Finite Element Method
Table of Contents
- Differential Equation are continuous systems
- Computers can't solve continuous system. One idea is convert it to a discrete the problem. e.g the derivative of a function is the slope at a single point, however in discrete, the derivative is the slope of line joining two points.
- Another different approach to this is the idea Galerkin had. He had the idea of trial functions. Close enough functions to approximate the actual solutions.
- At that time he used 2 or 3 trial functions, but nowadays, we have computers so, we use 1000s of simple functions. This way the problem of solving the diffrential equation is changed to finding the coefficients with which to multiply the trial functions.
FEM was initially developed for Structural Problems. And in those problems the motion is very small. A bridge deflects ever so slightly. But on the other hand in fluid dynamics, a river flows, and thus it's a completely different problem. Much difficult.
- For Solid Mechanics, we are good.
- For Fluid mechanics, we have much work to do.
- For Gas mechanics, we have much much work to do.
Also Wavelets! People think why don't we use wavelets as trial functions?
1. Wikipedia - PageDump
To solve a problem, the FEM subdivides a large system into smaller, simpler parts that are called finite elements.
The finite element method formulation of a boundary value problem finally results in a system of algebraic equations
he simple equations that model these finite elements are then assembled into a larger system of equations that models the entire problem. The FEM then uses
variational methods from the calculus of variations to approximate a solution by minimizing an associated error function
Accurate representation of complex geometry
- Inclusion of dissimilar material properties
- Easy representation of the total solutionCapture of local effects.
each subdomain represented by a set of element equations to the original problem
To explain the approximation in this process, the Finite element method is commonly introduced as a special case of Galerkin method.
The process, in mathematical language, is to construct an integral of the inner product of the residual and the weight functions and set the integral to zero. In simple terms, it is a procedure that minimizes the error of approximation by fitting trial functions into the PDE. The residual is the error caused by the trial functions, and the weight functions are polynomial approximation functions that project the residual.
The process eliminates all the spatial derivatives from the PDE, thus approximating the PDE locally with
a set of algebraic equations for steady state problems,
a set of ordinary differential equations for transient problems.
A finite element method is characterized by a
variational formulation
, a discretization strategy , one or more solution algorithms , and post-processing procedures .
A discretization strategy is understood to mean a clearly defined set of procedures that cover (a) the creation of finite element meshes, (b) the definition of basis function on reference elements (also called shape functions) and (c) the mapping of reference elements onto the elements of the mesh.
We define a new operator or map ϕ ( u , v ) {\displaystyle φ (u,v)} by using integration by parts
Depending on the author, the word "element" in the "finite element method" refers either to the triangles in the domain, the piecewise linear basis function, or both.
an author interested in curved domains might replace the triangles with curved primitive
s, and so might describe the elements as being curvilinear
The finite element method is not restricted to triangles
but can be defined on quadrilateral subdomains
Higher-order shapes (curvilinear elements) can be defined with polynomial and even non-polynomial shapes (e.g. ellipse or circle)
Mesh adaptivity may utilize various techniques, the most popular are:
- moving nodes (r-adaptivity)
- refining (and unrefined) elements (h-adaptivity)
- changing order of base functions (p-adaptivity)combinations of the above (hp-adaptivity).
primary advantage of this choice of basis is that the inner products
will be zero for almost all j , k {\displaystyle j,k}
As we have discussed before, most of the entries of L {\displaystyle L} and M {\displaystyle M} are zero because the basis functions v k {\displaystyle vk} have small support.
Such matrices are known as sparse matrices, and there are efficient solvers for such problems (much more efficient than actually inverting the matrix.) In addition, L {\displaystyle L} is symmetric and positive definite, so a technique such as the conjugate gradient method is favored.
In general, the finite element method is characterized by the following process. One chooses a grid for Ω {\displaystyle Ω } . In the preceding treatment, the grid consisted of triangles, but one can also use squares or curvilinear polygons. Then, one chooses basis functions. In our discussion, we used piecewise linear basis functions, but it is also common to use piecewise polynomial basis functions.
For second-order elliptic boundary value problems, piecewise polynomial basis function that is merely continuous suffice (i.e., the derivatives are discontinuous.) For higher-order partial differential equations, one must use smoother basis functions