Systematic Refinement Research Articles

Learning by doing: systematic abstraction refinement for hybrid control synthesis

The synthesis of discrete event controllers for given continuous dynamics is studied within the hybrid system theory and its applications. A common approach involves the generation of a discrete abstraction of the continuous plant model, thus transforming the hybrid control problem into a purely discrete one that is then addressable using methods from the discrete event systems theory. In previous work, conditions were derived guaranteeing that successful synthesis on the abstraction level would provide a solution for the underlying hybrid problem. If synthesis failed, however, the abstraction was in need of refinement. This resulted in an iterative procedure alternating abstraction refinement with trial controller synthesis. The authors now use a temporal decomposition of the control problem to extract relevant diagnostic information when the synthesis step fails. In contrast with standard unfocused and global refinement strategies, the new iteration ‘learns by doing’ in the synthesis step and implements a refinement that is tailored to the particular hybrid plant and specification at hand.

Global optimization and modeling techniques for planar multilayered dielectric structures

We present a multilevel global optimization strategy for synthesizing planar multilayered dielectric structures. A low discrepancy sequence of sample points with uniform variable space coverage allows a global-level search while systematic refinement using gradient-based techniques identifies optima at the local level. Since efficient local optimization is important for this method, a fast calculation approach based on mode matching is presented; this also facilitates the compact derivation of analytical gradients. The approach is compared with genetic and simulated annealing algorithms through an antireflection coating design. The method proves to be competitive in terms of its performance, nonadaptive algorithm, and ability to track local solutions.

Using patterns for the refinement and translationof UML models: A controlled experiment

This paper presents a controlled experiment, conducted at the University of Kaiserslautern which evaluates an approach known as SORT, for the systematic refinement and translation of UML Diagrams. Specifically, the experiment investigates the effects of SORT, with respect to the mapping of object-oriented UML design models to source code, by comparing the effects of different approaches to such mappings (SORT and ad-hoc1) on the quality attributes understandability, verifiability, and effort (time). The experimental results demonstrate that OO systems developed by applying SORT are more understandable and verifiable. In summary, SORT can help to improve the quality of software systems, but its application alone does not guarantee quality.

Uncertainties in the simulation of catalytic distillation process: A systematic grid refinement study

Computational fluid dynamics (CFD) methods are gradually finding applications in the simulation of distillation processes. In the discretization of the governing differential equations for the mass- and the energy-balance, finite volume method (FVM) is commonly employed. Upwind differencing scheme (UDS) and central differencing scheme (CDS) are most commonly employed in FVM to approximate the values of the variables on the control volume faces. The problems that arise while using these approximation methods are discussed here. These may be overcome by using a bounded linear deferred correction scheme (LDC). When solutions on more than one grid sizes are available, and the order of the numerical scheme is known, the exact solution may be predicted by using Richardson extrapolation technique. If the order is unknown, then the order of the scheme may be determined from grid refinement studies. However, for the determined order of the scheme to be meaningful, sufficiently fine grids must be used for these studies.

Verification and validation of impinging round jet simulations using an adaptive FEM

AbstractThis paper illustrates the use of an adaptive finite element method as a means of achieving verification of codes and simulations of impinging round jets, that is obtaining numerical predictions with controlled accuracy. Validation of these grid‐independent solution is then performed by comparing predictions to measurements. We adopt the standard and accepted definitions of verification and validation (Technical Report AIAA‐G‐077‐1998, American Institute of Aeronautics and Astronautics, 1998; Verification and Validation in Computational Science and Engineering. Hermosa Publishers: Albuquerque, NM, 1998). Mesh adaptation is used to perform the systematic and rigorous grid refinement studies required for both verification and validation in CFD. This ensures that discrepancies observed between predictions and measurements are due to deficiencies in the mathematical model of the flow. Issues in verification and validation are discussed. The paper presents an example of code verification by the method of manufactured solution. Examples of successful and unsuccessful validation for laminar and turbulent impinging jets show that agreement with experiments is achieved only with a good mathematical model of the flow physics combined with accurate numerical solution of the differential equations. The paper emphasizes good CFD practice to systematically achieve verification so that validation studies are always performed on solid grounds. Copyright © 2004 John Wiley & Sons, Ltd.

