Basis Functions Of Degree Research Articles

Locking-free isogeometric discretizations of linear plane Timoshenko rods: LAS elements

We make use of linear plane curved Timoshenko rods as a model problem to study how to overcome shear and membrane locking in NURBS-based discretizations. In this work, we propose lumped-assumed-strain (LAS) elements, a projection-based assumed-strain treatment that removes shear and membrane locking for a very broad range of slenderness ratios. For a NURBS patch with basis functions of degree p and Cp−1 continuity across element boundaries, the space of the assumed strains is defined on a B-spline patch with the same elements, but using basis functions of degree p−1 and Cp−2 continuity across element boundaries. The assumed strains are obtained by performing a L2 projection of the compatible strains at the patch level in which the consistent mass matrix is substituted with the lumped mass matrix. Our numerical investigations suggest that LAS elements need either the same or slightly finer mesh resolutions than B̄ elements in order for the relative errors in L2 norm of the unknowns and the stress resultants to be all below 1%. However, LAS elements are significantly more computationally efficient than B̄ elements for a given mesh. The use of C1-continuous quadratic LAS elements with 2 Gauss–Legendre quadrature points per element exhibit superconvergence of the stress resultants for most slenderness ratios, making it a particularly cost-effective choice of obtaining accurate results. When compared with linear Lagrange B̄ elements with 2 Gauss–Legendre quadrature points per element, C1-continuous quadratic LAS elements with 2 Gauss–Legendre quadrature points per element require far fewer elements in order to obtain relative errors in L2 norm of the unknowns and the stress resultants smaller than 1%.

Efficient and robust quadratures for isogeometric analysis: Reduced Gauss and Gauss–Greville rules

This work proposes two efficient quadrature rules, reduced Gauss quadrature and Gauss–Greville quadrature, for isogeometric analysis. The rules are constructed to exactly integrate one-dimensional B-spline basis functions of degree p, and continuity class Cp−k, where k is the highest order of derivatives appearing in the Galerkin formulation of the problem under consideration. This is the same idea we utilized in Zou et al. (2021), but the rules therein produced negative weights for certain non-uniform meshes. The present work improves upon Zou et al. (2021) in that the weights are guaranteed to be positive for all meshes. The reduced Gauss quadrature rule is built element-wise according to the element basis degree and smoothness. The Gauss–Greville quadrature rule combines the proposed reduced Gauss quadrature and Greville quadrature Zou et al. (2021). Both quadrature rules involve many fewer quadrature points than the full Gauss quadrature rule and avoid negative quadrature weights for arbitrary knot vectors. The proposed quadrature rules are stable and accurate, and they can be constructed without solving nonlinear equations, therefore providing efficient and easy-to-use alternatives to full Gauss quadrature. Various numerical examples, including curved shells, demonstrate that they achieve good accuracy, and for p=5 and 6 eliminate locking.

Weighted isogeometric collocation based on Spline Projectors

In this paper, we propose a novel version of weighted isogeometric collocation that is especially suited for adaptive THB-spline refinement.It is well known that the choice of the collocation nodes is crucial to ensure stability and good approximation properties, especially when adaptive refinement is performed. In order to address this issue, we make use of a particular class of locally supported quasi-interpolant schemes to propose the new method of Weighted Isogeometric Collocation based on Spline Projectors (WICSP).We show that WICSP performs well in the case of tensor-product spline discretizations, both with respect to the rate of convergence and computational complexity. In particular, we observe experimentally an optimal rate of convergence for odd degree basis functions and obtain a dimension-independent computational complexity O(np) for matrix-free applications, similar to other approaches such as weighted quadrature (Calabrò et al., 2017) and clustered collocation (Montardini et al., 2017).We explore how these results extend to the case of adaptively refined THB-spline discretizations. We observe that WICSP is compatible with THB-spline refinement, exhibiting good accuracy and low computational costs. In fact, we get a complexity of O(np2d) for matrix assembly and O(npd) for the matrix-free case. This compares well with the available methods for isogeometric THB-spline discretizations.

