In this study, the authors present a uniform algebraic trigonometric tension B‐spline‐based differential quadrature method combined with an optimized hybrid block method to numerically solve the ...Rosenau–KdV–RLW equation. The discrete mass and energy have been calculated, showing that they are conserved, thus indicating the efficiency and accuracy of the present approach. The method outperforms other methods and does not require linearization, allowing it to be directly implemented for solving nonlinear partial differential equations.
We propose three novel consistent specification tests for quantile regression models which generalize former tests in three ways. First, we allow the covariate effects to be quantile‐dependent and ...nonlinear. Second, we allow parameterizing the conditional quantile functions by appropriate basis functions, rather than parametrically. We are thereby able to test for general functional forms, while retaining linear effects as special cases. In both cases, the induced class of conditional distribution functions is tested with a Cramér–von Mises type test statistic for which we derive the theoretical limit distribution and propose a bootstrap method. Third, a modified test statistic is derived to increase the power of the tests. We highlight the merits of our tests in a detailed MC study and two real data examples. Our first application to conditional income distributions in Germany indicates that there are not only still significant differences between East and West but also across the quantiles of the conditional income distributions, when conditioning on age and year. The second application to data from the Australian national electricity market reveals the importance of using interaction effects for modeling the highly skewed and heavy‐tailed distributions of energy prices conditional on day, time of day and demand.
A simply and effectively computational optimization for porosity-dependent isogeometric analysis of functionally graded (FG) sandwich nanoplates is proposed for the first time. Porosity-dependent ...material properties are defined via the modified power law function. The distribution of ceramic volume fraction is approximated by using the multi-patch B-spline basis functions through the thickness direction. This approach ensures smoothly and continuously vary material properties across each layer, and automatically satisfies the C0-continuity at each layer interfaces. To consider length scale effects, the Eringen’s nonlocal elasticity theory is used to model porous FG sandwich nanoplates. Based on a combination of NURBS formulations and four variables refined plate theory, governing equations of the nanoplates are derived and employed to obtain natural frequencies of the porous FG sandwich nanoplates. The present approximation is easy to satisfy the requirement of at least third order derivatives of basis functions in approximate formulations of nanoplates. To save computational costs, an adaptive hybrid evolutionary firefly algorithm is used. Continuous design variables including the thickness of each layer and the ceramic volume fraction at control points are considered for constraint optimization problems. New results are performed and considered as benchmark results for further studies on the porous FG sandwich nanoplates.
GeoPDEs (http://rafavzqz.github.io/geopdes) is an Octave/Matlab package for the solution of partial differential equations with isogeometric analysis, first released in 2010. In this work we present ...in detail the new design of the package, based on the use of Octave and Matlab classes. Compared to previous versions the new design is much clearer, and it is also more efficient in terms of memory consumption and computational time.
Powell‐Sabin B‐splines are employed to model progressive fracturing in a fluid‐saturated porous medium. These splines are defined on triangles and are 𝒞1‐continuous throughout the domain, including ...the crack tips, so that crack initiation can be evaluated directly at the tip. On one hand, the method captures stresses and fluid fluxes more accurately than when using standard Lagrange elements, enabling a direct assessment of the fracture criterion at the crack tip and ensuring local mass conservation. On the other hand, the method avoids limitations for discrete crack analysis which adhere to isogeometric analysis. A crack is introduced directly in the physical domain. Due to the use of triangles, remeshing and crack path tracking are straightforward. During remeshing transfer of state vectors (displacement, fluid pressure) is required from the old to the new mesh. The transfer is done using a new approach which exploits a least‐squares fit with the energy balance and conservation of mass as constraints. The versatility and accuracy to simulate free crack propagation are assessed for mode‐I and mixed‐mode fracture problems.
Generalized B-splines (GB-splines) are a special class of Tchebycheff B-splines that are smooth piecewise function with sections in more general spaces. GB-splines allow for an exact representation ...of conic sections as well as transcendental curves and thus they become very attractive for geometrical modeling and numerical simulation. In this paper, we present overlapping Schwarz preconditioners for elliptic problems discretized with isogeometric analysis based on GB-splines. An h-analysis of the proposed preconditioners shows an optimal convergence rate bound that is scalable in the number of subdomains and that is linear in the ratio between subdomain and overlap sizes. Numerical results in two- and three-dimensional tests confirm this analysis and also illustrate the good convergence properties of the preconditioner with respect to the discretization parameters, the domain deformation and the jumping coefficients.
•Generalized B-splines (GB-splines) are a special class of Tchebycheff B-splines that are smooth piecewise function with sections in more general spaces.•We present overlapping additive Schwarz (OAS) preconditioners for elliptic problems discretized with isogeometric analysis based on GB-splines.•We prove that the 2-level OAS preconditioners have an optimal convergence rate bound.
