Lagrange Basis Research Articles

Stability analysis of the marching-on-in-time boundary element method for electromagnetics

The Time Domain Integral Equation method for electromagnetics is an appealing computational method for many applications in industry. However, its applicability has long been suffering from instabilities. A rigorous analysis of the variational formulation is imperative to the successful design of stable and robust numerical schemes. In this paper, an established functional framework and stability theorem will be extended to the differentiated version of the electric field integral equations, which can be discretized more efficient and is more often used in engineering literature. The extended stability theorem, combined with efficiency requirements, will give guidelines on the choice of test and basis functions of the space–time Petrov–Galerkin scheme. A discrete equivalence with the collocation method results in the recommendation to choose the quadratic spline basis function in the standard Marching-on-in-Time scheme. Computational experiments confirm that the quadratic spline basis functions have superior stability characteristics compared to the conventional quadratic Lagrange basis functions in time.

On variable step Hermite–Birkhoff solvers combining multistep and 4-stage DIRK methods for stiff ODEs

Variable-step (VS) 4-stage k-step Hermite---Birkhoff (HB) methods of order p = (k + 2), p = 9, 10, denoted by HB (p), are constructed as a combination of linear k-step methods of order (p ? 2) and a diagonally implicit one-step 4-stage Runge---Kutta method of order 3 (DIRK3) for solving stiff ordinary differential equations. Forcing a Taylor expansion of the numerical solution to agree with an expansion of the true solution leads to multistep and Runge---Kutta type order conditions which are reorganized into linear confluent Vandermonde-type systems. This approach allows us to develop L(a)-stable methods of order up to 11 with a > 63°. Fast algorithms are developed for solving these systems in O (p2) operations to obtain HB interpolation polynomials in terms of generalized Lagrange basis functions. The stepsizes of these methods are controlled by a local error estimator. HB(p) of order p = 9 and 10 compare favorably with existing Cash modified extended backward differentiation formulae of order 7 and 8, MEBDF(7-8) and Ebadi et al. hybrid backward differentiation formulae of order 10 and 12, HBDF(10-12) in solving problems often used to test higher order stiff ODE solvers on the basis of CPU time and error at the endpoint of the integration interval.

A limiter-based well-balanced discontinuous Galerkin method for shallow-water flows with wetting and drying: One-dimensional case

An important part in the numerical simulation of tsunami and storm surge events is the accurate modeling of flooding and the appearance of dry areas when the water recedes. This paper proposes a new algorithm to model inundation events with piecewise linear Runge–Kutta discontinuous Galerkin approximations applied to the shallow water equations. This study is restricted to the one-dimensional case and shows a detailed analysis and the corresponding numerical treatment of the inundation problem.The main feature is a velocity based “limiting” of the momentum distribution in each cell, which prevents instabilities in case of wetting or drying situations. Additional limiting of the fluid depth ensures its positivity while preserving local mass conservation. A special flux modification in cells located at the wet/dry interface leads to a well-balanced method, which maintains the steady state at rest. The discontinuous Galerkin scheme is formulated in a nodal form using a Lagrange basis. The proposed wetting and drying treatment is verified with several numerical simulations. These test cases demonstrate the well-balancing property of the method and its stability in case of rapid transition of the wet/dry interface. We also verify the conservation of mass and investigate the convergence characteristics of the scheme.

Computing the topology of an arrangement of parametric or implicit algebraic curves in the Lagrange basis

The study of arrangements of geometric objects is a problem of great interest in combinatorial and computational geometry, and in CAGD, with applications in computer-aided manufacturing, geometric modeling, robotics, NC-machining, molecular modeling, etc. [2, 3] We present a new algebraic method for computing the topology of the arrangement of a finite family of planar algebraic curves. The curves are given either by their implicit equations or by polynomial parametrizations. As far as we know, none of the previous works on this problem include the treatment of parametrized curves. The input of the method is assumed to be given as Lagrange data, i.e., by the values of the polynomials defining the curves, at a suitable set of nodes. The output is a graph representing critical points of the curves, all the intersection points between curves, and segments joining some of these points isotopic to the arcs of the curves. The method follows a sweep-line algorithm. To find critical points, intersections and self-intersections we use Bezout matrices. We take advantage of the existence of a basis of the nullspace of a Bezout matrix that has a nice and useful structure (see [4]). We use algebra by values techniques, which means that all computations (Bezout matrices, polynomial derivatives, etc.) are done in the Lagrange basis [5, 1]. This approach reduces the computational cost of manipulating high degree polynomials, and avoids conversion between different basis, which is known to be a numerically unstable procedure. To compute the roots of the determinant of a Bezout matrix we apply linearization, i.e., we use a straightforward procedure to build (in the Lagrange basis) a companion matrix pencil whose generalized eigenvalues are the roots of the aforementioned determinant. The algorithm has been tested using Maple 17, showing that it is accurate and quite efficient.

