Local refinement with hierarchical B-spline structures is an active topic of research in the context of geometric modeling and isogeometric analysis. By exploiting a multilevel control structure, we ...show that truncated hierarchical B-spline (THB-spline) representations support interactive modeling tools, while simultaneously providing effective approximation schemes for the manipulation of complex data sets and the solution of partial differential equations via isogeometric analysis. A selection of illustrative 2D and 3D numerical examples demonstrates the potential of the hierarchical framework.
In this article, we focused on solving numerically the coupled nonlinear Schrödinger equations (NLFSEs) with Caputo fractional derivative in time. The discrete schemes are constructed by using the ...trigonometric B‐spline collocation method and the non‐polynomial B‐spline method for space discretization respectively, while the
L1‐formula is applied in time discretization. The Von Neumann approach is applied to examine the stability of the proposed methods. For validating the accuracy and efficiency of the presented schemes, numerical tests are compared with the exact solution.
The construction of classical hierarchical B-splines can be suitably modified in order to define locally supported basis functions that form a partition of unity. We will show that this property can ...be obtained by reducing the support of basis functions defined on coarse grids, according to finer levels in the hierarchy of splines. This truncation not only decreases the overlapping of supports related to basis functions arising from different hierarchical levels, but it also improves the numerical properties of the corresponding hierarchical basis — which is denoted as truncated hierarchical B-spline (THB-spline) basis. Several computed examples will illustrate the adaptive approximation behavior obtained by using a refinement algorithm based on THB-splines.
► A normalized hierarchical tensor–product B-spline basis is proposed. ► The construction relies on a truncation mechanism of coarse basis functions. ► The local refinement property is illustrated by an adaptive approximation strategy. ► Sparsity and condition numbers of the related matrices are studied experimentally.
ABSTRACT
Quantile estimates are generally interpreted in association with the return period concept in practical engineering. To do so with the peaks‐over‐threshold (POT) approach, combined ...Poisson‐generalized Pareto distributions (referred to as PD‐GPD model) must be considered. In this article, we evaluate the incorporation of non‐stationarity in the generalized Pareto distribution (GPD) and the Poisson distribution (PD) using, respectively, the smoothing‐based B‐spline functions and the logarithmic link function. Two models are proposed, a stationary PD combined to a non‐stationary GPD (referred to as PD0‐GPD1) and a combined non‐stationary PD and GPD (referred to as PD1‐GPD1). The teleconnections between hydro‐climatological variables and a number of large‐scale climate patterns allow using these climate indices as covariates in the development of non‐stationary extreme value models. The case study is made with daily precipitation amount time series from southeastern Canada and two climatic covariates, the Arctic Oscillation (AO) and the Pacific North American (PNA) indices. A comparison of PD0‐GPD1 and PD1‐GPD1 models showed that the incorporation of non‐stationarity in both POT models instead of solely in the GPD has an effect on the estimated quantiles. The use of the B‐spline function as link function between the GPD parameters and the considered climatic covariates provided flexible non‐stationary PD‐GPD models. Indeed, linear and nonlinear conditional quantiles are observed at various stations in the case study, opening an interesting perspective for further research on the physical mechanism behind these simple and complex interactions.
Using statistical tools like the cross‐wavelet analysis illustrated in the figure, common features of variability are found between precipitation extreme events and the Artic Oscillation index at the Upper Stewiacke station located in Nova Scotia (Canada). Using this index as covariate, we developed non‐stationary Poisson‐generalized Pareto models, which allow observing conditional quantiles with concave form. The proposed models are more flexible than classical extreme value non‐stationary models which often used prior assumption of linear dependence.
Summary
The paper introduces a novel multiresolution scheme to topology optimization in the framework of the isogeometric analysis. A new variable parameter space is added to implement ...multiresolution topology optimization based on the Solid Isotropic Material with Penalization approach. Design density variables defined in the variable space are used to approximate the element analysis density by the bivariate B‐spline basis functions, which are easily obtained using k‐refinement strategy in the isogeometric analysis. While the nonuniform rational B‐spline basis functions are used to exactly describe geometric domains and approximate unknown solutions in finite element analysis. By applying a refined sensitivity filter, optimized designs include highly discrete solutions in terms of solid and void materials without using any black and white projection filters. The Method of Moving Asymptotes is used to solve the optimization problem. Various benchmark test problems including plane stress, compliant mechanism inverter, and 2‐dimensional heat conduction are examined to demonstrate the effectiveness and robustness of the present method.
Summary
An enhancement of the extended B‐spline‐based implicit material point method (EBS‐MPM) is developed to avoid pressure oscillation and volumetric locking. The EBS‐MPM is a stable implicit MPM ...that enables the imposition of arbitrary boundary conditions thanks to the higher‐order EBS basis functions and the help of Nitsche's method. In particular, by means of the higher‐order EBS basis functions, the EBS‐MPM can suppress the cell‐crossing errors caused by material points crossing the background grid boundaries and can avoid both the stress oscillations arising from inaccurate numerical integration and the ill‐conditioning of the resulting tangent matrices. Although the higher‐order EBS basis functions are known to avoid volumetric locking, the problem of pressure oscillation has not yet been resolved. Therefore, to suppress pressure oscillation due to quasi‐incompressibility, we propose the incorporation of the F‐bar projection method into the EBS‐MPM, which is compatible with the higher‐order EBS basis functions. Three representative numerical examples are presented to demonstrate the capability of the proposed method in suppressing both the pressure oscillation and volumetric locking. The results of the proposed method are compared to those of the finite element method with F‐bar elements and those of isogeometric analysis with quadratic NURBS elements.
Smooth spline functions such as B-splines and NURBS are already an established technology in the field of computer-aided design (CAD) and have in recent years been given a lot of attention from the ...computer-aided engineering (CAE) community. The advantages of local refinement are obvious for anyone working in either field, and as such, several approaches have been proposed. Among others, we find the three strategies Classical Hierarchical B-splines, Truncated Hierarchical B-splines and Locally Refined B-splines. We will in this paper present these three frameworks and highlight similarities and differences between them. In particular, we will look at the function space they span and the support of the basis functions. We will then analyse the corresponding stiffness and mass matrices in terms of sparsity patterns and conditioning numbers. We show that the basis functions in general do not span the same space, and that conditioning numbers are comparable. Moreover we show that the weighting needed by the Classical Hierarchical basis to maintain partition of unity has significant implications on the conditioning numbers.
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