Stochastic material modelling and multi-fidelity uncertainty quantification of macroscopic bone tissue

In this thesis, the simulation of macroscopic bone tissue is studied in a probabilistic framework, where special emphasis is given to how to model and generate random material tensors that are symmetric and positive definite (SPD). The goal of this thesis is to discuss and propose a novel procedure—known as the fully controlled stochastic material modelling approach—for modelling random second-order SPD material tensors that extends the capability of the existing, random scaling-only, procedure by parametrizing/controlling the directional uncertainty, given that a specific class of material invariance for the entire random ensemble is prescribed. In addition, a scenario of fluctuating material symmetry around the Fréchet mean of random SPD tensors is developed, in which the mean tensor belongs to a higher order of material symmetry, whereas each realisation belongs to a lower order.

Another focus of this thesis is on using a variance reduction technique known as the multilevel Monte Carlo method (MLMC) to efficiently assess the impact of material uncertainty on system response by determining statistics like mean and variance. As the convergence of the classical MLMC algorithm strongly depends on the solution magnitude, a novel scale-invariant version of the multilevel Monte Carlo procedure—called the scale-invariant multilevel Monte Carlo method (SMLMC)—is proposed.
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Finally, a 2D and 3D model of steady-state heat conduction in a human proximal femur is investigated as an example, with a spatially constant random thermal conductivity tensor. Furthermore, the random elastic tensor/matrix field of a 2D linear elastic material model is simulated.