dense_generalized_alpha_integrator Derived Type

private subroutine initialize_dense_integrator(this, mass, damping, stiffness, rho_infinity)

Copies dense matrices and computes generalized-alpha coefficients once.

Arguments

private subroutine integrator_solve(this, forces, dt, displacement, velocity, acceleration)

Advances through a force history with one column per time point. The number of steps is size(forces, 2) - 1; only the final state is returned.

Arguments

private subroutine dense_integrator_step(this, force_current, force_next, dt, displacement, velocity, acceleration)

Advances a linear structural system by one generalized-alpha step. Let the input state be indexed by n and the output state by n+1. The method solves the weighted equilibrium

$$M[(1-\alpha_m)a_{n+1}+\alpha_m a_n] + C[(1-\alpha_f)v_{n+1}+\alpha_f v_n] + K[(1-\alpha_f)u_{n+1}+\alpha_f u_n] = (1-\alpha_f)f_{n+1}+\alpha_f f_n.$$

The high-frequency spectral radius $\rho_\infty$ determines

$$\alpha_m=\frac{2\rho_\infty-1}{\rho_\infty+1},\qquad \alpha_f=\frac{\rho_\infty}{\rho_\infty+1},\qquad \gamma=\tfrac12+\alpha_f-\alpha_m,\qquad \beta=\tfrac14(1+\alpha_f-\alpha_m)^2.$$

With $u^*=u_n+h v_n+h^2(\tfrac12-\beta)a_n$ and $v^*=v_n+h(1-\gamma)a_n$, solve $A a_{n+1}=b$, where

$$A=(1-\alpha_m)M+(1-\alpha_f)\gamma h C +(1-\alpha_f)\beta h^2 K,$$

$$b=(1-\alpha_f)f_{n+1}+\alpha_f f_n-\alpha_m M a_n -C[(1-\alpha_f)v^*+\alpha_f v_n] -K[(1-\alpha_f)u^*+\alpha_f u_n].$$

Finally, $u_{n+1}=u^*+\beta h^2 a_{n+1}$ and $v_{n+1}=v^*+\gamma h a_{n+1}$. At $\rho_\infty=1$, this is the average-acceleration (trapezoidal) Newmark scheme.

The LU factorization of A is reused whenever the time step is unchanged.

Arguments