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Abstract
Triangular pyramid and Quadrangular pyramid elements for partial double-layer spherical reticulated shells of pyramidal system are investigated in the present study. Macro programs for six typical partial double-layer spherical reticulated shells of pyramidal system are compiled by using the ANSYS Parametric Design Language (APDL). Internal force analysis of six spherical reticulated shells is carried out. Distribution regularity of the stress and displacement are studied. A shape optimization program is proposed by adopting the sequence two-stage algorithm (RDQA) in FORTRAN environment based on the characteristics of partial double-layer spherical reticulated shells of pyramidal system and the ideas of discrete variable optimization design. Shape optimization is achieved by considering the objective function of the minimum total steel consumption, global and locality constraints. The shape optimization of six spherical reticulated shells is calculated with the span of 30m~120m and rise to span ratio of 1/7~1/3. The variations of the total steel consumption along with the span and rise to span ratio are discussed with contrast to the results of shape optimization. The optimal combination of main design parameters for six spherical reticulated shells is investigated, i.e., the number of the optimal grids. The results show that: (1) The Kiewitt and Geodesic partial double-layer spherical reticulated shells of triangular pyramidal system should be preferentially adopted in large and medium-span structures. The range of rise to span ratio is from 1/6 to 1/5. (2) The Ribbed and Schwedler partial double-layer spherical reticulated shells of quadrangular pyramidal system should be preferentially adopted in small-span structures. The rise to span ratio should be 1/4. (3) Grids of the six spherical reticulated shells can be optimized after shape optimization and the total steel consumption is optimized to be the least.

Key Words
pyramidal system; APDL; parametric modeling; RDQA; shape optimization

Address
J. Wu: Geotechnical and Structural Engineering Research Center, Shandong University, Ji\'nan 250061, Shandong, China; Institute of Engineering Mechanics, Shandong Jianzhu University, Ji\'nan 250101, Shandong, China X.Y. Lu: Institute of Engineering Mechanics, Shandong Jianzhu University, Ji\'nan 250101, Shandong, China S.C. Li: Geotechnical and Structural Engineering Research Center, Shandong University, Ji\'nan 250061, Shandong, China D.L. Zhang: Shandong Agriculture and Engineering University Ji\'nan 250100, Shandong, China Y.G. Xue, L.P. Li and Y.G. Xue: Geotechnical and Structural Engineering Research Center, Shandong University, Ji\'nan 250061, Shandong, China

Abstract
The goal of this study is to investigate computational convergence of optimal solutions, with respect to optimality criteria (OC) method and methods of moving asymptotes (MMA) as optimization model for non-linear programming of material topology optimization using an acceleration method that makes design variables rapidly move toward almost 0 and 1 values. 99 line topology optimization MATLAB code uses loop vectorization and memory pre-allocation as properly exploiting the strengths of MATLAB and moves portions of code out of the optimization loop so that they are only executed once as restructuring the program. Numerical examples of a simple beam under a lateral load and a given material density limitation provide merits and demerits of the present OC and MMA for 99 line topology optimization code of continuous material topology optimization design.

Key Words
acceleration method; OC; MMA; MATLAB; topology optimization; convergence

Address
Dongkyu Lee and Nguyen Hong Chan: Department of Architectural Engineering, Sejong University, Seoul, 143-747, Korea Soomi Shin: Research Institute of Industrial Technology, Pusan National University, Busan, 609-735, Korea

Abstract
High powered computers and engineering computer systems allow designers to routinely simulate complex physical phenomena. The presented work deals with the analysis of two finite element method optimization techniques (First Order Method-FOM and Subproblem Approximation Method-SAM) implemented in the individual Design Optimization module in the Ansys software to analyze the behavior of real problems. A design optimization is a difficult mathematical process, intended to find the minimum or maximum of an objective function, which is mostly based on iterative procedure. Using optimization techniques in engineering designs requires detailed knowledge of the analyzed problem but also an ability to select the appropriate optimization method. The methods embedded in advanced computer software are based on different optimization techniques and their efficiency is significantly influenced by the specific character of a problem. The efficiency, robustness and accuracy of the methods are studied through strictly convex two-dimensional optimization problem, which is represented by volume minimization of two bars‟ plane frame structure subjected to maximal vertical displacement limit. Advantages and disadvantages of the methods are described and some practical tips provided

Key Words
design optimization; First Order Method; Subproblem Approximation Method; feasible/infeasible design space; robustness; accuracy

Address
Filip Fedorik and Mikko Malaska, Structural Engineering and Construction Technology Research Group, University of Oulu, P.O. Box 4200, FI-90014, Oulu, Finland Jiří Kala, Institute of Structural Mechanics, Brno University of Technology, Veveří 331/95, 60200 Brno, Czech Republic Antti Haapala, Wood Material Science, University of Eastern Finland, P.O. Box 111, FI-80101, Joensuu, Finland

