Localized construction of curved surfaces from polygon meshes: A simple and practical approach on GPU

We present a method for refining n-sided polygons on a given piecewise linear model by using local computation, where the curved polygons generated by our method interpolate the positions and normals of vertices on the input model. Firstly, we construct a Bézier curve for each silhouette edge. Secondly, we employ a new method to obtain C¹ continuous cross-tangent functions that are constructed on these silhouette curves. An important feature of our method is that the cross tangent functions are produced solely by their corresponding facet parameters. Gregory patches can therefore be locally constructed on every polygon while preserving G¹ continuity between neighboring patches. To provide a flexible shape control, several local schemes are provided to modify the cross-tangent functions so that the sharp features can be retained on the resultant models. Because of the localized construction, our method can be easily accelerated by graphics hardware and fully run on the Graphics Processing Unit (GPU).     

Triangular bubble spline surfaces

We present a new method for generating a Gⁿ-surface from a triangular network of compatible surface strips. The compatible surface strips are given by a network of polynomial curves with an associated implicitly defined surface, which fulfill certain compatibility conditions. Our construction is based on a new concept, called bubble patches, to represent the single surface patches. The compatible surface strips provide a simple Gⁿ-condition between two neighboring bubble patches, which are used to construct surface patches, connected with Gⁿ-continuity. For n≤2, we describe the obtained Gⁿ-condition in detail. It can be generalized to any n≥3. The construction of a single surface patch is based on Gordon–Coons interpolation for triangles.Our method is a simple local construction scheme, which works uniformly for vertices of arbitrary valency. The resulting surface is a piecewise rational surface, which interpolates the given network of polynomial curves. Several examples of G⁰, G¹ and G²-surfaces are presented, which have been generated by using our method. The obtained surfaces are visualized with reflection lines to demonstrate the order of smoothness.     

Multi-degree reduction of Bézier curves using reparameterization

L₂-norms are often used in the multi-degree reduction problem of Bézier curves or surfaces. Conventional methods on curve cases are to minimize , where and are the given curve and the approximation curve, respectively. A much better solution is to minimize , where is the closest point to point , that produces a similar effect as that of the Hausdorff distance. This paper uses a piecewise linear function L(t) instead of t to approximate the function φ(t) for a constrained multi-degree reduction of Bézier curves. Numerical examples show that this new reparameterization-based method has a much better approximation effect under Hausdorff distance than those of previous methods.     

Loop subdivision surfaces interpolating -spline curves

This paper presents a novel method for defining a Loop subdivision surface interpolating a set of popularly-used cubic B-spline curves. Although any curve on a Loop surface corresponding to a regular edge path is usually a piecewise quartic polynomial curve, it is found that the curve can be reduced to a single cubic B-spline curve under certain constraints of the local control vertices. Given a set of cubic B-spline curves, it is therefore possible to define a Loop surface interpolating the input curves by enforcing the interpolation constraints. In order to produce a surface of local or global fair effect, an energy-based optimization scheme is used to update the control vertices of the Loop surface subjecting to curve interpolation constraints, and the resulting surface will exactly interpolate the given curves. In addition to curve interpolation, other linear constraints can also be conveniently incorporated. Because both Loop subdivision surfaces and cubic B-spline curves are popularly used in engineering applications, the curve interpolation method proposed in this paper offers an attractive and essential modeling tool for computer-aided design.     

Extensions of the witness method to characterize under-, over- and well-constrained geometric constraint systems

This paper describes new ways to tackle several important problems encountered in geometric constraint solving, in the context of CAD, and which are linked to the handling of under- and over-constrained systems. It presents a powerful decomposition algorithm of such systems.Our methods are based on the witness principle whose theoretical background is recalled in a first step. A method to generate a witness is then explained. We show that having a witness can be used to incrementally detect over-constrainedness and thus to compute a well-constrained boundary system. An algorithm is introduced to check if anchoring a given subset of the coordinates brings the number of solutions to a finite number.An algorithm to efficiently identify all maximal well-constrained parts of a geometric constraint system is described. This allows us to design a powerful algorithm of decomposition, called W-decomposition, which is able to identify all well-constrained subsystems: it manages to decompose systems which were not decomposable by classic combinatorial methods.     

Global existence and nonexistence for degenerate parabolic equations with nonlinear boundary flux

In this paper, by constructing various kinds of sub- and super-solutions and using the basic properties of M-matrix, we give the necessary and sufficient conditions of global existence for nonnegative solutions to a degenerate parabolic system with completely coupled boundary conditions, which generalize the recent results of, for instance, Cui [Z. Cui, Critical curves of the non-Newtonian polytropic filtration equations coupled with nonlinear boundary conditions, Nonlinear Anal. 68 (2008) 3201–3208], Zhou–Mu [J. Zhou, C. Mu, Algebraic criteria for global existence or blow-up for a boundary coupled system of nonlinear diffusion equations, Appl. Anal. 86 (2007) 1185–1197] etc.     

