Structural Optimization 4, 156-164 (1992) Structml Optimization © Spdnger-Verlag 1992 On the convergence quality of minimum-weight design of large space frames under multiple dynamic constraints* O.G. McGee and K.F. Phan Department of Civil Engineering, The Ohio State University, Columbus, Ohio 43210, USA Abstract Weight optimization of large space frames under dynamic constraints requires adaptable design search techniques because of the enormous number of sizing variables, the highly nonlinear frequency constraint surfaces, and the multiple points of local minima. With modern OC procedures, based on alternately satisfying the constraints (scaling) and applying the Kuhn-Tucker (optimality) condition (resizing), the convergence to weight min- ima is oftentimes oscillatory. Even though the convergence rate for OC methods is initially fast, it gradually slows near local ex- trema, mainly because the selection of appropriate step sizes (or move limits) in the redesign phase becomes increasingly difficult. The focus here is to create specific criteria based on past scaled designs to *'damp out" the oscillatory convergence propensity of OC recursive methods. Several OC recursive strategies, which are frequently worked to resize and to evaluate the Lagrange multi- pliers, are steered by the present approach to design large space frames under multiple dynamic constraints. Besides this, the de- sign iteration histories obtained by the various recursive strategies are compared graphically. On the whole, the present OC approach achieves a smooth upper-bound convergence to weight minima, as it quickly dissolves the (sometimes violent) oscillations of scaled weights in the iteration history. Most of all, the method elimi- nates the need for adjustments of internal parameters during the redesign phase. 1 Introduction The pure sizing optimization of large space frames having nonstruetural mass with sp ecified gauge restrictions and nat- ural frequency constraints is a purposeful class of design prob- lem. One practical connection involves controlling critical ranges of natural frequencies to avoid potential resonant con- ditions either reverberating from connected equipment and actuators, or ensuing from narrow-band forced excitations. Be that as it may, the preliminary design of practical space frames can feature hundreds of independent sizing variables and thousands of dependent ones. More importantly, reso- nant proviso near the lowest dynamical modes may in fact govern the multidisciplinary optimization issues linked with the preliminary design. Indeed, the frequency constraints as- sociated with these critical modes are highly nonlinear in the sizing variable space. Because of the extreme nonlinearity therein, a design search must locally record previous design iterates, and hence, adaptively select new search paths to assure a smooth convergence to weight minima. *Presented in part at the Third Air Force/NASA Syrup. on "Recent Advances in multidisciplinary analysis and optimization" in San Frar~- cisco, California, Sept. 24-26, 1990 The optimality criteria (OC) approach to structural synthesis was introduced approximately two decades ago (Venkayya 1971). Since then, many algorithms using OC procedures have been proposed [see the paper by Venkayya (1978) for an excellent review]. Most of the OC methods pro- posed have added to a conceptual understanding of practical optimization techniques for large-scale structural problems. Modern OC algorithms (Khot 1985; Grandhi and Venkayya 1986; Venkayya and Tischler 1983; Fleury and Sander 1979; Kolonay et a l. 1988; McGee and Phan 1988, 1989, 1991; Phan and McGee 1991a, 1991b; McGee et al. 1990; Bathe 1982) involve first choosing a suitable initial design, and then, al- ternately satisfying multiple constraints (scaling) and apply- ing the Kuhn-Tueker (optimality) condition (resizing) to in- directly reach an optimum solution. Meanwhile, a number of reeursive procedures are available (Khot 1985; Grandhi and Venkayya 1986; Venkayya and Tischler 1983) for rudi- mentary estimates of the sizing variables. For computational ease, several uncoupled recursive approaches have been pro- posed to estimate the Lagrange multipliers that correspond to the critical or potentially critical constraints (Khot 1985; Grandhi and Venkayya 1986; Venkayya and Tischler 1983). Nevertheless, the convergence quality of existing OC re- cursive schemes (Khan el al. 1978, 1979; Syed el al. 1980; Khan and Willmert 1981; Khan 1984; Khot 1985; Granhdi and Venkayya 1986; Venkayya and Tischler 1983) depends on the selection of a proper step size (which is similar to a move limit in classical mathematical programming meth- ods). Without a clear reading of the step size, the conver- gence to a local optimum solution is many times oscillatory. Even though the convergence rate for OC methods is initially rapid, it gradually slows near local extrema, mainly because the reading of suitable step sizes in the redesign phase be- comes increasingly difficult. Until recently (McGee and Phan 1988, 1989, 1991; Phan and McGee 1991a, 1991b; McGee et al. 1990; Bathe 1982), this reading was an arbitrary process with much experimentation done on the same problem. Ideally, a methodology that uses specific criteria based on past scaled designs to smooth the oscillatory convergence propensity of OC recursive methods would be quite benefi- cial to practical space frame applications. The focus here is to propose such a design method that for the first time can self-operatively optimize large space frames using various OC recursive design approaches. In this paper, a closed-form scal- ing procedure is united with an adaptable redesign strategy in which linear extrapolates of past scaled designs are cou- pled with automatically tuned OC recursive formulae. These