Analysis of self-equilibrated networks through cellular modeling
Omar Aloui, David Orden, Nizar Bel Hadj Ali, Landolf Rhode-Barbarigos
AAnalysis of self-equilibrated networks through cellular modeling
O. ALOUI a , D. ORDEN b , N. BEL HADJ ALI c,d , L. RHODE-BARBARIGOS a * a Department of Civil, Architectural and Environmental Engineering, University of Miami, Coral Gables, FL 33146, USA b Departamento de Física y Matemáticas, Universidad de Alcalá. Ctra. Madrid-Barcelona, Km. 33,600, 28805, Alcalá de Henares, Spain c Ecole Nationale d’Ingénieurs de Gabès, University of Gabès, Rue Omar Ibn-Elkhattab 6029, Gabès, Tunisia d LASMAP, Ecole Polytechnique de Tunisie, University of Carthage, B.P. 743, La Marsa 2078, Tunisia
Abstract
Network equilibrium models represent a versatile tool for the analysis of interconnected objects and their relationships. They have been widely employed in both science and engineering to study the behavior of complex systems under various conditions, including external perturbations and damage. In this paper, network equilibrium models are revisited through graph-theory laws and attributes with special focus on systems that can sustain equilibrium in the absence of external perturbations (self-equilibrium). A new approach for the analysis of self-equilibrated networks is proposed; they are modeled as a collection of cells, predefined elementary network units that have been mathematically shown to compose any self-equilibrated network. Consequently, the equilibrium state of complex self-equilibrated systems can be obtained through the study of individual cell equilibria and their interactions. A series of examples that highlight the flexibility of network equilibrium models are included in the paper. The examples attest how the proposed approach, which combines topological as well as geometrical considerations, can be used to decipher the state of complex systems. Introduction
Since their inception, networks have been serving as a powerful tool to model a wide range of engineering problems. In 1736, Leonhard Euler used graph representations to prove that the problem of the seven bridges of Königsberg has no solution [1]. This laid the ground for the emergence of network theory and predefined the concept of topology. However, network applications did not remain confined to the study of topological properties of systems but quickly evolved to incorporate a comprehensive description of their equilibrium states. The integration of equilibrium in networks led to the discovery of network equilibrium models. This turned out to be very influential in the study of electrical networks especially with the advances generated by the work of Kirchhoff on the “node and mesh” rules for electrical circuits [2-3]. Network equilibrium models have also been used in the analysis of mechanical and structural systems. In 1864, Maxwell formulated the counting conditions to determine the rigidity of bar-jointed frameworks based on the topology of the underlying network [4]. Maxwell’s work was further refined by Calladine, Pellegrino, Roth, Whitely and Connelly [5-9], where the network structure of the framework described the equilibrium between conjugate variables: forces and displacements. These conjugate uantities are attributed to the network nodes and edges that have to satisfy nodal equilibrium and geometric compatibility. Moreover, the analogy between the analysis of electrical networks and mechanical systems was also recognized as nodal equilibrium and geometric compatibility relations in structural frameworks are reflected in Kirchhoff current and voltage laws. Hähnle and Firestone provided a complete set of analogies between electrical and mechanical systems where forces are treated as currents and displacements are considered as voltages, allowing researchers to explain electrical phenomena by referring to mechanical systems and vice-versa [10,11]. Network equilibrium models have also been adopted in the analysis of electrical and power networks [12-14], telecommunication networks [15-17], transportation networks [18-20], as well as supply chain networks [21-23]. The wide range of physical and engineering systems that are depicted through network models underlines the value of using network representation to model such systems, as the variables involved in the related problems can be attributed to the network components (nodes and edges) and the relations between these variables can be described. It is thus widely recognized that a significant portion of physical, engineering and mathematical problems lie within the scope of network theory [24-28]. However, there is often dim interest in studying the abstract topological and algebraic properties of network equilibrium models when the focus is on specific context. Nevertheless, understanding the abstract properties of network equilibrium models is critical for their application, as it provides a platform for studying interdependent models as well as a common language for interdisciplinary collaborations. Analogies can thus be drawn making possible to adopt solutions already developed in other fields. One can find some interest on the abstract properties of network problems in the work of Roth, who applied algebraic topology concepts to study the existence of a solution to the network equilibrium problem [29-31]. Branin built upon Roth’s work to study the topological structure of the network or the linear graph and the associated algebraic structure, setting up ground rules for network analogies and discussing the interpretations of these rules in electrical, mechanical, and structural systems [32,33]. More recently, Reinschke provided a comprehensive description of the network equilibrium models, starting from an abstract model for the variable attributes of the network components and the network elements relations (NER) between them [34]. This paper extends Reinschke’s work on network equilibrium models by focusing on a special class of network models, referred to as the self-equilibrated network models, with a novel approach for the analysis of their topological and algebraic properties based on elementary units called cells. Self-equilibrated network models present a redundancy in their elements that can be explored in science and engineering applications that require a certain degree of damage tolerance. The paper is organized as follows: Section 2 includes a review of network equilibrium models through the description of their network laws and attributes (for more details, see appendix A), as well as of their properties and equilibrium state. Self-equilibrated networks are described in Section 3 through the definition of their constitutive cells and their interactions, as well as their impact in the network attributes. In Section 4, the equilibrium in examples of self-equilibrated network models is studied considering also the effects of external perturbation and damage (element loss). Section 5 concludes the paper with a discussion for the use of the proposed model. . Network equilibrium models
Let 𝐺𝐺 ( 𝑉𝑉 , 𝐸𝐸 ) be a graph that describes the set of nodes 𝑉𝑉 and the set of edges 𝐸𝐸 of a network and 𝑛𝑛 𝑣𝑣 be the number of vertices and 𝑛𝑛 𝑒𝑒 the number of edges in the graph. The graph is equipped with a set of node and edge attributes that satisfy the equilibrium conditions referred to, in graph theory, as circuit laws and cut-set laws. In this paper, node attributes are referred to as potential attributes; they are denoted by a 𝑛𝑛 𝑣𝑣 × 1 vector 𝑝𝑝 and they satisfy circuit laws. Note that each component 𝑝𝑝 𝑖𝑖 represents the value of the potential at node 𝑣𝑣 𝑖𝑖 which in turn is a 𝑑𝑑 × 1 vector where 𝑑𝑑 represents the dimensionality of the problem. Edge attributes are referred to as flow attributes; they are denoted by a 𝑛𝑛 𝑒𝑒 × 1 vector 𝑓𝑓 and they satisfy cut-set laws. The properties of edge attributes and node attributes along with the description of cut-set laws and circuit laws are discussed in detail in Appendix A. 2.1. Network equilibrium In this section, network equilibrium is described in terms of the topology of the graph and its different attributes. A network equilibrium model is thus given by the flow and potential attributes, and the interrelations between them that can be expressed by [34]: 𝑟𝑟 ( 𝑓𝑓 , 𝑝𝑝 ) = 0 (1) where 𝑓𝑓 is the 𝑛𝑛 𝑒𝑒 × 1 vector (indexed by the edges of 𝐺𝐺 ) of the values that the flow takes on each edge and 𝑝𝑝 represents the 𝑛𝑛 𝑣𝑣 × 1 vector (indexed by the nodes of 𝐺𝐺 ) of the values that the potential takes on the nodes. Equation 1 can thus be used to model voltage-current relations in electrical circuits or constitutive relations, such as force-displacement relations, in solid mechanics. In constitutive relations, flow and potential are related through the impedance of the electrical circuit branch (inductance, capacitance, resistance, etc.) or the stiffness of the structural member. Similar concepts can be found in transportation networks. Figure 1 represents a network equilibrium model with edge and end-nodes equipped with flow and potential attributes. In Figure 1, 𝑢𝑢 and 𝑣𝑣 are the end-nodes of the edge ( 𝑢𝑢 , 𝑣𝑣 ) . Nodes 𝑢𝑢 and 𝑣𝑣 are attributed an intrinsic potential represented by the value the potential function 𝑝𝑝 takes on 𝑢𝑢 and 𝑣𝑣 and an independent potential 𝑝𝑝 𝑒𝑒 . This incurs the potential difference variable 𝛿𝛿𝑝𝑝� ( 𝑢𝑢 , 𝑣𝑣 ) � on the edge ( 𝑢𝑢 , 𝑣𝑣 ) and 𝛿𝛿𝑝𝑝 𝑒𝑒 . Edge ( 𝑢𝑢 , 𝑣𝑣 ) is also attributed the intrinsic flow represented by the value of the flow function 𝑓𝑓 on ( 𝑢𝑢 , 𝑣𝑣 ) and the independent flow 𝑓𝑓 𝑒𝑒 . Independent flow variables can be interpreted as independent current source in electrical circuits or perturbations to the element stresses in structural members such as thermal expansion. In transportation networks, 𝑓𝑓 𝑒𝑒 can be used to model perturbation to the flow of goods due to external agents such as a change in edge capacity. The independent potential variable accounts for independent voltage sources in electrical circuits, external loads or displacements applied to the nodes of a structure or changes in the stock due to creation of new quantities of goods and/or additional traveling agents in transportation systems. 