An adaptive multi-level substructuring method for efficient modelingof complex structures

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An adaptive method for modeling complex structures with maximum accuracy with a given number of degrees of freedom is presented. T h e method uses a multi-level substructuring approach, in which the structure is partitioned into a number of substructures, each of which is itself composed of substructures, and so on. This approach lends itself to efficient representation of localized response in particular, and also to parallelization. Convergerrce using this method is signifc;tnt,ly faster t,hari when the model is rcfiried u ~ i i forrrily, i n terms of the number of degrees of freedom required to achieve a given level of accuracy. A numerical example is presented. I n t r o d u c t i o n Many aerospace and naval structures possess such complex geometry that any models tha t represent them with a reasonable level of detail must have an extremely high number of degrees of freedom. Finite element models with tens or hundreds of thousands of degrees of freedom are used increasingly frequently for static structural analysis, but for harmonic response over a range in frequency, such models can require computation that is prohibitively expensive. In the structural acoustic analysis of naval structures, lowfrequency modeling can be done using a fairly crude model of an entire structure, because elastic wavelengths are long enough tha t fine details are inconsequential. High-frequency modeling can be done successfiilly by modeling the region around the excitation in detail, but treating portions further from the excitation as if they were rigid, because of the localization of the response. IIowever, i n the middle frequency range, response can be sigriificant over the entire structure, and can involve short enough elastic wavelengths tha t the entire structure must be modeled with a prohibitively high level of detail. For problems such as these, extending the frequency range that can be handled will require methods that can achieve increased accuracy with a given number of degrees of freedom. Because of the camplexity of the behavior tha t is of int.erest, i t does not seem feasible to reliably predict a priori how model order 'Assistant Professor, Aerospace Engineering & Engineering hlechanics. Member A IAA. tGraduate Research Assistant. can be reduced to a minimum without dcgrading accuracy. Guyan reduction' and substructuring methods2 are used frequently in dynamic structural analysis, but primarily a t very low frequencies where the analyst can comfortably rely on intuition. For higher frequencies, it quickly becomes attractive t o shift the burden of model reduction to the computer as much as possible, so an automated adaptive method is of interest. Adaptive methods have been used in more and more areas in which finite elements are used over the past decade,3' but they have found limited application in structural analysis. Existing adaptive methods refine the finite element model by subdividing elements or by increasing the degree of the polynomial interpolation in elements so tha t the solution of partial differential equations is approximated more accurately. However, in structural analysis, a crude model must be refined so tha t it represents the geometric detail more accurately. Even though the response is governed by partial differential equations over members of the structure, for practical purposes the response can be considered t o be governed by an extremely high-order discrete model which represents all of the geometric detail of the struct,ure very accurately. If further refinement beyond this high-order detailed model is necessary, existing adaptive methods could be appropriately used. However, for many problems refinement of the model to represent the gecmetric detail of the structure requires so many degrees of freedom that existing adaptive methods would never