Finite difference method

Abstract: Three different finite difference schemes for solving the heat equation in one space dimension with boundary conditions containing integrals over the interior of the interval are considered. The schemes are based on the forward Euler, the backward Euler and the Crank-Nicolson methods. Error estimates are derived in maximum norm. Results from a numerical experiment are presented.

Abstract: A P-SV hybrid method is developed for calculating synthetic seismograms involving two-dimensional localized heterogeneous structures. The finite difference technique is applied in the heterogeneous region and generalized ray theory solutions from a seismic source are used in the finite difference initiation process. The seismic motions, after interacting with the heterogeneous structures, are propagated back to the Earth's surface analytically with the aid of the Kirchhoff method. Anomalous long-period SKS-SPdKS observations, sampling a region near the core-mantle boundary beneath the southwest Pacific, are modeled with the hybrid method. Localized structures just above the core-mantle boundary, with lateral dimensions of 250 to 400 km, can explain even the most anomalous data observed to date if S velocity drops up to 30% are allowed for a P velocity drop of 10%. Structural shapes and seismic properties of those anomalies are constrained from the data since synthetic waveforms are sensitive to the location and lateral dimension of seismic anomalies near the core-mantle boundary. Some important issues, such as the density change and roughness of the structures and the sharpness of the transition from the structures to the surrounding mantle, however, remain unresolved due to the nature of the data.

Abstract: The key objective of this work is the design of an unconditionally stable, robust, efficient, modular, and easily expandable finite element-based simulation tool for cardiac electrophysiology. In contrast to existing formulations, we propose a global–local split of the system of equations in which the global variable is the fast action potential that is introduced as a nodal degree of freedom, whereas the local variable is the slow recovery variable introduced as an internal variable on the integration point level. Cell-specific excitation characteristics are thus strictly local and only affect the constitutive level. We illustrate the modular character of the model in terms of the FitzHugh–Nagumo model for oscillatory pacemaker cells and the Aliev–Panfilov model for non-oscillatory ventricular muscle cells. We apply an implicit Euler backward finite difference scheme for the temporal discretization and a finite element scheme for the spatial discretization. The resulting non-linear system of equations is solved with an incremental iterative Newton–Raphson solution procedure. Since this framework only introduces one single scalar-valued variable on the node level, it is extremely efficient, remarkably stable, and highly robust. The features of the general framework will be demonstrated by selected benchmark problems for cardiac physiology and a two-dimensional patient-specific cardiac excitation problem. Copyright © 2009 John Wiley & Sons, Ltd.

Abstract: Equations describing small deformations of thin shells of arbitrary shape are written in terms of the displacements of the middle surface and rotation of the normals to the middle surface. The theory accounts for transverse shear deformations. By then using simple bilinear approximations of the displacement and rotation fields within finite elements of the shell, a consistent discrete model is obtained; the model provides complete interelement compatibility, and applies to the analysis of thin shells of arbitrary shape. A discrete equivalent of the Kirchhoff hypothesis is then introduced, which greatly improves convergence rates, and which forces the finite element model to convergence to the continuous Kirchhoff model. Numerical examples are included.