Model Order Reduction in Geometrically Nonlinear Analyses

In aircraft and aerospace structures, nonlinear structural phenomena can be observed under certain loading and boundary conditions. Traditional structural and aeroelastic analyses techniques have been historically developed with linearization approximations which inevitably produces undesirable deviations in the solutions of nonlinear systems. Dedicated nonlinear solution techniques (for ex: in the finite element framework) can be utilized to improve the accuracy, however, they tend to be computationally expensive. Researchers at DLR in collaboration with TU Delft aim to develop and extend nonlinear solution techniques with model order reduction to improve computational efficiency while retaining the accuracy in the analyses.

Why is structural nonlinearity relevant in aeroelasticity?

Imagine a flexible wing structure undergoing very large deflections or components in a spacecraft subjected to large amplitude vibrations. In either case, we observe a variance in the structural stiffness as a function of the deflections. These are generally categorized as geometrically nonlinear behaviour (see Figure 1) and cannot be correctly simulated using linear analysis techniques which, by extension, also directly impacts our ability to accurately simulate the aeroelastic behaviour of such structures.

Fig 1: Incremental depiction of geometrically nonlinear behaviour in a 2-D flat plate

What is model order reduction?

Model order reduction aims at reducing model complexities and thereby, reducing numerical simulation time. It is in fact a widely used approach in linear analyses where a subset of the structural vibration mode shapes is utilized to obtain a reduced set of linear finite element (FE) equations. This implies that instead of solving n finite element equations, we only solve m equations (where m << n). The same cannot be directly applied when nonlinearity is introduced into the system since the deformation behaviour of the system no longer remains the same.

How do we solve it for nonlinear systems?

The obvious, yet not so straightforward, solution is to eliminate the linearization approximations in formulating the equations of motion. The necessary adaptations include: (1) consideration of a complete nonlinear strain model, (2) computation of nonlinear structural stiffness as a function of the displacements. Doing this for an FE model requires an iterative approach with step-wise loading. This considerably increases the computational times, especially when working under dynamic loading conditions. Our preliminary investigations have been conducted using a reduced order model (ROM) based on the Koiter-Newton reduction technique [1,2]. The highlight of this method is that unlike many mode shape-based approaches, the need for computing derivatives of the mode shapes is eliminated. Promising results are seen in structures subjected to the stretching induced nonlinearity. A reduction of up to 92 % in the simulation time can be demonstrated for the flat plate test case with good solution accuracy (see Figure 2).

Fig 2: Nonlinear response analysis of a flat plate

Ongoing studies

When investigating cantilevers, it is observed that up to about 15 % tip deflection (measure of halfspan) it is possible to capture the nonlinear response with an error margin of 2-3 %. Beyond that the ROM must be updated in order to account for the large rotations in the geometry [3] and to correctly obtain the deflection behaviour (see Figure 4). Ongoing studies aim to investigate the application of such methods for studying nonlinear structural and aeroelastic characteristics of wing-like geometries and its potential benefits.

Fig 3: Nonlinear state of deflection in a slender wingbox

Fig 4: Deflection profile along the centre line in a slender wingbox at different loading conditions [3,4]