Potential systems for PDEs having several conservation laws

A generalization of the usual procedure for constructing potential systems for systems of partial differential equations with multidimensional spaces of conservation laws is considered. More precisely, for the construction of potential systems with a multi-dimensional space of local conservation laws, instead of using only basis conservation laws, their arbitrary linear combinations are used that are inequivalent with respect to the equivalence group of the class of systems or symmetry group of the fixed system. It appears that the basis conservation laws can be equivalent with respect to groups of symmetry or equivalence transformations, or vice versa; in this sense the number of independent linear combinations of conservation laws can be grater than the dimension of the space of conservation laws. The first possibility leads to an unnecessary, often cumbersome, investigation of equivalent systems, the second one makes possible that a great number of inequivalent potential systems are missed. Examples of all these possibilities are given.     

A novel nonsmooth dynamics method for multibody systems with friction and impact based on the symplectic discrete format

As multibody systems often involve unilateral constraints, nonsmooth phenomena, such as impacts and friction, are common in engineering. Therefore, a valid nonsmooth dynamics method is highly important for multibody systems. An accuracy representation of multibody systems is an important performance indicator of numerical algorithms, and the energy balance can be used efficiently evaluate the performance of nonsmooth dynamics methods. In this article, differential algebraic equations (DAEs) of a multibody system are constructed using the D'Alembert's principle, and a novel nonsmooth dynamics method based on symplectic discrete format is proposed. The symplectic discrete format can maintain the energy conservation of a conservative system; this property is expected to extend to nonconservative systems with nonsmooth phenomena in this article. To evaluate the properties of the proposed method, several numerical examples are considered, and the results of the proposed method are compared with those of Moreau's midpoint rule. The results demonstrate that the solutions obtained using the proposed method, which is based on the symplectic discrete format, can realize a higher solution accuracy and lower numerical energy dissipation, even under a large time step.     

Conservation laws for 3-dimensional compressible Euler equations

The generators of infinitesimal symmetry transformations for the Euler equations, and the corresponding set of adjoint variables are derived. The associated conservation laws are then discussed. A detailed analysis of 1-dimensional flows brings into evidence the connections with current alternative approaches to conservation laws.     

Time finite element based Moreau‐type integrators

With the postulation of the principle of virtual action, we propose, in this paper, a variational framework for describing the dynamics of finite dimensional mechanical systems, which contain frictional contact interactions. Together with the contact and impact laws formulated as normal cone inclusions, the principle of virtual action directly leads to the measure differential inclusions commonly used in the dynamics of nonsmooth mechanical systems. The discretization of the principle of virtual action in its strong and weak variational form by local finite elements in time provides a structured way to derive various time‐stepping schemes. The constitutive laws for the impulsive and nonimpulsive contact forces, ie, the contact and impact laws, are treated on velocity‐level by using a discrete contact law for the percussion increments in the sense of Moreau. Using linear shape functions and different quadrature rules, we obtain three different stepping schemes. Besides the well‐established Moreau time‐stepping scheme, we can present two alternative integrators referred to as symmetric and variational Moreau‐type stepping schemes. A suitable benchmark example shows the superiority of the newly proposed integrators in terms of energy conservation properties, accuracy, and convergence.     

Conservation properties of a time FE method. Part IV: Higher order energy and momentum conserving schemes

In the present paper a systematic development of higher order accurate time stepping schemes which exactly conserve total energy as well as momentum maps of underlying finite‐dimensional Hamiltonian systems with symmetry is shown. The result of this development is the enhanced Galerkin (eG) finite element method in time. The conservation of the eG method is generally related to its collocation property. Total energy conservation, in particular, is obtained by a new projection technique. The eG method is, moreover, based on objective time discretization of the used strain measure. This paper is concerned with particle dynamics and semi‐discrete non‐linear elastodynamics. The related numerical examples show good performance in presence of stiffness as well as for calculating large‐strain motions. Copyright © 2005 John Wiley & Sons, Ltd.     

