Relaxed Boundary Conditions Research Articles

Response of circular flexible foundations subjected to horizontal and rocking motions

A methodology using modal analysis is developed to evaluate dynamic vertical displacements of a circular flexible foundation resting on soil media subjected to horizontal and rocking motions. The influence of the soil reaction forces on the foundation is considered by introducing modal impedance functions, which can be determined by an efficient procedure with ring elements. The displacements of the foundation can then be easily solved by modal superposition. Parametric studies for modal responses of the flexible foundation indicate that the coupled response of the foundation is significantly influenced by relative stiffness among the foundation and the soil medium, vibration frequency range, foundation mass, and boundary contact conditions. The welded boundary condition should be considered to predict the coupling response while the relaxed boundary condition may be used to predict approximately the vertical displacements. As a foundation with a relative stiffness ratio more than three, it is found that the foundation can be considered as rigid to calculate coupling displacements. For a slightly flexible foundation, considerations of three modes are sufficient enough to obtain accurate foundation responses. Moreover, at low frequencies, the coupling effect due to higher mode can be neglected.

Sailfish: A flexible multi-GPU implementation of the lattice Boltzmann method

We present Sailfish, an open source fluid simulation package implementing the lattice Boltzmann method (LBM) on modern Graphics Processing Units (GPUs) using CUDA/OpenCL. We take a novel approach to GPU code implementation and use run-time code generation techniques and a high level programming language (Python) to achieve state of the art performance, while allowing easy experimentation with different LBM models and tuning for various types of hardware. We discuss the general design principles of the code, scaling to multiple GPUs in a distributed environment, as well as the GPU implementation and optimization of many different LBM models, both single component (BGK, MRT, ELBM) and multicomponent (Shan–Chen, free energy). The paper also presents results of performance benchmarks spanning the last three NVIDIA GPU generations (Tesla, Fermi, Kepler), which we hope will be useful for researchers working with this type of hardware and similar codes. Program SummaryProgram title: SailfishCatalogue identifier: AETA_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AETA_v1_0.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, N. IrelandLicensing provisions: GNU Lesser General Public License, version 3No. of lines in distributed program, including test data, etc.: 225864No. of bytes in distributed program, including test data, etc.: 46861049Distribution format: tar.gzProgramming language: Python, CUDA C, OpenCL.Computer: Any with an OpenCL or CUDA-compliant GPU.Operating system: No limits (tested on Linux and Mac OS X).RAM: Hundreds of megabytes to tens of gigabytes for typical cases.Classification: 12, 6.5.External routines: PyCUDA/PyOpenCL, Numpy, Mako, ZeroMQ (for multi-GPU simulations), scipy, sympyNature of problem:GPU-accelerated simulation of single- and multi-component fluid flows.Solution method:A wide range of relaxation models (LBGK, MRT, regularized LB, ELBM, Shan–Chen, free energy, free surface) and boundary conditions within the lattice Boltzmann method framework. Simulations can be run in single or double precision using one or more GPUs.Restrictions:The lattice Boltzmann method works for low Mach number flows only.Unusual features:The actual numerical calculations run exclusively on GPUs. The numerical code is built dynamically at run-time in CUDA C or OpenCL, using templates and symbolic formulas. The high-level control of the simulation is maintained by a Python process.Additional comments:!!!!! The distribution file for this program is over 45 Mbytes and therefore is not delivered directly when Download or Email is requested. Instead a html file giving details of how the program can be obtained is sent. !!!!!Running time:Problem-dependent, typically minutes (for small cases or short simulations) to hours (large cases or long simulations).

Analysis of the Cahn–Hilliard Equation with a Relaxation Boundary Condition Modeling the Contact Angle Dynamics

We analyze the Cahn–Hilliard equation with a relaxation boundary condition modeling the evolution of an interface in contact with the solid boundary. An L ∞ estimate is established which enables us to prove the global existence of the solution. We also study the sharp interface limit of the system. The dynamic of the contact point and the contact angle are derived and the results are compared with the numerical simulations.

