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Many of us think FEA is used for structural problems and CFD is used for fluid dynamics problems. But I have read that the above statement is wrong. So please explain the exact usage of those two.Also please give the basic difference between them.

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CFD (computational fluid dynamics) includes any numerical method used to solve fluid flow problems.

FEA (finite element analysis) is one numerical method for solving partial differential equations, independent of what the equations are modelling.

It is true that FEA is the most popular method for solving computational mechanics problems.

There are several different approaches used to solve CFD problems, one of which is FEA - though the CFD community often describe FEA as an "unstructured grid method" instead of "FEA", in contrast to the regular "structured grids" used in finite difference solution methods.

Many partial differential equations can be classified as "elliptic", "parabolic", or "hyperbolic" depending on how the solution propagates to different parts of the domain over time. Most structural analysis problems are elliptic, though problems involving time-dependent heat transfer are parabolic. Even for differential equations which strictly speaking are hyperbolic, if the speed of sound in the material is large compared with the speed of the response of the structure, it is often useful to use an approximate parabolic or elliptic equation instead (for example by assuming the speed of sound is infinite).

The solutions of differential equations for fluid flow often don't fit into this neat mathematical classification, and have a mixture of elliptic and hyperbolic behaviour in different parts of the domain (and the boundary between the different types of behaviour is unknown before the solution has been found). So, even when finite element methods are used in CFD, much of the detail of the numerical algorithms is very different from the FEA methods used for structural analysis problems.

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First I would like to clear the doubt that the difference between FEA and CFD is that FEA is for structural applications and CFD for fluid dynamical applications, is wrong.

Computational Fluid Dynamics (CFD) refers to the use of the numerical techniques to solve fluid dynamical problems. When I say 'numerical techniques', I am referring to a very broad range of techniques, including, but not limited to, Finite Difference Methods, Finite Element Methods, Finite Volume Methods, Polynomial Fitting, Spectral Methods, Boundary Element Methods and so on. Even though the basic philosophy is the same i.e. discretize a system with infinite degrees of freedom into a finite system, these are all different techniques with different mathematical foundations. For instance, Finite Difference Methods are based upon discretization of the differential form of governing equations and work by approximating derivative via truncation of Taylor Series expansions.

The order of accuracy of FDM depends on the highest order of the Taylor series expansion terms that are eliminated. FDM is the most intuitive of these methods to understand. There are compact schemes such as Pade Schemes which allows one to improve the accuracy of Finite Differences for the same numerical stencil. The FVM and FEM method on the other hand involve discretization of the integral form of the PDE. Spectral Methods involve discretization of the grid into a finite set of points and representing the solution as a linear combination of periodic functions (for instance, a Fourier Series). In CFD the governing PDEs are of course the Navier-Stokes equations (Navier–Stokes equations).

Finite Element Method(FEM), also known as Finite Element Analysis(FEA) is a specific numerical technique that, of course, solves a continuous problem stated in the form of a PDE, by discretizing the problem into a finite number of nodal points but it does so by first multiplying the differential form of the governing equation(PDE) with an arbitrary weighting function and using Integration by parts and the Divergence Theorem to obtain, what is known as the 'Weak Form' of the governing equation, and then formulating a system of linear equations by approximating the solution field as a linear combination of a finite number of basic functions, each of which are pairwise orthogonal and altogether satisfy partition of unity. The beauty of the Finite Element Method and the weak form is that it allows us to approximate our quantity of interest using functions that have 'weaker' continuity requirements while at the same time maintaining compatibility at element interfaces.

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