To reduce this problem we designed line-focused probes. The aperture of the probes was 30 mm X 10 mm, the focal distance 45 mm, the center frequency 550 kHz. To perfonn basic experiments it proved advantageous to work in a first step in a both-sided probe arrangement. In this way misinterpretations caused by specular reflections are excluded. The experimental arrangement is shown is Fig. 8. The specimen, a 2 mm thick CFC plate was introduced between the probes. The axial and lateral distance as well as the alignment angle could be varied. As an example. Fig. 8 shows on the right side two A-scans corresponding to two different Lamb wave modes generated at an angle of incidence of 0° and about 30°, resp.  [c.845]

Abstract. Systems with multiple time scales, and with forces which can be subdivided into long and short range components are frequently encountered in computational chemistry. In recent years, new, powerful and efficient methods have been developed to reduce the computational overhead in treating these problems in molecular dynamics simulations. Numerical reversible integrators for dealing with these problems called r-RESPA (Reversible Reference System Propagator Algorithms) are reviewed in this article. r-RESPA leads to considerable speedups in generating moleculcir dynamics trajectories with no loss of accuracy. When combined with the Hybrid Monte Carlo (HMC) method and used in the Jump-Walking and the Smart-Walking algorithms, r-RESPA is very usefnl for the enhanced sampling of rough energy landscapes in biomolecules.  [c.297]

Reversible Reference System Propagator Algorithms (r-RESPA) 299  [c.299]

In conventional MD the forces are recomputed after each time step. The force calculations account for as much as 95% of the CPU time in an MD simulation. In systems with long range forces, the force computation becomes the major bottleneck to the computation. When using the direct pairwise evaluation, the computational effort required to compute the long-range Coulomb forces on N interacting particles is of order N. A variety of strategies, such as the fast multipole method and the particle-particle-mesh Ewald method, have been introduced to reduce the computational effort in calculating the forces. Building on earlier reference system propagator algorithm (RESPA) based integrators,[13, 14, 15, 16] a class of new reversible and symplectic integrators have been invented that greatly reduces the intrinsic multiple time scale problem. By using a reversible Trotter factorization of the classical propagator[17] one can generate simple, accurate, reversible and symplectic integrators that allow one to integrate the fast motions using small time steps and the slow degrees of freedom using large time steps. [17] This approach allows one to split the propagator up into a fast part, due to the high frequency vibrations, and slow parts, due to short range, intermediate range, and long range forces, in a variety of ways. These new integrators, called reversible reference system algorithms (r-RESPA), require for the treatment of all-atom force fields no more CPU time than constrained dynamics and often lead to even larger improvements in speed. Although r-RESPA is quite simple to implement, there are many ways to factorize the propagator. A recent paper shows how to avoid bad strategies. [18] Applications of these methods to Car-Parrinello ab initio molecular dynamics has resulted in speedups by a factor of approximately five in semiconductor materials. [19, 20, 21] There has been significant progress in recent years to apply these methods to systems of biological relevance. [22, 23, 24, 25, 26]  [c.299]

Reversible Reference System Propagator Algorithms (r-RESPA) 301  [c.301]

Reversible Reference System Propagator Algorithms (r-RESPA) 303  [c.303]

The aforementioned factorizations of the classical propagator can be used to generate efficient reversible and symplectic integrators for systems with long and short range forces and for systems in which the degrees of freedom ( an be subdivided into fast and slow subsets. All of the methods described below are called Reference System Propagator Algorithms (RESPA) a name that we gave to our initial attempts to use an underlying reference system propagator for the motion. This early effort resulted in non-reversible integrators. [13, 14, 15, 16] If the Liouville operator of the system is decomposed into a reference system part, iL ef, and a correction part , i6L,  [c.303]

Reversible Reference System Propagator Algorithms (r-RESPA) 305  [c.305]

This procedure is very cost efficient when the fast (or light) particles are the dilute component because then one only has to update the forces on the heavy particles (the expensive part of the computation) every large time step in.stead of every small time step as would be the case in the straightforward application of the Verlet integrator. For example when applied to a system containing 64 particles of mass 1 dissolved in 800 solvent atoms of mass 100, the CPU time for the full simulation took only slightly longer than it would if the complete system was made up of heavy particles. [14] In contrast, application of the usual Verlet integrator using the small time step required for the light particles but evaluating all the forces after each one of these small time steps required approximately ten times the CPU time used in the RESPA integrator. The same accuracy was achieved in these two different treatments.  [c.305]

