Euler-Lagrange Partial Differential Equations Research Articles

Euler–Lagrange equation for gradient-type Lagrangian and related conservation laws

AbstractVariational calculus with gradient-type variations has often been neglected, although it proves to be suitable for certain concrete problems governed by several evolution variables. These kinds of variations lead to Euler–Lagrange partial differential equations controlled by the right-hand member. In this context, we also introduce anti-trace Euler–Lagrange partial differential equations that are suitable for some innovative ideas. Also, some applications are provided for the theoretical results derived in the paper.

On equilibrium equations and their perturbations using three different variational formulations of nonlinear electroelastostatics

We derive the equations of nonlinear electroelastostatics using three different variational formulations involving the deformation function and an independent field variable representing the electric character – considering either the electric field , the electric displacement or the electric polarization . The first variation of the energy functional results in the set of Euler–Lagrange partial differential equations, which are the equilibrium equations, boundary conditions and certain constitutive equations for the electroelastic system. The partial differential equations for obtaining the bifurcation point have also been found using the bilinear functional based on the second variation. We show that the well-known Maxwell stress in a vacuum is a natural outcome of the derivation of equations from the variational principles and does not depend on the formulation used. As a result of careful analysis, it is found that there are certain terms in the bifurcation equation that appear to be difficult to obtain using ordinary perturbation-based analysis of the Euler–Lagrange equation. From a practical viewpoint, the formulations based on and result in simpler equations and are expected to be more suitable for analysing problems of stability as well as post-buckling behaviour.

Electrical Performance Design for Circuit Breakers Based on Multi-domain Sequential Coupling

This paper presents an electrical performance design method of mechatronic circuit breakers (CB) based on multi-domain sequential coupling (MSC). Firstly, the attraction force of electromagnetic trip mechanism is obtained by Maxwell formula and Ampere’s circuital law. The multi-body kinematics and dynamics model of CB are built by recursive Euler–Lagrange partial differential equations using generalized coordinates. The electrical performance optimization model of CB is built to obtain the optimal opening angle, clearance and over-travel of the contact. The MSC piecewisely combines one field with another field according to the variation and sensitivity of the unstable field to perform multiphysics simulation such as magnetohydro-dynamics (MHD). The physical experiment of rated short-circuit capacity is carried out to verify the proposed method using infrared thermogram. The peak voltage, Joule integral and on–off time of the CB are optimized and shortened with maximum change of ratio 8.13% which proves that the CB has better reliability with longer mean time between failures.

Lagrange力学に基づく局所エネルギー保存型数値解法導出法と線形波動方程式に対する無反射境界条件への応用(理論, 科学技術計算と数値解析研究部会)

Lagrange力学に基づく局所エネルギー保存型数値解法導出法と線形波動方程式に対する無反射境界条件への応用(理論, 科学技術計算と数値解析研究部会)

Interruption performance design of variable freedom mechanism triggered by electro-mechanical-magnetic coupling

Coupling mechanism plays an important role in transmitting, motivating and actuating mechanical functions. However, it is difficult to obtain the transient dynamics performance of mechanism with variable degree of freedom precisely. Therefore, an interruption performance design method of variable freedom mechanism triggered by electro-magneto-thermo coupling is put forward. The Euler-Lagrange partial differential equations of variable freedom mechanism are built using generalized coordinates. Degree of freedom reduction rules are proposed to merge transformation or rotation constraints and obtain the total degrees of freedom of variable freedom mechanism at each transient status. Bivariate interpolating is employed to determine the electro-mechanical-magnetic coupled Lorentz force. Dynamics performance is simulated by iteration of linear algebraic equations using implicit predictor-corrector integration method. The design parameters such as stiffness and pre-tightening force of trigger spring, permissible dimension deviations and hole-shaft fit tolerance are determined and improved using the sensitivity analysis of simulation results. The pneumatic mechanical endurance and thermal infrared temperature rise experiments are accomplished to determine the infrared radiation energy distribution and transient working status of components. It gives an auxiliary thermo-visual approach for transient performance design of coupling mechanism.

Lagrangian approach to deriving energy-preserving numerical schemes for the Euler–Lagrange partial differential equations

We propose a Lagrangian approach to deriving energy-preserving finite difference schemes for the Euler–Lagrange partial differential equations. Noether’s theorem states that the symmetry of time translation of Lagrangians yields the energy conservation law. We introduce a unique viewpoint on this theorem: “the symmetry of time translation of Lagrangians derives the Euler–Lagrange equation and the energy conservation law, simultaneously.” The proposed method is a combination of a discrete counter part of this statement and the discrete gradient method. It is also shown that the symmetry of space translation derives momentum-preserving schemes. Finally, we discuss the existence of discrete local conservation laws.

An adaptive meshfree method for phase-field models of biomembranes. Part I: Approximation with maximum-entropy basis functions

We present an adaptive meshfree method to approximate phase-field models of biomembranes. In such models, the Helfrich curvature elastic energy, the surface area, and the enclosed volume of a vesicle are written as functionals of a continuous phase-field, which describes the interface in a smeared manner. Such functionals involve up to second-order spatial derivatives of the phase-field, leading to fourth-order Euler–Lagrange partial differential equations (PDE). The solutions develop sharp internal layers in the vicinity of the putative interface, and are nearly constant elsewhere. Thanks to the smoothness of the local maximum-entropy (max-ent) meshfree basis functions, we approximate numerically this high-order phase-field model with a direct Ritz–Galerkin method. The flexibility of the meshfree method allows us to easily adapt the grid to resolve the sharp features of the solutions. Thus, the proposed approach is more efficient than common tensor product methods (e.g. finite differences or spectral methods), and simpler than unstructured C0 finite element methods, applicable by reformulating the model as a system of second-order PDE. The proposed method, implemented here under the assumption of axisymmetry, allows us to show numerical evidence of convergence of the phase-field solutions to the sharp interface limit as the regularization parameter approaches zero. In a companion paper, we present a Lagrangian method based on the approximants analyzed here to study the dynamics of vesicles embedded in a viscous fluid.

