User:Deuxoursendormis/Sandbox

Dynamic Relaxation
The dynamic relaxation is a computation modeling, which can be used for the form-finding of cable and fabric structures. The dynamic relaxation method is base on a discretized continuum in which the mass is assumed to be lumped at given nodes. The system oscillates about the equilibrium position under the influence of loads. The iterative process is achieved by simulating a pseudo-dynamic process in time, this iteration are base on an update of the geometry time by time.

Main equations use
Using the relation of the second Newton's laws by considering the node $$i$$, at the time $$t$$, in the $$x$$ direction:
 * $$R_{ix}(t)=M_{i}A_{ix}(t)\frac{}{}$$

Where:
 * $$R$$ is the residual force
 * $$M$$ is the mass
 * $$A$$ is the acceleration

By realizing a double numerical integration of the acceleration (here by central finite difference form ), a relation between the speed $$V$$, the geometry $$X$$ and the residuals is obtained:
 * $$V_{ix}(t+ \frac {\Delta t} {2})=V_{ix}(t- \frac {\Delta t} {2})+\frac{\Delta t}{M_i}R_{ix}(t)$$
 * $$X_i(t+ \Delta t)=X_i(t- \Delta t)+\Delta t \times V_{ix}(t+ \frac {\Delta t} {2})$$

Where:
 * $$\Delta t$$ is the time interval between two updates.

Using a sum of the forces at the node, permit to obtain the relation between the residuals and the geometry:
 * $$R_{ix}(t+ \Delta t)=P_{ix}(t+ \Delta t)+\sum \frac {T_m(t+ \Delta t)}{l_m(t+ \Delta t)} \times (X_j(t+ \Delta t)-X_i(t+ \Delta t))$$

Where:
 * $$P$$ is the applied load component
 * $$T$$ is the tension in link $$m$$ between nodes $$i$$ and $$j$$
 * $$l$$ is the length of the link.

The sum is realizing on all the link of the node. By repeating the use of the relation between the residuals and the geometry and then the relation between the geometry and the residual, the pseudo-dynamic process is simulated.

Step of the iteration
1. Define the initial kinetic energy and all nodal velocity components to zero:
 * $$E_k(t=0)=0\frac{}{}$$
 * $$V_i(t=0)=0\frac{}{}$$

2. Compute the geometry set and the applied load component:
 * $$X_i(t=0)\frac{}{}$$
 * $$P_i(t=0)\frac{}{}$$

3. Compute the residual:
 * $$T_m(t)\frac{}{}$$
 * $$R_i(t)\frac{}{}$$

4. Reset the residuals of constrained nodes to zero 5. Update velocity and coordinates:
 * $$V_i(t+ \frac {\Delta t}{2})\frac{}{}$$
 * $$X_i(t+\Delta t)\frac{}{}$$

6. Return to step 3 until the structure is in static equilibrium

Damping
In the dynamic relaxation method, there is a possibility to use damping to enhance the method by reducing the number of iteration. There are two method of damping: The viscous damping possesses the advantage to stay close to the reality of a cable which effectively possesses a viscous comportment. Moreover it is easy to realize because the speed is already computed. The kinetic energy damping is an artificial damping which differs from the reality but offer a drastic reduction of the number of iteration. However it needs to compute the kinetic energy and to detect peak of this energy, after detecting this energy peak the geometry has to be updated to this position.
 * The viscous damping, which assume that cable possess a viscous comportment.
 * The kinetic energy damping, this consist in at each time a kinetic energy peak (which means a position of equilibrium) is detected, the geometry is update to this position and the speed is reset.