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What is the Finite Element Method?

By James Doehring
Updated May 21, 2024
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The finite element method is a tool for computing approximate solutions to complex mathematical problems. It is generally used when mathematical equations are too complicated to be solved in the normal way, and some degree of error is tolerable. Engineers commonly use the finite element method because they are concerned with designing products for practical applications and do not need perfect solutions. The finite element method can be adapted to varying requirements for accuracy and can reduce the need for physical prototypes in the design process.

One application of the finite element method is the modeling of complex physical deformations in materials. The damage that a car experiences from a front-end collision is one example of a complicated deformation. Deformations in one area depend on deformations in other areas—the collision needs to be modeled over many different steps in time to see what the end result will be. This large number of steps makes it impractical to model such a problem by hand. A computer that uses the finite element method, however, could solve this problem with a high degree of accuracy.

Moreover, deformations of real-world materials, like many other physical phenomena, are complicated effects. One problem with modeling such effects using accurate mathematical equations is that they would be too complicated to be solved with current knowledge. Numerical methods in mathematics, therefore, are used to approximate more complicated equations by using simpler equations over many different steps. In the finite element method, a mesh is created to model changes over space using many small, simpler elements. The degree of error resulting from this simplification depends on the number of total elements in the mesh.

For the finite element method to produce meaningful results, a set of boundary conditions needs to be set up with the problem. These essentially define what sort of conditions the model needs to respond to. In the car example, the boundary conditions would be the forces inflicted on the car by the external object. Boundary conditions can be point forces, distributed forces, thermal effects like temperature changes or applied heat energy, or positional constraints. Without boundary conditions, it is impossible to set up a problem, because the model would have little to respond to.

One advantage of the finite element method is that it is easy to produce detailed visualizations of a problem. Once a model has been fully solved, this information can be transferred into a picture. Specific stresses in different mesh elements, for example, can be assigned different colors. Visualizations allow engineers to intuitively identify weak points in a design, and they may use this information to create a new design. Visualization software is an essential part of many finite element computer programs.

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