Discrete Lagrangian mechanics for nonseparable nonsmooth systems

We consider event‐driven schemes for the simulation of nonseparable mechanical systems subject to holonomic unilateral constraints. Systems are modeled in discrete time using variational integrator (VI) theory, by which equations of motion follow from discrete variational principles. For smooth dynamics, VIs are known to exactly conserve a discrete symplectic form and a modified Hamiltonian function. The latter of these conservation laws can play a pivotal role in stabilizing the energy behavior of collision simulations. Previous efforts to leverage modified Hamiltonian conservation have been limited to integrators using the Störmer–Verlet method on separable, nonsmooth Hamiltonian mechanical systems. We generalize the existing approach to the family of all VIs applied to nonseparable, potentially nonconservative Lagrangian mechanical systems. We examine the properties of the resulting integrators relative to other structured collision simulation methods in terms of conserved quantities, trajectory errors as a function of initial condition, and required computation time. Interestingly, we find that the modified collision Verlet algorithm (MCVA) using the Störmer–Verlet integrator defined as a composition method leads to the best accuracy. Although relative to this method, the VI‐based generalized MCVA method offers computational savings when collisions are particularly sparse. Copyright © 2015 John Wiley & Sons, Ltd.     

Conservation laws, duality and symmetry loss in solid mechanics

The paper deals with conservation laws which are not of the pure divergence type and thus do not provide a path-independent integral for use in Fracture Mechanics. It is shown that Duality is the right tool to re-establish the symmetry between equations and to provide conservation laws of the pure divergence type. The loss of symmetry of some energetic expressions is exploited to derive a new method for solving some inverse problems. In particular, the earthquake inverse problem is solved analytically. Dedicated to George Herrmann.     

On the stability of the equilibrium state and small oscillations of non-holonomic systems

The stability of small oscillations of non -holonomic systems is discussed. In contrast to Neimark and Fufaev, we have shown here that a non-holonomic system does indeed possess an isolated equilibrium state under certain conditions. We provide conditions under which there exists asymptotic stability of small oscillations of a non-holonomic system about its equilibrium position. The connection between a holonomic and non-holonomic system is also discussed. The results obtained here can be useful for the control of a non-holonomic system     

Adjoint shape design sensitivity analysis of molecular dynamics for lattice structures using GLE impedance forces

We presented a shape design sensitivity analysis method for lattice structures using a generalized Langevin equation (GLE) to overcome the difficulty of discrete nature in atomic systems. Taking advantage of the GLE forces, the perturbed atomistic region is treated as the GLE impedance forces and the shape design problem of discrete atomic variations is converted into a non-shape problem with GLE impedance forces. We developed an adjoint variable method in order to improve the computational efficiency for molecular dynamics (MD) with many design variables. Due to the translational symmetry in lattice structure, the size of the time history kernel function that accounts for the boundary effects of reduced systems could be reduced to that of a single atoms DOFs. In numerical examples, the convergent characteristic of shape sensitivity according to the amount of shape variations is investigated in MD systems. Also, the results of the derived shape sensitivity turn out to be more accurate and efficient, compared with those of the finite difference ones.     

Pattern evocation and geometric phases in mechanical systems with symmetry

This paper is concerned with the relation between the dynamics of a given Hamiltonian system with a given symmetry group and its reduced dynamics. We illustrate the process of visualization of reduced orbits using the double spherical pendulum. In this process of visualization, one sees certain patterns when the dynamics is viewed relative to rotating frames with certain critical angular velocities. By using the reduced dynamics, we also explain these patterns. We show that if the motion on the phase space reduced by a continuous symmetry group at a given momentum level is periodic, then there is a uniformly rotating frame, that is, a one-parameter group motion, relative to which the unreduced trajectory is periodic with the same period. If the continuous symmetry group of the system is Abelian, which corresponds to the system having cyclic variables, we derive an explicit expression for the required angular velocity in terms of the dynamic phase (an average of the mechanical connection) and the geometric phase (the holonomy of the mechanical connection). We show that one can also find such a frame if the reduced orbit is quasi-periodic and a KAM (Kolmogorov-Arnold-Moser) condition is satisfied The almost periodic case is also discussed. An important aspect of this procedure is how to use it in the presence of discrete symmetries. We show that, under appropriate conditions, the visualized orbit has, relative to a suitable uniformly rotating frame, the same temporal behavior and discrete symmetries as the reduced orbit. Since these spatio-temporal patterns are not apparent with repect to most frames, we call the phenomenon pattern evocation     

