Methods for the identification of material parameters in distributed models for flexible structures

Theoretical and numerical results are presented for inverse problems involving estimation of spatially varying parameters such as stiffness and damping in distributed models for elastic structures such as Euler-Bernoulli beams. An outline of algorithms used and a summary of computational experiences are presented

Identification of an unknown spatial load distribution in a vibrating beam or plate from the final state

The theoretical and numerical determination of a space-dependent load distribution in a simply supported non-homogeneous Euler-Bernoulli beam and Kirchhoff-Love plate is investigated. The uniqueness of a solution to this inverse source problem is proved, whilst counter examples are constructed to discuss the conditions under which uniqueness holds. A convergent and stable iterative algorithm is proposed for the recovery of the unknown load source and a stopping criterion is also given. Several one-dimensional numerical experiments are considered to investigate the properties of the proposed iterative procedure

Numerical recovery of material parameters in Euler-Bernoulli beam models

A fully Sinc-Galerkin method for recovering the spatially varying stiffness parameter in fourth-order time-dependence problems with fixed and cantilever boundary conditions is presented. The forward problems are discretized with a sinc basis in both the spatial and temporal domains. This yields an approximation solution which converges exponentially and is valid on the infinite time interval. When the forward methods are applied to parameter recovery problems, the resulting inverse problems are ill-posed. Tikhonov regularization is applied and the resulting minimization problems are solved via a quasi-Newton/trust region algorithm. The L-curve method is used to determine an appropriate value of the regularization parameter. Numerical results which highlight the method are given for problems with both fixed and cantilever boundary conditions

The identification of a distributed parameter model for a flexible structure

A computational method is developed for the estimation of parameters in a distributed model for a flexible structure. The structure we consider (part of the RPL experiment) consists of a cantilevered beam with a thruster and linear accelerometer at the free end. The thruster is fed by a pressurized hose whose horizontal motion effects the transverse vibration of the beam. The Euler-Bernoulli theory is used to model the vibration of the beam and treat the hose-thruster assembly as a lumped or point mass-dashpot-spring system at the tip. Using measurements of linear acceleration at the tip, it is estimated that the parameters (mass, stiffness, damping) and a Voight-Kelvin viscoelastic structural damping parameter for the beam using a least squares fit to the data. Spline based approximations to the hybrid (coupled ordinary and partial differential equations) system are considered; theoretical convergence results and numerical studies with both simulation and actual experimental data obtained from the structure are presented and discussed

Physics-informed neural networks for solving forward and inverse problems in complex beam systems

This paper proposes a new framework using physics-informed neural networks (PINNs) to simulate complex structural systems that consist of single and double beams based on Euler-Bernoulli and Timoshenko theory, where the double beams are connected with a Winkler foundation. In particular, forward and inverse problems for the Euler-Bernoulli and Timoshenko partial differential equations (PDEs) are solved using nondimensional equations with the physics-informed loss function. Higher-order complex beam PDEs are efficiently solved for forward problems to compute the transverse displacements and cross-sectional rotations with less than 1e-3 percent error. Furthermore, inverse problems are robustly solved to determine the unknown dimensionless model parameters and applied force in the entire space-time domain, even in the case of noisy data. The results suggest that PINNs are a promising strategy for solving problems in engineering structures and machines involving beam systems

Development of an online progressive mathematical model of needle deflection for application to robotic-assisted percutaneous interventions

A highly flexible multipart needle is under development in the Mechatronics in Medicine Laboratory at Imperial College, with the aim to achieve multi-curvature trajectories inside biological soft tissue, such as to avoid obstacles during surgery. Currently, there is no dedicated software or analytical methodology for the analysis of the needle’s behaviour during the insertion process, which is instead described empirically on the basis of experimental trials on synthetic tissue phantoms. This analysis is crucial for needle and insertion trajectory design purposes. It is proposed that a real-time, progressive, mathematical model of the needle deflection during insertion be developed. This model can serve three purposes, namely, offline needle and trajectory design in a forward solution of the model, when the loads acting on needle from the substrate are known; online, real-time identification of the loads that act on the needle in a reverse solution, when the deflections at discrete points along the needle length are known; and the development of a sensitivity matrix, which enables the calculation of the corrective loads that are required to drive the needle back on track, if any deviations occur away from a predefined trajectory. Previously developed mathematical models of needle deflection inside soft tissue are limited to small deflection and linear strain. In some cases, identical tip path and body shape after full insertion of the needle are assumed. Also, the axial load acting on the needle is either ignored or is calculated from empirical formulae, while its inclusion would render the model nonlinear even for small deflection cases. These nonlinearities are a result of the effects of the axial and transverse forces at the tip being co-dependent, restricting the calculation of the independent effects of each on the needle’s deflection. As such, a model with small deflection assumptions incorporating tip axial forces can be called “quasi-nonlinear” and a methodology is proposed here to tackle the identification of such axial force in the linear range. During large deflection of the needle, discrepancies between the shape of the needle after the insertion and its tip path, computed during the insertion, also significantly increase, causing errors in a model based on the assumption that they are the same. Some of the models developed to date have also been dependent on existing or experimentally derived material models of soft tissue developed offline, which is inefficient for surgical applications, where the biological soft tissue can change radically and experimentation on the patient is limited. Conversely, a model is proposed in this thesis which, when solved inversely, provides an estimate for the contact stiffness of the substrate in a real-time manner. The study and the proposed model and techniques involved are limited to two dimensional projections of the needle movements, but can be easily extended to the 3-dimensional case. Results which demonstrate the accuracy and validity of the models developed are provided on the basis of simulations and via experimental trials of a multi-part 2D steering needle in gelatine.Open Acces