Partial differential equations (PDEs) are the language of choice for describing continuous physics. By continuous physics, I mean any theory where physical quantities are represented by continuously varying functions - quantities such as temperature, strain and electromagnetic fields. Many of the fundamental concepts of science are described by PDE's, such as expressing heat flow in terms of the gradient of the temperature, or expressing the net source of momentum as the divergence of a stress tensor.
Finite element analysis (FEA) is the most powerful method for numerically solving PDEs. FEA has a decided advantage over finite difference methods in representing complicated geometries and boundary conditions, which are often essential in realistic scientific problems. FEA software is usually highly specialized: packages are available that provide specific solutions in areas such as solid mechanics, heat transfer, and electromagnetics....and they are expensive. The software world is changing: FEA software is now available to do more of the drudge work than ever before... and prices are much lower. State-of-the-art finite element software accepts wider ranges of equations than before. There are now solutions that let you solve generalized problems, beyond the predefined equations, which specialized applications were written to handle. The most up-to-date software can automatically generate element meshes, perform error analysis, refine the mesh as needed, select iteration methods for nonlinear problems, and adaptively select time steps in dynamic problems to optimize speed and stability of solutions. As a bonus, most of the leading symbolic math packages like Macsyma, Maple and Mathematica can generate PDEs automatically, solve some cases symbolically, and transfer the equations to a finite element program for numerical solving.
Prices for FEA software used to start above ten thousands dollars. Today's new generation of FEA software, available for personal computers and workstations at drastically lower prices, means that solving PDEs can be a routine procedure for scientists who are not experts in numerical analysis and have modest budgets. It's a very exciting time for scientists who want to solve PDEs.
--This article was submitted by Richard Petti, President of Macsyma Inc., developers of Macsyma, a symbolic mathematics program, and PDEase, a new add-on for performing finite element analysis.
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