Choice Of Basis Functions Research Articles

Density Functional Theory and Beyond for Band-Gap Screening: Performance for Transition-Metal Oxides and Dichalcogenides.

The performance of a wide variety of commonly used density functionals, as well as two screened hybrid functionals (HSE06 and TB-mBJ), on predicting electronic structures of a large class of en vogue materials, such as metal oxides, chalcogenides, and nitrides, is discussed in terms of band gaps, band structures, and projected electronic densities of states. Contrary to GGA, hybrid functionals and GGA+U, both HSE06 and TB-mBJ are able to predict band gaps with an appreciable accuracy of 25% and thus allow the screening of various classes of transition-metal-based compounds, i.e., mixed or doped materials, at modest computational cost. The calculated electronic structures are largely unaffected by the choice of basis functions and software implementation, however, might be subject to the treatment of the core electrons.

Simulation of structural dynamics using a non-polynomial method

In structural dynamics, the modeling of steady-state thin plate bending is an important but, especially at high frequencies, computationally challenging problem. When solving the displacement of an elastic thin plate, a fourth order partial differential equation (Kirchhoff's plate equation) needs to be solved. In addition, two boundary conditions are needed in order to uniquely solve the problem. Polynomial methods, such as the finite element method (FEM) and discontinuous Galerkin method (DGM), are generally used to solve the plate dynamics. At higher frequencies the computational burden of a low order FEM becomes rapidly unbearable. Consequently, non-polynomial modeling methods are investigated because of their capability to solve the problem more efficiently than the standard FEM. The non-polynomial method used in this study is called the ultra weak variational formulation (UWVF). The UWVF uses finite element meshes and it is essentially an upwind DGM with a special choice of basis functions. To date, the UWVF has been successfully used in electromagnetism, acoustics and linear elasticity. We shall show, using a mixture of theory and numerical examples, that the UWVF is feasible for thin plate problems. For these problems the UWVF basis consists of plane wave and evanescent (corner) wave functions.

Regional Sea Level Reconstruction in the Pacific Ocean

As a result of the spatial and temporal sampling shortcomings of tide gauge and satellite altimetry data, forming a sea level record that has the quality and duration necessary to compare past to present sea level is a challenge. Here, we build on the recently developed method for reconstructing sea level involving cyclostationary empirical orthogonal functions (CSEOFs) and, in particular, examine how the reconstruction varies as a result of the domain used to compute the CSEOF basis functions. While not affecting the ability to capture the dominant climate signals, the choice of basis function is found to have a significant effect on the regional distribution of the trend in sea level. We also demonstrate the use of a CSEOF reconstructed sea level dataset for climate monitoring, focusing on climate signals in the Pacific Ocean.

The Ultra Weak Variational Formulation Using Bessel Basis Functions

AbstractWe investigate the ultra weak variational formulation (UWVF) of the 2-D Helmholtz equation using a new choice of basis functions. Traditionally the UWVF basis functions are chosen to be plane waves. Here, we instead use first kind Bessel functions. We compare the performance of the two bases. Moreover, we show that it is possible to use coupled plane wave and Bessel bases in the same mesh. As test cases we shall consider propagating plane and evanescent waves in a rectangular domain and a singular 2-D Helmholtz problem in an L-shaped domain.

표면반사율 모델링을 위한 새로운 N차원 기저함수

일반적으로 표면반사율과 분광반사율을 N차원의 칼라 코드로부터 정확히 복원하기 위해서는 N개의 기저함수가 필요하다. 전형적인 렌더링 응용에서 벡터의 덧셈, 스칼라 곱셈 및 성분별 곱셈에 대한 벡터 연산이 이질동형이라고 가정하고 광원의 중첩, 광원-표면간 상호간섭 및 상호반사와 같은 물리적인 연산을 모델링하지만 벡터 연산이 물리적인 현상을 그대로 반영하는 것은 아니다. 그러나 만약 기저함수가 특성함수로써 제한된다면 표면반사율과 분광반사율의 사상 결과 및 벡터들은 렌더링에서 물리적인 연산인 이질이형을 유지하게 된다. 본 논문은 새로운 N차원의 특성함수를 제안하고 N차원의 기저함수로 근사화된 먼셀 칼라 칩에 대하여 제안한 알고리즘의 정확성을 평가할 것이다. The N basis functions are typically chosen so that Surface reflectance functions(SRFs) and spectral power distributions (SPDs) can be accurately reconstructed from their N-dimensional vector codes. Typical rendering applications assume that the resulting mapping is an isomorphism where vector operations of addition, scalar multiplication, component-wise multiplication on the N-vectors can be used to model physical operations such as superposition of lights, light-surface interactions and inter-reflection. The vector operations do not mirror the physical. However, if the choice of basis functions is restricted to characteristic functions then the resulting map between SPDs/SRFs and N-vectors is anisomorphism that preserves the physical operations needed in rendering. This paper will show how to select optimal characteristic function bases of any dimension N (number of basis functions) and also evaluate how accurately a large set of Munsell color chips can approximated as basis functions of dimension N.

