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Concept
Dynamic relaxation
Summary
Dynamic relaxation is a numerical method, which, among other things, can be used to do "form-finding" for cable and fabric structures. The aim is to find a geometry where all forces are in equilibrium. In the past this was done by direct modelling, using hanging chains and weights (see Gaudi), or by using soap films, which have the property of adjusting to find a "minimal surface".
The dynamic relaxation method is based on discretizing the continuum under consideration by lumping the mass at nodes and defining the relationship between nodes in terms of stiffness (see also the finite element method). The system oscillates about the equilibrium position under the influence of loads. An iterative process is followed by simulating a pseudo-dynamic process in time, with each iteration based on an update of the geometry, similar to Leapfrog integration and related to Velocity Verlet integration.
Considering Newton's second law of motion (force is mass multiplied by acceleration) in the direction at the th node at time :
Where:
is the residual force
is the nodal mass
is the nodal acceleration
Note that fictitious nodal masses may be chosen to speed up the process of form-finding.
The relationship between the speed , the geometry and the residuals can be obtained by performing a double numerical integration of the acceleration (here in central finite difference form), :
Where:
is the time interval between two updates.
By the principle of equilibrium of forces, the relationship between the residuals and the geometry can be obtained:
where:
is the applied load component
is the tension in link between nodes and
is the length of the link.
The sum must cover the forces in all the connections between the node and other nodes.
By repeating the use of the relationship between the residuals and the geometry, and the relationship between the geometry and the residual, the pseudo-dynamic process is simulated.
- Set the initial kinetic energy and all nodal velocity components to zero:
- Compute the geometry set and the applied load component:
Official source
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