Numerical Basis Functions Research Articles

Prediction model of aircraft hinge moment: Compressed sensing based on proper orthogonal decomposition

By hinge moment, we mean the aerodynamic torque exerted on the rudder shaft by the airflow passing through the aircraft control surface, with obtaining high-precision results often relying on wind tunnel tests. Due to the complex aerodynamic balance insulation and installation errors that must be considered in cryogenic wind tunnels, the main method for calculating hinge moments is to directly integrate surface pressure distribution information. However, it is usually difficult to arrange enough pressure taps, resulting in the accuracy failing to meet expectations. Combining the sparse wind tunnel test data and low-precision computational fluid dynamics results, this paper introduces the compressed sensing based on proper orthogonal decomposition (CS-POD) method and presents the sub-Ma model and the full-Ma model for predicting hinge moments. The number of sensors and sensor positions are determined based on the sparsity of the numerical simulations and basis functions. Then, the CS algorithm solves the basis coefficients. Finally, the hinge moments are obtained by integrating the reconstruction pressure distribution which is calculated by linearly combining the basis functions and basis coefficients. The result shows that the full-Ma model exhibits higher prediction accuracy with approximately five sensors under subsonic and transonic cases, reducing the relative error of the sub-Ma model by 2–10 times, even at high angles of attack. The mean reconstruction accuracy for the hinge moments is 97.6%, and for the normal forces, it is 94.3%. Therefore, adding relevant terms when the number of samples is small can effectively improve modeling accuracy.

Atomic Electronic Structure Calculations with Hermite Interpolating Polynomials.

We have recently described the implementation of atomic electronic structure calculations within the finite element method with numerical radial basis functions of the form χμ(r) = r-1Bμ(r), where high-order Lagrange interpolating polynomials (LIPs) were used as the shape functions Bμ(r). In this work, we discuss how χμ(r) can be evaluated in a stable manner at small r and also revisit the choice of the shape functions Bμ(r). Three kinds of shape functions are considered: in addition to the continuous LIPs, we consider the analytical implementation of first-order Hermite interpolating polynomials (HIPs) that are continuous, as well as numerical implementations of n-th order ( continuous) HIPs that are expressed in terms of an underlying high-order LIP basis. Furnished with the new implementation, we demonstrate that the first-order HIPs are reliable even with large numbers of nodes and that they also work with nonuniform element grids, affording even better results in atomic electronic structure calculations than LIPs with the same total number of basis functions. We demonstrate that discontinuities can be observed in the spin-σ local kinetic energy τσ in small LIP basis sets, while HIP basis sets do not suffer from such issues; however, either set can be used to reach the complete basis set limit with smooth τσ. Moreover, we discuss the implications of HIPs on calculations with meta-GGA functionals with a number of recent meta-GGA functionals, and we find most Minnesota functionals to be ill-behaved. We also examine the potential usefulness of the explicit control over the derivative in HIPs for forming numerical atomic orbital basis sets, but we find that confining potentials are still likely a better option.

Meta-GGA Density Functional Calculations on Atoms with Spherically Symmetric Densities in the Finite Element Formalism.

Density functional calculations on atoms are often used for determining accurate initial guesses as well as generating various types of pseudopotential approximations and efficient atomic-orbital basis sets for polyatomic calculations. To reach the best accuracy for these purposes, the atomic calculations should employ the same density functional as the polyatomic calculation. Atomic density functional calculations are typically carried out employing spherically symmetric densities, corresponding to the use of fractional orbital occupations. We have described their implementation for density functional approximations (DFAs) belonging to the local density approximation (LDA) and generalized gradient approximation (GGA) levels of theory as well as Hartree-Fock (HF) and range-separated exact exchange [Lehtola, S. Phys. Rev. A 2020, 101, 012516]. In this work, we describe the extension to meta-GGA functionals using the generalized Kohn-Sham scheme, in which the energy is minimized with respect to the orbitals, which in turn are expanded in the finite element formalism with high-order numerical basis functions. Furnished with the new implementation, we continue our recent work on the numerical well-behavedness of recent meta-GGA functionals [Lehtola, S.; Marques, M. A. L. J. Chem. Phys. 2022, 157, 174114]. We pursue complete basis set (CBS) limit energies for recent density functionals and find many to be ill-behaved for the Li and Na atoms. We report basis set truncation errors (BSTEs) of some commonly used Gaussian basis sets for these density functionals and find the BSTEs to be strongly functional dependent. We also discuss the importance of density thresholding in DFAs and find that all of the functionals studied in this work yield total energies converged to 0.1 μEh when densities smaller than 10-11a0-3 are screened out.

