Optimization of structures using convex model superposition

Optimization of structures using convex model superposition Chris P. Pantelides, Sara Ganzerli Department of Civil and Environmental Engineering, University of C/Wz, &;/f Za&e C;Yy, C/faA ^772, U&4 Email: chris@civil. Utah, edu and S. Ganzerli@m. cc. Utah, edu Abstract A new approach is presented for implementing a multidimesional convex model for the optimal design of structures subjected to bounded but uncertain loads. This is a non-probabilistic method for including uncertainty in the design. In previous studies, a two-stage optimization process was used to predict the effect of uncertainties, modeled using convex theory, on the response constraints. Here, a superposition method is used to find the response of structures with load uncertainties. The convex model is implemented on the effect of the uncertain loads, i.e. the displacements and stresses. The advantage of the method is that one optimization process is eliminated, and the constraints do not need to be expressed explicitly as a function of the design variables. Introduction Most structural design problems include uncertainties in both the demand on the structure in terms of loads, and the capacity of the structure in terms of resistance. A new method for modeling uncertainties in optimal structural design is the convex model (Pantelides and Ganzerli [1]). The uniform bound convex model was implemented for structural optimization, through an antioptimization process that finds the worst condition produced on the constraints by a certain load condition (Elishakoff et al. [2]). The anti-optimization process is solved at each step of the structural optimization, therefore the two optimization processes are nested. Lombardi et al. proposed the use of the antioptimization technique in the study of composite materials [3], and Barbieri et al. used the anti-optimization technique for obtaining the optimal shape and size of trusses [4]. Lombardi [5, 6] proposed the "two step" optimization method, in which the two optimization processes are executed independently. In this paper,

196 Computer Aided Optimum Design of Structures the second optimization process is eliminated entirely, using instead a superposition method of convex models. Formulation of the Problem The objective function to be minimized is the volume or the cost of the structure, f (A, P), where A is the vector of the design variables or areas of the structural members; P is the vector of the design parameters or uncertain loads. The optimal design problem can be stated as minimize f (A, P) A such that gj(a, P)>0 ; j = 1...ng (1) where g/a) are the constraints, and ng is the number of constraints to be satisfied by the optimal design. In general, the constraints may include axial or bending stresses for the structural members, joint displacements, and member buckling loads. The parameters that characterize the design, such as loads, are often uncertain. In the previous work [2, 5, 6] a uniform bound convex model was implemented to include these uncertainties. In the uniform bound convex model, each uncertain parameter is constrained to vary between an upper and lower bound. The set of parameters that describe the problem is assumed to belong to a bounded set Cp. The aim is to find the most critical values of the design parameters P, that belong to Cp. This problem has been solved by Elishakoff et al. [2] using an anti-optimization process, in which the two optimizations are nested. The method employs two optimization procedures: the first is the minimization of the structural volume under stress and displacement constraints; the second optimization is the process of finding the maximum structural response for uncertain loads. In the "two step" approach, presented by Lombardi [5, 6], the two-optimization processes are made independent of each other, reducing the computational effort of the entire procedure. In the "two step" approach the minimization of the function f(a, P) and the maximization of the constraint functions over the convex domain are carried out in two separate steps. Both approaches lead to the same results; the "two step" approach is computationally more efficient. The innovative idea of the approach presented here is to eliminate the second step in the procedure, i.e., to be able to solve the design problem with only one optimization process. The optimization is implemented only for the traditional task of structural design, while taking into account the uncertainties that affect the design parameters, i.e. the loads. This can be achieved provided that superposition can be used in the optimal structural design. For example, consider a structure which is loaded with two loads, PI and fj, that are uncertain in magnitude. Let the nominal values be /V and P ^ respectively. Assume that the bounds of these uncertainties are known. Furthermore, assume that PI deviates from its nominal value by (Pi * f/), and PI deviates from its nominal