Pattern evocation and geometric phases in mechanical systems with symmetry

This paper is concerned with the relation between the dynamics of a given Hamiltonian system with a given symmetry group and its reduced dynamics. We illustrate the process of visualization of reduced orbits using the double spherical pendulum. In this process of visualization, one sees certain patterns when the dynamics is viewed relative to rotating frames with certain critical angular velocities. By using the reduced dynamics, we also explain these patterns. We show that if the motion on the phase space reduced by a continuous symmetry group at a given momentum level is periodic, then there is a uniformly rotating frame, that is, a one-parameter group motion, relative to which the unreduced trajectory is periodic with the same period. If the continuous symmetry group of the system is Abelian, which corresponds to the system having cyclic variables, we derive an explicit expression for the required angular velocity in terms of the dynamic phase (an average of the mechanical connection) and the geometric phase (the holonomy of the mechanical connection). We show that one can also find such a frame if the reduced orbit is quasi-periodic and a KAM (Kolmogorov-Arnold-Moser) condition is satisfied The almost periodic case is also discussed. An important aspect of this procedure is how to use it in the presence of discrete symmetries. We show that, under appropriate conditions, the visualized orbit has, relative to a suitable uniformly rotating frame, the same temporal behavior and discrete symmetries as the reduced orbit. Since these spatio-temporal patterns are not apparent with repect to most frames, we call the phenomenon pattern evocation     

Mei symmetry of discrete mechanico-electrical systems

Mei symmetry of discrete mechanico-electrical systems on a uniform lattice is investigated. The definition and criterion of the discrete analog of Mei symmetry for the system are presented. The condition of existence of the discrete Mei conserved quantity induced directly by Mei symmetry as well as its form is given. Finally, an example is discussed to illustrate these results.     

Conformal invariance of Mei symmetry for discrete Lagrangian systems

The conformal invariance of the Mei symmetry and the conserved quantities are investigated for discrete Lagrangian systems under the infinitesimal transformation of the Lie group. The difference Euler–Lagrange equations on regular lattices of the discrete Lagrangian systems are presented via the transformation operators in the space of the discrete variables. The conformal invariance of the Mei symmetry is defined for the discrete Lagrangian systems. The criterion equations and the determining equations are proposed. The conserved quantities of the systems are derived from the structure equation governing the gauge function. Two examples are given to illustrate the application of the results.     

Reduction of discrete element models by Karhunen–Loève transform: a hybrid model approach

The term “discrete element method” (DEM) in engineering science comprises various approaches to model physical systems by agglomerates of free particles. While shapes, sizes and properties of particles may vary, in most DEM models, particles are not confined by constraints, but subject to applied forces derived from potential fields and/or contact laws. This general approach allows for widespread use of DEM models for physical phenomena including gas dynamics, granular flow, fracture and impact analysis. However, its characteristic feature, combining particle restraints and forces into applied forces, does not only provide for flexible adaption of DEM to different physics, but also creates the most limiting restriction: Evaluation of the applied forces for each particle is computational expensive restraining the time sequence and sample size for numerical analyses. As an ansatz to circumvent this obstacle for a class of DEM models, we propose a model order reduction method based on coherency in the dynamics of particles. While initial flexibility of DEM is conserved, computational effort can be reduced significantly.     

Assumed strain finite element methods for conserving temporal integrations in non‐linear solid dynamics

This paper presents a new class of assumed strain finite elements to use in combination with general energy‐momentum‐conserving time‐stepping algorithms so that these conservation properties in time are preserved by the fully discretized system in space and time. The case of interest corresponds to nearly incompressible material responses, in the fully non‐linear finite strain elastic and elastoplastic ranges. The new elements consider the classical scaling of the deformation gradient with an assumed Jacobian (its determinant) defined locally through a weighted averaging procedure at the element level. The key aspect of the newly proposed formulation is the definition of the associated linearized strain operator or B‐bar operator. The developments presented here start by identifying the conditions that this discrete operator must satisfy for the fully discrete system in time and space to inherit exactly the conservation laws of linear and angular momenta, and the conservation/dissipation law of energy for elastic and inelastic problems, respectively. Care is also taken of the preservation of the relative equilibria and the corresponding group motions associated with the momentum conservation laws, and characterized by purely rotational and translational motions superimposed to the equilibrium deformed configuration. With these developments at hand, a new general B‐bar operator is introduced that satisfies these conditions. The new operator not only accounts for the spatial interpolations (e.g. bilinear displacements with piece‐wise constant volume) but also depends on the discrete structure of the equations in time. The aforementioned conservation/dissipation properties of energy and momenta are then proven to hold rigorously for the final numerical schemes, unconditionally of the time step size and the material model (elastic or elastoplastic). Different finite elements are considered in this framework, including quadrilateral and triangular elements for plane problems and brick elements for three‐dimensional problems. Several representative numerical simulations are presented involving, in particular, the use of energy‐dissipating momentum‐conserving time‐stepping schemes recently developed by the author and co‐workers for general finite strain elastoplasticity in order to illustrate the properties of the new finite elements, including these conservation/dissipation properties in time and their locking‐free response in the quasi‐incompressible case. Copyright © 2007 John Wiley & Sons, Ltd.     

