Optimised Spatial Structures

Msc Architectural Computation | Bartlett School of Architecture, UCL

Optimised Spatial Structures Load Path Optimisation of Predominantly Compression Networks using Algebraic 3D Graphic Statics and Genetic Algorithm Paris Nikitidis

This dissertation is submitted in partial fulfillment of the requirements for the degree of Master of Science in Architectural Computation from the Bartlett School of Architecture, University College London, September 2019

Optimised Spatial Structures

I, Paris Nikitidis, confirm that the work presented in this thesis is my own. Where information has been derived from other sources, I confirm that this has been indicated in the thesis.

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Abstract Algebraic constructions of polyhedral reciprocal diagrams for 3D Graphic Statics (3DGS) and optimization of load paths of compression-only thrust networks have been previously studied and explored (​Hablicsek, Akbarzabeh, Guo, 2019​; ​Liew et al., 2019​). The paper explores the statical indeterminacy of the reciprocal dual in 3DGS to optimize the structure load path for weight minimization using an evolutionary optimization process.

Graphic statics is a unique approach that enables interactive manipulation of both form and forces, with mechanisms for the visualization of internal forces. This force-driven structural design allows designers to discover new structural typologies and design possibilities that are not only spatially complex but also constrained to be in static equilibrium. The field of computational form-finding has made able for architects and engineers to design these efficient structures, by enabling the incorporation of structural and design criteria in the design process, we can generate efficient and effective results that satisfy boundary and load conditions.

Contributions of this research are the development of a form-finding tool for 3DGS, a computational method address for designers and engineers that incorporate design criteria and load path optimization.

The framework presented gives the ability to the user to select and assign values of force densities to all edges of a thrust network to achieve desired design results.

Keywords Three-dimensional Graphic Statics · 3DGS · Polyhedral reciprocal diagrams · Linear Algebra · Equilibrium matrix · Load Path · Optimisation algorithm · Genetic Algorithm

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Acknowledgments I would like to express my sincere gratitude and thanks to my supervisor, that helped me carry on this research and dissertation.

Vishu Bhooshan, for his constant essential support in the development of my research questions, computational methods, and visualization approaches. His contributions to this research by way of critical thinking, questioning and general conversation, but also for sharing his time, knowledge and expertise in guiding and helping me achieve great results. His help has been invaluable and most enjoyable to me. I thank him for my personal development as a researcher and computational designer for the past year. I consider him a personal mentor, and the period we worked together, the most influential in my career.

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Table of Contents Abstract Keywords

2 2

Acknowledgements

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Table of Contents

4

Table of Figures 1. Introduction

5 6

Contribution and Organization

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Notations Table

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Table

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2. Literature Review

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Graphic Statics

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ForceDensityMethods

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3D Graphic Statics

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Load Optimization

15

3. Construction

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Connectivity of elements

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Equilibrium Matrix

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Indeterminacy

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Solving Matrix/Equilibrium Solutions

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Moore-Penrose Inverse

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RREF

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4. Algorithm Implementation

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Constructing the dual

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Optimization

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5. Results and Analysis Experiments - Different Primals

25 25

MPI + GA:

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RREF + GA:

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6. Discussion and Conclusion

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Comparing results

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Future Work - implementing MME

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Conclusion

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7. References

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Table of Figures Fig. 1​: 2D Graphic Statics examples and applications. Control over the shape of the top chord in combination with the force constraint on the bottom chord allows for an exploration of free-form constant force trusses, (​Van Mele, Block, 2014​). Right, parameter-driven explorations of the relation between form and forces, the same structural principles can result in different architectural shapes.

Fig. 2​: ​2D and 3D corresponding diagram in two basic geometries.

Fig. 3​: ​Corresponding colors for every corresponding element in primal and dual.

Fig. 4​: ​A Closed group of polyhedral, Annotations and description of properties. Indication of elements.

Fig. 5​: ​Left, Closed group of 6 pyramids, forming a cube, with GFP as the external polyhedron, Right, dual reciprocal diagram.

Fig. 6​: ​Left, Closed group of 4 tetrahedrons, forming an isosceles tetrahedral, with GFP as the external polyhedron, Right, dual reciprocal diagram.

Fig. 7​: ​Connectivity of elements in primal. Top Left, Vertices-Edges, Top Right Vertex-Edge Matrix. Middle Left Faces-Edges. Middle Right Face-Edge Matrix. Bottom Left, Faces-Cells, Bottom Right, Faces-Cells Matrix ​(​Hablicsek, Akbarzabeh, Guo, 2019​).

Fig. 8​: ​Top, Detailed description of primal in equilibrium matrix ​A​, with annotations and elements, Bottom, corresponding A matrix [ 24 x 12 ].

Fig. 9​: ​Force diagrams as a primal with the external cell as its GPF, the user input parameters to explore a variety of compression-and-tension combined forms in equilibrium. Left, From the top, primal, a reciprocal diagram computed by the ​RREF method, User specified edge in compression, and in tension.