𝑟𝑟 ( 𝑓𝑓 , 𝑝𝑝 ) represents the flow-potential relations. Figure 1: Edge and end-node attribute representation in a network equilibrium model. 𝑝𝑝 ( 𝑢𝑢 ): potential values in nodes 𝑢𝑢 , 𝑣𝑣 . 𝛿𝛿𝑝𝑝 ( 𝑢𝑢 , 𝑣𝑣 ) : potential difference values between nodes 𝑢𝑢 and 𝑣𝑣 . 𝑝𝑝 𝑢𝑢𝑒𝑒 : independent potential variable in node 𝑢𝑢 . 𝛿𝛿𝑝𝑝 𝑒𝑒 : independent potential difference value between nodes 𝑢𝑢 and 𝑣𝑣 . 𝑓𝑓 ( 𝑢𝑢 , 𝑣𝑣 ) : flow values on edge ( 𝑢𝑢 , 𝑣𝑣 ) . 𝑓𝑓 𝑒𝑒 : independent flow value on edge ( 𝑢𝑢 , 𝑣𝑣 ) Since the flow space ℱ and the potential difference space 𝛿𝛿𝛿𝛿 are orthogonal complements of each other, and ℱ and 𝛿𝛿𝛿𝛿 are associated to the cycle space 𝒞𝒞 and the cut-set space (bond space) ℬ , for the determination of whether an edge attribute 𝑓𝑓 is a flow it suffices to verify its orthogonality with a basis of the cut-set space ℬ . Let 𝐵𝐵 be the matrix formed by vectors ( 𝑏𝑏 , 𝑏𝑏 , … , 𝑏𝑏 𝑟𝑟 ) 𝑇𝑇 , then the cut law can be expressed in matrix form as: 𝐵𝐵 ( 𝑓𝑓 + 𝑓𝑓 𝑒𝑒 ) = 0 (2) Analogously the circuit law can be expressed as: 𝐶𝐶 ( 𝛿𝛿𝑝𝑝 + 𝛿𝛿𝑝𝑝 𝑒𝑒 ) = 0 (3) where 𝐶𝐶 is the collection of the cycle space bases. The network equilibrium model in a one-dimensional space is thus described by: �𝐵𝐵 ( 𝑓𝑓 + 𝑓𝑓 𝑒𝑒 ) = 0 𝐶𝐶 ( 𝛿𝛿𝑝𝑝 + 𝛿𝛿𝑝𝑝 𝑒𝑒 ) = 0 𝑟𝑟 ( 𝑓𝑓 + 𝑓𝑓 𝑒𝑒 , 𝑝𝑝 + 𝑝𝑝 𝑒𝑒 ) = 0 𝑐𝑐𝑢𝑢𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑙𝑙𝑙𝑙𝑙𝑙𝑐𝑐𝑐𝑐𝑐𝑐𝑟𝑟𝑐𝑐𝑢𝑢𝑐𝑐𝑐𝑐 𝑙𝑙𝑙𝑙𝑙𝑙𝑐𝑐𝑓𝑓𝑙𝑙𝑓𝑓𝑙𝑙 𝑙𝑙𝑛𝑛𝑑𝑑 𝑝𝑝𝑓𝑓𝑐𝑐𝑐𝑐𝑛𝑛𝑐𝑐𝑐𝑐𝑙𝑙𝑙𝑙 𝑟𝑟𝑐𝑐𝑙𝑙𝑙𝑙𝑐𝑐𝑐𝑐𝑓𝑓𝑛𝑛𝑐𝑐 (4) 2.2. Network equilibrium models in higher dimensions Networks that are embedded in higher dimensions can be used to describe a wide range of physical and engineering systems as well as model one-dimensional systems with interdependencies. Therefore, this section discusses the dimensionality of the network equilibrium models. The dimensionality of the flow variables can be described as interconnected networks of the same topology and same embedding (potentials) with each network being endowed with a one-dimensional component of flow and flow-potential relations that describe the inter-dependence of flow components. Conversely, the dimensionality of the potential variables introduces the concept of direction (described by vectors) to the network models. In this paper, directions refer to the generalization of the concept of orientation in higher dimensions. For instance, in DC electrical circuits (which correspond to one-dimensional networks), flow orientation is set by convention from higher voltage to lower voltage (negative potential difference). In structural systems, direction is set by a position vector given by a unit vector obtained by dividing the “potential ifference” by the length of the edge. Consequently, direction is an inherent part of the description of the network equilibrium. Consider an elementary cut Δ 𝑢𝑢 = [{ 𝑢𝑢 }, 𝑉𝑉 { 𝑢𝑢 } ⁄ ] in the network 𝐺𝐺 ( 𝑉𝑉 , 𝐸𝐸 ) equipped with a potential 𝑝𝑝 and a flow 𝑓𝑓 . The cut law is given by: � 𝑓𝑓� ( 𝑢𝑢 , 𝑣𝑣 ) � = 0 𝑣𝑣∈𝑉𝑉 ( 𝑢𝑢 , 𝑣𝑣 ) ∈𝐸𝐸 (5) Note that in Equation 5, the direction is already incorporated in the flow 𝑓𝑓 ( 𝑢𝑢 , 𝑣𝑣 ) described by 𝑓𝑓 ( 𝑢𝑢 , 𝑣𝑣 ) = ‖𝑓𝑓 ( 𝑢𝑢 , 𝑣𝑣 ) ‖ . 𝑐𝑐⃗ ( 𝑢𝑢 , 𝑣𝑣 ) as ‖𝑓𝑓 ( 𝑢𝑢 , 𝑣𝑣 ) ‖ represents the magnitude of the flow and 𝑐𝑐⃗ ( 𝑢𝑢 , 𝑣𝑣 ) corresponds to a unit vector that depicts the direction of the edge ( 𝑢𝑢 , 𝑣𝑣 ) . Now, let 𝑑𝑑 ( 𝑢𝑢 , 𝑣𝑣 ) be a distance function between nodes 𝑢𝑢 and 𝑣𝑣 . In one dimension, the distance is defined as 𝑑𝑑 ( 𝑢𝑢 , 𝑣𝑣 ) =| 𝑝𝑝 ( 𝑢𝑢 ) − 𝑝𝑝 ( 𝑣𝑣 )| = �𝛿𝛿𝑝𝑝� ( 𝑢𝑢 , 𝑣𝑣 ) �� . In higher dimensions, the distance 𝑑𝑑 is recognized as the Euclidean distance (L2 norm) where 𝑑𝑑 ( 𝑢𝑢 , 𝑣𝑣 ) = �𝛿𝛿𝑝𝑝� ( 𝑢𝑢 , 𝑣𝑣 ) �� . In two dimensions, 𝑑𝑑 ( 𝑢𝑢 , 𝑣𝑣 ) = �𝛿𝛿𝑝𝑝 ( 𝑢𝑢 , 𝑣𝑣 ) 𝑥𝑥2 + 𝛿𝛿𝑝𝑝 ( 𝑢𝑢 , 𝑣𝑣 ) 𝑦𝑦2 . The flow density 𝜔𝜔 𝑓𝑓 ( 𝑢𝑢 , 𝑣𝑣 ) is defined as the quantity ‖𝑓𝑓 ( 𝑢𝑢 , 𝑣𝑣 ) ‖‖𝛿𝛿𝛿𝛿 ( 𝑢𝑢 , 𝑣𝑣 ) ‖ . By introducing the flow density and the distance function, the cut laws can be expressed as: � 𝑓𝑓� ( 𝑢𝑢 , 𝑣𝑣 ) � = 0 𝑣𝑣∈𝑉𝑉 ( 𝑢𝑢 , 𝑣𝑣 ) ∈𝐸𝐸 ⇔ � �𝑓𝑓� ( 𝑢𝑢 , 𝑣𝑣 ) �� . 𝑐𝑐⃗ ( 𝑢𝑢 , 𝑣𝑣 ) = 0 𝑣𝑣∈𝑉𝑉 ( 𝑢𝑢 , 𝑣𝑣 ) ∈𝐸𝐸 ⇔ � �𝑓𝑓� ( 𝑢𝑢 , 𝑣𝑣 ) �� . 𝛿𝛿𝑝𝑝� ( 𝑢𝑢 , 𝑣𝑣 ) ��𝛿𝛿𝑝𝑝� ( 𝑢𝑢 , 𝑣𝑣 ) �� = 0 𝑣𝑣∈𝑉𝑉 ( 𝑢𝑢 , 𝑣𝑣 ) ∈𝐸𝐸 ⇔ � 𝜔𝜔 𝑓𝑓 � ( 𝑢𝑢 , 𝑣𝑣 ) � . 𝛿𝛿𝑝𝑝� ( 𝑢𝑢 , 𝑣𝑣 ) � = 0 𝑣𝑣∈𝑉𝑉 ( 𝑢𝑢 , 𝑣𝑣 ) ∈𝐸𝐸 ⇔ � 𝜔𝜔 𝑓𝑓 � ( 𝑢𝑢 , 𝑣𝑣 ) � . �𝑝𝑝 ( 𝑢𝑢 ) − 𝑝𝑝 ( 𝑣𝑣 ) � = 0 𝑣𝑣∈𝑉𝑉 ( 𝑢𝑢 , 𝑣𝑣 ) ∈𝐸𝐸 (6) Using the flow density allows to redefine the equilibrium problem in terms of a scalar quantity reducing the number of variables. A multidimensional network problem can thus be simplified as interrelated networks of the same topology that share the same flow density. When all elementary cut-sets are considered, the cut laws can be described in matrix form as: 𝛿𝛿𝛿𝛿 . �𝜔𝜔 𝑓𝑓 + 𝜔𝜔 𝑓𝑓 𝑒𝑒 � = 0 ⇔ �𝐵𝐵 𝑑𝑑𝑐𝑐𝑙𝑙𝑑𝑑 ( 𝛿𝛿𝑝𝑝 𝑥𝑥1 ) 𝐵𝐵 𝑑𝑑𝑐𝑐𝑙𝑙𝑑𝑑 ( 𝛿𝛿𝑝𝑝 𝑥𝑥2 ) ⋮𝐵𝐵 𝑑𝑑𝑐𝑐𝑙𝑙𝑑𝑑 ( 𝛿𝛿𝑝𝑝 𝑥𝑥𝑥𝑥 ) � �𝜔𝜔 𝑓𝑓 + 𝜔𝜔 𝑓𝑓 𝑒𝑒 � = 0 ⇔ ( 𝕀𝕀 𝑥𝑥 ⨂𝐵𝐵 ). � 𝑑𝑑𝑐𝑐𝑙𝑙𝑑𝑑 ( 𝛿𝛿𝑝𝑝 𝑥𝑥1 ) 𝑑𝑑𝑐𝑐𝑙𝑙𝑑𝑑 ( 𝛿𝛿𝑝𝑝 𝑥𝑥2 ) ⋮ 𝑑𝑑𝑐𝑐𝑙𝑙𝑑𝑑 ( 𝛿𝛿𝑝𝑝 𝑥𝑥𝑥𝑥 ) � �𝜔𝜔 𝑓𝑓 + 𝜔𝜔 𝑓𝑓 𝑒𝑒 � = 0 ⇔ ( 𝕀𝕀 𝑥𝑥 ⨂𝐵𝐵 ). 𝑑𝑑𝑐𝑐𝑙𝑙𝑑𝑑 ( 𝛿𝛿𝑝𝑝 ). �𝐽𝐽 𝑥𝑥 , ⨂𝕀𝕀 𝑚𝑚 ��𝜔𝜔 𝑓𝑓 + 𝜔𝜔 𝑓𝑓 𝑒𝑒 � = 0 (7) where 𝐽𝐽 𝑥𝑥 , is an all-ones 𝑑𝑑 × 1 vector, and 𝛿𝛿𝑝𝑝 is the 𝑑𝑑𝑛𝑛 𝑒𝑒 × 1 vector of potential differences, ordered such that all the components of the same dimension are grouped together. These components are denoted 𝛿𝛿𝑝𝑝 𝑥𝑥 , 𝛿𝛿𝑝𝑝 𝑥𝑥 , … , 𝛿𝛿𝑝𝑝 𝑥𝑥 𝑑𝑑 . Note that 𝛿𝛿𝑝𝑝 can be expressed as 𝛿𝛿𝑝𝑝 = ( 𝕀𝕀 𝑥𝑥 ⨂𝐵𝐵 𝑇𝑇 ). 𝑝𝑝 , where 𝑝𝑝 is the vector of potential values where the components of each dimensions 𝑋𝑋 , 𝑋𝑋 , … , 𝑋𝑋 𝑥𝑥 are grouped together. 𝕀𝕀 𝑥𝑥 is the 𝑑𝑑 × 𝑑𝑑 identity matrix. 𝐵𝐵 is the 𝑛𝑛 𝑣𝑣 × 𝑛𝑛 𝑒𝑒 matrix that groups all the elementary cut-sets. 𝜔𝜔 𝑓𝑓 and 𝜔𝜔 𝑓𝑓 𝑒𝑒 are 𝑛𝑛 𝑒𝑒 × 1 vectors of the internal flow densities and independent flow densities respectively. 𝑑𝑑𝑐𝑐𝑙𝑙𝑑𝑑 ( 𝑢𝑢 ) is the function that takes a 𝑛𝑛 vector 𝑢𝑢 and returns a 𝑛𝑛 × 𝑛𝑛 diagonal matrix 𝑈𝑈 . The matrix 𝐴𝐴 𝛿𝛿𝛿𝛿 is known in the analysis of pin-jointed frameworks as the equilibrium matrix, where a more comprehensible form can be expressed as: 𝐴𝐴 𝛿𝛿𝛿𝛿 = �𝐵𝐵 𝑑𝑑𝑐𝑐𝑙𝑙𝑑𝑑 ( 𝐵𝐵 𝑇𝑇 𝑋𝑋 ) 𝐵𝐵 𝑑𝑑𝑐𝑐𝑙𝑙𝑑𝑑 ( 𝐵𝐵 𝑇𝑇 𝑋𝑋 ) ⋮𝐵𝐵 𝑑𝑑𝑐𝑐𝑙𝑙𝑑𝑑 ( 𝐵𝐵 𝑇𝑇 𝑋𝑋 𝑥𝑥 ) � (8) with ( 𝑋𝑋 , 𝑋𝑋 , … , 𝑋𝑋 𝑥𝑥 ) being the components of each dimension of 𝑝𝑝 . Note that this form contains redundant rows. The redundancy created by the topology of the graph can be omitted by using 𝐵𝐵 𝑟𝑟 instead of 𝐵𝐵 , where the rows represent the basis of the cut-set space. However, the dimensionality of the problem might also create redundant rows. Circuit laws can also be described based on the potential density 𝜔𝜔 𝛿𝛿𝛿𝛿 ( 𝑢𝑢 , 𝑣𝑣 ) = ‖𝛿𝛿𝛿𝛿 ( 𝑢𝑢 , 𝑣𝑣 ) ‖‖𝑓𝑓 ( 𝑢𝑢 , 𝑣𝑣 ) ‖ , with the circuit law for a given cycle 𝐶𝐶 becoming: � 𝛿𝛿𝑝𝑝� ( 𝑢𝑢 , 𝑣𝑣 ) � = 0 ( 𝑢𝑢 , 𝑣𝑣 ) ∈𝐶𝐶 ⇔ � �𝛿𝛿𝑝𝑝� ( 𝑢𝑢 , 𝑣𝑣 ) �� . 𝑐𝑐⃗ ( 𝑢𝑢 , 𝑣𝑣 ) = 0 ( 𝑢𝑢 , 𝑣𝑣 ) ∈𝐶𝐶 ⇔ � �𝛿𝛿𝑝𝑝� ( 𝑢𝑢 , 𝑣𝑣 ) �� . 𝑓𝑓� ( 𝑢𝑢 , 𝑣𝑣 ) ��𝑓𝑓� ( 𝑢𝑢 , 𝑣𝑣 ) �� = 0 ( 𝑢𝑢 , 𝑣𝑣 ) ∈𝐶𝐶 ⇔ � 𝜔𝜔 𝛿𝛿𝛿𝛿 � ( 𝑢𝑢 , 𝑣𝑣 ) � . 𝑓𝑓� ( 𝑢𝑢 , 𝑣𝑣 ) � = 0 ( 𝑢𝑢 , 𝑣𝑣 ) ∈𝐶𝐶 (9) onsidering all independent cycles of the graph, the circuit law can be expressed in matrix form as: 𝐴𝐴 𝑓𝑓 . �𝜔𝜔 𝛿𝛿𝛿𝛿 + 𝜔𝜔 𝛿𝛿𝛿𝛿 𝑒𝑒 � = 0 ⇔ ⎝⎛𝐶𝐶 𝑑𝑑𝑐𝑐𝑙𝑙𝑑𝑑 ( 𝑓𝑓 𝑥𝑥1 ) 𝐶𝐶 𝑑𝑑𝑐𝑐𝑙𝑙𝑑𝑑 ( 𝑓𝑓 𝑥𝑥2 ) ⋮𝐶𝐶 𝑑𝑑𝑐𝑐𝑙𝑙𝑑𝑑�𝑓𝑓 𝑥𝑥𝑥𝑥 �⎠⎞ �𝜔𝜔 𝛿𝛿𝛿𝛿 + 𝜔𝜔 𝛿𝛿𝛿𝛿 𝑒𝑒 � = 0 ⇔ ( 𝕀𝕀 𝑥𝑥 ⨂𝐶𝐶 ). ⎝⎛ 𝑑𝑑𝑐𝑐𝑙𝑙𝑑𝑑 ( 𝑓𝑓 𝑥𝑥1 ) 𝑑𝑑𝑐𝑐𝑙𝑙𝑑𝑑 ( 𝑓𝑓 𝑥𝑥2 ) ⋮ 𝑑𝑑𝑐𝑐𝑙𝑙𝑑𝑑�𝑓𝑓 𝑥𝑥𝑥𝑥 �⎠⎞ �𝜔𝜔 𝛿𝛿𝛿𝛿 + 𝜔𝜔 𝛿𝛿𝛿𝛿 𝑒𝑒 � = 0 ⇔ ( 𝕀𝕀 𝑥𝑥 ⨂𝐶𝐶 ). 𝑑𝑑𝑐𝑐𝑙𝑙𝑑𝑑 ( 𝑓𝑓 ). �𝐽𝐽 𝑥𝑥 , ⨂𝕀𝕀 𝑚𝑚 ��𝜔𝜔 𝛿𝛿𝛿𝛿 + 𝜔𝜔 𝛿𝛿𝛿𝛿 𝑒𝑒 � = 0 (10) where 𝑓𝑓 is the 𝑑𝑑𝑛𝑛 𝑒𝑒 × 1 vector of flow values, ordered so that all components of the same dimension are grouped together. 𝐴𝐴 𝛿𝛿𝛿𝛿 and 𝐴𝐴 𝑓𝑓 define the static equilibrium of the model. In other words, if the system is in equilibrium, Equations 7 and 10 have to be satisfied. This allows finding the equilibrium flows in a network when the potential is known, and vice versa. When a perturbation occurs to the flows or the potentials, the new equilibrium is governed by the network element relation 𝑟𝑟 ( 𝑓𝑓 + 𝑓𝑓 𝑒𝑒 , 𝑝𝑝 + 𝑝𝑝 𝑒𝑒 ) = 0 which can be expressed using the flow and potential densities as 𝑟𝑟�𝜔𝜔 𝑓𝑓 + 𝜔𝜔 𝑓𝑓 𝑒𝑒 , 𝜔𝜔 𝛿𝛿𝛿𝛿 + 𝜔𝜔 𝛿𝛿𝛿𝛿𝑒𝑒 � = 0 or 𝑟𝑟�𝜔𝜔 𝑓𝑓 + 𝜔𝜔 𝑓𝑓 𝑒𝑒 , 𝑝𝑝 + 𝑝𝑝 𝑒𝑒 � = 0 . The use of flow densities is advised for networks with higher dimensions. In the study of self-equilibrated networks, special interest is given to flow variables and the flow space ℱ , with space ℱ referring to the actual flow space in one-dimensional applications and to the flow density space in higher dimensions. Self-equilibrated network models