Uncertainty Quantification for Electromagnetic Systems Using ASGC and DGTD Method

In this paper, an adaptive hierarchical sparse grid collocation (ASGC) method combined with the discontinuous Galerkin time-domain method is leveraged to quantify the impacts of random parameters on the electromagnetics systems. The ASGC method approximates the stochastic observables of interest using interpolation functions over a set of collocation points determined by the Smolyak's algorithm integrated with an adaptive strategy. Instead of resorting to a full-tensor product sense, the Smolyak's algorithm constructs the collocation points in a hierarchical scheme with the interpolation level. Enhanced by an adaptive strategy, the Smolyak's algorithm will sample more points along important dimensions with sharp variations or discontinuities, resulting in a nonuniform sampling scheme. To flexibly handle different stochastic systems, either piecewise linear or Lagrange polynomial basis functions are applied. With these strategies, the number of collocation points is significantly reduced. The statistical knowledge of stochastic observables including the expected value, variance, probability density function, and cumulative distribution function are presented. The accuracy and robustness of the algorithm are demonstrated by various examples.

Comparison of mixed [formula omitted]-BEM (stabilized and non-stabilized) for frictional contact problems

We present boundary integral equation procedures for contact problems (with Tresca or Coulomb friction) which are based on mixed formulations where besides the displacement also the traction on the contact boundary part appears (as a Lagrange multiplier). This approach allows for an easy and efficient way to perform an hp-BE method by the use of biorthogonal basis functions. This is especially suited for applying the semi-smooth Newton (SSN) method which is a very efficient solver superior to standard algorithm like Uzawa. With an adaptive algorithm we perform locally mesh refinements and increase of polynomial degrees for the BE solution—thus correctly representing the contact phenomena. We also present as stabilized version of our mixed hp-BEM scheme with Gauss–Lobatto–Lagrange basis which circumvents the discrete inf–sup condition. Numerical results supporting our theory are reported.

Backward Error of Polynomial Eigenvalue Problems Solved by Linearization of Lagrange Interpolants

This article considers the backward error of the solution of polynomial eigenvalue problems expressed as Lagrange interpolants. One of the most common strategies to solve polynomial eigenvalue problems is to linearize, which is to say that the polynomial eigenvalue problem is transformed into an equivalent larger linear eigenvalue problem, and solved using any appropriate eigensolver. Much of the existing literature on the backward error of polynomial eigenvalue problems focuses on polynomials expressed in the classical monomial basis. Hence, the objective of this article is to carry out the necessary extensions for polynomials expressed in the Lagrange basis. We construct one-sided factorizations that give simple expressions relating the eigenvectors of the linearization to the eigenvectors of the polynomial eigenvalue problem. Using these relations, we are able to bound the backward error of an approximate eigenpair of the polynomial eigenvalue problem relative to the backward error of an approximate eigenpair of the linearization. We develop bounds for the backward error involving both the norms of the polynomial coefficients and the properties of the Lagrange basis generated by the interpolation nodes. We also present several numerical examples to illustrate the numerical properties of the linearization and develop a balancing strategy to improve the accuracy of the computed solutions.

