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Computational Heat Transfer
Published in Greg F. Naterer, Advanced Heat Transfer, 2018
The spectral element method is a type of weighted residual finite element method that approximates the governing equations by a weak formulation. Similarly to the weighted residual method, the differential equation is multiplied by an arbitrary basis function and integrated over the domain. The interpolation and basis functions are normally high-order polynomials up to 10th order. The high-order accuracy leads to a rapid convergence of the method. Since there are a large number of integrations due to the high-order accuracy, a very efficient integration procedure must be used. Sun and Li (2010) applied a spectral element method to coupled heat conduction and radiation problems. The solution domain for the radiation equation of transfer and energy equation was discretized by spectral elements. The results showed a high accuracy and exponential convergence of results, even though a relatively small number of nodes were used in the simulations.
Computational Numerical Methods in Engineering
Published in Abul Hasan Siddiqi, Mohamed Al-Lawati, Messaoud Boulbrachene, Modern Engineering Mathematics, 2017
Abul Hasan Siddiqi, Mohamed Al-Lawati, Messaoud Boulbrachene
The spectral element method represents a special case of Galerkin methods in which the finite dimensional space of test functions is made of continuous piecewise algebraic polynomials of high degree on each element of a partition of the computational domain. For ease of exposition, we will focus only on the one-dimensional problem: Find u(x) such that: () {−u″+u=fu(0)=u(1)=0.
Introduction to Spectral Finite Element Formulation
Published in Srinivasan Gopalakrishnan, Elastic Wave Propagation in Structures and Materials, 2023
A spectral element method is essentially a finite element method formulated in the frequency domain. However, the method of implementation is quite different. The basic differences between the SFEM and conventional FEM are highlighted in the following paragraphs.
Numerical investigations on model order reduction to SEM based on POD-DEIM to linear/nonlinear heat transfer problems
Published in Numerical Heat Transfer, Part B: Fundamentals, 2021
Yazhou Wang, Guoliang Qin, Kumar K. Tamma, Dean Maxam, David Tae
and are the Legendre-Gauss-Lobatto (LGL) points respectively defined as the zeros of and Usually in the spectral element, we use the same polynomial degree in both ξ and η directions, namely we set Figure 2 shows the physical element and the spectral element as well as the distributions of LGL points of order N = 7. The LGL points can be mapped to the physical element using Eq. (4), yielding a high-order computational mesh in the physical element ultimately. Figure 3 shows the distributions of elements and LGL points at different degrees in the physical coordinates. Here we can see that the spectral element method is an h/p-type method; hence the numerical accuracy can be improved by either refining the elements or increasing the polynomial degree, and especially high-order schemes (shape function of large degree) can be easily obtained and programmed owing to the general formulation of shape functions.
Direct numerical simulation of transitional flow in a finite length curved pipe
Published in Journal of Turbulence, 2018
Amirreza Hashemi, Paul F. Fischer, Francis Loth
In this work, the spectral element method (SEM) was employed using an open-source code, nek5000. This Navier–Stokes equation solver is a high fidelity parallel code which breaks the computational domain in to a number of elements. The SEM code represents velocity and pressure as Nth-order tensor-product polynomials within each of K computational mesh cells-bricks. The total number of grid points is approximately KN3. The polynomials can be differentiated to compute derived quantities and provide for accurate high Reynolds number solutions with minimal numerical dissipation and dispersion. In turbulent flows, fluctuations are present with many different wavelengths that propagate in all directions. DNS simulations require high-resolution grid density to capture the entire range of these turbulent wavelengths. The computational mesh or node spacing size must be less than the wavelength or the number of nodes in each direction should be more than wavenumbers of the travelling waves. The performance advantage of the SEM is a rapid exponential convergence in space with scalable multigrid solvers and parallel processing.
A hybrid discontinuous spectral element method and filtered mass density function solver for turbulent reacting flows
Published in Numerical Heat Transfer, Part B: Fundamentals, 2020
Jonathan Komperda, Zia Ghiasi, Dongru Li, Ahmad Peyvan, Farhad Jaberi, Farzad Mashayek
Although low-order finite volume and finite difference methods have become the primary choice of many commercial computational fluid dynamics codes, and have been proven successful in simulating a wide variety of problems in engineering design, many researchers seek to benefit from higher order methods [40]. The motivation for high-order discretization schemes stemmed from the need to reduce errors in simulations while improving computational efficiency and minimizing numerical dissipation [41]. Recent advances in computing have brought high-order spectral element methods to light. The spectral element method is a high-order technique that combines the geometric flexibility of finite elements with the high accuracy of infinitely differentiable spectral functions [42]. Spectral methods are discretization schemes for the solution of partial differential equations in weak form. They employ high-order Lagrange interpolants in conjunction with particular quadrature rules, such as Gauss–Legendre or Chebyshev–Gauss. It is assumed that the solution can be expressed as a series of polynomial basis functions, which can approximate the solution accurately as the polynomial degree tends to infinity [43]. Patera and Maday pioneered the use of spectral element methods [44–47]. Later, Kopriva extended the spectral element method to staggered-grid spectral methods, known as STAC3M, and placed the method in a discontinuous Galerkin framework, known as dGSEM [42, 48–50]. The discontinuous spectral element method (DSEM) was then presented by Jacobs et. al. [51, 52] for the simulation of three-dimensional (3-D) compressible flows.