Classification of the methods (Классификация методов)

• Finite difference – метод сеток (метод конечных разностей)

• Finite elements – метод конечных элементов (используется в Quickfield)

• Finite volumes – метод конечных объемов

• Method of moments (method of spatial equations) – метод моментов

• Boundary equations – метод граничных уравнений

• Integro-differential – интегрально-дифференциальный метод

Hybrid method combines several methods (Boundary equations+ Finite elements)

Method of moments (Метод моментов)

This method is quite simple but is not very accurate. It is usually applied to solving magnetic problems. The basis of this method is a system of a several basic equations.

H_m – magnetic field induced by the magnetized object

H_c – magnetic field induced by the current (may be calculated independently on solving the general problem by Biot-Savart Law or another)

Here and then we shall assume, that we work inside isotropic medium.

The general system may consist of volume which if field with magnetize material (область M). may be ferromagnetic (H=10²–10³) or a material with smaller magnetic permeability. Another part of the volume is the field with the primary sources (область J) (usually currents, but also it may be permanent magnet)

These two volumes are surrounded by air or vacuum. In principle the domain may be infinitely big (tend to infinity). Method of moments hasn’t boundaries, so in this case it is more accurate, than other methods.

The first relation is some kind of the Coulomb Law for the magnetic field. So, we use magnetization instead of charges.

We have changed a variable. Now is not field intensity the unknown value, but magnetization.

Discretization of the problem domain (Дискретизация проблемной области)

We should split the brick into basic elements (tetrahedrons, prisms or parallelepipeds)

Algebraic equation system (Алгебраическая система уравнений)

• The magnetization of each element is considered constant (the simplest approach). More complex approximation schemes are not used.

• To form a system of equations the method is collocations is used (we must choose some central point inside the element and the magnetic field which induced by all parts of the magnetic system is calculated just for this central point. And then, it’s assumed, that everywhere inside the element the magnetic field intensity has the same value)

Finite element method (Метод конечных элементов)

Main steps:

1. Problem formulation – problem domain, equation, boundary conditions, material properties.

2. Discretization of the problem domain.

3. Approximation of the unknown function.

4. Approximation of the solved equation and the boundary conditions.

5. Solution of the algebraic equation system (generally – nonlinear).

6. Post-processing.

Discretization (Дискретизация)

The element type choice

Examples of the mesh (triangles and quadrangles)

Linear approximation

The number of free parameters is the same!

In principle we can express coefficients a, b, c in terms of the nodal potentials U₁, U₂, U₃

In linear approximation the vertices are the same with the nodal.