Variational integrators for electric circuits

Variational integrators are symplectic-momentum preserving integrators that are based on a discrete variational formulation of the underlying system. So far, variational integrators have been mainly developed and used for a wide variety of mechanical systems. In this work, we develop a variational integrator for the simulation of electric circuits. An appropriate variational formulation is presented to model the circuit from which the equations of motion are derived. Finally, a corresponding time-discrete variational formulation provides an iteration scheme for the simulation of the electric circuit. In this way, a variational integrator is constructed that gains several advantages. A comparison to standard integration techniques shows that even for simple LCR circuits a better long-time energy behavior and frequency preservation can be obtained. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Variational integrators of higher order for flexible multibody systems

In this paper we present a time stepping scheme which is based on a variational integrator. This higher-order time stepping scheme includes constraints and a viscoelastic material formulation. A variational integrator is structure-preserving which results from using a discrete variational principle. Therefore, a variational integrator always takes the form of discrete EULER-LAGRANGE equations or the equivalent position-momentum equations. In this framework, we consider the motion of a flexible rope with non-holonomic constraints by the LAGRANGE-multiplier technique. The time stepping scheme is derived from a space-time discretization of HAMILTON's principle. The space discretization is based on one-dimensional linear LAGRANGE polynomials, whereas the time discretization is based on higher-order polynomials and higher-order quadrature rules. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Variational integrators of higher order for constrained dynamical systems

The variational integrators presented in [5] are applied to systems with holonomic constraints, yielding constrained higher order variational integrators that are an extension of the constrained Galerkin methods in [4]. The construction of the integrators bases on a discrete version of Hamilton's principle. The inheritance of qualitative properties associated to the solution of the dynamical system to the discrete solution is analysed. Furthermore, the convergence order of the integrators and the computational efficiency is investigated numerically. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Higher-order energy consistent time integrators for nonlinear thermoviscoelastodynamics

An advantage of the temporal fe method is that higher-order accurate time integrators can be constructed easily. A further important advantage is the inherent energy consistency if applied to equations of motion. The temporal fe method is therefore used to construct higher-order energy-momentum conserving time integrators for nonlinear elastodynamics (see Ref. [1]). Considering finite motions of a flexible solid body with internal dissipation, an energy consistent time integration is also of great advantage (see the references [2, 3]). In this paper, we show that an energy consistent time integration is also advantageous for dynamics with dissipation arising from conduction of heat as well as from a viscous material. The energy consistency is preserved by using a new enhanced hybrid Galerkin (ehG) method. The obtained numerical schemes satisfy the energy balance exactly, independent of their accuracy and the used time step size. This guarantees numerical stability. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Mechanical integrators for mixed elements in nonlinear elastodynamics

Enhanced assumed strain (EAS) elements (see, for example, [1]) are well-known to exhibit improved convergence behavior, especially in the context of bending dominated situations and in the incompressible limit. In the present work we focus on the application of EAS elements to nonlinear elastodynamics. In particular, we aim at the design of energy-momentum schemes for the stable numerical integration of the semi-discrete equations of motion. For this purpose we make use of the notion of a G-equivariant discrete derivative introduced by Gonzalez [2] in the framework of general finite-dimensional Hamiltonian systems with symmetry. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Variational integrators for open-loop and closed-loop optimal control of mechanical systems

Mechanical systems have structural properties, e.g. symplecticity, symmetry, and a specific energy behavior, which get lost in standard integration methods. Therefore, symplectic integration methods are used in simulation and control of mechanical systems. This paper combines two methods of the class of structure-preserving control methods, namely a recently developed feedback control method and open loop optimal control based on variational integrator discretization. The combination is applied to the benchmark example of a cart pendulum system. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Variational approach for nonlinear oscillators

We propose a novel variational approach for limit cycles of a kind of nonlinear oscillators. Some examples are given to illustrate the effectiveness and convenience of the method. The obtained results are valid for the whole solution domain with high accuracy.     

Variational approach for nonlinear oscillators

We propose a novel variational approach for limit cycles of a kind of nonlinear oscillators. Some examples are given to illustrate the effectiveness and convenience of the method. The obtained results are valid for the whole solution domain with high accuracy.     

