Order Basis Functions Research Articles

Isogeometric analysis of thin Reissner–Mindlin shells: locking phenomena and B-bar method

We propose a local type of B-bar formulation, addressing locking in degenerated Reissner–Mindlin shell formulation in the context of isogeometric analysis. Parasitic strain components are projected onto the physical space locally, i.e. at the element level, using a least-squares approach. The formulation allows the flexible utilization of basis functions of different orders as the projection bases. The introduced formulation is much cheaper computationally than the classical $$\bar{B}$$ method. We show the numerical consistency of the scheme through numerical examples, moreover they show that the proposed formulation alleviates locking and yields good accuracy even for slenderness ratios of $$10^5$$ , and has the ability to capture deformations of thin shells using relatively coarse meshes. In addition it can be opined that the proposed method is less sensitive to locking with irregular meshes.

Influences of surface effects on large deflections of nanomembranes with arbitrary shapes by the coupled BE-RBFs method

This paper aims to analyze large deflections of nanomembranes including surface effects with arbitrary shapes. The nonlinear differential equations of nanomembranes are formulated by using the nonlinear kinematic relations and the surface elasticity theory of Gurtin–Murdoch. The principle of virtual work is applied to establish the three governing partial differential equations of nanomembranes. The coupled boundary element-radial basis functions (BE-RBFs) method is developed to solve the complicated nonlinear problem of nanomembranes. The proposed methodology is based on the analog equation method in conjunction with radial basis functions in order that the boundary line integrals and boundary elements are only involved. The validation and accuracy of the present method are evaluated by comparing the obtained results with those available from other numerical solutions. The proposed formulation can provide the numerical results that correspond to the experimental findings of the monolayer circular graphene membrane by specifying the proper surface properties. The influences of the surface elastic constants and residual surface stress on large deflection responses of nanomembranes are evidently investigated. Moreover, some numerical results of the present formulations could serve as a benchmark for the numerical evaluation of future research. Finally, the interesting results of large deflection analysis of the various nanomembrane shapes using the coupled BE-RBFs method are highlighted.

A New Hybrid Finite Element Method: Electromagnetic Propagation in Bent Waveguides

A new hybrid finite element method (HFEM) was developed, using basis functions of first and second order, to analyse the electromagnetic wave propagation in three types of waveguides: disk waveguides, ring waveguides and cylindrical waveguides. For validation purposes, results obtained with the HFEM method were compared with those obtained with a recursive finite element method (RFEM) and with two different analytical methods. The domain discretization effect on the results of the new HFEM method was examined using either a uniform distribution of nodes or a Delaunay distribution method. The results show that the Delaunay method, used in the mesh definition, is important, not only to assure the reliability of the method but also to significantly reduce the computation time and memory consumption. Also, the influence of the distance between the fictitious boundary and the waveguide core was analysed showing an important effect on the result numerical accuracy. In conclusion, results show that the new hybrid finite element method presented has a significant advantage on calculation time over the recursive finite element method.

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.

Multiscale Finite Element Method for heat transfer problem during artificial ground freezing

In this work, we consider the heat transfer problem during artificial ground freezing. We present the mathematical model and the fine grid approximation for heterogeneous porous media, where freezing pipes are considered as line source terms. Mathematical model is described by a two-phase Stefan problem. Fine grid approximation is performed using a finite element method. The main goal of the work is the construction of the coarse grid approximation using the Generalized Multiscale Finite Element Method (GMsFEM). In GMsFEM, we solve local spectral problem in order to determine main heat flow characteristics in the heterogeneous porous media and construct special additional multiscale basis functions in order to perform an accurate approximation of the line source terms (freezing pipes). We present numerical results for two heat transfer model problems in the two and three-dimensional formulations and investigate accuracy of the proposed multiscale methods for different numbers of the multiscale basis functions.

