Determination of an Extremal in Two-Dimensional Variational Problems Based on the RBF Collocation Method

This paper introduces a direct method derived from the global radial basis function (RBF) interpolation over arbitrary collocation nodes occurring in variational problems involving functionals that depend on functions of a number of independent variables. This technique parameterizes solutions with an arbitrary RBF and transforms the two-dimensional variational problem (2DVP) into a constrained optimization problem via arbitrary collocation nodes. The advantage of this method lies in its flexibility in selecting between different RBFs for the interpolation and parameterizing a wide range of arbitrary nodal points. Arbitrary collocation points for the center of the RBFs are applied in order to reduce the constrained variation problem into one of a constrained optimization. The Lagrange multiplier technique is used to transform the optimization problem into an algebraic equation system. Three numerical examples indicate the high efficiency and accuracy of the proposed technique.     

Parallel finite element simulations of incompressible viscous fluid flow by domain decomposition with Lagrange multipliers

A parallel approach to solve three-dimensional viscous incompressible fluid flow problems using discontinuous pressure finite elements and a Lagrange multiplier technique is presented. The strategy is based on non-overlapping domain decomposition methods, and Lagrange multipliers are used to enforce continuity at the boundaries between subdomains. The novelty of the work is the coupled approach for solving the velocity–pressure-Lagrange multiplier algebraic system of the discrete Navier–Stokes equations by a distributed memory parallel ILU (0) preconditioned Krylov method. A penalty function on the interface constraints equations is introduced to avoid the failure of the ILU factorization algorithm. To ensure portability of the code, a message based memory distributed model with MPI is employed. The method has been tested over different benchmark cases such as the lid-driven cavity and pipe flow with unstructured tetrahedral grids. It is found that the partition algorithm and the order of the physical variables are central to parallelization performance. A speed-up in the range of 5–13 is obtained with 16 processors. Finally, the algorithm is tested over an industrial case using up to 128 processors. In considering the literature, the obtained speed-ups on distributed and shared memory computers are found very competitive.     

非匹配网格上广义Stokes问题的区域分裂解法及事后误差估算

发展了一种广义Stokes问题的无覆盖区域分裂解法。子域交界面上的约束条件是通过引入一Lagrange乘子而得到弱满足的,在有限元离散子域的交界处网格可以是非匹配的。应用Petrov Galerkin方法解每个子域上的广义Stokes问题,而交界面上的Lagrange乘子则通过共轭梯度法迭代求解,各变量均由线性函数离散。对上述区域分裂解法,还构造了基于求解当地问题的误差事后估算方法。各变量的当地误差估算器定义在二阶非连续鼓包(bump)函数的空间中。最后给出了基于事后误差估算值的自适应网格上的数值结果。     

A coupled edge-/face-based smoothed finite element method for structural-acoustic problems

The edge-based smoothed finite element method (ES-FEM) and the face-based smoothed finite element method (FS-FEM) developed recently have shown great efficiency in solving solid mechanics problems with triangular and tetrahedral meshes. In this paper, a coupled ES-/FS-FEM model is extended to solve the structural-acoustic problems consisting of a plate structure interacting with the fluid medium. Three-node triangular elements and four-node tetrahedral elements are used to discretize the two-dimensional (2D) plate and three-dimensional (3D) fluid, respectively, as they can be generated easily and even automatically for complicated geometries. The field variable in each element is approximated using the linear shape functions, which is exactly the same as that in the standard FEM. The gradient field of the problem is obtained particularly using the gradient smoothing operation over the edge-based and face-based smoothing domains in 2D and 3D, respectively. The gradient smoothing technique can provide a proper softening effect to the model, effectively solve the problems caused by the well-known “overly-stiff” phenomenon existing in the standard FEM, and hence significantly improve the accuracy of the solution for the coupled systems. Intensive numerical studies have been conducted to verify the effectiveness of the coupled ES-/FS-FEM for structural-acoustic problems.     

Hyperbolic Divergence Cleaning for the MHD Equations

In simulations of magnetohydrodynamic (MHD) processes the violation of the divergence constraint causes severe stability problems. In this paper we develop and test a new approach to the stabilization of numerical schemes. Our technique can be easily implemented in any existing code since there is no need to modify the solver for the MHD equations. It is based on a modified system in which the divergence constraint is coupled with the conservation laws by introducing a generalized Lagrange multiplier. We suggest a formulation in which the divergence errors are transported to the domain boundaries with the maximal admissible speed and are damped at the same time. This corrected system is hyperbolic and the density, momentum, magnetic induction, and total energy density are still conserved. In comparison to results obtained without correction or with the standard “divergence source terms,” our approach seems to yield more robust schemes with significantly smaller divergence errors.     

