Quadratic Spline Research Articles

Resolved‐particle fixed bed CFD with microkinetics for ethylene oxidation

The incorporation of an ethylene oxidation microkinetic model into fixed‐bed CFD is studied using two different approaches. The first is based on mapping pre‐calculated reaction rates into quadratic splines under steady‐state conditions without any further assumptions or simplification of the elementary steps. The second approach uses conventional reaction engineering assumptions such as quasi‐equilibrium (QE) and hybrid steady state (HSS) to reduce the kinetic model and generate lumped rate expressions. Both kinetic models are implemented for illustrative resolved‐particle CFD simulations in a randomly packed bed of 120 spheres. It is shown that the QE and HSS assumptions are not valid throughout the range of reacting conditions. Comparison of the temperature and species profiles for these two approaches shows that the strong gradients inside the bed cause significant deviations in the reduced model compared to using the splines method with full microkinetics, which produces accurate results without increasing the computational time. © 2016 American Institute of Chemical Engineers AIChE J, 63: 87–94, 2017

Gauss–Galerkin quadrature rules for quadratic and cubic spline spaces and their application to isogeometric analysis

We introduce Gaussian quadrature rules for spline spaces that are frequently used in Galerkin discretizations to build mass and stiffness matrices. By definition, these spaces are of even degrees. The optimal quadrature rules we recently derived (Bartoň and Calo, 2016) act on spaces of the smallest odd degrees and, therefore, are still slightly sub-optimal. In this work, we derive optimal rules directly for even-degree spaces and therefore further improve our recent result. We use optimal quadrature rules for spaces over two elements as elementary building blocks and use recursively the homotopy continuation concept described in Bartoň and Calo (2016) to derive optimal rules for arbitrary admissible numbers of elements. We demonstrate the proposed methodology on relevant examples, where we derive optimal rules for various even-degree spline spaces. We also discuss convergence of our rules to their asymptotic counterparts, these are the analogues of the midpoint rule of Hughes et al. (2010), that are exact and optimal for infinite domains.

Hemoglobin Concentration and Risk of Incident Stroke in Community-Living Adults.

In previous observational studies, hemoglobin concentrations have been associated with an increased risk of stroke. However, these studies were limited by a relatively low number of stroke events, making it difficult to determine whether the association of hemoglobin and stroke differed by demographic or clinical factors. Using Cox proportional hazards analysis and Kaplan-Meier plots, we examined the association of baseline hemoglobin concentrations with incident stroke in the Reasons for Geographic and Racial Differences in Stroke (REGARDS) study, a cohort of black and white adults aged ≥45 years. A total of 518 participants developed stroke over a mean 7±2 years of follow-up. There was a statistically significant interaction between hemoglobin and sex (P=0.05) on the risk of incident stroke. In Cox regression models adjusted for demographic and clinical variables, there was no association of baseline hemoglobin concentration with incident stroke in men, whereas in women, the lowest (<12.4 g/dL) and highest (>14.0 g/dL) quartiles of hemoglobin were associated with higher risk of stroke when compared with the second quartile (12.4-13.2 g/dL; quartile 1: hazard ratio, 1.59; 95% confidence interval, 1.09-2.31; quartile 2: referent; quartile 3: hazard ratio, 0.91; 95% confidence interval, 0.59-1.38; quartile 4: hazard ratio, 1.59; 95% confidence interval, 1.08-2.35). Similar results were observed in models stratified by hemoglobin and sex and when hemoglobin was modeled as a continuous variable using restricted quadratic spline regression. Lower and higher hemoglobin concentrations were associated with a higher risk of incident stroke in women. No such associations were found in men.

Medical Monitoring Model of Internet of Things Based on the Adaptive Threshold Difference Algorithm

There are still problems such as low detection accuracy and poor noise immunity in the application of the standard threshold difference algorithm in the signal detection of electrocardiosignal (ECG), in this paper, a medical monitoring model based on the adaptive threshold difference is proposed. First we use a nonlinear filter to filter the P wave and T wave which are low frequency in ECG signal. Then complex wave QRS will be tested. Then the algorithm will be more accuracy through the detection of the R-R interval length and the adjustment of threshold. Finally, the ECG signal will be test with quadratic spline wavelet twice, and the error judgment will be known through adaptive threshold difference. In the simulation experiments, after judging error by wavelet transformation and making the standard threshold difference algorithm optimize adaptively, algorithm showed excellent detection accuracy with and without noise.

