Novel Method to Generate and Optimize Reticulated Structures of a Non Convex Conception Domain

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1 , pp Novel Method to Generate and Optimize Reticulated Structures of a Non Convex Conception Domain Zineb Bialleten 1*, Raddouane Chiheb 2, Abdellatif El Afia 3 and Rdouan Faizi 4 Ecole Nationale Supérieure d'informatique et d'analyse des Systèmes(ENSIAS), Mohamed V University, Rabat, Morocco 1 zineb.biallaten@um5s.net.ma 2 r.chiheb@um5s.net.ma 3 a.elafia@um5s.net.ma 4 r.faizi@um5s.net.ma Abstract Reticulated structures are increasingly represented in industry. They are lighter than traditional structures which allows weight saving. So they can be the best choice when material gain offsets the production cost which is an optimization purpose. In this paper, we propose an algorithm to generate automatically a reticulated structure. We particularly study non-convex conception domains which can be decomposed into a finite number of convex sub-domains, specially polygons. This solution generates a structure with an optimum form based on an algorithm that works on convex design domains. Keywords: Reticulated Structure; Convex decomposition; Structural Optimization, Structure Conception 1. Introduction Optimization is a concern of the various actors in many areas including consultancies in mechanical disciplines. An optimization problem seeks to find the better solution to a problem respecting a number of constraints. For shape optimization, the objective is to seek the best structure that ensures efficient performance at minimum cost: minimum weight, minimum volume, minimum deformation energy or other. According to G. Allaire [1], we distinguish between three categories of shape optimization problems. One that seeks the best dimensions of a structure. Another that change only the coordinates of the structure borders without changing its topology. And last, which is the most interesting, since it gives a possibility to modify the initial topology of the structure without restrictions to find the best possible shape: it is the topology optimization. Thus, structures in continuum and those created using discrete components are not treated with the same methods [2]. These optimization methods could be manual or automatic. However, the manual ones depends on designer skills which is inefficient so searchers proposed techniques to optimize a structure automatically. And by examining the results of a large part of optimization solutions we note that in the majority of cases the result is a reticulated structure which is a set of cylindrical bars interconnected at their extremities. For example, structures of suspension bridges. In this context, several studies have investigated the use of homogenization theory [1, 3]. Homogenization method "considers that a shape optimization is a material distribution problem: at any point in space putting a vacuum or a material"[4]. One of methods which used the concept of homogenization was developed by Chiheb & Panasenko in [5]. They * Corresponding Author ISSN: IJSEIA Copyright c 2017 SERSC

2 created a tool to generate automatically a reticulated structure from a convex conception domain. Although design domains are not all convex. That's why the convexity of the conception domain is a constraint that reduce the perimeter of use of this algorithm. Therefore we thought to expand the use of the previous algorithm to generate automatically a reticulated structure from a non-convex conception domain whatever its complexity. Our idea is to decompose the conception domain into a finite number of convex sub-domains. Then using the algorithm [5] to generate the structure from each sub-domain. And combine these structures to create one global reticulated structure. This paper is organized as follows: We present the main idea of our technique in Section 2. In Section 3, we explain in detail steps of the new algorithm. Next in Section 4, we discuss the formulation of the optimization problem. And in Section 5, we show some industrial examples. Finally, we conclude in Section Algorithm's Idea We propose an algorithm to generate a reticulated structure automatically from a conception domain defined by its boundary. Firstly, the designer start by giving area where the structure is to be placed. Whatever the shape drawn by the designer, the structure to generate must be totally contained in this area. Next, if the conception domain is non-convex then it will be decomposed into the minimum number of convex sub-domains using an algorithm of convex decomposition. Then a discretization process is lunched. This process aims to designate a set of points representing the boundaries of all sub-domains. These points are the initial nodes of the reticulated structure. Non-convex domain Decomposition Generation Fusion Figure 1. Steps of New Algorithm Afterward, each part of decomposition step will be handled separately. A structure is generated per sub-domain using the algorithm proposed in [5]. Finally, all partial structures are merged to create a global reticulated structure. So the main idea of our new algorithm is to study the possibility of decomposing a non- convex domain in a finite set of convex sub-domains. To accomplish this step we have to study existing techniques of convex decomposition of forms. 18 Copyright c 2017 SERSC

