30,943 research outputs found
A survey of partial differential equations in geometric design
YesComputer aided geometric design is an area
where the improvement of surface generation techniques
is an everlasting demand since faster and more accurate
geometric models are required. Traditional methods
for generating surfaces were initially mainly based
upon interpolation algorithms. Recently, partial differential
equations (PDE) were introduced as a valuable
tool for geometric modelling since they offer a number
of features from which these areas can benefit. This work
summarises the uses given to PDE surfaces as a surface
generation technique togethe
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Automatic, computer aided geometric design of free-knot, regression splines
A new algorithm for Computer Aided Geometric Design of least squares (LS) splines with variable knots, named GeDS, is presented. It is based on interpreting functional spline regression as a parametric B-spline curve, and on using the shape preserving property of its control polygon. The GeDS algorithm includes two major stages. For the first stage, an automatic adaptive, knot location algorithm is developed. By adding knots, one at a time, it sequentially "breaks" a straight line segment into pieces in order to construct a linear LS B-spline fit, which captures the "shape" of the data. A stopping rule is applied which avoids both over and under fitting and selects the number of knots for the second stage of GeDS, in which smoother, higher order (quadratic, cubic, etc.) fits are generated. The knots appropriate for the second stage are determined, according to a new knot location method, called the averaging method. It approximately preserves the linear precision property of B-spline curves and allows the attachment of smooth higher order LS B-spline fits to a control polygon, so that the shape of the linear polygon of stage one is followed. The GeDS method produces simultaneously linear, quadratic, cubic (and possibly higher order) spline fits with one and the same number of B-spline regression functions. The GeDS algorithm is very fast, since no deterministic or stochastic knot insertion/deletion and relocation search strategies are involved, neither in the first nor the second stage. Extensive numerical examples are provided, illustrating the performance of GeDS and the quality of the resulting LS spline fits. The GeDS procedure is compared with other existing variable knot spline methods and smoothing techniques, such as SARS, HAS, MDL, AGS methods and is shown to produce models with fewer parameters but with similar goodness of fit characteristics, and visual quality
Comparison of parametric, orthogonal, and spline functions to model individual lactation curves for milk yield in Canadian Holsteins
Test day records for milk yield of 57,390 first lactation Canadian Holsteins were analyzed with a linear model that included the fixed effects of herd-test date and days in milk (DIM) interval nested within age and calving season.
Residuals from this model were analyzed as a new variable and fitted with a five parameter model, fourth-order Legendre polynomials, with linear, quadratic and cubic spline models with three knots. The fit of the models was
rather poor, with about 30%-40% of the curves showing an adjusted R-square lower than 0.20 across all models. Results underline a great difficulty in modelling individual deviations around the mean curve for milk yield. However, the Ali and Schaeffer (5 parameter) model and
the fourth-order Legendre polynomials were able to detect two basic shapes of individual deviations among the mean curve. Quadratic and, especially, cubic spline functions had better fitting performances but a poor predictive
ability due to their great flexibility that results
in an abrupt change of the estimated curve when data are missing. Parametric and orthogonal polynomials seem to be robust and affordable under this standpoint
Improving convergence in smoothed particle hydrodynamics simulations without pairing instability
The numerical convergence of smoothed particle hydrodynamics (SPH) can be
severely restricted by random force errors induced by particle disorder,
especially in shear flows, which are ubiquitous in astrophysics. The increase
in the number NH of neighbours when switching to more extended smoothing
kernels at fixed resolution (using an appropriate definition for the SPH
resolution scale) is insufficient to combat these errors. Consequently, trading
resolution for better convergence is necessary, but for traditional smoothing
kernels this option is limited by the pairing (or clumping) instability.
Therefore, we investigate the suitability of the Wendland functions as
smoothing kernels and compare them with the traditional B-splines. Linear
stability analysis in three dimensions and test simulations demonstrate that
the Wendland kernels avoid the pairing instability for all NH, despite having
vanishing derivative at the origin (disproving traditional ideas about the
origin of this instability; instead, we uncover a relation with the kernel
Fourier transform and give an explanation in terms of the SPH density
estimator). The Wendland kernels are computationally more convenient than the
higher-order B-splines, allowing large NH and hence better numerical
convergence (note that computational costs rise sub-linear with NH). Our
analysis also shows that at low NH the quartic spline kernel with NH ~= 60
obtains much better convergence then the standard cubic spline.Comment: substantially revised version, accepted for publication in MNRAS, 15
pages, 13 figure
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