Order Of Interpolation Polynomials Research Articles

3D frequency-domain airborne EM forward modelling using spectral element method with Gauss–Lobatto–Chebyshev polynomials

ABSTRACTThe spectral element method (SEM) based on high-order complete orthogonal polynomials is an accurate and efficient numerical method for electromagnetic modelling due to its spectral accuracy and exponential convergence. The SEM combines the flexibility of the finite-element method and the high accuracy of the spectral method. In this paper, we introduce SEM into three-dimensional frequency-domain airborne electromagnetic forward modelling. Starting from Maxwell's equations, we obtain a vector Helmholtz equation for the electric field. We use the Galerkin method to discretise the Helmholtz equation, in which the curl-conforming Gauss–Lobatto–Chebyshev polynomials are used as basis functions. The GLC polynomials help to derive the analytical expressions of entries in the system matrix and thus guarantee the modelling accuracy. Finally, we use the direct solver MUMPS to solve for the electric field and calculate the magnetic field by interpolation. For numerical experiments, we first compare our results with the semi-analytical solutions of a homogeneous half-space to verify the accuracy of our algorithm. We then analyse the characteristics of SEM by assuming different orders of interpolation polynomials and meshes. We also compare our method with the finite element method and SEM based on Gauss–Lobatto–Legendre polynomials. The results show that SEM is an efficient and effective method for electromagnetic modelling, it can deliver very accurate results and is less sensitive to mesh quality than the finite element method.

Derived Nodes Approach for Improving Accuracy of Machining Stability Prediction

Machining process dynamics can be described by state-space delayed differential equations (DDEs). To numerically predict the process stability, diverse piecewise polynomial interpolation is often utilized to discretize the continuous DDEs into a set of linear discrete equations. The accuracy of discrete approximation of the DDEs generally depends on how to deal with the piecewise polynomials. However, the improvement of the stability prediction accuracy cannot be always guaranteed by higher-order polynomials due to the Runge phenomenon. In this study, the piecewise polynomials with derivative-continuous at joint nodes are taken into consideration. We develop a recursive estimation of derived nodes for interpolation approximation of the state variables, so as to improve the discretization accuracy of the DDEs. Two different temporal discretization methods, i.e., second-order full-discretization and state-space temporal finite methods, are taken as demonstrations to illustrate the effectiveness of applying the proposed approach for accuracy improvement. Numerical simulations prove that the proposed approach brings a great improvement on the accuracy of the stability lobes, as well as the rate of convergence, compared to the previous recorded ones with the same order of interpolation polynomials.

A comparative study of finite element and spectral element methods in seismic wavefield modeling

The computational accuracy and efficiency of finite element method and spectral element method (SEM) are investigated thoroughly in time-domain elastic wavefield modeling. The diagonal mass matrices of the FEM and SEM free from matrix inversion are compared comprehensively by making full use of the mass-lumped technique and quadrature rules. We investigate the FEM and SEM based, respectively, on quadrilateral with the polynomials of degrees one and two, and on triangular grids with the polynomials of degrees one and three. Generally, the numerical solutions based on quadrilateral grids have a higher precision than those computed on triangular grids when the same order of polynomials is used. The FEM has a comparable accuracy to the SEM with the same number of interpolant points. In view of the triangular and quadrilateral SEMs, the former suffers from larger computational costs and relatively lower accuracy compared with the latter. Furthermore, the convergence study proves that the triangular SEM produces consistently larger errors than the quadrilateral SEM for any order and element sizes. However, the triangular SEM can adapt to arbitrary complex geometries effectively. In terms of efficiency, the FEM has an efficiency comparable with the SEM on condition that the order of interpolation polynomials is identical. In addition, a perfectly matched layer (PML) boundary condition in variational form is deduced. By introducing four intermediate variables in frequency domain, the PML avoids convolution calculation and obtains an exact solution through inverse Fourier transform in time domain. The numerical examples verify the validity and effectiveness of the PML.

HAMILTONIAN AS ERROR INDICATOR IN THE P-VERSION OF FINITE ELEMENT METHOD

This paper presents the Hamiltonian-based error analysis applied to two-dimensional elasostatic problems. The accuracy enhancement is achieved by using the p-version of finite element method. The results show that the Hamiltonian error has faster rates of convergence at lower order of interpolation polynomials to compare with the energy error, and the Hamiltonian error clearly indicates great error reductions at a certain polynomial order. This can not only obtain an accurate enough solution but also save extra computational time. Another strategy is presented by computing the residual of the Hamiltonian-based governing equations. Relative values of residuals between elements can provide an index of selecting the best polynomial orders. Illustrative examples show the validities of the two approaches.

On efficiency improvement of demolished concrete, recycled aggregate and recycled aggregate concrete testing

Recycled aggregate (RA) has been used in various construction applications around the world, mainly as sub-grade, roadwork and unbound materials, but not in high-grade applications. This paper examines relationships among demolished concrete (DC) characteristics, properties of their RA and strength of their recycled aggregate concrete (RAC). A Vandermonde matrix for interpolation polynomials coefficient estimation is used to give detailed mathematical relationships among pairs of samples, which can be used to determine redundant tests. Different orders of interpolation polynomials are used for comparison, hence the best-fit equations with the lowest fitting errors from different orders of polynomials can be found.

Optimization of Aggregate Testing Using Polynomial Regression

Recycled Aggregate (RA) has only been used as sub-grade applications, roadwork and unbound materials because it is believed that RA has lower strength and poorer quality than normal aggregate. This paper assesses RA quality by proposing a new testing procedure. The new procedure is developed by using correlation information among the standard twenty three tests commonly employed in the construction industry. Samples of RA from ten demolition sites were obtained with service life ranging from ten to forty years. One additional set of samples was specifically collected from the Tuen Mun Area 38 Recycling Plant, Hong Kong for testing. The characteristics of these eleven sets of samples were then compared with normal aggregate samples. A Vandermonde matrix for interpolation polynomials coefficient estimation is employed to yield detailed mathematical relationships between results of two particular tests, from that redundant tests can be identified. Different orders of interpolation polynomials are used for comparison, hence the best-fit equations with lowest fitting errors from different orders of polynomials can be found. This paper shows that four out of twenty three tests are required in the new testing procedure.

Optimal aggregate testing using Vandermonde polynomials and spectral methods

Recycled aggregate (RA) has been used in various construction applications around the world mainly as sub-grade, roadwork and unbound materials, but not in higher-grade applications. The major barrier encountered is the variation of quality within RA, which causes lower strength, and poorer quality. This work studies the relationships among six parameters describing the characteristics of RA: (i) particle size distribution, (ii) particle density, (iii) porosity and absorption, (iv) particle shape, (v) strength and toughness, and (vi) chemical composition. Samples of RA from 10 demolition sites were obtained with service life ranging from 10 to 40 years. One additional set of samples was specifically collected from the Tuen Mun Area 38 Recycling Plant. The characteristics of these eleven sets of samples were then compared with normal aggregate samples. A Vandermonde matrix for interpolation polynomial coefficient estimation is used to give detailed mathematical relationships among pairs of samples, which can be used to work out redundant tests. Different orders of interpolation polynomials are used for comparison, hence the best-fit equations with the lowest fitting errors from different orders of polynomials can be found. Fitting error distributions are then studied by using spectral methods such as power spectra and bispectra. From that, the best equations for result estimations can be obtained. This study reveals that there is strong correlation among test parameters, and by measuring two of them: either “particle density” or “porosity and absorption” or “particle shape” or “strength and toughness”, and “chemical content”, it is sufficient to study RA.