3 3. Algorithm's Details 3.1. Convex Decomposition Many algorithms are efficiently applied to simple that on complex shapes [6]. Therefore, we assume that a technique applied to a convex conception domain can be extended to be applied to a non-convex domain if we can divide it into convex subdomains. So, let Ω be a non-convex domain, Ω will be decomposed into a finite number of nonoverlapping convex parts: Ω = / is convex and = Ø, i j (1) In literature we find a variety of methods for decomposing a shape. These techniques differ according to the expected purpose of the decomposition and the required properties of the components resulting from this decomposition. All these techniques aim to decompose a form into a finite number of simple parts and among the simple shapes targeted we find the convex shapes [7]. For convex decomposition, we find algorithms for decomposing shapes into strictly convex sub-domains [7]. These algorithms cannot be used on any type of shape and are specially designed for decomposing polygons. On the other hand, we find algorithms that propose to decompose a shape into sub-shapes that are approximately convex. These algorithms can be used with a wider class of shapes, they are more efficient and more performed. So, in this work we are interested in the second category of methods. We need an algorithm to decompose an arbitrary conception domain into a minimum number of non-overlopping convex parts. To achieve our goal, we first determined the candidate algorithms to meet our constraints [6, 9-11]. We have developed a comparative study to choose the best algorithm from those selected. This choice was based on a set of criteria: Results must be convex Results intersect only on borders The nature of the shapes on which the authors tested their algorithm Minimizing the number of resulting parts Minimizing the length of contours Number and type of parameters used for each algorithm After the comparison, we deduced that the technique proposed by Liu & al. [11] is close to satisfy our constraints. In fact, the results of the DuDe [11] solution are more efficient comparing to those of the other algorithms. DuDe is a binary linear method to decompose complex shapes with an important number and different sizes of holes. It works on overlopping objects and divide them into non-overlopping cuts. Resulting cuts are near-convex and their number is minimal. Copyright c 2017 SERSC 19

4 Figure 2. Results of Convex Decomposition using DuDe 3.2. Discretization After decomposing the conception domain into convex sub-domains, we discretize his boundary. This boundary is composed of all borders of resulting components: is a convex sub-domain and n is the number of sub-domains. Discretization aims to represent the boundary by a finite number of points which will represent the extremities of segments to generate. A large number of discretization points provide a better approximation of the design domain by approaching the continuous domain: For each i [1,n] ; { / } (3) By partitioning, we created a set of adjacent components with shared contours. Therefore, the points belonging to a shared boundary are treated commonly for each of these areas during generating step. Moreover, the user chooses the number of discretization points for a component and this number is the same for all sub-domains allowing a uniform discretization Generating a Reticulated Structure Per sub-domain As we have already indicated, automatic generation of a reticulated structure from a convex conception domain is a subject that has already been dug by Chiheb & Panasenko [5]. They exploited the property of a convex domain saying that a segment that ends belong to the convex area is included in this field. So they get to generate all possible segments between the nodes of the discretized boundary of conception domain. The original algorithm of Chiheb & Panasenko [5] start with calling a procedure to link discretization points in pairs. Each link is a segment. This one cannot be accepted if a discretization point (other than its extremities) belongs to him: is a segment if (4) (p is a discretization point) (2) 20 Copyright c 2017 SERSC

5 This test eliminates redundancy in the results. Next, for each created segment, another procedure is called to detect intersection points between this segment and the others that have already been generated. Subsequently, the intersection points are sorted according to their Euclidean distances to extremities of segments to which they belong. Finally, a connectivity matrix is produced to be used for describing the equation of equilibrium. Algorithm 1 shows the resolution steps of [5] and an illustrative example is presented in Figure 3. Algorithm 1: GenerateReticulateStructure_ConvexDomain(D) Input: convex conception domain D Output: Connectivity matrix M 1. Discretize D --> {p k }; 2 For each pi of {p k } 3 For each p j of {p k } such that i j 4. Create segments linking {p k } ; 5. Find_Intersection_Points( ) ; 6. Sort_Points({p I }) ; //p I is an intersection point 7. Genarate_Connectivity_Matrix() --> M; Figure 3. Generating Reticulated Structure in Convex Conception Domain (a) Conception Domain (b) Discretization Step (c) Generated Structure For our new algorithm, all these subroutines should be applied on each sub-domain of decomposition step. Yet, if this application is sequential then the algorithm may be greedy and time-consuming depending on the number of sub-domains and the number of discretization points. Therefore, we suggest to solve this problem by designing the program to be run in parallel in this step. Parallelism is to involve several processors for performing a calculation in order to reduce the running time and optimize the use of material resources. To implement this mechanism we partition a program into tasks which are grouped in process and every process is treated by a processor. For our case, each sub-domain of decomposition step will be handled by an independent process and the list of discretization points is a resource shared between different processes Merging Step Actually merging partial solutions is a fusion of local connectivity matrices into a global matrix. Each process of step 3 gives a reticulated structure with a connectivity matrix describing relations between the nodes of this structure. Copyright c 2017 SERSC 21

