Stable long-term simulation of dynamically loaded elastomers

A well–known problem in long–term simulations of flexible solid bodies is the restriction to small time steps of standard time integrators in order to obtain a stable simulation. One approach is to achieve exact energy conservation while simulating a nonlinear elastic body (see Reference [1] and the references therein). Additionally, total linear and total angular momentum is conserved for a free motion. This approach leads to a qualitatively improved solution, because the approximated time evolution exactly fulfills the same physical laws as the exact time evolution. Incorporating energy dissipation, the energy conserving time integration is extended to an energy consistent time integration. It turned out that such an energy consistent time integration is also of great advantage when computing finite motions of flexible solid bodies with material dissipation (see References [2,3]). This paper points out that an energy consistent time discretisation is also advantageous for dynamic finite deformation thermoviscoelasticity under dynamic loads. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Galerkin-based Time Integrators for Geometrically Nonlinear Elasto-Plastodynamics

Computational modelling often demands the incorporation of dynamic effects. The consideration of computational elastoplastodynamics requires in particular advanced numerical techniques to satisfy the classical balance laws like balance of linear and angular momentum or the laws of thermodynamics. Regarding the thermodynamical consistency of elasto-plastodynamics, the conservation of the total energy in the elastic case and the fulfilment of the dissipation inequality in the plastic case are very important. In this contribution we deal with a mechanically and thermodynamically consistent Galerkin-based time-stepping scheme for finite elasto-plastodynamics. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Symmetric multistep methods over long times

Summary. For computations of planetary motions with special linear multistep methods an excellent long-time behaviour is reported in the literature, without a theoretical explanation. Neither the total energy nor the angular momentum exhibit secular error terms. In this paper we completely explain this behaviour by studying the modified equation of these methods and by analyzing the remarkably stable propagation of parasitic solution components.Mathematics Subject Classification (1991): 65L06, 65P10     

Energy and momentum conserving methods of arbitrary order for the numerical integration of equations of motion

Summary In Part I of this work, numerical methods were derived for the solution of the equations of motion of a single particle subject to a central force which conserved exactly the energy and momenta. In the present work, the methodology of Part I is extended, in part, to motion of a system of particles in that the energy and linear momentum are conserved exactly. In addition, the angular momentum will be conserved to one more order of accuracy than in conventional methods. Exact conservation of the total angular momentum results only for the lowest order numerical approximation, which is equivalent to the discrete mechanics presented elsewhere.     

On the existence of weak solutions to the equations of non‐stationary motion of heat‐conducting incompressible viscous fluids

This paper is concerned with the equations of non‐stationary motion in 3D of heat‐conducting incompressible viscous fluids with temperature‐dependent viscosity. The conservation of internal energy includes the usual dissipation term. We prove the existence of a ‘weak solution with defect measure’ to the system of PDEs under consideration. Our method of proof is based on a regularization of the equations of conservation of momentum. Copyright © 2006 John Wiley & Sons, Ltd.     

Conserving algorithms for the dynamics of Hamiltonian systems on lie groups

Summary Three conservation laws are associated with the dynamics of Hamiltonian systems with symmetry: The total energy, the momentum map associated with the symmetry group, and the symplectic structure are invariant under the flow. Discrete time approximations of Hamiltonian flows typically do not share these properties unless specifically designed to do so. We develop explicit conservation conditions for a general class of algorithms on Lie groups. For the rigid body these conditions lead to a single-step algorithm that exactly preserves the energy, spatial momentum, and symplectic form. For homogeneous nonlinear elasticity, we find algorithms that conserve angular momentum and either the energy or the symplectic form.     

自由下落猫的非完整运动规划最优控制研究

研究猫在自由下落时姿态运动规划问题．自由落体的猫在空中转体运动由于角速度不可积,姿态运动方程呈现为非完整形式．当系统角动量为0时,导出由两个对称刚体组成的自由下落猫的非完整姿态运动方程．利用该非完整方程系统的控制问题可转化为无漂移系统的非完整运动规划问题．基于Ritz近似理论,给出自由落体猫姿态运动规划的Gauss-Newton算法．最后对自由落体猫作了数值仿真实验,仿真结果验证了该算法的有效性．     

Gauging magnetorotational instability

Previously (Z. Angew. Math. Phys. 57:615–622, 2006), we examined the axisymmetric stability of viscous resistive magnetized Couette flow with emphasis on flows that would be hydrodynamically stable according to Rayleigh’s criterion: opposing gradients of angular velocity and specific angular momentum. A uniform axial magnetic field permeates the fluid. In this regime, magnetorotational instability (MRI) may occur. It was proved that MRI is suppressed, in fact no instability at all occurs, with insulating boundary conditions, when a term multipling the magnetic Prandtl number is neglected. Likewise, in the current work, including this term, when the magnetic resistivity is sufficiently large, MRI is suppressed. This shows conclusively that small magnetic dissipation is a feature of this instability for all magnetic Prandtl numbers. A criterion is provided for the onset of MRI.     