AN OPEN-SOURCE MATLAB IMPLEMENTATION FOR ELASTIC ANALYSES OF HETEROGENEOUS MATERIALS USING THE EXTENDED MULTISCALE FINITE ELEMENT METHOD

The extended multiscale finite element method (EMsFEM) has been more and more widely used in the multiscale analysis of the mechanical properties of heterogeneous materials and structures, since it has no assumptions of scale separation and periodicity compared to the asymptotic homogenization method and the representative volume element (RVE) method. To further facilitate the promotion and application of the EMsFEM, this paper discourses the basic ideas, principles, and implementations of this method in detail, and releases the corresponding MATLAB code for free use by researchers. On the work of this paper, scholars can carry out further research on different multiscale computation problems. In the EMsFEM, numerical basis functions play a key role as a bridge, establishing a connection between the microscopic heterogeneous properties of materials and their macroscopic equivalent mechanical quantities. In this paper, four approaches are introduced to construct the numerical shape functions and their implementation processes are also presented. Finally, some typical numerical examples are given for verifying the effectiveness of the EMsFEM. The MATLAB code can be download from the website https://github.com/hliu-whu/EMsFEM.

Fast and robust all-electron density functional theory calculations in solids using orthogonalized enriched finite elements

We present a computationally efficient approach to perform systematically convergent real-space all-electron Kohn-Sham DFT calculations for solids using an enriched finite element (FE) basis. The enriched FE basis is constructed by augmenting the classical FE basis with atom-centered numerical basis functions, comprising of atomic solutions to the Kohn-Sham problem. Notably, to improve the conditioning, we orthogonalize the enrichment functions with respect to the classical FE basis, without sacrificing the locality of the resultant basis. In addition to improved conditioning, this orthogonalization procedure also renders the overlap matrix block-diagonal, greatly simplifying its inversion. Subsequently, we use a Chebyshev polynomial based filtering technique to efficiently compute the occupied eigenspace in each self-consistent field iteration. We demonstrate the accuracy and efficiency of the proposed approach on periodic unit-cells and supercells. The benchmark studies show a staggering $130\times$ speedup of the orthogonalized enriched FE basis over the classical FE basis. We also present a comparison of the orthogonalized enriched FE basis with the LAPW+lo basis, both in terms of accuracy and efficiency. Notably, we demonstrate that the orthogonalized enriched FE basis can handle large system sizes of $\sim$10,000 electrons.

Theoretical approaches for doubly-excited Rydberg states in quasi-two-electron systems: two-electron dynamics far away from the nucleus

ABSTRACT We report on recent progress regarding the theoretical treatment of doubly-excited Rydberg states of atoms with two valence electrons. Building upon state-of-the-art techniques, we demonstrate that an approach based on configuration interaction with exterior complex scaling (CI-ECS) is capable of describing high Rydberg states () and does not rely on the assumptions of R-matrix multichannel quantum-defect theory. We show that CI-ECS allows the simultaneous, accurate treatment of a range of doubly-excited Rydberg states broader than previously possible. The implementation of the method with numerical basis functions is explained in detail. As an example, CI-ECS calculations of the doubly-excited states of Mg and Sr are presented, and excellent agreement is found with available experimental data. The CI-ECS method opens new perspectives for studying collective two-electron dynamics in doubly-excited Rydberg states of alkaline-earth metal atoms, and is also of interest for the field of Rydberg-based quantum simulation.

Study of LCAO-MO calculation by using completely numerical basis functions

Choice of basis functions is decisive in LCAO-MO method to calculate accurately MOs of a molecule under complicated electric/magnetic potential. For such situation, calculating basis functions analytically is generally difficult. To obtain them, the new numerical calculating method based on matrix operation and the new sample point system based on FCC lattice were proposed. Further LCAO-MO calculation using basis functions calculated by this method was applied for small molecules under inner electric potential and showed reasonable results. Finally, two challenges to apply this method for arbitrary magnetic potential are clarified.

