Heterogeneous asynchronous time integrators built from the energy method for coupling newmark and α-schemes

The time integration procedure selected in computational structural dynamics must possess at least the stability and accuracy properties required for the convergence to the exact solution. Other desired properties are the unconditional stability for linear dynamics, second-order of accuracy, high frequency dissipation capabilities, self-starting, no overshoot, one step method and no inure than one set of implicit equations to be solved for each tinie step (single-step-single-solve format). In linear dynamics, the stability is classically assessed by a spectral study of the amplification matrix, whereas physical energy bounds are preferred in nonlinear dynamics. Popular a-schemes (HHT-α, WBZ-α, CH-α) are second-order accurate and provides numerical dissipation for spurious high frequencies due to the flute element discretization. To go beyond the staildard approach based on the same time integration scheme (homogeneous time integration scheme) and the same time step for all the finite elements of the mesh (synchronous time integration), the purpose of tins paper is to describe a general methodology for building Heterogeneous (different time integration schemes such as Newmark or a-schemes) Asynchronous (different time steps) Time Integrators (HATI) for computational dynamics. The key point for building the HATI methods is to cancel the interface pseudo-energy as introduced by Hughes in the so-called clergy method employed for proving the stability of implicit-explicit algorithms in its pioneer works on heterogeneous time integrators. By canceling the pseudo-energy at the interface between subdomains and assumillg a linear time variation of the Lagrange multipliers at the coarse time scale, the HATI method, called BCG-macro method, is derived. It can handle any dissipative cm-schemes (HHT-α, WBZ-cmα, CH-α), while preserving the secoud-order of accuracy when adopting different time steps. In addition to the energy argument (cancelation of the interface pseudo-energy), the stability and order of accuracy is proved by the spectral study of the amplification matrix

How do wave packets spread? Time evolution on Ehrenfest time scales

We derive an extension of the standard time dependent WKB theory which can be applied to propagate coherent states and other strongly localised states for long times. It allows in particular to give a uniform description of the transformation from a localised coherent state to a delocalised Lagrangian state which takes place at the Ehrenfest time. The main new ingredient is a metaplectic operator which is used to modify the initial state in a way that standard time dependent WKB can then be applied for the propagation. We give a detailed analysis of the phase space geometry underlying this construction and use this to determine the range of validity of the new method. Several examples are used to illustrate and test the scheme and two applications are discussed: (i) For scattering of a wave packet on a barrier near the critical energy we can derive uniform approximations for the transition from reflection to transmission. (ii) A wave packet propagated along a hyperbolic trajectory becomes a Lagrangian state associated with the unstable manifold at the Ehrenfest time, this is illustrated with the kicked harmonic oscillator.Comment: 30 pages, 3 figure

Numerical simul tion of droplet impact erosion : dang van fatigue approach

The aim of this work is to understand the erosion mechanism caused by repeated water droplets impingement on a metallic structure, and then perform numerical simulations of the damage. When a high velocity water droplet with small diameter impacts a rigid surface, interaction is driven by inertial effects. Upon impact, the “water-hammer” pressure appears by inertial effect at the center of the contact though the maximum pressure occurs on the envelope of the contact area. Lateral jetting occurs by compression when the wave front travelling inside droplet overtakes the contact area. Concerning the structure, erosion is due to fatigue crack- ing. First, material grains are weakened during an “incubation” phase. After a large number of impacts, micro-cracks emerge and lead to ejection or fracture of grains, what is called “am- plification” phase. Numerical simulation including rigid solid allows to locate the most loaded zones of the area, by observing the pressure and mainly the impulse. A 2-way coupling compu- tation with fluid-structure interaction at macroscopic scale allows to confirm the fatigue-based mechanism by observing the hydrostatic stress. Finally, erosion program developed with Dang Van criterion provides the location of the most eroded zones of the structure during a loading cycle. They locate at the edge of jetting zone, which shows the influence of microjets in the erosion mechanism

Mixed-mode stress intensity factors for graded materials

AbstractIn this paper, we present a general method for the calculation of the various stress intensity factors in a material whose constitutive law is elastic, linear and varies continuously in space. The approach used to predict the stress intensity factors is an extension of the interaction integral method. For this type of material, we also develop a systematic method to derive the asymptotic displacement fields and use it to achieve better-quality results. A new analytical asymptotic field is given for two special cases of graded materials. Numerical examples focus on materials with space-dependent Young modulus

Block circulant matrices with circulant blocks, weil sums and mutually unbiased bases, II. The prime power case