Equivariant Epsilon Constants, Discriminants and Étale Cohomology

Let $L/K$ be a finite Galois extension of number fields. We formulate and study a conjectural equality between an element of the relative algebraic K-group $K_0(\mathbb{Z}[\mathrm{Gal}(L/K)], \mathbb{R})$ which is constructed from the equivariant Artin epsilon constant of $L/K$ and a sum of structural invariants associated to $L/K$. The precise conjecture is motivated by the requirement that a special case of the equivariant refinement of the Tamagawa Number Conjecture of Bloch and Kato (as formulated by Flach and the second-named author) should be compatible with the functional equation of the associated L-function. We show that, more concretely, our conjecture also suggests a completely systematic refinement of the central approach and results of classical Galois module theory. In particular, the evidence for our conjecture that we present here already strongly refines many of the main results of Galois module theory.

Multiple Scales via Galerkin Projections: Approximate Asymptotics for Strongly Nonlinear Oscillations

The method of multiple scales and the related method of averaging are commonly used to study slowly modulated oscillations. If the system of interest is a slightly perturbed harmonic oscillator, then these techniques can be applied easily. If the unperturbed system is strongly nonlinear (though possibly conservative), then these methods can run into difficulties due to the impossibility of carrying out required analytical operations in closed form. In this paper, we abandon the requirement of closed form analytical treatment at all stages. Instead, Galerkin projections are used to obtain approximate realizations of the method of multiple scales. This paper adapts recent work using similar ideas for approximate realizations of the method of averaging. A key contribution of the present work is in the systematic identification and removal of secular terms in the general nonlinear case, a procedure that is more difficult than for the perturbed harmonic oscillator case, and that is unnecessary for averaging. A strength of the present work is that the heuristics (Galerkin) and asymptotics (multiple scales) are kept distinct, leaving room for systematic refinement of the former without compromising the asymptotic features of the latter.

Sloshing in Rectangular and Cylindrical Tanks

To validate an existing finite volume computational method, featuring a novel scheme to capture the temporal evolution of the free surface, fluid motions in partially filled tanks were simulated. The purpose was to compare computational and experimental results for test cases where measurements were available. Investigations comprised sloshing in a rectangular tank with a baffle at 60% filling level and in a cylindrical tank at 50% filling level. The numerical study started with examining effects of systematic grid refinement and concluded with examining effects of three-dimensionality and effects of variation of excitation period and amplitude. Predicted time traces of pressures and forces compared favorably with measurements.

Adsorption of Water in Activated Carbons: Effects of Pore Blocking and Connectivity

We present a simulation study of the adsorption mechanism of water in a realistic carbon structure. Water molecules are modeled using a recently developed fixed-point charge water model optimized to the vapor−liquid coexistence properties.1 Reverse Monte Carlo techniques2 are used to generate a realistic porous carbon model composed of graphitic microcrystals consisting of rigid basal plates. Arrangements of the carbon plates are driven by a systematic refinement of simulated carbon−carbon radial distribution functions to match experimentally measured radial distribution functions. The adsorption of water in activated (having oxygenated surface groups) and nonactivated (graphitic) carbon is investigated using the grand canonical Monte Carlo simulation method. The adsorption behavior is found to be strongly dependent on the presence of activated sites. No appreciable adsorption occurs in the graphitic carbon until the pressure approaches the bulk gas saturation pressure. Effects of the surface site density, surface site position, and the magnitude of the surface site electrostatic charge on the adsorption behavior are investigated. Water adsorption is found to increase as the activated site density increases. The presence of water vapor in porous carbons is shown to dramatically affect the connectivity of the available pore space. Finally, cluster calculations are performed to investigate the growth mechanism of water clusters in activated carbons.

Code Verification by the Method of Manufactured Solutions

Verification of Calculations involves error estimation, whereas Verification of Codes involves error evaluation, from known benchmark solutions. The best benchmarks are exact analytical solutions with sufficiently complex solution structure; they need not be realistic since Verification is a purely mathematical exercise. The Method of Manufactured Solutions (MMS) provides a straightforward and quite general procedure for generating such solutions. For complex codes, the method utilizes Symbolic Manipulation, but here it is illustrated with simple examples. When used with systematic grid refinement studies, which are remarkably sensitive, MMS produces strong Code Verifications with a theorem-like quality and a clearly defined completion point.