Another important application of this strategy was to the vibrational relaxation of a stiff diatomic molecule dissolved in a Lennard-Jones solvent. As is typical of such problems, the frequency of the oscillator can be an order of magnitude or more larger than the typical frequencies found in the spectral density of the solvent. Thus very small time steps are required to to integrate the equations of motion, but because there are very few accepting solvent modes at the frequency of the oscillator, its vibrational relaxation time will be very long, largely occurring by a multiphonon mechanism. In the past it was not practicable to simulate these processes directly. Using a form of r-RESPA modified for the specific case of an oscillator dissolved in a slow solvent, we have been able to reduce the CPU time required for these calculations by factors of ten in many cases making possible the direct simulation of. such energy transfer problem.s. [34] When this strategy has been applied to the calculation of the IR, and Raman spectrum of crystalline buckminster-  [c.305]

The switching function s(a ) was taken to be a sigmoidal function (usually a cubic spline) whose inflection point (switching point) and skin-depth can be optimized. The short range force Fs(x) = s(x)F x) deflnes the time step to be used in a molecular dynamics calculation. In the velocity Verlet integrator one must compute the full force after each time step. If only the short range force were present, the CPU cost would be small because each particle would only interact with its nearest neighbors. It is the long range force Fi(x) = (1 — s x))F x) which is costly to calculate. We introduced this strategy into the r-RESPA propagator factorization, [17] and as with the non-reversible RESPA, we showed that this can significantly reduce the CPU eost of the simulation.  [c.306]

Reversible Reference System Propagator Algorithms (r-RESPA) 307 Thus the propagator in Eq. (27) produces the following dynamics algorithm  [c.307]

The Applications of RESPA to Proteins and Chemical Systems  [c.308]

Reversible Reference System Propagator Algorithms (r-RESPA) 309  [c.309]

Reversible Reference System Propagator Algorithms (r-RESPA) 311  [c.311]

The development of these systems first led to the of differential rcsp. multi-differential probes. These probes offer the highest possible sensitivity towards gradual influences, but  [c.307]

With the piezo-based air-bome probes, air coupled non-destructive testing by generation and detection of Lamb waves has become feasable [8,9]. However, because of the large differences of the wave velocities of waves propagating in air and Lamb waves, resp., the efficiency of Lamb wave excitation is critically dependent on the angle of incidence. It was shown both theoretically and experimentally, that the echo amplitude is very sensitive to the alignment of the transducers [8] using probes with plane apertures, misalignments of 0.6° reduce the amplitude by about 50 %.  [c.845]

Miehelsen H A, Rettner C T and Auerbaeh D J 1993 The adsorption of hydrogen at eopper surfaees A model system for the study of aetivated adsorption Surface Reacf/onsed R J Madix (Berlin Springer) p 123  [c.918]

Figure Bl.15.9. The ENDOR spectrum of the perinaphthenyl radical in mineral oil taken at room temperature. Iw" j/fi = 1M MU/, jn Figure Bl.15.9. The ENDOR spectrum of the perinaphthenyl radical in mineral oil taken at room temperature. Iw" j/fi = 1M MU/, jn<t kfU /ti = 5,12 MU/,are the hyperfme coupling constants for the protons in the position O sin l Oi rcspeuiivi-ly, J l = 0,631 niT iinJ n /(, 4fi J=0 l83 mT.
Lemma 3. Let p M be a probability measure and let X and Y be disjoint sets which are dx- resp. dy-almost invariant with respect to p. Moreover suppose that f (X) n Y = 0 and f Y) n A = 0. Then X UY is 6xuy-almost invariant with respect to p where  [c.106]

The invariant measure corresponding to Aj = 1 has already been shown in Fig. 6. Next, we discuss the information provided by the eigenmeasure U2 corresponding to A2. The box coverings in the two parts of Fig. 7 approximate two sets Bi and B2, where the discrete density of 1 2 is positive resp. negative. We observe, that for 7 > 4.5 in (15) the energy E = 4.5 of the system would not be sufficient to move from Bi to B2 or vice versa. That is, in this case Bi and B2 would be invariant sets. Thus, we are exactly in the situation illustrated in our Gedankenexperiment in Section 3.1.  [c.112]

The third eigenmeasure 1 3 corresponding to A3 provides information about three additional almost invariant sets on the left hand side in Fig. 8 we have the set corresponding to the oscillation C D, whereas on the fight hand side the two almost invariant sets around the equilibria A and B are identified. Again the boxes shown in the two parts of Fig. 8 approximate two sets where the diserete density of 1/3 is positive resp. negative. In this case we can use Proposition 2 and the fact that A and B are symmetrically lelated to conclude that for all these almost invariant sets 5 > A3 = 0.9891.  [c.113]