A robust multi‐sampling rate digital predistortion for ultra‐broadband power amplifiers

AbstractIn this letter, a robust multisampling rate digital predistortion (DPD) scheme is proposed to linearize ultra‐broadband power amplifiers (PAs) under the limited DAC sampling rate. In the ultra‐broadband scenarios, it is no longer affordable to employ DACs with 3‐5 times over‐sampling rate, which will inevitably lead to spectrum aliasing for the sake of high order basis function and compromise the DPD performance. In order to improve the modeling accuracy, the input signal with low sampling rate will be up‐sampled firstly to generate nonaliasing basis functions. Then, after filtering and down‐sampling operation, the coefficients are extracted with low sampling rate. Moreover, the conventional behavioral model is modified with the linear‐spline basis functions to improve the stability and complexity. Measurements on Doherty PA and class‐AB PA with 100 MHz, 200 MHz, and 500 MHz bandwidth input signal at 2.1 GHz carrier frequency show the EVM can be improved to 0.71%, 0.94%, and 1.86% respectively.

Numerical approximation of the 3D hydrostatic Navier–Stokes system with free surface

In this paper we propose a stable and robust strategy to approximate the 3D incompressible hydrostatic Euler and Navier–Stokes systems with free surface. Compared to shallow water approximation of the Navier–Stokes system, the idea is to use a Galerkin type approximation of the velocity field with piecewise constant basis functions in order to obtain an accurate description of the vertical profile of the horizontal velocity. Such a strategy has several advantages. It allowsto rewrite the Navier–Stokes equations under the form of a system of conservation laws with source terms,the easy handling of the free surface, which does not require moving meshes,the possibility to take advantage of robust and accurate numerical techniques developed in extensive amount for Shallow Water type systems.Compared to previous works of some of the authors, the three dimensional case is studied in this paper. We show that the model admits a kinetic interpretation including the vertical exchanges terms, and we use this result to formulate a robust finite volume scheme for its numerical approximation. All the aspects of the discrete scheme (fluxes, boundary conditions, ...) are completely described and the stability properties of the proposed numerical scheme (well-balancing, positivity of the water depth, ...) are discussed. We validate the model and the discrete scheme with some numerical academic examples (3D non stationary analytical solutions) and illustrate the capability of the discrete model to reproduce realistic tsunami waves propagation, tsunami runup and complex 3D hydrodynamics in a raceway.

Data‐Driven Inference of Thermosphere Composition During Solar Minimum Conditions

AbstractMass density variations can deviate from the expected behavior caused by temperature due to changes in the composition. Such deviations can be especially significant during solar minimum conditions. Model‐data differences are typically resolved through temperature corrections while overlooking the role of errors in lower boundary composition. In this work, we use a data‐driven methodology to simultaneously estimate thermosphere composition and temperature contributions to model‐data differences. The methodology uses modal decomposition to extract high‐dimensional, reduced order basis functions for the covariance of the neutral thermospheric species and temperature. The extracted basis functions are combined with CHAllenging Minisatellite Payload and Gravity Recovery And Climate Experiment mass density measurements using a nonlinear least squares solver. We demonstrate the methodology using the Naval Research Laboratory's empirical Mass Spectrometer and Incoherent Scatter (MSIS) model to derive the high‐dimensional basis functions. We characterize and quantify the contribution of temperature and lower boundary effects with oxygen and helium since the two species have a direct impact on drag and orbit prediction through gas‐surface interactions and mass density. We analyze the month of December in 2008, based on the work of Thayer et al. (2012), and estimate that lower boundary composition errors contribute approximately 50% of the model‐data differences.

High-order compact finite difference scheme for option pricing in stochastic volatility jump models

We derive a new high-order compact finite difference scheme for option pricing in stochastic volatility jump models, e.g. in Bates model. In such models the option price is determined as the solution of a partial integro-differential equation. The scheme is fourth order accurate in space and second order accurate in time. Numerical experiments for the European option pricing problem are presented. We validate the stability of the scheme numerically and compare its performance to standard finite difference and finite element methods. The new scheme outperforms a standard discretisation based on a second-order central finite difference approximation in all our experiments. At the same time, it is very efficient, requiring only one initial $LU$-factorisation of a sparse matrix to perform the option price valuation. Compared to finite element approaches, it is very parsimonious in terms of memory requirements and computational effort, since it achieves high-order convergence without requiring additional unknowns, unlike finite element methods with higher polynomial order basis functions. The new high-order compact scheme can also be useful to upgrade existing implementations based on standard finite differences in a straightforward manner to obtain a highly efficient option pricing code.