Computer Aided Optimum Design of Structures 197 value by (02 * f/), where Pi and P2 are percent deviations. These loads will create stresses and displacements that will be used as constraints in the optimal design. The deviations of the load magnitudes from their nominal values are assumed to be represented by a uniform bound convex set. For two uncertain parameters, one can represent the bounded convex domain as a rectangle. This concept can be generalized to N number of uncertainties where the convex domain is an N- degree multidimensional box. Without loss of generality, the convex set when only two loads are considered, can be described as <p <p/ CP=a,, where P. and P- represent the lower and the upper bounds of the magnitude of the load Pj respectively, and P is the nominal value of the load. The quantities P. and P" can also be expressed as a function of the percentages of the uncertainties as follows vl-pj-fijpj (3a) P» = pj+f3jpj (3b) where Py is the percentage of uncertainty that affects the nominal value of the magnitude of the load Pj. As an illustrative example, assume that a structure has two loads PI and?->, and two degrees of freedom; the displacements and the internal forces of the structure, which is assumed to respond in the linear range, can be expressed as a linear function of the loads as [7] >0 F(P,P ) = SBX(P,Pj) = SBK-N (4b) where X are the global displacements, K* is the inverse of the global stiffness matrix, B is the transpose of the statics matrix, and S is the element stiffness matrix.

198 Computer Aided Optimum Design of Structures To obtain the worst possible case of the effect of the uncertain but bounded loads on the structural response, it is desirable to implement the convex model on the effect (forces and displacements). This can be done by evaluating the nominal displacements and forces, when the structure is loaded with PI and Pj, applied separately. First, the displacements and forces due to PI are expressed as (5a) ) (5b) Similarly, the displacements and forces due to P^ can be expressed using eqn (5) if subscript 1 is replaced by subscript 2. Second, the convex model is implemented on the displacements (forces) to maximize the structural response due to the uncertain loads. This is achieved by adding the magnitude of the displacements (forces) obtained by loading the structure with only PI multiplied by its percent deviation, i.e. Pi, and the magnitude of the displacements (forces) obtained by loading the structure with P^ multiplied by its percent deviation, i.e. Pi, to the magnitude of the nominal displacements and forces given by eqn (4). The maximum convex displacements and forces due to the uncertain loads are given as X,on=X( 7>o, ^ )±ZP; X( P.O ) Fcon=F( P*,Pf)± Zpi F ( Pf ) (6a) (6b) where, the summation in this case extends from i=l to 2; if the nominal displacements are positive the plus sign must be used, otherwise the minus sign is used. The same sign convention is true for the forces. In the optimal structural design, the convex displacements and forces represented by eqn (6) are used directly in the evaluation of the constraints gj(a, P) in eqn (1). If the uncertainty in the nominal value of the loads can be represented by a convex set, it is also true that the displacements and internal forces produced by the uncertain loads belong to a convex set. Recall the definition for the linear combination of convex sets. If Aj(i = l,...,r) is a collection of convex sets in Ln, then A(A)=Z4Aj ; 4>0 (7) i=l is a linear combination of the sets A; (i = l,...,r). Ln is a finite n-dimensional normed linear space. According to a convex set theorem [8], the linear combination A(/l) in eqn (7) is convex for each set of values X, with A., 0 (/ =!,...,/ ).

Computer Aided Optimum Design of Structures 199 The displacements and forces due to the convex portion of the loads, P are obtained by a linear combination of the convex portion of the loads and the elements of the K* and SBK* matrices. These matrix elements are independent of the uncertainties in the loads, and since the absolute value of the displacements and the forces is considered in eqn (6), these can be thought of as the positive coefficients 1; in eqn (7). Furthermore, the convex portion of the loads, ftjpj, in eqn (3) represents elements belonging to the convex set. The linear combination of the convex portions contained in the convex displacements and forces in eqn (6) is convex because the coefficients (5^ are percentages, hence positive numbers. Thus, the expressions for displacements and forces in eqn (6) are the convex model-based responses, and can be used directly in the optimization of the structure. The structural optimization has as its objective the minimization of the structural volume, which is related to the cost to be minimized. The optimization algorithm was stated in eqn (1). The constraints are derived from eqn (6), which incorporates the convex model superposition to account for the uncertainties in the loads. The use of eqn (6) does not create non-linearity because the optimization algorithm used (i.e. the method of feasible directions) is independent of the derived equations. The program D.O.T. [9] was used for the optimization process. The structural analysis is performed for a structure which has a response that remains in the linear range of the material properties. In addition, geometric non-linearities have not been considered. Ten-bar Truss Example A ten bar aluminum truss was chosen for the purpose of comparison of the method presented here with the methods presented in [2, 5, 6]. The ten-bar aluminum truss is shown in Figs. 1 and 2. The modulus of elasticity of the truss members is 10* ksi. The length of the horizontal and vertical members is 360 in. The loading condition is composed of three loads: PI is a vertical load at degree of freedom (DOF) 6, P^ is a vertical load at DOF 8, and PS is a horizontal load at DOF7. The objective function to be minimized in the optimization process is the volume of the structure. The design variables are the areas of the members; constraints are imposed on stresses and displacements. The minimum cross sectional area is set at 0.1 inl The limit on the stresses is 25 ksi for all members except for member 9, for which the maximum allowable stress in this case is set at 75 ksi. No distinction is made between the limit on the compressive or tensile stresses. The allowable displacement is set at 5 in. The displacement of DOF 8 controls the design for this loading condition. Two optimal designs are performed with different loading conditions and with two different sets of constraints; they will be referred to as CASE 1 and CASE 2. In CASE 1, the nominal value of the loads is Pj = P^ = 100 kip, and f, = 400 kip. Constraints are imposed on the stresses for all members, and on the vertical displacement of P,

