For the elements an analysis of the finite element method strang pdf a poset, see compact element. Stokes differential equations used to simulate airflow around an obstruction. Typical problem areas of interest include structural analysis, heat transfer, fluid flow, mass transport, and electromagnetic potential. The analytical solution of these problems generally require the solution to boundary value problems for partial differential equations.
The finite element method formulation of the problem results in a system of algebraic equations. The method yields approximate values of the unknowns at discrete number of points over the domain. To solve the problem, it subdivides a large problem into smaller, simpler parts that are called finite elements. The simple equations that model these finite elements are then assembled into a larger system of equations that models the entire problem.
FEM then uses variational methods from the calculus of variations to approximate a solution by minimizing an associated error function. The global system of equations has known solution techniques, and can be calculated from the initial values of the original problem to obtain a numerical answer. To explain the approximation in this process, FEM is commonly introduced as a special case of Galerkin method.
The process, in mathematical language, is to construct an integral of the inner product of the residual and the weight functions and set the integral to zero. In simple terms, it is a procedure that minimizes the error of approximation by fitting trial functions into the PDE. The residual is the error caused by the trial functions, and the weight functions are polynomial approximation functions that project the residual.
These equation sets are the element equations. They are linear if the underlying PDE is linear, and vice versa. Algebraic equation sets that arise in the steady state problems are solved using numerical linear algebra methods, while ordinary differential equation sets that arise in the transient problems are solved by numerical integration using standard techniques such as Euler’s method or the Runge-Kutta method. This spatial transformation includes appropriate orientation adjustments as applied in relation to the reference coordinate system.
The process is often carried out by FEM software using coordinate data generated from the subdomains. FEA as applied in engineering is a computational tool for performing engineering analysis.
It includes the use of mesh generation techniques for dividing a complex problem into small elements, as well as the use of software program coded with FEM algorithm. In applying FEA, the complex problem is usually a physical system with the underlying physics such as the Euler-Bernoulli beam equation, the heat equation, or the Navier-Stokes equations expressed in either PDE or integral equations, while the divided small elements of the complex problem represent different areas in the physical system.