Basis Functions Of Degree Research Articles

Dynamic models for Large Eddy Simulation of compressible flows with a high order DG method

The impact of dynamic models for applications to LES of compressible flows is assessed in the framework of a numerical model based on high order discontinuous finite elements. The projections onto lower dimensional subspaces associated with lower degree basis functions are used as LES filter, along the lines proposed in Variational Multiscale templates. Comparisons with DNS results available in the literature for plane and constricted channel flows at Mach numbers 0.2, 0.7 and 1.5 show clearly that the dynamic models are able to improve the prediction of most key features of the flow with respect to the Smagorinsky models employed so far in a VMS-DG context.

A novel extension to the polynomial basis functions describing Bezier curves and surfaces of degree n with multiple shape parameters

The construction of Bézier curves using shape control parameters is one of the most popular areas of research in computer aided geometric design (CAGD). A class of new polynomial basis functions with n−1 local shape control parameters is presented here to allow the construction of Bézier curves with n local shape control parameters, which is an extension to the classical Bernstein basis functions of degree n. The properties of the proposed basis functions and the corresponding piecewise polynomial curve with n−1 local shape control parameters are analyzed. This analysis shows that the new class of polynomial functions meets the conditions required for both C0, C1 and C2 continuity as well as G0, G1 and G2 continuity. Some curve design applications are then discussed and an extended application for surface design is also presented. The modeling examples illustrate that the new extension provides not only a better approximation and mathematical description of Bézier curves, but allows the shape parameters to be altered, making it a valuable method for the design of curves and surfaces.

Legendre Galerkin method for weakly singular Fredholm integral equations and the corresponding eigenvalue problem

In this paper, we consider Galerkin method for weakly singular Fredholm integral equations of the second kind and its corresponding eigenvalue problem using Legendre polynomial basis functions of degree ≤n. We obtain the convergence rates for the approximated solution and iterated solution in weakly singular Fredholm integral equations of the second kind and also obtain the error bounds for the approximated eigenelements in the corresponding eigenvalue problem. We illustrate our results with numerical examples.

An analysis of the spectrum of the discontinuous Galerkin method

We derive explicit expressions for the eigenvalues (spectrum) of the discontinuous Galerkin spatial discretization applied to the linear advection equation. We show that the eigenvalues are related to the subdiagonal [p/p+1] Padé approximation of exp(−z) when pth degree basis functions are used. We derive an upper bound on the eigenvalue with the largest magnitude as (p+1)(p+2). We demonstrate that this bound is not tight and prove that the asymptotic growth rate of the spectral radius is slower than quadratic in p. We also analyze the behavior of the spectrum near the imaginary axis to demonstrate that the spectral curves approach the imaginary axis although there are no purely imaginary eigenvalues.

The k-Version of Finite Element Method for Initial Value Problems: Mathematical and Computational Framework

This paper presents a mathematical and computational framework for initial value problems (IVP) in which the numerical approximations can be of higher order global differentiability in space and time and the resulting computational processes are unconditionally stable. This is accomplished using Hk, p scalar product spaces containing basis functions of degree p = (p1,p2), p1 and p2 being the degrees of local approximation in space and time and order k = (k1, k2), k1 and k2 being orders of the scalar product space in space and time and ensuring that the integral forms are space-time integral forms that are space-time variationally consistent (STVC). It is shown that order of the scalar product space k (in space and time) is an intrinsically important independent parameter in all finite element computations for IVP in addition to the discretization length h and the degree p of the local approximations, thus in all finite element computations all quantities are dependent on h, p and k. Hence, we have k-version of finite element method and associated k, hk, pk and hpk processes in addition to h, p and hp processes for IVP. Space-time meshes as well as space-time, time marching approaches are discussed and it is demonstrated that space-time, time marching processes are superior in all aspects compared to space-time meshes. Significant features of the proposed mathematical and computational framework are presented and discussed mathematically in context with Galerkin method, Petrov Galerkin method, Weighted Residual method, Galerkin method with weak form and Least Squares Processes.

