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There are six equality constraints for forces Qi for member i; therefore, Eq. (26) fact includes 6n equality constraints.
For the equality constraints in Eq. (25), since the global equilibrium equations are only functions of Qi, =0, and =0. =1 (s=1, . . . ,6n),if force Qs is connected to displacement j, and =0 otherwise. Note that Qs is the sth component of the force vector Q for all structural members. For the equality constraints in Eq. (26) and _hi e /_Qs=_si; s=1, . . . ,6n, where _si=0, if Qs is not a force component of member i; otherwise it is a unit direction vector, with its element corresponding to Qs as one, and all others as
and =_si; s=1, . . . ,6n, where =0, if Qs is not a force component of member i; otherwise it is a unit direction vector, with its element corresponding to Qs as one, and all others as zero.
For the inequality constraints in Eq. (27)
and =0 (q=1, . . . ,m). It can be seen that once the areas A, nodal displacements r, and member forces Q are available, Eqs. (25)–(31) can be evaluated directly for constraints and their derivatives.
Implementation
A SQP algorithm in SNOPT package is used to solve optimization problems (Gill et al. 2002). It is a stand-alone Windows-based program that uses text files to communicate with the commercial package ANSYS (Swanson Analysis Systems, Inc. 2002). To use the algorithm, cost and constraint functions and their gradients need to be provided. For all the formulations, ANSYS is used as a black box and the data from its output files are used to evaluate the functions and their derivatives externally to ANSYS. A system-level command is used to execute the stand-alone program ANSYS. Sparsity of all the function gradients is exploited while using SNOPT with alternate formulations. For CF, member areas, moments of inertia and other data are sent to ANSYS which performs structural analysis and writes nodal displacements and member forces in the output file. In the alternative formulations, no equilibrium equations are solved as nodal displacements are also sent to ANSYS and it calculates and outputs member forces.
Conventional Formulation
In the conventional formulation, ANSYS is used to analyze the structure and is called again to calculate the sensitivity information by the analytical method (Arora 1995). The constraint gradients are evaluated using the direct differentiation method (Wang and Arora 2005). In this process, additional assembly of the global stiffness matrix and its decomposition are not needed. ANSYS is restarted with the new sensitivity load vectors to evaluate the gradients. The sensitivity load vectors are assembled external to ANSYS using data in its output file and the element matrices. Therefore, ANSYS is called twice for one evaluation of both functions and their derivatives, which makes the implementation a little tedious. In addition, each call for function evaluation during line search requires complete structural analysis. This implementation is similar to that for the trusses in Wang and Arora (2005)where more details are given.
Alternate Formulation 1
As matrices and for member i are fixed and independent of A and r, the constraint functions and their derivatives in Eqs. (19)–(24) can be easily implemented using the current A and r. and contain information about elastic modulus E, member connectivity information, such as transformation between the local and global coordinate systems, and member length L. All these can be read from the ANSYS input or output file. Another possibility is to evaluate the equilibrium equality constraints in Eq. (19) and stress constraints in Eq. (20) by directly using the member forces and stresses in ANSYS output file, instead of using A and r to calculate them. Derivatives of constraint functions are evaluated external to ANSYS, using the matrices and , and the member connectivity information. Note that as matrices and in Eqs. (19)–(24) are simple in form, they are programmed in user subroutines.