The finite element method approximates complex mathematical problems with a certain degree of error. Engineers use it to design practical products and reduce the need for physical prototypes. It models physical strains in materials and uses a mesh to simplify equations. Boundary conditions must be set for meaningful results, and visualizations help identify weaknesses in a design.
The finite element method is a tool for calculating approximate solutions to complex mathematical problems. It is generally used when mathematical equations are too complicated to solve in the normal way and a certain degree of error is tolerable. Engineers commonly use the finite element method because they are designing products for practical applications and do not need perfect solutions. The finite element method can be adapted to different accuracy requirements 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 strains in materials. The damage a car sustains from a head-on collision is an example of a complicated deformation. Deformations in one area depend on deformation in other areas: the collision has to be modeled in many different steps over time to see what the end result will be. This large number of steps makes it impractical to manually model such a problem. A computer using the finite element method, however, could solve this problem with a high degree of accuracy.
Furthermore, real-world material deformations, 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 solve with current knowledge. Numerical methods in mathematics, therefore, are used to approximate more complicated equations using simpler equations in many different steps. In the finite element method, a mesh is created to model changes in space using many smaller, simpler elements. The degree of error resulting from this simplification depends on the total number of elements in the mesh.
For the finite element method to produce meaningful results, a set of boundary conditions with the problem must be set. They essentially define what kind of conditions the model has 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 such as changes in temperature or applied thermal energy, or location constraints. Without boundary conditions, it is impossible to set up a problem, because the model would have little to answer for.
An advantage of the finite element method is that it is easy to produce detailed visualizations of a problem. Once a model has been fully resolved, this information can be transferred into an image. For example, specific stresses in different mesh elements can be assigned different colors. Visualizations allow engineers to intuitively identify weaknesses in a design and can use this information to create a new design. Visualization software is an essential part of many finite element computer programs.
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