Finite-element models are an essential tool in the design and verification of launch vehicle and spacecraft structures. They are initially developed to define the force and deflection (stiffness) relationships needed to form loads analysis structural dynamic models. Then, once the working environment is defined in terms of forces, accelerations, and enforced displacements, they are used to determine the impact on structural integrity. The physical relationships between applied loads and displacements are governed by physics and empirical rules; however, the corresponding equations become impossible to solve for structures with complex shapes and boundary conditions. The finite-element method provides a way to generate numerical solutions. Essentially, it breaks a complex system down into a manageable (finite) number of elements. A curve, for example, could be drawn as a series of steps; the smaller the steps, the smoother the curve—but the more information required. In terms of loads analysis, the finite-element method approximates the continuous deformation of a structure, which is unknown, as a combination of mathematical shape functions defined over segments of the structure. In this way, an approximation to the deformation function can be derived by numerically solving a matrix of scale factors for the shape functions.
Finite-element model of a satellite. Colors represent different types of elements. The finite-element method approximates the continuous deformation of a structure as a combination of mathematical shape functions defined over segments of the structure. |
Modern finite-element tools simplify assembly of the matrix equations, but they still require significant engineering judgment. Often, the engineer needs to predict local stress for features that are only a few millimeters in size. If one were to subdivide a structure as large as a launch vehicle to this level of fidelity, the stress prediction would quickly become too complex for even the most advanced computer; many tens of millions of equations would be needed. Thus, depending on the scale of the feature of interest, more or less refinement may be needed.
In essence, the challenge in using the finite-element method is in understanding the inherent assumptions of the underlying theory and the assumed shape functions. In addition, because the finite-element solution is approximate, errors creep in from various sources. These include numerical errors, discretization errors, theoretical simplifications, boundary condition errors, uncertainty in material properties, etc. Even the variability in manufacturing tolerances can lead to significant errors when dealing with precision structures. For this reason, finite-element models must be correlated with test data to ensure the validity of the predictions.
Because of the significant engineering judgment involved in developing finite-element models, Aerospace is intimately involved in the process for Air Force programs. With proper application, the finite-element method is a powerful analytic tool in the development of loads analysis models and the assessment of structural integrity.
—E. K. Hall II