An Approximate Solution of Functionally Graded Timoshenko Beam Using B-Spline Collocation Method

Collocation methods are popular in providing numerical approximations to complicated governing equations owing to their simplicity in implementation. However, point collocation methods have limitations regarding accuracy and have been modified upon with the application of B-spline approximations. The present study reports the stress and deformation behavior of shear deformable functionally graded cantilever beam using B-spline collocation technique. The material grading is along the beam height and varies according to power law. Poisson’s ratio is assumed to be a constant. The equations are derived using virtual work principle in the framework of Timoshenko beams to obtain a unified formulation for such beams. A sixth order basis function is used for approximation and collocation points are generated using Greville abscissa. Deformation and stresses; bending (axial) stresses and transverse (shear) stresses, and position of neutral axis are studied for a wide range of power law index values. The results are reported along the beam cross-section and beam length.

Scattered data approximation of fully fuzzy data by quasi-interpolation

Fuzzy quasi-interpolations help to reduce the complexity of solving a linear system of equations compared with fuzzy interpolations. Almost all fuzzy quasi-interpolations are focused on the form of $widetilde{f}^{*}:mathbb{R}rightarrow F(mathbb{R})$ or $widetilde{f}^{*}:F(mathbb{R})rightarrow mathbb{R}$. In this paper, we intend to offer a novel fuzzy radial basis function by the concept of source distance. Then, we will construct a fuzzy linear combination of such basis functions in order to introduce a fully fuzzy quasi-interpolation in the form of $widetilde{f}^{*}:F(mathbb{R})rightarrow F(mathbb{R})$. Also the error estimation of the proposed method is proved in terms of the fully fuzzy modulus of continuity which will be introduced in this paper. Finally some examples have been given to emphasize the acceptable accuracy of our method.

Finite element methods for the linear regularized 13-moment equations describing slow rarefied gas flows

The paper is concerned with the numerical solution of the linear steady-state regularized 13-moment equations in two space dimensions. To facilitate the understanding of the specific challenges, the equations are first divided into two subsystems before the full system is approached. The arising problems mainly stem from the complicated saddle-point structure as well as the non-standard nature of the boundary conditions. A continuous interior penalty method is presented and the pronounced advantages of utilizing high order basis functions in this setting are illustrated. To render the presented approach more efficient, a hybridization technique is presented that originates in the discontinuous Galerkin method.

A unified framework for the multi-scale computational homogenisation of 3D-textile composites

This paper extends the applications of a novel and fully automated multi-scale computational homogenisation framework, originally proposed by the authors (Ullah et al. (2017)) for unidirectional and 2D-textile composites, to 3D-textile composites. 3D-textile composites offer many advantages over 2D-textile composites but their highly complicated and unpredictable post-cured geometries make their design very challenging. Accurate computational models are therefore essential to the development of these materials. The computational framework described in this paper possesses a variety of novel features which have never been tried for this class of composites and can potentially help to fully automatise and improve their design process. A unified approach is used to impose the representative volume element boundary conditions, which allows convenient switching between linear displacement, uniform traction and periodic boundary conditions. The computational framework is implemented using hierarchic basis functions of arbitrary polynomial order, which allows one to increase the order of approximation without changing the finite element mesh. The yarns' principal directions, required for the transversely isotropic material model are calculated using a potential flow analysis along these yarns. This feature is very useful for 3D-textile composites and can accurately determine fibres’ directions even in the case of very deformed yarns. A numerical example from literature consisting of a 3D-orthogonal woven composite is used to demonstrate the correct implementation and performance of the developed computational framework. Also, the developed computational framework is used to perform a comparative study of the homogenised mechanical properties of five 3D-textile composites with different yarn architectures.

