Dynamic Relaxation

Given that our main interest is in long-term tectonic modeling, high-frequency vibrations are not relevant to the overall deformation pattern. A strong and efficient damping is necessary to achieve quasi-static solutions of the dynamic equation. Complementarily, force amplification might be needed to accelerate the transient process to achieve equilibrium. Therefore, we either damp or amplify the total net force in the discretized nodal momentum equation,

$$m_a{\mathbf{a}}_a={\mathbf{f}}_a,$$

according to the direction of velocity (Cundall, 1989):

$$m{a}_i=({\mathbf{f}}_{damped}{)}_i=f_i-0.8\text{sgn}(u_i)|f_i|,$$

where subscript $i$ denotes the $i$-th component of a vector and $\text{sgn}$ denotes the signum function. The motivation for the choice of damping/amplification is based on the simple observation that in an under-damped oscillator, the direction of force is always opposite to the velocity direction, while in an over-damped system, the direction of the force is parallel to the velocity direction. We found that this choice of damping/amplification accomplishes the design goals satisfactorily (i.e., robustly and economically).