Six-DOF vibrations of partial contact sliders in hard disk drives

A six-degree-of-freedom slider dynamic simulator is developed to analyze the slider’s motion in the vertical, pitch, roll, yaw, length and width directions. The modified time-dependent Reynolds equation is used to model the air bearing and a new second order slip model is used for a bounded contact air bearing pressure. The simulator considers the air bearing shear acting on the air bearing surface and the slider–disk contact and adhesion. Simulation results are analyzed for the effects of the disk surface micro-waviness and roughness, skew angle, slider–disk friction and micro-trailing pad width on the vertical bouncing, down-track and off-track vibrations of a micro-trailing pad partial contact slider.     

Intermolecular force and surface roughness models for air bearing simulations for sub-5 nm flying height sliders

When the spacing between the slider and the disk is less than 5 nm, the intermolecular forces between the two solid surfaces can no longer be assumed to be zero. The model proposed by Wu and Bogy (ASME J Trib 124:562–567, 2002) can be view as a flat slider–disk intermolecular force model. The contact distance between the slider and disk needs to be considered in this model when the slider-disk spacing is in the contact regime. To get more accurate intermolecular force effects on the head disk interface, the slider and disk surface roughness need to be considered, when the flying height is comparable to the surface RMS roughness value or when contact occurs. With the intermolecular force model and asperity model implemented in the CML air bearing program, the effect of intermolecular adhesion stress on the slider at low flying height is analyzed in the static flying simulation. It is found that the intermolecular adhesion stress between the slider and the disk has slight effect on the slider-disk interface for a flying slider.     

Simulation of the head disk interface for discrete track media

This paper investigates the effect of discrete tracks on the steady state flying behavior of sub ambient proximity sliders. A finite element based air bearing simulator is used to simulate the flying characteristics of sliders over a grooved disk surface. Sliders flying over discrete track disks “see” a disk surface that consists of ridges and grooves. The air bearing pressure build-up for sliders flying over discrete track disks is different from that for sliders flying over plane disks. Low air bearing pressure can be expected for those regions of the slider that are positioned over grooves, while high air bearing pressure exists over ridges. The air bearing characteristics are determined for several pico and femto-type air bearing sliders flying over discrete track disks. An empirical equation is obtained describing the loss of flying height of a slider flying over discrete track disks.     

Monogamy and polygamy for multi-qubit W-class states using convex-roof extended negativity of assistance and Rényi-$$\alpha $$ entropy

We present some new analytical polygamy inequalities satisfied by the x-th power of convex-roof extended negativity of assistance with \(x\ge 2\) and \(x\le 0\) for multi-qubit generalized W-class states. Using Rényi-\(\alpha \) entropy (R\(\alpha \)E) with \(\alpha \in [(\sqrt{7}-1)/2, (\sqrt{13}-1)/2]\), we prove new monogamy and polygamy relations. We further show that the monogamy inequality also holds for the \(\mu \)th power of Rényi-\(\alpha \) entanglement. Moreover, we study two examples in multipartite higher-dimensional system for those new inequalities.     

Stability of properties of Kolmogorov complexity under relativization

Assume that a tuple of binary strings \(\bar a\) = 〈a ₁ ..., a _n 〉 has negligible mutual information with another string b. Does this mean that properties of the Kolmogorov complexity of \(\bar a\) do not change significantly if we relativize them to b? This question becomes very nontrivial when we try to formalize it. In this paper we investigate this problem for a special class of properties (for properties that can be expressed by an ?-formula). In particular, we show that a random (conditional on \(\bar a\)) oracle b does not help to extract common information from the strings a _i .     

Hybridized Discontinuous Galerkin Method for Elliptic Interface Problems: Error Estimates Under Low Regularity Assumptions of Solutions

New hybridized discontinuous Galerkin (HDG) methods for the interface problem for elliptic equations are proposed. Unknown functions of our schemes are \(u_h\) in elements and \(\hat{u}_h\) on inter-element edges. That is, we formulate our schemes without introducing the flux variable. We assume that subdomains \(\Omega _1\) and \(\Omega _2\) are polyhedral domains and that the interface \(\Gamma =\partial \Omega _1\cap \partial \Omega _2\) is polyhedral surface or polygon. Moreover, \(\Gamma \) is assumed to be expressed as the union of edges of some elements. We deal with the case where the interface is transversely connected with the boundary of the whole domain \(\overline{\Omega }=\overline{\Omega _1\cap \Omega _2}\). Consequently, the solution u of the interface problem may not have a sufficient regularity, say \(u\in H^2(\Omega )\) or \(u|_{\Omega _1}\in H^2(\Omega _1)\), \(u|_{\Omega _2}\in H^2(\Omega _2)\). We succeed in deriving optimal order error estimates in an HDG norm and the \(L^2\) norm under low regularity assumptions of solutions, say \(u|_{\Omega _1}\in H^{1+s}(\Omega _1)\) and \(u|_{\Omega _2}\in H^{1+s}(\Omega _2)\) for some \(s\in (1/2,1]\), where \(H^{1+s}\) denotes the fractional order Sobolev space. Numerical examples to validate our results are also presented.     