Rectification of El Niño–Southern Oscillation into Climate Anomalies of Decadal and Longer Time Scales: Results from Forced Ocean GCM Experiments

Abstract To better understand the causes of climate change in the tropical Pacific on the decadal and longer time scales, the rectification effect of ENSO events is delineated by contrasting the time-mean state of two forced ocean GCM experiments. In one of them, the long-term mean surface wind stress of 1950–2011 is applied, while in the other, the surface wind stress used is the long-term mean surface wind stress of 1950–2011 plus the interannual monthly anomalies over the period. Thus, the long-term means of the surface wind stress in the two runs are identical. The two experiments also use the same relaxation boundary conditions, that is, the SST is restored to the same prescribed values. The two runs, however, are found to yield significantly different mean climate for the tropical Pacific. The mean state of the run with interannual fluctuations in the surface winds is found to have a cooler warm pool, warmer thermocline water, and warmer eastern surface Pacific than the run without interannual fluctuations in the surface winds. The warming of the eastern Pacific has a pattern that resembles the observed decadal warming. In particular, the pattern features an off-equator maximum as the observed decadal warming. The spatial pattern of the time-mean upper-ocean temperature differences between the two experiments is shown to resemble that of the differences in the nonlinear dynamic heating, underscoring the role of the nonlinear ocean dynamics in the rectification. The study strengthens the suggestion that rectification of ENSO can be a viable mechanism for climate change of decadal and longer time scales.

A blister test for interfacial adhesion of large-scale transferred graphene

A blister test and associated analysis was developed to characterize the interfacial adhesion between graphene and substrates to which it has been transferred. In this study, chemical vapor deposition grown graphene had been transferred to a highly polished copper substrate from its seed foil. The graphene/photoresist composite film was pressurized with deionized water through a nominally 1-mm hole in the copper and the deflection of the membrane was measured by a full field interference method. The deflection profiles compared well with those obtained from a linear plate model that accounted for the initial strain in the membrane and a relaxed boundary condition at the edge of the blister. This was used to calculate the energy release rate as a function of delamination growth to obtain fracture resistance curves for the graphene/copper interface. To the best of our knowledge, these are the first measurements of fracture resistance curves for adhesive interactions between transferred graphene and a copper substrate. The delamination path of the graphene/photoresist sample was confirmed by Raman spectroscopy. The measured adhesion energy for the graphene/copper interface was higher than that of a photoresist/copper interface, but slightly lower than previous measurements for as-grown graphene on copper foil.

Generalized characteristic relaxation boundary conditions for unsteady compressible flow simulations

We develop numerical boundary conditions for the compressible Navier–Stokes equations based on a generalized relaxation approach (GRCBC), which hinges on locally one-dimensional characteristic projection at the computational boundaries, supplemented with available information from the flow exterior. The basic idea is to estimate the amplitude of incoming characteristic waves through first-order one-sided finite-difference approximations which involve the value of the reference flow state at the first exterior (ghost) point. Unlike other characteristic-based relaxation methods, the present one requires minimal user-supplied input, including the reference flow state, which may be totally or partially known, and in general may vary both in space and time. Furthermore, it can be applied to any type of computational boundary, either inflow or outflow, either subsonic or supersonic. The method is theoretically predicted to convey reduced reflection of waves at computational boundaries compared to other ones, and to have better properties of frequency response to injected disturbances. Numerical tests confirm the improvement of the nonreflecting performance, and demonstrate high degree of flexibility, also for problems with non-trivial far-field boundary conditions (e.g. flows in rotating reference frames) and for the artificial stimulation of subsonic turbulent boundary layers.

