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A shape-finding approach of tree-like structure based on grouping strategy and generalized inverse matrix theory
Abstract
Unstable link models in equilibrium transfer their loads only through axial forces and the generalized inverse matrix theory can be used in the shape-finding analysis of unstable link structures. According to this principle and the generalized inverse matrix theory, a shape-finding approach for tree-like structure is proposed, which forms a progressive process to update the shape of branching model until the potential energy of the model reaches minimum. In the proposed approach, members of the branching model are grouped and the total length of the members in the same group are constrained. The total length can be redistributed among the members in the same group in the shape-finding process. The redistribution of length is particularly useful for the shape-finding of tree-like structure as it can avoid the generation of inefficient members in the shape-found structure. Changing the grouping mode of members is a way to generate multiple schemes of structural shapes. Numerical examples illustrate the features of the approach and validate its effectiveness.
... The inverse hanging method coupled with multi-objective optimization was proposed by Su [11] to conduct the form-finding analysis of free-form grid structures. Cai [12,13] introduced the shape generation strategies and topology optimization methods for truss structures. Wang [14] proposed a shape optimization algorithm, which is suitable for irregular grid shell structures. ...
... In the form-finding analysis, the ideal force density is the Euler stability buckling capacity of each member of the supporting structure divided by its geometric length as Eq. (12). The members of branching columns will be instability simultaneously under the vertical load of the optimized structure, and the buckling capacity of branching columns will be maximized. ...
The shape generation is one of the key problems in the research of grid structures and branching columns. The existing algorithms are only applicable to a single grid structure or branching column, but the interaction between grid structures and branching columns should be sufficiently considered in the design process. According to this research gap, a form-finding and intelligent optimization algorithm is presented for the grid structures with branching columns in this study. Firstly, the force density method is used to generate the shape of integral structures and the bending moment is removed completely. Since the members of branching columns are mainly subjected to axial compression, the buckling capacity can be the determining factor in the design. Secondly, the corresponding iterative optimization is utilized based on the form-finding analysis of branching columns by using the updated force density method (UFDM), which is combined with the Euler’s buckling theory. The buckling capacity of branching columns can be maximized by using the UFDM. The intelligent section selecting algorithm is also added to the iterative optimization, which automatically selects the appropriate sectional size according to its internal force and material design strength. The applicability of the proposed algorithm is verified by analyzing numerical examples which come from actual projects.
... In the field of the numerical creation of reasonable forms for branching structure, scholars also focus on two directions: form-finding and structural optimization techniques. Typical form-finding methods include the graphic statics [35], the inverted hanging recursive method [12,36], the force density method [37], the element-clustered method [38,39], the quasi-mechanism method [40], and others. The methods based on optimization techniques include bi-directional evolutionary structural optimization (BESO) [41], extended evolutionary structural optimization (EESO) [42], sensitivity analysis [43], and others. ...
Branching and free-form structures are widely used in large-span buildings. Their shapes are the main factors that affect their mechanical performances. Many studies have been carried out on the morphology of single structural systems, but less on hybrid structures. However, both of them often appear in the same building. In order to reflect the cooperative bearing of substructures in the optimization process of branch-supported free-form surface structures , this paper proposes a holistic shape optimization method. The proposed method extracts design variables based on the structural modeling process, and uses the coordinates in the parametric domain to realize a mathematical description of the positional relationship between substructures. Then, the sensitivity analysis method is used to adjust the position of design variables to reduce the overall strain energy, realizing the integrated shape optimization of this hybrid structure. The effectiveness of the method is validated through several numerical examples. The results show that the overall stiffness of the optimized structure has been significantly improved, and the process of integrated optimization is more convenient. Furthermore, the way of adjusting design variables directly affects the shape and mechanical performance of the optimized structure. This feature serves as a valuable design tool that can provide multiple feasible solutions for architectural and structural design.
... Form-finding is one of the research hotspots of branching structures [3,4]. In branching structures form-finding aims to rationally arrange the spatial coordinates of each connection node, such that all branch components are only subjected to axial loads. ...