Symmetric and symplectic exponential integrators for nonlinear Hamiltonian systems

This letter studies symmetric and symplectic exponential integrators when applied to numerically computing nonlinear Hamiltonian systems. We first establish the symmetry and symplecticity conditions of exponential integrators and then show that these conditions are extensions of the symmetry and symplecticity conditions of Runge–Kutta methods. Based on these conditions, some symmetric and symplectic exponential integrators up to order four are derived. Two numerical experiments are carried out and the results demonstrate the remarkable numerical behaviour of the new exponential integrators in comparison with some symmetric and symplectic Runge–Kutta methods in the literature.     

Constraint preserving integrators for general nonlinear higher index DAEs

Summary. In the last few years there has been considerable research on numerical methods for differential algebraic equations (DAEs) where is identically singular. The index provides one measure of the singularity of a DAE. Most of the numerical analysis literature on DAEs to date has dealt with DAEs with indices no larger than three. Even in this case, the systems were often assumed to have a special structure. Recently a numerical method was proposed that could, in principle, be used to integrate general unstructured higher index solvable DAEs. However, that method did not preserve constraints. This paper will discuss a modification of that approach which can be used to design constraint preserving integrators for general nonlinear higher index DAEs. Received August 25, 1993 / Revised version received April 7, 1994     

Variational iteration technique for solving nonlinear equations

In this paper, we use the variational iteration technique to suggest some new iterative methods for solving nonlinear equations f(x)=0. We also discuss the convergence criteria of these new iterative methods. Comparison with other similar methods is also given. These new methods can be considered as an alternative to the Newton method. We also give several examples to illustrate the efficiency of these methods. This technique can be used to suggest a wide class of new iterative methods for solving system of nonlinear equations.     

Explicit Gautschi-type integrators for nonlinear multi-frequency oscillatory second-order initial value problems

Numerical Algorithms - The main theme of this paper is explicit Gautschi-type integrators for the nonlinear multi-frequency oscillatory second-order initial value problems of the form $y^{prime...     

Addition in a class of nonlinear Stieltjes integrators

A product integral formula is established for the generation of an evolution system by the sum of two Stieltjes integrators. The class of Stieltjes integrators considered is the class recently introduced and studied by J.V. Herod.     

Variational method for precise asymptotic formulas for nonlinear eigenvalue problems

We consider the nonlinear eigenvalue problem motivated by the perturbed elliptic sine-Gordon equation where p >1 is a constant and λ ∈ R is an eigenvalue parameter. Our aim is to clarify the asymptotic relationship between L ^p+1-norm, L ^∞-norm of the solutions and λ when these norms are large. To this end, we consider an associated variational problem with this equation on a manifold new parameter), and obtain a solution pair . We establish the precise asymptotic formulas for .     

Variational formulations for nonlinear wave propagation and unsteady transonic flow

A variational principle developed recently for constrained vector fields is applied to nonlinear waves and to unsteady, transonic flow. The mathematical conditions on the admissible functional form of the speed of propagation is compatible with physical considerations for the first systems and the treatment of the mixed derivative is indicative for a proper way to discretize the second for numerical calculations. The variational formulation provides a framework for stability analysis and finite element approximations for the nonlinear systems considered.

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Zusammenfassung Ein Variationsprinzip, das in letzter Zeit für Vektorfelder mit einschränkenden Bedingungen entwickelt worden ist, wird auf nichtlineare Wellen und auf nichtstationäre transonische Strömung angewendet. Die mathematische Bedingung für die zulässige funktionale Form der Fortpflanzungsgeschwindigkeit ist verträglich mit den physikalischen Bedingungen für das erste System, und die Behandlung der gemischten Ableitung deutet an, wie das zweite System für numerische Behandlung diskretisiert werden soll. Die variationale Formulierung gibt einen Rahmen für eine Stabilitätsuntersuchung und für eine Näherung durch finite Elemente für das nichtlineare System.

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Work partially supported by Grant AFOSR-73-2561.     

Variational iteration technique for solving a system of nonlinear equations

In this paper, we use the variational iteration technique to suggest and analyze some new iterative methods for solving a system of nonlinear equations. We prove that the new method has fourth-order convergence. Several numerical examples are given to illustrate the efficiency and performance of the new iterative methods. Our results can be viewed as a refinement and improvement of the previously known results.