A Design Method for Optimal Truss Structures with Certain Redundancy Based on Combinatorial Rigidity Theory

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1 0 th Word Congress on Structura and Mutidiscipinary Optimization May 9 -, 03, Orando, Forida, USA A Design Method for Optima Truss Structures with Certain Redundancy Based on Combinatoria Rigidity Theory Rie Kohta, Makoto Yamakawa, Naoki Katoh 3, Yoshikazu Araki and Makoto Ohsaki 5 Kyoto University, Kyoto, Japan, se.kohta@archi.kyoto-u.ac.jp Tokyo Denki University, Tokyo, Japan, myamakawa@mai.dendai.ac.jp 3 Kyoto University, Kyoto, Japan, naoki@archi.kyoto-u.ac.jp Kyoto University, Kyoto, Japan, araki@archi.kyoto-u.ac.jp Hiroshima University, Hiroshima, Japan, ohsaki@hiroshima-u.ac.jp Abstract The Truss Topoogy Design (TTD) probem deas with the seection of optima configuration for pin-jointed trusses, in particuar, the optimization of the connectivity of the nodes by the members, in which voume and/or compiance are minimized. In genera, it is known that such truss structures are staticay determinate and not redundanty rigid, that is, if just one member is damaged or ost, the entire structure cannot support oads. Therefore, it is important to take redundancy of structures into consideration in the TTD. We present a new practica design method for finding a redundant TTD based on combinatoria rigidity theory. In this paper, we define, as redundancy, the margin of the number of members unti the coapse of the entire structure when some components are damaged or ost. A truss structure is said to be a -edge-rigid truss if we need to remove at east two members from the truss structure so that the structure becomes non-rigid. We can find a -edge-rigid TTD by using a method based on combinatoria rigidity theory. The present method enabes to find an approximatey optima TTD with ow computationa cost. In numerica exampes, we compute redundanty rigid truss structures, in which objective vaue of the soution is about one percent greater than that of the ower bound. Therefore, we can concude that the method is effective to design an optima redundant truss structure. Keywords: Truss Topoogy Optimization, Redundancy, Combinatoria Rigidity Theory. Introduction It is very important to take redundancy into consideration in structura design, because it heps us to design against the force beyond our estimation. Generay it is difficut to ensure the safety against such unknown future forces. In particuar, there is a concern over progressive coapse, which is the coapse of the whoe structure triggered by the partia damage. In order to overcome these dangers, we must consider redundancy of structura design. This paper deas with the Truss Topoogy Design (TTD) probem with redundancy. There is not many researches on redundancy in the TTD athough truss structures designed by the TTD are generay staticay determinate and do not have redundancy. In the foowing subsections, we provide some backgrounds for this study...truss Topoogy Design (TTD) probem The TTD probem deas with the seection of optima configuration for pin-jointed trusses, in particuar, the optimization of the connectivity of the nodes by the members, in which voume and/or compiance are minimized. Ground structure method is known as the most popuar method of TTD. In this method, a ground structure with a sufficienty arge number of members is given as the initia configuration (as shown in Figure ). Then, the optima topoogy is obtained by soving the optimization probem in which design variabes are the cross section areas of the members, and the members with no cross section area are removed. It is known that the optima truss structure by the method is staticay determinate [3]. Staticay determinate truss structures are not redundanty rigid, that is, if just one member is damaged or ost, the entire structure cannot support oads. Figure : Ground structure Figure : Optima truss structure

