A study was performed to evaluate the accuracy achievable in the use of submodeling in finite-element modeling of the mechanical response of a structural system that includes components embedded in a larger structure that is subjected to a large transient load. The specific system studied was a simplified model of a “smart” projectile containing a substructure that supported an electronic-circuit board on which were mounted two capacitors and an eight-lead integrated circuit (see figure). The main body or shell of the projectile was represented as a cylindrical ring supporting the substructure. The dimensions of the various components were chosen to be typical of “smart” munitions. The transient load condition was represented by a velocity-versus-time boundary condition, typical of the velocity versus time of a projectile at launch, imposed at the lower surface of the cylindrical ring.

The Global Model includes the domain of interest represented by the submodel. The mesh of the global model is much coarser than that of the submodel.
In submodeling, the model of a structural system comprises a global model and local model denoted the submodel. The global model represents the entire structure and contains a coarse representation of a domain of interest (e.g., the integrated circuit, the capacitors, and the supporting substructure in this study). The global model is refined enough (that is, its computational mesh is fine enough) to enable accurate calculation of the displacement on the boundary of the local domain of interest. Subsequently, the solution obtained from the global model along the boundary of the domain of interest is applied as a displacement boundary condition on the submodel.

The submodel is a highly refined model of the domain of interest (that is, its computational mesh is much finer than that of the global model). The main motivation for submodeling is that it takes much less computation time than does modeling of the entire structure using a single mesh fine enough for the domain of interest. The primary assumption in submodeling is that the structural details of the submodel do not significantly affect the global model. In most practical applications, there are no known a priori methods for determining the validity of this assumption.

In the study, computational simulations to determine the deformations and stresses in the domain of interest during the first 1.4 ms of launch time were performed by use of (1) a fully refined baseline model of the entire structure, including the domain of interest; (2) a global-model/submodel combination utilizing the displacement boundary condition sampled at 1,000 subintervals of the 1.4-ms launch interval; and (3) a global-model/submodel combination utilizing the displacement boundary condition sampled at 100 subintervals of the 1.4-ms launch interval.

The two different time discretizations of the displacement boundary condition were used because time-dependent fidelity of boundary conditions had previously been found to affect results in submodeling and, hence, it was felt necessary to examine the effect in the system of this study. The results from the two submodel cases were found to be good approximations of those of the global model, but in the 100-point case, the stress peaks and valleys were found to overshoot and undershoot those of the baseline model. In the 1,000-point case, as expected, the stress peaks and valleys overshot and undershot by smaller amounts; that is, they approximated those of the baseline model more closely.

It was known a priori that submodeling might introduce spurious high-frequency content into the solution. The study included an examination of this effect, leading to the conclusion that through proper care in selection of the frequency of the global-model output, one can minimize adverse effects. It was shown that for the case studied, sufficient accuracy in the stress results of the submodel can be realized if one considers the desired frequency response range. If one subsequently samples the global loading finely enough, any high-frequency content introduced lies above the frequency range of interest.

This work was done by Brian M. Powers and David A. Hopkins of the Army Research Laboratory.


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Study of Submodeling of a Small Component in a Structure

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This article first appeared in the April, 2009 issue of Defense Tech Briefs Magazine.

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