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.
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.
This paper presents a new, highly effective approach for optimal smooth trajectory planning of high-speed pick-and-place parallel robots. The pick-and-place path is decomposed into two orthogonal ...coordinate axes in the Cartesian space and quintic B-spline curves are used to generate the motion profile along each axis for achieving C4-continuity. By using symmetrical properties of the geometric path defined, the proposed motion profile becomes essentially dominated by two key factors, representing the ratios of the time intervals for the end-effector to move from the initial point to the adjacent virtual and/or the via-points on the path. These two factors can then be determined by maximizing a weighted sum of two normalized single-objective functions and expressed by curve fitting as functions of the width/height ratio of the pick-and-place path, so allowing them to be stored in a look-up table to enable real-time implementation. Experimental results on a 4-DOF SCARA type parallel robot show that the residual vibration of the end-effector can be substantially reduced thanks to the very continuous and smooth joint torques obtained.
Extended B‐spline‐based implicit material point method Yamaguchi, Yuya; Moriguchi, Shuji; Terada, Kenjiro
International journal for numerical methods in engineering,
15 April 2021, Letnik:
122, Številka:
7
Journal Article
Recenzirano
An implicit material point method (MPM) is enhanced by extended B‐splines with the aim of achieving numerical stability and properly imposing boundary conditions. The standard B‐spline basis ...functions capable of suppressing the cell‐crossing error are used for the approximation over the domain, whereas extended B‐splines (EBS) are active for boundary cells, each of which is occupied by a small physical domain. The basis consisting of EBS plays a central role to avoid both stress oscillation arising from inaccurate numerical integration and ill‐conditioning of their resulting tangent matrices, both of which are caused by the boundary cells containing smaller number of material points than interior ones. Besides the standard material points for the interior region, boundary points are introduced to explicitly represent the boundary geometries and their movements, and to weakly impose an arbitrary boundary condition with the Nitsche's method. The Nitsche's terms in the linearized weak form are decomposed into normal and tangential directions to deal with both slip and nonslip boundary conditions. Four case studies are made to demonstrate the performance and capability of the proposed method, named EBS‐MPM, in stably solving the quasi‐static equilibrium problems for hyperelastic bodies and properly applying arbitrary boundary conditions.
On n‐dimensional quadratic B‐splines Raslan, K. R.; Ali, Khalid K.
Numerical methods for partial differential equations,
March 2021, 2021-03-00, 20210301, Letnik:
37, Številka:
2
Journal Article
Recenzirano
In this paper, we present new constructions of the n‐dimensional quadratic B‐splines method. The quadratic B‐splines method format is displayed in a single, two, and three‐dimensional format. These ...constructions are of utmost importance in solving differential equations in their various dimensions, which have applications in many fields of science. Some numerical examples are also presented, through which we can show the accuracy of this construction by giving some comparisons of our results using this new construction and between the results in other works. Some figures also give in this paper for the solutions.
Hierarchical B-splines, which possess the local refinement capability, have been recognized as a useful tool in the context of isogeometric analysis. However, similar as for tensor-product B-splines, ...isogeometric simulations with hierarchical B-splines face a big computational burden from the perspective of matrix assembly, particularly if the spline degree p is high. To address this issue, we extend the recent work (Pan et al., 2020) – which introduced an efficient assembling approach for tensor-product B-splines – to the case of hierarchical B-splines. In the new approach, the integrand factor is transformed into piecewise polynomials via quasi-interpolation. Subsequently, the resulting elementary integrals are pre-computed and stored in a look-up table. Finally, the sum-factorization technique is adopted to accelerate the assembly process. We present a detailed analysis, which reveals that the presented method achieves the expected complexity of O(pd+1) per degree of freedom (without taking sparse matrix operations into account) under the assumption of mesh admissibility. We verify the efficiency of the new method by applying it to an elliptic problem on the three-dimensional domain and a parabolic problem on the four-dimensional domain in space–time, respectively. A comparison with standard Gaussian quadrature is also provided.
Summary
The isogeometric formulation of the boundary element method (IgA‐BEM) is investigated within the adaptivity framework. Suitable weighted quadrature rules to evaluate integrals appearing in ...the Galerkin BEM formulation of 2D Laplace model problems are introduced. The proposed quadrature schemes are based on a spline quasi‐interpolation (QI) operator and properly framed in the hierarchical setting. The local nature of the QI perfectly fits with hierarchical spline constructions and leads to an efficient and accurate numerical scheme. An automatic adaptive refinement strategy is driven by a residual‐based error estimator. Numerical examples show that the optimal convergence rate of the Galerkin solution is recovered by the proposed adaptive method.
We derive a collection of fundamental formulas for quantum B-splines analogous to known fundamental formulas for classical B-splines. Starting from known recursive formulas for evaluation and quantum ...differentiation along with quantum analogues of the Marsden identity, we derive quantum analogues of the de Boor–Fix formula for the dual functionals, explicit formulas for the quantum B-splines in terms of divided differences of truncated power functions, formulas for computing divided differences of arbitrary functions by quantum integrating certain quantum derivatives of these functions with respect to the quantum B-splines, closed formulas for the quantum integral of the quantum B-splines over their support, and finally a 1/q-convolution formula for uniform q-B-splines.
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