Discrete symmetries of chaotic strings

Chaotic strings are particular classes of coupled map lattices that can serve as models for vacuum fluctuations in stochastically quantized field theories. In this article we look at two important properties of chaotic strings, namely (i) the behaviour under discrete symmetry transformations and (ii) the loss of ergodic behaviour in certain coupling regions. We show that several of the chaotic string dynamics can be transformed into each other by simple discrete coordinate transformations. We investigate how expectation values converge in the various coupling parameter regions and single out those stable zeros of the correlation function that correspond to ergodic states with well-defined convergence properties.     

LARGE DISPLACEMENT ELASTOPLASTIC ANALYSIS OF SEMIRIGID STEEL FRAMES

A mathematical programming approach for the large displacement analysis of elastoplastic plane frames with flexible, partial-strength beam-to-column connections is proposed. The incremental formulation is developed within the framework of a discrete model, piecewise linear yield surfaces and large displacement theory. It is based on the fundamental relations of equilibrium, compatibility and constitutive laws, all expressed in incremental form. The irreversible plasticity laws are used in a stepwise holonomic format. A feature of the final mathematical programming problem, known as a non-linear complementarity problem, is that the governing relations exhibit symmetry as a result of the use of fictitious quantities and non-linear residuals. The iterative predictor–corrector computational scheme adopted has the ability of tracing historically a sequence of plastic hinge activations and/or unloadings well beyond any critical point. Numerical examples are presented to illustrate and validate the accuracy of the approach.     

3D dynamics of discrete element systems comprising irregular discrete elements—integration solution for finite rotations in 3D

An algorithm for transient dynamics of discrete element systems comprising a large number of irregular discrete elements in 3D is presented. The algorithm is a natural extension of contact detection, contact interaction and transient dynamics algorithms developed in recent years in the context of discrete element methods and also the combined finite‐discrete element method. It complements the existing algorithmic procedures enabling transient motion including finite rotations of irregular discrete elements in 3D space to be accurately integrated. Copyright © 2002 John Wiley & Sons, Ltd.     

一类高阶KdV类型水波方程的多辛Euler-box格式

高阶KdV类型水波方程作为一类重要的非线性方程有着许多广泛的应用前景．本文主要研究高阶KdV类型水波方程的多辛Euler-box格式．首先,通过正则变换,构造了高阶KdV方程的多辛结构,并得到该系统的多辛守恒律、局部能量守恒律和动量守恒律．然后,我们利用Euler-box格式对高阶KdV方程进行离散,并基于Hamilton空间体系的多辛理论研究了该系统的离散Euler-box格式．我们证明该格式满足离散多辛守恒律,并且给出该格式的向后误差分析．最后,数值算例结果表明该多辛离散格式具有较好的长时间数值稳定性．     

Stability analysis and controller synthesis for hybrid dynamical systems

Wherever continuous and discrete dynamics interact, hybrid systems arise. This is especially the case in many technological systems in which logic decision-making and embedded control actions are combined with continuous physical processes. Also for many mechanical, biological, electrical and economical systems the use of hybrid models is essential to adequately describe their behaviour. To capture the evolution of these systems, mathematical models are needed that combine in one way or another the dynamics of the continuous parts of the system with the dynamics of the logic and discrete parts. These mathematical models come in all kinds of variations, but basically consist of some form of differential or difference equations on the one hand and automata or other discrete-event models on the other hand. The collection of analysis and synthesis techniques based on these models forms the research area of hybrid systems theory, which plays an important role in the multi-disciplinary design of many technological systems that surround us. This paper presents an overview from the perspective of the control community on modelling, analysis and control design for hybrid dynamical systems and surveys the major research lines in this appealing and lively research area.