Boundary conditions for molecular dynamics simulation of crystalline solids are considered with the objective of eliminating the reflection of phonons. A variational formalism is presented to construct boundary conditions that minimize total phonon reflection. Local boundary conditions that involve a few neighbors of the boundary atoms and limited number of time steps are found using the variational formalism. Their effects are studied and compared with other boundary conditions such as truncated exact boundary conditions or by appending border atoms where artificial damping forces are applied. In general it is found that, with the same cost or complexity, the variational boundary conditions perform much better than the truncated exact boundary conditions or by appending border atoms with empirical damping profiles. Practical issues of implementation are discussed for real crystals. Application to brittle fracture dynamics is illustrated.