Fully numerical calculations on atoms with fractional occupations and range-separated exchange functionals

A recently developed finite element approach for fully numerical atomic structure calculations [S. Lehtola, Int. J. Quantum Chem. 119, e25945 (2019)] is extended to the description of atoms with spherically symmetric densities via fractionally occupied orbitals. Specialized versions of Hartree-Fock as well as local density and generalized gradient approximation density functionals are developed, allowing extremely rapid calculations at the basis set limit on the ground and low-lying excited states even for heavy atoms. The implementation of range-separation based on the Yukawa or complementary error function (erfc) kernels is also described, allowing complete basis set benchmarks of modern range-separated hybrid functionals with either integer or fractional occupation numbers. Finally, computation of atomic effective potentials at the local density or generalized gradient approximation levels for the superposition of atomic potentials (SAP) approach [S. Lehtola, J. Chem. Theory Comput. 15, 1593 (2019)] that has been shown to be a simple and efficient way to initialize electronic structure calculations is described. The present numerical approach is shown to afford beyond microhartree accuracy with a small number of numerical basis functions, and to reproduce literature results for the ground states of atoms and their cations for $1 \leq Z \leq 86 $. Our results indicate that the literature values deviate by up to 10 {\mu}Eh from the complete basis set limit. The numerical scheme for the erfc kernel is shown to work by comparison to results from large Gaussian basis set calculations from the literature. Spin-restricted ground states are reported for Hartree-Fock and Hartree-Fock-Slater calculations with fractional occupations for $1 \leq Z \leq 118$.

Comparison of interactions between aluminium hydride oxide surfaces and three energetic plasticizers: DFT calculations

Abstract α- and γ-alumina (α-Al2O3 and γ-Al2O3) are used to model the surface oxide layer of AlH3 to investigate the micro-mechanism for the interactions of three energetic plasticizers: Bis-2,2-dinitropropyl formal (BDNPF), N-butyl-N-(2-nitroxyethyl) nitramine (BuNENA) and nitroglycerin (NG) with AlH3 surface oxidation layers. First principle density functional theory (DFT) calculations with all-electron double numerical basis functions (DND) are employed. It is found that three energetic plasticizers can be strongly adsorbed by the oxide film of AlH3 and release a large amount of heat. Moreover, BuNENA and NG containing -ONO2 groups easily undergo barrierless decomposition reactions on the α-Al2O3 (0001) surface and spontaneously produces NO2 radicals, causing significant changes in the structures of BuNENA and NG molecules as well as of the α-Al2O3 (0001) surface. Furthermore, NG can also undergo a barrierless decomposition reaction on the γ-Al2O3 (110) surface. However, BuNENA does not decompose on the γ-Al2O3 (110) surface, and it is not easy for BDNPF to decompose on either α-Al2O3 (0001) or γ-Al2O3 (110) surfaces; hence, it remains relatively stable. According to these results, it is suggested that the order of compatibility among the three energetic plasticizers with AlH3 is BDNPF > BuNENA > NG.

Adjoint-based airfoil optimization with adaptive isogeometric discontinuous Galerkin method

In this work, an adjoint-based airfoil shape optimization algorithm is developed based on the adaptive isogeometric discontinuous Galerkin method for compressible Euler equations to investigate the significance of each design variable of airfoil B-spline parameterization. We first parameterize the airfoil by B-spline curve approximation with some control points viewed as design variables, and build the B-spline representation of the flow field with the curve to apply the goal-oriented h-adaptive isogeometric DG method for flow solution. Then we compute and employ the discrete adjoint solutions for both multi-target error estimation in adaptive mesh refinement. With the isogeometric nature, not only all the geometrical cells but also the numerical basis functions can be analytically expressed by the design variables, indicating that the numerical solutions and objective could be differentiable with respect to those variables. Consequently, the gradient is totally computed in an accurate approach, and the sensitivity analysis is thus improved, by reducing the spatial discretization error and introducing the analytical expression of derivative, to reveal the key parameters for optimization in an intuitive and efficient manner. Although the SQP optimization algorithm is adopted in the paper, the given accurate gradient can be applied to any gradient-based optimization algorithm. The proposed algorithm is demonstrated on RAE2822 airfoil with inviscid transonic flow, where the shape is optimized to minimize the drag coefficient at a constrained lift and airfoil area. The numerical results show that the drag is much more sensitive to the design variables near the tailing edge at the beginning but sensitivity is reduced when optimal.