Right, From the top, primal, a reciprocal diagram computed by the ​MPI method, User specified edge in compression, and in tension.

Fig. 10​: ​Computational flow and method diagram. Description of previously implemented methods by (​Hablicsek, Akbarzabeh, Guo, 2019​)​.

Fig. 11​: ​Computational flow and method diagram. Description of contribution methods. Additional steps implemented in this research are highlighted with accent color (magenta).

Fig. 12​: ​Catalog of geometries, categorized by the method used and indication about load paths, Aq product and convergence time for each. Primal geometries increasing in internal faces from left to right.

Fig. 13​: ​Comparison of export graphs of the optimization process for the two employed methods. Left Aq product and optimization iterations, Middle, ​φ and optimization iterations, Right, Convergence rate and iterations.

Fig. 14​: ​Comparison graph between convergence time of two methods used, convergence time and internal faces.

Fig. 15​: ​Computational flow and method diagram. Description of future contributions. Future steps intended to be developed within the scope of this research are highlighted with accent color (green).

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1. Introduction From Gaudi’s graphic statics and hanging models to Frei Otto’s physical model based form-finding and analysis, has emerged the need for a better more intuitive method that empowers a structurally-informed design process connecting architectural intent and a structural requirement. Recent computational methods for form-finding of structural networks based on the Force Density Method (​Linkwitz, 1971; 2014​; Schek, 1974​), Dynamic Relaxation and Particle-spring Systems (​Adriaenssens et al., 1975​; ​Kilian, Ochsendorf, 2005​; ​Bhooshan, Veeneddaal, Block, 2014​), Thrust Network Analysis (​Block, 2009​) address these notions by developing appropriate frameworks that bridge the gap between designers and efficient structures. Many finite element analysis approaches (​Bletzinger, Ramm, 1993​) can analyze geometries and output a significant amount of numerical feedback, but lack immediate insight into the internal force distribution and do not allow the designer to explore the form geometry whilst being aware of the forces.

One of the most unique properties of computational graphic statics is that the form of the structure can be modified through the manipulation of the force diagrams. Structural optimization is a powerful technique to determine the optimal geometry to design efficient systems. Force density method and in extension Thrust Network Analysis and Graphic Statics have proven to be very useful form-finding tools, where the geometry is being manipulated based on the given connectivity of elements and boundary conditions.

Notable examples of computational form-finding tools are Frei’s Otto “natural structures” (​Otto, 1973​), compression-only funicular structure Armodilo Vault from Block Research Group (​Block et al., 2017​).

3D Graphical statics (3DGS) is a recent development of graphic statics in three dimensions based on remarkable historical research by Rankine and Maxwell (​Akbarzadeh, Van Mele, Block, 2015​). The basic principals, properties, and application of 3DGS were extensively demonstrated in the work of Juney L. and Akbarzabeh M. (​2016​). Important mention in the design project MycoTree, a spatial branching structure developed by KIT and BRG for the “Beyond Mining - Urban Growth” exhibition. (​Juney, Van Mele, Block, 2018​)

Contribution and Organization This paper will try to establish a relation between the topology of form, load paths of force and material density/efficiency. By applying different objective functions and manipulating specific edges on the form polyhedrons, the optimization of load paths will produce material-efficient and light reciprocal structures that address the design criteria.

The contributions of this study can be stated as the development of a framework, used in the early design phase, that facilitates the structurally-informed exploration of architectural geometries, using algebraic formulated methods of construction. The computational method gives the ability to manipulate the geometry of the dual form diagram based on design criteria and optimal load distribution. The development was oriented to maximize the freedom of the user to choose from the edges of the network and traverse the form. The ability to allocate certain values of force densities on the edges, while the structure is in equilibrium, offers flexibility in the process to address desired design results. No

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commercially available tools addressing material optimization while using algebraic descriptions, proposed for designers.

The outlines of this paper start with section ​1. Introduction​, where the basic notions are acquainted.

Section ​2. Literature Review​, explains the theoretical framework of the research. It provides a clear description of the properties of form-finding methods by unfolding the characteristics of Graphic statics and other Force-Driven methods. Algebraic formulations and constructions in 3D Graphic statics are presented with illustrated examples to help with understanding and gaining intuition. An explanation of the load optimization of structures is presented, by describing the calculation and properties of load paths.

Section ​3. Construction gives details for the input primal, closed group of polyhedral meshes, and the procedure to analyze and prepare the primal geometry. Also, it expands on the creation of the connectivity matrices by presenting an example of a simple primal. It describes the constraints on the creation process of the equilibrium matrix, as well as, the degrees of (in)determinacies of the systems.

The methodology offers two different numerical methods for solving the equilibrium equations, the properties, and the development of each are explained, alongside the mathematical formulas. Section ​4.

Algorithmic implementation presents the computational implementation of the algebraic formulation of 3DGS and the construction of the dual. The objective functions and the implementation of user criteria in the optimization are explained, together with the important learning variables of the evolutionary algorithm. In Section ​5. Results and Analysis​, the form-finding, analysis, and manipulation of form diagrams are presented through a catalog of optimized geometries, together with a comparison of the two aforementioned methods. The final discussion of the outcome of this research and the future goals for this research are listed in Section ​6. Discussion and Conclusion​. Section ​7 References​.