A Neumann series of Bessel functions representation for solutions of Sturm–Liouville equations

A Neumann series of Bessel functions (NSBF) representation for solutions of Sturm–Liouville equations and for their derivatives is obtained. The representation possesses an attractive feature for applications: for all real values of the spectral parameter \(\omega \) the estimate of the difference between the exact solution and the approximate one (the truncated NSBF) depends on N (the truncation parameter) and the coefficients of the equation and does not depend on \(\omega \). A similar result is valid when \(\omega \in {\mathbb {C}}\) belongs to a strip \(\left| \hbox {Im }\omega \right| <C\). This feature makes the NSBF representation especially useful for applications requiring computation of solutions for large intervals of \(\omega \). Error and decay rate estimates are obtained. An algorithm for solving initial value, boundary value or spectral problems for the Sturm–Liouville equation is developed and illustrated on a test problem.     

Shock response analysis of hard disk drive using flexible multibody dynamics formulation

Recent development of the shock analysis on the HDD is briefly reviewed. A flexible multi-body dynamics formulation is developed to simulate the shock response of the HDD. If one component in the HDD is changed, only mode shapes and frequencies of that component should be re-calculated and then used to obtain the system’s response. Steady state Reynolds equation is solved to obtain the air pressure on the slider and disk for various slider positions. An air pressure table is formed and used to model the non-linear air bearing during the simulation. Responses of flying height for different direction and shock duration time are analyzed. Results show that the flying state of the slider is more sensitive to the shock with shorter duration time.     

Optimal quantum thermometry by dephasing

Decoherence often happens in the quantum world. We try to utilize quantum dephasing to build an optimal thermometry. By calculating the Cramér–Rao bound, we prove that the Ramsey measurement is the optimal way to measure the temperature for uncorrelated probe particles. Using the optimal measurement, the metrological equivalence of product and maximally entangled state of initial quantum probes always holds. Contrary to frequency estimation, the optimal temperature estimation can be obtained in the case \(\nu <1\), not \(\nu >1\). For the general Zeno regime (\(\nu =2\)), uncorrelated product states are the optimal choice in typical Ramsey spectroscopy setup. In order to improve the resolution of temperature, one should reduce the characteristic time of dephasing factor \(\gamma (t)\propto t^2\), and the power \(\nu <1\) appears after it. Under the imperfect condition, maximally entangled state can perform better than product state. Finally, we investigate other environmental influence on the measurement precision of temperature. Based on it, we define a new way to measure non-Markovian effect.     

Compatibility of observables on effect algebras

We compare different notions of simultaneous measurability (compatibility) of observables on lattice \(\sigma \)-effect algebras and more generally, on \(\sigma \)-effect algebras that can be covered by \(\sigma \)-MV-algebras. We prove that every \(\sigma \)-MV-algebra is the range of a \(\sigma \)-additive observable, and we compare the following notions of compatibility of observables: joint measurability, coexistence, joint measurability of binarizations, coexistence of binarizations, smearings of the same observable. We prove that if there is a faithful state on the effect algebra, then any two standard observables that are smearings of the same (sharp) observable admit a generalized joint observable.     

Counter machines and crystallographic structures

One way to depict a crystallographic structure is by a periodic (di)graph, i.e., a graph whose group of automorphisms has a translational subgroup of finite index acting freely on the structure. We establish a relationship between periodic graphs representing crystallographic structures and an infinite hierarchy of intersection languages \(\mathcal {DCL}_d,\,d=0,1,2,\ldots \), within the intersection classes of deterministic context-free languages. We introduce a class of counter machines that accept these languages, where the machines with d counters recognize the class \(\mathcal {DCL}_d\). An intersection of d languages in \(\mathcal {DCL}_1\) defines \(\mathcal {DCL}_d\). We prove that there is a one-to-one correspondence between sets of walks starting and ending in the same unit of a d-dimensional periodic (di)graph and the class of languages in \(\mathcal {DCL}_d\). The proof uses the following result: given a digraph \(\Delta \) and a group G, there is a unique digraph \(\Gamma \) such that \(G\le \mathrm{Aut}\,\Gamma ,\,G\) acts freely on the structure, and \(\Gamma /G \cong \Delta \).     

