Adaptive formfinding method for formfixed spatial network structures
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Abstract
An effective formfinding method for formfixed spatial network structures is presented in this paper. The adaptive formfinding method is introduced along with the example of designing an ellipsoidal network dome with bar length variations being as small as possible. A typical spherical geodesic network is selected as an initial state, having bar lengths in a limit group number. Next, this network is transformed into the ellipsoidal shape as desired by applying compressions on bars according to the bar length variations caused by transformation. Afterwards, the dynamic relaxation method is employed to explicitly integrate the node positions by applying residual forces. During the formfinding process, the boundary condition of constraining nodes on the ellipsoid surface is innovatively considered as reactions on the normal direction of the surface at node positions, which are balanced with the components of the nodal forces in a reverse direction induced by compressions on bars. The node positions are also corrected according to the fixedform condition in each explicit iteration step. In the serial results of time history, the optimal solution is found from a time history of states by properly choosing convergence criteria, and the presented formfinding procedure is proved to be applicable for formfixed problems.
Keywords
Formfinding Spatial network structure Formfixed Structural optimization Dynamic relaxation Explicit integrationIntroduction

Stiffness matrix methods are based on using the standard elastic and geometric stiffness matrices that were adapted from structural analysis. These methods account for the material properties in computation, which may lead to difficulty in operations of matrices and control of (stable) convergence.

Geometric stiffness methods are material independent, with only a geometric stiffness. However, these methods are applied in their linear form and produce results that are not constructionally practicable (Barnes 1977); thus, they can serve only as a preliminary result.

Force density methods. Because the ratio of force to length is a central unit in mathematics, force density methods can be considered as subtypes under the category of geometric stiffness methods. Similar to geometric stiffness methods, additional iterations are necessary for uniform or geodesic networks or shapedependent loading, making the method nonlinear (Barnes 1977; Haber and Abel 1982; Tan 1989; Lewis 2008; Koohestani 2014). Recently, the socalled thrust network analysis derived from the force density method has been used to find the shape of a discrete membrane restrained to a given geometric limitation (Block 2009; Marmo and Rosati 2017).

Dynamic equilibrium or relaxation methods that solve the problem of dynamic equilibrium to arrive at a steadystate solution are equivalent to the static solution of static equilibrium. As adapted from explicit time series integration, the time step parameters are required to control stability and convergence (Barnes 1977; Lewis 1989; Baraff and Witkin 1998). The main advantage of the dynamic relaxation method is that no assembled structural stiffness matrix is required; hence, it is suitable for highly nonlinear problems (Topping and Ivanyi 2007). The iterations in dynamic relaxation methods simulate the physical evolutionary process of the structures with feasible geometric configurations. Furthermore, with the development of the computer technique, the time and resource consumption in iterations and results storage has been significantly reduced. Such dynamic relaxation methods are becoming more popular (Olsson 2012; Bagrianski and Halpern 2014) and will be the basis for developing the method to be presented in this paper.
In this case, the dynamic relaxation method will be applied, thus avoiding inverse of stiffness matrices of complicated and varied geometries; the typical method needs to be adapted to easily update the positiondependent boundary conditions. Therefore, the objective of this paper is to, based on the dynamic relaxation method, construct an adaptive form finding method for a bar network on a formfixed surface with boundary conditions updated in each time step.
Objective and framework
To demonstrate the framework of the presenting method, the example of designing an ellipsoid shaped geodesic network dome is employed. The lengths of semiprincipal axes of the ellipsoid are a = 15 m, b = 11 m, and c = 12 m (height). The target is to obtain a geodesic network dome in this desired ellipsoidal shape, having bars with as few length variations as possible, and each bar length should be approximately 3 m, for the purpose of economy and convenience of construction.
Geodesic network
Typical geodesic domes with different frequencies
Frequency (V)  Geodesic dome  Number of length groups  Number of bars (struts, beams)  Number of nodes (joints, hubs) 