The aim of this study is to improve the numerical solution of the modified equal width wave equation. For this purpose, finite difference method combined with differential quadrature method with ...Rubin and Graves linearizing technique has been used. Modified cubic B‐spline base functions are used as base function. By the combination of two numerical methods and effective linearizing technique high accurate numerical algorithm is obtained. Three main test problems are solved for various values of the coefficients. To observe the performance of the present method, the error norms of the single soliton problem which has analytical solution are calculated. Besides these error norms, three invariants are reported. Comparison of the results displays that our algorithm produces superior results than those given in the literature.
An algebraic characterization of B-splines Kamont, Anna; Passenbrunner, Markus
Journal of mathematical analysis and applications,
08/2023, Letnik:
524, Številka:
1
Journal Article
Recenzirano
Odprti dostop
B-splines of order k can be viewed as a mapping N taking a (k+1)-tuple of increasing real numbers a0<⋯<ak and giving as a result a certain piecewise polynomial function. Looking at this mapping N as ...a whole, basic properties of B-spline functions imply that it has the following algebraic properties: (1) N(a0,…,ak) has local support contained in the interval a0,ak; (2) N(a0,…,ak) allows refinement, i.e. for every a∈∪j=0k−1(aj,aj+1) we have that if (α0,…,αk+1) is the increasing rearrangement of the points {a0,…,ak,a}, the ‘old’ function N(a0,…,ak) is a linear combination of the ‘new’ functions N(α0,…,αk) and N(α1,…,αk+1); (3) N is translation and dilation invariant. It is easy to see that derivatives of N(a0,…,ak) satisfy properties (1)–(3) as well.
Let F be a mapping taking (k+1)-tuples of increasing real numbers to some generalized function. In this paper we show that under some additional mild condition on the size of the supports of F(a0,…,ak) relative to the interval a0,ak, properties (1)–(3) are already sufficient to deduce that F(a0,…,ak) is a non-zero multiple of (some derivative of) a B-spline function. However, and somewhat surprisingly, we explicitly give examples of choices of F satisfying (1)–(3) but are not of this form.
This paper addresses fluid‐driven crack propagation in a porous medium. Cohesive interface elements are employed to model the behaviour of the crack. To simulate hydraulic fracturing, a fluid ...pressure degree of freedom is introduced inside the crack, separate from the fluid degrees of freedom in the bulk. Powell‐Sabin B‐splines, which are based on triangles, are employed to describe the geometry of the domain and to interpolate the field variables: displacements and interstitial fluid pressure. Due to their C1$\mathcal {C}^1$‐continuity, the stress and pressure gradient are smooth throughout the whole domain, enabling a direct assessment of the fracture criterion at the crack tip and ensuring local mass conservation. Due to the use of triangles, crack insertion and remeshing are straightforward and can be done directly in the physical domain. During remeshing a mapping of the state vector (displacement and interstitial fluid pressure) is required. For this, a new methodology is exploited based on a least‐square fit with the energy balance and mass conservation as constraints. The accuracy to model free crack propagation is demonstrated by two numerical examples, including crack propagation in a plate with two notches.
The basic idea of this article is to investigate the numerical solutions of Gardner Kawahara equation, a particular case of extended Korteweg‐de Vries equation, by means of finite element method. For ...this purpose, a collocation finite element method based on trigonometric quintic B‐spline basis functions is presented. The standard finite difference method is used to discretize time derivative and Crank–Nicolson approach is used to obtain more accurate numerical results. Then, von Neumann stability analysis is performed for the numerical scheme obtained using collocation finite element method. Several numerical examples are presented and discussed to exhibit the feasibility and capability of the finite element method and trigonometric B‐spline basis functions. More specifically, the error norms L2$$ {L}_2 $$ and L∞$$ {L}_{\infty } $$ are reported for numerous time and space discretization values in tables. Graphical representations of the solutions describing motion of wave are presented.
The basic idea of this article is to investigate the numerical solutions of Gardner Kawahara equation, a particular case of extended Korteweg‐de Vries equation, by means of finite element method. For this purpose, a collocation finite element method based on trigonometric quintic B‐spline basis functions is presented. The standard finite difference method is used to discretize time derivative and Crank–Nicolson approach is used to obtain more accurate numerical results. Then, von Neumann stability analysis is performed for the numerical scheme obtained using collocation finite element method. Several numerical examples are presented and discussed to exhibit the feasibility and capability of the finite element method and trigonometric B‐spline basis functions. More specifically, the error norms L2$$ {L}_2 $$ and L∞$$ {L}_{\infty } $$ are reported for numerous time and space discretization values in tables. Graphical representations of the solutions describing motion of wave are presented.
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