High-order Lagrange Polynomials Research Articles

Component-wise modelling of core and skin in sandwich structures using a single refined beam finite element

This study presents an efficient component-wise representation of both core and skin components of a sandwich structure using one-dimensional (1D) finite elements. The approach considers a core made of hexagonal cells and is based on the Carrera Unified Formulation (CUF). CUF enables the modeling of complex structures as 1D beam models by accurately defining their cross-sectional geometry using high-order Lagrange polynomials. This method ensures the exact representation of both geometry and deformation characteristics. Unlike traditional 1D finite elements definition, in this study the structural beam axis is aligned with the thickness of the sandwich panel, while the cross-section extends across its width. In this way, both core and skins are described with their actual geometry, without the need of adopting any homogenization. Static linear analyses demonstrate that the proposed CUF-based advanced 1D modeling achieves structural response accuracy comparable to that of three-dimensional (3D) FEM models implemented in Abaqus. Notably, this is accomplished with a significant reduction in the degrees of freedom, highlighting the computational efficiency of CUF framework. These results underline the potential of advanced CUF-based 1D modeling for applications involving sandwich structures with any core geometry.

Buckling Analysis of Laminated Composite Beams by Using an Improved First Order Formulation

In this work, a finite element model based on an improved first-order formulation (IFSDT) is developed to analyze buckling phenomenon in laminated composite beams. The formulation has five independent variables and takes into account thickness stretching. Three-dimensional constitutive equations are employed to define the material properties. The Trefftz criterion is used for the stability analysis. The finite element model is derived from the principle of virtual work with high-order Lagrange polynomials to interpolate the field variables and to prevent shear locking. Numerical results are compared and validated with those available in literature. Furthermore, a parametric study is presented.

Aiding and Opposing Re-circulating Mixed Convection Flows in a Square Vented Enclosure

A numerical study is performed to investigate mixed convective flow in a vertically vented square enclosure with heating or cooling one of the vertical walls and the other three walls are insulated. This study is conducted to analyse and clarify the intricate interaction between the buoyancy-induced flow and the pressure-driven external flow in a vertical upward laminar air jet for both assisting and opposing buoyancy situations. The approach of spectral-element that is introduced as a finite-element method with high-order Lagrange polynomials as interpolating functions, is applied to discretise unsteady Navier-Stokes, continuity and energy equations, and for calculating the velocity and temperature fields. The numerical solutions are obtained for air having a Prandtl number of (Pr = 0.7) and for buoyancy parameters which cover the whole mixed convection region, ranging from pure forced convection with a Reynolds number ranging between (10 ⩽Re ⩽350) to pure free convection with a Richardson number ranging between (0 ⩽Ri ⩽30), with a single dimensionless enclosure size. The results reveal that the flow and thermal patterns can be significantly influenced by both aiding and opposing buoyancies. Also, the mean heat transfer rates increase with increasing either the buoyancy force Richardson number or the forced flow Reynolds number in assisting flows. However, in opposing flows, it is found that the mean heat transfer rates decrease with increasing the buoyancy effects for small to moderate (Ri ⩽2) but, they begin rising for higher Richardson number when the free convection starts domination the flow.

Spectral-element simulations of elastic wave propagation in exploration and geotechnical applications

We apply the spectral-element method (SEM), a high-order finite-element method (FEM) to simulate seismic wave propagation in complex media for exploration and geotechnical problems. The SEM accurately treats geometrical complexities through its flexible FEM mesh and accurately interpolates wavefields through high-order Lagrange polynomials. It has been a numerical solver used extensively in earthquake seismology. We demonstrate the applicability of SEM for selected 2D exploration and geotechnical velocity models with an open-source SEM software package SPECFEM2D. The first scenario involves a marine survey for a salt dome with the presence of major internal discontinuities, and the second example simulates seismic wave propagation for an open-pit mine with complex surface topography. Wavefield snapshots, synthetic seismograms, and peak particle velocity maps are presented to illustrate the promising use of SEM for industrial problems.

On accuracy and performance of high-order finite volume methods in local mean energy model of non-thermal plasmas

In this paper, a high-order finite volume method is employed to solve the local energy approximation model equations for a radio-frequency plasma discharge in a one-dimensional geometry. The so called deferred correction technique, along with high-order Lagrange polynomials, is used to calculate the convection and diffusion fluxes. Temporal discretization is performed using backward difference schemes of first and second orders. Extensive numerical experiments are carried out to evaluate the order and level of accuracy as well as computational efficiency of the various methods implemented in the work. These tests exhibit global convergence rate of up to fourth order for the spatial error, and of up to second order for the temporal error.