Numerical Methods and Comparison for the Dirac Equation in the Nonrelativistic Limit Regime

We analyze rigorously error estimates and compare numerically spatial/temporal resolution of various numerical methods for the discretization of the Dirac equation in the nonrelativistic limit regime, involving a small dimensionless parameter \(0<\varepsilon \ll 1\) which is inversely proportional to the speed of light. In this limit regime, the solution is highly oscillatory in time, i.e. there are propagating waves with wavelength \(O(\varepsilon ^2)\) and O(1) in time and space, respectively. We begin with several frequently used finite difference time domain (FDTD) methods and obtain rigorously their error estimates in the nonrelativistic limit regime by paying particular attention to how error bounds depend explicitly on mesh size h and time step \(\tau \) as well as the small parameter \(\varepsilon \). Based on the error bounds, in order to obtain ‘correct’ numerical solutions in the nonrelativistic limit regime, i.e. \(0<\varepsilon \ll 1\), the FDTD methods share the same \(\varepsilon \)-scalability on time step and mesh size as: \(\tau =O(\varepsilon ^3)\) and \(h=O(\sqrt{\varepsilon })\). Then we propose and analyze two numerical methods for the discretization of the Dirac equation by using the Fourier spectral discretization for spatial derivatives combined with the symmetric exponential wave integrator and time-splitting technique for temporal derivatives, respectively. Rigorous error bounds for the two numerical methods show that their \(\varepsilon \)-scalability is improved to \(\tau =O(\varepsilon ^2)\) and \(h=O(1)\) when \(0<\varepsilon \ll 1\). Extensive numerical results are reported to support our error estimates.     

Effect of thermal pole tip protrusion and disk roughness on slider disk contacts

Ultra-high areal density for hard disk drives requires a stable head disk interface at a flying height lower than 8 nm. At such a low flying height, small flying height variations may cause slider/disk contacts. Slider/disk contacts can also occur when a write-current is applied to the write coil since the flying height between slider and disk can be affected by the thermal expansion of the pole tip. In this paper, we investigate the vibration characteristics of sliders during thermally induced contacts using laser Doppler vibrometry. We perform a parametric study of contact events using disks with different surface roughness and lubricant thicknesses, and analyze the slider motion statistically. For a given write current, we observe that the slider vibrations increase with disk roughness and lubricant thickness.     

Decoupled,Unconditionally Stable,Higher Order Discretizations for MHD Flow Simulation

We propose, analyze, and test a new MHD discretization which decouples the system into two Oseen problems at each timestep yet maintains unconditional stability with respect to the time step size, is optimally accurate in space, and behaves like second order in time in practice. The proposed method chooses a parameter \(\theta \in [0,1]\), dependent on the viscosity \(\nu \) and magnetic diffusivity \(\nu _m\), so that the explicit treatment of certain viscous terms does not cause instabilities, and gives temporal accuracy \(O(\Delta t^2 + (1-\theta )|\nu -\nu _m|\Delta t)\). In practice, \(\nu \) and \(\nu _m\) are small, and so the method behaves like second order. When \(\theta =1\), the method reduces to a linearized BDF2 method, but it has been proven by Li and Trenchea that such a method is stable only in the uncommon case of \(\frac{1}{2}< \frac{\nu }{\nu _m} < 2\). For the proposed method, stability and convergence are rigorously proven for appropriately chosen \(\theta \), and several numerical tests are provided that confirm the theory and show the method provides excellent accuracy in cases where usual BDF2 is unstable.     

Engineering design optimization using an improved local search based epsilon differential evolution algorithm

Many engineering problems can be categorized into constrained optimization problems (COPs). The engineering design optimization problem is very important in engineering industries. Because of the complexities of mathematical models, it is difficult to find a perfect method to solve all the COPs very well. \(\varepsilon \) constrained differential evolution (\(\varepsilon \)DE) algorithm is an effective method in dealing with the COPs. However, \(\varepsilon \)DE still cannot obtain more precise solutions. The interaction between feasible and infeasible individuals can be enhanced, and the feasible individuals can lead the population finding optimum around it. Hence, in this paper we propose a new algorithm based on \(\varepsilon \) feasible individuals driven local search called as \(\varepsilon \) constrained differential evolution algorithm with a novel local search operator (\(\varepsilon \)DE-LS). The effectiveness of the proposed \(\varepsilon \)DE-LS algorithm is tested. Furthermore, four real-world engineering design problems and a case study have been studied. Experimental results show that the proposed algorithm is a very effective method for the presented engineering design optimization problems.     

