327 research outputs found
Data visualization using rational spline interpolation
AbstractA smooth curve interpolation scheme for positive, monotonic, and convex data has been developed. This scheme uses piecewise rational cubic functions. The two families of parameters, in the description of the rational interpolant, have been constrained to preserve the shape of the data. The rational spline scheme has a unique representation. The degree of smoothness attained is C1
Algorithm xxxx: HiPPIS A High-Order Positivity-Preserving Mapping Software for Structured Meshes
Polynomial interpolation is an important component of many computational
problems. In several of these computational problems, failure to preserve
positivity when using polynomials to approximate or map data values between
meshes can lead to negative unphysical quantities. Currently, most
polynomial-based methods for enforcing positivity are based on splines and
polynomial rescaling. The spline-based approaches build interpolants that are
positive over the intervals in which they are defined and may require solving a
minimization problem and/or system of equations. The linear polynomial
rescaling methods allow for high-degree polynomials but enforce positivity only
at limited locations (e.g., quadrature nodes). This work introduces open-source
software (HiPPIS) for high-order data-bounded interpolation (DBI) and
positivity-preserving interpolation (PPI) that addresses the limitations of
both the spline and polynomial rescaling methods. HiPPIS is suitable for
approximating and mapping physical quantities such as mass, density, and
concentration between meshes while preserving positivity. This work provides
Fortran and Matlab implementations of the DBI and PPI methods, presents an
analysis of the mapping error in the context of PDEs, and uses several 1D and
2D numerical examples to demonstrate the benefits and limitations of HiPPIS
A sharp interface isogeometric strategy for moving boundary problems
The proposed methodology is first utilized to model stationary and propagating cracks. The crack face is enriched with the Heaviside function which captures the displacement discontinuity. Meanwhile, the crack tips are enriched with asymptotic displacement functions to reproduce the tip singularity. The enriching degrees of freedom associated with the crack tips are chosen as stress intensity factors (SIFs) such that these quantities can be directly extracted from the solution without a-posteriori integral calculation.
As a second application, the Stefan problem is modeled with a hybrid function/derivative enriched interface. Since the interface geometry is explicitly defined, normals and curvatures can be analytically obtained at any point on the interface, allowing for complex boundary conditions dependent on curvature or normal to be naturally imposed. Thus, the enriched approximation naturally captures the interfacial discontinuity in temperature gradient and enables the imposition of Gibbs-Thomson condition during solidification simulation.
The shape optimization through configuration of finite-sized heterogeneities is lastly studied. The optimization relies on the recently derived configurational derivative that describes the sensitivity of an arbitrary objective with respect to arbitrary design modifications of a heterogeneity inserted into a domain. The THB-splines, which serve as the underlying approximation, produce sufficiently smooth solution near the boundaries of the heterogeneity for accurate calculation of the configurational derivatives. (Abstract shortened by ProQuest.
Estimation of Space Deformation Model for Non-stationary Random Functions
Stationary Random Functions have been successfully applied in geostatistical
applications for decades. In some instances, the assumption of a homogeneous
spatial dependence structure across the entire domain of interest is
unrealistic. A practical approach for modelling and estimating non-stationary
spatial dependence structure is considered. This consists in transforming a
non-stationary Random Function into a stationary and isotropic one via a
bijective continuous deformation of the index space. So far, this approach has
been successfully applied in the context of data from several independent
realizations of a Random Function. In this work, we propose an approach for
non-stationary geostatistical modelling using space deformation in the context
of a single realization with possibly irregularly spaced data. The estimation
method is based on a non-stationary variogram kernel estimator which serves as
a dissimilarity measure between two locations in the geographical space. The
proposed procedure combines aspects of kernel smoothing, weighted non-metric
multi-dimensional scaling and thin-plate spline radial basis functions. On a
simulated data, the method is able to retrieve the true deformation.
Performances are assessed on both synthetic and real datasets. It is shown in
particular that our approach outperforms the stationary approach. Beyond the
prediction, the proposed method can also serve as a tool for exploratory
analysis of the non-stationarity.Comment: 17 pages, 9 figures, 2 table
Total Positivity of the Cubic Trigonometric Bézier Basis
Within the general framework of Quasi Extended Chebyshev space, we prove that the cubic trigonometric Bézier basis with two shape parameters λ and μ given in Han et al. (2009) forms an optimal normalized totally positive basis for λ,μ∈(-2,1]. Moreover, we show that for λ=-2 or μ=-2 the basis is not suited for curve design from the blossom point of view. In order to compute the corresponding cubic trigonometric Bézier curves stably and efficiently, we also develop a new corner cutting algorithm
Weighted Quasi Interpolant Spline Approximations: Properties and Applications
Continuous representations are fundamental for modeling sampled data and
performing computations and numerical simulations directly on the model or its
elements. To effectively and efficiently address the approximation of point
clouds we propose the Weighted Quasi Interpolant Spline Approximation method
(wQISA). We provide global and local bounds of the method and discuss how it
still preserves the shape properties of the classical quasi-interpolation
scheme. This approach is particularly useful when the data noise can be
represented as a probabilistic distribution: from the point of view of
nonparametric regression, the wQISA estimator is robust to random
perturbations, such as noise and outliers. Finally, we show the effectiveness
of the method with several numerical simulations on real data, including curve
fitting on images, surface approximation and simulation of rainfall
precipitations
Unstructured Grid Generation Techniques and Software
The Workshop on Unstructured Grid Generation Techniques and Software was conducted for NASA to assess its unstructured grid activities, improve the coordination among NASA centers, and promote technology transfer to industry. The proceedings represent contributions from Ames, Langley, and Lewis Research Centers, and the Johnson and Marshall Space Flight Centers. This report is a compilation of the presentations made at the workshop
Assessment of the effects of soil variability in the modeling of liquefiable soils
La licuación del suelo puede tener consecuencias catastróficas en términos de daños estructurales y pérdida de vidas humanas. Una de las principales incertidumbres al momento de realizar modelos para predecir la respuesta del suelo ante el fenómeno de licuación es la variabilidad espacial de las propiedades del suelo causada por su naturaleza heterogénea. Esta investigación tiene como objetivo evaluar los efectos de la variabilidad espacial de un depósito de suelos licuables. Para desarrollar este análisis, se implementó un modelo determinístico de elementos finitos. El suelo se modeló utilizando un modelo constitutivo de múltiples superficies de fluencia, el cual fue calibrado empleando ensayos triaxiales cíclicos. Posteriormente, se desarrolló una evaluación estocástica del problema empleando el Método de Elementos Finitos Aleatorios (RFEM), en el que se modeló la densidad relativa como un campo aleatorio gaussiano correlacionado espacialmente, considerando la variabilidad esperada en condiciones experimentales y en las condiciones in situ del suelo. Al final, se determinaron los efectos de la variabilidad espacial del suelo comparando los resultados de las simulaciones determinísticas y estocásticas con resultados experimentales de centrífuga.MaestríaMagister en Ingeniería Civi
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