Getting Started with Robust Finite Element Method and Solvers

Key Takeaways

• The finite elements on a surface are interconnected through nodes. The finite elements take any arbitrary shape to ensure that the problem region is completely covered until the boundary.

• FEM represents the problem region through a set of equations that correspond to each finite element. FEM solvers find solutions to these matrix equations either by using direct solvers or indirect solvers.

• The indirect solvers are also called iterative solvers, as they depend on numerical iterations for finding a solution set.  It requires constraints for the convergence of iterations. As the number of iterations increases, the accuracy of the solution also increases. 

In FEM, the problem surface is decomposed into finite elements called mesh.

Most engineering problems can be resolved by solving algebraic, ordinary, or partial differential equations. In electrical engineering, math function formulations are so intense that finding solutions by considering all the engineering and real-world constraints or parameters is a tedious job. 

Robust Finite Element Method (FEM) captured the attention of electrical engineers due to its ease and reliability in solving complex math problems by breaking down a complicated  system into finite elements. The robust FEM application resolves parameter dependent problems with solutions that uniformly converge for all parameters at any value of time, t. Initially, the application of FEM was limited to waveguides within the mechanical domain, but it picked up in popularity quickly,  and now finds applications in fields such as stress analysis, thermal analysis, electromagnetics, modulation, sensor analysis, etc. 

Where Can We Apply Robust FEM?

FEM is a numerical technique used to solve complex math functions related to engineering. Often, confusion arises regarding which numerical method should be used for solving a particular problem. 

FEM divides the problem surface into small regions called mesh elements. These finite elements cover the entire region within the boundary, and each of these elements are connected with ‘nodes’.  This process of decomposing the region into finite elements is called discretization. The density of the elements is varied according to the application. 

The function formulation is based on the fundamentals of energy principles, or applied mathematics, onto the finite elements. The parameter-driven functions formulated within the elements are approximated using the interpolation function or shape function. The element-wise equations are assembled together to represent the entire 1-D, 2-D, or 3-D problem region in a matrix form. FEM solvers are computational algorithms that are used to solve the matrix equations. After all these steps, the problem is resolved with accurate mathematical solutions. A sign of robust FEM is when the accuracy of the solution remains uncompromised as the system constraints or parameters move to its limits.   

For obtaining solutions using FEM, the system should satisfy the following requirements:

1. The mathematical function formulated should address all the continuity and boundary conditions.

2. The function should be convergent, otherwise the accuracy of the results will be compromised. When the function converges uniformly for all values of the parameter, it showcases a robust FEM application.

3. The functions formulated should be computationally tractable. 

If the system is satisfying the above requirements, you can go ahead with FEM.

Selection of FEM Solvers

The accuracy of the FEM solution is determined by the discretization process, interpolation function, and the FEM solvers. The selection of FEM solvers depends on the size of the FEM model, type of analysis, and the accuracy of the solution expected. FEM solvers are used for solving the set of matrix equations that represent any FEM model. 

The basic matrix equations of the FEM model are:

where Kis the stiffness matrix, fis the force vectors, and uis the unknown matrix.

There exists both direct solvers and iterative solvers for finding the unknown matrix. The method of inverting the stiffness matrix K.i.e K-1for determining the unknowns is the procedure followed in the direct solvers. However, the large dimension, presence of zeroes in the stiffness matrix, and memory requirements for taking Kat once limit the application of the direct solvers. The Gauss elimination method and the LU or Cholesky factorization method, are examples of direct solvers.

Iterative solvers initialize uwith approximate values u0and substitute Au0in equation ❲1❳ where Ais the sparse matrix. The general rule to start the iterative method  is such that

Then equation ❲1❳ reduces to 

The iterative process continues with equation ❲4❳ until equation ❲5❳ is satisfied

Substitute n=0 in equation ❲4❳ and you will get Au1. Increment the n value until equation ❲5❳ is satisfied. 

The indirect solvers require less memory space compared to the direct solvers. The number of iterations plays a key role in bringing down the convergence tolerance. The accuracy of iterative solvers depends on the convergence tolerance. As the iteration increases, the tolerance band decreases. Usually, constraints based on the physical conditions are set to converge the iterations. A good initial guess can also improve the convergence rate of the FEM solver. The Gauss-seidel algorithm is an example of the iterative method. 

Application of FEM in Electrical Engineering

The versatility and flexibility of the FEM numerical method paves the way for solving the simultaneous algebraic equations or ordinary and partial differential equations in electrical engineering. The direct solvers and iterative solvers are equally suited for solving time-dependent, linear, and non-linear systems. However, the need for large memory limits the direct solvers, and iterative solvers are chosen in most of the analyses. Some of the analysis in electrical engineering where FEM can be applied are:

1. Magnetostatic analysis- The magnetic field related quantities such as flux, flux density, field intensity, and inductances are used to analyze the design of electric motors, generators, and permanent magnet devices.

2. Electromagnetic analysis- The two sub-classification in this analysis are: 

i❳ Transient electromagnetic analysis-The time functions of magnetic quantities such as flux density, field intensity, and current densities are analyzed for transient and steady-state analysis of electromagnetic devices.

ii❳ Time-harmonic electromagnetic analysis- The magnetic fields and electric fields co-exist in systems carrying alternating currents or alternating flux. The quantities such as currents, voltage, the heat generated, flux density, and impedance are used to formulate the FEM functions in this type of analysis. 

3. Electrostatic analysis- This is the type of FEM analysis applied to fuses and power transmission lines. 

4. Thermal analysis- The temperature distribution, heat transfer, and heat losses are the general quantities considered in thermal analysis. Heat sink design is the major application domain of this type of analysis.

Your aim might be  to conduct a piecewise analysis of an electrical system that can be mathematically represented by a set of ordinary or partial differential equations. The solutions for these engineering problems obtained by applying the robust FEM technique are reliable, accurate, and align well with real-world physical conditions. 

Working within FEM techniques will encourage and improve your capacities for resolving electromagnetics, thermal, and static necessities in your designs. The main impedance is: when?