Dynamic Relaxation

Introduction

Dynamic relaxation is a method commonly used for form finding of shell structures. It was introduced by Alistair Day [1] and solves the nodal position $\mathbf{x}_a$ for each node $a$ in the structure in such a way that nodal equilibrium is fulfilled

$\mathbf{p}_a + \sum\limits_b \mathbf{f}_{ab} = \mathbf{0}$,

in which $\mathbf{p}_a$ is the sum of all the applied external loads and the support reaction if the node is restrained, and $\mathbf{f}_{ab}$ the internal force in member $ab$ ending in node $a$.

As long as the position is incorrect a out of balance or residual force appears at each node

$\mathbf{p}_a + \sum\limits_b \mathbf{f}_{ab} = \mathbf{r}_a$.

By introducing a small time step $\delta t$ and a fictitious mass $m_a$ at each node, we can trace the structure’s (fictitious) motion in time as it moves towards the equilibrium position.

To update the position $\mathbf{x}_a$ for node $a$, the following three operations are performed consecutively

$\ddot{\mathbf{x}}_a \gets \mathbf{r}_a/m_a$,
$\dot{\mathbf{x}}_a \gets \mu \cdot \dot{\mathbf{x}}_a + \delta t \cdot \ddot{\mathbf{x}}_a$,
$\mathbf{x}_a \gets \mathbf{x}_a + \delta t \cdot \dot{\mathbf{x}}_a$,

where $\ddot{\mathbf{x}}_a$ and $\dot{\mathbf{x}}_a$ are the acceleration and velocity of the node, respectively, and $\mu$ is a damping parameter introduced to ensure numerical stability.

Fictitious mass

The mass is fictitious and adjusted to adjusted to optimise the rate of convergence [2] and usually taken as

$m = \frac{\left(\delta t\right)^2}{2}S$,

where $S$ is the largest direct stiffness occuring at the node during the simulation. Further strategies on choosing the fictitious mass is discussed in [3], including using different masses in different directions, effectively turning the mass in to a vector $\mathbf{m}_a=\left(m_{a,x}, m_{a,y}, m_{a,z}\right)$.

Axial stiffness

If there only exists axial stiffness in the system, the internal force in member $ab$ is found from

$\mathbf{f}_{ab} = \frac{T_{ab}}{l_{ab}}\left(\mathbf{x}_b-\mathbf{x}_a\right)$,

where $T_{ab}$ and $l_{ab}$ are the tension in member $ab$ and the current length, respectively. $\left(\mathbf{x}_b-\mathbf{x}_a\right)/l_{ab}$ is a unit vector in the direction from node $a$ to node $b$.

It should be noted that so far we have discussed pure static equilibrium. No assumptions have been made regarding the material properties of the members which might be linear or non-linear elastic or be subject to creep or plastic deformation. The structure may be statically determinate or indeterminate or even a mechanism, provided that it is in equilibrium. The structure may have undergone a large deformation from some initial state.

In order to determine the form found geometry we need further information regarding the tensions $T_{ab}$ and their relationship with the current lengths $l_{ab}$. The simplest relationship is linear elastic,

$T_{ab} = k_{ab} \left(l_{ab}-l_{0,ab}\right)$,

in the case of a member with unstressed length $l_{0,ab}$. The constant $k_{ab} = (EA)_{ab}/l_{0,ab}$ in which $(EA)_{ab}$ is equal to the Young’s modulus times the cross-sectional area of the member. However, during form finding we can postulate any relationship between tension and length, including inextensible members whose length cannot change and members with a constant tension.

References

[1] A. S. Day. An introduction to dynamic relaxation. The Engineer, 219:218–221, 1965.
[2] M. R. Barnes, “Form finding and analysis of tension structures by dynamic relaxation,” International Journal of Space Structures, vol. 14, no. 2, pp. 89–104, 1999.
[3] M. Rezaiee-pajand, M. Kadkhodayan, J. Alamatian, and L. Zhang. A new method of fictitious viscous damping determination for the dynamic relaxation method. Computers & Structures, 89 (9–10):783–794, 2011.