Question

When you use a numerical method like the finite element method, you divide a large body into an equivalent system of smaller bodies called elements. This is done because using an analytical method on the larger body may require a solution to a boundary value partial differential equation; a task that might be difficult or impossible to solve. The question: why does dividing a large object into smaller objects help at all? If you divide a rectangle into tiny rectangles and ask each of those tiny rectangles the same question that you were asking the large rectangle (which would require the solution to a boundary value differential equation), what good have you done? Please explain.

Answer 1

Finite element method is a numerical method firstly you need to understand that so what is a numerical method lets cover that ;

Numericalmethod : is a mathematical tool designed to solve numerical problems. The implementation of a numerical method with an appropriate convergence check in a programming language is called a numerical algorithm.

In our case we use numerical method term because we are trying to solve partial differantial equations with the help of finite element methods. So basically we are doing some math. in order to solve some equations.

Types of Finite Element Methods:

1. Linear systems
2. Non-linear systems

So what are the basic equations for these systems;

1. [K]*{U}={R}
2. [K(u)]*{U}={R}

in first equation we have stiffness matrix (K), unknown variables matrix (U) and right hand side vector (R). Stiffness matrix is not depend on unknown variables thats why we call this Linear system (but not just for that reason etc.)

in second equation we have stiffness matrix which is depend on unknown variables .

Both of these equations’ stiffness matrix’s includes partial differantial equations so we are are trying to solve these equations with the help of fem.

Steps of Finite Element Analysis

1. Discretization: divide into pieces a.k.a mesh, nodes,elements.
2. Property of a generic element: shape functions, element formulations.
3. Assembly of elements: global stiffness matrix and right hand side vector
4. Applying boundary conditions : loads and conditions.
5. Solving the system: gauss elimination, decomposition methods etc.
6. Post processing: stress, strain and reaction forces calculations.

All in all meaning of finite element method is part or product will be divided into small pieces (elements) in order to solve linear or non-linear equations. The more small pieces better result can be achieved but after some point it can be meaningless to increase order of elements or number of elements because results will be converged at some point

One of the purposes of meshing is to actually make the problem solvable using Finite Element. By meshing, you break up the domain into pieces, each piece representing an element. You need these elements to be able to apply Finite Element since Finite Element is all about having a basis local to an Element and stitching a bunch of local solutions together to build the global one. If you did not mesh and just assumed some basis that covered the whole domain, that would be a Spectral Method.

One other aspect of meshing is accuracy of your solution. It can be shown that your global solution accuracy is a function of element width and the smallest angle at some element’s vertices. Due to these contributors to accuracy, it's important to be careful about how you implement your mesh to ensure angles within elements are larger (smaller angles hurts accuracy) and that you get it sufficiently refined to get the accuracy you wish.

If you want to try and avoid all of this, there are approaches out there called Meshless Methods which basically solve problems based on particle clouds and typically using some nearest neighbor algorithms, radial basis functions, and potentially collocation approaches. Something that might be interesting to look into!