Form-finding analysis has been deeply investigated with the extensive application of branching structures. In addition to the form-finding analysis, another important scientific problem to be resolved is determining the optimal topology of branching structures. The topological study of branching structure is inseparable from form-finding analysis because they are coupled together. An intelligent design algorithm for branching structures based on updated force density method is proposed to deeply investigate the topological form of branching structures. The intelligent design algorithm comprises three sub-algorithms, namely, form-finding algorithm based on force density method, length optimization algorithm based on updated force density method, and topology optimization algorithm. The proposed algorithm can automatically select the effective component to support the load while decreasing the sectional area of the inefficient component. Thus, topology optimization can be achieved by “killing” inefficient components. The component section can be determined in two ways, namely, fully stressed design algorithm and section library selection. The component length can be optimized through the proposed method to maximize the structural stability. The proposed method and research results can lay a foundation for the intelligent design of branching structures.
... Xu [13] performed form-finding and shape optimization of branching columns through graphic statics. The grouping strategy and generalized inverse matrix theory was adopted by Tu [14] for the form-finding analysis of branching structures. ...
Grid structures and branching columns have been widely adopted due to their high structural efficiency and novel appearance. High structural efficiency must be achieved through a strict form-finding analysis. Existing algorithms are mainly applicable for grid structures or branching columns only. However, the form-finding of upper grid structures and the lower branching columns are mutually affected. A form-finding algorithm is needed for grid structures supported by branching columns. In the present work, an efficient form-finding algorithm suitable for grid structures is initially proposed, and the feasibility of this algorithm is verified through three examples. Then, the form-finding algorithm for grid structures is integrated with the form-finding algorithm of branching columns. Results indicated that the proposed algorithm can be used for form-finding analysis of various grid structures supported by branching columns. This work was conducted using a general finite element analysis program and can be easily mastered by engineering designers while avoiding tedious programming.
To realize the intelligent optimization for the tree structures supporting freeform surface roof or bearing uneven load, a new shape optimization method based on the back-propagation (BP) neural network is proposed. This method enables the intelligent positioning of hierarchical nodes through a recursive approach from top-down, with the aim of satisfying the zero bending moment and structural stability. Using three-dimensional tree structures as an example, this study provides a detailed description of the implementation method and steps of intelligent shape optimization, along with a comparative analysis with the reverse-hang recursive approach. Results indicate that the proposed approach effectively addresses the challenge of locating load-bearing centers in tree structures with uneven loads or freeform surface roofs. It not only demonstrates universality for tree structures under complex engineering conditions, but also enhances efficiency and intelligence in the structure optimization design.
With the rapid development of machine learning (ML), it is regarded as an effective approach for the shape-finding method of tree-like structures, for optimizing their structural forms. Considering that the existing shape-finding methods are do not easily solve the shape-finding problem of tree-like structures under a non-uniform load or supporting an irregular surface roof, the combination of a shape-finding concept and ML opens a new direction for shape-finding design. This paper presents a novel ML approach by combining a new shape-finding concept with a backpropagation and particle swarm optimization neural network. The core concept is to locate the load-bearing centre of each shape-finding unit by executing a positioning program, which finds the coordinates of the optimized nodes to achieve shape-finding design. Taking a tree-like structure with bottom two-bifurcate and upper four-bifurcate types as a shape-finding example, this paper elaborates the ML shape-finding method and specific steps in detail. Moreover, different shape-finding cases are analysed and compared with the inverse-hang recursive method. Results show that the ML shape-finding method can not only solve the unsolvable problem of shape-finding design of tree-like structures under a non-uniform load or supporting an irregular surface roof but also has higher efficiency than the compared method.
This research explores innovations in structural design and construction through the generative design technique BESO (Bi-directional Evolutionary Structural Optimization)[1]and the application of robotic fabrication to produce efficient and elegant spatial structures. The innovative pavilion discussed in this paper demonstrates a design and fabrication process and thecollaborationbetween architecture and engineering research groups through a series of small-scale test models and a full-scale model of topologically optimized spatial structures. The focus of this work is the use of a modified BESO technique to optimize the structure which features branches of various sizes, inspired by Gaudi’s Sagrada Familia Bacilica, and the introduction of large-scalerobotic 3D printing developed at RMIT University.The advantages of the new design and construction process are efficient material usage and elegant structural forms.
