Dynamic relaxation
Encyclopedia
Dynamic relaxation is a numerical method, which, among other things, can be used 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.
direction at the 
th node at time 
:
Where:
is the residual force
is the nodal mass
is the nodal acceleration
Note that fictitious modal 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.
2. Compute the geometry set and the applied load component:
3. Compute the residual:
4. Reset the residuals of constrained nodes to zero
5. Update velocity and coordinates:
6. Return to step 3 until the structure is in static equilibrium
There are two methods of damping:
The advantage of viscous damping is that it represents the reality of a cable with viscous properties. Moreover it is easy to realize because the speed is already computed.
The kinetic energy damping is an artificial damping which is not a real effect, but offers a drastic reduction in the number of iterations required to find a solution. However, there is a computational penalty in that the kinetic energy and peak location must be calculated, after which the geometry has to be updated to this position.
Tensile structure
A tensile structure is a construction of elements carrying only tension and no compression or bending. The term tensile should not be confused with tensegrity, which is a structural form with both tension and compression elements....
. 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
Soap film
Soap films are thin layers of liquid surrounded by air. For example, if two soap bubbles enters in contact, they merged and a thin film is created in between. Thus, foams are composed of a network of films connected by Plateau borders...
, which have the property of adjusting to find a "minimal surface
Minimal surface
In mathematics, a minimal surface is a surface with a mean curvature of zero.These include, but are not limited to, surfaces of minimum area subject to various constraints....
".
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
Mechanical equilibrium
A standard definition of static equilibrium is:This is a strict definition, and often the term "static equilibrium" is used in a more relaxed manner interchangeably with "mechanical equilibrium", as defined next....
position under the influence of loads. An iterative process is followed by simulating a pseudo-dynamic
Dynamics (mechanics)
In the field of physics, the study of the causes of motion and changes in motion is dynamics. In other words the study of forces and why objects are in motion. Dynamics includes the study of the effect of torques on motion...
process in time, with each iteration based on an update of the geometry.
Main equations use
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 accelerationNote that fictitious modal 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.
Iteration Steps
1. Set the initial kinetic energy and all nodal velocity components to zero:2. Compute the geometry set and the applied load component:
3. Compute the residual:
4. Reset the residuals of constrained nodes to zero
5. Update velocity and coordinates:
6. Return to step 3 until the structure is in static equilibrium
Mechanical equilibrium
A standard definition of static equilibrium is:This is a strict definition, and often the term "static equilibrium" is used in a more relaxed manner interchangeably with "mechanical equilibrium", as defined next....
Damping
It is possible to make dynamic relaxation more computationally efficient (reducing the number of iterations) by using damping.There are two methods of damping:
- Viscous damping, which assumes that connection between the nodes has a viscous force component.
- Kinetic energy damping, where the coordinates at peak kinetic energy are calculated (the equilibrium position), then updates the geometry to this position and resets the velocity to zero.
The advantage of viscous damping is that it represents the reality of a cable with viscous properties. Moreover it is easy to realize because the speed is already computed.
The kinetic energy damping is an artificial damping which is not a real effect, but offers a drastic reduction in the number of iterations required to find a solution. However, there is a computational penalty in that the kinetic energy and peak location must be calculated, after which the geometry has to be updated to this position.
Further reading
- W J LEWIS, TENSION STRUCTURES: Form and behaviour, London, Telford, 2003
- D S WAKEFIELD, Engineering analysis of tension structures: theory and practice, Bath, Tensys Limited, 1999
- H.A. BUCHHOLDT, An introduction to cable roof structures, 2nd ed, London, Telford, 1999