Finite element analysis (FEA) is a numerical method for solving differential equations with partial derivatives, as well as integral equations arising in applied physics. The analysis is widely used for solving problems in mechanics of deformable solid body, heat transfer, fluid dynamics, and electrodynamics.
The name of the analysis shows its essence. The area, in which you look for solution of differential equations, is divided into a finite number of subdomains (elements). In each of the elements, the view of the approximating function is selected randomly.
In the simplest case, this is a first degree polynomial. Out of its element, approximating the function is zero. Function values at the boundaries of the elements (nodes) are the solutions of the problem and are not known in advance. Approximating functions ratios are usually of equal values of functions on the borders between the elements (nodes). Then these coefficients are expressed through the function values in nodes of elements. A system of linear algebraic equations is formed. The number of equations is equal to the number of unknown values in nodes, where the original system is directly proportional to the number of elements and is limited only by the capabilities of the computer. Since each element is associated with a limited number of neighboring elements, the system of linear equations has a sparse look that makes its solution easier.
In matrix terms, the so-called rigidity matrix (or Dirichlet matrix) and mass matrix are constructed. Then, these matrices are put in boundary conditions (for example, under Neumann condition there is no change in the matrix, and under Dirichlet conditions, the matrix rows and columns corresponding to the boundary nodes are erased, because the effect of boundary conditions to the respective component of the solution is known). After that, a system of linear equations is constructed and solved with one of the known methods.
In terms of computational mathematics, the idea of the finite element analysis is that minimizing variational functional tasks is carried out on the set of functions, each of which is defined in its subdomain, for numerical analysis system allows us to consider it as one of the specific branches of diacoptic, a common method of systems investigation by there dismemberment.
Finite element analysis has more complex implementations than finite difference method. The finite element analysis, however, has a number of benefits in the form of real-world tasks: free form treatment area; the grid can be made more rare in locations where special precision is not necessary.
For a long time widely FEA distribution was hampered by a lack of automatic partitioning algorithms for “almost equilateral triangles (error, depending on the method of variation, is inversely proportional to the sine of the acute, or obtuse angle in split). However, this task was successfully solved (algorithms based on Delaunay Triangulation), which made it possible to create a fully automatic finite element CAD.
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