In contemporary architectural design new and experimental forms are constantly searched for and in the interaction between structural and architectural design a readable structural function adding to the architectural experience can be of interest. Cross-laminated timber plate elements show a high stiffness to weight ratio, and provide a wide range of possible structural applications. They have been analysed in structural systems for multi-storey residential buildings and in tensegric assemblies forming plane roof surfaces and single curved structures [1,2] based on a reduced number of element shapes. The application in faceted structures is possible by combining plate elements with linear edge joints or nodal joints, where the relation between wish for complexity in overall shape and desired production rationality and assembling simplicity decide the geometric character. The current work takes a plane roof structure with square elements as a point of departure for development of 3-dimensional plate structures and a study of interplay and result of combinations with branching column structures. By modifying square elements into rhombuses and combining them four together simple pyramid roof units and repetitive structures with rigid behaviour are easily obtainable. By varying the rhomboid proportions different pitches of the roof unit are obtained. By regarding physical modelling the effects of development from 2-dimensional to 3-dimensional units are studied concerning assembly stiffness and preconditions for interplay with supporting column structures. Branching column structures [3] can support the plates either at the centre of plates, which then cantilever, or at inter-plate nodes. Optimizing studies are applied on plate assembly geometry and design of supporting column structures. Using Evolutionary Computation (EC) methods [4], geometric relationships between the shells and branching columns are optimized for least weight. Parameters of shell geometry and the related branching column geometry and topology are explored. Examples of systems with good performance under load are shown and compared in terms of efficiency and stiffness.
The shapes of trees are complex and fractal-like, and they have a set of physical, mechanical and biological functions. The relation between them always draws attention of human beings throughout history and, focusing on the relation between shape and structural strength, architects have designed a number of treelike structures, referred as dendriforms. The replication and adoption of the treelike patterns for constructing architectural structures have been varied in different time periods based on the existing and advanced knowledge and available technologies. This paper, by briefly discussing the biological functions and the mechanical properties of trees with regard to their shapes, overviews and investigates the chronological evolution and advancements of dendriform and arboreal structures in architecture referring to some important historical as well as contemporary examples.
Branching structures are based on geometric systems that expand through bifurcation without returning to form closed cells. In this sense, branching structures resemble the structure of trees that branch continually outward. In architectural engineering, these forms can be used either as tension or compression systems. Numerous built examples have been produced since the initial inspiring studies made by Frei Otto in the early 1960's. Form finding techniques based on models have been used in the past to study these forms. Although thread models can be effective in the study of force paths, they cannot distinguish between tension and compression and have no way to take member buckling into account. But buckling does have an influence on appropriate geometry of a compression system. Also, minimal paths (or pseudo minimal paths based on surface tension thread models) have been used to explore possible geometries for branching structures. In this paper, both surface tension thread models dipped in water, and weighted string models are shown in comparison with ideal tension and compression forms found with a computational method based on Genetic Algorithms. The same computational model is used to find geometries with minimal overall member length. Both 2D and 3D geometries are derived. Peer Reviewed http://deepblue.lib.umich.edu/bitstream/2027.42/58599/1/pvb_IASS07.pdf
Owing to the development of architectural design method and technologies for fabrication and construction, we have more chance to design a complex shape for long-span structures [1]. For realizing a rational form of free-surface, the load path to the support structure is a significant factor. Hence, optimization of support structure is equivalently important as optimization of the shell surface. Tree-type support structure, first proposed by German architect/engineer Frei Otto in 1960, was applied to Stuttgart airport. Since then, more and more tree-type support structure were built, for instance the London Stansted airport, the Lisbon Oriente station and the Mumbai Tote restaurant. However, the existing tree-type support structures were designed based mainly on aesthetic factor rather than on physical rationality. The idea of biomimetics [2] , which designs artifacts by mimicking the characteristics of organisms like the artificial life [3] , is not new; however, recent development of computer technology enabled us to apply it to various fields of engineering including architectural design, construction, planning and environment [4,5]. In order to achieve a tree-type support structure considering the mechanical rationality, various methods have been proposed. Rational tree-type structures were obtained by Kolodziejczyk [6] through 'wet silk model method' for reverse-hanging experiment, and by von Buelow [7] through 'dry silk model method'. Moreover, von Buelow [8] investigated the method that generates rational tree-type structure by utilizing genetic algorithm. Hunt [9] proposed a tree-type structure that transmits loads with axial force only by transforming rigid connections into pins. Cui and Jiang [10] minimized the strain energy of structure using the sensitivity coefficients, and presented a method for simultaneously optimizing the shape, topology and sectional area of skeletal structures including tree-type structures. Among the methods of biomimetics and artificial life, the fractal geometry proposed by Mandelbrot [11] can express the tree shape in nature. The L-system is a typical application of fractal geometry to generate tree-type [12] structures. In this study, we propose a method of optimizing a tree-type support structure, considering the aesthetic effect, to evolve the structure to a rationalized form. The initial shape is generated using the fractal geometry [13] , and the B-spline curve is used to generate the boundaries and cave of the free-form shell. It is shown in the numerical examples that the strain energy of overall structure can be reduced by optimizing the shape of the tree-type structure.
