The finite element method (FEM) is a numerical approach for approximating solutions to differential equations. It is useful when a problem is too complex to solve exactly, especially when its geometry, materials, or boundary conditions are complicated.
Many engineering models are expressed using partial differential equations (PDEs). Examples include equations describing structural deformation, heat transfer, fluid flow, mass transport, and electromagnetic potential. These equations describe quantities that vary continuously across a domain.
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What is the finite element method?
FEM is a numerical method that approximates solutions to differential equations by dividing a domain into simpler finite elements and combining their equations.
What does a mesh represent?
A mesh is the discretized numerical domain: elements divide the region, and nodes describe their connections and locations.
What is an element equation?
It is a local equation that approximates the original governing equations over an individual finite element.
What does assembly accomplish?
Assembly systematically combines the element equations according to shared nodes and connectivity to create a global system for the whole domain.
How are FEM and FEA related?
FEM is the numerical method; its practical application to engineering analysis is called finite element analysis, or FEA.
Quiz questions
What does the finite element method primarily approximate?Answer: Numerical solutions to differential equations over a continuous domain.
In deriving a weak form for a second-order boundary-value problem, what does integration by parts commonly accomplish?Answer: It transfers a derivative from the unknown function to the test function.
Why do finite element matrices often contain many zero entries?Answer: Basis functions with local support do not overlap when their nodes are far apart.
How does the p-version of FEM typically improve an approximation?Answer: It increases the polynomial degree of basis functions while keeping the mesh structure.
Which statement describes a common difference between FEM and the finite difference method?Answer: FEM uses element-based approximations, while finite differences approximate derivatives on a grid.