An inherently consistent reproducing kernel gradient smoothing framework toward efficient Galerkin meshfree formulation with explicit quadrature

A reproducing kernel gradient smoothing framework with explicit quadrature rules is proposed for efficient Galerkin meshfree formulation. In this framework, the meshfree smoothed gradients are formulated via a reproducing kernel representation of the standard gradients of field variables. It is interesting to find that this reproducing kernel construction of smoothed gradients of meshfree shape functions enables an identical satisfaction of the integration constraint derived from Galerkin meshfree formulation. In other words, the integration consistency is an inherent property of the proposed reproducing kernel gradient smoothing methodology regardless of integration schemes. Consequently, without violation of the integration constraint, the conventional normal low order Gauss quadrature rules in finite element analysis now can be used to properly integrate the meshfree stiffness matrix simply through replacing the standard meshfree gradients with the reproducing kernel smoothed gradients at Gauss points. In order to efficiently compute the reproducing kernel smoothed gradients, a set of explicit quadrature rules referring to 2D triangular and 3D tetrahedral integration cells are systematically developed. The total number of sample points associated with these quadrature rules is minimized from a global point of view through introducing an equivalent number of sample points, which implies as many as sample points are shared by neighboring integration cells. The proposed methodology recovers the stabilized conforming nodal integration when linear basis function is employed, while the focus of the present work is multidimensional higher order basis functions such as quadratic and cubic ones. Superior convergence, accuracy as well as efficiency performances of the proposed reproducing kernel gradient smoothing framework are thoroughly demonstrated by a series of numerical examples.

Analysis of Finite Antenna Arrays Using the Characteristic Modes of Isolated Radiating Elements

This paper proposes an efficient method for analyzing the radiation, scattering, and port properties of finite antenna arrays based on the theory of characteristic modes. Using the characteristic modes of isolated radiating elements, we obtain the electrical properties of a finite antenna array, which is different from the traditional modal analysis method for the entire array. The modal excitation coefficients of radiating elements in an array environment are derived by considering the mutual coupling effect that is characterized from the perspective of modal coupling between different elements. The characteristic modes of isolated radiating elements are analyzed using the method of moments. The proposed method can handle the array problems with arbitrary spacing between elements and arbitrary excitation distributions. Since the characteristic currents are used as higher order basis functions and the periodicity of the arrays is considered, the computational efficiency could be significantly improved. Numerical experiments are provided to demonstrate the proposed method.

Evaluation of 4-D Reaction Integrals Via Double Application of the Divergence Theorem

The use of the method of moments to solve surface integral equations is one of the most popular numerical techniques in electromagnetic modeling and analysis. This method requires the accurate and efficient numerical evaluation of iterated surface integrals over both source and testing domains. In this paper, we propose a scheme for evaluating these 4-D interaction integrals between pairs of arbitrarily positioned and oriented elements. The approach is based on applying the surface divergence theorem twice, once on the source and once on the test domain. When the integrations are reordered as two outer contour integrals plus two inner radial integrals, the initial radial integrations provide significant smoothing of the underlying singular integrands. The method is numerically validated for static and dynamic kernels arising in the electric field integral equation, i.e., for kernels with 1/R singularities, and linear basis functions. The proposed formula to evaluate 4-D reaction integrals can be extended to different kernels and to different elements, e.g., to curved or volumetric elements, and to basis functions of higher order.

Multi-dimensional spectral gap optimization of order parameters (SGOOP) through conditional probability factorization.

