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Two Dimensional Solid Elements: General Requirements for an Element

General Requirements for an Element

Finding an appropriate displacement shape function or interpolation function has been a subject of extensive study since the development of the finite element analysis method. While new types of elements are being introduced on a regular basis, some basic requirements characterize a good approximation for the displacement. Among those requirements are isotropy, ability to model rigid body motions and constant strains, and element compatibility. An isotropic interpolation function is a function that would not favor a direction over the other. An interpolation function that has the form u = a_1 + a_2X_1 + a_3X_2^2 would clearly produce different results if the structure is rotated such that the coordinates X_1 and X_2 are switched. However, an interpolation function of the form u=a_1 + a_2X_1 + a_3X_2 is isotropic. Another basic requirement is that the displacement interpolation function should be able to model rigid body motion. An interpolation function that has the form u = a_1X_1 + a_2X_2 cannot model a constant displacement, and thus, cannot be acceptable. Such interpolation function, however, can model constant strain. Another major requirement for the interpolation functions is ensuring the compatibility between elements. All the elements presented in the previous section ensure that if two elements of the same type share the same nodes along a boundary, then the displacement along that boundary (the shared side) between the elements is continuous. This was ensured by choosing interpolation or shape functions that guarantee that the displacement on the boundaries (sides) of the element is completely determined by the nodes on that side, and thus, the elements that share that side along with the associated nodes share the same displacement across that side. The displacement interpolation functions presented so far are thus truly C_0 interpolation functions. In some applications, more (or less) stringent requirements can be considered, and the user should be aware of the capabilities or lack thereof of the elements being used.

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