Flamelet models, which enable the storing of precomputed detailed chemistry into lookup tables, are widely used in combustion simulations. They allow the computation of accurate results at low ...computational cost, but standard implementations can lead to numerical problems due to a non-smooth representation, and their applicability is limited by memory requirements. Here, the methods used by a newly developed and optimised lookup table generator based on B-spline interpolation are presented. The creation of smooth representations of flamelet solutions requiring less than one fifth of the number of points in each direction compared to the non-smooth representations of standard lookup tables based on linear interpolation is shown to be possible. The new B-spline interpolation based tables are also applied within a large-eddy simulation of the Swirling Methane/Hydrogen Flame 1 and the results are compared to simulations using lookup tables based on linear interpolation or optimised artificial neural networks. Better performance of the B-spline interpolation based tables with respect to physical accuracy and numerical performance is demonstrated.
In this article, the three-dimensional conformal time derivative generalized q- deformed Sinh–Gordon equation is discussed analytically and numerically using the (G′/G)-expansion approach and finite ...element method respectively. The analytical method successfully extracts the solutions, as well as the appropriate constraint requirements for the existence of solutions, placed on parameters. The extended cubic B-spline technique is also used to present the numerical findings. We also provide numerous figures to demonstrate the various solitons propagation patterns. The proposed equation has opened up new options for describing physical systems that have lost their symmetry. This work is a continuation of series of works we are doing on this point.
Simplified models are widely applied in finite element computations regarding mechanical and structural problems. However, the simplified model sometimes causes many deviations in the finite element ...analysis (FEA) of structures, especially in the non-designed structures which have undergone unknowable deformation features. Hence, a novel FEA methodology based on the parametric model by approximating three-dimensional (3D) feature data is proposed to solve this problem in the present manuscript. Many significant and effective technologies have been developed to detect 3D feature information accurately, e.g., terrestrial laser scanning (TLS), digital photogrammetry, and radar technology. In this manuscript, the parametric FEA model combines 3D point clouds from TLS and the parametric surface approximation method to generate 3D surfaces and models accurately. TLS is a popular measurement method for reliable 3D point clouds acquisition and monitoring deformations of structures with high accuracy and precision. The B-spline method is applied to approximate the measured point clouds data automatically and generate a parametric description of the structure accurately. The final target is to reduce the effects of the model description and deviations of the FEA. Both static and dynamic computations regarding a composite structure are carried out by comparing the parametric and general simplified models. The comparison of the deformation and equivalent stress of future behaviors are reflected by different models. Results indicate that the parametric model based on the TLS data is superior in the finite element computation. Therefore, it is of great significance to apply the parametric model in the FEA to compute and predict the future behavior of the structures with unknowable deformations in engineering accurately.
In this paper, we use quadratic B-splines to reconstruct an approximating function by using the integral values of successive subintervals, rather than the usual function values at the knots. It is ...called integro quadratic spline interpolation. Compared to the other existing methods, our method can tackle integro interpolation problem from the integral values on arbitrary successive subintervals. The general approximation error is studied and the super convergence property is also derived when the interval is equally partitioned. Moreover, it can work successfully without any boundary condition. Numerical experiments show our method is easy to implement and effective.
Feedrate optimization (FO) and servo error pre-compensation (SEP) are often performed independently to improve the accuracy and speed, respectively, of computer-controlled manufacturing machines. ...However, this independent approach leads to excessive tradeoff between speed and accuracy. To address this issue, the authors have proposed a new concept of simultaneous FO and SEP (or FOSEP) where SEP is integrated into FO, yielding large reductions in motion time without sacrificing positioning accuracy relative to independent FO and SEP. However, in their prior work, the authors used linear programming to achieve FOSEP resulting in the following: (i) inaccuracy in enforcing nonlinear constraints and (ii) poor computational efficiency for long toolpaths. To address these two problems, this paper proposes a new approach for FOSEP using windowed sequential linear programming (SLP). The use of SLP improves the accuracy of FOSEP in enforcing nonlinear constraints; however, it lowers the computational efficiency of FOSEP. Windowing addresses the problem of low computational efficiency by applying SLP to FOSEP in small overlapping batches. A downside of windowed SLP is that it may lead to infeasibility in the optimization. This problem is resolved by smoothly switching between the optimal solution obtained using windowed SLP and a backup conservative solution in case of impending infeasibility. The proposed windowed SLP with smooth switching approach for FOSEP is validated in simulations where it significantly improves the accuracy and computational efficiency of FOSEP while guaranteeing feasibility. The practical benefits of the proposed approach for FOSEP is demonstrated in experiments on a 3D printer where it achieves up to 25% reduction in cycle time without sacrificing printing quality relative to the conventional approach of independent FO then SEP, both applied to a long toolpath.