Limited area NWP and regional climate modeling: a test of the relaxation vs Eta lateral boundary conditions

With very few exceptions, just about all limited area models (LAMs) used in operational NWP and regional climate modeling use the Davies (Q J R Meteorol Soc 102:405–418, 1976) relaxation lateral boundary conditions (LBCs), even though they make no effort to respect the basic mathematics of the problem. While in the early stages of the primitive equation LAM development in the seventies numerous schemes have been proposed and tested, LAM communities have eventually for the most part settled on the relaxation LBCs with few questions asked. An exception is the Eta model used extensively at NCEP and several other centers, in which the Mesinger (Contrib Atmos Phys 50:200–210, 1977) LBCs are used, designed and based on knowledge available before the introduction of the relaxation scheme. They prescribe variables along the outermost row of grid points only; all of them at the inflow points and one less at the outflow points where the tangential velocity components are extrapolated from inside of the model domain. Additional schemes are in place to suppress separation of gravity-wave solutions on C-subgrids of the model’s E-grid. A recent paper of Veljovic et al. (Meteor Zeitschrift 19:237–246, 2010) included three 32-day forecasts done with both the Eta and the relaxation LBCs and the comparison of some of their verification results. Here we extend this experiment by three additional forecasts to arrive at an ensemble of six members run with both schemes, and present a more complete discussion of results. We in addition show results of one of these forecasts in which the linear change of relaxation coefficients was replaced by the change following the recommendation of Lehmann (Meteorol Atmos Phys 52:1–14, 1993). We feel that the results of our two verification schemes strongly suggest the advantage of the Eta over the conventional relaxation scheme, thereby raising doubts as to the justification for its use.

定常歩容に漸近する２脚ロボットの自在歩容生成

In this paper, we propose a method to generate various walking gait patterns for biped robots in real-time using an approximated dynamics model with two point masses representing motion of the feet and a linear inverted pendulum representing the horizontal motion of the torso. The motion of the inverted pendulum is calculated from the trajectories of the feet and the desired ZMP. Supposing these trajectories for the current gait and the next cyclic gait are given, the initial state of the inverted pendulum of the cyclic gait is calculated, and the desired ZMP trajectory of the current gait is modified so that the current gait converges to the cyclic gait using the concept called “the divergent component of the motion” as a relaxed boundary condition. Using these techniques, real-time gait generation is achieved which requires a comparatively small amount of modification of the desired ZMP trajectory while guaranteeing the continuity of the gait generation.

Interaction between an edge dislocation and a circular inclusion with interface slip and diffusion

We investigate in detail the transient response induced by an edge dislocation near a circular elastic inclusion with simultaneous interface slip and diffusion. A rigorous solution to the interaction problem is derived in series form. As the time approaches infinity, our solution just recovers the classical one derived by Srolovitz et al. (Acta Metall 1984;32:1079) for fully relaxed boundary conditions. In addition, we observe that the edge dislocation will induce a uniform rigid-body rotation in the inclusion as the time approaches infinity. When the dislocation is far away from the inclusion, simple asymptotic expressions of the glide and climb forces on the dislocation are also obtained. Furthermore, five extreme cases for the imperfect interface are discussed; in particular, we derive approximate closed-form expressions of the decaying internal stress field within the inclusion and the image force on the dislocation for long-range stress relaxations when the interface diffusion occurs much faster than the interface slip and vice versa. Some interesting physical behaviors are observed.

Snow contribution to springtime atmospheric predictability over the second half of the twentieth century

A set of global atmospheric simulations has been performed with the ARPEGE-Climat model in order to quantify the contribution of realistic snow conditions to seasonal atmospheric predictability in addition to that of a perfect sea surface temperature (SST) forcing. The focus is on the springtime boreal hemisphere where the combination of a significant snow cover variability and an increasing solar radiation favour the potential snow influence on the surface energy budget. The study covers the whole 1950–2000 period through the use of an original snow mass reanalysis based on an off-line land surface model and possibly constrained by satellite snow cover observations. Two ensembles of 10-member AMIP-type experiments have been first performed with relaxed versus free snow boundary conditions. The nudging towards the monthly snow mass reanalysis significantly improves both potential and actual predictability of springtime surface air temperature over Central Europe and North America. Yet, the impact is confined to the lower troposphere and there is no clear improvement in the predictability of the large-scale atmospheric circulation. Further constraining the prescribed snow boundary conditions with satellite observations does not change much the results. Finally, using the snow reanalysis only for initializing the model on March 1st also leads to a positive impact on predicted low-level temperatures but with a weaker amplitude and persistence. A conditional skill approach as well as some selected case studies provide some guidelines for interpreting these results and suggest that an underestimated snow cover variability and a misrepresentation of ENSO teleconnections may hamper the benefit of an improved snow initialization in the ARPEGE-Climat model.