200 Computer Aided Optimum Design of Structures 360 in. 360 in. 360 in. Load Element Number Node Number Figure 1. Ten-bar Aluminum Truss (1 in. = 25.4 mm) 1 Degree of Freedom Node Number Figure 2. Ten-bar Aluminum Truss Displacements

Computer Aided Optimum Design of Structures 201 DOF 8. In the second optimal design, CASE 2, the nominal value of the loads is PI= P2= 100 kip, and PS = 0 kip. Constraints are imposed only on the stresses. The optimal design of CASE 1 under nominal loads is presented in Table 1, and that for CASE 2 under the nominal load condition is presented in Table 2. Table 1. CASE 1 - Optimal Volume Designs with Stress and Displacement Constraints for the 10-bar Aluminum Truss Quantity Cross-Sectional Area Member Number (1) 1 2 3 4 5 6 7 8 9 10 Volume (in.*)' Nominal Case (2) 4.0354 4.0354 12.0996 3.8651 11.2637 2.7580 1.5986x10* (in.y Convex Model (3) 4.4605 0.10 6.8217 14.0409 4.3126 12.3827 4.6343 1.9506x10* Table 2. CASE 2 - Optimal Volume Designs with Stress Constraints for the 10-bar Aluminum Truss Quantity Cross-Sectional Area Member Number (1) 1 2 3 4 5 6 7 8 9 10 Volume (in.y in. = 25.4 mm Nominal Case (2) 7.8933 0.10 8.0945 3.9045 0.1033 5.7969 5.5072 3.6698 0.1445 1.4965x10* Convex Mod el (3) 8.7097 0.10 8.8903 4.3098 6.3503 6.0949 4.6587 0.1445 1.6775x104

202 Computer Aided Optimum Design of Structures Subsequently, uncertainties in the loads are considered, and the optimal designs for the two cases are obtained using the superposition method presented in Eq. (6). In CASE 1 and CASE 2, all the loads are assumed to be uncertain by 10 % of their nominal value. Therefore P/ = P/ = 90 kip, and P" = P^=ilO kip. In CASE 1, load PS is also considered with upper and lower bounds of P" = 440 kip, and P/ = 360 kip. In both CASE 1 and 2, the volume of the convex model design is higher than the volume of the nominal case. The results presented here are in excellent agreement with the ones obtained by the "antioptimization" method [2], and the "two step" method [5, 6]. Conclusions An optimal structural design under uncertain loads is presented. A new method was introduced to implement the well known uniform bound convex model. Previous studies, proposed a solution of this problem using the anti-optimization process and the "two step" method. These methods employ two optimization procedures: thefirstis the minimization of the structural volume under stress and displacements constraints; the second optimization is the process of finding the maximum structural response for the uncertain loads. These two processes are nested in the anti-optimization method, whereas in the "two step" optimization they can be executed separately. The present method allows for performance of the structural optimization, taking into account the uncertainties in the loads, without the need for a second optimization process. The second optimization process is eliminated by using a convex model superposition method. As opposed to the convex model methods previously adopted, the convex model superposition method does not require the expression of the constraints as explicit functions of the design variables and design parameters. This benefit becomes important for large structures with a large number of uncertain loads. Acknowledgements Financial support by the National Science Foundation under Grant No. CMS- 9522654 is gratefully acknowledged. References [1] Pantelides, C. P., and Ganzerli, S., Design of trusses under uncertain loads using convex models. ASCE J. of Structural Engineering, 124 (3), 318-329, 1998.

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