Equations of motion of rotating bodies in general relativity in the post-Newtonian approximation

Equations of the translational and rotational motion of two bodies possessing intrinsic angular momentum are obtained by the Einstein—Infeld—Hoffmann method in the post-Newtonian approximation. The results agree with the Kerr metric expressed in a harmonic system of coordinates with symmetry of the spatial components of the metric with respect to its indices and with a conservation law for the total angular momentum that is the sum of the orbital and spin angular momenta, and they give the correct passage to the limit to the equation of motion of a test particle with spin.All-Russia Institute of Experimental Physics. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 101, No. 1, pp. 123–135, October, 1994.     

重建极性连续统理论的基本定律和原理(Ⅶ)——增率型

目的是建立微极连续统增率型的较为完整的运动方程,边界条件和能率方程.为此,先给出较为完整的变形梯度及其逆的定义.接着推导出各种应力率和偶应力率间的新关系式.最后,作为一种特殊情形得到连续统力学的耦合的增率型运动方程、边界条件和能率方程.     

重建极性连续统理论的基本定律和原理(Ⅵ)——质量和惯性守恒定律

重建极性连续统理论的耦合型质量和惯性的守恒定律和局部守恒方程以及跳变条件.为此推导出新的变形梯度、线元、面元和体元的物质导数,并给出广义Reynolds输运定理.把这些结果和作者以前推导出的耦合型动量、动量矩和能量的基本定律和有关原理结合在一起就大体上构成极性连续统理论相当完整的耦合型基本定律、局部守恒和均衡方程及原理体系.从此体系可以根据常用的局部化方法给出耦合型的非局部质量和惯性守恒方程以及动量、动量矩和能量均衡方程.     

Mass,momentum and energy conservation laws in zero-pressure gas dynamics and δ-shocks: II

We study δ-shocks in a one-dimensional system of zero-pressure gas dynamics. In contrast to well-known papers (see References) this system is considered in the form of mass, momentum and energy conservation laws. In order to define such singular solutions, special integral identities are introduced which extend the concept of classical weak solutions. Using these integral identities, the Rankine–Hugoniot conditions for δ-shocks are obtained. It is proved that the mass, momentum and energy transport processes between the area outside the of one-dimensional δ-shock wave front and this front are going on such that the total mass, momentum and energy are independent of time, while the mass and energy concentration processes onto the moving δ-shock wave front are going on. At the same time the total kinetic energy transforms into total internal energy.     

Well posedness for the system modelling the motion of a rigid body of arbitrary form in an incompressible viscous fluid

In this paper, we consider the interaction between a rigid body and an incompressible, homogeneous, viscous fluid. This fluid-solid system is assumed to fill the whole space ℝ ^d , d = 2 or 3. The equations for the fluid are the classical Navier-Stokes equations whereas the motion of the rigid body is governed by the standard conservation laws of linear and angular momentum. The time variation of the fluid domain (due to the motion of the rigid body) is not known a priori, so we deal with a free boundary value problem. We improve the known results by proving a complete wellposedness result: our main result yields a local in time existence and uniqueness of strong solutions for d = 2 or 3. Moreover, we prove that the solution is global in time for d = 2 and also for d = 3 if the data are small enough. Patricio Cumsille’s research was partially supported by CONICYT-FONDECYT grant (No. 3070040) and Takéo Takahashi’s research was partially supported by Grant (JCJC06 137283) of the Agence Nationale de la Recherche.     