Variational treatment of electron–polyatomic-molecule scattering calculations using adaptive overset grids

The Complex Kohn variational method for electron-polyatomic molecule scattering is formulated using an overset grid representation of the scattering wave function. The overset grid consists of a central grid and multiple dense, atom-centered subgrids that allow the simultaneous spherical expansions of the wave function about multiple centers. Scattering boundary conditions are enforced by using a basis formed by the repeated application of the free particle Green's function and potential, $\hat{G}^+_0\hat{V}$ on the overset grid in a "Born-Arnoldi" solution of the working equations. The theory is shown to be equivalent to a specific Pad\'e approximant to the $T$-matrix, and has rapid convergence properties, both in the number of numerical basis functions employed and the number of partial waves employed in the spherical expansions. The method is demonstrated in calculations on methane and CF$_4$ in the static-exchange approximation, and compared in detail with calculations performed with the numerical Schwinger variational approach based on single center expansions. An efficient procedure for operating with the free-particle Green's function and exchange operators (to which no approximation is made) is also described.

Large-scale all-electron density functional theory calculations using an enriched finite-element basis

We present a computationally efficient approach to perform large-scale all-electron density functional theory calculations by enriching the classical finite element basis with compactly supported atom-centered numerical basis functions that are constructed from the solution of the Kohn-Sham (KS) problem for single atoms. We term these numerical basis functions as enrichment functions, and the resultant basis as the enriched finite element basis. The enrichment functions are compactly supported through the use of smooth cutoff functions, which enhances the conditioning and maintains the locality of the basis. The integrals involved in the evaluation of the discrete KS Hamiltonian and overlap matrix in the enriched finite element basis are computed using an adaptive quadrature grid based on the characteristics of enrichment functions. Further, we propose an efficient scheme to invert the overlap matrix by using a block-wise matrix inversion in conjunction with special reduced-order quadrature rules to transform the discrete Kohn-Sham problem to a standard eigenvalue problem. Finally, we solve the resulting standard eigenvalue problem by using a Chebyshev polynomial based filtering technique to compute the relevant eigenspectrum. We demonstrate the accuracy, efficiency and parallel scalability of the proposed method on semiconducting and heavy-metallic systems of various sizes, with the largest system containing 8694 electrons. We obtain accuracies in the ground-state energies that are within $\sim 1$mHa with ground-state energies obtained from classical finite element as well as gaussian basis sets. We observe a $50-300$ and $\sim 8$ fold reduction in the overall computational time when compared to classical finite element and gaussian basis, respectively. We also observe good parallel scalability up to 384 processors for a benchmark system comprising of 280-atom silicon nano-cluster.

Modal wavefront estimation from its slopes by numerical orthogonal transformation method over general shaped aperture.

Wavefront estimation from the slope-based sensing metrologies zis important in modern optical testing. A numerical orthogonal transformation method is proposed for deriving the numerical orthogonal gradient polynomials as numerical orthogonal basis functions for directly fitting the measured slope data and then converting to the wavefront in a straightforward way in the modal approach. The presented method can be employed in the wavefront estimation from its slopes over the general shaped aperture. Moreover, the numerical orthogonal transformation method could be applied to the wavefront estimation from its slope measurements over the dynamic varying aperture. The performance of the numerical orthogonal transformation method is discussed, demonstrated and verified by the examples. They indicate that the presented method is valid, accurate and easily implemented for wavefront estimation from its slopes.

Efficient modeling of photonic crystals with local Hermite polynomials

Developing compact algorithms for accurate electrodynamic calculations with minimal computational cost is an active area of research given the increasing complexity in the design of electromagnetic composite structures such as photonic crystals, metamaterials, optical interconnects, and on-chip routing. We show that electric and magnetic (EM) fields can be calculated using scalar Hermite interpolation polynomials as the numerical basis functions without having to invoke edge-based vector finite elements to suppress spurious solutions or to satisfy boundary conditions. This approach offers several fundamental advantages as evidenced through band structure solutions for periodic systems and through waveguide analysis. Compared with reciprocal space (plane wave expansion) methods for periodic systems, advantages are shown in computational costs, the ability to capture spatial complexity in the dielectric distributions, the demonstration of numerical convergence with scaling, and variational eigenfunctions free of numerical artifacts that arise from mixed-order real space basis sets or the inherent aberrations from transforming reciprocal space solutions of finite expansions. The photonic band structure of a simple crystal is used as a benchmark comparison and the ability to capture the effects of spatially complex dielectric distributions is treated using a complex pattern with highly irregular features that would stress spatial transform limits. This general method is applicable to a broad class of physical systems, e.g., to semiconducting lasers which require simultaneous modeling of transitions in quantum wells or dots together with EM cavity calculations, to modeling plasmonic structures in the presence of EM field emissions, and to on-chip propagation within monolithic integrated circuits.

