Home    Introduction    Theory    Numerical    Results    Conclusions

Numerical

In the weak statement above, once H matrix is completed through calculation, now nodal velocities for each point can be obtained.

By multiplying the inverse H matrix on both sides, theoretically we can calculate all the nodal velocities at a certain time step. In our group topic, different from Cartesian coordinate case, since we need to simulate on the forming of encapsulated void structure under the surface, we needed to select the three dimensional evolution. But, to make the calculation simple and efficiently reflect the shape changes of the void structure, axial symmetry coordinate was chosen. Due to the coordinate change, the term for H matrix and the force term has been altered.    

In the coordinate, x value represents the radius of the structure at a specific height, and the y value represents the height in the cylindrical structure. When it comes to the force term, γ is the surface tension, l is the element length, Θ is the angle with respect to the x direction. Below shows the H matrix elements in the Axial-symmetry cylindrical coordinate.

Based on the nodal velocities for all points, by using Euler method, we update the position of each nodal points after the certain time step. And doing this closed loop calculation, we can simulate the structural evolution on the surface groove. In order to check its difference of shape change depending on the initial geometry, we input different initial element configuration by coding. Following is the Euler method.

Yn+1 = Yn + hf(tn,yn)

Y’(t)=f(t,y(t)) & Y(to)=Yo

tn+1=tn + h (h is the time step)

Matlab code: m file