Final answer:
The shape functions in the natural coordinate system determine the local approximation of the solution within each finite element in finite element analysis. The isoparametric mapping equations define the relationship between the physical coordinates and the natural coordinates.
Step-by-step explanation:
Shape Functions in Natural Coordinates
In the context of finite element analysis, the shape functions in the natural coordinate system determine the local approximation of the solution within each finite element.
These shape functions are typically polynomial functions that depend on the coordinates (ξ, η) of the natural coordinate system.
The number and order of shape functions depend on the type of element being used.
Isoparametric Mapping Equations
The isoparametric mapping equations define the relationship between the physical coordinates (x, y) and the natural coordinates (ξ, η).
These equations take the form:
x = x(ξ, η)
y = y(ξ, η)
where x and y are the physical coordinates and ξ and η are the natural coordinates.
The specific form of these equations depends on the element shape and the chosen mapping strategy.