What are shape functions and how are they used in FEA?

[chandran]

I know that shape function is a function that will give the displacements inside an element if its displacement at all the node locations of the element are known.

What is linear ,polynomial in shape functions. If i say as linear somebody else may say polynomial etc.

I am not able to visualize this shape functions at all and i am stuck with this in fea. Can anybody throw some light.

 

[PerennialII]

In FEM the approximate solution is typically constructed using polynominals of various degrees (and hence we've different order elements), since the mathematical construction and evaluation of integrals numerically are pretty straightforward when using them. Other bases such as harmonic ones are used seldom, but in principle the method can be constructed with pretty much whatever functions.

We can present the shape / basis functions $\phi_{i}$ for the solution $v(x)$ (of the FEA problem in question that is), say for a 1D problem, as

$$v(x)=\sum_{i=1}^n \eta_{i}\phi_{i}(x)$$

where $\eta_{i}=v(x_{i})$ (the nodal values), $x$ belongs to the solution domain, the summation is carried over the elements in the discretization (and the shape functions work element by element, satisfying the necessary conditions such as boundary conditions at element nodes), and the FEA minimization problem can be then formulated as finding $u \in V$ for

$$F(u) \leq F(v), \forall v \in V$$

where $V$ is the space for $v$.

Typically nowadays most problems are solved either using linear or quadratic elements (polynomial sense)(quads preferred often if we're considering e.g. typical structural mechanics elliptic problems, if the computational cost is not an issue), although in many applications use of for example the p - FEM is beneficial (where you adaptively decide the interpolation order depending on your solution in either a priori / a posteriori sense).