Session: 01-01-01: General Topics of Aerospace Structures

Paper Number: 137554

137554 - Beam Cross-Sectional Analysis for Shape Optimization 

Aerospace structures that have one principal geometric dimension much larger than the others, such as helicopter rotors or high-aspect wings, are frequently modeled as beams. A beam model involves decomposing a three-dimensional (3-D) problem into two sequential sub-problems: two-dimensional (2-D) analysis and one-dimensional (1-D) analysis. The 2-D analysis computes the behavior of the "cross-section" -- the plane defined by two axes separate from that of the direction in-line with the largest dimension. The 1-D analysis utilizes the cross-sectional properties defined over the "beam axis" -- a topologically 1-D entity embedded in 3-D space. Both 1-D analysis and 2-D analysis are necessary to accurately model a structure. With the existence of geometrically exact 1-D analysis, all geometrical nonlinearities of a beam axis' behavior can be captured, given a small strain assumption. This leaves all other approximations to the 2-D analysis; meaning a major driver of accuracy of a beam model is the computation of realistic cross-sectional behavior. In simple cases (such as isotropic, homogeneous, symmetric cross-sections), the cross-sectional properties have analytical solutions, but in general the cross-section will posses complicated behavior that cannot easily be computed analytically. There has been considerable effort to produce tools that can compute cross-sectional properties of composite and anisotropic materials of arbitrary shape, such as Variational Asymptotic Beam Sectional analysis (VABS) and BEam Cross-Sectional Analysis Software (BECAS). These tools can be paired with an appropriate 1-D analysis tool, such as GEBT, GXBeam, etc., to compute the behaviour of a structure.

Existing beam tools are quite good at forward analysis. However, since the creation of these cross-sectional analysis tools, there have been significant advances in the field of structural optimization. Gradient-based approaches utilizing the adjoint method have emerged as the leading approach to optimization with many design variables. Unfortunately, without access to derivative information with respect to design variables, previously implemented cross-sectional analysis approaches are not well suited to be utilized in specific types of design optimizations, such as shape optimization of a beam's cross-section. This means existing tools force an optimization to use sub-optimal approaches such as costly finite-difference methods for derivative information or abandoning gradient based methods altogether. As a result, while there exist many examples of structural optimization utilizing 1-D beam models, the general case of optimizing a cross-sectional shape of a beam using 2-D cross-sectional analysis tools is not tractable with these methods. This presentation will demonstrate a software package to perform cross-sectional analysis and 1-D beam analysis tasks, with an eye towards performing 2-D cross-sectional shape optimization. This software package uses an approach to  cross-sectional analysis based on the analysis of an under-constrained finite element problem. The cross-sectional analysis is implemented using an open-source partial differential equation solver called FEniCSx, which can provide derivative information with respect to various design parameters. Forward analysis results showing similarity to existing cross-sectional analysis tools and background theory will be presented. Preliminary work on connecting the 2-D analysis with a full optimization of a beam structure will be presented.

Presenting Author: Joshua Krokowski University of California San Diego

Presenting Author Biography: PhD Student in Large-Scale Design Optimization (LSDO) Lab advised by Professor John Hwang. Present research focus on finite element analysis of structures used in large-scale optimization. Past experience as a mechanical engineer designing, building, and testing equipment used in nuclear decommissioning.