Notations Table To create a clear description of the paper, it is necessary to denote the geometric and algebraic object into specific notations. Objects symbolizing calculation matrices and vectors, topological data, primal and reciprocal diagrams. The table below encompasses all the notations used in the paper.

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Table Topology Description Γ Γ†

Primal diagram Dual, reciprocal

v e f c

Number of vertices in Γ Number of edges in Γ Number of faces in Γ Number of cells in Γ

v† e† f† c†

Number of vertices in Γ† Number of edges in Γ† Number of faces in Γ† Number of cells in Γ†

Ce×v Ce×f Cf×c

Edge–vertex connectivity matrix of Γ Edge–face connectivity matrix of Γ Face-cell connectivity matrix of Γ

A A+ Arref

Equilibrium matrix Moore–Penrose inverse of A Reduced Row Echelon form of A

nˆi Nx Ny Nz

Unit normal vector of face fi Diagonal matrix of the x-coords of nˆi Diagonal matrix of the y-coords of the nˆi Diagonal matrix of the z-coords of the nˆi

q

Solution of the equilibrium equation

ξ ζ

Parameter for the Moore–Penrose Inverse method Parameter for RREF method

r φ

Rank of A Load Path

Matrices and Vectors

Parameters

Other

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2. Literature Review

Fig.1

Graphic Statics In graphic statics, the geometry of the structure and its equilibrium are represented with two diagrams in a reciprocal relation, the form and force diagram, ​Γ and Γ† ​respectively (​Maxwell, 1864​). ​The form diagram contains details about the length of each member, the placement of the supports and the applied loads, while the equilibrium and the magnitude of the forces are represented by the force diagram. This reciprocity of the diagrams means that they are in a topologically dual and geometrically dependent relationship. ​Figure 1 left, shows different examples of form and force diagrams, on a given boundary and load conditions. Form finding methods using graphic statics are essentially the geometric construction of these two reciprocal diagrams for various geometries, loading cases, and boundary conditions (​Van Mele, Block, 2014​).

2D graphic statics was initially established and practiced in the late nineteenth century (​Juney, Van Mele, Block, 2017​). It included step by step instructions for the geometric construction of the reciprocal diagrams. In recent practices of Graphic statics, it was recognized that the combination with computational methods was able to lead to innovative design tools allowing the exploration of

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sophisticated and structural efficient design solutions. ​Figure 1 right, illustrate examples of the application of graphic statics in architectural geometries. The nodes and polygons that represent the geometry of the form (truss) have reciprocal polygons and nodes in the force domain (​Maxwell, 1864​). Every node in the geometry corresponds to a polygon in the force domain and every polygon in the geometry corresponds to a node in the force domain and every line representing the line of action of each truss member corresponds to a reciprocal line in the force domain. In ​Figure 2 top, the corresponding elements between primal and dual are highlighted in red color. This method of mapping used by Maxwell resulted in perpendicular reciprocal lines. The use of a different mapping, like a hyperbola, is introduced by (​Cremona, 1872​) and results in the reciprocal lines being parallel. This reciprocal relation means that the properties like mapping are both-ways. The length of each of the lines in the force domain is proportional to the axial force in the reciprocal line representing the truss member​ ​(​Culmann, 1864​; ​Wolfe, 1921​).

Fig.2 The use of computational methods has significantly accelerated the construction of the reciprocal diagrams and establish a relationship between the form of a structure and its geometrical equilibrium of forces. Together with an interactive environment, this has proven to be very innovative and helpful for designers and engineers to gain intuition on the distribution of forces of structures. An approach for direct construction, together with interactive manipulation of reciprocal diagrams of 2D Graphic statics has been

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developed in CAD environment - RhinoVault (​Rippman, 2016​), and in a Mesh Modeling environment MayaVault (​Bhooshan et al., 2018​). This was made possible through the formulation of algebraically-constrained equations that describe the topological and geometrical relationships. “Graphic statics has been used with success to find global and local equilibrium of forces separately in the force diagram and so enable intricate designs of compression-only structural forms.” (​Akbarzadeh, Van Mele, Block, 2014​).

The advantages of an algebraic approach to graphical analysis of structures using graph-theoretical properties of reciprocal graphs have been noted in T. Van Mele ​(​2014​). ​His research and computational framework have been recognized for the significance and possibilities provided by a useful and fast tool for the real-time, interactive computational exploration of two-dimensional graphic statics through different practical examples.

ForceDensityMethods The duality between the geometry of a network and its internal forces is a historical concept, first explained by (​Maxwell, 1864​). He called this relationship reciprocal and defined it as follows: “Two plane figures are reciprocal when they consist of an equal number of lines so that corresponding lines in the two figures are perpendicular and corresponding lines which converge to a point in one figure form a closed polygon in the other.” This means that the equilibrium of a node in the first diagram is represented by a closed polygon in the second diagram and vice versa. This being the primary principle of Graphic statics.

