In finite element method (FEM), the basis functions are used to approximate the solution of a partial differential equation over a finite element domain. These basis functions are defined on a set of discrete nodes within the element and are typically expressed as polynomials.

The construction of basis functions in FEM involves the following steps:

Select a shape function: A shape function is a function that describes the variation of the solution within an element. The choice of shape function depends on the geometry of the element and the nature of the problem being solved. Common shape functions include linear, quadratic, and cubic functions.

Define the nodal values: The nodal values are the values of the solution at the nodes within the element. These values are used to determine the coefficients of the basis functions.

Construct the basis functions: The basis functions are constructed by combining the shape functions with the nodal values. For example, the linear shape function can be multiplied by the nodal values to construct a linear basis function.

Ensure completeness and continuity: The basis functions must be complete, meaning that they span the space of all possible solutions, and they must be continuous, meaning that they are smooth and have no discontinuities at the element boundaries. This can be achieved by selecting appropriate shape functions and ensuring that the nodal values are consistent across the element boundaries.

Assemble the global system: Once the basis functions are constructed for each element, they are assembled into a global system of equations that describes the entire domain. This system is solved using numerical methods to obtain the solution to the partial differential equation.

1. Identify the elements that make up the problem.

2. Define a set of nodes and elements to connect them.

3. Choose an appropriate interpolation method and use nodal points to construct the basis functions.

4. Apply the chosen interpolation method to the nodal points to form the basis functions.

5. Apply the boundary conditions to the basis functions.

6. Assemble the basis functions together to form the global system of equations.

7. Solve the system of equations to obtain the solution.

If you are starting with FEA, I would suggest trying some simple elements that have exact analytic solution, like straight springs (you can use these in 1D, 2D and 3D) or beams (in 2D or 3D). You can derive basis functions for these from analytic solutions, but of course you can also try different basis functions and compare the results.