Finally, the information on the remaining almost invariant sets in the neighborhood of the equilibria C and D ean be extracted using the eigenvalue A4 with the eigenmeasure P4 (see Fig. 9). In the two parts of Fig. 9 we show again the boxes, whieh approximate two sets, where the diserete density of 1/4 is positive resp. negative. Let us denote by Y the union of the boxes around equilibrium B in the first part of the figure and by X the boxes around D. (We ignore the isolated box in the left lower corner, which we regard as a numerical artifact.) We now use Lemma 3 to derive a lower bound for 5x-Numerically we obtain the values 4(W) = 0.3492 and v4 Y) = 0.1508. Note that i 4(X U y) = 0.5 and A4 - - 1 = 2Sxuy (using again the symmetry and Corollary 4) whieh leads to the estimate  [c.113]

In this context the velocity Verlet integrator is equivalent to taking the reference system to be the dynamical system with all of the forces turned off that is, the ideal gas system. In some cases the reference system can be solved analytically, and we refer to these methods as the Numerical Analytical Propagator Algorithm (NAPA). The development of symplectic, reversible RESPA (r-RESPA) integration methods grew out of our earlier attempts to devise multiple time scale integrators based on the generation of the dynamics of a reference system and, in principle, exact correction to it.[13, 14, 15, 16] The latter, being non-reversible, guided us in the direction of analyzing the structure of the classical propagator and the use of the symmetric Trotter factorization. In fact in the development of r-RESPA and r-NAPA we have adopted many of many of the strategies used in our earlier non-reversible RESPA (nr-RESPA).[17] All of these r-RESPA integrators are also symplectic. First we tr( at the problem where there are fast and slow degrees of freedom (or light and heavy particles). Then we treat the case where the forces can be subdivided into short and long range components. Finally, we show how the long and short range force factorizations can be combined with the and slow factorization yielding a speedup which is approximately the product of the sp( edups achieved when these factorizations are used separately. There are many variations on the theme introduced here.  [c.303]

In many cases the dynamical system consists of fast degrees of freedom, labeled x, and slow degrees of freedom, labeled y. An example is that of a fluid containing polyatomic molecules. The internal vibrations of the molecules are often very fast compared to their translational and orientational motions. Although this and other systems, like proteins, have already been treated using RESPA,[17, 34, 22, 23, 24, 25, 26] another example, and the one we focus on here, is that of a system of very light particles (of mass m) dissolved in a bath of very heavy particles (mass M).[14] The positions of the heavy particles are denoted y and the positions of the light particles rire denoted by X. In this case the total Liouvillian of the system is  [c.304]

Another immediate application of r-RESPA is to the case when the force can be subdivided into a short range part and a long range part. One way for effectuating this break up is to introduce a switching function, s x) that is unity at short inter-particle separations and 0 at large inter-particle separations. We introduced this strategy in our earlier non-reversible RESPA paper [15] where we expressed the total force as.  [c.306]

It is worth calling to attention one difference between the force subdi-vi.sion used in r-RESPA[17] and the one used in the original non-reversible RESPA.[15] In the non-reversible RESPA paper we included the value of the long range force at the beginning of the time interval into the reference system equation of motion which was then integrated for n small time steps. We then solved the correction equation involving the difference between the true i orce and the reference system force for one large time step. This was shown to lead to a more stable integration scheme with much smaller long time drift than when the long range force was not introduced into the reference set of equations. Unfortunately, in the r-RESPA factorization there is no way to introduce the long range force at the beginning of the interval into the reference system propagator because that would remove reversibility. Strategies are being developed to implement such effects in new reversible integrators [37].  [c.307]

The preceding breakup for light and heavy particles can be combined with breaking the forces up into short and long range forces in r-RESPA[17] in a similar manner to what was done in non-reversible RESPA. [16] We can then further factorize the three propagators appearing in Eq. (22) by using the fact orization used to generate the velocity Verlet integrator with the forces  [c.307]

See pages that mention the term RECP : [c.206]    [c.192]    [c.289]    [c.309]    [c.348]    [c.412]    [c.531]    [c.531]    [c.531]    [c.794]    [c.840]    [c.332]    [c.663]    [c.1144]    [c.2762]    [c.735]    [c.753]    [c.6]    [c.6]    [c.304]    [c.306]    [c.310]    [c.310]   
Computational chemistry (2001) -- [ c.84 , c.262 , c.367 ]