In network equilibrium models, cut laws and circuit laws are always defined by linear systems as attested by Equations 7 and 10. The cut laws and the circuit laws can thus be expressed as: 𝐴𝐴 𝛿𝛿𝛿𝛿 . 𝜔𝜔 𝑓𝑓 = 𝐹𝐹 (11) 𝐴𝐴 𝑓𝑓 . 𝜔𝜔 𝛿𝛿𝛿𝛿 = 𝑃𝑃 (12) where the effects of the external perturbations to the system are lumped into the vectors 𝐹𝐹 and 𝑃𝑃 . The solutions to Equations 11 and 12 admit two parts, a homogeneous solution that depends solely on the topology of the system and the values assigned to the other variable type, and a particular solution that depends also on the external perturbation. Algebraically, the homogeneous solution for the flow density 𝜔𝜔 𝑓𝑓 and the potential density 𝜔𝜔 𝛿𝛿𝛿𝛿 corresponds to the nullspace of the matrices 𝐴𝐴 𝛿𝛿𝛿𝛿 and 𝐴𝐴 𝑓𝑓 . Self-equilibrium occurs when the system is in a non-trivial equilibrium state in the absence of external perturbations ( 𝐹𝐹 = 0 and 𝑃𝑃 = 0 ). A self-equilibrated network is thus a network where the cut-laws have a non-zero homogeneous solution reflecting that the system can be in a state of self-equilibrium in the absence of external erturbations. Let Ω 𝑓𝑓 be the collection of the 𝑐𝑐 basis vectors of the null-space of 𝐴𝐴 𝛿𝛿𝛿𝛿 and 𝛼𝛼 ∈ℝ 𝑠𝑠 be a vector of 𝑐𝑐 real coefficients. The flow density solution can be expressed as: 𝜔𝜔 𝑓𝑓 = 𝜔𝜔 𝛿𝛿𝑓𝑓 + 𝜔𝜔 ℎ𝑓𝑓 = 𝜔𝜔 𝛿𝛿𝑓𝑓 + Ω 𝑓𝑓 𝛼𝛼 (13) In this case, cut laws admit an infinity of solutions governed by the nullspace of the flow equilibrium matrix. This is important when studying the redundancy of the network and its ability to sustain damage. The flow equilibrium matrix 𝐴𝐴 𝛿𝛿𝛿𝛿 is a 𝑑𝑑𝑛𝑛 𝑣𝑣 × 𝑛𝑛 𝑒𝑒 matrix where each column corresponds to an edge and each row describes an elementary cut of a node in a given dimension. The nullspace of 𝐴𝐴 𝛿𝛿𝛿𝛿 exists if, and only if, 𝐴𝐴 𝛿𝛿𝛿𝛿 has redundant columns (edges). Therefore, the existence of a flow mode in the network reflects that the network has more edges than required for flow admission. In other words, each vector in Ω 𝑓𝑓 corresponds to a different flow path inside the network. The different flow modes correspond to different independent cycles of the network graph when the potential is one-dimensional. For electrical circuits, this indicates the existence of duplicate components and that the loss of one component does not necessarily lead to the failure of the electrical circuit as a whole. In structural systems, the existence of multiple flow modes indicates that the structure is indeterminate having multiple load paths, while in transportation networks, multiple flow modes reflect the existence of multiple distribution paths from one point to another. Analogously, the potential equilibrium matrix 𝐴𝐴 𝑓𝑓 can have a nullspace depending on its rank. In this case, for a given topology and flow vector, the circuit laws admit an infinite number of solutions governed by the nullspace of the potential equilibrium matrix 𝐴𝐴 𝑓𝑓 . Let Ω 𝛿𝛿𝛿𝛿 be the collection of the 𝑐𝑐 basis vectors of 𝐴𝐴 𝑓𝑓 , then: 𝜔𝜔 𝛿𝛿𝛿𝛿 = 𝜔𝜔 𝛿𝛿𝛿𝛿𝛿𝛿 + 𝜔𝜔 ℎ𝛿𝛿𝛿𝛿 = 𝜔𝜔 𝛿𝛿𝛿𝛿𝛿𝛿 + Ω 𝛿𝛿𝛿𝛿 𝛼𝛼 (14) The existence of these infinite solutions reflects that a given flow could be associated to multiple potentials, hence the potential can continuously change without affecting the flow. An example of this feature can be found in the structural analysis of self-stressed networks, where the existence of potential modes may indicate the existence of infinitesimal mechanism in the structure. 3.1. Cellular structure of self-equilibrated networks As established in previous sections, the algebraic structure of the cut laws and circuit laws solutions forms a vector space. Consequently, the behavior of the network equilibrium model subject to external perturbations and/or damage (i.e., member removal) can be predicted by analyzing the topology of the network and the potential at each node. De Guzmán and Orden mathematically proved that all self-equilibrated frameworks are composed of elementary cells [38], while Aloui et al. developed a bio-inspired generative approach to design and analyze self-stressed frameworks embedded in two-dimensional and three-dimensional spaces by decomposing the underlying graph to elementary units called cells [35-37]. They showed that the basis for the flow space ℱ can be escribed solely by getting the cellular structure of the graph. In this paper, the approach is generalized to any arbitrary potential dimension. Let 𝐺𝐺 ( 𝑉𝑉 , 𝐸𝐸 ) be a graph that describes the set of nodes 𝑉𝑉 and the set of edges 𝐸𝐸 of a system. Consider 𝑓𝑓 as the flow variable attributed to the edges, 𝜔𝜔 as the corresponding flow density, and 𝑝𝑝 as the potential attributed to the nodes. The graph 𝐺𝐺 is embedded in a 𝑑𝑑 -dimensional space. A cell is defined as the complete graph on 𝑑𝑑 + 2 nodes that has a one-dimensional flow space. Figure 2 shows the cell topology in a one-, two-, three- and four-dimensional space. Figure 2: Illustration of cell topologies in a one-, two-, three- and four-dimensional space.
Complex self-equilibrated frameworks can be obtained through cellular multiplication and the mechanisms of adhesion and fusion (Figure 3) [35-37]. Adhesion represents the combination of two cells without removing any shared edges (the underlying graphs are glued together), while fusion refers to the combination of two cells with the removal of one or more of their shared edges. Consequently, adhesion increases the flow space dimension ℱ and the number of flow modes in the network, while fusion reduces the dimension of the flow space ℱ of the network and the number of flow modes. It should also be noted that in cellular morphogenesis, there is a distinction between cells and unicellular organisms, with cells having always the same topological structure (a complete graph), while unicellular organisms represent network structures with one flow mode. Unicellular organisms are required to obtain a complete description of the flow space. Figure 3: Illustration of the adhesion and fusion mechanisms using two-dimensional cells.
The cellular structure of a self-equilibrated network model refers to the series of cells and unicellular organisms that through adhesion compose the network. A decomposition algorithm for self-equilibrated network models, that gives a cellular structure of a 𝑑𝑑 -dimensional network model based on 𝑑𝑑 -dimensional cells was proposed in [37]. Figure 4 shows the cellular structure of two examples of network models that have the same underlying graph but are embedded in two different dimensions ( 𝑑𝑑 = 1, 𝑑𝑑 = 2 ). Figure 4: Cellular structure of a one-dimensional embedding (left) and two-dimensional embedding (right) of the same network.
Flow modes and space Flow modes are defined as the vector basis for the flow space ℱ of a self-equilibrated network. They thus represent the vector solution for the flow variables that satisfy the cut laws. In cellular orphogenesis, flow modes have a more direct interpretation with every flow mode corresponding to a cell or unicellular organism composing the network. In previous work, Aloui et al. proposed an analytical solution for the flow mode of a two-dimensional and a three-dimensional cell [33, 34]. They showed that the flow densities of the cells in two dimensions and three dimensions can be obtained through the product of the signed volume of two specific oriented two-simplices and three-simplices, respectively. In this paper, using algebraic geometry, a generalization of the analytical solutions for the flow mode for cells of an arbitrary dimension is proposed. A detailed proof of the generalization can be found in Appendix B. It is shown that, in a 𝑑𝑑 -dimensional space, this result can be generalized to the product of the signed volumes of two specific oriented 𝑑𝑑 -simplices each defined by an ordered set of 𝑑𝑑 + 1 nodes in the cell. Let 𝜔𝜔 𝑖𝑖𝑖𝑖 be the flow density of the edge ( 𝑣𝑣 𝑖𝑖 , 𝑣𝑣 𝑖𝑖 ) , 𝑆𝑆 𝑖𝑖 = �𝑣𝑣 𝑖𝑖 , ( 𝑣𝑣 𝑘𝑘 ) � and 𝑆𝑆 𝑖𝑖 = �𝑣𝑣 𝑖𝑖 , ( 𝑣𝑣 𝑘𝑘 ) � the two ordered sets of vertices representing the 𝑑𝑑 -simplices at 𝑣𝑣 𝑖𝑖 and 𝑣𝑣 𝑖𝑖 , and 𝑉𝑉 ( 𝑆𝑆 ) the function that returns the oriented volume of the oriented simplex 𝑆𝑆 = �𝑣𝑣 𝛿𝛿 , … 𝑣𝑣 𝛿𝛿𝑥𝑥+1 � where { 𝛿𝛿 , 𝛿𝛿 , … , 𝛿𝛿 𝑥𝑥+1 } is a specific order of the nodes. 𝑃𝑃 𝑘𝑘 = ( 𝑃𝑃 , … , 𝑃𝑃 𝑥𝑥𝛿𝛿 ) is the d-dimensional potential associated to 𝑣𝑣 𝛿𝛿 . The signed volume 𝑉𝑉 ( 𝑆𝑆 ) is thus given by: 𝑉𝑉 ( 𝑆𝑆 ) = 1 𝑑𝑑 ! �� 𝑃𝑃 … 𝑃𝑃 𝑥𝑥𝛿𝛿 𝑃𝑃 … 𝑃𝑃 𝑥𝑥𝛿𝛿 ⋮ ⋮ ⋮ ⋮ 𝑃𝑃 𝑑𝑑+1 … 𝑃𝑃 𝑥𝑥𝛿𝛿 𝑑𝑑+1 �� (15) Consequently, the flow density 𝜔𝜔 𝑖𝑖𝑖𝑖 can be obtained as: 𝜔𝜔 𝑖𝑖𝑖𝑖 = 𝑉𝑉 �𝑣𝑣 𝑖𝑖 , ( 𝑣𝑣 𝛿𝛿 ) � 𝑉𝑉 �𝑣𝑣 𝑖𝑖 , ( 𝑣𝑣 𝛿𝛿 ) � (16) By applying Equation 16 on all edges �𝑣𝑣 𝑖𝑖 , 𝑣𝑣 𝑖𝑖 � ∈ 𝐸𝐸 , one can obtain the analytical expression of the flow mode for a cell in a 𝑑𝑑 -dimensional space. Once an analytical solution for the flow density modes of the cells is obtained, a basis for the flow space ℱ can be constructed considering the cellular structure of the network model. Each cell composing the network has its own flow mode and represents a component of the basis of the flow space ℱ . However, for the basis to be complete flow modes corresponding to unicellular structures have to also be considered. Flow modes corresponding to unicellular structures can be calculated using the fusion principles. When two cells undergo fusion (removal of one or more shared edges), the resulting network will have one flow density mode. Since each cell has one flow density mode, fusion can be thought of as finding the specific linear combination of flow modes of the two cells that cancels the flow density in the removed edges. Since every flow mode is defined to a constant, finding this specific combination is always possible when a single edge needs to be removed. However, when the number of removed edges is larger or equal to two, the potentials attributed to the nodes become degenerate collapsing to a lower dimensional space. Figures 7 and 8 show the ellular structures and flow density spaces ℱ of the examples presented in the previous section (Figure 4) with flow density modes grouped into matrix Ω . Figure 5: Cellular structure and flow density space ℱ of a network embedded in a one-dimensional space. Figure 6: Cellular structure and flow density space ℱ of a network embedded in a two-dimensional space. Equilibrium and damage analysis in self-equilibrated networks
In this section, equilibrium and damage in self-equilibrated networks are discussed through a series of examples. The examples, selected for their generality and brevity, represent network applications from different fields to highlight the applicability of the method in different contexts. 4.1.
