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)     

Material models in principal stretches for elastodynamics

In the present work we deal with the conserving integration of elastic bodies undergoing finite deformations. In particular, we make use of constitutive laws formulated in terms of principal stretches. Most material models for hyperelastic isotropic materials are described in terms of principal stretches (Simo and Taylor [1]), like the Neo–Hooke material which is a special case of the Ogden material, or in invariants. The main advantage of principal stretches is the fact that they can be measured directly, which means that the numerical results can be compared easily with experimental ones, see for example, Ogden [2]. Moreover, it is advantageous to describe viscoelastic material behaviour (e.g. rubberlike materials) in terms of principal stretches. Concerning the discretization in space we apply the enhanced assumed strain (EAS) method, see Simo and Armero [3]. For the discretization in time we aim at numerical integrators which inherit fundamental conservation laws from the underlying continuous system. In particular, we propose an energy and momentum conserving time–stepping scheme which relies on the notion of a discrete gradient (or derivative) in the sense of Gonzalez [4]. The proposed approach starts from our previous developments in [5]. Numerical examples demonstrate the advantageous properties of the present formulation. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Natural co-ordinates for control applications

Natural coordinates have emerged to be well-suited for both rigid and flexible multibody dynamics. Especially the combination of structural elements and energy-momentum consistent time stepping schemes leads to superior numerical stability as well as an automatable assembly, resulting in both excellent run-time behaviour as well as moderate modelling effort (see [1]). Incorporation of modern methods for finite-element simulations, such as mortar methods for contact or domain decomposition both for structural elements as well as continuum elements is straightforward ([2]). Augmentation techniques allow a systematic integration of both mechanical and non-mechanical quantities for simulation (see [3] and [4]), which makes this approach suitable especially for emulation and simulation of mechatronic systems. We will present an approach for evaluating forward control strategies with flexible multibody systems. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A finite element method for contact/impact

Ideas from the analysis of differential-algebraic equations are applied to the numerical solution of frictionless contact/impact problems in solid mechanics. Index-one and two formulations for dynamic contact–impact within the context of the finite element method are derived. The resulting equations are shown to stabilize the kinematic fields at the contact interface, at the expense of a small energy loss, which is shown to decrease consistently with mesh refinement. This energy dissipation is shown to be necessary for the establishment of persistent contact. A Newmark-type time integration scheme is derived from the proposed formulation, and shown to yield excellent results in modeling the transition to contact/impact.     

Dynamical models of molecular chains and efficient integration algorithms

Chains of point masses and chains of rigid bodies are used to model biological polymers. To investigate their dynamics we propose a method which allows an efficient realization of the constraints jointly with a simple and accurate integration of the free rigid body motion. The method is quite effective to evolute the geodesic flow of a rigid body chain and the global performance depends on the computational complexity of the algorithms used to compute the interaction forces. Our approach is suitable to describe a chain of rigid bodies immersed in a thermal bath. In the method we propose, the constraints are realized by hard springs whose elastic constant is set to maximize the energy dissipation rate of a Runge–Kutta integrator scheme. Moreover the use of local Lagrangian coordinates is introduced using the possibility of a continuous change of chart, such that the distance from the coordinate singularities is the highest possible. For a chain of point masses the numerical results are checked with another method where the constraints are exactly realized by means of Lagrangian coordinates. When the chain is subject to regular interactions potentials plus a thermal bath the exact and approximate constraints realization provide comparable results.     

有限温度下线性谐振子晶格的分子动力学模拟

基于双向界面条件和声子热浴,提出了一种新的热流输入方法,该方法未引入任何耗散因子或经验参数,能在局域的空间和时间上实现有限温度下的原子模拟.对于一维线性谐振子晶格,采用双向界面条件作为系统的边界,目的是为了让热流能从外界输入系统,同时允许内部的波动自由地传出,从而实现系统中能量的动态平衡.通过数值计算发现,双向界面条件能让正方向的波完整地输入,同时还能抑制反方向的波的输入,因此,边界条件可以起到行波的二极管的作用.声子热浴的正则模态能很好地描述原子的热振动,通过推导可将正则模态分解为正方向和反方向的输入波,取正方向的波来构造热源项.数值算例表明,热流输入方法对于线性谐振子链非常有效,系统能快速地达到预期的温度,并且能够维持在稳定的状态,同时,还能很好地处理有限温度下的非热运动.     

