# Dynamic relaxation

Dynamic relaxation

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[1].

## Main equations use

Considering Newton's second law of motion (force is mass multiplied by acceleration) in the x direction at the ith node at time t:

$R_{ix}(t)=M_{i}A_{ix}(t)\frac{}{}$

Where:

R is the residual force
M is the nodal mass
A 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 V, the geometry X and the residuals can be obtained by performing a double numerical integration of the acceleration (here in central finite difference form[2]), :

$V_{ix}\left(t+ \frac {\Delta t} {2}\right) = V_{ix} \left(t- \frac {\Delta t} {2}\right) + \frac{\Delta t}{M_i}R_{ix}(t)$
$X_i(t+ \Delta t)=X_i(t)+\Delta t \times V_{ix} \left(t+ \frac {\Delta t} {2}\right)$

Where:

Δt 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:

$R_{ix}(t+ \Delta t)=P_{ix}(t+ \Delta t)+\sum \frac {T_m(t+ \Delta t)}{l_m(t+ \Delta t)} \times (X_j(t+ \Delta t)-X_i(t+ \Delta t))$

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 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:

$E_k(t=0)=0\frac{}{}$
$V_i(t=0)=0\frac{}{}$

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

It is possible to make dynamic relaxation more computationally efficient (reducing the number of iterations) by using damping[3]. 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.

• 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

## References

1. ^ W J LEWIS, TENSION STRUCTURES: Form and behaviour, London, Telford, 2003
2. ^ D S WAKEFIELD, Engineering analysis of tension structures: theory and practice, Bath, Tensys Limited, 1999
3. ^ W J LEWIS, TENSION STRUCTURES: Form and behaviour, London, Telford, 2003

Wikimedia Foundation. 2010.

### Look at other dictionaries:

• Dynamic mechanical analysis — Dynamic Mechanical Analyzer Acronym DMA Classification Thermal analysis Manufacturers Bose Electroforce Group, Mettler Toledo, Netzsch Instruments, PerkinElmer, TA Instruments, Triton Technology Ltd, Gabo Other techniques Related Isothermal… …   Wikipedia

• Dynamic nuclear polarisation — Dynamic nuclear polarization (DNP)[1] [2] results from transferring spin polarization from electrons to nuclei, thereby aligning the nuclear spins to the extent that electron spins are aligned. Note that the alignment of electron spins at a given …   Wikipedia

• Dynamic mechanical analysis — Viscoanalyseur Pour les articles homonymes, voir AMD et DMA. Un viscoanalyseur ou analyseur mécanique dynamique (AMD) fait partie de la famille des appareils de DMA ou DMTA (en anglais Dynamic Mechanical Thermal Analysis). Cet instrument… …   Wikipédia en Français

• Gas dynamic laser — (GDL) is laser based on differences in relaxation velocities of molecular vibrational states.The laser medium gas has such properties that an energetically lower vibrational state relaxes faster than a higher vibrational state,thus a population… …   Wikipedia

• List of numerical analysis topics — This is a list of numerical analysis topics, by Wikipedia page. Contents 1 General 2 Error 3 Elementary and special functions 4 Numerical linear algebra …   Wikipedia

• Mathematical optimization — For other uses, see Optimization (disambiguation). The maximum of a paraboloid (red dot) In mathematics, computational science, or management science, mathematical optimization (alternatively, optimization or mathematical programming) refers to… …   Wikipedia

• Bates method — Alternative medicine / fringe therapies William Bates and his assistant. Claims The need for eyeglasses can be reversed by relaxation. Related fields Ophthalmo …   Wikipedia

• Sophrology — was created by Dr. Alfonso Caycedo in the 1960s.It is a branch of neurological medicine that studies the human consciousness and its positive modifications through practice of specific methodology, inspired in great far eastern yogic sources.*SOS …   Wikipedia

• The Art of Seeing — is a 1942 book by Aldous Huxley, which contains an explanation and discussion of the Bates Method for better eyesight.Huxley’s own sightIn the preface to the book, Huxley describes how, at the age of sixteen, he had a violent attack of keratitis… …   Wikipedia

• List of mathematics articles (D) — NOTOC D D distribution D module D D Agostino s K squared test D Alembert Euler condition D Alembert operator D Alembert s formula D Alembert s paradox D Alembert s principle Dagger category Dagger compact category Dagger symmetric monoidal… …   Wikipedia