A Sparse Interpolation Algorithm for Dynamical Simulations in Computational Chemistry

In this paper we present a new implementation of Smolyak's sparse grid interpolation algorithm designed for dynamical simulations. The implementation is motivated by an application to quantum chemistry where the goal is to simulate photo-induced molecular transformations. A molecule conforms to a geometry that minimizes its potential energy, and many molecules have multiple potential energy minima. These geometries correspond to local minima of the molecule's potential energy surface, and one can simulate how a molecule transitions from one geometry to another by following the steepest descent path, or reaction path, on potential energy surfaces. Molecular vibrations and thermal fluctuations cause randomness in dynamics, so one must follow several paths simultaneously to more accurately simulate possible reaction paths. Current algorithms for reaction path following are too computationally burdensome for molecules of moderate size, but Smolyak's interpolation algorithm offers a cheap surrogate for potential energy surfaces. While current implementations of Smolyak's algorithm are not designed to simultaneously follow multiple reaction paths efficiently, our implementation of Smolyak's algorithm achieves this efficiency by recursively defining Lagrange basis polynomials and making use of an efficient reformulation of Smolyak's algorithm. In this paper we describe our new implementation of Smolyak's algorithm and compare performance times to MATLAB's Sparse Grid Interpolation Toolbox to demonstrate its computational savings. We also present dynamical simulations for the photoisomerization of 2-butene as an example of our reaction path following method.

The Electrochemical Machining Analysis of Aeroengine Blade Based on Isogeometric Method

Electrochemical machining is an important method for the blade in aeroengine. Analysis of electric field in machining gap is the basis of cathode design. For solving the low precision problem for sensitive boundary which is caused by traditional numerical analysis method, this paper proposed a method of promoting the analysis precision by using the isogeometric analysis. The NURBS basis functions are used to replace the Lagrange basis function for establishing the solving equations of the electrochemical machining gap. The problem of the noninterpolation feature belonging to the NURBS basis functions, which could bring the error for imposing Dirichlet boundary condition, is settled. At last, the superiority, including precision and the rate of convergence, of the isogeometric analysis is proved by the comparison test.

Three-stage Hermite–Birkhoff solver of order 8 and 9 with variable step size for stiff ODEs

Variable-step (VS) $$3$$3-stage Hermite---Birkhoff (HB) methods HB$$(p=k+1)$$(p=k+1) of order $$p=8$$p=8 and 9 are constructed as a combination of linear $$k$$k-step methods of order $$(p-2)$$(p-2) and a diagonally implicit one-step $$3$$3-stage Runge---Kutta method of order 3 (DIRK3) for solving stiff ordinary differential equations. Forcing a Taylor expansion of the numerical solution to agree with an expansion of the true solution leads to multistep and Runge---Kutta type order conditions which are reorganized into linear confluent Vandermonde-type systems. This approach allows us to develop A-stable methods of order up to 5 and A($$\alpha $$?)-stable methods of order up to 10. Fast algorithms are developed for solving these systems in O$$(p^2)$$(p2) operations to obtain HB interpolation polynomials in terms of generalized Lagrange basis functions. The stepsize of these methods are controlled by a local error estimator. When programmed in C++, HB($$p$$p) of order $$p=8$$p=8 and 9 compare favorably with existing Cash modified extended backward differentiation formulae of order 7 and 8, MEBDF(7---8), in solving problems often used to test higher order stiff ODE solvers on the basis of CPU time and error at the endpoint of the integration interval.

Algebraic multigrid techniques for discontinuous Galerkin methods with varying polynomial order

We present a parallel algebraic multigrid (AMG) algorithm for the implicit solution of the Darcy problem discretized by the discontinuous Galerkin (DG) method that scales optimally for regular and irregular meshes. The main idea centers on recasting the preconditioning problem so that existing AMG solvers for nodal lower order finite elements can be leveraged. This is accomplished by a transformation operator which maps the solution from a Lagrange basis representation to a Legendre basis representation. While this mapping function must be user supplied, we demonstrate how easily it can be constructed for somepopular finite element representations includingquadrilateral/hexahedral and triangular/tetrahedral DG formulations. Furthermore, we show that the mapping does not depend on the Jacobian transformation between reference and physical space and so it can be constructed with very limited mesh information. Parallel performance studies demonstrate the versatility of this approach.