Asymptotic stability of finite energy in Navier Stokes-elastic wave interaction

We consider a model of fluid-structure interaction in a bounded domain Ω∈ℝ² where Ω is comprised of two open adjacent sub-domains occupied, respectively, by the solid and the fluid. This leads to a study of Navier Stokes equation coupled on the interface to the dynamic system of elasticity. The characteristic feature of this coupled model is that the resolvent is not compact and the energy function characterizing balance of the total energy is weakly degenerated. These combined with the lack of mechanical dissipation and intrinsic nonlinearity of the dynamics render the problem of asymptotic stability rather delicate. Indeed, the only source of dissipation is the viscosity effect propagated from the fluid via interface. It will be shown that under suitable geometric conditions imposed on the geometry of the interface, finite energy function associated with weak solutions converges to zero when the time t converges to infinity. The required geometric conditions result from the presence of the pressure acting upon the solid.     

轴对称弹性体的有限元分析

轴对称弹性力学问题的有限元分析长期以来都是采用三角圆环有限元和线性形状函数.由于积分困难,常用近似积分求得刚度矩阵,这种近似积分对于靠近旋转对称轴的元素,误差很大,所以,长期以来,被认为不满意的办法.也有用精确积分计算刚度矩阵的.但本文指出,这种积分只适用于有中孔的轴对称体.对于实心的轴对称体而言,这种刚度矩阵都不收敛,计算是无效的.本文提出了一种新的形状函数,当径向座标r接近于零时,这种形状函数的径向位移u自然地接近于零.如果用这种新的形状函数,则由此计算求得的刚度矩阵,不论三角圆环有限元的位置是否靠近轴线.都是存在的.这种有限元,就能用于计算实心的轴对称体的问题.     

On dynamic finite element analysis of viscoplastic thin-walled structures with non-local damage: A phase-field approach

[EN] In this work, a nonlocal damage model is proposed for dynamic analysis of viscoplastic shell structures using the phase-field approach. A phase-field variable on the mid surface is introduced to characterize the nonlocal damage as well as the transition between undamaged and damaged phase. The total free energy in [1] is modified as a sum of Helmholtz free-energy and Ginzburg-Landau one. The latter is defined as a function of the phase-field variable and its corresponding gradient. This enhancement gives rise to an introduction of gradient parameters in terms of a substructure-related intrinsic length-scale. The evolution of the phase-field based damage variable can be found from the minimum principle of the dissipation potential [3]. The performance of the proposed model is demonstrated through numerical results of a plate with a circular hole. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Diffuse Interface Methods for Multiple Phase Materials: An Energetic Variational Approach

In this paper, we introduce a diffuse interface model for describing the dynamics of mixtures involving multiple (two or more) phases. The coupled hydrodynamical system is derived through an energetic variational approach. The total energy of the system includes the kinetic energy and the mixing (interfacial) energies. The least action principle (or the principle of virtual work) is applied to derive the conservative part of the dynamics, with a focus on the reversible part of the stress tensor arising from the mixing energies. The dissipative part of the dynamics is then introduced through a dissipation function in the energy law, in line with Onsager's principle of maximum dissipation. The final system, formed by a set of coupled time-dependent partial differential equations, reflects a balance among various conservative and dissipative forces and governs the evolution of velocity and phase fields. To demonstrate the applicability of the proposed model, a few two-dimensional simulations have been carried out, including (1) the force balance at the three-phase contact line in equilibrium, (2) a rising bubble penetrating a fluid-fluid interface, and (3) a solid particle falling in a binary fluid. The effects of slip at solid surface have been examined in connection with contact line motion and a pinch-off phenomenon.     

On numerical schemes for phase-field models for electrowetting with electrolyte solutions

We present an energy-stable, decoupled discrete scheme for a recent model (see [1]) supposed to describe electrokinetic phenomena in two-phase flow with general mass densities. This model couples momentum and Cahn–Hilliard type phase-field equations with Nernst–Planck equations for ion density evolution and an elliptic transmission problem for the electrostatic potential. The transport velocities in our scheme are based on the old velocity field updated via a discrete time integration of the force densities. This allows to split the equations into three blocks which can be treated sequentially: The phase-field equation, the equations for ion transport and electrostatic potential, and the Navier–Stokes type equations. By establishing a discrete counterpart of the continuous energy estimate, we are able to prove the stability of the scheme. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A phase field model for fracture