Equilibrium of self-equilibrated networks in the absence of external perturbation Consider the electrical circuit illustrated in Figure 7. This circuit is known as the Wheatstone bridge and it is frequently used in electrical engineering for the precise measurement of an unknown electrical resistance through the balancing of two “arms” of a bridge circuit. It is also often used along with an operational amplifier to measure physical parameters such as temperature or strain, while variations of the bridge can also measure capacitance, inductance and impedance. Here, the topology of the circuit and its equilibrium state are studied through the analysis of the cellular structure of the network corresponding to the circuit (Figure 7).
Figure 7: Wheatstone bridge electrical circuit and its topology.
The network equilibrium model corresponding to Wheatstone bridge is one-dimensional, with each node 𝑣𝑣 𝑖𝑖 being associated with a scalar voltage potential 𝑉𝑉 𝑖𝑖 . Moreover, flow attributes in this model corresponds to the currents 𝐼𝐼 𝑖𝑖𝑖𝑖 in each edge �𝑣𝑣 𝑖𝑖 , 𝑣𝑣 𝑖𝑖 � . Figure 8 shows the cells composing the network and the corresponding flow density modes calculated through the expressions developed in Section 3.2. Figure 8: Cellular structure and flow density modes for the network equilibrium model corresponding to Wheatstone bridge electrical circuit.
The dimension of the flow density space ℱ is three, given by the number of cells and unicellular structures composing the network. Any current density that can flow in the circuit is thus a linear combination of the three flow density modes given in Figure 8. This implies that the circuit is edundant and can withstand the loss of up to two edges provided that the damage does not affect the current or the voltage source. With each edge �𝑣𝑣 𝑖𝑖 , 𝑣𝑣 𝑖𝑖 � characterized by its own impedance 𝑅𝑅 𝑖𝑖𝑖𝑖 , the network equilibrium model of Wheatstone bridge is described by: Cut-set laws ( current laws ): (17) � 𝐼𝐼 𝑖𝑖𝑖𝑖𝑖𝑖 𝑠𝑠 . 𝑡𝑡 . ( 𝑖𝑖 , 𝑖𝑖 ) ∈𝐸𝐸 = 0 , 1 ≤ 𝑐𝑐 ≤ Circuit laws (v oltage laws ): � �𝑉𝑉 𝑖𝑖 − 𝑉𝑉 𝑖𝑖 � ( 𝑖𝑖 , 𝑖𝑖 ) ∈∁ = 0 , ∁ ∈ [[1,2,3], [1,2,4], [1,3,4]] Flow and potential relation (
Ohm’s law ): 𝑟𝑟 ( 𝛿𝛿𝑉𝑉 , 𝐼𝐼 ) = 𝛿𝛿𝑉𝑉 − 𝑅𝑅𝐼𝐼 = 0 𝐸𝐸 represents the set of edges of the circuit network. ∁ refers to a cycle in the cycle space of the network. 𝑅𝑅 is a diagonal matrix where each diagonal entry is the impedance 𝑅𝑅 𝑖𝑖𝑖𝑖 of the corresponding edge �𝑣𝑣 𝑖𝑖 , 𝑣𝑣 𝑖𝑖 � . In a matrix form, the self-equilibrium of the circuit is described as: Cut-set laws ( current laws ): (18) 𝐵𝐵 𝐼𝐼 = �
1 1 1 0 0 01 0 0 − − − −
1 0 1 � ⎝⎜⎜⎛𝐼𝐼 𝐼𝐼 𝐼𝐼 𝐼𝐼 𝐼𝐼 𝐼𝐼 ⎠⎟⎟⎞ =0 Circuit laws (v oltage laws ): 𝐶𝐶 𝐼𝐼 = � −
1 0 1 0 01 0 −
1 0 1 00 1 −
1 0 0 1 � ⎝⎜⎜⎛𝑉𝑉 − 𝑉𝑉 𝑉𝑉 − 𝑉𝑉 𝑉𝑉 − 𝑉𝑉 𝑉𝑉 − 𝑉𝑉 𝑉𝑉 − 𝑉𝑉 𝑉𝑉 − 𝑉𝑉 ⎠⎟⎟⎞ =0 Flow and potential relation (
Ohm’s law ): 𝑟𝑟 ( 𝛿𝛿𝑉𝑉 , 𝐼𝐼 ) = ⎝⎜⎜⎛𝑉𝑉 − 𝑉𝑉 𝑉𝑉 − 𝑉𝑉 𝑉𝑉 − 𝑉𝑉 𝑉𝑉 − 𝑉𝑉 𝑉𝑉 − 𝑉𝑉 𝑉𝑉 − 𝑉𝑉 ⎠⎟⎟⎞ − ⎝⎜⎜⎛𝑅𝑅 𝑅𝑅 𝑅𝑅 𝑅𝑅 𝑅𝑅
00 0 0 0 0 𝑅𝑅 ⎠⎟⎟⎞ ⎝⎜⎜⎛𝐼𝐼 𝐼𝐼 𝐼𝐼 𝐼𝐼 𝐼𝐼 𝐼𝐼 ⎠⎟⎟⎞ = 0 The self-equilibrium of the circuit in Equation 18 can thus be used to explain the principle of the Wheatstone bridge which is based on null deflection. 4.2.
Equilibrium of self-equilibrated networks under external perturbation The example of a self-stressed three-dimensional pin-jointed framework consisting of elements in compression (bars) and elements in tension (cables) is analyzed using network equilibrium modeling. Self-equilibrated axially loaded structures, also known as tensegrity structures, have been proposed for a variety of applications in science and engineering from cellular modeling to obotics. Moreover, they are statically indeterminate structures (i.e. they contain multiple load paths) with often multiple self-stress states. Here, the proposed framework consists of two three-dimensional cells combined through adhesion. For simplicity, all bars and cables are assumed to have the same cylindrical cross-sectional areas of cm and cm , respectively. They also have the same material properties with a Young modulus 𝐸𝐸 𝑏𝑏𝑏𝑏𝑟𝑟 = 𝐺𝐺𝑃𝑃𝑙𝑙 and 𝐸𝐸 𝑐𝑐𝑏𝑏𝑏𝑏𝑐𝑐𝑒𝑒 = 30 𝐺𝐺𝑃𝑃𝑙𝑙 . Figure 9 illustrates the configuration and the type of elements in the structure while Figure 10 shows its corresponding underlying graph along with the three-dimensional cellular structure.
Figure 9: Configuration of the structure embedded in a three-dimensional space along with its cellular structure.
Figure 10: Underlying abstract graphs for the structure illustrated in Figure 9.
The cellular structure of this three-dimensional network model reveals that the structure is composed of two cells. Consequently, the flow density space has two flow density modes. The cells and their corresponding flow density modes are shown in Figure 11. The flow density variable in this case corresponds to the self-stress inside the structure. Any self-stress state of the structure will thus be a linear combination of these two self-stress modes. The model can thus explain and decipher statical indeterminacy in a tensegrity structure.
Figure 11: Flow density modes of the structure in the initial configuration
The structure is initially in equilibrium under the effect of prestress introduced by applying a relative shortening
𝛥𝛥𝑐𝑐𝑐𝑐 of the cables of −3 (Figure 12). This state of self-equilibrium is described by the vector of internal forces 𝑓𝑓 which can be expressed as a linear combination of the flow modes. Figure 12: Illustration of the new equilibrium configuration under prestress and the corresponding flow modes
Now, consider that an external load
𝐹𝐹⃗ = − 𝑧𝑧⃗ 𝐾𝐾𝐾𝐾 is applied to the structure at node 𝑣𝑣 with the vertical displacements of nodes 𝑣𝑣 , 𝑣𝑣 and 𝑣𝑣 blocked generating the reactions 𝑅𝑅�⃗ =4.8 𝑧𝑧⃗ 𝐾𝐾𝐾𝐾 , 𝑅𝑅�⃗ = 2.6 𝑧𝑧⃗ 𝐾𝐾𝐾𝐾 and
𝑅𝑅�⃗ = 2.6 𝑧𝑧⃗ 𝐾𝐾𝐾𝐾 (Figure 13). The equilibrium of the structure implies that the sum of applied forces and reactions is equal to zero and that the sum of the moments with respect to a point in space is zero. The network equilibrium model for this system is given by the nodal equilibrium and geometric compatibility equations representing cut-set and circuit laws, respectively, while Hooke’s laws applied at each element of the structure correspond to the potential-flow relations (Equation 19).
Figure 13: Illustration of the new equilibrium configuration under perturbation and the corresponding flow modes.
Cut laws can be determined by isolating the nodes through elementary cut-sets. Circuit laws have to be expressed in a basis of the cycle space 𝒞𝒞 . Since cycles can be viewed as one-dimensional cells and unicellular organisms, a basis for the cycle space 𝒞𝒞 can be found through the cellular structure of the corresponding graph. Figure 14 illustrates the cycle space basis for the example. Cut-set laws (nodal equilibrium): (19) � 𝑓𝑓⃗ 𝑖𝑖𝑖𝑖𝑖𝑖 𝑠𝑠 . 𝑡𝑡 .( 𝑖𝑖 , 𝑖𝑖 ) ∈𝐸𝐸 = 𝐹𝐹 𝑖𝑖 , 1 ≤ 𝑐𝑐 ≤ Circuit laws (geometric compatibility): � �𝛿𝛿𝑃𝑃�⃗ 𝑖𝑖 − 𝛿𝛿𝑃𝑃�⃗ 𝑖𝑖 � ( 𝑖𝑖 , 𝑖𝑖 ) ∈∁ = 0 , 𝐶𝐶 ∈ [{1,2,3}, {1,3,4}, {1,4,5}, {2,3,4}, {2,4,5}, {2,5,6}, {3,4,5}, {3,5,6}, {4,5,6}]
Flow and potential relation (Hooke’s law): 𝑓𝑓 ( 𝑖𝑖 , 𝑖𝑖 ) − 𝐾𝐾 ( 𝑖𝑖 , 𝑖𝑖 ) ��𝑃𝑃�⃗ ( 𝑖𝑖 , 𝑖𝑖 ) � − �𝑃𝑃�⃗ ( 𝑖𝑖 , 𝑖𝑖 ) ��������������� 𝛿𝛿𝑐𝑐 ( 𝑖𝑖 , 𝑗𝑗 ) =𝑐𝑐 ( 𝑖𝑖 , 𝑗𝑗 ) −𝑐𝑐 ( 𝑖𝑖 , 𝑗𝑗 ) = 0, ∀ ( 𝑐𝑐 , 𝑗𝑗 ) ∈ 𝐸𝐸 Figure 14: Cycle-space basis based on one-dimensional cellular structure of the network.