Originally graphic statics was developed by Karl Culmann at the ETH Zurich (​Culmann, 1866​), to serve as a comprehensive method to analyze and design structures graphically in two dimensions through geometry and drawing techniques, which are naturally familiar to architects and engineers. ​Block (2009) presented a framework that extended this two-dimensional approach in his dissertation “Thrust Network Analysis - Exploring Three-dimensional Equilibrium” by combining graphic statics and the Force Density Method, providing a highly controlled and intuitive form-finding process for funicular shells.

The Force Density Method (FDM) was developed in response to the requirement: “ the form is obtained in a purely geometric manner by ensuring equilibrium of forces, but requiring assumptions on the members’ force densities, defined as the force divided by the length of the member ” (​Linkwitz, 1971​; ​Schek, 1974​).

Form-finding methods used in the design of funicular structures fall into three main categories (​Veenendaal and Block 2012​; ​Bhooshan et al., 2018​): ● Stiffness matrix methods, which are based on the material to be used in the fabrication phase of the project. ● Geometric stiffness methods, which requires only boundary conditions and design load scenario These methods are Thrust network analysis (TNA) and force density method FDM. ● Dynamic equilibrium methods, geometric stiffness methods, which helps in the design explorations.

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3D Graphic Statics In 3DGS, the form and the force diagrams are polyhedral diagrams. A closed force polyhedron whose faces are perpendicular to loads of the node represents the equilibrium of each node of the form with its applied loads. The area of each face of the force polyhedron represents the magnitude of the force in the corresponding member of the form polyhedron. ​Figure 2 bottom, presents three-dimensional geometries with their reciprocal diagram, faces are highlighted to illustrate the measurement of the forces applied.

Fig.3 A recent paper presented an algebraic method to construct reciprocal polyhedral diagrams of 3D Graphic Statics (​Hablicsek, Akbarzabeh, Guo, 2019​). ​The input diagram can serve as both form and force.

Algebraic equations describe the topological and geometrical relationships explain the construction of the compatible reciprocal diagram. It also explains the process of developing the algebraic constraints and the equilibrium equations for these diagrams. The paper provides additional methods for determining the geometric/static degrees of (in)determinacy of the reciprocal diagrams. “For indeterminate cases, the deliberate control of the edge lengths allows exploring and manipulating a variety of solutions in equilibrium without breaking the reciprocity between two diagrams”​ (​Hablicsek, Akbarzabeh, Guo, 2019​).

There is a lack of design tools that encompass the algebraic methods of 3DGS, and especially in a real-time interactive environment. A recently developed, commercially available computational framework

by Masoud Akbarzadeh and Andrei Nejur from Polyhedral Structures Lab (PSL) for form-finding in 3DGS (​PolyFrame, 2018​) is an innovative tool for the construction of reciprocal polyhedral diagrams of 3D Graphic Statics for conceptual structural design purposes. It enables designers to use the intuitive aspect of 3DGS but serves mostly as a structural design tool addressing engineers.

This section describes the properties of the reciprocal polyhedral diagrams in the context of the 3D Graphic Statics and sets the basic concepts for understanding the approach to construct these diagrams.

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Fig.4

In 3DGS, both form and force diagrams are made of a closed group of polyhedral cells. There is a need to define an external polyhedron by locating all the external faces, faces which do not have an equal flipped face in the group, and the rest of the cells are inside the external polyhedron.

Closed group: To further explain a closed group of cells, we note some of their properties illustrated in Figure 4​, and listed below as: ● Edges share an identical vertex with its adjacent edges, ​e​i and ​e​i+1 share ​v​i ● Face share an identical edge with its adjacent faces, ​f​i and ​f​i+1 share ​e​i ● Cells share an identical face with its adjacent cells, ​c​i and ​c​i+1 share ​f​i+1.

The force diagram of 3DGS is to be distinguished into 2 categories: global force polyhedron (GFP), and nodal force polyhedrons (NFP). A GFP is the static equilibrium of externally applied loads and reaction forces., while an NFP is the equilibrium of forces grouping together at a node in the form diagram. Each cell in a group of cells of the form diagram can act as the GFP, but as noted in (​Hablicsek, Akbarzabeh, Guo, 2019​) ​“..if GFP is the external polyhedron, the force diagram can represent a compression/tension-only structural form”. Different cells can be chosen as a GFP and it will result in an equilibrium solution with compressive and tensile members.

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As mentioned in section ​2. 3D Graphic Statics​, the reaction forces in the force diagram correspond to the area of faces in the closed polyhedral group, force diagram. While the form diagram represents reaction forces together with its applied loads and location of the supports with open cells. ​Figure 4 and ​Figure 5 illustrate a primal force diagram with its external polyhedron as GFP and the corresponding dual form diagram with its applied reaction forces.