Connections between the discontinuous Galerkin method and high‐order flux reconstruction schemes

SUMMARYWith high‐order methods becoming more widely adopted throughout the field of computational fluid dynamics, the development of new computationally efficient algorithms has increased tremendously in recent years. One of the most recent methods to be developed is the flux reconstruction approach, which allows various well‐known high‐order schemes to be cast within a single unifying framework. Whilst a connection between flux reconstruction and the more widely adopted discontinuous Galerkin method has been established elsewhere, it still remains to fully investigate the explicit connections between the many popular variants of the discontinuous Galerkin method and the flux reconstruction approach. In this work, we closely examine the connections between three nodal versions of tensor‐product discontinuous Galerkin spectral element approximations and two types of flux reconstruction schemes for solving systems of conservation laws on quadrilateral meshes. The different types of discontinuous Galerkin approximations arise from the choice of the solution nodes of the Lagrange basis representing the solution and from the quadrature approximation used to integrate the mass matrix and the other terms of the discretization. By considering both linear and nonlinear advection equations on a regular grid, we examine the mathematical properties that connect these discretizations. These arguments are further confirmed by the results of an empirical numerical study. Copyright © 2014 John Wiley & Sons, Ltd.

Linearization of Lagrange and Hermite interpolating matrix polynomials

This paper considers interpolating matrix polynomials P (λ) in Lagrange and Hermite bases. A classical approach to investigate the polynomial eigenvalue problem P (λ)x = 0 is linearization, by which the polynomial is converted into a larger matrix pencil with the same eigenvalues. Since the current linearizations of degree n Lagrange polynomials consist of matrix pencils with n + 2 blocks, they introduce additional eigenvalues at infinity. Therefore, we introduce new linearizations which overcome this. Initially, we restrict to Lagrange and barycentric Lagrange matrix polynomials and give two new and more compact linearizations, resulting in matrix pencils of n+ 1 and n blocks for polynomials of degree n. For the latter, there is a one-to-one correspondence between the eigenpairs of P (λ) and the eigenpairs of the pencil. We also prove that these linearizations are strong. Moreover, we show how to exploit the structure of the proposed matrix pencils in Krylov-type methods, so that in this case we only have to deal with linear system solves of matrices of the original matrix polynomial dimension. Finally, we generalize for multiple interpolation and introduce new linearizations for Hermite Lagrange and barycentric Hermite matrix polynomials. Again, we can show that the linearizations are strong and that there is a one-to-one correspondence of the eigenpairs.

An isogeometric analysis for elliptic homogenization problems

A novel and efficient approach which is based on the framework of isogeometric analysis for elliptic homogenization problems is proposed. These problems possess highly oscillating coefficients leading to extremely high computational expenses while using traditional finite element methods. The isogeometric analysis heterogeneous multiscale method (IGA-HMM) investigated in this paper is regarded as an alternative approach to the standard finite element heterogeneous multiscale method (FE-HMM) which is currently an effective framework to solve these problems. The method utilizes non-uniform rationalB-splines (NURBS) in both macro and micro levels instead of standard Lagrange basis. Besides the ability to describe exactly the geometry, it tremendously facilitates high-order macroscopic/microscopic discretizations thanks to the flexibility of refinement and degree elevation with an arbitrary continuity level provided by NURBS basis functions. Furthermore, the nearly optimal quadrature rule for IGA (Auricchio et al., 2012) introduced recently is utilized to reduce significantly the number of micro problems, which is the main factor contributing into the computational cost in heterogeneous multiscale method. A priori error estimates of the discretization error coming from macro and micro meshes and optimal micro refinement strategies for macro/micro NURBS basis functions of arbitrary orders are derived. An efficient coupling between degrees of macro and micro basis functions is introduced. Numerical results show the excellent performance of the proposed method.

Emulation to simulate low-resolution atmospheric data

Climate model development, testing, and analysis involve running the model extensively to tune the subgrid- scale parameters that provide closure to the system. This process demands substantial time and computational resources even for typical spatial resolutions and becomes in feasibly expensive for high-resolution studies. This paper presents alternative, computationally feasible methods to emulate the simulations within acceptable error bounds. This strategy can be easily implemented to obtain an ensemble of model runs. The paper outlines three approximation strategies: (1) interpolation with Lagrange basis functions, (2) least-squares (LS) approximation, and (3) interpolation with radial basis functions. As a proof of concept, a suite of relevant physical quantities are evaluated at unknown grid points of parameters, space, and time. The values obtained by emulation are compared against the simulated values to check the feasibility of the method. The advantages and shortcomings of the above-mentioned approximation schemes are discussed, including the savings of time and computational resources.