Asymptotic symmetries of Rindler space at the horizon and null infinity

We investigate the asymptotic symmetries of Rindler space at null infinity and at the event horizon using both systematic and ad hoc methods. We find that the approaches that yield infinite-dimensional asymptotic symmetry algebras in the case of anti--de Sitter and flat spaces only give a finite-dimensional algebra for Rindler space at null infinity. We calculate the charges corresponding to these symmetries and confirm that they are finite, conserved, and integrable, and that the algebra of charges gives a representation of the asymptotic symmetry algebra. We also use relaxed boundary conditions to find infinite-dimensional asymptotic symmetry algebras for Rindler space at null infinity and at the event horizon. We compute the charges corresponding to these symmetries and confirm that they are finite and integrable. We also determine sufficient conditions for the charges to be conserved on-shell, and for the charge algebra to give a representation of the asymptotic symmetry algebra. In all cases, we find that the central extension of the charge algebra is trivial.

Chiral gravity, log gravity, and extremal CFT

We show that the linearization of all exact solutions of classical chiral gravity around the ${\mathrm{AdS}}_{3}$ vacuum have positive energy. Nonchiral and negative-energy solutions of the linearized equations are infrared divergent at second order, and so are removed from the spectrum. In other words, chirality is confined and the equations of motion have linearization instabilities. We prove that the only stationary, axially symmetric solutions of chiral gravity are BTZ black holes, which have positive energy. It is further shown that classical log gravity---the theory with logarithmically relaxed boundary conditions---has finite asymptotic symmetry generators but is not chiral and hence may be dual at the quantum level to a logarithmic conformal field theories (CFT). Moreover we show that log gravity contains chiral gravity within it as a decoupled charge superselection sector. We formally evaluate the Euclidean sum over geometries of chiral gravity and show that it gives precisely the holomorphic extremal CFT partition function. The modular invariance and integrality of the expansion coefficients of this partition function are consistent with the existence of an exact quantum theory of chiral gravity. We argue that the problem of quantizing chiral gravity is the holographic dual of the problem of constructing an extremal CFT, while quantizing log gravity is dual to the problem of constructing a logarithmic extremal CFT.

Horizontal convection in water heated by infrared radiation and cooled by evaporation: scaling analysis and experimental results

An experimental study of horizontal convection with a free surface has been conducted. Fresh water was heated from above by an infrared lamp placed at one end of a tank, and cooled by evaporation as the water moved away from the heat source. The heat radiated from the lamp was absorbed in a thin (less than 1 mm) layer next to the surface, and then advected and diffused away from the lamp region. Latent heat loss dominated the surface cooling processes and accounted for at least 80% of the energy loss. The velocity and temperature fields were recorded with PIV technology, thermometers and an infrared camera. In similarity with previous horizontal convection experiments the measurements showed a closed circulation with a gradually cooling surface current moving away from the lamp. Below the surface current the water was stably stratified with a comparatively thick and slow return current. The thickness and speed, and hence the mass transport, of the surface-and the return current increased with distance from the lamp.The latent cooling at the free surface gives a heat flux which increases with the temperature difference between the surface water and the air above it. Hence the surface temperature relaxes towards an equilibrium value, for which the heat flux is zero. The main new result is a scaling law, taking into account this relaxation boundary condition for the surface temperature. The new scaling includes a (relaxation) length scale for the surface temperature, equivalent to the distance the surface current travels before it has lost the heat that was gained underneath the lamp. The length scale increases with the forcing strength and the (molecular) thermal diffusivity but decreases with the strength of the relaxation. Numerical simulations of this problem for a shallow tank have also been performed. The velocity and temperature in the laboratory and numerical experiments agree with the scaling laws in the upper part of the tank, but not in the lower.