2  2  26  65  
4  6  250  91  
6  9  555  196  
8  19  980  341 
Based on the characteristics of the geodesic network, the design target is considered to have similar distribution of bars of a geodesic dome at a frequency of 6 V with a spherical radium of 15 m, where bar lengths in nine groups vary from 2.439 to 3.429 m. Thus, this network (designated as initial state or zero state in the following content) is transformed to an ellipsoidal geodesic network by scaling in x, y, and z directions with factors of 15/15, 11/15, and 12/15, respectively (designated as first state). The next step is to find a new node position on the current ellipsoidal surface that allows the bar lengths to “return” to the corresponding lengths of the geodesic network before transformation. To achieve this goal, the adaptive formfinding procedure is conducted, as presented in the following section.
Formfinding procedure
In the first state, bars are given precompressions calculated from the changing lengths while being transformed from the initial state, F_{1} = EA(L_{1 −} L_{0})/L_{0}. It can be predicted that, after releasing the bar forces, the network form will expand along the surface to cover more surface than a halfdome in the end; thus, the “overdesign” part will be cut off properly to fit the target network.
In the formfinding procedure, the precompressions in bars are the only loadings considered on the network structure. Regarding the nodal forces induced by releasing the precompressed bars, their components in the directions of surface normal at the nodes should be balanced with the reaction forces from the surface constraining boundary conditions. The residual forces of the nodal forces and the reaction forces, therefore, are the components of the nodal forces in the tangent plane of the surface at nodes and can be calculated according to the nodal forces and the positions of the nodes. In this manner, for any state of the network, the boundary conditions are considered in the calculation of the residual forces without recognizing the degrees of freedom that are fixed or not fixed. Moreover, the operations on vectors and matrices are unified for different geometries.
Procedure and formulations
Geometry state
Boundary conditions/constraints update
Explicit integration
The relaxation of the bar forces initiates the structural dynamic time history process. The state variables of a new step will be explicitly integrated from the current state according to the governing ordinary differential equations. Typically, the integration implementation uses either the explicit classic 4th order Runge–Kutta Method (Baraff and Witkin 1998) or the Central Finite Difference Method (Barnes 1999). Here, a simple conditionally stable explicit method based on the Modified Trapezoidal Rule Method (Pezeshk and Camp 1995) is used.
Considering the constraints again here, the resulting coordinates x_{t+Δt}, will position outside from the ellipsoid surface because the nodal accelerations cause the displacements and velocities on the tangent directions of previous positions x_{ t }. Therefore, the coordinates will be projected back to the constraint of the ellipsoid surface, and the velocities will be updated by removing the component on newly normal directions \({\tilde{\mathbf{n}}}_{t + \Delta t}\) updated via Eqs. (12) and (13). The updated coordinates and velocities will be used for updating other state variables and for integration in the next iteration.
Convergence criteria
 (1)
small variations in the displacements between successive iterations (\(\left\ {{\mathbf{x}}_{t + \Delta t}  {\mathbf{x}}_{t} } \right\ < {\varvec{\upvarepsilon}}\));
 (2)
small variations of the bar forces (or bar lengths) between successive iterations (\(\left\ {{\mathbf{F}}_{t + \Delta t}  {\mathbf{F}}_{t} } \right\ < {\varvec{\upvarepsilon}}\));
 (3)
small values of the residual forces (\(\left\ {{\mathbf{R}}_{t}  {\mathbf{P}}_{t} } \right\ < {\varvec{\upvarepsilon}}\));
 (4)
small values of the kinetic energy \(\left( {\left\ {\frac{1}{2}{\mathbf{M}}\text{diag} \left( {\left\ {{\mathbf{v}}_{t} } \right\} \right)^{2} } \right\ < {\varvec{\upvarepsilon}}} \right)\);
 (5)
maximum number of iterations (or maximum time duration) reached.
The dynamic relaxation process may not have a numerical convergence that allows the structure to achieve a still or rest state because no damping is introduced, as the equation of motion Eq. (16) represents an undamped structure system to simplify the calculation. However, with a preset maximum number of iterations, the calculation will “converge”, even if no damping exists in the system.
After the calculation converged, the optimal (best) solution of the network will be selected according to the convergence criteria that the calculation achieved. Recalling the objective description of the design example, the state of minimum residual forces of the entire calculation duration will be selected as the optimal solution. The maximum acceptable errors have been decided according to not only numerical considerations but also structural design ones. If no numerical convergence is achieved in the relevant time durations (converged according to criterion 5), the state of minimum residual forces may not be at the end of the time history; as a result, the state variables must be evaluated if the network is desired. Otherwise, the time duration must be extended for another new calculation, or the almosttarget state from current calculation is used as the initial state for the new calculation.
Solutions and evaluation
Optimal solution
Representative states during the calculation
Time step  Network  Max. residual force  Bar length variation distribution  

