diabayes.forward_models.inertial_springblock#

diabayes.forward_models.inertial_springblock(t: Float, v: Float, v_partials: Variables, variables: Variables, dstate: Float[Array, '...'], constants: InertialSpringBlockConstants) → Float[source]#

An inertial spring-block loading formulation

\[\frac{\mathrm{d} \mu}{\mathrm{d} t} = \left[ \frac{\partial v}{\partial \mu} \right]^{-1} \left( \frac{1}{M} \left[ k \left( v_{lp} t - x \right) - \mu \right] - \frac{\partial v}{\partial \theta} \frac{\mathrm{d} \theta}{\mathrm{d} t} - \dots \right)\]

The acceleration term in the classical inertial spring-block formulation is decomposed into its partial derivatives, avoiding the need for solving a second-order ODE. These partial derivatives (v_partials) are computed using the JAX autodiff framework.

Notes

This formulation is rather stiff, and for certain parameter values could lead to extremely small time steps necessary to maintain numerical accuracy. It is recommended to use a conventional (non-inenrtial) springblock formulation for basic velocity-steps and slide-hold-slide simuilations. Inertia is only really needed for stick-slip simulations.

Parameters:

  • t (Float) – Current value of time [s]

  • v (Float) – Instantaneous fault slip rate [m/s]

  • v_partials (Variables) – The partial derivatives of slip rate to the relevant variables (friction and state variables)

  • variables (Variables) – The instantaneous values of the variables: friction (mu), slip (slip), and other state variables (not used)

  • dstate (Array) – The time derivatives of the state variables. The radiation term is v_partials @ dstate (excluding mu)

  • constants (InertialSpringBlockConstants) – The spring-block constants containing the mass term M, the stiffness k, and the load-point velocity v_lp

Returns:

dmu – The rate of change of the friction coefficient [1/s]

Return type:

Float

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