On refined analysis of bifurcation buckling for the axially compressed circular cylinder

We present extensive numerical results of bifurcation buckling analysis of the axially compressed circular cylinder. The analysis is based on the modified displacement version of the non-linear theory of thin elastic shells developed by Opoka and Pietraszkiewicz [Opoka, S., Pietraszkiewicz, W., 2009. On modified displacement version of the non-linear theory of thin shells. International Journal of Solids and Structures, 46, 3103–3110.]. To solve the buckling problem we apply the separation of variables and expansion of all fields into Fourier series in circumferential direction, with subsequent accurate calculations of eigenvalues of determinants of corresponding 8 × 8 complicated matrices. The numerical analysis of the buckling load is performed for the cylinders with length-to-diameter ratio in the range (0.05, 60), with eight sets of incremental work-conjugate boundary conditions analogous to those used in the literature and partly summarized in the book by Yamaki [Yamaki, N., 1984. Elastic Stability of Circular Cylindrical Shells. Elsevier, Amsterdam], and additionally with six sets of boundary conditions not discussed in the literature yet. The results allow us to formulate several important conclusions, such as: (a) omission in the non-linear BVP small terms of the order of error introduced by the error of constitutive equations leads to overestimated buckling loads for long cylinders with clamped boundaries; (b) for some relaxed boundary conditions the buckling load decreases for short cylinders with decrease of the cylinder length; (c) the results for additional six sets of boundary conditions reveal existence of several new cases, in which by relaxing geometric boundary conditions the buckling load falls down to about one half of the classical value in a wide range of the cylinder length-to-diameter ratios.

Numerical analysis of the influence of tilt of crucibles on interface shape and fields of temperature and velocity in the unidirectional solidification process

We carried out calculations to investigate the influence of tilt of crucibles on the melt–crystal interface shape and fields of temperature and velocity of the melt and/or crystal by three-dimensional global and melt–crystal analyses. It was found that flow velocity was larger in the case of a fixed boundary condition of edge of the interface than that in the case of a relaxed condition. Furthermore, deflection of the interface with a fixed boundary condition was smaller than that without the fixed boundary condition. These results indicate that we should use three-dimensional global analysis with a relaxed boundary condition to investigate the influence of tilt of crucibles on the interface shape and flow velocity.

Boundary-Value Problems for Hyperbolic Equations Related to Steady Granular Flow

Boundary value problems for steady-state flow in elastoplasticity are a topic of mathematical and physical interest. In particular, the underlying PDE may be hyperbolic, and uncertainties surround the choice of physically appropriate stress and velocity boundary conditions. The analysis and numerical simulations of this paper address this issue for a model problem, a system of equations describing antiplane shearing of an elastoplastic material. This system retains the relevant mathematical structure of elastoplastic planar flow. Even if the flow rule is associative, two significant phenomena appear: (i) For boundary conditions suggestive of granular flow in a hopper, in which it seems physically natural to specify the velocity everywhere along a portion of the boundary, no such solutions of the equations exist; rather, we construct a solution with a shear band (velocity jump) along part of the boundary, and an appropriate relaxed boundary condition is satisfied there. (ii) Rigid zones appear inside deforming regions of the flow, and the stress field in such a zone is not uniquely determined. For a nonassociative flow rule, an extreme form of nonuniqueness—both velocity and stress—is encountered.

