RATTLE method

The constrained equations of motion in cartesian eoordinates can be solved by the SHAKE or (the essentially equivalent) RATTLE method (see [8]) which requires the solution of a non-linear system of equations in the Lagrange multiplier funetion A. The equivalent formulation in local coordinates ean still be integrated by using the explicit Verlet method. 

Also used for hardness determination are impact abrasion and rattle methods. In impact abrasion methods (impact abrasion hardness), three abrasive techniques are used a freely falling abrasive grain flow—the Scott tower method (Fig. 4.4.3a and b), pressure abrasion—the Macken-sen-Zeiss blower method (Fig. 4.4.7), and ultrasonic drilling—the Sonic Mill method (Fig. 4.4.18). 

Rattling method. In many countries, the United States in particular, the previously mentioned rattler test is used for measurement of abrasive wear of mouldings. In the United States, it is widely used for testing paving stone. By this method, data on abrasion resistance and also edge-preservation resistance (Fig. 4.4.19) are obtained. The rattling is carried in mills... 

Considered as a basic engineering parameter of rock materials, next to crushing and bending strength, is surface resistance to destructive mechanical action, defined as hardness or abrasiveness. Bohme s disc is generally recommended for rock materials, specifically for stone elements exposed to abrasive action and for rocks to be processed into concrete and terrazzo aggregate, whereas Deval s rattle method is recommended for railroad aggregate. 

To integrate these equations of motion [173,174] one can use a velocity verlet algorithm [117] propagating similarly the nuclei positions and the electronic degrees of freedom Cj(G). The orthonormality conditions can be taken into account by the SHAKE and RATTLE method [173, 175, 176]. Constant ionic temperature can also be imposed through the use of thermostats [169,170,173,177]. 

A method that automatically preserves both the constraint and hidden constraint is the RATTLE method [12] which consists of the following steps ... 

Given the above discussion and derivation, the proof of symplecticness of the RATTLE method (4.30)-(4.34) is now a formality. For details refer to [229]. 

The concept of a symplectic method is easily extended to systems subject to holonomic constraints [22]. For example the RATTLE discretization is found to be a symplectic discretization. Since SHAKE is algebraically equiva lent to RATTLE, it, too, has the long-term stability of a symplectic method. 

Both methods are time-reversible. For A = 0, they reduce to SHAKE and RATTLE. In contrast to SHAKE and RATTLE, the time step is not restricted by the largest eigenvalue of A. 

Our discussion so far has considered the use of SHAKE with the Verlet algorithm Versions have also been derived for other integration schemes, such as the leap-froj algorithm, the predictor-corrector methods and the velocity Verlet algorithm. In the cast of the velocity Verlet algorithm, the method has been named RATTLE [Anderson 1983]... 

I can rattle on forever about all the tricks and methods one could tiy but I guess you pretty much get the picture by now. So let s finish of with a few quick tips to remember. 

Evans BE, Rattle KE, Bock MG, DiPardo RM, Freidinger RM, Whitter WL, Lundell GF, Yeber DF, Anderson PS, Chang RS, et al. Methods for drug discovery development of potent, selective, orally effective CCK antagonists. J Med Chem 1988 31 2235— 2246. 

The first authentic article of aluminium was a rattle for the infant destined later to become Emperor Napoleon hi. In 1854, an aluminium medal was struck and presented to him, and he both authorised and financed experiments to manufacture the metal on a larger scale. He had visions of supplying his troops with helmets and breastplates of aluminium, but its price of over 100 per lb. rendered the proposition hopeless. After the invention of the Hall-H roult electrolytic process the price fell to about 85 per ton in 1914. The world consumption in 1938 was 550,000 tons, and by 1941 it was close upon one million tons. It was stated in October 1939, that bullet-proof duralumin armour was among Germany s new methods of warfare on the Western Front. 

Not until 1855 did a French chemist, Henri Etienne Sainte-Claire Deville (1818-81), work out an adequate method for preparing reasonably pure aluminum in moderate quantities. Even then, it was far more expensive than steel, so that it was used for ostentation, as, for instance, for the rattle of Napoleon Ill s infant son, or the cap at the top of the Washington Monument. 

Higher Order Symplectic Methods SHAKE and RATTLE... 

The term SHAKE normally refers to both the discretization mentioned above and the component-wise solution of the nonlinear equations (these methods were suggested together in the paper [322]). Using the traditional SHAKE approach we solve the Eqs. (4.27)-(4.29) using the iterative procedure of Sect. 4.3.3. If RATTLE is used for discretization, the iterative method of Sect. 4.3.3 can again be used to solve (4.30)-(4.32), followed by a subsequent solve of Eqs. (4.33)-(4.34). 

While SHAKE/RATTLE is a robust procedure the sequential, iterative character of nonlinear solvers needed for its implementation can lead inefficiencies when it is implemented on a parallel computer. Therefore, a number of the alternative methods avoid iteration. We briefly summarize a few of the popular alternative schemes here. 

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