Strain gradient elasto-plasticity model: 3D isogeometric implementation and applications to cellular structures

In the present work, we combine Mindlin’s strain gradient elasticity theory and Gudmundson–Gurtin–Anand strain gradient plasticity theory to form a unified framework. The gradient plasticity model is enriched by including the gradient of elastic strains into the expression of the internal virtual work and free energy. This augments the modelling capabilities by incorporating elasticity-related length scales along with plasticity-related energetic and dissipative ones. The strong form governing equations are derived via the principle of virtual work addressing a complete set of boundary conditions. The fourth-order boundary value problem of the gradient elasto-plasticity model is then formulated in a variational form within an H2 Sobolev space setting. Conforming Galerkin discretizations for numerical results are obtained utilizing an isogeometric approach with NURBS basis functions of degree p≥2 providing Cp−1-continuity. The implementation follows a viscoplastic constitutive framework and adopts the backward Euler time integration scheme. A set of benchmark examples is considered to illustrate convergence properties and to accomplish parameter studies. It is shown that the elastic length scale parameter controls the slope of the elastic part and causes an additional hardening in the plastic part of the material response curves. Finally, an illustrative example is considered in order to demonstrate the applicability of both the continuum model and the numerical method in capturing the size-dependent torsion response of cellular structures.

Superconvergent isogeometric collocation method with Greville points

A superconvergent isogeometric collocation method that employs the conventional Greville abscissae as collocation points is proposed. Firstly, a basis transformation is presented for isogeometric basis functions, which preserves the exact geometry representation characterized by isogeometric analysis. More specifically, the geometry is exactly represented by the transformed isogeometric basis functions and the Greville points that function as control points in the current formulation. Meanwhile, the proposed basis transformation enables a very straightforward formulation of the consistency conditions. It also turns out that the transformed isogeometric basis functions span the same solution space as the standard isogeometric basis functions. Subsequently, with the aid of these consistency conditions, an accuracy analysis is presented for multi-dimensional isogeometric collocation formulation and the basis degree discrepancy issue is theoretically identified. In order to resolve this issue, a recursive gradient formulation is further introduced to effectively construct the gradients of transformed isogeometric basis functions. It is shown that these recursive gradients meet additional higher order consistency conditions. Consequently, a superconvergent isogeometric collocation method is finally established, which utilizes the recursive gradients to replace the conventional gradients of isogeometric basis functions and directly collocates at Greville points. The superconvergence of the proposed method regarding odd degree basis functions is then proved, which is also well manifested by numerical results.

Numerical scheme for the far-out-of-equilibrium time-dependent Boltzmann collision operator: 1D second-degree momentum discretisation and adaptive time stepping

Study of far-from-equilibrium thermalisation dynamics in quantum materials, including the dynamics of different types of quasiparticles, is becoming increasingly crucial. However, the inherent complexity of either the full quantum mechanical treatment or the solution of the scattering integral in the Boltzmann approach, has significantly limited the progress in this domain. In our previous work we had developed a solver to calculate the scattering integral in the Boltzmann equation. The solver is free of any approximation (no linearisation of the scattering operator, no close-to-equilibrium approximation, full non-analytic dispersions, full account of Pauli factors, and no limit to low order scattering) [1]. Here we extend it to achieve a higher order momentum space convergence by extending to second degree basis functions. We further use an adaptive time stepper, achieving a significant improvement in the numerical performance. Moreover we show adaptive time stepping can prevent intrinsic instabilities in the time propagation of the Boltzmann scattering operator. This work makes the numerical time propagation of the full Boltzmann scattering operator efficient, stable and minimally reliant on human supervision.

Isogeometric Free Vibration Analysis of Curved Euler–Bernoulli Beams with Particular Emphasis on Accuracy Study

An isogeometric free vibration analysis is presented for curved Euler–Bernoulli beams, where the theoretical study of frequency accuracy is particularly emphasized. Firstly, the isogeometric formulation for general curved Euler–Bernoulli beams is elaborated, which fully takes the advantages of geometry exactness and basis function smoothness provided by isogeometric analysis. Subsequently, in order to enable an analytical frequency accuracy study, the general curved beam formulation is particularized to the circular arch problem with constant radius. Under this circumstance, explicit mass and stiffness matrices are derived for quadratic and cubic isogeometric formulations. Accordingly, the coupled stencil equations associated with the axial and deflectional displacements of circular arches are established. By further invoking the harmonic wave assumption, a frequency accuracy measure is rationally attained for isogeometric free analysis of curved Euler–Bernoulli beams, which theoretically reveals that the isogeometric curved beam formulation with [Formula: see text]th degree basis functions is [Formula: see text]th order accurate regarding the frequency computation. Numerical results well confirm the proposed theoretical convergence rates for both circular arches and general curved beams.