Anisotropic optical flow algorithm based on self-adaptive cellular neural network

An anisotropic optical flow estimation method based on self-adaptive cellular neural networks (CNN) is proposed. First, a novel optical flow energy function which contains a robust data term and an anisotropic smoothing term is projected. Next, the CNN model which has the self-adaptive feedback operator and threshold is presented according to the Euler–Lagrange partial differential equations of the proposed optical flow energy function. Finally, the elaborate evaluation experiments indicate the significant effects of the various proposed strategies for optical flow estimation, and the comparison results with the other methods show that the proposed algorithm has better performance in computing accuracy and efficiency.

Reformulating the full-Stokes ice sheet model for a more efficient computational solution

Abstract. The first-order or Blatter-Pattyn ice sheet model, in spite of its approximate nature, is an attractive alternative to the full Stokes model in many applications because of its reduced computational demands. In contrast, the unapproximated Stokes ice sheet model is more difficult to solve and computationally more expensive. This is primarily due to the fact that the Stokes model is indefinite and involves all three velocity components, as well as the pressure, while the Blatter-Pattyn discrete model is positive-definite and involves just the horizontal velocity components. The Stokes model is indefinite because it arises from a constrained minimization principle where the pressure acts as a Lagrange multiplier to enforce incompressibility. To alleviate these problems we reformulate the full Stokes problem into an unconstrained, positive-definite minimization problem, similar to the Blatter-Pattyn model but without any of the approximations. This is accomplished by introducing a divergence-free velocity field that satisfies appropriate boundary conditions as a trial function in the variational formulation, thus dispensing with the need for a pressure. Such a velocity field is obtained by vertically integrating the continuity equation to give the vertical velocity as a function of the horizontal velocity components, as is in fact done in the Blatter-Pattyn model. This leads to a reduced system for just the horizontal velocity components, again just as in the Blatter-Pattyn model, but now without approximation. In the process we obtain a new, reformulated Stokes action principle as well as a novel set of Euler-Lagrange partial differential equations and boundary conditions. The model is also generalized from the common case of an ice sheet in contact with and sliding along the bed to other situations, such as to a floating ice shelf. These results are illustrated and validated using a simple but nontrivial Stokes flow problem involving a sliding ice sheet.

A variational method for the recovery of dense 3D structure from motion

The purpose of this study is to investigate a variational formulation of the problem of three-dimensional (3D) interpretation of temporal image sequences based on the 3D brightness constraint and anisotropic regularization. The method allows movement of both the viewing system and objects and does not require the computation of image motion prior to 3D interpretation. Interpretation follows the minimization of a functional with two terms: a term of conformity of the 3D interpretation to the image sequence first-order spatio-temporal variations, and a term of regularization based on anisotropic diffusion to preserve the boundaries of interpretation. The Euler–Lagrange partial differential equations corresponding to the functional are solved efficiently via the half-quadratic algorithm. Results of several experiments on synthetic and real image sequences are given to demonstrate the validity of the method and its implementation.

Motion tracking as spatio-temporal motion boundary detection

The purpose of this study is to investigate tracking of moving objects in a sequence of images by detecting the surface generated by motion boundaries in the space–time domain. Estimation of this spatio-temporal surface is formulated as a Bayesian image partitioning problem. Minimization of the resulting energy functional seeks a solution biased toward smooth closed surfaces which coincide with motion boundaries, have small area, and partition the image into regions of contrasting motion activity. The Euler–Lagrange partial differential equations of minimization are expressed as level set evolution equations to obtain a topology independent and numerically stable algorithm. The formulation does not require estimation of the image motion field or assume a known background. It allows multiple non-simultaneous independent motions to occur and can account for camera motion without prior estimation of this motion. The analysis assumes short-range image motion. With moving cameras, it assumes that this short-range motion varies smoothly everywhere except across motion boundaries.

Efficient Multiresolution Counterparts to Variational Methods for Surface Reconstruction

Variational methods have been employed with considerable success in computer vision, particularly for surface reconstruction problems. Formulations of this type require the solution of computationally complex Euler–Lagrange partial differential equations (PDEs) to obtain the desired reconstructions. Further, the calculation of reconstruction error covariances for such approaches are usually neglected.In this paper we describe a computationally efficient multiscale approach to surface reconstruction which differs fundamentally from other multiresolution methods that are used to solve the Euler–Lagrange PDEs. Instead, we interpret the variational problem as a statistical estimation problem in order to define a nearby, but slightlydifferent, multiscale estimation problem that admits efficient solutions for both surface reconstructionandthe calculation of error statistics. In particular, the membrane and thin-plate variational models for surfaces are interpreted as 1/f2prior statistical models for the surface and its gradients, respectively. Such 1/f2behavior is then achieved using a recently introduced class of multiresolution models that admits algorithms with constant per-pixel computational complexity.