Abstract
This study presents optimizing structural topology patterns using regularization of Heaviside function. The present method needs not filtering process to typical SIMP method. Using the penalty formulation of the SIMP approach, a topology optimization problem is formulated in co-operation, i.e., couple-signals, with design variable values of discrete elements and a regularized Heaviside step function. The regularization of discontinuous material distributions is a key scheme in order to improve the numerical problems of material topology optimization with 0 (void)-1 (solid) solutions. The weak forms of an equilibrium equation are expressed using a coupled regularized Heaviside function to evaluate sensitivity analysis. Numerical results show that the incorporation of the regularized Heaviside function and the SIMP leads to convergent solutions. This method is tested using several examples of a linear elastostatic structure. It demonstrates that improved optimal solutions can be obtained without the additional use of sensitivity filtering to improve the discontinuous 0-1 solutions, which have generally been used in material topology optimization problems.

Key Words
optimization; topology patterns; SIMP; filtering process, regularized Heaviside function

Address
Dongkyu Lee, Department of Architectural Engineering, College of Engineering, Sejong University, 143-747, Seoul, Korea Soomi Shin, Research Institute of Industrial Technology, Pusan National University, 609-735, Busan, Korea

Abstract
The main objective of this research is to present the procedures of combining topology, shape & sizing optimization for truss structure by employing strain energy as objective function under the constraints of volume fractions which yield more general solution than that of total weight approach. Genetic Algorithm (GA) is used as searching engine for the convergence solution. A number of algorithms from previous research are used for evaluating the feasibility and stability of candidate to accelerate convergence and reduce the computational effort. It is followed by solving problem for topology & shape optimization and topology, shape & sizing optimization of truss structure to illustrate the feasibility of applying the objective function of strain energy throughout optimization stages.

Key Words
truss optimization; topology optimization; strain energy; Kinematic stability; genetic algorithm

Address
Xuan-Hoang Nguyen and Jaehong Lee: Department of Architectural Engineering, Sejong University, 98 Gunja Dong, Gwangjin Gu, Seoul 143-747, Republic of Korea

Abstract
In this paper, the Multi-Swarm Fruit Fly Optimization Algorithm (MFOA) is presented for structural damage identification using the first several natural frequencies and mode shapes. We assume damage only leads to the decrease of element stiffness. The differences on natural frequencies and mode shapes of damaged and intact state of a structure are used to establish the objective function, which transforms a damage dentification problem into an optimization problem. The effectiveness and accuracy of MFOA are demonstrated by three different structures. Numerical results show that the MFOA has a better capacity for structural damage identification than the original Fruit Fly Optimization Algorithm (FOA) does.

Key Words
damage identification; multi-swarm fruit fly optimization algorithm; non-destructive techniques; frequency domain

Address
S. Li and Z.R. Lu: damage identification; multi-swarm fruit fly optimization algorithm; non-destructive techniques; frequency domain

Abstract
The suspended dome system is a new structural form that has become popular in the construction of long-span roof structures. Suspended dome is a kind of new pre-stressed space grid structure that has complex mechanical characteristics. In this paper, an optimum topology design algorithm is performed using the enhanced colliding bodies optimization (ECBO) method. The length of the strut, the cable initial strain, the cross-sectional area of the cables and the cross-sectional size of steel elements are adopted as design variables and the minimum volume of each dome is taken as the objective function. The topology optimization on lamella dome is performed by considering the type of the joint connections to determine the optimum number of rings, the optimum number of joints in each ring, the optimum height of crown and tubular sections of these domes. A simple procedure is provided to determine the configuration of the dome. This procedure includes calculating the joint coordinates and steel elements and cables constructions. The design constraints are implemented according to the provision of LRFD-AISC (Load and Resistance Factor Design-American Institute of Steel Constitution). This paper explores the efficiency of lamella dome with pin-joint and rigid-joint connections and compares them to investigate the performance of these domes under wind (according to the ASCE 7-05), dead and snow loading conditions. Then, a suspended dome with pin-joint single-layer reticulated shell and a suspended dome with rigid-joint single-layer reticulated shell are discussed. Optimization is performed via ECBO algorithm to demonstrate the effectiveness and robustness of the ECBO in creating optimal design for suspended domes.

