Calculating Electromagnetic Field Problems Using Finite Element Algorithm

The finite element algorithm is a widely used numerical computation method in engineering and scientific fields, particularly suitable for solving electromagnetic field problems. By discretizing complex continuous systems into a finite number of simple elements, the finite element algorithm efficiently handles complex geometries and boundary conditions. In code implementation, this typically involves mesh generation using libraries like Gmsh or MATLAB's PDE toolbox to create triangular or quadrilateral elements.

Boundary condition processing is crucial in electromagnetic field computations. The first-type boundary condition (Dirichlet boundary condition) is commonly used to specify exact values of electric or magnetic potential on boundaries. For instance, in electrostatic problems, conductor surface potentials can be set as fixed values, which in programming terms translates to assigning constant values to specific node indices in the solution vector. The second-type boundary condition (Neumann boundary condition) describes normal derivatives on boundaries, such as magnetic field strength distributions, which require special treatment in stiffness matrix assembly through surface integral calculations.

The core concept of the finite element method involves dividing the solution domain into small elements (like triangles or quadrilaterals) and using interpolation functions within each element to approximate the actual field distribution. Key implementation steps include: defining shape functions for each element type, calculating local stiffness matrices through Gaussian quadrature integration, and assembling global matrices using connectivity information. The final global linear equation system Ax=b is solved using numerical solvers like conjugate gradient or direct LU decomposition methods.

This method applies not only to static fields (like electrostatic and magnetostatic fields) but also extends to time-harmonic and transient field computations. By adjusting element shapes and densities through adaptive mesh refinement algorithms, the method flexibly meets different accuracy requirements, making it a powerful tool in electromagnetic field analysis. Modern implementations often incorporate parallel computing techniques for large-scale problems and may include post-processing modules for field visualization and result validation.