The inverse problem of recovering an unsteady linear load for an elastic rod of finite length

The main purpose of the paper is to obtain solutions for new non-stationary inverse problems for elastic rods. The objective of this study is to develop and implement new methods, approaches and algorithms for solving non-stationary inverse problems of rod mechanics. The direct non-stationary problem for an elastic rod consists in determining elastic displacements, which satisfies a given equation of non-stationary oscillations in partial derivatives and some given initial and boundary conditions. The solution of inverse retrospective problems with a completely unknown space-time law of load distribution is based on the method of influence functions. With its application, the inverse retrospective problem is reduced to solving a system of integral equations of the Volterra type of the first kind in time with respect to the sought external axial load of the elastic rod. To solve it, the method of mechanical quadratures is used in combination with the Tikhonov regularisation method.

A Symmetric Prior for the Regularisation of Elastic Deformations: Improved Anatomical Plausibility in Nonlinear Image Registration

Nonlinear registration is critical to many aspects of Neuroimaging research. It facilitates averaging and comparisons across multiple subjects, as well as reporting of data in a common anatomical frame of reference. It is, however, a fundamentally ill-posed problem, with many possible solutions which minimise a given dissimilarity metric equally well. We present a novel regularisation method that aims to selectively drive solutions towards those which would be considered anatomically plausible by penalising unlikely lineal, areal and volumetric deformations. In addition, our penalty is symmetric in the sense that geometric expansions and contractions are penalised equally, which encourages inverse-consistency. We demonstrate that our method is able to significantly reduce volume and shape distortions compared to state-of-the-art elastic (FNIRT) and plastic (ANTs) registration frameworks. Crucially, this is achieved whilst matching or exceeding the registration quality of these methods, as measured by overlap scores of labelled cortical regions. Furthermore, extensive use of GPU parallelisation has allowed us to implement what is a highly computationally intensive optimisation strategy while maintaining reasonable run times of under half an hour.

Rheology of Un-Sieved Concentrated Domestic Slurry: A Wide Gap Approach

Information on the rheology of domestic slurries is essential in designing pipeline transportation in novel sanitation systems. As concentrated slurries in their original collected state have wide particle size distribution, with particles up to 2 mm, a wide gap rheometer is used to acquire the rheograms. Rheograms obtained from a wide gap rheometer require a method to convert the rotational velocity to the shear rate, and this method must be robust to noisy data and yield stress in the slurry. For this purpose, a Tikhonov regularisation method is chosen as it suits the criteria the best. Using this, the rheograms are obtained for various total suspended solids (TSS) concentrations of slurries. A Herschel-Bulkley rheological model is used to represent the rheology of the slurries. The influence of the change in concentration of the slurries is represented through its influence on the Herschel-Bulkley parameters. The consistency index K exponentially increases with the concentration. The yield stress τ y , is 0 at low concentrations, and above 2.0% TSS (wt./wt.) exponentially increases with the concentration. The behaviour index n , is 1 at low concentrations, and above 2.6% TSS (wt./wt.) it decreases in an inverse power law with the concentration to reach a sort of plateau.

Tikhonov Regularisation Method for Simultaneous Inversion of the Source Term and Initial Data in a Time-Fractional Diffusion Equation

The inverse problem of identifying the time-independent source term and initial value simultaneously for a time-fractional diffusion equation is investigated. This inverse problem is reformulated into an operator equation based on the Fourier method. Under a certain smoothness assumption, conditional stability is established. A standard Tikhonov regularisation method is proposed to solve the inverse problem. Furthermore, the convergence rate is given for an a priori and a posteriori regularisation parameter choice rule, respectively. Several numerical examples, including one-dimensional and two-dimensional cases, show the efficiency of our proposed method.