Three-dimensions finite element methods

In reality, heat is conducted in all three spatial dimensions. While specific building simulation codes can model the transient and steady-state two-dimensional temperature distribution in building structures using finite-difference or finite-elements methods, conduction is normally modeled one-... 

Z. Ge, T. X. Wu, R. Lu, et al.. Comprehensive three-dimension dynamic modeling of liquid crystal devices using finite element method, J. Display Technology, 1, 194 (2005). 

The finite element method has developed into the most important approach for solving many engineering problems. Its premier advantage is the ability to readily model complex spatial geometries associated with material structures. This is in both two dimensions as covered in this work and three dimensions which are not considered in fliis work. A very extensive literature exists on this subject and many authors have written complete books on the subject. Obviously a single chapter can not hope to cover in depfli the many important aspects of this subject. It is hoped that the material in this chapter provides the reader with a deeper appreciation for the approach and wifli some computer code that can be used for some of the simpler PDEs. A search of finite element resources in a library or on the web will result in a multitude of resources. 

Adolf et al [89] focused primarily on epoxy systems. They consider the key to the success of a constitutive model to be its choice of strain measure and the inclusion of free-energy-accelerated relaxations. The model only requires linear properties (i.e., properties that may be predictable by the methods developed in this book) for materials prior to their synthesis, since nonlinear behavior arises naturally from the formalism. Thermal properties and epoxy curing are also treated by their model. The authors have also attempted to treat failure by identifying a critical hydrostatic tension consistent with glassy failure. The model has been validated with a wide variety of types of material tests. The finite element simulations are performed in three dimensions. These authors have, thus far, done only a limited amount of preliminary work with heterophasic systems, but they report that the results were encouraging. 

To study the dynamic behavior of the BZ gels, we numerically integrate Eqs (8.1 -8.3) in two [1, 2] or three [3] dimensions using our recently developed gLSM. This method combines a finite-element approach for the spatial discretization of the elastodynamic equations and a finite-difference approximation for the reaction and diffusion terms. We used the gLSM approach to examine 2D confined films and 3D bulk samples here, we briefly discuss the more general 3D formulation [3]. 

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