The main purpose of this article is to facilitate the implementation of spacetime finite element methods in four-dimensional space. Generally speaking, researchers and engineers have previously relied on their intuition and ability to visualize two-and three-dimensional spaces. Unfortunately these attributes do not naturally extend to four-dimensional space, and therefore the development of numerical methods in this context is a unique and difficult challenge. In order to construct a finite element method in four-dimensional space, it is necessary to create a numerical foundation, or equivalently a numerical infrastructure. This foundation should include a collection of suitable elements (usually hypercubes, simplices, or closely related polytopes), numerical interpolation procedures
The usage of lasers in ocean studies is widespread. Each step of laser beam propagation through the ocean is a major topic to be analyzed, often independently. The objective of this study is to focus on the interaction of laser beams with the air-ocean interface; specifically, the modeling and analysis of the effects of a gravity wave perturbed ocean surface on laser propagation. The directional energy spectrum of Neumann with the Fourier series expansion is used in a Monte Carlo simulation of the gravity wave perturbed ocean surface model. Beam tracing with the ABCD matrix approach is used for the laser beam propagation analysis rather than using ray tracing like found in similar studies. Specific parameters are used in the model to output not only a qualitative model but also numerical and realistic results. The main purposes of this study are implementing a numerical model to see the effects of the ocean surface on laser propagation and analyzing the feasibility of using the beam tracing approach in such a model.
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