Shell-crossing in quasi-one-dimensional flow

Blow-up of solutions for the cosmological fluid equations, often dubbed shell-crossing or orbit crossing, denotes the breakdown of the single-stream regime of the cold-dark-matter fluid. At this instant, the velocity becomes multi-valued and the density singular. Shell-crossing is well understood in one dimension (1D), but not in higher dimensions. This paper is about quasi-one-dimensional (Q1D) flow that depends on all three coordinates but differs only slightly from a strictly 1D flow, thereby allowing a perturbative treatment of shell-crossing using the Euler--Poisson equations written in Lagrangian coordinates. The signature of shell-crossing is then just the vanishing of the Jacobian of the Lagrangian map, a regular perturbation problem. In essence the problem of the first shell-crossing, which is highly singular in Eulerian coordinates, has been desingularized by switching to Lagrangian coordinates, and can then be handled by perturbation theory. Here, all-order recursion relations are obtained for the time-Taylor coefficients of the displacement field, and it is shown that the Taylor series has an infinite radius of convergence. This allows the determination of the time and location of the first shell-crossing, which is generically shown to be taking place earlier than for the unperturbed 1D flow. The time variable used for these statements is not the cosmic time $t$ but the linear growth time $\tau \sim t^{2/3}$. For simplicity, calculations are restricted to an Einstein--de Sitter universe in the Newtonian approximation, and tailored initial data are used. However it is straightforward to relax these limitations, if needed.Comment: 9 pages; received 2017 May 24, and accepted 2017 June 21 at MNRA

The coupled dual boundary element-scaled boundary finite element method for efficient fracture mechanics

A novel numerical method is presented for applications to general fracture mechanics problems in engineering. The coupled dual boundary element-scaled boundary finite element method (DBE-SBFEM) incorporates the numerical accuracy of the SBFEM and the geometric versatility of the DBEM. Background theory, detailed derivations and literature reviews accompany the extensions made to the methods constituents necessary for their coupling as part of the present work. The coupled DBE-SBFEM, its constituent components and their application to linear elastic fracture mechanics are critically assessed and presented with numerical examples to demonstrate both method convergence and improvements over previous work. Further, a proof of concept demonstrates an alternative formation of the DBEM that both negates the need for hyper-singular integration and lends itself to a wider variety of imposed boundary conditions. Conclusions to this work are drawn and further recommendations for research in this area are made