Biorthogonal bases with local support and approximation properties

We construct locally supported basis functions which are biorthogonal to conforming nodal finite element basis functions of degree p in one dimension. In contrast to earlier approaches, these basis functions have the same support as the nodal finite element basis functions and reproduce the conforming finite element space of degree p - 1. Working with Gaus-Lobatto nodes, we find an interesting connection between biorthogonality and quadrature formulas. One important application of these newly constructed biorthogonal basis functions are two-dimensional mortar finite elements. The weak continuity condition of the constrained mortar space is realized in terms of our new dual bases. As a result, local static condensation can be applied which is very attractive from the numerical point of view. Numerical results are presented for cubic mortar finite elements.

THE k-VERSION OF FINITE ELEMENT METHOD FOR SELF-ADJOINT OPERATORS IN BVP

In this paper a new mathematical and computational framework for boundary value problems described by self-adjoint differential operators is presented. In this framework, numerically computed solutions, when converged, possess the same degree of global smoothness in terms of differentiability up to any desired order as the theoretical solutions. This is accomplished using spaces Ĥk,p that contain basis functions of degree p and order k - 1 (or the order of the space k). It is shown that the order of space k is an intrinsically important independent parameter in all finite element computational processes in addition to the discretization characteristic length h and the degree of basis functions p when the theoretical solutions are analytic. Thus, in all finite element computations, all quantities of interest (e.g., quadratic functional, error or residual functional, norms and seminorms, error norms, etc.) are dependent on h, p as well as k. Therefore, for fixed h and p, convergence of the finite element process can also be investigated by changing k, hence k-convergence and thus the k-version of finite element method. With h, p, and k as three independent parameters influencing all finite element processes, we now have k, hk, pk, and hpk versions of finite element methods. The issue of minimally conforming finite element spaces is reexamined and it is demonstrated that the definition of currently believed minimally conforming space which permit weak convergence of the highest-order derivatives of the dependent variables appearing in the bilinear form is not justifiable mathematically or from physics view point. A new criterion is proposed for establishing the minimally conforming spaces which is more in agreement with the physics and mathematics of the BVP. Significant features and merits of the proposed mathematical and computational framework are presented, discussed, illustrated, and substantiated mathematically as well as numerically with the Galerkin and least-squares finite element formulations for self-adjoint boundary-value problems.

A discontinuous quadratic finite element method for one-dimensional diffusion problems

In this paper, we discuss the discontinuous quadratic finite element method for one-dimensional diffusion problems. We prove the stability for the polynomial basis functions of degree = 2, which is indicated by numerical experiments in the recent paper of Babuška, Baumann and Oden [1].

Analysis of chebyshev finite element method for linear differential equations using equal interval elements

In this paper, the authors present an analysis of the finite element solution to linear differential equations using new Chebyshev polynomial basis functions of degree r (1≤ r < ∞) in the interval [0, h]. The advantage of this technique over our recent result in [3] is that the finite element solutions obtained for the class of linear differential equations is suitable for computer automation.

Timoshenko beam finite elements using trigonometric basis functions

A Timoshenko beam finite element is developed using trigonometric basis functions. The properties and performance of these new elements are explored through a series of illustrative problems that treat both straight and curved geometries. Both linear and nonlinear strain-displacement relations are employed in the formulation and comparison is made to results obtained from full, completely reduced, and partially reduced integration of polynomial basis functions of degree one through three. The results obtained in this work indicate that the trigonometric basis functions are a competitive alternative to polynomial basis functions with regard to accuracy and convergence and most importantly they possess a freedom from shear and membrane locking. The trigonometric basis functions also allow the recovery of rigid-body motions in the case of straight and circular arc curved beams.

Higher order Chebyshev basis functions for two‐point boundary value problems

AbstractCubic basis functions in one dimension for the solution of two‐point boundary value problems are constructed based on the zeros of Chebyshev polynomials of the first kind. A general formula is derived for the construction of polynomial basis functions of degree r, where 1 ≤r &lt; ∞. A Galerkin finite element method using the constructed basis functions for the cases r = 1, 2 and 3 is successfully applied to three different types of problem including a singular perturbation problem.