Completeness of generating systems for quadratic splines on adaptively refined criss-cross triangulations

Hierarchical generating systems that are derived from Zwart–Powell (ZP) elements can be used to generate quadratic splines on adaptively refined criss-cross triangulations. We propose two extensions of these hierarchical generating systems, firstly decoupling the hierarchical ZP elements, and secondly enriching the system by including auxiliary functions. These extensions allow us to generate the entire hierarchical spline space – which consists of all piecewise quadratic C1-smooth functions on an adaptively refined criss-cross triangulation – if the triangulation fulfills certain technical assumptions. Special attention is dedicated to the characterization of the linear dependencies that are present in the resulting enriched decoupled hierarchical generating system.

Computationally efficient incorporation of microkinetics into resolved-particle CFD simulations of fixed-bed reactors

A method is developed to couple microkinetics with fluid flow in a fixed bed and transport inside the catalyst particles using computational fluid dynamics. Initially, the microkinetics model is solved for a wide range of different temperatures, and partial pressures. Next, reaction rates are mapped into quadratic multivariate splines. Splines coefficients are then imported into our user-defined function, and consequently the reaction rates are evaluated at each iteration simultaneously with the CFD simulations. This method has been applied to our solid particle model to investigate the effects of fluid flow, transport and elementary reaction steps on each other for ethylene and methanol partial oxidations. Reaction rates of all elementary steps as well as species surfaces sites and compositions are evaluated inside the particle. The suggested method can couple complex reaction mechanisms with detailed CFD simulations without increasing the computational time compared with global kinetics methods.

Efficient simulation and rendering of realistic motion of one-dimensional flexible objects

In gross motion of flexible one-dimensional (1D) objects such as cables, ropes, chains, ribbons and hair, the assumption of constant length is realistic and reasonable. The motion of the object also appears more natural if the motion or disturbance given at one end attenuates along the length of the object. In an earlier work, variational calculus was used to derive natural and length-preserving transformation of planar and spatial curves and implemented for flexible 1D objects discretized with a large number of straight segments. This paper proposes a novel idea to reduce computational effort and enable real-time and realistic simulation of the motion of flexible 1D objects. The key idea is to represent the flexible 1D object as a spline and move the underlying control polygon with much smaller number of segments. To preserve the length of the curve to within a prescribed tolerance as the control polygon is moved, the control polygon is adaptively modified by subdivision and merging. New theoretical results relating the length of the curve and the angle between the adjacent segments of the control polygon are derived for quadratic and cubic splines. Depending on the prescribed tolerance on length error, the theoretical results are used to obtain threshold angles for subdivision and merging. Simulation results for arbitrarily chosen planar and spatial curves whose one end is subjected to generic input motions are provided to illustrate the approach.

Option pricing in jump diffusion models with quadratic spline collocation

In this paper, we develop a robust numerical method in pricing options, when the underlying asset follows a jump diffusion model. We demonstrate that, with the quadratic spline collocation method, the integral approximation in the pricing PIDE is intuitively simple, and comes down to the evaluation of the probabilistic moments of the jump density. When combined with a Picard iteration scheme, the pricing problem can be solved efficiently. We present the method and the numerical results from pricing European and American options with Merton’s and Kou’s models.

Interpolation properties of C1 quadratic splines on hexagonal cells

Let Δn be a cell with a single interior vertex and n boundary vertices v1,…,vn. Say that Δn has the interpolation property if for every z1,…,zn∈R there is a spline s∈S21(Δn) such that s(vi)=zi for all i. We investigate under what conditions does a cell fail the interpolation property. The question is related to an open problem posed by Alfeld, Piper, and Schumaker in 1987 about characterization of unconfinable vertices.For hexagonal cells, we obtain a geometric criterion characterizing the failure of the interpolation property. As a corollary, we conclude that a hexagonal cell such that its six interior edges lie on three lines fails the interpolation property if and only if the cell is projectively equivalent to a regular hexagonal cell. Along the way, we obtain an explicit basis for the vector space S21(Δn) for n≥5.

