Assembly of Global Matrices and Boundary Conditions In the finite element method (FEM), we consider a physical domain divided into smaller elements. To anal...
Assembly of Global Matrices and Boundary Conditions
In the finite element method (FEM), we consider a physical domain divided into smaller elements. To analyze the behavior of a system, we assemble global matrices and impose boundary conditions on the element level. This allows us to perform analyses in various regions of the domain simultaneously, thereby obtaining a more accurate solution.
Assembly of Global Matrices:
Global matrices are collections of coefficients that describe the behavior of individual elements in the domain. They represent the influence of each element on neighboring elements. For example, in a 2D element, the global matrix would consist of 4 elements, with each element influencing the others.
Assembly of Boundary Conditions:
Boundary conditions specify specific values that the elements on the edge or boundary of the domain should satisfy. These conditions can include continuous values, such as temperature or pressure, or they can be more complex, such as prescribed displacements or velocities.
How it is Done:
To assemble the global matrices and impose boundary conditions, we use techniques such as:
Finite difference methods: For 1D and 2D elements, we assemble global matrices using finite differences.
Fourier series: For periodic boundary conditions, we use Fourier series expansions.
Dirichlet boundary conditions: For boundary conditions involving fixed values, we use Dirichlet functions.
Benefits of Global Assembly:
Global assembly provides several benefits, including:
Computational efficiency: It allows us to analyze the entire domain simultaneously, reducing the number of element evaluations.
Accuracy: By considering the global behavior of the system, it provides more accurate solutions.
Versatility: It can be applied to various physical and mathematical problems.
Example:
Consider a 1D heat equation with periodic boundary conditions. We can assemble the global matrix using finite differences and impose the Dirichlet boundary condition (fixed temperature). The solution would provide the temperature distribution in the domain over time