Simulation Tools for Biomechanical Applications with PGD-Based Reduced Order Models

Numerical simulation tools are generally used in all modern engineering fields, especially those having difficulties in performing large number of practical experiments, such as biomechanics. Among the computational methods, Finite Element (FE) is an essential tool. Nowadays, the fast-growing computational techniques, from the upgrading hardware to the emerging of novel algorithm, have already enabled extensive applications in biomechanics, including mechanical analysis from musculoskeletal or cardiovascular system in macro scale to cell structures or tissue behaviours in micro scale. For applications that require fast response and/or multiple queries, Reduced Order Modelling (ROM) methods have been developed based on existing methods such as FE, and have eventually enabled real-time numerical simulation for a large variety of engineering problems. In this thesis, several novel computational techniques are developed to explore the capability of Proper Generalised Decomposition (PGD), which is an important approach of ROM. To assess the usability of the PGD-based ROM for biomechanical applications, a real human femur bone is chosen to study its mechanical behaviour as an example. Standard image-based modelling procedure in biomechanics is performed to create an FE model which is then validated with in vitro experimental results. As a major contribution, a non-intrusive scheme of the PGD framework is developed and implemented using commonly-used industrial software such as Matlab and Abaqus. It uses Abaqus as an external FE solver, which is called by in-house Matlab codes implementing the PGD algorithms. An example code is available at https://github.com/xizou/NIPGD. This scheme takes advantages of the maturity, robustness and availability of existing FE solvers, and demonstrates a great potential for being applied to industrial projects. To solve parametrised partial differential equations with a parameter space subjected to physical or geometric constraints, a novel strategy is proposed. This strategy provides an approach that collects the most correlated parameters, and then separates them into 2D/3D spaces, instead of separating the parameter space into tensor products of 1D spaces in a Cartesian fashion as it is done in conventional PGD framework. Inspired by the fast-developing methods of isogeometric analysis, it is interesting to borrow the isogeometric idea to exploit the ways of discretising the parameter space inside the PGD framework. The high continuity of B-spline shape functions enables more accurate results for the computation of sensitivities with respect to the parameters. A classical mechanical problem is investigated with orthotropic materials in 2D, with the intention of further application in biomechanics. In addition, an exploration of the generalisation of PGD to nonlinear problems in solid mechanics is presented as another main contribution. Following the large strain theory, Picard linearisation is used to establish a consistent PGD framework within total Lagrange formulation. As a preliminary example, the St.Venant-Kirchhoff constitutive model is adopted. A practical example of the femur bone simulation is provided, the material parameters are obtained through an identification problem using the PGD vademecum, and in a further step, another PGD vademecum is generated for real-time simulation accounting for various loading locations.