Fig.5

Fig.6

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Load Optimization The problem of optimizing truss structures for mass minimization has been historically studied (​Michell, 1904​). Various researches have addressed the optimization of network-type structures such as trusses, grid shells and thrust networks. Researchers have focused on applications for the optimization of building frames or with the improvement of structural trusses (​Liew et al., 2019​). More specifically, the work by (​Mazurek, Baker, Tort, 2011​), and (​Beghini et al. 2014​) showed that for truss structures, optimizing for the minimal load path ​φ results in the minimal volume solution for a given stress level σ for the edges in the truss. Based on Maxwell’s formula on load paths (​Maxwell, 1864​), the function is subject to the volume, cross-sectional area, and length of every edge, and can be written as:

min ∑ V i = min ∑ Ai li = min i

i

1 σ

∑ ||f i || li i

3. Construction Connectivity of elements The relation between the components of each diagram should be described algebraically by multiple connectivity matrices for the ​v​, ​v† ​vertices, ​e​, ​e† edges, ​f​, ​f† faces, and ​c​, c† cells of the form and force diagram respectively. ​Figure 3 shows the corresponding elements between the two diagrams. The connectivity matrices will define the topological and geometrical relationships between the primal and the dual diagram and the construction method.

In order to construct the connectivity matrices, there is a need to perform some initial actions on the primal, a group of connected polyhedral meshes with information about the external loads. External loads are represented by the GFP global force polyhedral. The main action required is a distinction between internal and external elements, with external being the elements adjacent to the GFP and internal the rest. An arbitrary numbering system for the elements is required to perform with a selection of one internal face, edge and vertex for between every adjacent cell.

Figure 7 demonstrates the connectivity matrices ​Ce×v​, ​Ce×f​, ​Cf×c with corresponding visual representations of the elements used. A simple primal is used, made of a cube subdivided into 6 faces and the volume center, each cell is a pyramid with a square base of the face.

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Ce×v ​Edge - Vertex: Connectivity between the edges and vertices of the primal need to be stored in a matrix of internal edges and all vertices, with values following the rule: ​If v​ ertex ​v​i is the start point of edge ​e​i value ​= +1 ​Else​ If​ vertex ​v​i is the endpoint of edge ​e​i ​value = ​ -1 ​ Else ​ ​value = ​ 0

Ce×f ​Edge - Face: Connectivity between the edges and faces of the primal are stored in a Matrix of internal edges and internal faces, with values following the rule: ​ ​If​ edge ​e​i is an edge of face​ f​i value = ​ +1 ​ Else If​ edge ​e​i is an opposite edge of face ​f​i ​value = ​ -1 ​Else ​ ​value = ​ 0

Cf×c ​Face - Cell: Connectivity between the faces and cells of the primal are stored in a Matrix of internal faces and cells, with values following the rule: ​If​ face ​f​i has the same direction as cell ​c​i ​value = ​ +1 ​Else If​ face ​f​i has the opposite direction as cell​ c​i ​value = ​ -1 ​Else​ ​value = ​ 0

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 Fig.7

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Taking into consideration the reciprocal relation between the two diagrams we can establish the following correlations: ● The ​v vertices and ​e edges of the primal map to ​f† faces and ​c† cells of the dual so the connectivity matrix ​Ce×v e ​ dge-vertex is equal to connectivity matrix ​Cf†×c†​ of faces and cells. ● The matrix ​Ce×f edge-face can also describe the connectivity between the faces and edges of the dual complex and thus equals to the connectivity matrix ​Cf†×e†​ face-edge. ● Faces ​f and cells ​c of the primal correspond to edges ​e† and vertices ​v† ​of the dual, the matrix Cf×c ​face-cell is equal to matrix ​Ce†×v† edge-vertex. The direction of the edges of the dual is denoted in the ​Cf×c​ face-cell connectivity matrix.

Equilibrium Matrix

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Fig.8

Constructing the dual requires the calculation of the coordinate difference between vertices. That is achieved through the connectivity matrices and the relationship between components.

Moreover, specific requirements for the construction can be listed as: ● ●

Around every face ​f†​i of the dual, the sum of the coordinate differences of the edges has to be zero.

Around each internal edge e​i and its adjacent faces ​f​i−k in the primal, the sum of the normal vector of the faces ​nˆi multiplied by the length ​q​i of the corresponding edge ​e†​i in the dual diagram should be the zero vector.

Combing the above constraints we can construct the equilibrium matrix ​A​, by stacking the product of the diagonal normal matrices ​Nx, Ny, Nz and the edge-face matrix in each direction respectively. Solving the equation ​A ​* ​q = 0 provides a possible vector of force densities for the edges of the dual, and so a fully detailed description for the creation of the dual. ​Figure 8 demonstrates the elements of a primal closed group and the corresponding equilibrium matrix ​A ​[24 x 12].

Indeterminacy The equation ​A * ​q = 0 is a homogeneous system and has only two sets of solutions, first being q=0, which is not an acceptable solution in 3DGS, and second (column number of ​A​, ​f - rank ​r​) solutions. The dimension of the solutions ​q is equal to the dimension of the right null-space of ​A​. The dimension of the right null space is equal to ​f − ​r​. With ​r being the number of dependent equations which is equal to the rank of the equilibrium matrix ​A​. The degrees of indeterminacy gives insight into different internal force or force density distributions that can be calculated and lead to the structure being in equilibrium.