On the use of many-body perturbation theory and quantum-electrodynamics in molecular electronic structure theory

Many-body perturbation theory is firmly established in molecular electronic structure studies both as a method of calculation in its own right and in underpinning other approaches to the electron correlation problem, such as coupled electron pair approximations and various cluster expansions. The central pillar of the many-body perturbation theory is the linked diagram theorem obtained by Goldstone using the method of Feynman graphs to enumerate the terms of the perturbation series. Feynman had introduced his graphs in his formulation of quantum electrodynamics. Goldstone restricted his analysis to the non-relativistic many-body problem, so that interactions were taken to be instantaneous and the effects of relativity were ignored. In recent years, the growing interest in the treatment of relativistic and quantum electrodynamic effects in atoms and molecules has necessitated the re-introduction of physics that has been known for over forty years. A key to this development for molecular systems is a rigorous and robust implementation of the algebraic approximation for Dirac and Dirac-like equations. The algebraic approximation provides a representation of both the positive-energy and the negative-energy branches of the Dirac spectrum. Relativistic many-body perturbation theory, relativistic coupled pair approximations and relativistic coupled cluster theories can be formulated within the ‘no-virtual-pair’ approximation. Such formulations are restricted to the positive energy branch of the spectrum. However, the negative energy states make an essential contribution to the description of atomic and molecular electronic structure. The investigation of this contribution is facilitated by proper implementation of the algebraic approximation using formulations which are amenable to systematic refinement.

Error Corrections For X-RAY Powder Diffractometry

The errors involved in precision lattice parameter determination fall into three broad groups: errors that can be corrected by applying a multiplication factor to the determined parameter(s), errors arising from a constant shift in peak position, and errors arising from a Bragg angle dependent shift in peak position. The latter two sources of error can be corrected by applying analytical techniques to the lattice parameters determined from the measured Bragg peaks. The time and effort involved in this process can be greatly reduced by using computational methods based on extrapolation, least squares analysis with systematic error refinement and pure lattice refinement techniques. Programs for these purposes were evaluated using the Si and LaB6 crystallographic standards certified by NIST. A computer extrapolation method utilizing the function coscotq gave lattice parameters with a precision of 1:100,000 for Bragg peaks at angles greater than 60 °q, but significantly lower levels of precision were obtained when lower angle Bragg peaks were included in the extrapolation. Low levels of precision were obtained when either corrected or uncorrected peak positions were analyzed by least squares in combination with a refinement of selected systematic errors. By contrast, a lattice refinement method with no facility for dealing with systematic errors yielded lattice parameters with a precision better than 1:100,000 regardless of whether any prior corrections had been applied to eliminate errors in peak position. An even greater precision of 1:500,000 was obtained when this lattice refinement method was applied to peak positions corrected by a second order polynomial fit to data from an internal standard.Les erreurs impliquées dans la détermination précise du paramètre de réseau se regroupent en trois grandes catégories, i.e., les erreurs qui peuvent être corrigées en appliquant un facteur de multiplication au(x) paramètre(s) déterminé(s), les erreurs qui proviennent d'un déplacement constant de la position du pic et les erreurs qui proviennent d'un déplacement de la position du pic qui dépend de l'angle de Bragg. Ces deux dernières sources d'erreurs peuvent être corrigées en appliquant des techniques analytiques aux paramètres du réseau déterminés à partir de la mesure des pics de Bragg. Le temps et l'effort impliqués dans ce procédé peuvent être grandement réduits en utilisant des méthodes de calcul basées sur l'extrapolation, l'analyse des moindres carrés avec raffinement de l'erreur systématique et des techniques de raffinement du réseau pur. On a évalué des logiciels ayant ces objectifs, en utilisant des étalons cristallographiques de Si et LaB6 certifiés par NIST. Une méthode d'extrapolation par ordinateur utilisant la fonction coscotq a donné des paramètres de réseau ayant une précision de 1:100,000 pour les pics de Bragg ayant des angles de plus de 60 °q. Cependant, on a obtenu des niveaux de précision significativement moins élevés lorsque l'on incluait dans l'extrapolation les pics de Bragg à angles moins élevés. On obtenait de faibles niveaux de précision quand la position des pics, corrigée ou non, était analysée par les moindres carrés en combinaison avec un raffinement d'erreurs systématiques choisies. En contraste, une méthode de raffinement du réseau, sans capacité de considération des erreurs systématiques, a produit des paramètres de réseau ayant une précision meilleure que 1:100,000, sans considération du fait que des corrections antérieures aient, ou non, été appliquées pour éliminer les erreurs de position des pics. On a obtenu une précision encore plus grande de 1:500,000 lorsque cette méthode de raffinement du réseau était appliquée à la position des pics corrigée par l'ajustement d'un polynôme du second ordre aux données d'un étalon interne.

