Boundary Value Formulation in Length

The SDEL algorithm allows the computation of atomically detailed trajectories connecting two known conformations of the molecule over long time scales. In contrast to normal and MTS molecular dynamics algorithms, step sizes can be increased easily by two or three orders of magnitude without 

The Onsager-Machlup action methodology has a critical disadvantage the total time of the trajectory is needed in advance. Also, low-resolution trajectories do not approach a physical limit when the step size increases, in contrast to SDEL as will be shown below. 

Like the Onsager-Machlup action method, the SDEL algorithm is based on the classical action. However, in this case the starting point is the action S parameterized according to the length of the trajectory  

Equation [34] generates a path in which the force is minimized in all directions except the tangential direction of the path. This is one of the definitions of the minimum energy path (MEP). ° Equation [34] suggests that SDEL provides a physically meaningful trajectory even at low resolution (large step sizes). 

The following steps are used to obtain an approximate trajectory using 

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The two equations of motion we discussed so far can be found in classical mechanics text books (the differential equations of motion as a function of the arch-length can be found in [5]). Amusingly, the usual derivation of the initial value equations starts from boundary value formulation while a numerical solution by the initial value approach is much more common. Shouldn t we try to solve the boundary value formulation first As discussed below the numerical solution for the boundary value representation is significantly more expensive, which explains the general preference to initial value solvers. Nevertheless, there is a subset of problems for which the boundary value formulation is more appropriate. For example boundary value formulation is likely to be efficient when we probe paths connecting two known end points. 

The model is formulated in terms of an integral equation which is solved with the condition at the boundary of the open crack and the bridging zone (.x — 0) that 6(0) = 5C. It is interesting to note that the structure of the rate-dependent problem is such that, aside from the material parameters, the solution is completely determined for a given crack velocity. For a given velocity, the value of the applied stress intensity factor, K, and the length of the cohesive zone, L, that maintains this condition is determined. Selected results are presented below. 

Formulation. We consider the flow of an incompressible liquid from (or into) a single straight-line fracture of length 2c, centered in a circular reservoir of radius R c, as shown in Figure 2-la. The pressure P(X,Y) assumed along the fracture -c < X < -tc, Y = 0 is the variable function Pref pt(X/c), where Pref is a reference level and pf is dimensionless. The pressure at the farfield boundary is a constant Pr. For a uniform isotropic medium, P(X,Y) satisfies the Dirichlet boundary value problem... 

Figure 12.30 compares the solution errors for the time of t = 10.0 which is approaching the steady state. In this case the errors are more uniformly distributed over the length of the line as one would expect for a time independent boundary value problem. Again one sees a slight advantage of using the second order formulation as opposed to the first order equation formulation. Also the comparison of the h-2h cases more clearly illustrates the expected factor of 4 difference in... 

Problem (3.29) with the formulated boundary conditions possesses a unique solution over the length of the entrance flow region 0 < x < Lx, but the value Lx, the unknown length of the entrance region, is to be chosen at a distance, where no further transformation of the flow field takes place. Lx can be easily adjusted in the course of the numerical performance. 

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