The variational formulation of brittle fracture as formulated for example by Francfort and Marigo in [1], where the total energy is minimized with respect to any admissible crack set and displacement field, allows the identification of crack paths, branching of preexisting cracks and even crack initiation without additional criteria. For its numerical treatment a continuous approximation of the model in the sense of Γ-convergence has been presented by Bourdin in [2]. In the regularized Francfort–Marigo model cracks are represented by an additional field variable (secondary variable) s∈[0,1] which is 0 if the material is cracked and 1 if it is undamaged. In this work, we reinterpret the crack variable as a phase field order parameter and address cracking as a phase transition problem. The crack growth is governed by the evolution equation of the order parameter which resembles the Ginzburg–Landau equation. The numerical treatment is done by finite elements combined with an implicit Euler scheme for the time integration. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Travelling fronts,pulses, and pulse trains in a 1D discrete reaction–diffusion system

Following an earlier work (briefly reviewed below) we investigate the temporal stability of an exact travelling front solution, constructed in the form of an integral expression, for a one-dimensional discrete Nagumo-like model without recovery. Since the model is a piecewise linear one with an on-site reaction function involving a Heaviside step function, a straightforward linearisation around the front solution presents problems, and we follow an alternative approach in estimating a ‘stability multiplier’ by looking at the variational problem as a succession of linear evolution of the perturbations, punctuated with ‘kicks’ of small but finite duration. Stability depends crucially on perturbations located at specific sites relative to the moving front (the ‘significant perturbations’, see below). Comparison is made with results of numerical integration of the reaction–diffusion system, whereby it appears likely that the travelling front is temporally stable for all relevant parameter values characterising the model. We modify the system by introducing a slow variation of a relevant recovery parameter and perform a leading order singular perturbation analysis to construct a pulse solution in the resulting model. In addition, we obtain a 1-parameter family of periodic pulse trains for the system, modelling re-entrant pulses in a one-dimensional ring of excitable cells.     

Dynamic finite deformation viscoelasticity and energy-consistent time integration

In the present work we deal with the conserving integration of viscoelastic bodies undergoing finite deformations. Isotropic viscoelastic materials can be described by using a symmetric viscous internal variable for measuring the inelastic strains, and the right Cauchy-Green tensor as measure of the total strain (see Reese & Govindjee [1]). Then, by using the unsymmetric product tensor , the purely elastic strains enter the isotropic free energy function. Alternatively, the i application of the symmetric right stretch tensor as internal variable allows to define a symmetric elastic strain tensor which enters the free energy (see also Miehe [2]). These two different approaches lead to different evolution equations for the viscous internal variable. In this lecture, both evolution equations are discretised by an ordinary midpoint rule at each Gauss point of a standard nonlinear displacement-based finite element in space. For discretising the semidiscrete Hamilton's equations of motion in time, we use numerical time integrators which preserve the fundamental conservation laws of the underlying system. In particular, we make use of a modified midpoint rule according to the discrete gradient method, proposed in Gonzalez [3]. Numerical examples demonstrate the difference between both formulations. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Phase-field approach to fracture for finite-deformation contact problems

The present contribution deals with a variationally consistent Mortar contact algorithm applied to a phase-field fracture approach for finite deformations, see [4]. A phase-field approach to fracture allows for the numerical simulation of complex fracture patterns for three dimensional problems, extended recently to finite deformations (see [2] for more details). In a nutshell, the phase-field approach relies on a regularization of the sharp (fracture-) interface. In order to improve the accuracy, a fourth-order Cahn-Hilliard phase-field equation is considered, requiring global C¹ continuity (see [1]), which will be dealt with using an isogeometrical analysis (IGA) framework. Additionally, a newly developed hierarchical refinement scheme is applied to resolve for local physical phenomena e.g. the contact zone (see [3] for more details). The Mortar method is a modern and very accurate numerical method to implement contact boundaries. This approach can be extended in a straightforward manner to transient phase-field fracture problems. The performance of the proposed methods will be examined in a representative numerical example. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Reconstructing Signals with Finite Rate of Innovation from Noisy Samples

A signal is said to have finite rate of innovation if it has a finite number of degrees of freedom per unit of time. Reconstructing signals with finite rate of innovation from their exact average samples has been studied in Sun (SIAM J. Math. Anal. 38, 1389–1422, 2006). In this paper, we consider the problem of reconstructing signals with finite rate of innovation from their average samples in the presence of deterministic and random noise. We develop an adaptive Tikhonov regularization approach to this reconstruction problem. Our simulation results demonstrate that our adaptive approach is robust against noise, is almost consistent in various sampling processes, and is also locally implementable.