# MoM Computational Resources Scaling

The usage of a dense matrix in the MoM implies a limit to the size of the problem that can be solved. The limit is determined by the available computational resources.

Although the MoM efficiently discretizes the model by only
requiring the bounding surface to be meshed, the method uses a dense matrix. As a result, the
memory scaling is proportional to N^{2} and CPU-time to N^{3}, where N is the
number of unknowns.

This is best illustrated by comparing the asymptotic behaviour of the memory and CPU-time
scaling of a model solved at one frequency and at double the frequency.

- At the higher frequency, the triangle patches are required to have half the edge lengths. The number of elements increases by a factor of four. The number of unknowns is proportional to the number of elements and the memory required to solve the problem increases by a factor of 16.
- When solving the problem at double the frequency, the simulation time increases by a factor of 64.

As the frequency and structure size increases, special techniques such as the multilevel fast multipole method, higher order basis functions and asymptotic techniques are required to obtain a solution efficiently.

Alternately, higher order basis functions for triangular elements could be used. Higher order
basis functions have more unknowns per element, but they allow larger mesh elements to be used.
The net result is that less memory is required.

Note: Use larger triangular elements provided the
larger triangles describe the model geometry accurately.

Higher order basis function
elements can be represented with curvilinear triangle patches that allow second order
descriptions of the triangular patch boundaries. The curvilinear elements allow a further
reduction in the number of elements required for an accurate representation of the model.