A semi-analytical solution for multilayer diffusion in a composite medium consisting of a large number of layers

Diffusion in a composite slab consisting of a large number of layers provides an ideal prototype problem for developing and analysing two-scale modelling approaches for heterogeneous media. Numerous analytical techniques have been proposed for solving the transient diffusion equation in a one-dimensional composite slab consisting of an arbitrary number of layers. Most of these approaches, however, require the solution of a complex transcendental equation arising from a matrix determinant for the eigenvalues that is difficult to solve numerically for a large number of layers. To overcome this issue, in this paper, we present a semi-analytical method based on the Laplace transform and an orthogonal eigenfunction expansion. The proposed approach uses eigenvalues local to each layer that can be obtained either explicitly, or by solving simple transcendental equations. The semi-analytical solution is applicable to both perfect and imperfect contact at the interfaces between adjacent layers and either Dirichlet, Neumann or Robin boundary conditions at the ends of the slab. The solution approach is verified for several test cases and is shown to work well for a large number of layers. The work is concluded with an application to macroscopic modelling where the solution of a fine-scale multilayered medium consisting of two hundred layers is compared against an “up-scaled” variant of the same problem involving only ten layers.

The Palu Metamorphic Complex, NW Sulawesi, Indonesia: Origin and evolution of a young metamorphic terrane with links to Gondwana and Sundaland

The Palu Metamorphic Complex (PMC) is exposed in a late Cenozoic orogenic belt in NW Sulawesi, Indonesia. It is a composite terrane comprising a gneiss unit of Gondwana origin, a schist unit composed of meta-sediments deposited along the SE Sundaland margin in the Late Cretaceous and Early Tertiary, and one or more slivers of amphibolite with oceanic crust characteristics. The gneiss unit forms part of the West Sulawesi block underlying the northern and central sections of the Western Sulawesi Province. The presence of Late Triassic granitoids and recycled Proterozoic zircons in this unit combined with its isotopic signature suggests that the West Sulawesi block has its origin in the New Guinea margin from which it rifted in the late Mesozoic. It docked with Sundaland sometime during the Late Cretaceous. U–Th–Pb dating results for monazite suggest that another continental fragment may have collided with the Sundaland margin in the earliest Miocene. High-pressure (HP) and ultrahigh-pressure (UHP) rocks (granulite, peridotite, eclogite) are found as tectonic slices within the PMC, mostly along the Palu–Koro Fault Zone, a major strike-slip fault that cuts the complex. Mineralogical and textural features suggest that some of these rocks resided at depths of 60–120 km during a part of their histories. Thermochronological data (U–Th–Pb zircon and 40Ar/39Ar) from the metamorphic rocks indicate a latest Miocene to mid-Pliocene metamorphic event, which was accompanied by widespread granitoid magmatism and took place in an extensional tectonic setting. It caused recrystallization of, and new overgrowths on, pre-existing zircon crystals, and produced andalusite–cordierite–sillimanite–staurolite assemblages in pelitic protoliths, indicating HT–LP (Buchan-type) metamorphism. The PMC was exhumed as a core complex at moderate rates (c. 0.7–1.0 mm/yr) accompanied by rapid cooling in the Plio-Pleistocene. Some of the UHP rocks were transported to the surface at significantly higher rates (⩾16 mm/yr). The results of our study do not support recent plate tectonic reconstructions that propose a NW Australia margin origin for the West Sulawesi block (e.g. Hall et al., 2009).