Arbitrary order recursive formulation of meshfree gradients with application to superconvergent collocation analysis of Kirchhoff plates

A general arbitrary order recursive gradient formulation is presented for meshfree approximation. According to this method, an nth order recursive meshfree gradient is formulated as an interpolation of the (n − 1)th order gradients by standard first order meshfree gradients, which finally can be expressed as a successive multiplication of standard first order meshfree gradients. This formulation avoids the complex and costly computation of conventional high order derivatives of meshfree shape functions. One crucial ingredient of the proposed methodology is that the resulting recursive meshfree gradients with a pth degree basis function not only meet the conventional pth order consistency conditions for standard gradients, but also satisfy (p + 1)th to (p + n − 1)th extra high order consistency conditions. This important property leads to superconvergent meshfree collocation algorithms and here we focus on the classical fourth order Kirchhoff plate problems. An accuracy analysis of the proposed recursive gradient meshfree collocation formulation for Kirchhoff plates reveals that superconvergence is simultaneously achieved for both even and odd degrees of basis functions. More specifically, two and four additional orders of accuracy are respectively gained by the proposed method for even and odd degree basis functions, compared with the standard meshfree collocation scheme. Furthermore, the extra high order consistency conditions of recursive meshfree gradients enable superconvergent meshfree collocation analysis of Kirchhoff plates using low order basis functions of less than 4th degree, while the standard meshfree collocation approach requires at least a 4th degree basis function to maintain convergence. The accuracy and efficiency of the proposed methodology are holistically demonstrated by numerical results.

Strong multipatch [formula omitted]-coupling for isogeometric analysis on 2D and 3D domains

The solution spaces of isogoemetric analysis (IGA) constructed from p degree basis functions allow up to Cp−1 continuity within one patch. However, for a multi-patch domain, the continuity is only C0 at the boundaries between the patches. In this study, we present the construction of basis functions of degree p≥2 which are C1 continuous across the common boundaries shared by the patches. The new basis functions are computed as a linear combination of the C0 basis functions on the multi-patch domains. An advantage of the proposed method is that for the new basis functions, the continuity within a patch is preserved, without additional treatment of the functions in the interior of the patch.We apply continuity constraints to the new basis functions to enforce C1 continuity, where the constraints are developed according to the concept of “matched Gk-constructions always yield Ck-continuous isogeometric elements” discussed in Groisser and Peters, (2015). However, for certain geometries, the over-constrained solution space will lead to C1 locking (Collin and Sangalli, 2016). We discuss and show the usage of partial degree elevation to overcome this problem. We demonstrate the potential of the C1 basis functions for IGA applications through several examples involving biharmonic equations.

Stability and error estimates for the successive-projection technique with B-splines in time

Abstract We study the successive projection technique with B-splines proposed by Takizawa and Tezduyar in 2014 (Computational Mechanics, vol. 53). The projection is considered for X -valued functions with a Banach space X . Stability and error estimates in the L ∞ ( 0 , T ; X ) norm are studied for B-spline basis functions of degree p = 1 , 2 , 3 , 4 . The quasi-uniformity of partition is always assumed and the projection is stable if p = 1 . We prove that, for p = 2 , 3 , 4 , the uniformity of partition is a sufficient condition for stability to hold. Furthermore, we infer from numerical experiments that stability holds at least for p = 5 , 6 , 7 . We also prove the error estimate using the spline-preserving property of the projector if the projection is stable.