Solving Matrix/Equilibrium Solutions The method employed for solving the equilibrium matrix in (​Hablicsek, Akbarzabeh, Guo, 2019​), consists of a matrix transformation to Reduced Row Echelon Form, which is a rank revealing the method that can be used to solve systems of linear equations. Through experimentation in the developing stage of this paper, there has been noticed a significant sensibility of the method to the values of the matrix ​A​, and thus a partial inability to compute correct results.

In more detail, the purpose of converting the matrix into RREF is to determine the number and index of the no-pivoting columns, corresponding to the independent edges. The sensitivity to the values compromises the result of the method by miss-calculating the numbers of no-pivoting columns, which is an inequality between the matrix rank. In the development of this work, it has been observed that rank revealing matrix decompositions are sensitive to a threshold since no floating-point matrix is rank deficient, meaning that matrices operations need a decimal description of the values. The typical

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threshold used is diagonal size times machine epsilon. To exceed this limitation it has been employed custom matrix operations methods and libraries that achieve calculations beyond the 5th decimal digit.

As stated in section ​2 Equilibrium Matrix​, the process of solving the equilibrium matrix requires a series of mathematical operations to pseudo-inverse matrix ​A ​and solve the equation system ​A​*​q​=0.

Here are the employed methods:

Moore-Penrose Inverse The equilibrium matrix A can be inverted by using the Moore–Penrose inverse (MPI) of A to solve the construction equation. The ​A+ ​ of ​A​ is used for its characteristic property ​AA+A​=​A​.

Any vector ​q​ resulted by the equation

​q​ = (I − ​A+ *​ ​A​) ​ξ solves the linear equation system (​Hablicsek, Akbarzabeh, Guo, 2019​).

Vector ​ξ is any [​f × 1] column vector (​Penrose, 1955​) and the user can choose each of the components.

For a well-distributed edges lengths and as a testing point in the search, assigning 1 to all components gives a dual solution. Moreover, for symmetrical primal diagrams, this approach can result in symmetrical q ​for certain values of ​ξ​, with most of them in a compression-only case. A notable contribution of this paper is the extension of this MPI method to give the ability to the user to specify certain edge lengths to particular edges of the dual.

RREF Algebraic 3D graphic statics presented with the RREF method to solve the construction equation. A user can specify the lengths of ​f ​− ​r ​independent edges to manipulate the geometry of the dual. The reduced row echelon form ​Arref of the matrix ​A reveals the rank and also the independent edges of the dual with a simple count into columns of ​Arref where there is or not a pivot. The user interacts with the components of the ζ [(​f ​− ​r​) × 1] vector that will determine the geometry of the dual. Property of the RREF method is that it has the same rank and column space so the equilibrium equation can be rewritten as Arref * q​ = 0.

Solution ​q ​can be computed with

q​ = −B​ζ B is an [​r​ ×(​f ​− ​r​)] matrix with independent columns of ​Arref.

Any positive and negative values can be assigned to ​ζ​, solutions are not guaranteed to have only positive or negative values. To address this issue the combination of this method with a GA optimization gives compression/tension only results.

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An alternative method explored in the development of this paper was the LU decomposition. In LU decomposition, matrix ​A ​is transformed in Row Echelon Form. Using LU decomposition with complete pivoting: the matrix ​A is separated into matrix L, unit-lower-triangular, and matrix U, upper-triangular, also, P and Q are permutation matrices of that partition. LU is a rank-revealing method and the eigenvalues (diagonal coefficients) of U are sorted in such a way that any zeros are at the end. The independent edges can be obtained by getting the last (​f ​− ​r​) pivoting coordinates of the Q matrix. While a q_​dependent​ ​vector can be calculated using the same equation as the ​RREF​ method.

This decomposition provides the generic approach to solving systems of linear equations, computing the rank, invertibility, inverse, kernel, and determinant. Continuing with back substitution to retrieve a ​q vector that contains the force densities of edges in the dual. The above method gives the user the ability to identify the independent edges and manipulate their length, changing the force densities, with all other edges being computed. The process concludes with the construction of an identity matrix [rank(​A​) ​r x (​f ​− ​r​)] whose product with the ​ζ vector of user inserted edge lengths gives the vector of the rest edges of the dual.

Several limitations were observed with the above method, which discarded the method from the final methodology. The most influential one being its sensitivity to the threshold. It requires constant adjustment based on the number of internal faces of every geometry.

The below ​Figure 9 shows ​two force diagrams as a primal with the external cell as its GPF, manipulated through input parameters to explore a variety of compression-and-tension combined forms in equilibrium.

Left, From the top, primal, a reciprocal diagram computed by the ​RREF method, User specified edge in compression, and in tension. Right, From the top, primal, a reciprocal diagram computed by the ​MPI method, User specified edge in compression, and in tension.

