On the uniqueness of singular solutions to boundary-initial value problems in linear elastodynamics

This investigation deals with certain generalizations of the classical uniqueness theorem for the second boundary-initial value problem in the linearized dynamical theory of not necessarily homogeneous nor isotropic elastic solids. First, the regularity assumptions underlying the foregoing theorem are relaxed by admitting stress fields with suitably restricted finite jump discontinuities. Such singularities are familiar from known solutions to dynamical elasticity problems involving discontinuous surface tractions or non-matching boundary and initial conditions. The proof of the appropriate uniqueness theorem given here rests on a generalization of the usual energy identity to the class of singular elastodynamic fields under consideration.

Following this extension of the conventional uniqueness theorem, we turn to a further relaxation of the customary smoothness hypotheses and allow the displacement field to be differentiable merely in a generalized sense, thereby admitting stress fields with square-integrable unbounded local singularities, such as those encountered in the presence of focusing of elastic waves. A statement of the traction problem applicable in these pathological circumstances necessitates the introduction of "weak solutions'' to the field equations that are accompanied by correspondingly weakened boundary and initial conditions. A uniqueness theorem pertaining to this weak formulation is then proved through an adaptation of an argument used by O. Ladyzhenskaya in connection with the first boundary-initial value problem for a second-order hyperbolic equation in a single dependent variable. Moreover, the second uniqueness theorem thus obtained contains, as a special case, a slight modification of the previously established uniqueness theorem covering solutions that exhibit only finite stress-discontinuities.

Evaluating boundary dependent errors in QM/MM simulations.

Hybrid quantum mechanics/molecular mechanics (QM/MM) simulations provide a powerful tool for studying chemical reactions, especially in complex biochemical systems. In most works to date, the quantum region is kept fixed throughout the simulation and is defined in an ad hoc way based on chemical intuition and available computational resources. The simulation errors associated with a given choice of the quantum region are, however, rarely assessed in a systematic manner. Here we study the dependence of two relevant quantities on the QM region size: the force error at the center of the QM region and the free energy of a proton transfer reaction. Taking lysozyme as our model system, we find that in an apolar region the average force error rapidly decreases with increasing QM region size. In contrast, the average force error at the polar active site is considerably higher, exhibits large oscillations and decreases more slowly, and may not fall below acceptable limits even for a quantum region radius of 9.0 A. Although computation of free energies could only be afforded until 6.0 A, results were found to change considerably within these limits. These errors demonstrate that the results of QM/MM calculations are heavily affected by the definition of the QM region (not only its size), and a convergence test is proposed to be a part of setting up QM/MM simulations.