Skeleton-stabilized immersogeometric analysis for incompressible viscous flow problems

A Skeleton-stabilized ImmersoGeometric Analysis technique is proposed for incompressible viscous flow problems with moderate Reynolds numbers. The proposed formulation fits within the framework of the finite cell method, where essential boundary conditions are imposed weakly using a Nitsche-type method. The key idea of the proposed formulation is to stabilize the jumps of high-order derivatives of variables over the skeleton of the background mesh. The formulation allows the use of identical finite-dimensional spaces for the approximation of the pressure and velocity fields in immersed domains. The stability issues observed for inf–sup stable discretizations of immersed incompressible flow problems are avoided with this formulation. For B-spline basis functions of degree k with highest regularity, only the derivative of order k has to be controlled, which requires specification of only a single stabilization parameter for the pressure field. The Stokes and Navier–Stokes equations are studied numerically in two and three dimensions using various immersed test cases. Oscillation-free solutions and high-order optimal convergence rates can be obtained. The formulation is shown to be stable even in limit cases where almost every elements of the physical domain is cut, and hence it does not require the existence of interior cells. In terms of the sparsity pattern, the algebraic system has a considerably smaller stencil than counterpart approaches based on Lagrange basis functions. This important property makes the proposed skeleton-stabilized technique computationally practical. To demonstrate the stability and robustness of the method, we perform a simulation of fluid flow through a porous medium, of which the geometry is directly extracted from 3D μCT scan data.

A novel extension of the Bézier model and its applications to surface modeling

The construction of Bézier curves and surfaces with shape parameters is one of the most popular areas of research in CAD. In this paper, we present a novel shape-adjustable generalized Bézier curve with multiple shape parameters and discuss its applications to surface modeling in engineering. Firstly, a class of new generalized Bernstein basis functions with explicit expressions is proposed, which is a natural extension of the classical Bernstein basis functions of degree n. Furthermore, the corresponding Bézier curves and surfaces with global and local shape parameters are constructed and their properties are also studied. The shapes of the curves and surfaces can be adjusted intuitively and foreseeably by changing the shape parameters. Secondly, in order to tackle the problem that most complex curves in engineering often cannot be constructed by using a single curve, we derive the necessary and sufficient conditions for C1 and C2 continuity of these new curves. Finally, using shape-adjustable generalized Bézier curves, we construct six different types of engineering surfaces with multiple shape parameters, including general cylinders, bilinear surfaces, ruled surfaces, swung surfaces, swept surfaces and rotation surfaces. The modeling examples show that the proposed methods are effective in geometric modeling.

Polar traveling ionospheric disturbances inferred with the B-spline method and associated scintillations in the Southern Hemisphere

A new method for analyzing travelling ionospheric disturbances (TIDs) is developed by using two B-spline basis functions of degree 4 on the Total Electron Content (TEC) data from the ground-based Global Positioning System (GPS) receivers. This method enhances the spatial resolution to about 0.1° (geographic latitude) × 0.1° (geographic longitude), which is useful in studying all scales (small, medium and large) TIDs. Using this method, we investigated TIDs and their associated scintillation on 18–19 July 2013 at Southern Hemisphere and found phase scintillation is more sensitive than amplitude scintillation to the TIDs at South Pole. To see the full impact of TIDs on scintillation, we have used a proxy phase scintillation index, calculated using geodetic GPS receivers over Antarctica. We have verified the presence of TIDs during these two days by using a Global Navigation Satellite System (GNSS)-TEC single station approach and SuperDARN slant range signals. Our results show the TEC fluctuations are associated with ionospheric scintillation. The shape of TIDs, their elongation and flattening along/across the geographic latitude/longitude, seems to be related to the magnitude and occurrence of ionospheric scintillations. Magnetospheric particle precipitation boost TEC gradients and generate stronger amplitude scintillation, however, large-scale plasma irregularities cause overall enhancement in magnitude of the phase scintillation index. Due to the high turbulence in the polar ionosphere, TIDs change their shapes quite quickly and/or may disappear in the background ionosphere. B-spline TIDs analysis method is very useful in identifying the visible as well as hidden TIDs parts in the polar ionosphere. For the first time, ionospheric scintillation has been investigated in the vicinity of TIDs at high- latitude in the southern hemisphere. Further, the presented B-spline TIDs analysis method is unique and simple in itself as it uses GPS receiver processed TEC data as the primary input. Our results show that at polar latitude it is not necessary that TIDs always appear near the high TEC regions. Usefulness of the B-spline TIDs detection method has been demonstrated in analyzing TIDs at all geographic locations and different solar activity conditions by comparing B-spline TIDs method produced results with the previous case studies.