In Equation 19, 𝐶𝐶 represents a cycle basis of the cycle space 𝒞𝒞 . 𝑃𝑃�⃗ 𝑖𝑖 is the position vector of node 𝑣𝑣 𝑖𝑖 . 𝛿𝛿𝑃𝑃�⃗ 𝑖𝑖 is the displacement of node 𝑣𝑣 𝑖𝑖 . 𝑃𝑃�⃗ ( 𝑖𝑖 , 𝑖𝑖 ) is the vector representation of the member ( 𝑐𝑐 , 𝑗𝑗 ) subjected to perturbations. 𝑓𝑓 ( 𝑖𝑖 , 𝑖𝑖 ) is the normal force of the member ( 𝑐𝑐 , 𝑗𝑗 ) . ‖ ⋅ ‖ is the Euclidean norm of a vector. 𝑙𝑙 ( 𝑖𝑖 , 𝑖𝑖 ) and 𝑙𝑙 ( 𝑖𝑖 , 𝑖𝑖 ) are the actual length and rest length of the member ( 𝑐𝑐 , 𝑗𝑗 ) and 𝐾𝐾 ( 𝑖𝑖 , 𝑖𝑖 ) = 𝐸𝐸 ( 𝑖𝑖 , 𝑗𝑗 ) 𝐴𝐴 ( 𝑖𝑖 , 𝑗𝑗 ) 𝑐𝑐 ( 𝑖𝑖 , 𝑗𝑗 ) is the normal stiffness of member ( 𝑐𝑐 , 𝑗𝑗 ) where 𝐸𝐸 ( 𝑖𝑖 , 𝑖𝑖 ) is its Young modulus and 𝐴𝐴 ( 𝑖𝑖 , 𝑖𝑖 ) is its cross section. In matrix form, the equilibrium of the structure is described by Equations 20. Cut-set laws (nodal equilibrium): (20) 𝐴𝐴 𝛿𝛿𝛿𝛿 𝜔𝜔 = 𝐹𝐹 𝐴𝐴 𝛿𝛿𝛿𝛿 is the equilibrium matrix calculated using equation 15 Circuit laws (geometric compatibility): ( 𝕀𝕀 𝑥𝑥 ⨂𝐶𝐶 ) 𝛿𝛿𝑃𝑃 = ( 𝕀𝕀 𝑥𝑥 ⨂𝐶𝐶 ) [( 𝕀𝕀 𝑥𝑥 ⨂𝐵𝐵 𝑇𝑇 ) ( 𝑃𝑃 − 𝑃𝑃 )] =0 𝐶𝐶 is the cycle space matrix, 𝐵𝐵 is the node to branch adjacency matrix, 𝛿𝛿𝑃𝑃 = �𝛿𝛿𝑋𝑋𝛿𝛿𝛿𝛿𝛿𝛿𝛿𝛿� and 𝑃𝑃 = �𝑋𝑋 𝛿𝛿 𝛿𝛿 � where 𝛿𝛿𝑋𝑋 , 𝛿𝛿𝛿𝛿 and 𝛿𝛿𝛿𝛿 are the vectors of the nodal displacements and 𝑋𝑋 , 𝛿𝛿 and 𝛿𝛿 are the vectors of the initial coordinates of the nodes. Flow and potential relation (Hooke’s law): 𝑓𝑓 − 𝐾𝐾 ( ‖𝑃𝑃 𝑣𝑣𝑒𝑒𝑐𝑐 ‖ 𝑣𝑣𝑒𝑒𝑐𝑐 − ‖𝑃𝑃 𝑣𝑣𝑒𝑒𝑐𝑐0 ‖ 𝑣𝑣𝑒𝑒𝑐𝑐 ) ��������������� 𝛿𝛿𝑐𝑐=𝑐𝑐−𝑐𝑐 = 0 is a
14 × 1 vector of the internal forces of the members. 𝐾𝐾 is the stiffness matrix of the structure. 𝑃𝑃 𝑣𝑣𝑒𝑒𝑐𝑐 =[ 𝑋𝑋 𝑣𝑣𝑒𝑒𝑐𝑐 𝛿𝛿 𝑣𝑣𝑒𝑒𝑐𝑐 𝛿𝛿 𝑣𝑣𝑒𝑒𝑐𝑐 ] and 𝑃𝑃 𝑣𝑣𝑒𝑒𝑐𝑐0 = [ 𝑋𝑋 𝑣𝑣𝑒𝑒𝑐𝑐0 𝛿𝛿 𝑣𝑣𝑒𝑒𝑐𝑐0 𝛿𝛿 𝑣𝑣𝑒𝑒𝑐𝑐0 ] where 𝑋𝑋 𝑣𝑣𝑒𝑒𝑐𝑐 , 𝛿𝛿 𝑣𝑣𝑒𝑒𝑐𝑐 and 𝛿𝛿 𝑣𝑣𝑒𝑒𝑐𝑐 are
14 × 1 vectors reflecting the components of the vector representations of the members in the new configuration and 𝑋𝑋 𝑣𝑣𝑒𝑒𝑐𝑐0 , 𝛿𝛿 𝑣𝑣𝑒𝑒𝑐𝑐0 and 𝛿𝛿 𝑣𝑣𝑒𝑒𝑐𝑐0 are
14 × 1 vectors reflecting the components of the vector representations of the members in the initial configuration. ‖⋅‖ 𝑣𝑣𝑒𝑒𝑐𝑐 is the row vector wise norm of a matrix. 𝑙𝑙 is a
14 × 1 vector representing the actual lengths of the members and 𝑙𝑙 represents the rest lengths of the members . When the structural system is subjected to perturbations, the nodal positions will adjust to the new equilibrium. Consequently, the flow density space will change. However, the cellular structure of the network remains the same implying that the network has always two flow density modes. The equilibrium state of the network can thus be described through a homogeneous solution describing the self-equilibrium in the absence of external loading and a particular solution which reflects the effect of the load [37]. Since flow-potential relations 𝑟𝑟 ( 𝑃𝑃 , 𝑓𝑓 ) are non-linear with respect to the configuration (geometry) of the structure 𝑃𝑃 , finding the equilibrium configuration of the structure under the effect of the external load requires the use of an appropriate numerical method [39-41]. In this paper, a dynamic relaxation algorithm [41] was employed to calculate the positions of the nodes and the internal forces in the elements of the structure under the effect of the external load 𝐹𝐹⃗ and the reactions
𝑅𝑅�⃗ , 𝑅𝑅�⃗ and 𝑅𝑅�⃗ considering all z-displacements on the basis nodes as blocked. In this analysis, self-weight was neglected and prestress in the cables was induced by elongation of 0.01 % of their rest length. Figure 13 illustrates the new equilibrium configuration and the new flow density modes. The model can thus be used to analyze self-stressable pin-jointed frameworks and explain their behavior under loading. 4.3. Equilibrium of self-equilibrated networks under damage In this section, the impact of damage (element removal) is assessed in a self-equilibrated network through the analysis of its cellular structure. Element removal can be thought of as the result of a fusion on the removed edge. Consequently, the number of cells and thus the number of flow modes decrease, reducing also the redundancy in the network. Element removal can thus be reflected by the fusion or the necrosis of cell. In this example, a two-dimensional network is analyzed. Figure 15 describes the network, its associated potential, cellular structure and corresponding flow density modes.
Figure 15: Illustration of the underlying graph of the network along with its associated cellular structure and flow density modes.
Let 𝜔𝜔 be the initial flow density that the network has under the potential values included in Figure 16. 𝜔𝜔 satisfies the self-equilibrium conditions and is a linear combination of the flow density modes provided in Figure 15: 𝜔𝜔 = 𝜔𝜔 { , , , } + 𝜔𝜔 { , , , } + 2 𝜔𝜔 { , , , } . Figure 16: Initial flow 𝑓𝑓 inside the network before damage. When an edge of the network is damaged (removed), the flow in the network can be adjusted according to where the damage occurs and to the system being modeled. Assume that edge (1,5) is removed. Topologically, the damage of edge (1,5) is described by the necrosis of cell {1,3,4,5} . The number of composing cells and thus the dimension of the flow space ℱ will decrease from three to two. The same effect occurs with the damage of edge (3,4) which can be described opologically by the fusions of cells {1,2,3,4} + {1,3,4,5} and cells {1,2,3,4} + {2,3,4,5} . Figure 17 shows the cellular structure of the damaged networks, along with their corresponding flow density modes. Figure 17: Cellular structure and flow density modes in the case of damage in edges (1,5) and (3,4).
Now, assume that the system being modeled has an additional constraint expressed by a desire to maintain flow at the same level before damage. Since the network has two redundant edges after damage, this is only possible for two edges. Considering the damage of edge (3,4), the flow in edges (1,5) and (2,5) can be kept the same with the new equilibrium given by: �− − � �𝛼𝛼 𝛼𝛼 � = � � (21) which gives 𝛼𝛼 = − and 𝛼𝛼 = − . Note that flow values are divided by the geometric length of the corresponding edge to get the flow densities. The new flow on the system is represented by Figure 18. Figure 18: Flow after damage of edge (3,4) and with the consideration of maintaining flow in edges (1,5) and (2,5) at the same level prior to damage.
In this example, the network and the constraint are chosen for simplicity. However, in a real system the designer or decision maker can choose any constraint or objective function to direct flow distribution with the flow problem, after damage being reduced into the identification of the appropriate 𝛼𝛼 and 𝛼𝛼 instead of optimizing with respect to the nine flow variables at each edge. Moreover, it should also be noted that there are systems, like the pin-jointed framework analyzed in the previous example, where the potential and flow values are closely related, adjusting simultaneously to a new equilibrium position after damage. In these cases, flow-potential relations can be used to simulate change and find the new equilibrium configuration of the network. However, changes in the number of flow density modes can already be identified through a review of the cellular structure of the network. Discussion
In summary, the paper focuses on self-equilibrated networks offering a new approach for their modeling and analysis. The proposed model is composed by a set of cut-set and circuit laws along with a set of flow and potential relations, while its analysis is conducted in terms of the cellular structure of the network. Cells refer to unitary network sub-systems that have a one-dimensional flow space with their dimension depending on the embedding of the network. An analytical expression of the flow mode of cells in a 𝑑𝑑 -dimensional space is provided. This expression combined with the cellular structure of a self-equilibrated network allows to analyze the equilibrium state in the network through the study of individual cell equilibria and their interactions. Self-equilibrated networks represent thus networks where cut-laws have a non-zero homogeneous solution. The system can thus be in a state of self-equilibrium in the absence of external perturbations with flow modes corresponding to different independent cycles of the network graph when the potential is one-dimensional. This implies damage-tolerance and resilience in the system as the network includes different flow paths. Given the wide range of physical and engineering systems that are depicted through network models, this modeling approach can thus have great impact in a variety of contexts. Here, the model is explored to describe the equilibrium in three network applications selected for their generality and brevity. The first example is the well-known Wheatstone bridge which is employed to measure electrical resistance as well as other physical parameters. It is shown that the model can be used to explain the functioning principle of Wheatstone bridge. The second example corresponds to a self-stressable pin-jointed structure composed of bars and cables. This type of tructures, also known as tensegrity, are statically indeterminate structures. It is shown that the model proposed can be used to explain their self-equilibrium as well as their behavior under loading. The third example focuses on the impact of damage (element removal) in a self-equilibrated network, where it is shown that the model can be used to identify changes in the number of flow density modes in the network through a simple review of its cellular structure. Since the cellular structure of the network is invariant of the load case and can accommodate the impact of damage (element removal) through cell fusion and/or necrosis, it provides an always topologically valid basis for the description for the network. Therefore, the proposed cellular approach represents a systematic and general approach for the analysis of self-equilibrated networks in science and engineering applications. Acknowledgements and Funding Statement
This material is based upon work supported by the
National Science Foundation , United States under Grant No. . David Orden has been partially supported by project
MTM2017-83750-P of the Spanish Ministry of Science (AEI/FEDER, UE), project
PID2019-104129GB-I00 of the Spanish Ministry of Science and Innovation, and by H2020-MSCA-RISE project, European Commission , while Nizar Bel Hadj Ali gratefully acknowledges the financial support of the Fulbright Visiting Scholar Program for the academic year 2018-2019. References [1] Euler, L., 1741. Solutio problematis ad geometriam situs pertinentis. Commentarii academiae scientiarum Petropolitanae, pp.128-140. [2] Kirchhoff, S., 1845. On the passage of an electric current through a plane, in particular through a circular one. Annalen der Physik , 140 (4), pp.497-514. [3] Kirchhoff, G., 1847. On the solution of the equations to which one is led in the investigation of the linear distribution of galvanic currents. Annalen der Physik , 148 (12), pp.497-508. [4]
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Let 𝐺𝐺 ( 𝑉𝑉 , 𝐸𝐸 ) be a graph that describes the set of nodes 𝑉𝑉 and the set of edges 𝐸𝐸 of a network and 𝑛𝑛 𝑣𝑣 be the number of vertices and 𝑛𝑛 𝑒𝑒 the number of edges in the graph. The graph is equipped with a set of node and edge attributes that satisfy the equilibrium conditions referred to, in graph theory, as circuit laws and cut-set laws. In this paper, node attributes are referred to as potential attributes; they are denoted by a 𝑛𝑛 𝑣𝑣 × 1 vector 𝑝𝑝 and they satisfy circuit laws. Note that each component 𝑝𝑝 𝑖𝑖 represents the value of the potential at node 𝑣𝑣 𝑖𝑖 which in turn is a 𝑑𝑑 × 1 vector where 𝑑𝑑 represents the dimensionality of the problem. Edge attributes are referred to as flow attributes; they are denoted by a 𝑛𝑛 𝑒𝑒 × 1 vector 𝑓𝑓 and they satisfy cut-set laws. 7.1. Network laws 7.1.1.
Cut-set laws A cut-set is a partition of the vertices of the graph 𝑉𝑉 into two disjoint sets 𝑆𝑆 and 𝑆𝑆̅ . Cut sets can be described in the case of planar graphs by a closed cutting curve Δ that divides the plane of the graph into two regions: inside and outside (Figure 1). Analogously, If the graph is not planar, then Δ represents a closed hyper-surface dimension 𝑑𝑑 − if the graph is embedded in a 𝑑𝑑 -dimnesional space. An orientation can be given to the cut Δ : inside outside ( Δ in Figure A.1) or the reverse ( Δ in Figure A.1). Figure A.1: Illustration of a cut-set.
