Finite Element Method
Since COMSOL is a finite element solver, a basic understanding of the finite
element method is necessary to work within the program. The in-depth complexities of
the program finite element method would take far too long to explain, but this section
provides a brief explanation as to what the finite element method is.
The most basic explanation of the finite element method is a technique for solving
a numerical solution to a boundary value problem described by a differential equation.
The differential equation itself is used to describe some aspect of engineering, for this
project, incompressible fluid flow. Specific boundary conditions determine what the
solutions to those equations must be on the boundary of the shape in question. In order to
numerically solve the problem, the geometry in which the equation will be solved must
be broken down into a stiffness matrix. The stiffness matrix is a series of points within
the boundaries of the geometry. The equations will be solved on each of these individual
points. Once the points are set in place, they are connected to one another. For a two
dimensional area these connections sometimes form rectangular mesh elements, other
times they form triangular mesh elements. COMSOL almost always uses a triangular
connectivity pattern. These points and their connections are what give the “mesh” its
visible mesh-like quality.
The finite element method is based on the idea that a continuous function may be
approximated by solving it at discrete points. In this case, those points are the individual
elements that are comprised of three nodes. For a simplified example, assume that there is
a governing equation for a two dimensional model with one unknown. There are two
functions necessary to solve this, and any other, finite element problem. The first is the
interpolation function. Each finite element has its own interpolation function to
approximate what is happening within that particular element, based on the governing
equation for the full model. The second necessary function is the shape function. Shape
functions are usually the coefficients in the interpolation function and are unique to each
node within the element. The solution of the shape function is such that if it were for each
node within one element, the value of the function will be one at one node and zero at
every other node in the element. There are as many shape functions within an element as
there are nodes. Each node will cause one of the shape functions to equal one at that
particular node and zero at every other node. Consequently, part of the interpolation
function is solved at each node individually. The solutions are then added together to find
the value of the variable in the interpolation function for that mesh element.
Since the solutions for finite element analysis are based these individual meshes
elements, the space between the nodes directly affects the accuracy of the solution. The
closer the nodes are, the smaller the mesh area of the mesh elements will be and the
smaller the distance across which the governing equation must be approximated. So, the
more refined the mesh, the better the solution. However, the more mesh elements present,
the more equations must be solved and the greater the computing power required to arrive
at a solution.