Crystal structure of the monoclinic perovskite Sr3.94Ca1.31Bi2.70O12

The crystal structure of the low-temperature oxidized form of Sr49.5Ca16.5Bi34O151 has been determined using a combination of neutron, synchrotron, and laboratory X-ray powder diffraction data. The structure is pseudo-orthorhombic; systematic absences and successful refinement indicated the true structure to be monoclinic, with space group P2l/n. Structural refinement using only neutron powder data yielded the lattice parameters a=8.38 898(29) Å, b=5.99 334(21) Å, c=5.89 586(20) Å, β=89.997(8)°, and V=296.43(3) Å3. This compound is a distorted perovskite phase [described in the perovskite ABO3 formula as Sr(Bi0.7Ca0.3)O3] with ordering of the M-site cations, resulting in the formula A2MM′O6. In this ordered structure, the A sites are solely occupied by Sr, the M sites mainly by Bi, while on the M′ sites Bi and Ca are distributed in an approximate ratio of 2:3. The MO6 and M′O6 octahedra share corners, and are tilted with respect to the neighboring layers with an angle of ∼15° around all three axes. The tilt system symbol is a+a−a− according to Glazer notation. All Bi ions are in the 5+ oxidation state.

Requirements for Mesh Resolution in 3D Computational Hemodynamics

Computational techniques are widely used for studying large artery hemodynamics. Current trends favor analyzing flow in more anatomically realistic arteries. A significant obstacle to such analyses is generation of computational meshes that accurately resolve both the complex geometry and the physiologically relevant flow features. Here we examine, for a single arterial geometry, how velocity and wall shear stress patterns depend on mesh characteristics. A well-validated Navier-Stokes solver was used to simulate flow in an anatomically realistic human right coronary artery (RCA) using unstructured high-order tetrahedral finite element meshes. Velocities, wall shear stresses (WSS), and wall shear stress gradients were computed on a conventional "high-resolution" mesh series (60,000 to 160,000 velocity nodes) generated with a commercial meshing package. Similar calculations were then performed in a series of meshes generated through an adaptive mesh refinement (AMR) methodology. Mesh-independent velocity fields were not very difficult to obtain for both the conventional and adaptive mesh series. However, wall shear stress fields, and, in particular, wall shear stress gradient fields, were much more difficult to accurately resolve. The conventional (nonadaptive) mesh series did not show a consistent trend towards mesh-independence of WSS results. For the adaptive series, it required approximately 190,000 velocity nodes to reach an r.m.s. error in normalized WSS of less than 10 percent. Achieving mesh-independence in computed WSS fields requires a surprisingly large number of nodes, and is best approached through a systematic solution-adaptive mesh refinement technique. Calculations of WSS, and particularly WSS gradients, show appreciable errors even on meshes that appear to produce mesh-independent velocity fields.

Convective film cooling over a representative turbine blade leading-edge

Computations are performed to simulate a discrete hole film cooling flow over an experimental test geometry representative of the leading edge of turbine blades. A multiblock pressure correction algorithm is used for the computations, and both low-Reynolds number and wall function k–ε models are used for turbulence closure. The flow through the coolant ducts, from the plenum to the blade surface, is resolved as a part of the computation by specifying the coolant mass flux in the plenum. A systematic grid refinement study is conducted with the finest grid consisting of approximately one million points. Next, the flowfield is examined ; key physical mechanisms resulting from the interactions between the cooling jets and the freestream are identified and their effect on the thermal field is compared with the experimentally observed thermal field. Finally, a study of geometric parametric variation is conducted to optimize the film cooling design. Nine different combinations of two parameters, namely, the relative stagger and the relative angle between the two rows of cooling holes are investigated for their effect on heat transfer on the blade surface.

