Numerical optimisation of worm locomotion on frictional substrates
(English) Optimal control theory provides a framework to determine optimal inputs for mechanical systems modeled as initial value problems. The resulting minimisation problem may be solved with known direct and indirect methods. This PhD thesis focuses on the development of novel structure preservin...
| Autor: | |
|---|---|
| Tipo de documento: | tese |
| Estado: | Versão publicada |
| Data de publicação: | 2024 |
| País: | España |
| Recursos: | CBUC, CESCA |
| Repositório: | TDR. Tesis Doctorales en Red |
| OAI Identifier: | oai:www.tdx.cat:10803/692243 |
| Acesso em linha: | http://hdl.handle.net/10803/692243 https://dx.doi.org/10.5821/dissertation-2117-415062 |
| Access Level: | Acceso aberto |
| Palavra-chave: | Optimal control Adjoint method Control Hamiltonian Structure-preserving discretisation Nonlinear elasticity Finite element Limbless locomotion Àrees temàtiques de la UPC::Matemàtiques i estadística 51 |
| Resumo: | (English) Optimal control theory provides a framework to determine optimal inputs for mechanical systems modeled as initial value problems. The resulting minimisation problem may be solved with known direct and indirect methods. This PhD thesis focuses on the development of novel structure preserving discretisation techniques for optimal control problems, with application to the mechanical systems and the locomotion of limbless organisms. Firstly, our research focuses on developing structure preserving time-integration schemes in the context of optimal control theory. We propose a novel discretisation that preserves the control Hamiltonian, an integral of the analytical Euler-Lagrange equations of the optimal control problem (OCP). Additionally, we introduce a discretisation technique that is capable to preserve control angular momentum maps resulting from the rotational symmetry of the underlying OCP. Next, we investigate the stability of numerical solutions when optimisation problems are discretised in time. We show that the numerical stability and the presence of numerical oscillations depends not only on the time-step size, but also on the parameters of the objective functional, which measures the amount of control input. Furthermore, we demonstrate with an illustrative example that these findings also carry over non-linear OCPs. Secondly, the numerical solution of the discretised OCP is tested with two strategies: monolithic and staggered approaches. The monolithic strategy solves all the optimality conditions for all time-steps as a single system of non-linear equations, and relies on a Newton-Raphson scheme, which guarantees quadratic rates of convergence in the vicinity of the optimal solution trajectory. The staggered strategy is based on the Forward-Backward Sweep Method (FBSM), where state and adjoint equations are solved separately, and the control equations provides an update of the control variables. Additionally, we device a hybrid solution strategy which combines the advantages of a conventional gradient-based FBSM with the individual Newton-based solution procedures once the solution is close to the optimal trajectory. Finally, we present a comprehensive framework for modeling the locomotion of limbless organisms on frictional substrates using both 2D and 3D continuum models based on Finite Element (FE) methods. For the 2D continuum model, muscle activity is simulated with an active stress approach, while the 3D continuum problem incorporates a multiplicative decomposition of the deformation gradient, which allows mimicking a broad range of locomotion patterns in 3D contractile elastic solids. We propose a two-field FE formulation based on positions and velocities. Governing partial differential equations are transformed into equivalent time-continuous differential-algebraic equations (DAEs). Next, the optimal locomotion strategies are studied in the framework of optimal control theory. We resort to adjoint-based methods and deduce the first-order optimality conditions, that yield a system of DAEs with two-point end conditions. The resulting discrete first-order optimality conditions form a non-linear programming problem that is solved efficiently with the FBSM. Lastly, some representative numerical examples are provided to compare the numerical performance of the developed computational framework for limbless locomotion. We evaluate the efficiency of structure preserving schemes and the robustness of monolithic, staggered and hybrid strategies. Our numerical experiments show that the predictions of the developed limbless locomotion are capable of simulating distinct locomotion patterns, and that the monolithic, staggered, and hybrid schemes yield very similar solutions. However, the hybrid approaches become more advantageous with regard to the computation time when the time-step decreases or the size of the problem increases. |
|---|