Stability of rootfinding for barycentric Lagrange interpolants

Computing the roots of a univariate polynomial can be reduced to computing the eigenvalues of an associated companion matrix. For the monomial basis, these computations have been shown to be numerically stable under certain conditions. However, for certain applications, polynomials are more naturally expressed in other bases, such as the Lagrange basis or orthogonal polynomial bases. For the Lagrange basis, the equivalent stability results have not been published. We show that computing the roots of a polynomial expressed in barycentric form via the eigenvalues of an associated companion matrix pair is numerically stable, and give a bound for the backward error. Numerical experiments show that the error bound is approximately an order of magnitude larger than the backward error. We also discuss the matter of scaling and balancing the companion matrix to bring it closer to a normal pair. With balancing, we are able to produce roots with small backward error.

The finite element simulation for the shallow impurity in quantum dots

The finite element method (FEM) has been implemented in this paper to investigate the electronic state of shallow hydrogenic impurities in spherical GaAs (Al,As) quantum dots (SQDs) by taking into account the finite value of realistic potential barrier height. The nonlinear partial differential equations have been discretized by means of Galerkin's weighted residue method assuming a uniform partition and using a quadratic Lagrange basis function for each finite element of the physical domain considered. The impurity binding energies have been calculated numerically by solving the governing equations and compared with the variational method. The results we have obtained are in excellent agreement with those published in the literature.

Design of Temporal Basis Functions for Time Domain Integral Equation Methods With Predefined Accuracy and Smoothness

A key parameter in the design of integral equation methods for transient electromagnetic scattering is the definition of temporal basis functions. The choice of temporal basis functions has a profound impact on the efficiency and accuracy of the numerical scheme. This paper presents a framework for the design of temporal basis functions with predefined accuracy and varying smoothness properties. The well-known shifted Lagrange basis functions naturally fit in this framework. New spline basis functions will be derived that have the same interpolation accuracy as shifted Lagrange basis functions and with the added advantage of being smooth. Numerical experiments show the positive influence of smoothness on the quadrature error in the numerical integration procedure. The global accuracy in time of the numerical scheme based on shifted Lagrange and spline basis functions has been experimentally analyzed. For a given interpolation error the experiments confirm the expected accuracy for the shifted Lagrange basis functions, but remarkably show a higher order of accuracy for the spline basis functions.

B-splines and NURBS based finite element methods for Kohn–Sham equations

This paper presents a B-splines and NURBS based finite element method for self-consistent solution of the Kohn–Sham equations [1,2] for electronic structure modeling of semiconducting materials. A Galerkin formulation is developed for the Schrödinger wave equation (SWE) that yields a complex-valued generalized eigenvalue problem. The nonlinear SWE that is embedded with a non-local potential as well as the nonlinear Hartree and exchange correlation potentials is solved in a self-consistent fashion. In the self-consistent solution procedure, a Poisson problem is integrated and solved as a function of the electron density that yields the local pseudopotential (for pseudopotential formulation) and the Hartree potential for SWE. Accuracy and convergence properties of the method are assessed through test cases and the superior performance of higher-order B-splines and NURBS basis functions as compared to the corresponding Lagrange basis functions is highlighted. Self-consistent solutions for semiconducting materials, namely, Gallium Arsenide (GaAs) and graphene are presented and results are validated via comparison with the planewave solutions.

Comparative Study of Three-Body Models for Continuum States

We discuss properties of three-body continuum states in the hyperspherical formalism. The associated coupled-channel system is solved on a Lagrange basis, and is complemented by the R-matrix theory for the treatment of scattering states. Three-body α + n + n phase shifts are determined under various conditions. Two approximations of the continuum are considered: the complex scaling and the pseudostate methods. Both are briefly outlined, and compared with the R-matrix approach for the dipole-strength distribution of 6 He. A fair agreement is obtained between the three methods. We also discuss the influence of the α + n and n + n interactions, as well as the effect of the Pauli principle. Some differences are obtained, but all calculations predict a broad 1 − resonance near 1 MeV.