2 .. Previous studies on redundancy of bar-and-joint framework A truss structure is a kind of bar-and-joint framework, which is a structure composed of rigid rods (bars) connected at their ends (joints). These frameworks are denoted by graphs such as Figure 3. A graph is said to be rigid if it has no nontrivia deformations preserving the edge engths. The rigidity of a graph has been studied in combinatoria rigidity theory, a fied of discrete mathematics. (a) (b) (c) Figure 3: (a) A rigid graph. (b) A non-rigid graph. (c) A redundanty rigid graph A. Garcia et a.(0) provide an O(n ) agorithm to augment a non-redundant rigid graph to a redundanty rigid graph by adding a minimum number of edges [], where a graph is redundanty rigid when the graph remains rigid even if any arbitrary edge is removed. C. Yu et a. (008) propose the concept of k-edge-rigid graphs[7]. The k-edge-rigid graph is sti rigid after deetion of any (k - ) edge(s), whie redundanty rigid graph is sti rigid after deetion of any one of edges. The redundanty rigid truss corresponds to a -edge-rigid graph by this definition. The method to generate a k-edge-rigid graph is proposed by N. Katoh et a. (0)[5]..3. Purpose of this paper We propose a new practica design method for finding a redundant TTD based on combinatoria rigidity theory in this paper. Since a truss structure can be abstracted to a graph, the rigidity of truss structures can be studied by combinatoria rigidity theory. Some important concepts about combinatoria rigidity theory and the agorithm find a minima covering proposed by Garcia [] are presented in section. The formuation of our probem for finding a redundant TTD and its approximate method are given in section 3. Finay, some numerica exampes are shown in section.. Combinatoria rigidity theory In this section, we expain the outine of this theory and the key concept of this paper, redundant range L(i,j)... Outine of combinatoria rigidity theory Combinatoria rigidity theory studies the rigidity of bar-joint frameworks from the viewpoint of graph theory. In fact, a bar-joint framework can be viewed as a graph G=(V,E) embedded on the pane where V and E denote the set of vertices which correspond to joints and that of edges which correspond to members, respectivey. Besides, V(G) and E(G) denote vertices and edges incuded in graph G, respectivey. The graph is caed rigid in R, if the corresponding framework aows ony trivia deformation, i.e., transation and rotation. In addition, a rigid graph with minima number of edges is caed a Laman graph, and the necessary and sufficient condition for a graph G=(V,E) to be a Laman graph is given as foows: E = V 3, (.a) E(X) X 3 for every X V with X V, (.b) where E(X) E denotes the set of edges in the subgraph induced by vertices X V, and E, V are the number of edges and vertices, respectivey. In addition, the notation E(X) is often used to stand for a graph G =(X,E(X)). For exampe, the graph iustrated in Figure (a) is a Laman graph: the equation (.a) is satisfied with E =5 and V =, and the subgraph E(X) in Figure (b) satisfies the equation (.b) with E(X) =3 and X =3. In the foowing, L denotes the set of Laman graphs.

3 X (a) (b) Figure : (a) A Laman graph. (b) A subgraph E(X) Based on this theory, Garcia et a. provide an O(n ) agorithm to augment a Laman graph G to a -edge-rigid graph (a redundanty rigid graph), by adding a minimum number of edges []. (a) (b) Figure 5: (a) A -edge-rigid graph. (b) A rigid graph An edge e of a rigid graph G is redundant if G e is sti rigid. A graph G is a -edge-rigid if G e is rigid for any edge e E(G) (see Figure 5(b)). In particuar, a generic circuit is a graph G such that G e is a Laman graph for any edge e E(G). Namey, G is a generic circuit if and ony if G e L, e G. () Generic circuits can be said to be minima -edge-rigid graphs. Since a truss structure can be abstracted to a graph, we can study its rigidity by combinatoria rigidity theory. In our study, we require that a truss is -edge-rigid. Then, we name this constraint as -edge-rigid constraint in this study... Redundant range L(i,j) In the foowing, an edge connecting vertices i and j is denoted by e = (i,j). When a new edge e = (i,j) is added to a Laman graph L, the graph L (i,j) contains a unique generic circuit, which is denoted by C(i,j) or C(e). Since this generic circuit contains the edge e = (i,j), the subgraph C(i,j) (i,j) is a Laman graph, which is caed a Laman subgraph. Garcia et a. denote this Laman subgraph by L(i,j) or L(e) []. Figures 6(a), 6(b) and 6(c) iustrate the concept of L(i,j). Each number in these figures represents the number of each vertex. In the Figure 6(b), the edges of the Laman subgraph L(,) are iustrated in bod straight ines. If we add to this Laman graph an edge connecting the vertex "" and the vertex "", then the edges of L(,) become redundant. In the same way, the edges of L(,5) iustrated in Figure 6(c) become redundant if we add the edge (,5) (a) (b) (c) Figure 6: (a) A Laman graph L. (b) L(,). (c) L(,5) Garcia et a. prove in their paper [] that, when the edges e, e,, e k are added to a Laman graph L, a subset of edges of L wi become redundant, which is denoted by L(e, e,,e k ). Then this set coincides with L(e ) L(e ) L(e k ): Lemma [] If L is a Laman graph, then L(e, e,,e k ) = L(e ) L(e ) L(e k ). 3

