Finite element method (FEM), or sometimes finite element analysis (FEA), is a numerical process used to solve partial differential equations within the fields of engineering and mathematics. Since many physics problems can be defined as partial differential equations, FEM is a common tool used in computer-aided engineering (CAE) simulation software to model structural analysis, heat transfer, electromagnetics and more.
What is the meaning of a finite element?
To solve an FEM analysis, the geometry of the problem must be broken down into smaller, simpler geometries called finite elements. This process is often called meshing, as it turns the larger geometry into a grid, or mesh, of smaller parts.
Once boundary conditions and material properties are selected, the differential equations that describe the physics of the system can be mimicked within each finite element using algebraic equations. This system of equations can then be solved iteratively until the value of an error function is minimized.
Smaller finite elements—also known as a finer mesh—will yield better estimates of the physics and capture local phenomena. However, the trade-off is that the simulation will be harder to solve computationally. To ensure a solution can be found that is both timely and accurate, the number and size of finite elements need to be optimized.
To get around these issues, simulation users will often create a mesh with finer elements near complex geometries and coarser elements where results are not expected to change much from element to element. These variable finite element sizes produce accurate results in areas of interest without making the simulation computationally expensive or time consuming.
When should FEM be used and what alternatives exist?
FEM can be used to simulate many physics problems, including computational fluid dynamics (CFD). However, for CFD the finite difference method (FDM) or finite volume method (FVM) are often selected instead. CFD problems tend to require a larger number of elements—making FEM too computationally expensive. FDM and FVM, however, use lower order approximations within each cell, making these methods more attractive in CFD scenarios.
On the other hand, FEM is more attractive in problems requiring fewer finite elements as it generally returns a higher quality of results and can accommodate geometries of higher complexity.