6 For a set of points {, a connectivity matrix is defined as a n n matrix M such as =1 when et are directly related otherwise =0. We note that giving a number to a point is crucial to identify each point and therefore to determinate their direct neighbors Figure 4. Example of a Connectivity Matrix So initially, each process of step 3 number the points discretizing the sub-domain that it treats. Then, the process gives a number to each new intersection point. It is obvious that for a given sub-domain numbers are pair wise distinct. However, if we consider the conception domain in all, we have several points with identical numbers. Moreover, we note that the points shared between multiple sub-domains have more than a number. It is because these numbers are assigned by handling only one sub-domain at a time; these numbers are local. Uniquely identify each node of the structure can be done by giving each node a global number. This can be achieved in two ways: Solution 1: Duplicates elimination After step 3, we number nodes of all partial structures. This numbering is global but nodes of shared borders will have more than a global number. So, before assigning a number to a node we check first if it was already considered. This check is based on the spatial coordinates of the node. Solution 2: Shared counter 3 In this option we propose to consider a global counter as a resource which is shared between processes that treat convex sub-domains in step 3. So global numbering must start at the beginning of step 3. First we number discretization points. Thereafter the shared counter is used by different processes with each detection of a new node. Although the first solution seems intuitive, but it can be expensive because it requires comparing all nodes in pairs to eliminate duplicates. That's why we chose the second option to implement our algorithm. After numbering we merge all connectivity matrices generated for each convex subdomain. This fusion is based on correspondence between the local and the global index. Then, the lines concerning a point about local connectivity matrices are copied successively in the global matrix respecting new numbers. The procedure of producing the global connectivity matrix is detailed in Algorithm Copyright c 2017 SERSC

7 Algorithm 2: CM_Fusion({M k }, {C k }, {p k }) Input: Connectivity matrices {M k }, decomposition cuts {C k }, discretization and intersection points {p k } Output: Global connectivity matrix M 1. Number all points 2 Initialize the matrix with 0 3. For each i in [1,N] do // N is the number of points 4. For each j in [1,i-1] do 5. check which C k contains pi --> {C l } ; 6. For each j in [i+1,n] do 7. Copy the value from the connectivity matrix of C l which contains pi respecting correspondence between local and global numbering ; M 1 = M 2 = M = Figure 5. Example of Connectivity Matrices Fusion Copyright c 2017 SERSC 23

8 3.5. Algorithm's Complexity The algorithm we propose in this work is divided into three main phases. For each of these phases, the programming type is selected carefully so as to minimize the complexity of the algorithm. The convex decomposition phase uses a 0-1 linear programming algorithm which is tested on 72 shapes and, according to its authors, its running time don't exceed 2 seconds. Next, a program with three sub-routines is called: Linking discretization points Finding intersection points between generating segments Giving the connectivity matrix; And it is obvious that the running time of this program depends on the number of discretization points. However, the program is not called once but it is executed for each of the components resulting from the decomposition phase (sub-domain). We have programmed this phase using the mechanism of parallelism. Then, the program will be processed by several processes where each process is responsible for creating a reticulated structure per sub-domain. A parallel execution allows the sharing of resources used during execution. These resources are used alternately by the different processes which reduce the running time of this phase. And then to combine the local solutions into a single global solution, we exploit the fact that the connectivity matrix is symmetric. Then the calculation is done for elements instead of such that n is the total number of the structure nodes. 4. Optimization Problem Formulation Conception and optimization of reticulated structures is a discrete topological optimization subject. It seeks the most optimal form of a structure in an environment where the structure can be subjected to mechanical or thermal forces of deformation influencing its stiffness. Our optimization objective is to find the bars transverse sections which minimizes energy of deformation of the reticulated structure under a constant weight constraint. With the automatic generation of the reticulated structure, sections of all the segments are the same is section of a segment and is his length. Optimum values of transverse sections are calculated with an iterative method. This calculation is based on the formula relating rigidity matrix and the deformation force. 5. Industrial Examples 5.1. The Mast A mast is a T-polygon conception domain. Its base is fixed and a load is applied on the upper part on both left and right sides. This case is a non convex conception domain. So we start with an initial reticulated structure generated using the tool presented in this paper. Then this structure is optimized by searching new values of transverse sections and by eliminating the bars with a negligible section. (5) 24 Copyright c 2017 SERSC

