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Alternate Formulation 2
It is seen that once the cross-sectional areas A, nodal displacements r and member nodal forces Q are available, Eqs. (25)–(31) can be used to evaluate the constraints and their derivatives. This requires values of the variables A and r and the matrices and Bi b for the ith member. Similar to AF1, the matrices and in Eqs. (26), (28), and (29) are calculated using the output data from ANSYS. In this study, the equilibrium constraints in Eqs. (25) and (26)are calculated directly using the current values of A, Q, and r. However, it is viable to evaluate Eq. (26) by directly using the member forces in ANSYS output file, instead of using A and r to calculate them.
In both the alternative formulations, only the cross-sectional properties and displacements are needed by the analysis code to calculate the member level quantities, such as forces, and to calculate the constraint functions and their derivatives. ANSYS is called only once for one evaluation of both functions and their
derivatives; therefore, no restart capability is needed. Basically AF1 and CF need similar calculations for gradient evaluations, except that no sensitivity analysis equations are solved in AF1. In AF2, the constraints in Eqs. (26)and (27) are both member-level calculations. Eq. (25)contains global equilibrium equations, which is in a simple sparse linear form and no assembly of global stiffness matrix is needed. The gradient calculation of functions in Eq. (25) is performed only once in the optimization process, since they are linear in variables. Therefore, the inclusion of forces as variables provides a decoupled representation of the problem functions, which makes the implementation of AF2 easier than AF1. The derivatives and in Eqs. (21), (23), (28), and (30) can be evaluated external to ANSYS, using appropriate relations in Eqs. (1)–(3).
Optimization Procedure
The step-by-step optimization procedure, illustrated in Fig. 2, is explained as follows
1. Define the optimization problem, including the objective function, optimization variables and constraints. Estimate initial values of the variables.
2. The optimization code calls the user-supplied subroutines, which calculate the objective and constraint functions and their derivatives. With the current values of the optimization variables, the user-supplied subroutines further call ANSYS to obtain the internal forces and stresses for each frame member. Constraint functions and their derivatives are evaluated explicitly using the member stiffness matrices and connectivity information. During the line search, ANSYS is called again to evaluate the problem functions.
3. Optimization variables are updated and the stopping criteria are checked for optimum solution
Role of ANSYS
Role of existing analysis software _ANSYS_ in different formulations is elaborated here for framed structures. The analysis software provides member connectivity information and direction cosines,
etc. Also if desirable, equality constraints in Eq. _19_ can be formed directly using the member forces in ANSYS output file. It may seems that use of the analysis program is not necessary in the alternative formulations, since the member-level matrices for a frame member are simple and explicit, and they can be directly programmed and calculated. However, the pre- and postprocessing capabilities of the existing codes are useful, especially for problems that are more complex. In those problems, it may not be possible or trivial to write the finite element matrices explicitly, and their coding may not be straightforward. This aspect will be investigated in future research, when structures that are more complex are considered, such as shells and plates.
In the conventional formulation, since the analysis program must be restarted to evaluate the displacement gradients, the program must have restart capability to make CF efficient; otherwise, the stiffness matrix and its decomposition will have to be regenerated which is inefficient. If the program uses iterative pro-cedures to solve the system of equations or some other approximate procedures, then that procedure must be repeated for sensitivity analysis which can be quite time consuming. It is noted here that although ANSYS is used in the present study, it can be replaced with any other analysis program with similar capabilities.