Spectral gap optimization of order parameters (SGOOP) [P. Tiwary and B. J. Berne, Proc. Natl. Acad. Sci. U. S. A. 113, 2839 (2016)] is a method for constructing the reaction coordinate (RC) in molecular systems, especially when they are plagued with hard to sample rare events, given a larger dictionary of order parameters or basis functions and limited static and dynamic information about the system. In its original formulation, SGOOP is designed to construct a 1-dimensional RC. Here we extend its scope by introducing a simple but powerful extension based on the notion of conditional probability factorization where known features are effectively washed out to learn additional and possibly hidden features of the energy landscape. We show how SGOOP can be used to proceed in a sequential and bottom-up manner to (i) systematically probe the need for extending the dimensionality of the RC and (ii) if such a need is identified, learn additional coordinates of the RC in a computationally efficient manner. We formulate the method and demonstrate its utility through three illustrative examples, including the challenging and important problem of calculating the kinetics of benzene unbinding from the protein T4L99A lysozyme, where we obtain excellent agreement in terms of dissociation pathway and kinetics with other sampling methods and experiments. In this last case, starting from a larger dictionary of 11 order parameters that are generic for ligand unbinding processes, we demonstrate how to automatically learn a 2-dimensional RC, which we then use in the infrequent metadynamics protocol to obtain 16 independent unbinding trajectories. We believe our method will be a big step in increasing the utility of SGOOP in performing intuition-free sampling of complex systems. Finally, we believe that the utility of our protocol is amplified by its applicability to not just SGOOP but also other generic methods for constructing the RC.

On Lototsky–Bernstein operators and Lototsky–Bernstein bases

We study generalizations of the classical Bernstein operators, the Lototsky–Bernstein operators, within the framework of non-polynomial spaces Un. It is shown that for any strictly increasing function p1(x)∈C[0,1] with p1(0)=0,p1(1)=1, there exist Lototsky–Bernstein operators that can fix p1(x) and still approximate all continuous functions uniformly on [0,1]. We study these special Lototsky–Bernstein operators, where the Lototsky–Bernstein basis functions of order n are generated by n specially chosen functions 0≤pi(x)≤1(1≤i≤n): all the pj(x)(j≥2) are determined recursively from p1(x). Concerning these functions, we derive two important results. Our first result shows that the average of the n functions pi(x)(1≤i≤n) converges uniformly; moreover, the convergence of pn(x) to x is also investigated. In addition, we give conditions on p1(x) ensuring all the pj(x)(j≥2) are increasing. Our second result says that the Lototsky–Bernstein operators Ln(f;x) are monotonic in n just as in the case of the classical Bernstein operators when f is (1,p1)-convex.

Nonconforming mixed finite elements for linear elasticity on simplicial grids

This paper introduces a new family of nonconforming mixed finite elements for solving the linear elasticity equations on simplicial grids. Besides, this paper describes the construction of the lowest order basis functions. The construction only involves simple computations due to the new explicit stress shape function spaces and the procedure applies for high order cases. Numerical experiments for four benchmark problems in mechanics indicate the robust locking‐free behavior and show that the lowest order nonconforming mixed method leads to smaller stress errors than the first and second order standard Galerkin methods for the nearly incompressible case.

Kirchhoff–Love shells within strain gradient elasticity: Weak and strong formulations and an [formula omitted]-conforming isogeometric implementation

A strain gradient elasticity model for shells of arbitrary geometry is derived for the first time. The Kirchhoff–Love shell kinematics is employed in the context of a one-parameter modification of Mindlin’s strain gradient elasticity theory. The weak form of the static boundary value problem of the generalized shell model is formulated within an H3 Sobolev space setting incorporating first-, second- and third-order derivatives of the displacement variables. The strong form governing equations with a complete set of boundary conditions are derived via the principle of virtual work. A detailed description focusing on the non-standard features of the implementation of the corresponding Galerkin discretizations is provided. The numerical computations are accomplished with a conforming isogeometric method by adopting Cp−1-continuous NURBS basis functions of order p≥3. Convergence studies and comparisons to the corresponding three-dimensional solid element simulation verify the shell element implementation. Numerical results demonstrate the crucial capabilities of the non-standard shell model: capturing size effects and smoothening stress singularities.