The kinematic analysis of the folding process is important in practical engineering applications of the mechanism of origami. This paper proposes an efficient methodology for tracing the folding process of origami in a three-dimensional space. This method directly controls the nodal coordinates in the origami model activated by external force on the vertices. The deforming path is obtained using an algorithm based on the generalized inverse theory. An improved type of origami unit is presented for the computational calculation. The results demonstrate that the computational efficiency is strongly related not only to the number of the unknowns, but also the singular value of the compatibility matrix of the origami model. Furthermore, the validity and versatility of the proposed method are confirmed through numerical examples, including general origami models, origami with gravity, origami with special boundary conditions, and curved-crease origami. The proposed method is validated to be feasible and efficient in analyzing the folding mechanism of origami structures for kinematic design in structural and mechanical applications.
Branching structures are beginning to draw the engineering designer’s attention for their novel appearance and high structural efficiency. Detailed form-finding analysis is important for branching structures and there are few applicable methods for this kind of analysis. A novel numerical method was proposed based on this research background. Double-element method was firstly introduced and the iterative program was proposed. Then the accuracy and reliability of the proposed method were validated by existing research results. The proposed method was utilized for form-finding analysis of three different kinds of tree structures. Results indicated that the proposed method was highly efficient in dealing with form-finding analysis of tree structures. In addition, the proposed method was easy and convenient to be adopted.
Owing to recent progress of computational methods and manufacturing technology, free-form shells are widely used for designing complex shapes of long-span roof structures. However, in most of studies on design and optimization of free-form shells, the shells have fixed and/or pin supports, and stiffness and deformation of the supporting structures are not considered.
Tree-type frame is an aesthetically and mechanically efficient structural system, which has been used as supporting structures for roofs of moderately large scale such as train stations and airport buildings. The natural forms including the shapes of trees can be modeled using the methods of bio-mimetics, or bio-inspired engineering. Among those methods, fractal geometry can be effectively used for generating tree-type frames.
In this paper, an optimization method is presented for design of tree-type frames as supporting structures of free-form shells. The boundary of the roof, projected to the horizontal plane, is divided into several parts, and are modeled using the B-spline curves of different orders. A unified algorithm for modeling complex external boundary as well as the internal boundary is presented utilizing the B-spline curves of orders with odd numbers.
The finite-element (FE) mesh is generated by constrained Delaunay triangulation, where the points along the boundary, the bottom node of support structure, and the intersection points between the type-type support structure and the upper shell surface are constrained to be included in the FE-mesh.
An algorithm for generating tree-type frame is presented based on iterative function system (IFS) of fractal geometry. The type of branching of the tree is selected from the list of preassigned types. Formulations of sensitivity analysis is presented for total strain energy with respect to the nodal coordinates including those constrained to move on the shell surface. The optimal solution to minimize the total strain energy is found using the steepest decent method.
It is demonstrated in the numerical examples that the nodal locations of the tree-type support structure can be optimized to minimize the total strain energy of the structure including the roof and support frame. It is shown that the strain energy due to bending deformation is mainly reduced through optimization; thus, an ideal load path utilizing axial forces to the ground is achieved. It is notable that the bending deformation of the upper shell roof can be reduced by optimizing only the nodal locations of the support structure. The present study explains how the IFS of fractal geometry can be combined with an optimization method to be applied to a practical engineering problem.