9 Figure 6. Optimum Reticulated Structure of a Mast 5.2. L-Shaped Bracket The L-shaped bracket is a problem topology optimization which is often studied in the literature. This structure is submitted to a vertical force at its upper right corner and its upper side is fixed. Using the optimizer of the structure we obtain structures in Figure 7 which shows the structure generated from this non convex conception domain. Figure 7. Optimum Reticulated Structure of a L-Shaped Bracket 6. Conclusion In this paper, we propose a tool to generate an optimum reticulated structure from a non-convex conception domain. We use a recent technique to decompose the conception domain into a minimum number of strict convex sub-domains. Then a reticulated structure is generated for each convex sub-domain using a program which runs in parallel to reduce the complexity of calculation. Those sub-structures are combined to create the required reticulated structure. Next the structure undergoes an optimization process to remove bars which have a negligible transverse section and therefore obtain the desired optimized structure. The key step of our solution is the convex decomposition. That is why we made a comparative study between a selection of methods in order to choose the method that suits us most. However, non-convex conception domains are not all decomposable into strict convex sub-domains. It is true that the chosen method of decomposition allow an Copyright c 2017 SERSC 25

10 approximate decomposition which makes all forms decomposable into convex subdomains with tolerance, but for our purpose we choose to respect the constraint of convexity without tolerance. So it would be interesting to treat this type of non-convex areas in a future work. Figure 8. Forms that Cannot be Decomposed into Exact Convex Parts References [1] G. Allaire, "La méthode d'homogénéisation pour l'optimisation topologique de structures élastiques". [2] P. Duysinx, "Optimisation topologique: du milieu continu à la structure élastique". PhD thesis, University of Liege, Belgium, (1996). [3] R. Chiheb, D. Cioranescu, A. El Janati and G. Panasenko, "Structures réticulées renforcées en élasticité", Comptes Rendus de l'académie des Sciences-Series I-Mathematics, vol. 326, no. 7, (1998), pp [4] G. Allaire, "Conception de Micro Mécanismes par Optimisation Topologique", Congrès Français de Mécanique, (2001). [5] R. Chiheb and G. Panasenko, "A novel algorithm for the conception and optimization of reticulate structures", Applicable Analysis, vol. 91, no. 5, (2012), pp [6] H. Liu, W. Liu and L. J. Latecki, "Convex shape decomposition", In Computer Vision and Pattern Recognition (CVPR), 2010 IEEE Conference, (2010) June. [7] J. M. Keil, "Decomposing a polygon into simpler components", SIAM Journal on Computing, vol. 14 no. 4, (1985), pp [8] J. M. Keil and J. Snoeyink, "On the time bound for convex decomposition of simple polygons", International Journal of Computational Geometry & Applications, vol. 12, no. 3, (2002), pp [9] J. M. Lien and N. M. Amato, "Approximate convex decomposition of polygons", In Proceedings of the twentieth annual symposium on Computational geometry, (2004) June. [10] Z. Ren, J. Yuan and W. Liu, "Minimum near-convex shape decomposition", Pattern Analysis and Machine Intelligence, IEEE Transactions on, vol. 35, no. 10, (2013), pp [11] G. Liu, Z. Xi and J.M. Lien, "Dual-space decomposition of 2d complex shapes", In Computer Vision and Pattern Recognition (CVPR), 2014 IEEE Conference, (2014) June. 26 Copyright c 2017 SERSC

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