A constraint sampling approach for multi-stage robust optimization

We propose a tractable approximation scheme for convex (not necessarily linear) multi-stage robust optimization problems. We approximate the adaptive decisions by finite linear combinations of prescribed basis functions and demonstrate how one can optimize over these decision rules at low computational cost through constraint randomization. We obtain a-priori probabilistic guarantees on the feasibility properties of the optimal decision rule by applying existing constraint sampling techniques to the semi-infinite problem arising from the decision rule approximation. We demonstrate that for a suitable choice of basis functions, the approximation converges as the size of the basis and the number of sampled constraints tend to infinity. The approach yields an algorithm parameterized in the basis size, the probability of constraint violation and the confidence that this probability will not be exceeded. These three parameters serve to tune the trade-off between optimality and feasibility of the decision rules and the computational cost of the algorithm. We assess the convergence and scalability properties of our approach in the context of two inventory management problems.

Tetrahedral-Based Vector Generalized Finite Element Method and Its Applications

Vector generalized finite element method was first introduced as a meshless method. Its application to practical problems was stymied by the fact that one needs information of intersection of the partition of unity domains with the boundaries of the problem domain. The framework presented in Tuncer 's work (IEEE Trans. Antennas Propag., vol. 58, no. 3, 887-899, Mar. 2010) overcame this bottleneck. In this letter, the framework is extended to tetrahedral meshes and applied to a number of electromagnetic problems. These examples are chosen so as to highlight some features of the method such as higher-order convergence, flexibility in the choice of basis functions, and use of different types of basis functions or mixed polynomial orders within a simulation.

Efficient numerical solution of the generalized Dirichlet–Neumann map for linear elliptic PDEs in regular polygon domains

A new and novel approach for analyzing boundary value problems for linear and for integrable nonlinear PDEs was recently introduced. For linear elliptic PDEs, an important aspect of this approach is the characterization of a generalized Dirichlet–Neumann map: given the derivative of the solution along a direction of an arbitrary angle to the boundary, the derivative of the solution perpendicularly to this direction is computed without solving on the interior of the domain. For this computation, a collocation-type numerical method has been recently developed. Here, we study the collocation’s coefficient matrix properties. We prove that, for the Laplace’s equation on regular polygon domains with the same type of boundary conditions on each side, the collocation matrix is block circulant, independently of the choice of basis functions. This leads to the deployment of the FFT for the solution of the associated collocation linear system, yielding significant computational savings. Numerical experiments are included to demonstrate the efficiency of the whole computation.

Study on dimension reduction methods for a fluid conveying tube

AbstractThe Application of Dimension Reduction by Approximate Inertial Manifolds to the dynamics of a fluid conveying tube doesn't perform as expected due to the presence of viscous damping terms, which lead to a cluster point of stable eigenvalues. We investigate methods to circumvent the problems introduced by this cluster point: For the spectral decomposition part of the invariant subspaces are computed using the Schur factorisation. Also a different choice of basis functions for the dominating modes avoids the difficulties. (© 2011 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

New strategies for full-wave simulations of large acoustic models

The simulation of large acoustic models using conventional low-order finite element or boundary element (BE) methods can be computationally demanding due to the requirement of wavelength dependent mesh density (5-10 elements per wavelength is a typical rule of thumb). A recent improvement of finite element type methods has focused on the efficient parallelization of problems to even thousands of processors. The discontinuous Galerkin (DG) methods, in time- and frequency-domain, have been particularly popular since DG methods retain the geometric flexibility of FE methods while allowing a more flexible choice of basis functions. The locality of DG basis makes them ideal for large-scale parallelization and the choice of basis can be easily extended to non-polynomial functions such as plane waves or Bessel functions. This study focuses on the three new strategies for solving large acoustic problems. First, the non-polynomial basis functions are used with the DG method in frequency-domain problems. Second, from element to element varying polynomial order is used for the wave equation in the time-domain. And third, cloud computing is utilized for efficient resource allocation with a fast multipole BE method for the acoustic Helmholtz equation.

