_{1}

The topology optimization method of continuum structures is adopted for the morphogenesis of dendriforms during the conceptual design phase. The topology optimization model with minimizing structural strain energy as objective and subject to structural weight constraint is established by the independent continuous mapping method (ICM) which is a popular and efficient method for the topology optimization of continuum structures. This optimization model is an optimization problem with a single constraint and can be solved by the iteration formula established based on the saddle condition. Taking the morphogenesis of a plane dendriform as an example, the influences on topologies of the dendriform are discussed for several factors such as the ratio of the reserved weight to the total weight, the stiffness and the geometry shape of the roof structure, the height of the design area, and so on. And several examples of application scenarios are presented, too. Numerical examples show that the proposed structural topology optimization method for the morphogenesis of dendriforms is feasible. It can provide diversiform topologies for the conceptual design of dendriforms.

Dendriforms, which was put forward first by a German Frei Otto in the 1960s, is a kind of bionic structure designed based on the shapes and mechanical characteristics of natural trees. The Stuttgart airport terminal (

achieved by a few bars. Many large-span space structures, such as airports, railway stations, public centers, and so on, adopted dendriforms as supported structures. For example, the Changsha railway station for high-speed trains in China (

The morphogenesis of the dendriforms is the most important problem during its structural design for its many branches and complicated form. The height, the layer, the number and the location of branches need to be designed. The suitable supported location of the roof structure needs to be determined. Thus, every component of the dendriform conforms to the optimal paths of transmitting loads; and the functional requirements of the building are also met.

There are three kinds of methods for the morphogenesis of dendriforms: the experimental methods, the geometric methods and the numerical methods. Experimental methods have the wet thread method, dry thread method, the beaded thread method and so on. The application of the experimental methods is restricted for their results influenced by model scales [

The more rational design can be achieved by the form finding method based on the topology optimization of continuum structures because the optimal topologies with skeleton forms can be obtained, and it is unnecessary to specify some prior data such as the height, the layers and the numbers of branches. But the ESO method [

The morphogenesis of dendriforms is used usually during the conceptual design phase. The usual way is to design a structural topology with maximum stiffness under the vertical loads acting on the roof structures. Thus, it can be formulated as a topology optimization problem of the continuum structure, namely: under the specified consumption of material, within the specified design area, optimizing the topology to maximize the structural stiffness under the specified roof loads. Because the maximum structural stiffness is equivalent to the minimum structural strain energy, the topology optimization model for generating a dendriform boils down to the topology optimization problem with minimizing structural strain energy objective subject to structural weight (or volume) constraint, as shown in Equation (1):

where,

For the topology optimization problem with minimizing structural strain energy as an objective subject to a specified weight constraint, the optimization model can be established by the ICM method as the following process. The discrete topology variables with values 0 or 1 are extended to the continuous topology variables with values in the interval [0, 1] by the approximation of the step function. The element weight and stiffness matrix are identified by the filter functions of weight and stiffness respectively [

, (2)

where

, (3)

where the power

Thus, the elemental strain energy can be expressed as:

where

The elemental weight can be expressed as:

Therefore, the topology optimization model established by the ICM method is written as:

To prevent the stiffness matrix to appear singular while the topology variable takes value 0, a small value

Because of the Equation (4) is an optimization problem with a single constraint, the constraint must to be taken as the equality constraint. Otherwise the problem will be an unconstrained problem and become a meaningless problem. Note

The augmented Lagrangian function of the problem is:

The saddle point for the above function taking the extremum condition is:

From it, we obtain:

Substitute Equation (10) into the equality constraint condition of Equation (7)

we obtain

Substitute Equation (12) into Equation (10), we have

Considering the interval constraints of topology design variables, namely

Update the active set, and return to Equation (13) to calculate

where

As showed in ^{5} MPa and the Poisson’s ratio 0.3. Under a specified constraint of the weight ratio, the optimal topology of the dendriform is obtained by minimizing the structural strain energy, namely maximizing the structural stiffness. The weight ratio is defined as the ratio of reserved weight to the initial total weight.

^{5} MPa) and the weight ratio (10%) are unchanged. In the cases that the height is small (Figures 5(a)-(c)), the main trunk of the dendriform will not appear. In the cases that the height is large enough (Figures 5(c)-(f)), the main trunk appears; and with the increase of the height of the design area, the optimal topologies of branches of the dendriforms are unchanged, only the height of the main trunk is increased.

Adopting dendriforms as bearing skeleton structures of walls, not only the loads acting on the walls can be transmitted effectively along the branches of dendriforms, but also a beautiful visual can be achieved. The morphogenesis of plane dendriforms can present diverse options.

As showed in the left figure of

The optimal topology is shown in the right figure of

and an arch structure is formed which are widely used in engineering for its wonderful mechanical performance.

As showed in the left figure of ^{2} is applied on the upper side of the cube. Along the upper side, a layer with the thickness of 0.5 m is taken as the roof structure, and is specified as non-design region. An area with sizes of 1m×1m in the middle part of the bottom of the cube is fixed and taken as the root of the dendriform. Structural material is steel, and the material properties are same with those in Example 1. The constraint and objective of the model are same with those in Example 2.

The optimal topology is shown in the right figure of

As showed in

edge with thickness of 0.3 m is used as the roof structure and is the non-design area. The structure is divided into four spans and 5 fixed points are set. Each fixed region has the width of 0.3m and is regarded as the roots of the dendriforms. Structural material is steel, and the material proper-ties are same with those in Example 1. The constraint and objective of the model are same with those in Example 2.

The optimal topology is shown in

1) It is showed from the numerical examples that it is feasible to generate dendriforms by the topology optimization method of continuum structures. Diverse options can be provided by the morphogenesis of dendriforms based on the topology optimization method during the conceptual design phase. Comparing with the dendriforms generating by the fractal methods which don’t consider the mechanical performance of dendriforms, the forms generated by the presented method achieve the maximum structural stiffness. Comparing with those methods based on mechanic models which need to specify design parameters such as the heights and the number of layers, branches at each node, and so on, the presented method can provide larger design space and seek more optimum topology of dendriforms because it is not necessary to specify design parameters.

2) The stiffness of the roof structures has significant effects on the optimal topology of dendriforms. Therefore, the roof structure should be analyzed together with the design area and be involved in the optimization process, and its stiffness should be simulated accurately.

3) The ratio of the structural weight has significant effects on the optimal topology of dendriforms. By setting a proper weight ratio to make the stiffness of the dendriform be similar to that of the roof structure, an ideal topology can be achieved.

4) The design region and the geometric shape of the roof structure have significant effects on the optimal topology of dendriforms. The parameters should be specified and simulated accurately during the conceptual design phase.

This work was supported by Natural Science Foundation in Hunan province of China (2016JJ6016) and by Department of Education in Hunan Province of China (15C0247).

Peng, X.R. (2016) Structural Topology Optimization Method for Morphogenesis of Dendriforms. Open Journal of Civil Engineering, 6, 526-536. http://dx.doi.org/10.4236/ojce.2016.64045