4 As iustrated in Figures 6(b) and 6(c), if we add both edges (,) and (,5), a edges of the Laman graph L become redundant. In other words, we can say that L(,) L(,5) = L((,), (,5)) is covering the whoe of L. In this way, every Laman graph can become a -edge-rigid graph by adding appropriate edges to the graph..3. Agorithm for finding a minima covering Garcia proposed the agorithm caed Find a minima covering to cacuate a set of edges, E = {e, e,,e k }, with minimum cardinaity, such that L(e, e,,e k ) = L(e ) L(e ) L(e k ) covers a the edges of a given Laman graph L. By using this agorithm, we can find a minimum number of edges we need to add. Degree of a vertex denotes the numbers of edges incident to the vertex. The agorithm Find a minima covering first chooses a vertex i of degree two or three which aways exists, and activates the agorithm Find a covering rooted in i, which finds h vertices j, j 3,, j h such that (i, j k ) covers a the edges of L, where a set of these vertices is denoted as k= L Extr(L). In the agorithm, if L has no vertices with degree two, we take a vertex k of degree three as i, find Extr(L) using the agorithm Find a covering rooted in i. After this, we rerun the agorithm by taking j Extr(L) as new i. At this point, the number of edges added is h. The agorithm then further reduces h edges to [h/] edges by repeatedy repacing consecutive three edges (i, j ), (i, j + ), (i, j + ) with two edges based on the combinatoria observation in the procedure Reduce a covering. In this procedure, we cacuate L(i, j ) and L(j +, j + ). Garcia showed that if L(i, j ) L(j +, j + ), then L(i, j ) L(j +, j + ) = L(i, j ) L(j, j + ) L(j, j + ), and otherwise L(i, j + ) L(j, j + ) = L(i, j ) L(j, j + ) L(j, j + ). For exampe, et us consider a Laman graph L iustrated in Figure 7(a) and the edges (,), (,3), (,). As iustrated in Figure 7(b), (c) and (d), L(,) L(,3) L(,) equas L. The procedure Reduce a covering first cacuates L(,) and L(3,) as iustrated in Figure 7(b) and (e). In this case, as L(,) L(3,) is not an empty set, L(,) L(3,) equas L(,) L(,3) L(,), i.e. the edges (,), (,3), (,) can be repaced by (,) and (3,). The fowchart of the agorithm Find a minima covering is iustrated in Figure 8. 3 (a) Graph L (b) L(,) in bod ine (c) L(,3) (d) L(,) 3 (e) L(3,) 3 (f) L(,) Figure 7: Graph L and some L(i,j)

5 start input: Laman graph L initiaize E = 0 Step: search for a vertex k with degree two or three with degree two is found take i = k with degree two is not found take i = k ( k is a vertex with degree three ) Step: activate Find a covering rooted in i take a new i the first vertex of Extr(L) Step3: activate Find a covering rooted in i add i to Extr(L) Step: Reduce a covering: reduce h edges to [h/] edges Step5: cacuate E output: E end Figure 8: Agorithm Find a minima covering start input: Laman graph L and a vertex i Initiaize Extr(L) = 0. A vertices except for i are unmarked. Expore a the verteces j of L Is j unmarked? yes no cacuate L(i, j) Mark a unmarked vertices in L(i, j). For each vertex j of Extr(L) in L(i, j), deete j from Extr(L). add j to Extr(L) oop output: Extr(L) = { j,, jh} end Figure 9: Agorithm Find a covering rooted in i 5