A surface blending approach for displacement features on freeform surfaces

Embedding a number of displacement features into a base surface is common in industrial product design and modeling, where displaced surface regions are blended with the unmodified surface region. The cubic Hermite interpolant is usually adopted for surface blending, in which tangent plane smoothness across the boundary curve is achieved. However, the polynomial degree of the tangent field curve obtained symbolically is considerably higher, and the reduction of the degree of a freeform curve is a non-trivial task. In this work, an approximation surface blending approach is proposed to achieve tangential continuity across the boundary curve. The boundary curve is first offset in the tangent field with the user-specified tolerance, after which it is refined to be compatible with the offset curve for surface blending. Since the boundary curve is offset in a three-dimensional (3D) space, the local self-intersection in the offset curve is addressed in a 2D space by approximately mapping the offset vectors in the respective tangent planes to the parameter space of the base surface. The proposed algorithm is validated using examples, and the normal vector deviation along the boundary curve is investigated.     

Computing the nearest polynomial with a zero in a given domain by using piecewise rational functions

For a real univariate polynomial f and a closed complex domain D whose boundary C is a simple curve parameterized by a univariate piecewise rational function, a rigorous method is given for finding a real univariate polynomial such that has a zero in D and is minimal. First, it is proved that the minimum distance between f and polynomials having a zero at α∈C is a piecewise rational function of the real and imaginary parts of α. Thus, on C, the minimum distance is a piecewise rational function of a parameter obtained through the parameterization of C. Therefore, can be constructed by using the property that has a zero on C and computing the minimum distance on C. We analyze the asymptotic bit complexity of the method and show that it is of polynomial order in the size of the input.     

Value of information in optimal flow-level scheduling of users with Markovian time-varying channels

In this paper, we design, characterize in closed-form, and evaluate a new index rule for Markovian time-varying channels, which gives rise to a simple opportunistic scheduling rule for flow-level scheduling in wireless downlink systems. For user channels, we employ the Gilbert–Elliot model with a flow-level interpretation: the channel condition follows a general two-state Markov chain with distinct probabilities of finishing the flow transmission. The index value of the bad channel condition takes into account both the one-period and the steady-state potential improvement of the service completion probability, while the good channel condition gets an absolute priority with the cμ-index (well-known to be throughput-optimal) as the tie-breaking rule. Our computational study confirms near-optimality of the proposed rule in most of the instances, and suggests that information about the channels steady state is often enough to achieve near-optimality.     

Output feedback design for saturated linear plants using deadzone loops

In this paper, we present an LMI-based synthesis approach on output feedback design for input saturated linear systems by using deadzone loops. Algorithms are developed for minimizing the upper bound on the regional L₂ gain for exogenous inputs with L₂ norm bounded by a given value, and for minimizing this upper bound with a guaranteed reachable set or domain of attraction. The proposed synthesis approach will always lead to regionally stabilizing controllers if the plant is exponentially unstable, to semi-global results if the plant is non-exponentially unstable, and to global results if the plant is already exponentially stable, where the only requirement on the linear plant is detectability and stabilizability. The effectiveness of the proposed techniques is illustrated with one example.     

Data reduction of large vector graphics

Fast algorithm for joint near-optimal approximation of multiple polygonal curves is proposed. It is based on iterative reduced search dynamic programming introduced earlier for the min-εproblem of a single polygonal curve. The proposed algorithm jointly optimizes the number of line segments allocated to the different individual curves, and the approximation of the curves by the given number of segments. Trade-off between time and optimality is controlled by the breadth of the search, and by the numbers of iterations applied.     

Constrained curve fitting on manifolds

When designing curves on surfaces the need arises to approximate a given noisy target shape by a smooth fitting shape. We discuss the problem of fitting a B-spline curve to a point cloud by squared distance minimization in the case that both the point cloud and the fitting curve are constrained to lie on a smooth manifold. The on-manifold constraint is included by using the first fundamental form of the surface for squared distance computations between the point cloud and the fitting curve. For the solution we employ a constrained optimization algorithm that allows us to include further constraints such as one-sided fitting or surface regions that have to be avoided by the fitting curve. We illustrate the effectiveness of our algorithm by means of several examples showing different applications.     

Stabilizing composite control for a class of linear systems modeled by singularly perturbed Ito differential equations

In this paper, the problem of robust H_∞ control is investigated for sampled-data systems with probabilistic sampling. The parameter uncertainties are time-varying norm-bounded and appear in both the state and input matrices. For the simplicity of technical development, only two different sampling periods are considered whose occurrence probabilities are given constants and satisfy Bernoulli distribution, which can be further extended to the case with multiple stochastic sampling periods. By applying an input delay approach, the probabilistic sampling system is transformed into a continuous time-delay system with stochastic parameters in the system matrices. By linear matrix inequality (LMI) approach, sufficient conditions are obtained, which guarantee the robust mean-square exponential stability of the system with an H_∞ performance. Moreover, an H_∞ controller design procedure is then proposed. An illustrative example is included to demonstrate the effectiveness of the proposed techniques.     