In our previous paper \cite{co1} we have shown that the theory of circulant matrices allows to recover the result that there exists $p+1$ Mutually Unbiased Bases in dimension $p$, $p$ being an arbitrary prime number. Two orthonormal bases $\mathcal B, \mathcal B'$ of $\mathbb C^d$ are said mutually unbiased if $\forall b\in \mathcal B, \forall b' \in \mathcal B'$ one has that $$| b\cdot b'| = \frac{1}{\sqrt d}$$ ($b\cdot b'$ hermitian scalar product in $\mathbb C^d$). In this paper we show that the theory of block-circulant matrices with circulant blocks allows to show very simply the known result that if $d=p^n$ ($p$ a prime number, $n$ any integer) there exists $d+1$ mutually Unbiased Bases in $\mathbb C^d$. Our result relies heavily on an idea of Klimov, Munoz, Romero \cite{klimuro}. As a subproduct we recover properties of quadratic Weil sums for $p\ge 3$, which generalizes the fact that in the prime case the quadratic Gauss sums properties follow from our results

Localization of quantum wave packets

We study the semiclassical propagation of squeezed Gau{\ss}ian states. We do so by considering the propagation theorem introduced by Combescure and Robert \cite{CR97} approximating the evolution generated by the Weyl-quantization of symbols $H$. We examine the particular case when the Hessian $H^{\prime\prime}(X_{t})$ evaluated at the corresponding solution $X_{t}$ of Hamilton's equations of motion is periodic in time. Under this assumption, we show that the width of the wave packet can remain small up to the Ehrenfest time. We also determine conditions for ``classical revivals'' in that case. More generally, we may define recurrences of the initial width. Some of these results include the case of unbounded classical motion. In the classically unstable case we recover an exponential spreading of the wave packet as in \cite{CR97}

Coherent States Expectation Values as Semiclassical Trajectories

We study the time evolution of the expectation value of the anharmonic oscillator coordinate in a coherent state as a toy model for understanding the semiclassical solutions in quantum field theory. By using the deformation quantization techniques, we show that the coherent state expectation value can be expanded in powers of $\hbar$ such that the zeroth-order term is a classical solution while the first-order correction is given as a phase-space Laplacian acting on the classical solution. This is then compared to the effective action solution for the one-dimensional \f^4 perturbative quantum field theory. We find an agreement up to the order \l\hbar, where \l is the coupling constant, while at the order \l^2 \hbar there is a disagreement. Hence the coherent state expectation values define an alternative semiclassical dynamics to that of the effective action. The coherent state semiclassical trajectories are exactly computable and they can coincide with the effective action trajectories in the case of two-dimensional integrable field theories.Comment: 20 pages, no figure

On the energy growth of some periodically driven quantum systems with shrinking gaps in the spectrum

We consider quantum Hamiltonians of the form H(t)=H+V(t) where the spectrum of H is semibounded and discrete, and the eigenvalues behave as E_n~n^\alpha, with 0<\alpha<1. In particular, the gaps between successive eigenvalues decay as n^{\alpha-1}. V(t) is supposed to be periodic, bounded, continuously differentiable in the strong sense and such that the matrix entries with respect to the spectral decomposition of H obey the estimate |V(t)_{m,n}|0, p>=1 and \gamma=(1-\alpha)/2. We show that the energy diffusion exponent can be arbitrarily small provided p is sufficiently large and \epsilon is small enough. More precisely, for any initial condition \Psi\in Dom(H^{1/2}), the diffusion of energy is bounded from above as _\Psi(t)=O(t^\sigma) where \sigma=\alpha/(2\ceil{p-1}\gamma-1/2). As an application we consider the Hamiltonian H(t)=|p|^\alpha+\epsilon*v(\theta,t) on L^2(S^1,d\theta) which was discussed earlier in the literature by Howland

Pulse-driven quantum dynamics beyond the impulsive regime

We review various unitary time-dependent perturbation theories and compare them formally and numerically. We show that the Kolmogorov-Arnold-Moser technique performs better owing to both the superexponential character of correction terms and the possibility to optimize the accuracy of a given level of approximation which is explored in details here. As an illustration, we consider a two-level system driven by short pulses beyond the sudden limit.Comment: 15 pages, 5 color figure

On the Geometry of Supersymmetric Quantum Mechanical Systems

We consider some simple examples of supersymmetric quantum mechanical systems and explore their possible geometric interpretation with the help of geometric aspects of real Clifford algebras. This leads to natural extensions of the considered systems to higher dimensions and more complicated potentials.Comment: 18 page