Finite difference method

Abstract: We discuss the foundations and state-of-the-art of the method of analytical regularization (MAR) (also called the semi-inversion method). This is a collective name for a family of methods based on conversion of a first-kind or strongly-singular second-kind integral equation to a second-kind integral equation with a smoother kernel, to ensure point-wise convergence of the usual discretization schemes. This is done using analytical inversion of a singular part of the original equation; discretization and semi-inversion can be combined in one operation. Numerous problems being solved today with this approach are reviewed, although in some of them, MAR comes in disguise.

Abstract: We outline an Eulerian framework for computing the thickness of tissues between two simply connected boundaries that does not require landmark points or parameterizations of either boundary. Thickness is defined as the length of correspondence trajectories, which run from one tissue boundary to the other, and which follow a smooth vector field constructed in the region between the boundaries. A pair of partial differential equations (PDEs) that are guided by this vector field are then solved over this region, and the sum of their solutions yields the thickness of the tissue region. Unlike other approaches, this approach does not require explicit construction of any correspondence trajectories. An efficient, stable, and computationally fast solution to these PDEs is found by careful selection of finite differences according to an up-winding condition. The behavior and performance of our method is demonstrated on two simulations and two magnetic resonance imaging data sets in two and three dimensions. These experiments reveal very good performance and show strong potential for application in tissue thickness visualization and quantification.

Abstract: In this paper, a numerical model based on the finite difference method is presented to predict tool and chip temperature fields in continuous machining and time varying milling processes. Continuous or steady state machining operations like orthogonal cutting are studied by modeling the heat transfer between the tool and chip at the tool—rake face contact zone. The shear energy created in the primary zone, the friction energy produced at the rake face—chip contact zone and the heat balance between the moving chip and stationary tool are considered. The temperature distribution is solved using the finite difference method. Later, the model is extended to milling where the cutting is interrupted and the chip thickness varies with time. The time varying chip is digitized into small elements with differential cutter rotation angles which are defined by the product of spindle speed and discrete time intervals. The temperature field in each differential element is modeled as a first-order dynamic system, whose time constant is identified based on the thermal properties of the tool and work material, and the initial temperature at the previous chip segment. The transient temperature variation is evaluated by recursively solving the first order heat transfer problem at successive chip elements. The proposed model combines the steady-state temperature prediction in continuous machining with transient temperature evaluation in interrupted cutting operations where the chip and the process change in a discontinuous manner. The mathematical models and simulation results are in satisfactory agreement with experimental temperature measurements reported in the literature.

Abstract: A method for solving boundary value problems (BVPs) is introduced using artificial neural networks (ANNs) for irregular domain boundaries with mixed Dirichlet/Neumann boundary conditions (BCs). The approximate ANN solution automatically satisfies BCs at all stages of training, including before training commences. This method is simpler than other ANN methods for solving BVPs due to its unconstrained nature and because automatic satisfaction of Dirichlet BCs provides a good starting approximate solution for significant portions of the domain. Automatic satisfaction of BCs is accomplished by the introduction of an innovative length factor. Several examples of BVP solution are presented for both linear and nonlinear differential equations in two and three dimensions. Error norms in the approximate solution on the order of 10-4 to 10-5 are reported for all example problems.