In this paper we present two unconditionally energy stable finite difference schemes for the Modified Phase Field Crystal (MPFC) equation, a sixth-order nonlinear damped wave equation, of which the purely parabolic Phase Field Crystal (PFC) model can be viewed as a special case. The first is a convex splitting scheme based on an appropriate decomposition of the discrete energy and is first order accurate in time and second order accurate in space. The second is a new, fully second-order scheme that also respects the convex splitting of the energy. Both schemes are nonlinear but may be formulated from the gradients of strictly convex, coercive functionals. Thus, both are uniquely solvable regardless of the time and space step sizes. The schemes are solved by efficient nonlinear multigrid methods. Numerical results are presented demonstrating the accuracy, energy stability, efficiency, and practical utility of the schemes. In * Currently at Haerbin Inst. Techn., China 1 arXiv:1208.2643v1 [math.NA] 8 Aug 2012 particular, we show that our multigrid solvers enjoy optimal, or nearly optimal complexity in the solution of the nonlinear schemes.which is significantly different from schemes we propose and analyze. In particular, theirs is not expected to be unconditionally solvable or energy stable. The authors did, however, give some evidence of the efficiency of their multigrid solver. We will conduct a similar study for our multigrid solver in this paper. The MPFC scheme in [10] is more or less the same as the first-order convex-splitting that we devised earlier in [18,19]. The first-order convex splitting scheme in our work [18,19] has two fundamental properties. It is unconditionally energy stable and unconditionally uniquely solvable. We rigorously analyzed this convex splitting scheme in [18,19], but we did not provide a practical solution strategy. That gap is filled here.In [8,21] we presented first and second-order accurate (in time) finite difference schemes for the purely parabolic PFC model, based on a convex splitting framework applied to the physical energy. The convexity splitting idea -in the context of first order (in time) convex splitting schemes -is generally credited to [7]. One of the main advances in [8] -and the more recent work in [14] -was the demonstration that the convex splitting framework can be extended to second-order (in time) methods as well, in a natural way. While the motivation in [8] was the application to the PFC model, the second-order convex splitting idea that we developed is, in fact, rather general.The main goal of this paper is to apply the first and second-order convex splitting framework to the MPFC equation. The principal challenge in doing so is that the extension to damped wave dynamics appears, at first sight, not so straightforward. This obstacle was first surmounted in [18,19], as already mentioned, where we extended the first-order convexity splitting idea for the MPFC model. Specifically, we showed the existence of a pseudo energy (different from the physical energy),...

The propose of this review was to summarize the advances in multi-scale surface technology of titanium implants to accelerate the osseointegration process. The several multi-scaled methods used for improving wettability, roughness, and bioactivity of implant surfaces are reviewed. In addition, macro-scale methods (e.g., 3D printing (3DP) and laser surface texturing (LST)), micro-scale (e.g., grit-blasting, acid-etching, and Sand-blasted, Large-grit, and Acid-etching (SLA)) and nano-scale methods (e.g., plasma-spraying and anodization) are also discussed, and these surfaces are known to have favorable properties in clinical applications. Functionalized coatings with organic and non-organic loadings suggest good prospects for the future of modern biotechnology. Nevertheless, because of high cost and low clinical validation, these partial coatings have not been commercially available so far. A large number of in vitro and in vivo investigations are necessary in order to obtain in-depth exploration about the efficiency of functional implant surfaces. The prospective titanium implants should possess the optimum chemistry, bionic characteristics, and standardized modern topographies to achieve rapid osseointegration.

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SUMMARYA negative stiffness damper (NSD) is proposed, and its performance on suppressing stay cable vibrations is investigated by numerical simulations and experimental tests. First, the NSD consists of two pressed springs and an oil damper. The two pressed springs are attached perpendicularly to the piston rod of the oil damper in symmetrical configuration. Mechanical model of this damper is derived according to the geometrical configuration and validated experimentally. Considering the simplified model of single-mode vibration of a cable, the cable frequency with the NSD is investigated theoretically by average method, and the lower limit on the pressed degree of two springs is proposed. Numerical examples of cable with NSD and oil damper under sinusoidal excitations are conducted. The NSD shows the superior reduction performance of both amplitudes in time history and peak values of frequency response. A series of single-mode and multi-mode cable vibration control experiments are carried out to evaluate mitigation performance achieved by the NSD. The results indicate that this NSD can provide larger additional modal damping ratio to the cable regardless of single-mode and multiple-mode vibration. Thus, this NSD achieves further reduction of the cable than the oil damper through negative stiffness behavior, instead of complex active or semi-active devices with real-time feedback.

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