6 3. Formuation of the probem 3.. Formuation of the origina probem In this subsection, the origina probem and its formuation wi be shown. First, we consider the ground structure ike Figure : the tota number of nodes is n and that of members is m= n C. The objective function to minimize is compiance, i.e., the work of the externa force. We consider four constraints: the equiibrium of force, upper bound of the tota voume, ower bound of the cross section area of each member and -edge-rigid constraint. This probem can be formuated as foows: Minimize P T U, (3.a) subject to K i U = P, (3.b) i I i I L i V U, (3.c) A L i (i I), (3.d) I e L, e I, (3.e) where P and U denote the vectors of externa force and dispacement of nodes,, K i, L i and A L i are respectivey the cross section area, the stiffness matrix, the ength and the ower bound of the cross section area of the member i, V U denotes the upper bound of the tota voume and I is a set of members in a topoogy. I can be referred to as the topoogy. Design variabes are both the topoogy I and the set of cross section areas in the topoogy { A L i (i I) }. In genera, it is difficut to find a -edge-rigid optima truss structure for the probem (3). The -edge-rigid constraint is a combinatoria constraint. For finding the exact optima soution, we may have to enumerate a possibe topoogies based on exact method. Because of its arge cacuation cost, it is difficut to sove arge-scae probems by the method. 3.. Approximation method for -edge-rigid TTD probem As mentioned above, the TTD probem with -edge-rigid constraint is difficut to sove. From a practica point of view, however, we may not need to find an exact optima soution of this probem. Thus, we find a -edge-rigid approximatey optima truss structure by using the method based on combinatoria rigidity theory. The proposed method consists of three steps: () as a reaxation probem, we sove the TTD probem without -edge-rigid constraint (step), after that, () we then find a -edge-rigid truss topoogy by adding a minimum number of members to the soution of step (step), and finay, (3) under the truss topoogy of step, we determine the optima cross section areas of the members of the truss structure (step3). The soution of step corresponds to the ower bound of the objective vaue of the goba optima soution, and that of step3 corresponds to the upper bound of the objective vaue. Hence, we can aso estimate the accuracy of the soution from the difference of objective vaues between the two soutions. In the foowing, each step is described in detai Step (ower bound of the origina probem) In this step, we sove probem (3) as a reaxation probem: we wi not consider -edge-rigid constraint and repace ower bound constraints of the cross section areas of the members by non-negative constraints. Therefore the soution of this probem () is the ower bound of the origina probem (3). Minimize P T U, (.a) subject to K i U = P, (.b) i I g i I g L i V U, (.c) 0 (i I g ), (.d) where I g denotes a set of members in a ground structure. Note that the number of eements of I g may be extremey arge depending on the size of the probem. Probem () is a convex programming probem and the goba optima soution can be found by the appropriate way (see Appendix). In genera, it is known that the optima soution of () is a staticay determinate truss structure under singe oading cases [3]. 6

7 3... Step (augmenting the rigidity) The optima truss structure in step is a staticay determinate truss and not redundanty rigid. Then in this step, we make a -edge-rigid truss topoogy by adding minimum number of edges to the topoogy of the optima truss of step. We use the agorithm Find a minima covering iustrated in Subsection 3.3 with a sight modification. Since the origina agorithm does not take the engths of edges into consideration, there is a possibiity that ong edges are added in the origina agorithm. This is undesirabe in terms of the increase of the tota voume by adding new edges. In the agorithm Find a covering rooted in i, the order of exporing a the vertices j of L is not be specified. Therefore, if there are two edges (i, j s ) and (i, j t ) such that L(i, j s )=L(i, j t ) as iustrated in Figure 0(b) and 0(c), depending on the order of exporation, either j s or j t wi be seected as Extr(L). When the vertex j s is expored before j t, j s wi be incuded in Extr(L), and vice versa. Then, we modify this order of exporation: the shorter the ength of an edge (i, j) is, the higher priority is given to the vertex to be expored than other vertices. By this modification, the shorter edge is seected as E, especiay in the case of E =, the shortest edge wi be picked. In the case E, however, it is not sure that the sum of the engths of edges is minimum. Here we just indicate that getting a set of shortest edges E is possibe by appying the agorithm with "tight vertex set" presented by N. Katoh [5]. For more detais, readers can refer to [5]. js js js jt jt jt i i (a) (b) (c) i Figure 0: (a) A Laman graph L. (b) L(i, j s ). (c) L(i, j t ) Step3 (upper bound of the origina probem) In this step, we determine the optima cross section areas of the members of the truss structure under the truss topoogy of step. This probem is formuated as foows: Minimize P T U, (5.a) subject to K i U = P, (5.b) i I i I L i V U, (5.c) A L i = A L (i I ), (5.d) where I denotes the set of members in the truss topoogy of step and a ower bounds of the cross section areas of the members equa A L for brevity. The optima vaue of Probem (5) is the upper bound of the one of Probem (3), because the optima soution of (5) satisfies a of the four constraints: the equiibrium of force, upper bound of the tota voume, ower bound of the cross section area of each member and -edge-rigid constraint.. Numerica Exampe The agorithm described in the previous section has been thoroughy examined in some numerica tests. We wi show the resuts beow... case In this exampe, we consider a cantiever beam iustrated in Figure. The eft side is pin-jointed to the wa and the vertica force of 0 [kn] is oaded at the ower right point. We give 0 0 vertices and 9,900 members that connect a pairs of vertices as the ground structure shown in Figure. The upper bound of the tota voume V U is.0 m 3, and in Step3, the ower bounds of the cross section areas of a members, A L i (i I ) are 5 cm. 7