Case-Study Inverse Thermal Analyses of Al2139 and Al2198 Electron Beam Welds

Case study inverse thermal analyses of A12139 and Al2198 electron beam welds are presented. These analyses represent a continuation of previous studies using laser beam welds, but provide accessibility to different regions of the parameter space for temperature histories than achievable using laser beams. For these analyses, a numerical methodology is employed, which is in terms of analytic functions for inverse thermal analysis of steady-state energy deposition in plate structures. The results of the case studies presented provide parametric representations of weld temperature histories, which can be adopted as input data to various types of computational procedures, such as those for prediction of solid-state phase transformations and their associated software implementations. In addition, these weld temperature histories can be used for construction of numerical basis functions that can be adopted for inverse analysis of welds corresponding to other process parameters or welding processes process conditions of which are within similar regimes.

MBPT calculation for the fine-structure intervals of principal series np(n=39) for Na

The fine-structure intervals of Na principal series np(n=39) are calculated by the many-body perturbation theory (MBPT) within the framework of relativity. To deal with the problem that a large set of continuum states is required in the MBPT calculation, an exponential potential is employed to generate a discrete, finite and nearly complete set of numerical basis functions. The zeroth-order wavefunctions and eignvalues are obtained by solving the relativistic Hartree-Fock (RHF) equation and the RHF equation with the action of a potential barrier. The basis set used in this work contains intermediate orbitals with the angular momentum l 6 and in an appropriate energy range, and most of them are the so called contracted orbitals. Encouraging results are obtained using this technique to calculate the second-order correlation corrections, combining the Breit effects in a first-order perturbation approach. Compared with other theoretical calculations, the present results are much close to the experimental results.

Case-Study Inverse Thermal Analyses of Al2198 Laser Welds

In this article, case-study inverse thermal analyses of Al2198 laser welds are presented. These analyses employ a numerical methodology, that is, in terms of analytic and numerical basis functions for inverse thermal analysis of steady-state energy deposition in plate structures. The results of the case studies presented provide parametric representations of weld temperature histories that can be adopted as input data to various types of computational procedures, such as those for prediction of solid-state phase transformations and their associated software implementations. In addition, these weld temperature histories can be used for construction of numerical basis functions that can be adopted for inverse analysis of welds corresponding to other process parameters or welding processes whose process conditions are within similar regimes.

Case-Study Inverse Thermal Analyses of Al2139 Laser Welds

Case study inverse thermal analyses of A12139 laser welds are presented. These analyses employ a numerical methodology that is in terms of analytic and numerical basis functions for inverse thermal analysis of steady state energy deposition in plate structures. The results of the case studies presented provide parametric representations of weld temperature histories that can be adopted as input data to various types of computational procedures, such as those for prediction of solid-state phase transformations and their associated software implementations. In addition, these weld temperature histories will be useful for construction of numerical basis functions that can be adopted for inverse analysis of welds corresponding to other process parameters or welding processes whose process conditions are within similar regimes.

Applications of the dirac-slater model to molecules

Applications of the Dirac Slater model to obtain relativistic energy levels and wave functions for molecules are reported. Results are presented for the uranyl ion UO2+2, calculated by a discrete-variational method using numerical free-atom basis functions. A simple moment polarized generalization of the Slater exchange potential is proposed for treating open-shell systems; results for a 5d atom are compared with corresponding nonrelativistic spin-polarized states.

Reduction of Azimuthal Amplitude Ripple in the Rotating-Mode Feed to a Radial Waveguide by Using a Crossed Dog-Bone Slot

The use of a crossed dog-bone slot in a rectangular-to-radial waveguide transformer is proposed as a feeder for a radial line slot antenna to reduce the amplitude ripple in the azimuthal direction because the beamwidth of the dog-bone slots is wider than that of straight slots in radiating into a radial waveguide. The transformer is analyzed and designed by using the method of moments with numerical eigenmode basis functions. By the use of the crossed dog-bone slot, the measured amplitude ripple in the azimuthal direction becomes 0.80 dB, which is 1.4 dB lower than that of the conventional crossed straight slot in the 5.8 GHz band.