Numerical simulation of premixed combustion using an enriched finite element method

In this paper we present a novel discretization technique for the simulation of premixed combustion based on a locally enriched finite element method (FEM). Use is made of the G-function approach to premixed combustion in which the domain is divided into two parts, one part containing the burned and another containing the unburned gases. A level-set or G-function is used to define the flame interface separating burned from unburned gases. The eXtended finite element method (X-FEM) is employed, which allows for velocity and pressure fields that are discontinuous across the flame interface. Lagrange multipliers are used to enforce the correct essential interface conditions in the form of jump conditions across the embedded flame interface. A persisting problem with the use of Lagrange multipliers in X-FEM has been the discretization of the Lagrange multipliers. In this paper the distributed Lagrange multiplier technique is adopted. We will provide results from a spatial convergence analysis showing good convergence. However, a small modification of the interface is required to ensure a unique solution. Finally, results are presented from the application of the method to the problems of moving flame fronts, the Darrieus–Landau instability and a piloted Bunsen burner flame.     

Particles on a string: Towards understanding a quantum mechanical divergence

The dynamics of two harmonically coupled solidbodies connected to a finitely extended mechanical field (string) is solved exactly and explicitly in the Lagrange formalism. For infinite length of strings the particles motion becomes damped, but not as a simple linearly damped harmonic oscillator. The model allows for a detailed discussion of its quantum mechanics, in particular of a previously recognized ultraviolet divergence.     

Could dynamical Lorentz symmetry breaking induce the superluminal neutrinos?

A toy fermion model coupled to the Lagrange multiplier constraint field is proposed. The possibility of superluminal neutrino propagation as a result of dynamical Lorentz symmetry breaking is studied.     

Generalized Hamiltonian formalism of (2+1)-dimensional non-linear \sigma-model in polynomial formulation

We investigate the canonical structure of the (2+1)-dimensional non-linear model in a polynomial formulation. A current density defined in the non-linear model is a vector field, which satisfies a formal flatness (or pure gauge) condition. It is the polynomial formulation in which the vector field is regarded as a dynamic variable on which the flatness condition is imposed as a constraint condition by introducing a Lagrange multiplier field. The model so formulated has gauge symmetry under a transformation of the Lagrange multiplier field. We construct the generalized Hamiltonian formalism of the model explicitly by using the Dirac method for constrained systems. We derive three types of the pre-gauge-fixing Hamiltonian systems: In the first system the current algebra is realized as the fundamental Dirac Brackets. The second one manifests the similar canonical structure as the Chern-Simons or BF theories. In the last one there appears an interesting interaction as the dynamic variables are coupled to their conjugate momenta via the covariant derivative. Received: 29 September 1998 / Published online: 14 January 1999     

A precise integration method for solving coupled vehicle–track dynamics with nonlinear wheel–rail contact

A new method is proposed as a solution for the large-scale coupled vehicle–track dynamic model with nonlinear wheel–rail contact. The vehicle is simplified as a multi-rigid-body model, and the track is treated as a three-layer beam model. In the track model, the rail is assumed to be an Euler-Bernoulli beam supported by discrete sleepers. The vehicle model and the track model are coupled using Hertzian nonlinear contact theory, and the contact forces of the vehicle subsystem and the track subsystem are approximated by the Lagrange interpolation polynomial. The response of the large-scale coupled vehicle–track model is calculated using the precise integration method. A more efficient algorithm based on the periodic property of the track is applied to calculate the exponential matrix and certain matrices related to the solution of the track subsystem. Numerical examples demonstrate the computational accuracy and efficiency of the proposed method.     

A Mixed Finite Element Method for Stationary Magneto-Heat Coupling System with Variable Coefficients

In this article, a mixed finite element method for thermally coupled, stationary incompressible MHD problems with physical parameters dependent on temperature in the Lipschitz domain is considered. Due to the variable coefficients of the MHD model, the nonlinearity of the system is increased. A stationary discrete scheme based on the coefficients dependent temperature is proposed, in which the magnetic equation is approximated by Nédélec edge elements, and the thermal and Navier–Stokes equations are approximated by the mixed finite elements. We rigorously establish the optimal error estimates for velocity, pressure, temperature, magnetic induction and Lagrange multiplier with the hypothesis of a low regularity for the exact solution. Finally, a numerical experiment is provided to illustrate the performance and convergence rates of our numerical scheme.     

On the spectral accuracy of a fictitious domain method for elliptic operators in multi-dimensions

This work is a continuation of the authors efforts to develop high-order numerical methods for solving elliptic problems with complex boundaries using a fictitious domain approach. In a previous paper, a new method was proposed, based on the use of smooth forcing functions with identical shapes, mutually disjoint supports inside the fictitious domain and whose amplitudes play the role of Lagrange multipliers in relation to a discrete set of boundary constraints. For one-dimensional elliptic problems, this method shows spectral accuracy but its implementation in two dimensions seems to be limited to a fourth-order algebraic convergence rate. In this paper, a spectrally accurate formulation is presented for multi-dimensional applications. Instead of being specified locally, the forcing function is defined as a convolution of a mollifier (smooth bump function) and a Lagrange multiplier function (the amplitude of the bump). The multiplier function is then approximated by Fourier series. Using a Fourier Galerkin approximation, the spectral accuracy is demonstrated on a two-dimensional Laplacian problem and on a Stokes flow around a periodic array of cylinders. In the latter, the numerical solution achieves the same high-order accuracy as a Stokes eigenfunction expansion and is much more accurate than the solution obtained with a classical third order finite element approximation using the same number of degrees of freedom.     