Let 𝑓𝑓 : 𝐸𝐸 → ℝ 𝑥𝑥 be a real valued function on the edges of the graph 𝐺𝐺 , where 𝑑𝑑 is the dimension of the problem. The values that 𝑓𝑓 takes on each edge 𝑐𝑐 ∈ 𝐸𝐸 are referred to as the edge attributes of the graph. By convention, an edge attribute is positive if it is defined on an edge 𝑐𝑐 ( 𝑢𝑢 , 𝑣𝑣 ) with 𝑢𝑢 ∈𝑆𝑆 𝑙𝑙𝑛𝑛𝑑𝑑 𝑣𝑣 ∈ 𝑆𝑆̅ , and negative if it is the opposite. In this paper, if Δ is a cut [ 𝑆𝑆 , 𝑆𝑆̅ ] , 𝑓𝑓 ( Δ ) will denote the sum of the edge attributes having one end node in 𝑆𝑆 and the other one in 𝑆𝑆̅ . Furthermore, 𝑓𝑓 obeys a cut-set law if, and only if, for any cut-set Δ = [ 𝑆𝑆 , 𝑆𝑆̅ ] , the edge attributes having one node in 𝑆𝑆 and the other node in 𝑆𝑆̅ sum up to zero, i.e., 𝑓𝑓 ( Δ ) = 0. Circuit laws circuit is a closed trail of the graph (i.e., a path with the first vertex being also the last one). A circuit 𝐶𝐶 is denoted by the set of nodes that forms it, taken in the order they are visited, and it can be represented by the set of edges in 𝐶𝐶 and an orientation (Figure A.2). Figure A.2: Illustration of a circuit
Let 𝑝𝑝 : 𝑉𝑉 → ℝ 𝑥𝑥 be a real valued function on the vertices of the graph 𝐺𝐺 , where 𝑑𝑑 is the dimension of the problem. The values that 𝑝𝑝 takes on each vertex are referred to as node attributes of the graph. Let 𝛿𝛿𝑝𝑝 : 𝐸𝐸 → ℝ be the function that returns the node-attribute difference between the end nodes 𝑢𝑢 and 𝑣𝑣 of the edge ( 𝑢𝑢 , 𝑣𝑣 ) , i.e., 𝛿𝛿𝑝𝑝� ( 𝑢𝑢 , 𝑣𝑣 ) � = 𝑝𝑝 𝑢𝑢 − 𝑝𝑝 𝑣𝑣 . Such a 𝑝𝑝 obeys a circuit law if, and only if, for any circuit 𝐶𝐶 in the graph, the node attributes differences 𝛿𝛿𝑝𝑝 sum up to zero over all the edges in the circuit, i.e., ∑ 𝛿𝛿𝑝𝑝� ( 𝑢𝑢 , 𝑣𝑣 ) � = 0 ( 𝑢𝑢 , 𝑣𝑣 ) ∈𝐶𝐶 . 7.2. Network attributes In addition to its topological structure (nodal connectivity), a graph can be equipped with node and edge attributes that satisfy any given relations governed by the interconnections between the nodes (topology of the graph). When equilibrium is considered, these relations represent a network equilibrium model. Reinschke describes two types of attributes that occur in real-world systems that can be attributed to nodes and edges of a network equilibrium model [A.1] referred to in this paper as flow variables and potential variables. 7.2.1.
Flow attributes 𝑓𝑓 A flow variable, by definition, is an edge attribute that satisfies a cut-set law on the graph. In Figure 1, 𝑓𝑓 is a flow variable. If Δ is considered, then 𝑓𝑓� (2,3) � + 𝑓𝑓� (1,3) � + 𝑓𝑓� (1,4) � + 𝑓𝑓� (3,7) � + 𝑓𝑓� (6,7) � = 0 . Examples of flow variables include currents in electrical systems, internal and external forces in structural systems, and fluxes of quantities in transportation problems. If the closed cut contains only one node, like Δ = � {7}, 𝑉𝑉 \{7} � in Figure A.1, it is referred to as an elementary cut. In structural systems, the flow variable for a single node refers to the nodal equilibrium of forces, while in electrical circuits, it can be interpreted as Kirchhoff 1 st law (current law). In transportation, the cut law for a single node may describe the flow conservation. 7.2.2. Potential attributes 𝑝𝑝 A potential variable is a node attribute that satisfies a circuit law on the graph. In Figure A.2, 𝑝𝑝 is a potential. If circuit [1,2,6,4,1] is considered, then the circuit law describes the relation: 𝛿𝛿𝑝𝑝� (1,3) � + 𝛿𝛿𝑝𝑝� (3,6) � + 𝛿𝛿𝑝𝑝� (6,4) � + 𝛿𝛿𝑝𝑝� (4,1) � = 0 . An elementary circuit for potential attributes represents a cycle. Circuit laws can be used to model Kirchhoff’s second law (voltage law) in electrical circuits or geometric compatibility relations in structures. Consequently, potential variables can represent voltages in electrical circuits, nodal displacements in structural systems, or quantities of goods at a given node in a transportation scheme. 7.3. Network topological and algebraic properties A network is described by a set of vertices 𝑉𝑉 and the connections 𝐸𝐸 between them. However, all information on the connectivity of the network can also be depicted using cut-sets or circuits. Let 𝐺𝐺 ( 𝑉𝑉 , 𝐸𝐸 ) be a directed graph where 𝑉𝑉 is the set of vertices, 𝑛𝑛 𝑣𝑣 the number of nodes, 𝐸𝐸 is the set of edges and 𝑛𝑛 𝑒𝑒 the number of edges. 7.3.1. Potential difference space 𝛿𝛿𝛿𝛿 and Bond space ℬ (cut-set space) Let 𝛿𝛿𝛿𝛿 be the space of all possible potential differences on the graph 𝐺𝐺 . Let 𝛿𝛿𝑝𝑝 , 𝛿𝛿𝛿𝛿 ∈ 𝛿𝛿𝛿𝛿 be two potential differences functions and 𝜆𝜆 ∈ ℝ a scalar. It can be seen that 𝛿𝛿𝛿𝛿 is closed under addition and multiplication by a scalar ( 𝛿𝛿𝑝𝑝 + 𝜆𝜆𝛿𝛿𝛿𝛿 ∈ 𝛿𝛿𝛿𝛿 ) . Consequently, the set of all possible potential differences on the graph constitutes a vector space. In this paper, potential differences associated with cut-sets are of special interest. Let Δ = [S, S � ] be a cut-set, then the potential difference associated to Δ is denoted 𝛿𝛿 p Δ and is defined as: 𝛿𝛿 p Δ � ( 𝑐𝑐 , 𝑗𝑗 ) � = � −
1 0 𝑐𝑐𝑓𝑓𝑐𝑐𝑓𝑓𝑐𝑐𝑓𝑓 ( 𝑐𝑐 , 𝑗𝑗 ) ∈ 𝛥𝛥 𝑙𝑙𝑛𝑛𝑑𝑑 𝑐𝑐𝑙𝑙𝑠𝑠𝑐𝑐 𝑓𝑓𝑟𝑟𝑐𝑐𝑐𝑐𝑛𝑛𝑐𝑐𝑙𝑙𝑐𝑐𝑐𝑐𝑓𝑓𝑛𝑛 ( 𝑐𝑐 , 𝑗𝑗 ) ∈ 𝛥𝛥 𝑙𝑙𝑛𝑛𝑑𝑑 𝑓𝑓𝑝𝑝𝑝𝑝𝑓𝑓𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑓𝑓𝑟𝑟𝑐𝑐𝑐𝑐𝑛𝑛𝑐𝑐𝑙𝑙𝑐𝑐𝑐𝑐𝑓𝑓𝑛𝑛 ( 𝑐𝑐 , 𝑗𝑗 ) ∉ 𝛥𝛥 (A.1) In the rest of the appendix, elementary cut-sets refer to cut-sets that isolate only one vertex 𝑣𝑣 and the associated potential difference will be denoted 𝛿𝛿𝑝𝑝 𝑣𝑣 . In an elementary cut-set, 𝛿𝛿𝑝𝑝 𝑣𝑣 can be simply expressed as: 𝛿𝛿𝑝𝑝 𝑣𝑣 � ( 𝑐𝑐 , 𝑗𝑗 ) � = 𝛿𝛿 𝑖𝑖𝑣𝑣 − 𝛿𝛿 𝑖𝑖𝑣𝑣 (A.2) where 𝛿𝛿 𝑖𝑖𝑖𝑖 is the Kronecker delta. A cut-set Δ can be described by a 𝑛𝑛 𝑒𝑒 vector 𝑏𝑏 Δ with components in { − . If 𝑏𝑏 Δ is indexed by the set of edges, 𝐸𝐸 , then 𝑏𝑏 Δ can be expressed as: 𝑏𝑏 ( 𝑖𝑖 , 𝑖𝑖 ) Δ = � −
1 0 𝑐𝑐𝑓𝑓𝑐𝑐𝑓𝑓𝑐𝑐𝑓𝑓 ( 𝑐𝑐 , 𝑗𝑗 ) ∈ 𝛥𝛥 𝑙𝑙𝑛𝑛𝑑𝑑 𝑐𝑐𝑙𝑙𝑠𝑠𝑐𝑐 𝑓𝑓𝑟𝑟𝑐𝑐𝑐𝑐𝑛𝑛𝑐𝑐𝑙𝑙𝑐𝑐𝑐𝑐𝑓𝑓𝑛𝑛 ( 𝑐𝑐 , 𝑗𝑗 ) ∈ 𝛥𝛥 𝑙𝑙𝑛𝑛𝑑𝑑 𝑓𝑓𝑝𝑝𝑝𝑝𝑓𝑓𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑓𝑓𝑟𝑟𝑐𝑐𝑐𝑐𝑛𝑛𝑐𝑐𝑙𝑙𝑐𝑐𝑐𝑐𝑓𝑓𝑛𝑛 ( 𝑐𝑐 , 𝑗𝑗 ) ∉ 𝛥𝛥 (A.3) Consequently, the cut-set vector 𝑏𝑏 Δ corresponding to Δ in Figure 1 is [0, − − − and 𝑏𝑏 corresponding to Δ = � {7}, 𝑉𝑉 \{7} � is [0,0,0,0,0,0,1,0,1, − . One can see that the cut-set vector is another way of expressing the potential difference function as a row vector whose oordinates are indexed with edges in 𝐸𝐸 . The components of this vector represent the values that the potential difference function takes on each edge ( 𝑐𝑐 , 𝑗𝑗 ) with respect to a cut Δ . The collection of all elementary cut-set row vectors corresponding to all the nodes in 𝑉𝑉 composes the matrix 𝐴𝐴 referred to in graph theory as the node-to-edge incidence matrix of the graph. Let 𝛿𝛿𝑝𝑝 be a potential difference function and 𝑝𝑝 𝑣𝑣 the associated potential at the node 𝑣𝑣 . Let 𝛿𝛿𝑝𝑝 𝑣𝑣 be the elementary potential difference functions associated to the node 𝑣𝑣 . Then for any ( 𝑐𝑐 , 𝑗𝑗 ) ∈ 𝐸𝐸 : � 𝑝𝑝 𝑣𝑣𝑣𝑣∈𝑉𝑉 δ p v � ( 𝑐𝑐 , 𝑗𝑗 ) � = � 𝑝𝑝 𝑣𝑣 �𝛿𝛿 𝑖𝑖𝑣𝑣 − 𝛿𝛿 𝑖𝑖𝑣𝑣 � 𝑣𝑣∈𝑉𝑉 = � 𝑝𝑝 𝑣𝑣 𝛿𝛿 𝑖𝑖𝑣𝑣𝑣𝑣∈𝑉𝑉 − � 𝑝𝑝 𝑣𝑣 𝛿𝛿 𝑖𝑖𝑣𝑣𝑣𝑣 = 𝑝𝑝 𝑖𝑖 − 𝑝𝑝 𝑖𝑖 = 𝛿𝛿𝑝𝑝� ( 𝑐𝑐 , 𝑗𝑗 ) � (A.4) Equation 4 proves that any potential difference can be expressed as a linear combination of the elementary potential differences associated with every node in the graph. Conversely, every linear combination of the elementary potential differences is also a potential difference. In fact, if ∑ 𝛼𝛼 𝑣𝑣 𝛿𝛿𝑝𝑝 𝑣𝑣𝑣𝑣∈𝑉𝑉 is a linear combination of the elementary potential difference states associated to every node, then the function 𝑝𝑝 on 𝑉𝑉 where 𝑝𝑝 ( 𝑣𝑣 ) = 𝛼𝛼 𝑣𝑣 is a potential since 𝛿𝛿𝑝𝑝 obeys a circuit law. Consequently, the potential difference space 𝛿𝛿𝛿𝛿 = 𝑐𝑐𝑝𝑝𝑙𝑙𝑛𝑛 (( 𝛿𝛿𝑝𝑝 𝑣𝑣 ) 𝑣𝑣∈𝑉𝑉 ) and the associated bond space ℬ correspond to the row space of the node-to-edge incidence matrix 𝐴𝐴 . The set of the linearly independent elementary potential differences constitute thus a vector basis for 𝛿𝛿𝛿𝛿 and they are referred to as the potential difference modes. 