Optimal design and optimal control of structures undergoing finite rotations and elastic deformations

In this work we deal with the optimal design and optimal control of structures undergoing large rotations. In other words, we show how to find the corresponding initial configuration and the corresponding set of multiple load parameters in order to recover a desired deformed configuration or some desirable features of the deformed configuration as specified more precisely by the objective or cost function. The model problem chosen to illustrate the proposed optimal design and optimal control methodologies is the one of geometrically exact beam. First, we present a non-standard formulation of the optimal design and optimal control problems, relying on the method of Lagrange multipliers in order to make the mechanics state variables independent from either design or control variables and thus provide the most general basis for developing the best possible solution procedure. Two different solution procedures are then explored, one based on the diffuse approximation of response function and gradient method and the other one based on genetic algorithm. A number of numerical examples are given in order to illustrate both the advantages and potential drawbacks of each of the presented procedures.

💡 Research Summary

This paper addresses the simultaneous optimal design and optimal control of structures that undergo large rotations combined with elastic deformations, using the geometrically exact beam as a representative model. The authors first formulate the problem in a non‑standard way by introducing Lagrange multipliers, which decouple the mechanical state variables (displacements and rotations) from the design variables (initial geometry, material parameters) and control variables (load magnitudes, directions, locations). This formulation embeds the equilibrium equations directly into the Lagrangian, allowing the objective function—typically a measure of deviation from a desired deformed shape or a performance criterion—to be differentiated with respect to all variables while preserving the constraints.

Two solution strategies are investigated. The first relies on a diffuse‑approximation (DA) of the response surface. A limited set of design‑control samples is generated, each requiring a full nonlinear finite‑element analysis of the beam. The resulting responses (e.g., nodal displacements, internal forces) are fitted with low‑order polynomial or radial‑basis functions, producing a smooth surrogate model f̂(d,c). Because f̂ is analytically differentiable, gradient‑based optimizers such as BFGS or nonlinear conjugate‑gradient can be employed, leading to rapid convergence when the surrogate accurately captures the true response. The main advantage is a dramatic reduction in the number of expensive finite‑element solves; the drawback is that surrogate errors may trap the algorithm in local minima, especially for highly nonlinear objectives.

The second strategy employs a genetic algorithm (GA), a population‑based meta‑heuristic that excels at global exploration. An initial population of design‑control vectors is randomly generated, evaluated using the exact finite‑element model, and then evolved through selection, crossover, and mutation. GA does not require gradient information and is robust against non‑convexity, making it suitable for complex objectives such as achieving a prescribed S‑shaped deformation. However, each individual evaluation demands a full nonlinear analysis, so the computational cost can be substantially higher than the DA‑based approach.

A series of numerical examples illustrate both methods. In simple cases—e.g., matching a straight‑line target shape—the DA‑gradient method converges within a few dozen function evaluations, delivering solutions in a fraction of the time required by GA. For more intricate targets involving curvature changes or multiple performance criteria, the surrogate’s fidelity deteriorates, and GA outperforms the gradient method by locating a better global optimum, albeit at a higher computational expense. Throughout the experiments, the Lagrange‑multiplier framework ensures that the equilibrium constraints are consistently enforced, allowing a fair comparison between the two algorithms.

The authors conclude that the Lagrangian formulation provides a versatile foundation for coupling design, control, and mechanics, and that the choice between surrogate‑based gradient optimization and evolutionary global search should be guided by problem complexity, dimensionality, and available computational resources. They suggest future work on dimensionality reduction for high‑dimensional design spaces, real‑time control via model order reduction, and experimental validation to confirm the predictive capability of the proposed methods.