Real-time physics-model-based simulation of the current density profile in tokamak plasmas

A new paradigm is presented to reconstruct the plasma current density profile in a tokamak in real-time. The traditional method of basing the reconstruction on real-time diagnostics combined with a real-time Grad–Shafranov solver suffers from the difficulty of obtaining reliable internal current profile measurements with sufficient spatial and temporal accuracy to have a complete picture of the profile evolution at all times. A new methodology is proposed in which the plasma current density profile is simulated in real-time by solving the first-principle physics-based equations determining its evolution. Effectively, an interpretative transport simulation similar to those run today in post-plasma shot analysis is performed in real-time. This provides real-time reconstructions of the current density profile with spatial and temporal resolution constrained only by the capabilities of the computational platform used and not by the available diagnostics or the choice of basis functions. The diagnostic measurements available in real-time are used to constrain and improve the accuracy of the simulated profiles. Estimates of other plasma quantities, related to the current density profile, become available in real-time as well. The implementation of the proposed paradigm in the TCV tokamak is discussed, and its successful use in plasma experiments is demonstrated. This framework opens up the possibility of unifying q profile reconstructions across different tokamaks using a common physics model and will support a wealth of applications in which improved real-time knowledge of the plasma state is used for feedback control, disruption avoidance, scenario monitoring and external disturbance estimation.

Theory of neural communication based on spatio-temporal coding

Pattern coding is a general concept for neural coding, which indicates that the objective meaning of information can be represented by spatio-temporal firing patterns of a group of neurons [1]. We introduce a feasible way through which spatio-temporal firing patterns represent complex information systematically. Provided that neural codes represent features in a vector space, neural communication channels can be characterized by independent pattern components constituting basis functions in the Hilbert space. Specific applied forms of the method, depending on the choice of basis functions, reduce to the traditional coding schemes, including rate coding, temporal coding, correlation coding, independent-spike coding, population coding, phase coding, and so on. In addition, it is suggested that the ordinary neural code in the brain might take a more elaborate form, in such a way that neural networks can send or receive complex information effectively and robustly through mutable spike trains. We discuss corresponding statistics of neural codes based on the theory. Finally, we present the scheme of a cortical module as the processing unit of communication and computation based on spatio-temporal coding. It may also be applied to modelling the stimulus-response of cortical neurons.

Physics based basis function for vibration analysis of high speed rotating beams

The natural frequencies of continuous systems depend on the governing partial differential equation and can be numerically estimated using the finite element method. The accuracy and convergence of the finite element method depends on the choice of basis functions. A basis function will generally perform better if it is closely linked to the problem physics. The stiffness matrix is the same for either static or dynamic loading, hence the basis function can be chosen such that it satisfies the static part of the governing differential equation. However, in the case of a rotating beam, an exact closed form solution for the static part of the governing differential equation is not known. In this paper, we try to find an approximate solution for the static part of the governing differential equation for an uniform rotating beam. The error resulting from the approximation is minimized to generate relations between the constants assumed in the solution. This new function is used as a basis function which gives rise to shape functions which depend on position of the element in the beam, material, geometric properties and rotational speed of the beam. The results of finite element analysis with the new basis functions are verified with published literature for uniform and tapered rotating beams under different boundary conditions. Numerical results clearly show the advantage of the current approach at high rotation speeds with a reduction of 10 to 33% in the degrees of freedom required for convergence of the first five modes to four decimal places for an uniform rotating cantilever beam.

Fokker-Planck equation with arbitrary dc and ac fields: Continued fraction method

The continued fraction method (CFM) is used to solve the Fokker-Planck equation with arbitrary dc and ac fields. With an appropriate choice of basis functions, the Fokker-Planck equation is converted into a set of linear algebraic equations with short-ranged coupling and then CFM is implemented to obtain numerical solutions with high efficiency. Both a proposed perturbative CFM and the numerically exact matrix CFM are used to study the nonlinear response of driven systems, with their results compared to assess the validity regime of the perturbative approach. The proposed perturbative CFM approach needs scalar quantities only and hence is more efficient within its validity regime. Two nonlinear systems of different nature are used as examples: molecular dipole (rotational Brownian motion) and particle in a periodic potential (translational Brownian motion). The associated full dynamics is presented in the compact form of hysteresis loops. It is observed that as the strength of an AC driving field increases, pronounced nonlinear effects are manifested in the deformation of the hysteresis loops.