Key Words
topology optimization; cable tension optimization; enhanced colliding bodies optimization; lamella dome; suspended dome; pin-joint dome; pre-stressed structure; double layer dome

Address
A. Kaveh: Centre of Excellence for Fundamental Studies in Structural Engineering, Iran University of Science and Technology, Narmak, Tehran, P.O. Box 16846-13114, Iran M. Rezaei: Road, Building and Housing Research Center, Tehran, P.O. Box 1145-1696, Iran

Abstract
There is a growing trend of considering uncertainty in optimization process since last few decades. In this regard, Robust Design Optimization (RDO) scheme has gained increasing momentum because of its virtue of improving performance of structure by minimizing the variation of performance and ensuring necessary safety and feasibility of constraint under uncertainty. In the present study, RDO of reinforced concrete folded plate and shell structure has been carried out incorporating uncertainty in the relevant parameters by Monte Carlo Simulation. Folded plate and shell structures are among the new generation popular structures often used in aesthetically appealing constructions. However, RDO study of such important structures is observed to be scarce. The optimization problem is formulated as cost minimization problem subjected to the force and displacements constraints considering dead, live and wind load. Then, the RDO is framed by simultaneously optimizing the expected value and the variation of the performance function using weighted sum approach. The robustness in constraint is ensured by adding suitable penalty term and through a target reliability index. The RDO problem is solved by Sequential Quadratic Programming. Subsequently, the results of the RDO are compared with conventional deterministic design approach. The parametric study implies that robust designs can be achieved by sacrificing only small increment in initial cost, but at the same time, considerable quality and guarantee of the structural behaviour can be ensured by the RDO solutions.

Key Words
robust design optimization; Monte Carlo simulation; folded plate structure; reinforced concrete shell; target reliability; parameter uncertainty

Address
Soumya Bhattacharjya, Subhasis Chakraborti and Subhashis Das: Department of Civil Engineering, Indian Institute of Engineering Science and Technology, Shibpur, West Bengal, India

Abstract
Spherical reticulated shells are widely applied in structural engineering due to their good bearing capability and attractive appearance. Parametric modeling of spherical reticulated shells is the basis of internal analysis and optimization design. In the present study, generation methods of nodes and the corresponding connection methods of rod elements are proposed. Modeling programs are compiled by adopting the ANSYS Parametric Design Language (APDL). A shape optimization method based on the two-stage algorithm is presented, and the corresponding optimization program is compiled in FORTRAN environment. Shape optimization is carried out based on the objective function of the minimum total steel consumption and the restriction condition of strength, stiffness, slenderness ratio, stability. The shape optimization of four typical Schwedler spherical reticulated shells is calculated with the span of 30 m~80 m and rise to span ratio of 1/7~1/2. Compared with the shape optimization results, the variation rules of total steel consumption along with the span and rise to span ratio are discussed. The results show that: (1) The left and right rod-Schwedler spherical reticulated shell is the most optimized and should be preferentially adopted in structural engineering. (2) The left diagonal rod-Schwedler spherical reticulated shell is second only to left and right rod regarding the mechanical behavior and optimized results. It can be applied to medium and small-span structures. (3) Double slash rod-Schwedler spherical reticulated shell is advantageous in mechanical behavior but with the largest total weight. Thus, this type can be used in large-span structures as far as possible. (4) The mechanical performance of no latitudinal rod-Schwedler spherical reticulated shell is the worst and with the second largest weight. Thus, this spherical reticulated shell should not be adopted generally in engineering.

Key Words
schwedler spherical reticulated shell; APDL; parametric modeling; shape optimization

Address
J. Wu, X.Y. Lu, S.C. Li and Y.G. Xue: Geotechnical and Structural Engineering Research Center, Shandong University, Ji\'nan 250061, Shandong, China; Institute of Engineering Mechanics, Shandong Jianzhu University, Ji\'nan 250101, Shandong, China X.Y. Lu: Institute of Engineering Mechanics, Shandong Jianzhu University, Ji\'nan 250101, Shandong, China S.C. Li, Z.H. Xu, L.P. Li: Geotechnical and Structural Engineering Research Center, Shandong University, Ji\'nan 250061, Shandong, China D.L. Zhang: Shandong Agriculture and Engineering University, Ji\'nan 250100, Shandong, China Y.G. Xue: Geotechnical and Structural Engineering Research Center, Shandong University, Ji\'nan 250061, Shandong, China

Abstract
A methodology based on Teaching Learning-Based Optimization (TLBO) algorithm is proposed for optimum design of reinforced concrete retaining walls. The objective function is to minimize total material cost including concrete and steel per unit length of the retaining walls. The requirements of the American Concrete Institute (ACI 318-05-Building code requirements for structural concrete) are considered for reinforced concrete (RC) design. During the optimization process, totally twenty-nine design constraints composed from stability, flexural moment capacity, shear strength capacity and RC design requirements such as minimum and maximum reinforcement ratio, development length of reinforcement are checked. Comparing to other nature-inspired algorithm, TLBO is a simple algorithm without parameters entered by users and self-adjusting ranges without intervention of users. In numerical examples, a retaining wall taken from the documented researches is optimized and the several effects (backfill slope angle, internal friction angle of retaining soil and surcharge load) on the optimum results are also investigated in the study. As a conclusion, TLBO based methods are feasible.

Key Words
cantilever retaining wall; reinforced concrete structures; Teaching-Learning Based optimization (TLBO); optimum design

Address
Rasim Temür and Gebrail Bekdaş: Department of Civil Engineering, Istanbul University, 34320 Istanbul, Turkey

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