In order to low-frequency stabilize the electric field integral equation (EFIE) when discretized with divergence conforming B-spline-based basis and testing functions in an isogeometric approach, we ...propose a corresponding quasi-Helmholtz preconditioner. To this end, we derive i) a loop-star decomposition for the B-spline basis in the form of sparse mapping matrices applicable to arbitrary polynomial orders of the basis as well as to open and closed geometries described by single-patch or multipatch parametric surfaces (as an example, nonuniform rational B-splines (NURBS) surfaces are considered). Based on the loop-star analysis, we show ii) that quasi-Helmholtz projectors can be defined efficiently. This renders the proposed low-frequency stabilization directly applicable to multiply-connected geometries without the need to search for global loops and results in better-conditioned system matrices compared with directly using the loop-star basis. Numerical results demonstrate the effectiveness of the proposed approach.
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.
An optimal version of spline dimensional decomposition (SDD) is unveiled for general high‐dimensional uncertainty quantification analysis of complex systems subject to independent but otherwise ...arbitrary probability measures of input random variables. The resulting method involves optimally derived knot vectors of basis splines (B‐splines) in some or all coordinate directions, whitening transformation producing measure‐consistent orthonormalized B‐splines equipped with optimal knots, and Fourier‐spline expansion of a general high‐dimensional output function of interest. In contrast to standard SDD, there is no need to select the knot vectors uniformly or intuitively. The generation of optimal knot vectors can be viewed as an inexpensive preprocessing step toward creating the optimal SDD. Analytical formulas are proposed to calculate the second‐moment properties by the optimal SDD method for a general output random variable in terms of the expansion coefficients involved. It has been shown that the computational complexity of the optimal SDD method is polynomial, as opposed to exponential, thus mitigating the curse of dimensionality by a discernible magnitude. Numerical results affirm that the optimal SDD method developed is more precise than polynomial chaos expansion, sparse‐grid quadrature, and the standard SDD method in calculating not only the second‐moment statistics, but also the cumulative distribution function of an output random variable. More importantly, the optimal SDD outperforms standard SDD by sustaining nearly identical computational cost.
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.
Motivated by risk assessment of coastal flooding, we consider time-consuming simulators with a spatial output. The aim is to perform sensitivity analysis (SA), quantifying the influence of input ...parameters on the output. There are three main issues. First, due to computational time, standard SA techniques cannot be directly applied on the simulator. Second, the output is infinite dimensional, or at least high dimensional if the output is discretized. Third, the spatial output is non-stationary and exhibits strong local variations.
We show that all these issues can be addressed all together by using functional PCA (FPCA). We first specify a functional basis, such as wavelets or B-splines, designed to handle local variations. Secondly, we select the most influential basis terms, either with an energy criterion after basis orthonormalization, or directly on the original basis with a penalized regression approach. Then FPCA further reduces dimension by doing PCA on the most influential basis coefficients, with an ad-hoc metric. Finally, fast-to-evaluate metamodels are built on the few selected principal components. They provide a proxy on which SA can be done. As a by-product, we obtain analytical formulas for variance-based sensitivity indices, generalizing known formula assuming orthonormality of basis functions.
•Metamodelling method for models with spatial output.•Sensitivity indices for model with spatial output.•Application on coastal flooding study case.
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