7.3.2. The flow space ℱ and the circuit space 𝒞𝒞 Let ℱ be the space of all possible flow functions on the graph 𝐺𝐺 . Let 𝑓𝑓 , 𝑑𝑑 ∈ ℱ be two flow functions and 𝜆𝜆 ∈ ℝ . The linear combination ( 𝑓𝑓 + 𝜆𝜆𝑑𝑑 ) obeys a cut law on the graph 𝐺𝐺 proving that ℱ is a vector space. Like potential differences, the flow functions associated to circuits are of interest for the description of the network. Let 𝐶𝐶 be a circuit in the graph 𝐺𝐺 . The flow 𝑓𝑓 𝐶𝐶 associated to the circuit 𝐶𝐶 is defined by: 𝑓𝑓 𝐶𝐶 � ( 𝑐𝑐 , 𝑗𝑗 ) � = � −
1 0 𝑐𝑐𝑓𝑓𝑐𝑐𝑓𝑓𝑐𝑐𝑓𝑓 ( 𝑐𝑐 , 𝑗𝑗 ) ∈ 𝐶𝐶 ( 𝑗𝑗 , 𝑐𝑐 ) ∈ 𝐶𝐶 ( 𝑐𝑐 , 𝑗𝑗 ) ∉ 𝐶𝐶 (A.5) Elementary circuits represent cycles in graph theory: closed paths where every vertex has exactly two neighbors. The space of all possible circuits is referred to as the circuit space 𝒞𝒞 in graph theory, or cycle space 𝒞𝒞 in some other works, which corresponds to the nullspace of the node-to-edge incidence matrix 𝐴𝐴 . The change in the denomination of the space can be justified by the fact that the circuit space 𝐶𝐶 can be studied through the set of independent cycles of the graphs. In fact, similar to cut-set vectors in the previous sections, every cycle 𝐶𝐶 corresponds to a 𝑛𝑛 𝑒𝑒 vector 𝑐𝑐 𝐶𝐶 with components in { − . The vector 𝑐𝑐 𝐶𝐶 is indexed by the set of edges 𝐸𝐸 and is defined by: ( 𝑖𝑖 , 𝑖𝑖 ) 𝐶𝐶 = � −
1 0 𝑐𝑐𝑓𝑓𝑐𝑐𝑓𝑓𝑐𝑐𝑓𝑓 ( 𝑐𝑐 , 𝑗𝑗 ) ∈ 𝐶𝐶 𝑙𝑙𝑛𝑛𝑑𝑑 𝑐𝑐𝑙𝑙𝑠𝑠𝑐𝑐 𝑓𝑓𝑟𝑟𝑐𝑐𝑐𝑐𝑛𝑛𝑐𝑐𝑙𝑙𝑐𝑐𝑐𝑐𝑓𝑓𝑛𝑛 ( 𝑐𝑐 , 𝑗𝑗 ) ∈ 𝐶𝐶 𝑙𝑙𝑛𝑛𝑑𝑑 𝑓𝑓𝑝𝑝𝑝𝑝𝑓𝑓𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑓𝑓𝑟𝑟𝑐𝑐𝑐𝑐𝑛𝑛𝑐𝑐𝑙𝑙𝑐𝑐𝑐𝑐𝑓𝑓𝑛𝑛 ( 𝑐𝑐 , 𝑗𝑗 ) ∉ 𝐶𝐶 (A.6) There is thus a direct association of the flow function 𝑓𝑓 𝐶𝐶 to the circuit vector 𝑐𝑐 𝐶𝐶 . The cycle space 𝒞𝒞 has a finite dimension 𝑐𝑐 and its bases correspond to 𝑐𝑐 cycle vectors, referred to as the flow modes. The determination of these vectors is discussed in the next section, through its relationship with the cut-set space ℬ and the node-to-edge incidence matrix of the graph 𝐴𝐴 . 7.3.3. Relationship between the potential space 𝛿𝛿𝛿𝛿 and the flow space ℱ Let 𝑓𝑓 be a function on the edges 𝐸𝐸 of the graph 𝐺𝐺 . 𝑓𝑓 is a flow if, and only, if it obeys a cut law on the graph 𝐺𝐺 . Since the set of elementary cuts ( 𝑏𝑏 𝑣𝑣 ) 𝑣𝑣∈𝑉𝑉 spans the cut-set space ℬ , 𝑓𝑓 has to obey a cut law on the cut-sets ( 𝑏𝑏 𝑣𝑣 ) 𝑣𝑣∈𝑉𝑉 in order to be a flow. These conditions can be expressed as: � 𝑏𝑏 𝑣𝑣 𝑓𝑓� ( 𝑐𝑐 , 𝑗𝑗 ) � = 0 ( 𝑖𝑖 , 𝑖𝑖 ) ∈𝐸𝐸 , ∀ 𝑣𝑣 ∈ 𝑉𝑉 (A.7) Since each 𝑏𝑏 𝑣𝑣 is associated to an elementary potential difference 𝛿𝛿𝑝𝑝 𝑣𝑣 , it can be seen that 𝑓𝑓 is a flow if, and only if, it is orthogonal to each elementary potential difference 𝛿𝛿𝑝𝑝 𝑣𝑣 , with the flow space ℱ representing the orthogonal complement of the potential difference space 𝛿𝛿𝛿𝛿 . Moreover, ℱ corresponds also to the nullspace of the node-to-edge incidence matrix 𝐴𝐴 . Algebraically, the flow modes can be determined through the basis of the nullspace of the incidence matrix A. However, the resulting flow modes are typically associated with complicated circuits in the graph when higher dimensions are considered. Aloui et al. (2019) [A.2-A.4] proposed a bio-inspired method for the study and design of self-equilibrated networks in two- and three-dimensional spaces in the context of tensegrity structures. In the proposed method, called cellular morphogenesis, flow modes are referred to as self-stress states corresponding to tensegrity cells (complete graphs on 𝑑𝑑 + 2 nodes) composing the overall structure. An algorithm was also provided for the determination of the tensegrity cells composing a given tensegrity structure, which is used in this paper for flow mode determination. Appendix B
The expression of the analytical solution for the flow mode of a cell resorts to algebraic geometry and the wedge product or exterior product ∧ . The reader is reminded here of some of the properties of wedge products used in this proof: • The wedge product, or exterior product, is an algebraic construction used in algebraic geometry to study areas, volumes and higher-dimensional analogues. • The wedge product of two vectors 𝑢𝑢 and 𝑣𝑣 is called a bivector 𝑢𝑢 ∧ 𝑣𝑣 , and it lives in a vector space called the exterior square distinct from the original vector space. • The wedge product 𝑢𝑢 ∧ 𝑢𝑢 = 0 . In general, the wedge product of linearly dependent vectors is always zero.
The wedge product is anticommutative 𝑢𝑢 ∧ 𝑣𝑣 = −𝑣𝑣 ∧ 𝑢𝑢 . • The wedge product is associative ( 𝑢𝑢 ∧ 𝑣𝑣 ) ∧ 𝑙𝑙 = 𝑢𝑢 ∧ ( 𝑣𝑣 ∧ 𝑙𝑙 ) = 𝑢𝑢 ∧ 𝑣𝑣 ∧ 𝑙𝑙 . • 𝑢𝑢 ∧ 𝑣𝑣 is also called a 2-blade and it represents the oriented area of the parallelogram defined by 𝑢𝑢 and 𝑣𝑣 and given the orientation described by Figure B.1. Figure B.1: Illustration of a 2-blade and its orientation. • A k-blade 𝑢𝑢 ∧ 𝑢𝑢 ∧ … ∧ 𝑢𝑢 𝑘𝑘 is a generalization of the 2-blade associated to a 𝑘𝑘 -dimensional oriented volume of a parallelotope. Figure B.2 shows an example of a 3-dimensional blade and its corresponding volume. Figure B.2: Illustration of a 2-blade and the orientation of the volume it defines. • It is common to consider a k-simplex instead of a parallelotope when the k-blade vectors are defined by two points in space. The 𝑘𝑘 -simplex and its orientation can then be entirely described by an ordered set ( 𝑃𝑃 𝑖𝑖 ) 𝑖𝑖≤𝑘𝑘 of 𝑘𝑘 + 1 points. The oriented volume can then be associated to the convex hull of the 𝑘𝑘 + 1 nodes (the smallest convex set containing the 𝑘𝑘 + 1 nodes of the k-simplex) endowed with the orientation of the ordered set of vertices. Consequently, the volume of the 𝑘𝑘 -simplex defined by the ordered set { 𝑃𝑃 , 𝑃𝑃 , … , 𝑃𝑃 𝑘𝑘 } is described using 𝑘𝑘 -blades by [B.1]: 𝑉𝑉 𝑘𝑘−𝑠𝑠𝑖𝑖𝑚𝑚𝛿𝛿𝑐𝑐𝑒𝑒𝑥𝑥 = 1 𝑘𝑘 ! �𝑃𝑃 𝑃𝑃 ��������⃗ ∧ 𝑃𝑃 𝑃𝑃 ��������⃗ ∧ … ∧ 𝑃𝑃 𝑃𝑃 𝑘𝑘 ���������⃗� (B.1) In a 𝑑𝑑 -dimensional space with an orthonormal basis ( 𝑥𝑥 ���⃗ , 𝑥𝑥 ����⃗ , … , 𝑥𝑥 𝑥𝑥 ����⃗ ) , the analytical value of the 𝑑𝑑 -blade defined by �𝑃𝑃 𝑃𝑃 ��������⃗ ∧ 𝑃𝑃 𝑃𝑃 ��������⃗ ∧ … ∧ 𝑃𝑃 𝑃𝑃 𝑥𝑥 ���������⃗� can be calculated by the determinant[B.1, B.2]: �𝑃𝑃 𝑃𝑃 ��������⃗ ∧ 𝑃𝑃 𝑃𝑃 ��������⃗ ∧ … ∧ 𝑃𝑃 𝑃𝑃 𝑘𝑘 ���������⃗� = � 𝑃𝑃 … 𝑃𝑃 𝑥𝑥0 𝑃𝑃 … 𝑃𝑃 𝑥𝑥1 ⋮ ⋮ ⋮ ⋮ 𝑃𝑃 … 𝑃𝑃 𝑥𝑥𝑥𝑥 � (B.2) In this paper, the wedge product is used to derive the analytical solution for the flow densities of the flow mode of a 𝑑𝑑 -dimensional cell. Let 𝐺𝐺 ( 𝑉𝑉 , 𝐸𝐸 ) be the complete graph on 𝑑𝑑 + 2 nodes associated to a d -dimensional cell. Let the set of ordered nodes ( 𝑣𝑣 𝑖𝑖 ) be the composing nodes of the graph 𝐺𝐺 . The graph 𝐺𝐺 is embedded in a 𝑑𝑑 -dimensional space where to each node 𝑣𝑣 𝑖𝑖 is associated a 𝑑𝑑 -dimensional potential 𝑝𝑝 𝑖𝑖 . In a complete graph on 𝑑𝑑 + 2 nodes, each vertex is adjacent to exactly 𝑑𝑑 + 1 nodes. The elementary cut law 𝑏𝑏 𝑣𝑣 𝑘𝑘 around a node 𝑣𝑣 𝑘𝑘 chosen arbitrarily from 𝑉𝑉 will give: � 𝜔𝜔 𝑘𝑘𝑖𝑖 𝑝𝑝 𝑘𝑘 𝑝𝑝 𝚤𝚤 ��������⃗ = 0 (B.3) Note that Eq (B.3) describes the nodal equilibrium at node 𝑘𝑘 and the vector 𝑝𝑝 𝑘𝑘 𝑝𝑝 𝚤𝚤 ��������⃗ represents the potential difference between nodes 𝑣𝑣 𝑖𝑖 and 𝑣𝑣 𝑘𝑘 . Since the flow space ℱ of a cell is one dimensional, it follows that all the flow densities of the members can be written as a function of one specific flow density, without loss of generality 𝜔𝜔 𝑘𝑘𝑚𝑚 ( 𝑠𝑠 ≠ 𝑘𝑘 ). The objective of this part is to write the flow density of an edge ( 𝑣𝑣 𝑘𝑘 , 𝑣𝑣 𝑛𝑛 ) ( 𝑛𝑛 ≠ 𝑠𝑠 ≠ 𝑘𝑘 ), 𝜔𝜔 𝑘𝑘𝑛𝑛 , as a function of the flow density of the edge ( 𝑣𝑣 𝑘𝑘 , 𝑣𝑣 𝑚𝑚 ) , 𝜔𝜔 𝑘𝑘𝑚𝑚 . Consider the ( 𝑑𝑑 − )-blade composed of the vectors 𝑝𝑝 𝑘𝑘 𝑝𝑝 𝚤𝚤 ��������⃗ where ≤ 𝑐𝑐 ≤ 𝑑𝑑 + 2, 𝑐𝑐 ≠ 𝑘𝑘 ≠ 𝑠𝑠 ≠ 𝑛𝑛 , and denoted: � 𝑝𝑝 𝑘𝑘 𝑝𝑝 𝚤𝚤 ��������⃗ (B.4) Note that this wedge product does not contain vectors 𝑝𝑝 𝑘𝑘 𝑝𝑝 𝑚𝑚 ����������⃗ and 𝑝𝑝 𝑘𝑘 𝑝𝑝 𝑛𝑛 ���������⃗ . Applying the wedge product of this ( 𝑑𝑑 − -blade to equation B.3 gives: � � 𝜔𝜔 𝑘𝑘𝑖𝑖 𝑝𝑝 𝑘𝑘 𝑝𝑝 𝚤𝚤 ��������⃗ � ∧ � � 𝑝𝑝 𝑘𝑘 𝑝𝑝 𝚤𝚤 ��������⃗ � = 0 (B.5) Given that the wedge product of linearly dependent vectors is zero ( 𝑢𝑢 ∧ 𝑢𝑢 = 0 ), the only terms that are left in the sum are those with 𝜔𝜔 𝑘𝑘𝑚𝑚 and 𝜔𝜔 𝑘𝑘𝑛𝑛 (all other products will have at least one repeated vector): 𝑘𝑘𝑚𝑚 �𝑝𝑝 𝑘𝑘 𝑝𝑝 𝑚𝑚 ����������⃗ ∧ � 𝑝𝑝 𝑘𝑘 𝑝𝑝 𝚤𝚤 ��������⃗ � + 𝜔𝜔 𝑘𝑘𝑛𝑛 �𝑝𝑝 𝑘𝑘 𝑝𝑝 𝑛𝑛 ���������⃗ ∧ � 𝑝𝑝 𝑘𝑘 𝑝𝑝 𝚤𝚤 ��������⃗ � = 0 (B.6) If ( 𝑝𝑝 𝑖𝑖 ) are linearly independent, then 𝜔𝜔 𝑘𝑘𝑛𝑛 can be easily expressed in terms of 𝜔𝜔 𝑘𝑘𝑚𝑚 . To simplify the notations, the wedge product in equation A.6 will be denoted by: �𝑝𝑝 𝑘𝑘 𝑝𝑝 𝑛𝑛 ���������⃗ ∧ � 𝑝𝑝 𝑘𝑘 𝑝𝑝 𝚤𝚤 ��������⃗ � = 𝑉𝑉 �𝑣𝑣 𝑘𝑘 , 𝑣𝑣 𝑛𝑛 , ( 𝑣𝑣 𝑖𝑖 ) � �� 𝑥𝑥 𝚤𝚤 ���⃗ 𝑥𝑥+2𝑖𝑖=1 � (B.7) The function 𝑉𝑉 is synonymous with the determinant in equation B.2. Consequently, the order of vertices inside the function has to respect the same order of appearance of indices in the wedge product. Since the set ( 𝑣𝑣 𝑖𝑖 ) is ordered, then the set of ordered vertices �𝑣𝑣 𝑘𝑘 , 𝑣𝑣 𝑚𝑚 , ( 𝑣𝑣 𝑖𝑖 ) � describes a set of vertices starting with vertex 𝑣𝑣 𝑘𝑘 , then vertex 𝑣𝑣 𝑚𝑚 , then the rest of the vertices following the same order of appearance in the ordered set ( 𝑣𝑣 𝑖𝑖 ) . Consequently, equation B.6 becomes: 𝜔𝜔 𝑘𝑘𝑚𝑚 𝑉𝑉 �𝑣𝑣 𝑘𝑘 , 𝑣𝑣 𝑚𝑚 , ( 𝑣𝑣 𝑖𝑖 ) � �� 𝑥𝑥 𝚤𝚤 ���⃗ 𝑥𝑥+2𝑖𝑖=1 � + 𝜔𝜔 𝑘𝑘𝑛𝑛 𝑉𝑉 �𝑣𝑣 𝑘𝑘 , 𝑣𝑣 𝑛𝑛 , ( 𝑣𝑣 𝑖𝑖 ) � �� 𝑥𝑥 𝚤𝚤 ���⃗ 𝑥𝑥+2𝑖𝑖=1 � = 0 ⇒ �𝜔𝜔 𝑘𝑘𝑚𝑚 𝑉𝑉 �𝑣𝑣 𝑘𝑘 , 𝑣𝑣 𝑚𝑚 , ( 𝑣𝑣 𝑖𝑖 ) � + 𝜔𝜔 𝑘𝑘𝑛𝑛 𝑉𝑉 �𝑣𝑣 𝑘𝑘 , 𝑣𝑣 𝑛𝑛 , ( 𝑣𝑣 𝑖𝑖 ) �� �� 𝑥𝑥 𝚤𝚤 ���⃗ 𝑥𝑥+2𝑖𝑖=1 � = 0 ⇒ 𝜔𝜔 𝑘𝑘𝑚𝑚 𝑉𝑉 �𝑣𝑣 𝑘𝑘 , 𝑣𝑣 𝑚𝑚 , ( 𝑣𝑣 𝑖𝑖 ) � + 𝜔𝜔 𝑘𝑘𝑛𝑛 𝑉𝑉 �𝑣𝑣 𝑘𝑘 , 𝑣𝑣 𝑛𝑛 , ( 𝑣𝑣 𝑖𝑖 ) � = 0 (B.8) If the potentials 𝑝𝑝 𝑖𝑖 , ∀ 𝑐𝑐 , are linearly independent, then function 𝑉𝑉 cannot be zero, leading to: 𝜔𝜔 𝑘𝑘𝑛𝑛 = − 𝑉𝑉 �𝑣𝑣 𝑘𝑘 , 𝑣𝑣 𝑚𝑚 , ( 𝑣𝑣 𝑖𝑖 ) �𝑉𝑉 �𝑣𝑣 𝑘𝑘 , 𝑣𝑣 𝑛𝑛 , ( 𝑣𝑣 𝑖𝑖 ) � 𝜔𝜔 𝑘𝑘𝑚𝑚 (B.9) The examination of the two- and three-dimensional cases of Equation B.9 shows that there is another way of expressing flow density. Taking flow densities variables as constants of multiplication: 𝜔𝜔 𝑘𝑘𝑚𝑚 = 𝑉𝑉 �𝑣𝑣 𝑘𝑘 , ( 𝑣𝑣 𝑖𝑖 ) � . 𝑉𝑉 �𝑣𝑣 𝑚𝑚 , ( 𝑣𝑣 𝑖𝑖 ) � (B.10) and replacing 𝜔𝜔 𝑘𝑘𝑚𝑚 in equation B.8 by it is expression in equation B.10 gives: 𝑉𝑉 �𝑣𝑣 𝑘𝑘 , ( 𝑣𝑣 𝑖𝑖 ) � . 𝑉𝑉 �𝑣𝑣 𝑚𝑚 , ( 𝑣𝑣 𝑖𝑖 ) � . 𝑉𝑉 �𝑣𝑣 𝑘𝑘 , 𝑣𝑣 𝑚𝑚 , ( 𝑣𝑣 𝑖𝑖 ) � + 𝜔𝜔 𝑘𝑘𝑛𝑛 𝑉𝑉 �𝑣𝑣 𝑘𝑘 , 𝑣𝑣 𝑛𝑛 , ( 𝑣𝑣 𝑖𝑖 ) � = 0 (B.11) Now, without loss of generality, assume that 𝑠𝑠 < 𝑛𝑛 < 𝑘𝑘 . The ordered set of vertices inside each function 𝑉𝑉 are reordered taking into consideration the changes to the values of 𝑉𝑉 . Since 𝑉𝑉 is a determinant, every permutation of a vertex will result in multiplying 𝑉𝑉 by -1: 𝑉𝑉 �𝑣𝑣 𝑘𝑘 , ( 𝑣𝑣 𝑖𝑖 ) � = 𝑉𝑉 �𝑣𝑣 𝑘𝑘 , ( 𝑣𝑣 𝑖𝑖 ) � (i) 𝑉𝑉 �𝑣𝑣 𝑚𝑚 , ( 𝑣𝑣 𝑖𝑖 ) � = ( − 𝑚𝑚+𝑛𝑛−2 𝑉𝑉 �𝑣𝑣 𝑛𝑛 , ( 𝑣𝑣 𝑖𝑖 ) � (ii) 𝑉𝑉 �𝑣𝑣 𝑘𝑘 , 𝑣𝑣 𝑚𝑚 , ( 𝑣𝑣 𝑖𝑖 ) � = ( − 𝑚𝑚−1 𝑉𝑉 �𝑣𝑣 𝑘𝑘 , ( 𝑣𝑣 𝑖𝑖 ) � (iii) 𝑉𝑉 �𝑣𝑣 𝑘𝑘 , 𝑣𝑣 𝑛𝑛 , ( 𝑣𝑣 𝑖𝑖 ) � = ( − 𝑛𝑛−2 𝑉𝑉 �𝑣𝑣 𝑘𝑘 , ( 𝑣𝑣 𝑖𝑖 ) � (iv) (B.12) In equation B.12 (i), there was no need to reorder the vertices. In equation B.12 (ii), node 𝑣𝑣 𝑛𝑛 is changed to the first position of the ordered set and node 𝑣𝑣 𝑚𝑚 is put back inside the remaining of the set in its order of appearance in the original set ( 𝑣𝑣 𝑖𝑖 ) . In equation B.12 (iii), 𝑣𝑣 𝑚𝑚 is put back inside the remaining of the set in its order of appearance in the original set ( 𝑣𝑣 𝑖𝑖 ) . In equation B.12 (iv), 𝑣𝑣 𝑛𝑛 is put back inside the remaining of the set in its order of appearance in the original set ( 𝑣𝑣 𝑖𝑖 ) . Now, with these changes, equation B.11 can be rewritten as: 𝑉𝑉 �𝑣𝑣 𝑘𝑘 , ( 𝑣𝑣 𝑖𝑖 ) � . ( − 𝑚𝑚+𝑛𝑛−2 𝑉𝑉 �𝑣𝑣 𝑛𝑛 , ( 𝑣𝑣 𝑖𝑖 ) � . ( − 𝑚𝑚−1 𝑉𝑉 �𝑣𝑣 𝑘𝑘 , ( 𝑣𝑣 𝑖𝑖 ) � + 𝜔𝜔 𝑘𝑘𝑛𝑛 ( − 𝑛𝑛−2 𝑉𝑉 �𝑣𝑣 𝑘𝑘 , ( 𝑣𝑣 𝑖𝑖 ) � = 0 ⇒ 𝑉𝑉 �𝑣𝑣 𝑘𝑘 , ( 𝑣𝑣 𝑖𝑖 ) � �𝑉𝑉 �𝑣𝑣 𝑛𝑛 , ( 𝑣𝑣 𝑖𝑖 ) � 𝑉𝑉 �𝑣𝑣 𝑘𝑘 , ( 𝑣𝑣 𝑖𝑖 ) � − 𝜔𝜔 𝑘𝑘𝑛𝑛 � = 0 𝜔𝜔 𝑘𝑘𝑛𝑛 = 𝑉𝑉 �𝑣𝑣 𝑘𝑘 , ( 𝑣𝑣 𝑖𝑖 ) � 𝑉𝑉 �𝑣𝑣 𝑛𝑛 , ( 𝑣𝑣 𝑖𝑖 ) � (B.13) which gives 𝜔𝜔 𝑘𝑘𝑛𝑛 as: 𝜔𝜔 𝑘𝑘𝑛𝑛 = 𝑉𝑉 �𝑣𝑣 𝑘𝑘 , ( 𝑣𝑣 𝑖𝑖 ) � 𝑉𝑉 �𝑣𝑣 𝑛𝑛 , ( 𝑣𝑣 𝑖𝑖 ) � (B.14) The consideration of all possible rearrangements of 𝑘𝑘 , 𝑠𝑠 𝑙𝑙𝑛𝑛𝑑𝑑 𝑛𝑛 leads thus to the same result. Given that 𝑘𝑘 , 𝑠𝑠 𝑙𝑙𝑛𝑛𝑑𝑑 𝑛𝑛 were chosen arbitrarily, this expression is valid for the flow density of any edge in the cell. Moreover, the geometric interpretation of this result reflects that each flow density 𝜔𝜔 𝑖𝑖𝑖𝑖 f an edge �𝑣𝑣 𝑖𝑖 , 𝑣𝑣 𝑖𝑖 � of a 𝑑𝑑 -dimensional cell can be calculated by the product of two oriented volumes of 𝑑𝑑 -simplexes 𝑆𝑆 𝑖𝑖 and 𝑆𝑆 𝑖𝑖 defined by the oriented set of nodes 𝑆𝑆 𝑖𝑖 = �𝑣𝑣 𝑖𝑖 , ( 𝑣𝑣 𝑘𝑘 ) � and 𝑆𝑆 𝑖𝑖 = �𝑣𝑣 𝑖𝑖 , ( 𝑣𝑣 𝑘𝑘 ) � . Figure B.3 illustrates the geometric interpretation of the result in two- and three-dimensions. Figure B.3: Illustration of the geometric interpretation of the flow mode solutions in two- and three-dimensions.
References [A.1]
Reinschke, K.J., 2001. On network models and the symbolic solution of network equations. International Journal of Applied Mathematics and Computer Science, 11, pp.237-269. [A.2]
Aloui, O., Orden, D. and Rhode-Barbarigos, L., 2018. Generation of planar tensegrity structures through cellular multiplication. Applied Mathematical Modelling, 64, pp.71-92. [A.3]