a b s t r a c tThe condition number of a matrix is commonly used for investigating the stability of solutions to linear algebraic systems. Recent meshless techniques for solving partial differential equations have been known to give rise to ill-conditioned matrices, yet are still able to produce results that are close to machine accuracy. In this work, we consider the method of fundamental solutions (MFS), which is known to solve, with extremely high accuracy, certain partial differential equations, namely those for which a fundamental solution is known. To investigate the applicability of the MFS, either when the boundary is not analytic or when the boundary data are not harmonic, we examine the relationship between its accuracy and the effective condition number.Three numerical examples are presented in which various boundary value problems for the Laplace equation are solved. We show that the effective condition number, which estimates system stability with the right-hand side vector taken into account, is roughly inversely proportional to the maximum error in the numerical approximation. Unlike the proven theories in literature, we focus on cases when the boundary and the data are not analytic. The effective condition number numerically provides an estimate of the quality of the MFS solution without any knowledge of the exact solution and allows the user to decide whether the MFS is, in fact, an appropriate method for a given problem, or what is the appropriate formulation of the given problem.
In this article, we calculate the seismic anisotropy resulting from melt redistribution during pure and simple shear deformation. Deformation strongly modifies the geometry of melts initially occupying three grain junctions. The initially isotropic fractional area of intergranular contact, contiguity, becomes anisotropic due to deformation. Consequently, the component of contiguity evaluated on the plane parallel to axis of maximum compressive stress decreases. During both modes of deformation, the trace of the contiguity tensor remains nearly unchanged. In the companion article [labeled DHM], we outline the numerical methods and present the synthetic micrographs from our numerical deformation experiments. In pure shear deformation, the principal contiguity directions remain stationary while they rotate during simple shear. The ratio between the principal components of the contiguity tensor decrease from 1 in an undeformed aggregate to 0.1 after 45% shortening in pure shear and to 0.3 after a shear strain of 0.75 in simple shear. In both pure and simple shear experiments, anisotropy in the shear wave velocity increases with the strain in a strongly nonlinear fashion. In pure shear deformation, the steady state microstructure produces nearly 3% anisotropy between shear waves vibrating perpendicular and parallel to the planes of melt films.
The microstructure of partially molten rocks strongly influences the macroscopic physical properties. Contiguity, a geometric parameter, is a tensorial quantity that describes the area fraction of intergranular contact in a partially molten aggregate. It is also a key parameter that controls the effective elastic strength of the grain network. As the shape of the grains evolves during deformation, so does the contiguity of each grain. In this article, we present the first set of numerical simulations of evolution of grain‐scale contiguity of an aggregate during matrix deformation using a fast multipole boundary elements method‐based model. We simulate a pure shear deformation of an aggregate of 1200 grains up to a shortening of 0.47 and a simple shear deformation of 900 grains up to a shear strain of 0.75, for solid‐melt viscosity ratios of 1 and 50. Our results demonstrate that the initially isotropic contiguity tensor becomes strongly anisotropic during deformation. We also observe that the differential shortening, the normalized difference between the major and minor axes of grains, is inversely related to the ratio between the principal components of the contiguity tensor. In pure shear, the principal components of the contiguity tensor remain parallel to the irrotational principal axes of the applied strain. In simple shear, however, the principal components of the contiguity tensor rotate continually during the course of deformation in this study. In the companion article we present the seismic anisotropy resulting from the anisotropic contiguity and the implications for the Earth's lithosphere‐asthenosphere boundary.
Seismic observations reveal a patchwork of thin and dense structures, named UltraLow Velocity Zones (ULVZs) atop the Earth's core mantle boundary. The high width to height ratio of the ULVZs, their spatial correlation with the edges of Large Low Shear Velocity Provinces (LLSVPs), and their preservation as distinct structures in the convecting mantle remains an enigmatic problem. In this article, we carry out a series of numerical simulations using Fast Multipole Boundary Elements Method (FMBEM) to address these questions and study the internal deformation within the ULVZs. Our results demonstrate that coupled flow between dense, low viscosity ULVZ patches and the LLSVP accumulates the ULVZ into stable piles along LLSVP corners, while coalescence and gravitational drainage leads to thin and wide ULVZs away from the corners. Deformation of the matrix is localized within the weaker ULVZ and the LLSVP edges, while the strain in the interior of the LLSVP remain uniform and low, explaining the observed localized anisotropy near LLSVP edges.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.