8 = 00 mm Figure : Design area (case ) Figure : Ground structure (case ) The foowings are the soutions of each step. Figure 3: Soution of Step (case ) Figure : Soution of Step (case ) Figure 5: Soution of Step3 (case ) In Figure, the dotted ine is the member added in Step. By adding this new member, the truss structure becomes -edge-rigid truss, in other words, it is sti rigid even if any one of the members is damaged or ost. The width of ines shown in Figure 5 represents the cross section area of each member. The maximum cross section area is about 8 times as arge as the minimum. The objective vaues of Step and Step3, denoted by f and f 3, are the ower and upper bounds of the objective vaue of the origina probem (3), respectivey. We achieve the resut that f 3 /f equas.008, that is, the upper bound is about just one percent greater than the ower bound. As a resut, it can be said that the optima truss structure of Step3 is an approximate soution cose to the optimum. In the above exampe, the difference between the ower and upper bounds is about one percent. However, the eve of approximation depends on the ower bound vaue of cross section areas, A L in (5.d). Figure 6 shows the reation between f 3 /f and A L. It is obvious that f 3 /f is a monotonicay increasing function of A L. Athough it is natura to determine the vaue of A L by the eve of approximation so that the stress of a members does not exceed an appropriate vaue, it is our future issue to determine the vaue of A L by considering the stress constraints. f3 / f the ower bound of section areas A L [cm ] Figure 6: Reation between and A L 8

9 .. cases and 3 (mutipe oads) We aso consider the TTD probems under mutipe oads, such as the figures iustrated in Figures 7 and 8. The eft side is pin-jointed to the wa and the vertica forces of 0 [kn] are oaded at the points indicated as downward arrows. The ower bounds of the cross section areas of a members, A L i (i I ) are 5 cm, and the upper bound of the tota voume V U is m 3. = 500 mm = 500 mm Figure 7: Ground structure (case ) Figure 8: Ground structure (case 3) The foowings are the soutions of each case. (a) (b) (c) Figure 9: Soutions in case of (a) Step, (b) Step, (c) Step3 (a) (b) (c) Figure 0: Soutions in case 3 of (a) Step, (b) Step, (c) Step3 As iustrated in Figures 9(b) and 0(b), we can achieve the -edge-rigid topoogies in Step by adding two members in case and three members in case 3, respectivey. In each case, the maximum cross section areas of the Step3 are cm and.9 0 cm, and they are about 6 times and 0 times as arge as the minimum cross section areas, respectivey. Regarding the objective vaues, we achieve the resut that f 3 /f equas.005 in case and.0088 in case 3, respectivey. 5. Concusion In this study, we achieve the foowing concusions.. We present a new practica design method based on combinatoria rigidity theory for finding an optima truss structure with redundancy, in which the truss structure is sti rigid even if any one of members is damaged or ost.. By this method, we can achieve upper and ower bound soutions for goba optima soution. In addition, its cacuation cost is O(n ), whie that of the method based on enumeration is O( n ). 3. In the numerica exampe of the ground structure with 00 vertices and 9,900 members, it is shown that the upper bound is about one percent greater than the ower bound, though the eve of reaxation depends on the vaue of the ower bound, A L. 9

10 Appendix Probem (3) is a convex programming probem and its decision variabes are cross section areas (i I g ) and dispacements U. This probem can be reformuated to a inear programming probem as foows [3,6]: Minimize P T U, (6.a) subject to V U T E b i U (i I g ), (6.b) L i where b i denotes the unit orientation vector of the member i and U is the scaed vaue of U as U =( f * / ) U; ( f * is the objective vaue at the optima soution of (6)). Acknowedgment We woud ike to acknowedge the support of The Kyoto University Foundation. References [] Y.Kanno and Y.Ben-Haim, Redundancy and Robustness, or When Is Redundancy Redundant?, Journa of Structura Engineering, 37, , 0. [] M.Ohsaki, N.Katoh, T.Kinoshita, S.Tanigawa, D.Avis and I.Streinu, Enumeration of optima pin-jointed bistabe compiant mechanisms with non-crossing members, Structura and Mutidiscipinary Optimization, 37, 65-65, 009 [3] M.Ohsaki, Optimization of Finite Dimensiona Structures, CRC Press, 00. [] A.Garcia and J.Teje, Augmenting the Rigidity of a Graph in R, Agorithmica, 59, 5-68, 0. [5] Y.Yoshinaka, N.Katoh and N.Kamiyama, Generating Redundanty Rigid Frameworks in -dimension, Asian Association for Agorithms and Computation, 0 [6] Aharon Ben-Ta, Arkadi Nemirovski, Lectures on Modern Convex Optimization, Society for Industria and Appied Mathematics, 987 [7] C.Yu and B.D.O.Anderson, Agent and Link Redundancy for Autonomous Formations, 7th IFAC Word Congress, , 008 0