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).

 

[tacman]

Shape functions are mathematical functions that are used in Finite Element Analysis (FEA) to represent the displacement of a finite element. They are used to calculate the deformation and stresses within an element based on the nodal displacements.

In FEA, a structure is divided into smaller elements, and the displacement at each node of these elements is known. However, the displacement within the element is not known. This is where shape functions come into play. They are used to interpolate the displacement within an element based on the nodal displacements.

Linear shape functions are simple functions that represent a linear variation of displacement within an element. They are used for elements with straight edges, such as triangles and quadrilaterals. Polynomial shape functions are more complex and can represent curved variations of displacement within an element. They are used for elements with curved edges, such as circles and ellipses.

To visualize shape functions, imagine a simple element with three nodes. Each node has a known displacement in the x and y directions. The shape function will take these nodal displacements as inputs and provide a mathematical expression for the displacement within the element. This displacement can then be used to calculate the stresses and strains within the element.

In summary, shape functions are essential in FEA as they allow us to calculate the displacement within an element based on the nodal displacements. They come in different forms, such as linear and polynomial, depending on the type of element being analyzed. Visualizing shape functions may be challenging at first, but with practice and understanding of their mathematical representation, it becomes easier to grasp their concept.