Uniform Closures: Order-Theoretically Reconstructing Logic Program Semantics and Abstract Domain Refinements

The notion of uniform closure operator is introduced, and it is shown how this concept surfaces in two different areas of application of abstract interpretation, notably in semantics design for logic programs and in the theory of abstract domain refinements. In logic programming, uniform closures permit generalization, from an order-theoretic perspective, of the standard hierarchy of declarative semantics. In particular, we show how to reconstruct the model-theoretic characterization of the well-known s-semantics using pure order-theoretic concepts only. As far as the systematic refinement operators on abstract domains are concerned, we show that uniform closures capture precisely the property of a refinement of being invertible, namely of admitting a related operator that simplifies as much as possible a given abstract domain of input for that refinement. Exploiting the same argument used to reconstruct the s-semantics of logic programming, we yield a precise relationship between refinements and their inverse operators: we demonstrate that they form an adjunction with respect to a conveniently modified complete order among abstract domains.

Citizen Participation in the Reform of Health Care Policy: A Case Example

The trend toward greater citizen participation in health care policy reform has its roots in the consumerism of the 1960s. This era witnessed the beginning of a dispersion of power in health care and an increase in the number and variety of stakeholders involved in the policy development process. Using the reform Ontario's long-term care policy as a case example, this paper offers observations about the benefits and challenges of participative policy-making. Despite the challenges and the paucity of hard evidence pointing to benefits, the author concludes that broad citizen participation in health care policy reform is a desirable goal. However, the capacity for genuine collaboration remains underdeveloped and requires more systematic refinement.

The Semi-Lattice of Piecewise Constant Controls for Non-Linear Systems: A Possible Foundation for Fuzzy Control

An examination of the literature on fuzzy control systems shows that the design of fuzzy controllers depends upon systematic refinement of a sequence of piecewise constant control laws rather than the use of results of the subject going under the title of fuzzy logic. In this paper, the lattice structure of piecewise constant stabilizing controls, with its natural refinement operation induced by a partial ordering ≤ and an associated meet operation π, is proposed as a foundation for the selection of control laws in fuzzy controllers.

Design and verification of the Rollback Chip using HOP

The use of formal methods in hardware design improves the quality of designs in many ways: it promotes better understanding of the design; it permits systematic design refinement through the discovery of invariants; and it allows design verification (informal or formal). In this paper we illustrate the use of formal methods in the design of a custom hardware system called the “Rollback Chip” (RBC), conducted using a simple hardware design description language called “HOP”. An informal specification of the requirements of the RBC is first given, followed by a behavioral description of the RBC stating its desired behavior . The behavioral description is refined into progressively more efficient designs, terminating in a structural description . Key refinement steps are based on system invariants that are discovered during the design, and proved correct during design verification. The first step in design verification is to apply a program called PARCOMP to derive a behavioral description from the structural description of the RBC. The derived behavior is then compared against the desired behavior using equational verification techniques. This work demonstrates that formal methods can be fruitfully applied to a nontrivial hardware design. It also illustrates the particular advantages of our approach based on HOP and PARCOMP. Last, but not the least, it formally verifies the RBC mechanism itself.

Efficient calculation of rovibrational eigenstates of sequentially bonded four-atom molecules

A new efficient and fully symmetry-adapted finite-basis variational method for calculating the rovibrational eigenstates (J≥0) of any sequentially bonded four-atom molecule is presented. The exact kinetic-energy operator T̂VR in valence coordinates is used in a scheme of successive basis-set contractions. The success of the method is demonstrated with new results for the molecules HCCH, HOOH, and HCNO, respectively linear, nonlinear, and quasilinear. The complexity of T̂VR contributes little to the computational cost, yielding a Hamiltonian matrix whose elements can all be cheaply calculated from products of arbitrarily accurate one-dimensional integrals; the dominant cost is that of matrix diagonalization. Matrices of up to 6000×6000 are used to obtain low-lying levels converged to small fractions of 1 cm−1, even for the difficult case of HOOH. The generalization to J≥0 allows the calculation of Π states for HCCH and HCNO and effective rotational constants for HCCH, all usefully converged. For the first time nine-dimensional rovibrational wave functions for four-atom systems are calculated without dynamical approximation and with basis sets well completed in all degrees of freedom. For HCCH it is straightforward to obtain hundreds of rovibrational (J=0,1,2) levels converged to better than 1.5 cm−1, opening up the possibility of the systematic refinement of the nuclear potential function V̂N.