Superconvergent gradient smoothing meshfree collocation method

A superconvergent meshfree collocation method with smoothed nodal gradients is presented. In this method, the first order smoothed gradients of meshfree shape function are constructed through a meshfree interpolation of the standard derivatives of meshfree shape function, while the second order smoothed gradients are computed through directly differentiating the first order smoothed gradients. It is noted that the second order smoothed gradients evaluated at meshfree nodes can be conveniently expressed as two successive first order gradient smoothing operations on the meshfree shape function, which facilitates a trivial numerical implementation. Subsequently, an employment of the second order smoothed gradients in the strong form of a given problem leads to the gradient smoothing meshfree collocation method purely using nodes as the collocation points. Based upon a local truncation error analysis, it is systematically shown that the proposed meshfree collocation method yields superconvergent solutions for odd degree basis functions. A key ingredient attributed to this superconvergence property is that the second order smoothed gradients meet the consistency conditions which go one order beyond the original basis degree of meshfree approximation. Another interesting fact is that the present formulation enables a convergent collocation scheme when the linear basis function is used in meshfree approximation, which is non-feasible in the conventional collocation formulation. The effectiveness of the proposed methodology is validated by numerical examples for both potential and elasticity problems. Numerical results well demonstrate the superconvergence and higher efficiency of the present gradient smoothing meshfree collocation method.

Skeleton-stabilized IsoGeometric Analysis: High-regularity interior-penalty methods for incompressible viscous flow problems

A Skeleton-stabilized IsoGeometric Analysis (SIGA) technique is proposed for incompressible viscous flow problems with moderate Reynolds number. The proposed method allows utilizing identical finite dimensional spaces (with arbitrary B-splines/NURBS order and regularity) for the approximation of the pressure and velocity components. The key idea is to stabilize the jumps of high-order derivatives of variables over the skeleton of the mesh. For B-splines/NURBS basis functions of degree k with Cα-regularity (0≤α<k), only the derivative of order α+1 has to be controlled. This stabilization technique thus can be viewed as a high-regularity generalization of the (Continuous) Interior-Penalty Finite Element Method. Numerical experiments are performed for the Stokes and Navier–Stokes equations in two and three dimensions. Oscillation-free solutions and optimal convergence rates are obtained. In terms of the sparsity pattern of the algebraic system, we demonstrate that the block matrix associated with the stabilization term has a considerably smaller bandwidth when using B-splines than when using Lagrange basis functions, even in the case of C0-continuity. This important property makes the proposed isogeometric framework practical from a computational effort point of view.

Shape preserving properties of univariate Lototsky–Bernstein operators

The main goal of this paper is to study shape preserving properties of univariate Lototsky–Bernstein operators Ln(f) based on Lototsky–Bernstein basis functions. The Lototsky–Bernstein basis functions bn,k(x)(0≤k≤n) of order n are constructed by replacing x in the ith factor of the generating function for the classical Bernstein basis functions of degree n by a continuous nondecreasing function pi(x), where pi(0)=0 and pi(1)=1 for 1≤i≤n. These operators Ln(f) are positive linear operators that preserve constant functions, and a non-constant function γnp(x). If all the pi(x) are strictly increasing and strictly convex, then γnp(x) is strictly increasing and strictly convex as well. Iterates LnM(f) of Ln(f) are also considered. It is shown that LnM(f) converges to f(0)+(f(1)−f(0))γnp(x) as M→∞. Like classical Bernstein operators, these Lototsky–Bernstein operators enjoy many traditional shape preserving properties. For every (1,γnp(x))-convex function f∈C[0,1], we have Ln(f;x)≥f(x); and by invoking the total positivity of the system {bn,k(x)}0≤k≤n, we show that if f is (1,γnp(x))-convex, then Ln(f;x) is also (1,γnp(x))-convex. Finally we show that if all the pi(x) are monomial functions, then for every (1,γn+1p(x))-convex function f, Ln(f;x)≥Ln+1(f;x) if and only if p1(x)=⋯=pn(x)=x.

