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NewmarkImplicitSolver

This component belongs to the category of integration schemes or ODE Solver.

This scheme is an implicit time integrator for dynamic system using the Newmark scheme. To compute the new position or new velocity, the NewmarkImplicitSolver is based on the following equations:

\[ x_{t+h}=x_t+h v_t+\frac{h^2}{2}((1-2\beta)a_t+2\beta a_{t+h}) \]
\[ v_{t+h}=v_t+h((1-\gamma)a_t+\gamma a_{t+h}) \]

Applied to a mechanical system where \(\small Ma_t+(r_MM+r_KK)v_t+Kx_t=f_{ext}\), we need to solve the following system:

\[ \tiny Ma_{t+h}+(r_MM+r_KK)v_{t+h}+Kx_{t+h}=f_{ext} \]
\[ \tiny Ma_{t+h}+(r_MM+r_KK)(v_t+h((1-\gamma)a_t+\gamma a_{t+h}))+K(x_t+hv_t+\frac{h^2}{2}((1-2\beta)a_t+2\beta a_{t+h}))=f_{ext} \]
\[ \tiny (M+h\gamma(r_MM+r_KK)+h^2\beta K)a_{t+h}=f_{ext}-(r_MM+r_KK)(v_t+h(1-\gamma)a_t)-K(x_t+hv_t+\frac{h^2(1-2\beta)}{2}a_t) \]
\[ \tiny ((1+h\gamma r_M)M+(h^2\beta +h\gamma r_K)K)a_{t+h}=f_{ext}-(r_MM+r_KK)v_t-Kx_t-(r_MM+r_KK)(h(1-\gamma)a_t)-K(hv_t+\frac{h^2(1-2\beta)}{2}a_t) \]
\[ \tiny ((1+h\gamma r_M)M+(h^2\beta+h\gamma r_K)K)a_{t+h}=a_t-(r_MM+r_KK)(h(1-\gamma)a_t)-K(hv_t+\frac{h^2(1-2\beta)}{2}a_t) \]

Sequence diagram

Usage

At each simulation step and each Newton Raphson iteration, the NewmarkImplicitSolver requires:

  • a LinearSolver to solve the linear system
  • and a MechanicalObject to store the state vectors.