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Â

Fig.9

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4. Algorithm Implementation All methods implemented for this research have been developed using the zSpace C++ Library, (​Bhoosan, 2018​), while for the visualization, all images and diagrams presented were made in Rhino 6 and Grasshopper (​Rutten, 2009​; ​McNeel, 2011​).

Constructing the dual

Fig.10

Fig.11

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The proposed method begins with the user input a closed group of polyhedral meshes as a primal. As explained in ​(​Hablicsek, Akbarzabeh, Guo, 2019​) first there is a need to define a Global-Force-Polyhedral that describes the external loads of the network, that can be either a cell of the primal or the external polyhedron. To produce predominant compression solution, this research explored only GFP as an external polyhedron. The next step in the process is the creation of the ​connectivity matrices of the primal (edge-vertex, edge-face, face-cell). They will define the relationship between the vertex-edge-face and cell between the primal and the dual. After finding a solution for the q, dual-edge lengths, Breadth-first search​, a graph tree search algorithm is employed to construct the dual, by starting from an arbitrary point in space and finding all paths to every other vertex using the face-cell matrix of the primal, ​which is equivalent to edge vertex of the dual​, and a diagonal matrix of faces normal vectors. In cases of primal with higher nullity than 1 (nullity>1) of the matrix, in this case, is equal to ​f - ​r​, there are multiple solutions.

Figure 10 illustrates the implemented methods and computational flow for algebraically constructing the dual diagram, while ​Figure 11 shows the presented methodology diagram of this research with the description of contribution methods and additional steps implemented, highlighted with accent color (magenta).

To define the load path we can denote that, in 3DGS for a given stress level is represented by boundary and connectivity conditions, while the minimum volume can be achieved by minimizing the product of absolute values of ​q edge force densities, multiplied by the corresponding ​f ​face area of the primal. In this paper denoted by ​φ​ (​Liew et al., 2019​).

φ​ ​=​ ∑ ||q i f area || i Optimization Solving the equilibrium equations requires insight into the nullspace of the matrix, and hence it can result in more than one solution for constructing the dual. In more detail, manipulating the length of edges of the dual by assigning values to the ​q vector will result in alternative form diagrams. This solution space can be explored to determine which solutions have lower force densities in the edges of the dual and thus achieving minimum material weight in the structure.

The objectives of the optimization are the minimization of the load path function ​φ ​of a given ​q​, ​φ​(​q​) and the minimization of the product of equilibrium matrix ​A and ​q​, that need to be equal or close to zero. The objective function is to be minimized with two sets of constraints: ● The calculated force densities ​q​. The vector must not contain zero values, which is not an acceptable solution and to further ensure that notion, the constraint has been set as ​q not having values from -1 to 1, q ​∄ [-1,1]. Also in order to minimize the search space, to prevent the

●

geometry to become too large, a value of ​q​_max has been used with a range of [10,20].

User-defined set of edges and corresponding values to generate solutions that meet design criteria.

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The optimization process uses a Genetic Algorithm (​Holland, 1975​; ​Sastry K. Goldberg D. Kendall G., 2005​), which has been found very appropriate based on this paper computational implementation. GA is commonly used for traversing a large search space and finding minimum solutions.

5. Results and Analysis Experiments - Different Primals Illustrated examples of the use of the methods described in this paper are presented in a catalog ​Figure 12​. Through explicit manipulation of edges of the dual, the geometries generated address different design criteria. The catalog presents resulting form diagrams from both ​MPI and ​RREF methods combined with the GA optimization algorithm for a visual explanation of the differences in the solutions. Important criteria of the construction were the minimization of the ​A​, q ​product that specifies the accuracy of the topological relationship, which is presented along with the examples. The ​φ ​value is noted for all geometries together with a comparison of the convergence time, to show computational cost​.

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Â

26 Fig.12

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Characteristics of each of the methods developed combined with GA for optimization can be stated as:

MPI + GA: Moore-Penrose Inverse method calculates the force densities of the dual with global manipulation. Assigning a desired values in the ​ξ ​vector, transform the geometry on a global scale, with specific edges not having exactly the proposed density but maintaining universal topology as well as a general distribution of the forces.

When combined with the GA method traverse the geometry to follow the user criteria with certain edges achieving requested results, while the topology and original symmetries in the force diagram are being carried on and updated to the new solution values. The method can produce results of compression-only structures.

RREF + GA: Reduced Row Echelon form method computes solution vector ​q only for dependent edges of the dual, while the user-defined ​ζ ​vector will correspond directly to the independent edges of the dual.

This method gives a clear manipulation of independent edges with the ability to set exact values of force densities. The produced results lack global distribution and corresponding topology, as force densities of adjacent edges, from independent ones, will be calculated according to the user-defined value.

The combination of RREF and GA manages to optimize the dual achieving design criteria. For instance, multiple user-specified edges can be set as goals to configure certain parts of geometry. Another criterion can be, global distribution and general scale of form. It was observed that convergence with both cases of criteria was not always able and most of the time it requires a high number of iterations. Moreover, the method is unable to result in a solution with only a single edge in tension and rest in compression, with neighboring edges being affected.