Snap-through buckling behavior of piezoelectric bimorph beams: I. Analytical andnumerical modeling

Piezoelectric structures are used in a variety of applications where instant response, highenergy conversion efficiency and accurate control are required. However, it is widely knownthat piezoelectric structures suffer from a series of drawbacks, among which the mostimportant is the small displacement capacity. A number of techniques have been used inorder to transform the micron-scale displacements of PZT layers into meaningfulmillimeter-scale ones. Non-linear mechanics belong to this category, providing thepossibility to transform a traditional bimorph linear output structure into a non-linearhigh displacement actuator with increased combination of force/displacementoutput. In the present work the analytical modeling and the subsequent analysis ofnon-linear actuators with enhanced characteristics in terms of displacement ispresented. The piezoelectric structure that is studied is a traditional bimorph structurewith two piezoelectric layers and an aluminum substrate. The main concept is toleverage non-linear mechanics, and more specifically snap-through buckling, so thatlarge displacements can be achieved with the transition of the structure from oneequilibrium position to another. During the development process the importanceof boundary conditions has been revealed and thus special attention has beenprovided to this issue. A modified analytical model was elaborated in order tocome up with a closed form solution including relaxed boundary conditions. Theexperimental verification of the analytical and numerical model is presented in part II.

TLM-based modeling and design exploiting space mapping

In this paper, we study the use of space-mapping (SM) techniques within the transmission-line matrix (TLM) method environment. Previous work on SM relies on an idealized coarse model in the design process of a computationally expensive fine model. For the first time, we examine the case when the coarse model is not capable of providing an ideal optimal response. We exploit a coarse-grid TLM solver with relaxed boundary conditions. Such a coarse model may be incapable of satisfying design specifications and traditional SM may fail. Our approach, which exploits implicit SM (ISM) and the novel output SM (OSM), overcomes this failure. Dielectric constant, an expedient preassigned parameter, is first calibrated to roughly align the coarse and fine TLM models. Our OSM scheme absorbs the remaining deviation between the implicitly mapped coarse-grid and fine-grid TLM responses. Because the TLM simulations are on a fixed grid, response interpolation is crucial. We also create a database system to avoid repeating simulations unnecessarily. Our optimization routine employs a trust region methodology. The TLM-based design of an inductive post, a single-resonator filter, and a six-section H-plane waveguide filter illustrate our approach. In a few iterations, our coarse-grid TLM surrogate, with approximate boundary conditions, achieves a good design of the fine-grid TLM model in spite of poor initial responses. Our results are verified with MEFiSTo simulations.

The Influence of Open versus Periodic Alongshore Boundaries on Circulation near Submarine Canyons

It is impractical to create gridded numerical models of coastal circulation with sufficient resolution around small topographic features, such as submarine canyons, and still have the alongshore boundaries placed beyond the decay distance of coastal trapped waves. Two solutions to this problem are to make the alongshore boundaries either open or periodic. Numerical simulations were performed with upwelling and downwelling winds to compare the effects of these different choices for boundary conditions. Several open boundary formulations were tried and three are discussed in detail. The offshore boundary was specified as “no gradient” for all variables with no serious effect. The “modified” Orlanski radiation condition is used for all variables at the alongshore boundaries, except the vertically integrated flow that has the strongest effect on the model solution. An alongshore pressure gradient, opposing the wind, develops in the model if the modified Orlanski radiation condition is applied to the barotropic flow, causing slower currents near the surface and deep undercurrents away from the shelf. The other cases, which combined either a radiation or a relaxation boundary condition with a local solution of the barotropic equations on the boundary, were at least initially similar to the periodic case but with slower alongshore flow. The initial impact of these differences on the circulation within the canyon was small. The models with the open boundaries were more stable (did not develop strong flow meanders) than the cases with periodic conditions as initial transients are not trapped, and amplified, within the domain. Thus, open cases, especially with the upwelling winds, could run for extended times.

Correction: Note on spectra from the relaxed boundary conditions [1]. By Z. K.-H. Chu

ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und MechanikVolume 82, Issue 7 p. 504-504 Correction Correction: Note on spectra from the relaxed boundary conditions [1]. By Z. K.-H. Chu First published: 12 June 2002 https://doi.org/10.1002/1521-4001(200207)82:7<504::AID-ZAMM1111504>3.0.CO;2-5AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat No abstract is available for this article. Volume82, Issue7July 2002Pages 504-504 RelatedInformation