Pure state ‘really’ informationally complete with rank-1 POVM

What is the minimal number of elements in a rank-1 positive operator-valued measure (POVM) which can uniquely determine any pure state in d-dimensional Hilbert space \(\mathcal {H}_d\)? The known result is that the number is no less than \(3d-2\). We show that this lower bound is not tight except for \(d=2\) or 4. Then we give an upper bound \(4d-3\). For \(d=2\), many rank-1 POVMs with four elements can determine any pure states in \(\mathcal {H}_2\). For \(d=3\), we show eight is the minimal number by construction. For \(d=4\), the minimal number is in the set of \(\{10,11,12,13\}\). We show that if this number is greater than 10, an unsettled open problem can be solved that three orthonormal bases cannot distinguish all pure states in \(\mathcal {H}_4\). For any dimension d, we construct \(d+2k-2\) adaptive rank-1 positive operators for the reconstruction of any unknown pure state in \(\mathcal {H}_d\), where \(1\le k \le d\).     

Simplified molecular gas film lubrication equation and accommodation coefficient effects

As the spacing between the flying head/slider and the rotating disk in hard disk drives (HDDs) continues to decrease, the interaction between the molecular gas and the surfaces of the disk and the head/slider becomes significant. The influence of surface accommodation coefficient (AC) is an important factor to govern the static characteristics of the head/slider. Starting from the polynomial logarithm fitting equations of Poisueille flow rate and Couette flow rate, a new simplified molecular gas film lubrication (MGL) equation is proposed to simulate the ultra-thin air bearing film in HDDs. The new MGL equation is simpler than that of the polynomial logarithm form of MGL equation. The new approach produces very good approximations for both Poisueille flow rate and Couette flow rate with very little differences to those based on the original MGL equation. The new simplified MGL equation is solved by using a meshless method, called least square finite difference (LSFD) method. Effects of ACs on the static characteristics of air bearing films in HDDs with ultra-low flying heights are investigated. Numerical results show that effects of ACs on the static characteristics are significant for the case of symmetric molecular interaction. On the other hand, effects of ACs at the disk surface on the static characteristics are significant for the case of non-symmetric molecular interaction, while effects of ACs at the slider surface on the static characteristics are weak.     

High Order Algorithm for the Time-Tempered Fractional Feynman–Kac Equation

We provide and analyze the high order algorithms for the model describing the functional distributions of particles performing anomalous motion with power-law jump length and tempered power-law waiting time. The model is derived in Wu et al. (Phys Rev E 93:032151, 2016), being called the time-tempered fractional Feynman–Kac equation named after Richard Feynman and Mark Kac who first considered the model describing the functional distribution of normal motion. The key step of designing the algorithms is to discretize the time tempered fractional substantial derivative, being defined as

$$\begin{aligned} {^S\!}D_t^{\gamma ,\widetilde{\lambda }} G(x,p,t)\!=\!D_t^{\gamma ,\widetilde{\lambda }} G(x,p,t)\!-\!\lambda ^\gamma G(x,p,t) \end{aligned}$$

with \(\widetilde{\lambda }=\lambda + pU(x),\, p=\rho +J\eta ,\, J=\sqrt{-1}\), where

$$\begin{aligned} D_t^{\gamma ,\widetilde{\lambda }} G(x,p,t) =\frac{1}{\varGamma (1-\gamma )} \left[ \frac{\partial }{\partial t}+\widetilde{\lambda } \right] \int _{0}^t{\left( t-z\right) ^{-\gamma }}e^{-\widetilde{\lambda }\cdot (t-z)}{G(x,p,z)}dz, \end{aligned}$$

and \(\lambda \ge 0\), \(0<\gamma <1\), \(\rho >0\), and \(\eta \) is a real number. The designed schemes are unconditionally stable and have the global truncation error \(\mathscr {O}(\tau ^2+h^2)\), being theoretically proved and numerically verified in complex space. Moreover, some simulations for the distributions of the first passage time are performed, and the second order convergence is also obtained for solving the ‘physical’ equation (without artificial source term).

     

$$\mathcal $$-FAINV: hierarchically factored approximate inverse preconditioners

Given a sparse matrix, its LU-factors, inverse and inverse factors typically suffer from substantial fill-in, leading to non-optimal complexities in their computation as well as their storage. In the past, several computationally efficient methods have been developed to compute approximations to these otherwise rather dense matrices. Many of these approaches are based on approximations through sparse matrices, leading to well-known ILU, sparse approximate inverse or factored sparse approximate inverse techniques and their variants. A different approximation approach is based on blockwise low rank approximations and is realized, for example, through hierarchical (\(\mathcal H\)-) matrices. While \(\mathcal H\)-inverses and \(\mathcal H\)-LU factors have been discussed in the literature, this paper will consider the construction of an approximation of the factored inverse through \(\mathcal H\)-matrices (\(\mathcal H\)-FAINV). We will describe a blockwise approach that permits to replace (exact) matrix arithmetic through approximate efficient \(\mathcal H\)-arithmetic. We conclude with numerical results in which we use approximate factored inverses as preconditioners in the iterative solution of the discretized convection–diffusion problem.