Max. (%)  Average (%)  Standard deviation (%)  
1  149.2 MN  26.7  15.0  7.3  
20  74.6 MN  27.0  14.1  7.0  
74  63.2 MN  25.3  10.7  6.5  
181  53.8 MN  22.7  5.5  5.2  
330  69.2 MN  45.3  11.8  10.0 
It is observed that in this calculation, no numerical convergence was achieved; nevertheless, the maximum residual force met the minimum value at the 181st step during the dynamic process. In the same step, the variation of the bar length reaches the point having minimum values of the average and standard deviation at the same time. Thus, the present procedure is proved to be applicable.
Discussion
In the procedure presented above, some key parameters should be noted, such as the initial state and the control of explicit integration and convergence. Avoiding lengthy parametric analyses, some discussions are introduced as follows.
Initial forces or impact pulse applied on bars will obviously affect the final state at end of time duration and the speed of dynamic evolution process. The induced higher accelerations on nodes will cause the nodes to oscillate at higher amplitudes. Therefore, the intermediate state will be missed in the time history records. In the present example, the initial bar forces (precompressions) are obtained from a spherical geodesic dome (as original bar lengths). If this radium of the geodesic dome is chosen with a large difference between the target ellipsoid semiprincipal axes, then the initial bar forces and the forces calculated during dynamic relaxation will be higher and induce time states with lower resolution. Thus, the initial geometry is suggested to be as closer to the target as possible.
In the explicit integration, as stated above, the studies on the explicit dynamic process revealed the time step length effect on the stability of integration. If the procedure is calculated in a materialindependent manner, then the time step also must be less than the dimension unit of the structure elements to ensure the explicit integration remains stable.
The convergence criteria chosen for the formfinding procedure are also important. In the example case presented, the target is to minimize length variations, which leads to the state of minimum residual force. The other convergence criteria (except the criterion 5) have reached 0state at the initial state and obviously are not applicable in this case. Finally, with the state of optimal solution found, the formfinding method and criterion chosen are proved to be effectively applicable.
Conclusions
 (1)
The dynamic relaxation method is applicable in formfinding process of formfixed spatial network structure, which avoids the inverse operations on complicated stiffness matrices in each step;
 (2)
During formfinding process, the boundary conditions can be applied as reaction forces through force equilibrium to avoid varying operations on vectors and matrices, especially for cases of timedependent varying boundary conditions;
 (3)
In spatial formfixed problem, the coordinates and velocities integrated from accelerations could be positioned out of the constraint form. In order to keep solution feasible, it is necessary to update the coordinates and velocities after each explicit integration;
 (4)
It is noticed that the choosing of initial state will affect a lot the states obtained in the time duration. It is better to choose the initial state closer as much as possible to the target state; the time step length is important for the stability of explicit integration, and for finding accurately the optimal solution; the convergence criteria chosen for time step iteration influence the target state characteristics. Therefore, the convergence criteria need to be chosen properly, especially in undamped systems.
Notes
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