Gauging magnetorotational instability

Previously (Z. Angew. Math. Phys. 57:615–622, 2006), we examined the axisymmetric stability of viscous resistive magnetized Couette flow with emphasis on flows that would be hydrodynamically stable according to Rayleigh’s criterion: opposing gradients of angular velocity and specific angular momentum. A uniform axial magnetic field permeates the fluid. In this regime, magnetorotational instability (MRI) may occur. It was proved that MRI is suppressed, in fact no instability at all occurs, with insulating boundary conditions, when a term multipling the magnetic Prandtl number is neglected. Likewise, in the current work, including this term, when the magnetic resistivity is sufficiently large, MRI is suppressed. This shows conclusively that small magnetic dissipation is a feature of this instability for all magnetic Prandtl numbers. A criterion is provided for the onset of MRI.     

The Vlasov-Maxwell-Boltzmann system near Maxwellians

Perhaps the most fundamental model for dynamics of dilute charged particles is described by the Vlasov-Maxwell-Boltzmann system, in which particles interact with themselves through collisions and with their self-consistent electromagnetic field. Despite its importance, no global in time solutions, weak or strong, have been constructed so far. It is shown in this article that any initially smooth, periodic small perturbation of a given global Maxwellian, which preserves the same mass, total momentum and reduced total energy (22), leads to a unique global in time classical solution for such a master system. The construction is based on a recent nonlinear energy method with a new a priori estimate for the dissipation: the linear collision operator L, not its time integration, is positive definite for any solutionf(t,x,v) with small amplitude to the Vlasov-Maxwell-Boltzmann system (8) and (12). As a by-product, such an estimate also yields an exponential decay for the simpler Vlasov-Poisson-Boltzmann system (24).     

Quantum three-body problems

A scheme for dealing with the quantum three-body problem is presented to separate the rotational degrees of freedom completely from the internal ones. In this method, the three-body Schrodinger equation is reduced to a system of coupled partial differential equations, depending only upon three internal variables. For arbitrary total orbital angular momentum / and the parity (− 1) ^l+λ (λ = 0 or 1), the number of the equations in this system isl = 1 −λ. By expanding the wavefunction with respect to a complete set of orthonormal basis functions, the system of equations is further reduced to a system of linear algebraic equations.     

On the motion of a rigid body immersed in a bidimensional incompressible perfect fluid

We consider the motion of a rigid body immersed in a bidimensional incompressible perfect fluid. The motion of the fluid is governed by the Euler equations and the conservation laws of linear and angular momentum rule the dynamics of the rigid body. We prove the existence and uniqueness of a global classical solution for this fluid–structure interaction problem. The proof relies mainly on weighted estimates for the vorticity associated with the strong solution of a fluid–structure interaction problem obtained by incorporating some viscosity.     

The properties of the steady motions of a body carrying a system of two-degree-of-freedom powered gyroscopes

The steady motions of a rigid body carrying several two-degree-of-freedom powered gyroscopes in a uniform external field are investigated. It is shown that when the installation scheme of the gyroscopes in the carrying body is collinear, the problem of determining the steady motions of the system and analysing their secular stability reduces for the most part to the previously solved, similar problem for a system with one gyroscope. It is established that when there is dissipation in the axes of the gyroscope frames, the system tends asymptotically to a state of rest if the absolute value of the total angular momentum of the system lies in the segment of possible absolute values of the angular momentum of the gyroscope rotors. The results of an analysis of the steady motions of a system carrying two gyroscopes with a non-collinear installation scheme are presented.     

Conservation properties of higher order time stepping schemes based on variational integrators

This paper deals with higher order accurate variational integrators for finite element systems. The variational integrator (VI) is based on higher order LAGRANGE polynomials as shape functions and a higher order GAUSSIAN quadrature rule. The goals of this paper are to implement a discrete gradient to preserve the balance of total energy and fulfill the constraints with the LAGRANGE multiplier method and a NEWTON-COTÊS quadrature rule. We show the calculation of bearing forces from the LAGRANGE multipliers, which are essential for the balance of total linear momentum and the balance of total angular momentum. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

重建极性连续统理论的基本定律和原理(Ⅰ)——微极连续统

在对现有微极连续统理论已进行过再研究的基础上重新建立较为完整的微极连续统理论的基本均衡定律和方程体系。在此重建的新体系中不但考虑了由于动量引起的附加动量矩、面力引起的附加面矩和体力引起的附加体矩,而且还考虑了微角速度引起的附加速度,从而可以建立起耦合型的动量、动量矩和能量的均衡定律。从这些新的基本均衡定律可以很自然地推导出相应的局部和非局部均衡方程。通过对比可以清楚地看到这些新结果较之现有的结果都完整。