Branching structures are applied increasingly in civil engineering due to their novel and beautiful configurations and reasonable force-bearing characteristics. As a complex space frame system, determination of the pattern of branches, namely form-finding analysis, is an important step in the design process of branching structures. In other words, the rationality of branch pattern will influence the mechanical behavior of branching structures directly. Based on the concept of form-active structures, and the analysis of the geometric and mechanical features of branching structures, a new form-finding method for branching structure, which was called inverse-hang recursive method, was proposed. The primary idea of this method can be illustrated as follow: finding the load-bearing centers of the branch roofs at each level, then let branches of each level point to their corresponding centers to generate the geometric form of branching structures progressively. According to this idea, the implementing procedure and some problems that deserve attention were discussed in details. At last, the inverse-hang recursive method was applied in the form-finding analysis of a practical branching structure. It can be seen that all the members of branching structures, which are generated by the proposed method, will only bear axial forces, like form-active structures; moreover, the proposed method is easy to carry out, and has more practicability than previous methods.
Cable and membrane structures belong to unstable structures, which mean the existence of rigid body displacements without strain and are usually used after the stabilization by the introduction of initial prestressing. An incremental procedure for the shapefinding analysis of the unstable link structures, is proposed by using the generalized inverse of the rectangular matrix which expresses the kinematic relation of unstable structures. The process of large displacement from the initial shape to the final stable one under the given loading condition can be numerically analyzed by the present procedure. The theoretical derivation of basic equations, illustrative examples and some discussions are presented in the paper.
In today's architecture there is an increasing trend towards lightweight construction, optimum structural performance, and more complex geometries and forms. In many cases, architects and engineers look to nature for inspiration on maximizing space, achieving the optimum force path, and interesting forms. The branching structure, or tree structure, is one such example of using natural form to satisfy architectural and structural requirements. This paper outlines a simple computer program that aids in the design of spatial tree structures: the user defines specific input parameters, and the program determines the equilibrium shape and member sizes. Several different geometries and loading patterns are investigated in a parametric study.
Based on thermal expansion theory, cable-sliding criterion equations are derived for the static analysis of cable-pulley systems. The friction between cable and pulley is considered in the equations. The cable-sliding criterion equations can govern the sliding motion between cable and pulley. In terms of the governing equations, a few program codes can be easily implemented using the APDL language of ANSYS software. The implemented program is further adopted in the following three examples: the structural behavior of a suspen-dome structure with sliding cable-strut joints; the constructional process of a suspen-dome structure with sliding cable-strut joints and the design of a tree structure. The analytical results show that the cable-sliding criterion equations are effective in describing the sliding motion between cable and pulley.
Form-finding is very important in the researches on the mechanical behavior of tree structures. Based on the researches of other form-finding methods of tree structures, a new form-finding method was proposed. Sliding cable element was used for simulation in order to reduce element bending moments, the general steps were given, and then a test was carried out to verify the proposed method. The conclusion was drawn from the internal force comparison that this method can effectively reduce element bending moments and the result is ideal. This method also shows such advantages as easy operation, no iteration or programming and very short consuming time. By optimizing the result with form-finding, this method can further reduce unit bending moments. By analyzing the effect of classification joint position changes on the result of form-finding and summarizing the rule, this method has the significance in engineering applications and provides some references for the analysis and design of tree structures.
A morphogenesis method is proposed for the topology and shape optimization of framed structures subject to spatial constraints. This combines direct elemental addition, or elimination, and free nodal shift, or restricted nodal shift related to the structures geometry. The optimization is based on elemental and nodal sensitivity information to generate or amend the structural topology and adjust the nodal positions to achieve a structure with minimum strain energy. In this method, the design parameters, such as supporting conditions, spatial constraints, etc, have significant influence on the final structural form; so various structural forms can be obtained by changing these design parameters in the project design phase. Several numerical examples are provided to illustrate the validity of this method and the mechanical behaviour of these structures. Results show that this can effectively reduce the structural bending moments and ensure sufficient structural stiffness.
An analytical method of structural behaviors of a hybrid structure which consists of cables and rigid structures is proposed and the following items are studied. (1) Introduction of the kinematic equation and the equilibrium equation, (2) Classification of stable and unstable hybrid structures. (3) Estimation of pre-stress modes and the introduction of pre-stress. (4) Stability under the introduction of pre-stress, (5) analytical method of stress and displacement under static loads, and (6) vibration analysis. In order to examine the validity of the proposed analytical method, the experiment on a hybrid structure model is reported, taking into consideration the introduction of pre-stress and the deformation under the static load.
Architectural and structural design of tree structure supporting free-surface shell roof using hanging upside-down model
- Eguchi
Proposal of design method of natural and rational tree-structure based on computational hanging upside-down simulation
- Eguchi