Quadratic spline collocation method for the time fractional subdiffusion equation

In this paper, exploiting the quadratic spline collocation (QSC) method, we numerically solve the time fractional subdiffusion equation with Dirichelt boundary value conditions. The coefficient matrix of the discretized linear system is investigated in detail. Theoretical analyses and numerical examples demonstrate the proposed technique can enjoy the global error bound with O(τ3+h3) under the L∞ norm provided that the solution v(x, t) has four-order continual derivative with respects to x and t, and it can achieve the accuracy of O(τ4+h4) at collocation points, where τ, h are the step sizes in time and space, respectively.

A Hybrid Algorithm Based on Optimal Quadratic Spline Collocation and Parareal Deferred Correction for Parabolic PDEs

Parareal is a kind of time parallel numerical methods for time-dependent systems. In this paper, we consider a general linear parabolic PDE, use optimal quadratic spline collocation (QSC) method for the space discretization, and proceed with the parareal technique on the time domain. Meanwhile, deferred correction technique is also used to improve the accuracy during the iterations. In fact, the optimal QSC method is a correction of general QSC method. Along the temporal direction we embed the iterations of deferred correction into parareal to construct a hybrid method, parareal deferred correction (PDC) method. The error estimation is presented and the stability is analyzed. To save computational cost, we find out a simple way to balance the two kinds of iterations as much as possible. We also argue that the hybrid algorithm has better system efficiency and costs less running time. Numerical experiments by multicore computers are attached to exhibit the effectiveness of the hybrid algorithm.

Option Pricing in Jump Diffusion Models with Quadratic Spline Collocation

Option Pricing in Jump Diffusion Models with Quadratic Spline Collocation

Linear Convergence of Descent Methods for the Unconstrained Minimization of Restricted Strongly Convex Functions

Linear convergence rates of descent methods for unconstrained minimization are usually proved under the assumption that the objective function is strongly convex. Recently it was shown that the weaker assumption of restricted strong convexity suffices for linear convergence of the ordinary gradient descent method. A decisive difference to strong convexity is that the set of minimizers of a restricted strongly convex function need be neither a singleton nor bounded. In this paper we extend the linear convergence results under this weaker assumption to a larger class of descent methods including restarted nonlinear CG, BFGS, and its damped limited memory variants, L-D-BFGS. For twice continuously differentiable objective functions we even obtain r-step superlinear convergence for the CG_DESCENT conjugate gradient method of Hager and Zhang, where r is greater than or equal to the rank of the Hessian at a minimizer. This is remarkable since the Hessian of a restricted strongly convex function need not have full rank. Furthermore we show that convex quadratic splines and objective functions of the unconstrained duals to some linearly constrained optimization problems are restricted strongly convex. In particular this holds for the regularized basis pursuit problem and its analogues for nuclear norm minimization and principal component pursuit.

An algorithm for [formula omitted] offset approximation based on circle approximation by [formula omitted] quadratic spline

We present a method for G2 approximation of offset curves using G2 quadratic spline approximation of circular arc. An algorithm for the approximation method is presented and is applied to some numerical examples.

Application of Cubic Box Spline Wavelets in the Analysis of Signal Singularities

Abstract In the subject literature, wavelets such as the Mexican hat (the second derivative of a Gaussian) or the quadratic box spline are commonly used for the task of singularity detection. The disadvantage of the Mexican hat, however, is its unlimited support; the disadvantage of the quadratic box spline is a phase shift introduced by the wavelet, making it difficult to locate singular points. The paper deals with the construction and properties of wavelets in the form of cubic box splines which have compact and short support and which do not introduce a phase shift. The digital filters associated with cubic box wavelets that are applied in implementing the discrete dyadic wavelet transform are defined. The filters and the algorithme à trous of the discrete dyadic wavelet transform are used in detecting signal singularities and in calculating the measures of signal singularities in the form of a Lipschitz exponent. The article presents examples illustrating the use of cubic box spline wavelets in the analysis of signal singularities.

