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3.5 Discretization Methods, Solution Algorithms, and Model Specifications 3.5.1 Finite Element and Control Volume Formulations

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Several kinds of computational algorithms exist to solve for the field variable in Eq. (9). One category is known as the finite element method (FEM), where the field variable is assumed to have a functional form or shape over discrete portions of the problem domain. In finite element, the governing equations are multiplied by a weight function and then integrated over an element. The weight function can have various forms. As an example, with the Galerkin method, the weight function is the shape function itself. Another category is known as the CV method, where the problem domain is divided instead into a multitude of small volume elements, each characterized by a single, representative value for each relevant field variable. The conservation laws and fluxes are enforced on each CV, where transport or exchange across adjoining boundaries are determined with finite‐difference estimates (usually, a truncated Taylor Series expansion based on unknown or estimated adjacent CV values) of the various derivatives in the governing equation. Whereas both of these numerical methods involve discretizing the problem domain into a multitude of elements or volumes that appear to be virtually the same, they are different as explained in detail in [7, 8]. Generally, more mathematics are involved with the FEM whereas the CV method, dealing with fluxes, can easily be associated with representations giving a physical significance to the problem.

Encyclopedia of Glass Science, Technology, History, and Culture

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