Decentralized damage identification using wavelet signal analysis embedded on wireless smart sensors

In this paper, a decentralized damage identification method using wavelet signal analysis tools embedded on wireless smart sensors (Imote2) has been proposed and experimentally validated. The damage identification analysis is decentralized by calculating discrete wavelet coefficients for acceleration in Imote2 sensors and transmitting the wavelet coefficients to a base station for damage identification through wavelet entropy indices. The wavelet entropy is modified to serve as a damage-sensitive signature that can be obtained both at different spatial locations and time stations to indicate existence of damage. It is known that wavelet-based approaches have clear advantages over Fourier transform-based ones for damage identification, since the wavelet transform allows for a wider choice of basis functions. This flexibility allows the wavelet transform to isolate changes in a signal that may be difficult to detect using other transform methods. To assess the reliability of the measurement signals, the wireless sensors have been compared with reference wired sensors. The proposed decentralized method for damage identification is verified via experimental tests using two laboratory structures: a three-story shear building structure and a three-dimensional truss bridge structure.

A meshless formulation of Euler-Bernoulli beam theory for prediction of large deformation behavior of fabrics

In this study, a meshless formulation of the Euler—Bernoulli theory with non-linear strain-displacement relationship, covering both axial and bending deformations, is presented for the prediction of large deformation behavior of fabrics. Fabrics are assumed to act like a sheet material with constant extensional and bending rigidity. The influences of the choice of basis functions and support sizes for both axial and bending deformations are outlined and it is verified by comparing deformations results with those published in the literature.

BOR-FDTD subgridding based on finite element principles

In this paper a recently developed provably passive and stable 3D FDTD subgridding technique, based on finite elements principles, is extended to body-of-revolution (BOR) FDTD. First, a suitable choice of basis functions is presented together with the mechanism to assemble them into an overall mesh consisting of coarse and fine mesh cells. Invoking appropriate mass-lumping concepts then leads to an explicit leapfrog time stepping algorithm for the amplitudes of the basis functions. Attention is devoted to provide the reader with insight into the updating equations, in particular at a subgridding boundary. Stability, grid reflection and dispersion are also discussed. Finally, some numerical examples for toroidal and cylindrical cavities demonstrate the stability and accuracy of the method.

The addition of fields on different meshes

Let T 1 and T 2 be two (arbitrarily unrelated) meshes of the same domain Ω ⊂ R d , 1 ⩽ d ⩽ 3 . Let V 1 and V 2 be the discrete function spaces induced from these meshes by equipping them with some choice of basis functions. This work discusses the following problem: given two functions f 1 ∈ V f and f 2 ∈ V g , compute f 1 + f 2. Despite the simple statement of the task, the general solution is not obvious, for the desired sum lies neither in V 1 nor in V 2 , and is not representable on either mesh. The problem is solved by efficiently constructing a supermesh T S of T 1 and T 2 which induces a superspace of both V 1 and V 2 . Once such a mesh is available, the desired quantity may be represented exactly (subject to roundoff).

Electronic structure of vertically coupled double quantum dots: Optimization of basis functions

We have performed a fully converged calculation for electronic structure of vertically coupled double quantum dots (DQDs) using linear combination of atomic orbitals. We find that to reach high accuracy of results and get a quick convergence, a special attention in choice of basis functions has to be paid, specially for small quantum dots. We also find the character of electronic states of DQDs evolves from atomic-like to molecular characteristic (bounding and anti-bounding) states as interdot distance decreases. We show that the electronic structure of DQDs can be engineered through properly DQD design. It is in sharp contrast to the natural molecules in which the geometric positions of composed atoms are fixed.

Characterizing the vibration of an elastically point supported rectangular plate using eigensensitivity analysis

Normalized frequencies are computed for a rectangular, isotropic plate resting on elastic supports. The normalized frequencies are determined using eigensensitivity analysis, which approximates the eigenparameters in a Mauclarin series, yielding an approximate closed-form expression. One benefit of the approximate closed-form expression is its computational efficiency and yet another is its application of re-analysis. Accuracy of the approximate expression is assessed by comparing results with the widely used Rayleigh–Ritz method using orthogonal polynomials and beam shape functions in both approaches. Consideration for a variety of edge conditions is given through a combination of simply supported, clamped and free boundary conditions. Results indicate that the accuracy of higher frequencies computed by the sensitivity approach is highly dependent upon choice of basis function.