# The finite element method for elasticity

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Finite element method (FEM) is used for most structural simulations in civil and petroleum engineerings. This method consists of solving the weak-form integral equations of the problem in question (e.g. Zienkiewicz and Taylor, 2000; Hughes, 2012). The studying body is divided into sub-elements those are connected by nodes of which the mechanical fields are obtained by solving a linear equation system. The later requires the construction and inversion of a rigidity matrix (for elasticity) that is a merge of those of the sub-elements. Inverting a very large rigidity matrix to solve a structural problem by FEM requires a large amount of time. A number of offline industrial softwares are developed on the basis of this method.

Here we provide an online simulator to illustrate the FEM technique, for academic purpose. We limit ourself to a basic structure with a small number of elements and a structured mesh to minimize the calculation time and adapt with an online version. Let consider for example a 2D ring (plane strain) with given inner and outer radii and the elastic properties (Young modulus and Poisson ratio). An uniform pressure is applied on the inner surface and the ourter surface is fixed. Exact analytical solution exists for this simple problem (Fjaer et al., 1992) but we will try to use FEM to solve it.

References
Zienkiewicz, O. C., & Taylor, R. L. (2000). The finite element method: solid mechanics (Vol. 2). Butterworth-heinemann.
Hughes, T. J. (2012). The finite element method: linear static and dynamic finite element analysis. Courier Corporation.
Fjaer, E., Horsrud, P., Raaen, A. M., Risnes, R., & Holt, R. M. (1992). Petroleum related rock mechanics (Vol. 33). Elsevier.

Input the parameters, then click Compute button to run the simulation.

Mesh's parameter

(This parameter defines the fineness of the mesh. Increasing this parameter allows increasing the accuracy of the results. However, a calculation using a smartphone or a personal computer with a parameter higher than 10 requires more than 1 minutes to finish.)

Inner pressure (MPa)

Young modulus (GPa)

Poisson ratio