Two-step extended RKN methods for oscillatory systems

In this paper, two-step extended Runge–Kutta–Nyström-type methods for the numerical integration of perturbed oscillators are presented and studied. The new methods inherit the framework of two-step hybrid methods and are adapted to the special feature of the true flows in both the internal stages and the updates. Based on the EN-trees theory [H.L. Yang, X.Y. Wu, X. You, Y.L. Fang, Extended RKN-type methods for numerical integration of perturbed oscillators, Comput. Phys. Comm. 180 (2009) 1777–1794], order conditions for the new methods are derived via the BBT-series defined on the set BT of branches and the BBWT-series defined on the subset BWT of BT. The stability and phase properties are analyzed. Numerical experiments show the applicability and efficiency of our new methods in comparison with the well-known high quality methods proposed in the scientific literature.     

Automatic calculation of supersymmetric renormalization group equations and loop corrections

SARAH is a Mathematica package for studying supersymmetric models. It calculates for a given model the masses, tadpole equations and all vertices at tree-level. This information can be used by SARAH to write model files for CalcHep/CompHep or FeynArts/FormCalc. In addition, the second version of SARAH can derive the renormalization group equations for the gauge couplings, parameters of the superpotential and soft-breaking parameters at one- and two-loop level. Furthermore, it calculates the one-loop self-energies and the one-loop corrections to the tadpoles. SARAH can handle all N=1 SUSY models whose gauge sector is a direct product of SU(N) and U(1) gauge groups. The particle content of the model can be an arbitrary number of chiral superfields transforming as any irreducible representation with respect to the gauge groups. To implement a new model, the user has just to define the gauge sector, the particle, the superpotential and the field rotations to mass eigenstates.

Program summary

Program title: SARAHCatalogue identifier: AEIB_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEIB_v1_0.htmlProgram obtainable from: CPC Program Library, Queen's University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 97 577No. of bytes in distributed program, including test data, etc.: 2 009 769Distribution format: tar.gzProgramming language: MathematicaComputer: All systems that Mathematica is available forOperating system: All systems that Mathematica is available forClassification: 11.1, 11.6Nature of problem: A supersymmetric model is usually characterized by the particle content, the gauge sector and the superpotential. It is a time consuming process to obtain all necessary information for phenomenological studies from these basic ingredients.Solution method: SARAH calculates the complete Lagrangian for a given model whose gauge sector can be any direct product of SU(N) gauge groups. The chiral superfields can transform as any, irreducible representation with respect to these gauge groups and it is possible to handle an arbitrary number of symmetry breakings or particle rotations. Also the gauge fixing terms can be specified. Using this information, SARAH derives the mass matrices and Feynman rules at tree-level and generates model files for CalcHep/CompHep and FeynArts/FormCalc. In addition, it can calculate the renormalization group equations at one- and two-loop level and the one-loop corrections to the one- and two-point functions.Unusual features: SARAH just needs the superpotential and gauge sector as input and not the complete Lagrangian. Therefore, the complete implementation of new models is done in some minutes.Running time: Measured CPU time for the evaluation of the MSSM on an Intel Q8200 with 2.33 GHz. Calculating the complete Lagrangian: 12 seconds. Calculating all vertices: 75 seconds. Calculating the one- and two-loop RGEs: 50 seconds. Calculating the one-loop corrections: 7 seconds. Writing a FeynArts file: 1 second. Writing a CalcHep/CompHep file: 6 seconds. Writing the LaTeX output: 1 second.     

An algorithm to generate all upper boundary points for in terms of minimal cuts

This paper discusses a stochastic-flow network from single-commodity case to multicommodity case. We propose a performance index, namely the probability that the upper bound of the system capacity is a given vector subject to the budget constraint, to evaluate the quality level for such a network. A simple approach based on minimal cuts is presented to generate the all upper boundary points for the demand subject to the budget B in order to evaluate the performance index.     

Fuzzy and possibilistic clustering for fuzzy data

The Fuzzy k-Means clustering model (FkM) is a powerful tool for classifying objects into a set of k homogeneous clusters by means of the membership degrees of an object in a cluster. In FkM, for each object, the sum of the membership degrees in the clusters must be equal to one. Such a constraint may cause meaningless results, especially when noise is present. To avoid this drawback, it is possible to relax the constraint, leading to the so-called Possibilistic k-Means clustering model (PkM). In particular, attention is paid to the case in which the empirical information is affected by imprecision or vagueness. This is handled by means of LR fuzzy numbers. An FkM model for LR fuzzy data is firstly developed and a PkM model for the same type of data is then proposed. The results of a simulation experiment and of two applications to real world fuzzy data confirm the validity of both models, while providing indications as to some advantages connected with the use of the possibilistic approach.     

MHD flow and heat transfer of a generalized Burgers’ fluid due to an exponential accelerating plate with the effect of radiation

This paper presents a research for the MHD flow and heat transfer of an incompressible generalized Burgers’ fluid due to an exponential accelerating plate with the effect of radiation. The fractional calculus approach is used to establish the constitutive relationship of the viscoelastic fluid. Exact analytic solutions are obtained for the velocity and temperature fields in integral and series form in terms of the G function by means of Fourier sine transform and Laplace transform. Moreover, the figures are plotted to show the effects of different parameters on the velocity and temperature fields.