1 - 10 of 10 Chapters
[Let us start by considering a simple one-dimensional differential equation, written in abstract form.]
[In most problems of mathematical physics the true solutions are nonsmooth, i.e., they are not continuously differentiable. Thus, we cannot immediately apply a Galerkin approach. For example, in the equation of static mechanical equilibrium]
[Classical techniques construct approximations from globally kinematically admissible functions, which we define as functions that satisfy the displacement boundary condition beforehand. Two main obstacles arise: (1) it may be very difficult to find a kinematically admissible function over the...
[As we have seen, the essential idea in the finite element method is to select a finite dimensional subspatial approximation of the true solution and form the following weak boundary problem]
[There are two main approaches to solving systems of equations resulting from numerical discretization of solid mechanics problems, direct and iterative. There are a large number of variants of each. Direct solvers are usually employed when the number of unknowns is not very large, and there are...
[Albeit a bit repetitive, we follow similar constructions to those used in the one-dimensional analysis of the preceding chapters. This allows readers a chance to contrast and compare the differences between one-dimensional and three-dimensional formulations.]
[Generally, the ability to change the boundary data quickly is very important in finite element computations. One approach to do this rapidly is via the variational penalty method.]
[As we have seen in the one-dimensional analysis, the essential idea in the finite element method is to select a finite dimensional subspatial approximation of the true solution and form the following weak boundary problem]
[We now give a brief introduction to time-dependent problems through the equations of elastodynamics for infinitesimal deformations]
[The finite element method is a huge field of study. This set of notes was designed to give students only a brief introduction to the fundamentals of the method.]
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