An arbitrary-order Runge–Kutta discontinuous Galerkin approach to reinitialization for banded conservative level sets

We present a Runge–Kutta discontinuous Galerkin method for solving conservative reinitialization in the context of the conservative level set method [18,19]. This represents an extension of the method recently proposed by Owkes and Desjardins [21], by solving the level set equations on the refined level set grid [9] and projecting all spatially-dependent variables into the full basis used by the discontinuous Galerkin discretization. By doing so, we achieve the full k+1 order convergence rate in the L1 norm of the level set field predicted for RKDG methods given kth degree basis functions when the level set profile thickness is held constant with grid refinement. Shape and volume errors for the 0.5-contour of the level set, on the other hand, are found to converge between first and second order. We show a variety of test results, including the method of manufactured solutions, reinitialization of a circle and sphere, Zalesak's disk, and deforming columns and spheres, all showing substantial improvements over the high-order finite difference traditional level set method studied for example by Herrmann [9]. We also demonstrate the need for kth order accurate normal vectors, as lower order normals are found to degrade the convergence rate of the method.

Spline-based DQM for multi-dimensional PDEs: Application to biharmonic and Poisson equations in 2D and 3D

The idea of differential quadrature is used to construct a new algorithm for the solution of differential equations. To determine the weighting coefficients of DQM, B-spline basis functions of degree r are used as test functions. The method is constructed on a set of points mixed from grid points and mid points of a uniform partition. Using the definition of B-splines interpolation as alternative, some error bounds are obtained for DQM. The method is successfully implemented on nonlinear boundary value problems of order m. Also the application of the proposed method to approximate the solution of multi-dimensional elliptic PDEs is included in the paper. As test problem, some examples of biharmonic and Poisson equations are solved in 2D and 3D. The results are compared with some existing methods to show the efficiency and performance of the proposed algorithm. Also some examples of time dependent PDEs are solved to compare the results with other existing spline based DQ methods.

Isogeometric approximation of cardiac electrophysiology models on surfaces: An accuracy study with application to the human left atrium

We consider Isogeometric Analysis in the framework of the Galerkin method for the spatial approximation of cardiac electrophysiology models defined on NURBS surfaces; specifically, we perform a numerical comparison between basis functions of degree p=1 and globally Ck-continuous, with k=0 or p-1, to find the most accurate approximation of a propagating front with the minimal number of degrees of freedom. We show that B-spline basis functions of degree p=1, which are Cp-1-continuous capture accurately the front velocity of the transmembrane potential even with moderately refined meshes; similarly, we show that, for accurate tracking of curved fronts, high-order continuous B-spline basis functions should be used. Finally, we apply Isogeometric Analysis to an idealized human left atrial geometry described by NURBS with physiologically sound fiber directions and anisotropic conductivity tensor to demonstrate that the numerical scheme retains its favorable approximation properties also in a more realistic setting.

Observations and modeling of ionospheric scintillations at South Pole during six X‐class solar flares in 2013

AbstractUsing two B‐spline basis functions of degree 4 and the ionospheric scintillation data from a Global Positioning Satellite System (GPS) scintillation receiver at South Pole, we reproduced ionospheric scintillation indices for the periods of the six X‐class solar flares in 2013. These reproduced indices have filled the data gaps, and they are serving as a smooth replica of the real observations. In either event, these modeled scintillation indices are minimizing the geometrical effects between GPS satellite and the receiver. Six X‐class solar flares have been studied during the summer and winter months, using the produced scintillation indices based on the observations from the GPS receiver at South Pole and the in situ plasma measurement from the associated passing of Defense Meteorological Satellite Program. Our results show that the solar flare peak suppresses the scintillation level and builds time‐independent scintillation patterns; however, after a certain time from the solar flare peak, complicated scintillation patterns develop at high‐latitude ionosphere and spread toward the polar cap boundary region. Substantial consistency has been found between moderate proton fluxes and scintillation enhancement.