The theory of generalized locally Toeplitz (GLT) sequences is a powerful apparatus for computing the asymptotic spectral distribution of square matrices A.sub.n arising from the discretization of ...differential problems. Indeed, as the mesh fineness parameter n increases to infinity, the sequence {A.sub.n}.sub.n often turns out to be a GLT sequence. In this paper, motivated by recent applications, we further enhance the GLT apparatus by developing a full theory of rectangular GLT sequences as an extension of the theory of classical square GLT sequences. We also provide two examples of application as an illustration of the potential of the theory presented herein. Key words. asymptotic distribution of singular values and eigenvalues, rectangular Toeplitz matrices, rectangular generalized locally Toeplitz matrices, discretization of differential equations, finite elements, tensor products, B-splines, multigrid methods AMS subject classifications. 15A18, 15B05, 47B06, 65N30, 15A69, 65D07, 65N55
We discuss the main components of the recently developed FEMB program package, which implements finite element methods with weighted B-splines for basic linear elliptic boundary value problems in two ...and three dimensions. A three-dimensional implementation without topological restrictions has not been available before. We describe in particular the mathematical background for the recursive quadrature/cubature over boundary cells and explain how to utilize the regular data structure of uniform B-splines efficiently. Considering the Lamé–Navier equations of linear elasticity as a typical example, we illustrate the performance of the main FEMB routines. The numerical tests confirm that, due to the new integration routines, the weighted B-spline solvers have become considerably more efficient.
Often when jointly modeling longitudinal and survival data, we are interested in a multivariate longitudinal measure that may not fit well by linear models. To overcome this problem, we propose a ...joint longitudinal and survival model that has a nonparametric model for the longitudinal markers. We use cubic B-splines to specify the longitudinal model and a proportional hazards model to link the longitudinal measures to the hazard. To fit the model, we use a Markov chain Monte Carlo algorithm. We select the number of knots for the cubic B-spline model using the Conditional Predictive Ordinate (CPO) and the Deviance Information Criterion (DIC). The method and model selection approach are validated in a simulation. We apply this method to examine the link between viral load, CD4 count, and time to event in data from an AIDS clinical trial. The cubic B-spline model provides a good fit to the longitudinal data that could not be obtained with simple parametric models.
For a class of compactly supported windows we characterize the frame property for a Gabor system {EmbTnag}m,n∈Z, for translation parameters a belonging to a certain range depending on the support ...size. We show that the obstructions to the frame property are located on a countable number of “curves.” For functions that are positive on the interior of the support these obstructions do not appear, and the considered region in the (a,b) plane is fully contained in the frame set. In particular this confirms a recent conjecture about B-splines by Gröchenig in that particular region. We prove that the full conjecture is true if it can be proved in a certain “hyperbolic strip.”
The selection of smoothing parameter is central to the estimation of penalized splines. The best value of the smoothing parameter is often the one that optimizes a smoothness selection criterion, ...such as generalized cross-validation error (GCV) and restricted likelihood (REML). To correctly identify the global optimum rather than being trapped in an undesired local optimum, grid search is recommended for optimization. Unfortunately, the grid search method requires a pre-specified search interval that contains the unknown global optimum, yet no guideline is available for providing this interval. As a result, practitioners have to find it by trial and error. To overcome such difficulty, we develop novel algorithms to automatically find this interval. Our automatic search interval has four advantages. (i) It specifies a smoothing parameter range where the associated penalized least squares problem is numerically solvable. (ii) It is criterion-independent so that different criteria, such as GCV and REML, can be explored on the same parameter range. (iii) It is sufficiently wide to contain the global optimum of any criterion, so that for example, the global minimum of GCV and the global maximum of REML can both be identified. (iv) It is computationally cheap compared with the grid search itself, carrying no extra computational burden in practice. Our method is ready to use through our recently developed
R
package
gps
(
≥
version 1.1). It may be embedded in more advanced statistical modeling methods that rely on penalized splines.
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