TOWARDS A MORE GENERAL TYPE OF UNIVARIATE CONSTRAINED INTERPOLATION WITH FRACTAL SPLINES

Recently, in [Electron. Trans. Numer. Anal. 41 (2014) 420–442] authors introduced a new class of rational cubic fractal interpolation functions with linear denominators via fractal perturbation of traditional nonrecursive rational cubic splines and investigated their basic shape preserving properties. The main goal of the current paper is to embark on univariate constrained fractal interpolation that is more general than what was considered so far. To this end, we propose some strategies for selecting the parameters of the rational fractal spline so that the interpolating curves lie strictly above or below a prescribed linear or a quadratic spline function. Approximation property of the proposed rational cubic fractal spine is broached by using the Peano kernel theorem as an interlude. The paper also provides an illustration of background theory, veined by examples.

Initial value problems with retarded argument solved by iterated quadratic splines

In this paper we propose a new iterative numerical method for initial value problems of first and second order involving retarded argument. The method uses a quadratic spline interpolation procedure activated at each iterative step. The convergence of this method of iterated splines is theoretically proven and tested on some numerical examples.

Curve network interpolation by C1 quadratic B-spline surfaces

In this paper we investigate the problem of interpolating a B-spline curve network, in order to create a surface satisfying such a constraint and defined by blending functions spanning the space of bivariate C1 quadratic splines on criss–cross triangulations. We prove the existence and uniqueness of the surface, providing a constructive algorithm for its generation. We also present numerical and graphical results and comparisons with other methods.

Midlife Alcohol Consumption and the Risk of Stroke in the Atherosclerosis Risk in Communities Study.

Alcohol consumption is common in the United States and may confer beneficial cardiovascular effects at light-to-moderate doses. The alcohol-stroke relationship remains debated. We estimated the relationship between midlife, self-reported alcohol consumption and ischemic stroke and intracerebral hemorrhage (ICH) in a biracial cohort. We examined 12,433 never and current drinkers in the Atherosclerosis Risk in Communities study, aged 45 to 64 years at baseline. Participants self-reported usual drinks per week of beer, wine, and liquor at baseline. We used multivariate Cox proportional hazards regression to assess the association of current alcohol consumption relative to lifetime abstention with incident ischemic stroke and ICH and modification by sex-race group. We modeled alcohol intake with quadratic splines to further assess dose-response relationships. One third of participants self-reported abstention, 39% and 24%, respectively, consumed ≤3 and 4 to 17 drinks/wk, and only 5% reported heavier drinking. There were 773 ischemic strokes and 81 ICH over follow-up (median≈22.6 years). For ischemic stroke, light and moderate alcohol consumption were not associated with incidence (hazard ratios, 0.98; 95% CI, 0.79-1.21; 1.06, 0.84-1.34), whereas heavier drinking was associated with a 31% increased rate relative to abstention (hazard ratios, 1.31; 95% CI, 0.92-1.86). For ICH, moderate-to-heavy (hazard ratios, 1.99; 95% CI, 1.07-3.70), but not light, consumption increased incidence. Self-reported light-to-moderate alcohol consumption at midlife was not associated with reduced stroke risk compared with abstention over 20 years of follow-up in the Atherosclerosis Risk in Communities study. Heavier consumption increased the risk for both outcomes as did moderate intake for ICH.

Stability analysis of the marching-on-in-time boundary element method for electromagnetics

The Time Domain Integral Equation method for electromagnetics is an appealing computational method for many applications in industry. However, its applicability has long been suffering from instabilities. A rigorous analysis of the variational formulation is imperative to the successful design of stable and robust numerical schemes. In this paper, an established functional framework and stability theorem will be extended to the differentiated version of the electric field integral equations, which can be discretized more efficient and is more often used in engineering literature. The extended stability theorem, combined with efficiency requirements, will give guidelines on the choice of test and basis functions of the space–time Petrov–Galerkin scheme. A discrete equivalence with the collocation method results in the recommendation to choose the quadratic spline basis function in the standard Marching-on-in-Time scheme. Computational experiments confirm that the quadratic spline basis functions have superior stability characteristics compared to the conventional quadratic Lagrange basis functions in time.