Finally, the results of this research can be concluded into two main categories based on the designer’s needs. First, ​general exploration​, generation of different form diagrams for a given force diagram. The more suitable method being the MPI together with the optimization approach that allows for a general manipulation, while also taking into consideration the global distribution and symmetry of the geometry.

Second, ​specified design,​ manipulation of certain edges. Using the RREF method enables the introduction of multiple design constraints or the optimal minimization of mass.

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6. Discussion and Conclusion Comparing results

Fig.13 An evolution for the optimization consisting of 50-100 generations was found to give an effective balance between convergence rate and accuracy, while for some specific user-defined constraints it requires more iterations or bigger population size to find a solution. A population size of 300 has been used in this research to give a broad initial diversity and local convergence to a solution is promoted. While cross-over ratio parameter CR = 0.5.

Following the suggested method to generate polyhedral reciprocal diagrams, designer or engineer can search for an optimal solution for an input aggregation, primal. This research provides a clear explanation of the context of graphic statics and a detailed description of methods to generate and manipulate dual form diagrams.

From the results of this work, and especially ​Figure 12​, it can be observed that there is space for possible exploration with different geometries and relations, that can generate interesting solutions. While the methodology of this paper provides a tool in exploring the solution space. ​Figure 13 is a collection of comparison graphs of the optimization process for the two employed methods. Left, ​A q product, Middle, φ and Right, Convergence rate for every iteration of the optimization. The comparison emphasizes and illustrates the advantages and disadvantages of each method, with the ​RREF method achieving better results in the clarity of the geometrical topology, minimal ​A q ​product values, but not converging into global optimal solutions with minimal ​φ​. The convergence rate was a comparison based on a “difficult” objective function with multiple dependent and independent edges being selected and assigned different values. Figure 13, Right shows that ​MPI was able to converge at a rate of 68.7% compared to ​RREF with 65.4%.

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Fig.14 Another important comparison was made on the convergence time based on the number of internal faces of the primal for the two methods. ​Figure 14 Illustrates average convergence time-points and it can be observed that the ​RREF method is performing much faster, with a critical point of intersection at 200 internal faces where the ​MPI​ method performing at a faster rate.

The data to construct the graphs in ​figure 13 and ​figure 14 were a collection of all the experiments, objective functions, and geometries exploited in this research. An average value was used to produce a more universal outcome.

Similar results have been achieved before with an important reference to the Algebraic Constructions of 3DGS ​(​Hablicsek, Akbarzabeh, Guo, 2019​)​, which demonstrated the initial process for an algebraic construction as well as an optimization method for the construction of the dual. Comparing the results was an initial intent of the research, originally, to recreate results and also expose the minor limitations of the aforementioned methods. Contribution of this research is an extension of the design usage with user input of specific edges, and not limited to only independent.

Future Work - implementing MME This process can be seen as a tool for designers and engineers and can be strengthened by the implementation in an interactive MME that will provide intuition and understanding of design variables for any change in the force distribution and easier manipulation of the diagrams to achieve the desired results (​Bhooshan et al., 2018​). A future goal of this research is the implementation in Maya 3D Modeling

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Software, that will be integrated into the current method. Mesh Modeling environment will give more freedom to the designer to generate primal groups, through the use of user-oriented mesh manipulation tools like. Conway operations and subdivision algorithms existing in Maya software can help designers produce easy and fast primal groups. The method will benefit from the circular process of redesigning to achieve desired results. Figure 15 shows the description of future contributions, with future steps intended to be developed within the scope of this research highlighted with accent color (green).

Fig.15

Conclusion This paper presented an exploration of the indeterminacy of reciprocal dual diagram in 3DGS and a load path optimization approach for minimizing the volume of predominantly compression thrust networks at a given constant stress level, while also addressing user design constraints. Based on the previous studies of Algebraic constructions of polyhedral reciprocal diagrams for 3D Graphic Statics (3DGS) (​Hablicsek, Akbarzabeh, Guo, 2019​; ​Liew et al., 2019​), this paper developed a framework that gives the ability to the user to select and assign values of force densities to all edges of a thrust network to achieve desired design results, which it the major the contribution.

The paper held a step-by-step description of the construction of the dual form diagram from a given close group of polyhedral. The algebraic approach of this research constructs the reciprocal diagram for force diagrams as the primal input. First by analyzing the primal and creating the equilibrium matrix and then with the two presented methods to solve the equation ​Aq ​= 0, which will provide the force densities ​q ​and construct the dual.

The Evolutionary optimization minimizes the Aq product, the load path ​φ ​and traverses the geometry to correspond to user-defined values for selected edges. A ​general exploration can be better achieved with the MPI combined with the optimization method that keeps properties of the original topology and symmetries of the force diagram while minimizing the objective function. A more ​specified design ​can be obtained with the RREF method combined with the evolutionary algorithm with more flexibility and freedom in manipulating the geometry but fewer compression-only solutions.

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