Topics in Computational Mechanics: Part 4

The geometries that we considered so far are simple. If we are to analyze a complex, real-world structure, we need a way to accurately model its geometry. Not to worry, we can map a simple element to a complex one using a function.

Today, we will study the isoparametric approach. It tells us how to define the function so that we can evaluate integrals fast and achieve optimal rate of convergence. It is key to making FEM practical and powerful.

Monte Carlo Simulations: How Big Is Your Heart?

Our final problem has no known exact solution. We want to find the area of the shape formed by,

$(x^{2} + y^{2} - r^{2})^{3} - a\,x^{2}y^{3} \leq 0$

The inequality has two parameters $r$ and $a$. They are quantities of length, so they take on a nonnegative value. Let’s try out some values and see what these parameters do. I have colored the resulting shapes in red. (Note, the scales are different.)

When $r = a$ (move along the diagonal, starting from bottom-left), we get the shape of a nice heart. When we increase $r$ while holding $a$ fixed, the heart morphs into a circle. On the other hand, when we increase $a$ while holding $r$ fixed, the heart turns into two petals. Well, I see bunny ears.

Clearly, the shape (i.e. area) of our heart depends on the radius $r$ and the ear length $a$. Can you guess the formula for the area $A = A(r, a)$?