent Area

This paper proves mathematically that the integral transforms between Whitham’s F-function and equivalent area are the inverse transforms of each other if and only if the slope of the equivalent area at the origin is zero. This mathematical fact contradicts the accepted unconditional inverse relation between Whitham’s F-function and equivalent area in the sonic boom research literature. Piecewise linear approximations of an F-function and of the second derivative of an equivalent area are used to derive numerical formulas for conversions between Whitham’s F-function and equivalent area. Numerical results are included to show convergence of the numerical conversions as the maximum length of the segments for piecewise linear approximations goes to zero. These numerical conversions are approximately the inverse transforms of each other when the second derivative of an equivalent area is continuous and the slope of the equivalent area at the origin is zero.

Whitham's F-function↗

Elastic interaction of a crack with a microcrack array. I - Formulation of the problem and general form of the solution. II - Elastic solution for two crack configurations (piecewise constant and linear approximations)

The elastic interactions of a two-dimensional configuration consisting of a crack with an array of microcracks located near the tip are studied. The general form of the solution is based on the potential representations and approximations of tractions on the microcracks by polynomials. In the second part, the technique is applied to two simple two-dimensional configurations involving one and two microcracks. The problems of stress shielding and stress amplification (the reduction or increase of the effective stress intensity factor due to the presence of microcracks) are discussed, and the refinements introduced by higher order polynomial approximations are illustrated.

Chudnovsky, A.↗

Piecewise-Planar Parabolic Reflectarray Antenna

The figure shows a dual-beam, dualpolarization Ku-band antenna, the reflector of which comprises an assembly of small reflectarrays arranged in a piecewise- planar approximation of a parabolic reflector surface. The specific antenna design is intended to satisfy requirements for a wide-swath spaceborne radar altimeter, but the general principle of piecewise-planar reflectarray approximation of a parabolic reflector also offers advantages for other applications in which there are requirements for wideswath antennas that can be stowed compactly and that perform equally in both horizontal and vertical polarizations. The main advantages of using flat (e.g., reflectarray) antenna surfaces instead of paraboloidal or parabolic surfaces is that the flat ones can be fabricated at lower cost and can be stowed and deployed more easily. Heretofore, reflectarray antennas have typically been designed to reside on single planar surfaces and to emulate the focusing properties of, variously, paraboloidal (dish) or parabolic antennas. In the present case, one approximates the nominal parabolic shape by concatenating several flat pieces, while still exploiting the principles of the planar reflectarray for each piece. Prior to the conception of the present design, the use of a single large reflectarray was considered, but then abandoned when it was found that the directional and gain properties of the antenna would be noticeably different for the horizontal and vertical polarizations.

Hodges, Richard↗

The Relation of Finite Element and Finite Difference Methods

Finite element and finite difference methods are examined in order to bring out their relationship. It is shown that both methods use two types of discrete representations of continuous functions. They differ in that finite difference methods emphasize the discretization of independent variable, while finite element methods emphasize the discretization of dependent variable (referred to as functional approximations). An important point is that finite element methods use global piecewise functional approximations, while finite difference methods normally use local functional approximations. A general conclusion is that finite element methods are best designed to handle complex boundaries, while finite difference methods are superior for complex equations. It is also shown that finite volume difference methods possess many of the advantages attributed to finite element methods.

Vinokur, M.↗

Helicopter rotor loads using discretized matched asymptotic expansions

The numerical practicality of a matched asymptotic expansion approach for the computation of unsteady three dimensional airloads on a helicopter rotor was improved. This effort utilizes a discretized repesentation of the doublet strength distribution and helical streamlines to decrease the computational requirements of the original analysis. The continuous variation of the doublet strength was approximated by piecewise constant or piecewise quadratic distributions, and the helical trajectory of a fluid particle was approximated by connected straight line segments. As a direct result of these simplified representations the computational time required for the execution of a typical flight condition was reduced by an order of magnitude with respect to the requirements of the original analysis. Airloads which were computed using the discretized method for a two bladed model rotor and a full scale four bladed rotor are in close agreement with measured results and airloads from the original asymptotic analysis. For conditions characterized by significant rotor/wake interaction the piecewise constant representation requires a reduced azimuth spacing to maintain acceptable accuracy.

Pierce, G. A.↗

Difference equation state approximations for nonlinear hereditary control problems

Discrete approximation schemes for the solution of nonlinear hereditary control problems are constructed. The methods involve approximation by a sequence of optimal control problems in which the original infinite dimensional state equation has been approximated by a finite dimensional discrete difference equation. Convergence of the state approximations is argued using linear semigroup theory and is then used to demonstrate that solutions to the approximating optimal control problems in some sense approximate solutions to the original control problem. Two schemes, one based upon piecewise constant approximation, and the other involving spline functions are discussed. Numerical results are presented, analyzed and used to compare the schemes to other available approximation methods for the solution of hereditary control problems.

Rosen, I. G.↗