A New High-order Time-kernel BIEM for the Burgers Equation

This paper presents a new high-order time-kernel boundary-integral-equation method (BIEM) for numerically solving transient problems governed by the Burgers equation. Instead of using high-order Lagrange polynomials such as quadratic and quartic interpolation functions, the proposed method employs integrated radial-basis-function networks (IRBFNs) to represent the unknown functions in boundary and volume integrals. Numerical implementations of ordinary and double integrals involving time in the presence of IRBFNs are discussed in detail. The proposed method is verified through the solution of diffusion and convection-diffusion problems. A comparison of the present results and those obtained by low-order BIEMs and other methods is also given.

High-order triangle-based discontinuous Galerkin methods for hyperbolic equations on a rotating sphere

High-order triangle-based discontinuous Galerkin (DG) methods for hyperbolic equations on a rotating sphere are presented. The DG method can be characterized as the fusion of finite elements with finite volumes. This DG formulation uses high-order Lagrange polynomials on the triangle using nodal sets up to 15th order. The finite element-type area integrals are evaluated using order 2 N Gauss cubature rules. This leads to a full mass matrix which, unlike for continuous Galerkin (CG) methods such as the spectral element (SE) method presented in Giraldo and Warburton [A nodal triangle-based spectral element method for the shallow water equations on the sphere, J. Comput. Phys. 207 (2005) 129–150], is small, local and efficient to invert. Two types of finite volume-type flux integrals are studied: a set based on Gauss–Lobatto quadrature points (order 2 N − 1) and a set based on Gauss quadrature points (order 2 N). Furthermore, we explore conservation and advection forms as well as strong and weak forms. Seven test cases are used to compare the different methods including some with scale contractions and shock waves. All three strong forms performed extremely well with the strong conservation form with 2 N integration being the most accurate of the four DG methods studied. The strong advection form with 2 N integration performed extremely well even for flows with shock waves. The strong conservation form with 2 N − 1 integration yielded results almost as good as those with 2 N while being less expensive. All the DG methods performed better than the SE method for almost all the test cases, especially for those with strong discontinuities. Finally, the DG methods required less computing time than the SE method due to the local nature of the mass matrix.

A nodal triangle-based spectral element method for the shallow water equations on the sphere

A nodal triangle-based spectral element (SE) method for the shallow water equations on the sphere is presented. The original SE method uses quadrilateral elements and high-order nodal Lagrange polynomials, constructed from a tensor-product of the Legendre–Gauss–Lobatto points. In this work, we construct the high-order Lagrange polynomials directly on the triangle using nodal sets obtained from the electrostatics principle [J.S. Hesthaven, From electrostatics to almost optimal nodal sets for polynomial interpolation in a simplex, SIAM Journal on Numerical Analysis 35 (1998) 655–676] and Fekete points [M.A. Taylor, B.A. Wingate, R.E. Vincent, An algorithm for computing Fekete points in the triangle, SIAM Journal on Numerical Analysis 38 (2000) 1707–1720]. These points have good approximation properties and far better Lebesgue constants than any other nodal set derived for the triangle. By employing triangular elements as the basic building-blocks of the SE method and the Cartesian coordinate form of the equations, we can use any grid imaginable including adaptive unstructured grids. Results for six test cases are presented to confirm the accuracy and stability of the method. The results show that the triangle-based SE method yields the expected exponential convergence and that it can be more accurate than the quadrilateral-based SE method even while using 30–60% fewer grid points especially when adaptive grids are used to align the grid with the flow direction. However, at the moment, the quadrilateral-based SE method is twice as fast as the triangle-based SE method because the latter does not yield a diagonal mass matrix.

A non-destructive evaluation of the material properties of a composite laminated plate

A non-destructive method for the evaluation of material properties of a rectangular, anisotropic, homogeneous plate with four free edges is presented. The method consists of two steps. In the first step, a certain number of the plate's natural frequencies are experimentally measured. In the second step, the plate rigidities are varied in a theoretical model, so that the calculated natural frequencies match as close as possible the corresponding experimental values. Two such models are presented, based on the Classical Lamination Theory and on a Higher Order Shear Deformation Theory. High order Lagrange polynomials are used as deflection functions and the Rayleigh-Ritz procedure is employed to arrive at the solution. The identification of the plate rigidities is done by means of an iterative Bayesian parameter estimation method, where possible measurement errors or rigidities' uncertainties can be taken into account.