Understanding Finite Element Method (FEM)

1. What is the finite element method (FEM)?

The FEM is a novel numerical method used to solve ordinary and partial differential equations. The method is based on the integration of the terms in the equation to be solved, in lieu of point discretization schemes like the finite difference method. The FEM utilizes  the  method  of  weighted  residuals  and  integration  by  parts  (Green-Gauss Theorem) to reduce second or derivatives to first order terms. The FEM has been used to solve a wider range of problems, and permits physical domains to be modeled directly using unstructured meshes typically based upon triangle so quadrilateral s in 2-D and tetrahedrons or hexahedral in 3-D.The solution domain is discretized into individual elements – these elements are operated upon individually and then solved globally using matrix solution techniques.

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2. What is the history of the FEM?

Early work on numerical solution of boundary – value problems can be traced to the use of finite differences schemes; South well used such methods in his book published in the mid1940’s. The beginnings of the finite element method actually stem from these early numerical methods and the frustration associated with attempting to use finite difference methods on more difficult, geometrically irregular problems. Beginning in the mid 1950s, efforts to solve continuum problems in elasticity using small, discrete “elements” to describe the overall behavior of simple elastic bars began to appear, and such techniques were initially applied to the aircraft industry. Actual coining of the term “finite element” appeared in a paper by Clough in 1960. The early use of finite elements lay in the application  to  structural – related   problems.   However,  others  soon  recognized  the versatility of the method and it sunder lying rich mathematical basis for application in non-structural areas. Since the early works, rapid growth in usage of the method has continued since the mid 1970s. Numerous articles and texts have been published, and new applications appear routinely in the literature.

3. Why should one use finite elements?

The versatility of the FEM, along with its rich mathematical formulation and robustness makes it an ideal numerical method for a wide range of problems. The ability to model complex geometries using unstructured meshes and employing elements that can be individually tagged makes the method unique. The ease of implementing boundary conditions as well as being able to use a wide family of element types is a definite advantage of the scheme over other methods. In addition, the FEM can be shown to stem from properly – posed functional minimization principles.

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4. Are there any recommended commercial FEM packages that are versatile in handling a wide range of problems?

Any of the well known and widely versatile FEM codes nowon the market are good – it just depends on how comfortable the user is with the I/O part of the program. ANSYS, is quite easy and very versatile – handling a wide range of problem classes including fluid flow (with turbulence), heat transfer, structural analysis, electrodynamics, and general PDEs including species transport, chemical reactions, and groundwater/porous media flows. For students ANSYS has rolled out special edition to learn & gain insights.

Visit https://www.ansys.com/en-in/academic

5. How long does it take for me to be able to use a FEM program?

Some programs allow you to solve problems quickly. It is always highly recommended that work out the example problems generally provided by most commercial software. It is also recommended that you spend time in genuinely going through the fundamentals and physics involved rather than be amused about the colorful plots produced by software. This essentially helps you to follow the area of Design analysis, simulation, validation & optimization practically from the career point of view.

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