Metric-affine variational principles in general relativity. I. Riemannian space-time

Within the confines of conventional general relativity, variational principles are analyzed in which the metric tensor and the asymmetric linear connection are varied independently. The constraint that space-time remain Riemannian is introduced by means of the Lagrange multiplier technique. The Lagrange multiplier which effects this constraint, the hypermomentum current, is closely related to the constraint force which keeps space-time Riemannian and should be a measure for the violation of the Riemannian constraint at the microscopic level.Alexander von Humboldt Fellow.     

弹性介质的Lagrange动力学与地震波方程

本文将地球介质看作是弹性介质, 从弹性体的Navier方程出发, 建立均匀弹性介质和非均匀弹性介质的分析动力学方程——Lagrange方程, 利用弹性介质的Lagrange方程导出匀弹性介质和非均匀弹性介质的地震波方程, 为用Lagrange分析动力学研究地球介质中地震波传播规律和解决地震勘探中的有关问题提供基础. 关键词： 地震勘探 弹性介质 Lagrange 方程 地震波方程     

Velocity–vorticity–helicity formulation and a solver for the Navier–Stokes equations

For the three-dimensional incompressible Navier–Stokes equations, we present a formulation featuring velocity, vorticity and helical density as independent variables. We find the helical density can be observed as a Lagrange multiplier corresponding to the divergence-free constraint on the vorticity variable, similar to the pressure in the case of the incompressibility condition for velocity. As one possible practical application of this new formulation, we consider a time-splitting numerical scheme based on an alternating procedure between vorticity–helical density and velocity–Bernoulli pressure systems of equations. Results of numerical experiments include a comparison with some well-known schemes based on pressure–velocity formulation and illustrate the competitiveness on the new scheme as well as the soundness of the new formulation.     

重心Lagrange插值配点法求解二维双曲电报方程

提出一种求解二维双曲电报方程的高精度重心Lagrange插值配点法.采用重心Lagrange插值构造包含时间和空间变量的近似函数.在给定Chebyshev-Gauss-Lobatto节点上,将多变量重心Lagrange插值近似函数代入双曲电报方程及其定解条件,得到离散代数方程组.包含狄里克雷和诺依曼边界条件的数值算例表明,本文方法程序实现方便并具有高精度,可应用于求解高维问题.     

Lagrange multiplier method used in BESⅢ

In this paper, the kinematic fitting with the Lagrange multiplier method has been studied for BESⅢ experiment. First we introduce the Lagrange multiplier method and implement kinematic constraints. Then we present the performance of the kinematic fitting algorithm. With the kinematic fitting, we can improve the resolution of track parameters and reduce the background.     

The matched asymptotic expansion (MAE) technique applied to acoustic radiation from vibrating surfaces

The MAE technique has been used in the field of fluid mechanics for many years. Only recently this technique has been applied to acoustic problems, where it has been found to be an excellent and powerful tool in analyzing either scattering and diffraction or radiation from moving rigid objects (propellers). The purpose of this paper is to very briefly review the MAE technique as applied to low frequency acoustics in general, and then apply the resulting approach to a series of progressively more difficult problems which are of interest to many underwater acousticians. The analysis is first applied to two problems with single degrees of freedom for structural vibrations: (1) a sphere, both velocity and force driven, and (2) a circular piston in infinite rigid baffle. These are classical problems and the solutions as obtained by the MAE technique are then compared to the exact classical solutions. The MAE solutions are then generalized to a more difficult problem, with two degrees of freedom for the surface vibration, where two concentric pistons in an infinite rigid baffle are vibrating and coupled via the fluid. For each of the problems analyzed, the structural wavelength a is assumed to be small compared to the fluid wavelength (i.e., ka ? 1). The inner region close to the vibrating structure, in which the fluid motion is effectively incompressible in nature, is governed by the Laplace equation while the outer solution is governed by the Helmholtz equation. The inner and outer solutions are obtained independently and are then joined together by the MAE matching procedure. A composite solution is then obtained from a combination of the inner and the outer solutions. Agreements with the exact theory for the radiated pressure, surface resistance and reactance are shown to be excellent.     

Gain ripple minimization in the wide-band SCISSOR Raman amplifier

A side coupled integrated spaced sequence (SCISSOR) of Silicon or Chalcogenide glass microrings of different diameter are considered as a wide-band Raman amplifier. Because the two-photon absorption (TPA) losses in Chalcogenide glass are less than that of Silicon, it is shown that Chalcogenide microrings are suitable for wide-band Raman amplifier in the telecommunication bandwidth (1.5 μm). The Lagrange unknown multiplier method is employed to minimize the gain ripple of the amplifier versus the pump and system parameters.