Medical Physics & Science Applications

Transcription

1 Power Conversion & Electromechanical Devices Medical Physics & Science Applications Transportation Power Systems 1-5: Introduction to the Finite Element Method Introduction Finite Element Method is used to determine solution to governing equations governing g equations are partial differential equations, describing the electric and magnetic fields Mesh volume subdivided into elements elements fit together to form a mesh solution potential is determined at each node of mesh (nodal solvers) solution potential is determined at each edge of mesh (edge solvers) 2

2 Model Reduction Physical Model specification - geometry, materials etc Electromagnetic Model reduction to active parts of the model Mathematical Model governing equations, boundary conditions, symmetry Numerical Model discretization (Finite Element Method) Algebraic model solve matrix equation to obtain potentials 3 Physical Model Geometry Materials Sources currents, charges, voltages Conducting strip line above dielectric substrate 4

3 Electrostatic Model 2-D approximation to strip line Only relevant parts are included Ignore: 3D effects conductors all charge has migrated to the surface irrelevant physical details (e.g. rounded corners) 5 Mathematical Model Utilise symmetry Include boundary conditions set V for unique solution This is an Electrostatic Model 6

4 Mathematical Model (2) Boundary Conditions are essential Options available are: Set potential (V) to value V = constant implies electric flux is normal to surface equivalent to Normal Electric Set potential derivative to value V = constant implies flux is tangential to surface n equivalent to Tangential Electric if constant = 0 If no condition applied, Tangential Electric is assumed in this model 7 Solution to Strip Line Potential solution is shown: 8

5 Numerical Model (Interpolation) Approximate solution by linear polynomials values at nodes are assumed to be correct the greatest error is where the slope changes the most 9 Numerical Model (Interpolation) 2 Improve interpolation by adding more elements 10

6 Numerical Model (Interpolation) 3 Improve interpolation by using quadratic interpolation 11 Numerical Model (Interpolation) 4 With linear interpolation, slope (field) is piecewise constant nodal averaging improves the field pattern value at node is average of field in adjacent elements field is then interpolated with linear polynomial 12

7 Numerical Model (2D Interpolation) Similarly in 2D, with bi-linear interpolation 13 Numerical Model (Modeller-Input) Geometric Modeller for easy model input finite element meshes are created automatically tetrahedral or mosaic meshes are created mesh size can be controlled complex models can easily be created 14

8 Tetrahedral mesh from Modeller 15 Algebraic Model This occurs within the analysis module there is no user intervention at this stage The following steps take place: mesh generation calculate matrix coefficients solve the matrix equation (giving potentials at nodes/edges) calculate fields by differentiating potentials these values are then passed to the Post-Processor 16

9 3-d modelling Considerations Physical Model only include: relevant geometric features material properties excitation conditions 17 3-d modelling Considerations (2) Magnetostatic Model: Non-magnetic and non-conducting materials are treated as air boundaries are sufficiently distant from the geometry source coils are not included in the mesh 18

10 3-d modelling Considerations (3) Mathematical Model: In regions with source currents, the field is split into H m and H s Source conductors do not form part of finite element model Regions containing source conductors use Reduced Magnetic Scalar Potential 19 3-d modelling Considerations (4) Mathematical Model (continued): Only 1/4 of the device is modelled Choice of potential in a region depends on that regions properties Magnetic scalar potential must be set at one point at least (for unique solution) If no potential value is set, software automatically sets one point to zero 20

11 Summary of Potentials TOSCA No Source Currents I s =0 μ 1 (iron regions) ψ Total (Scalar) Biot-Savart Conductors (Source Currents) exist I s 0 μ=1 (free space) φ Reduced (Scalar) 21 Summary of Potentials ELEKTRA, CARMEN, DEMAG Biot-Savart Conductors (Source Currents) exist I s 0 μ=1 (free space) Materials with conductivity 0 μ 1 (iron regions) Meshed Conductor in Circuit & Magnetic Regions with conductivity = 0 μ 1 (iron regions) A m A,V A Reduced (Vector) Total (Vector) + Electric Scalar Total (Vector) 22

12 Boundary Conditions Magnetic Scalar Potential Surfaces Constant potential implies flux is normal to surface (Normal Magnetic) φ = constant Zero derivative implies flux is tangential to surface (Tangential Magnetic) φ = 0 n The default value (if none is specified) is Tangential Magnetic 23 Boundary Conditions (2) Magnetic Vector Potential Surfaces: Current normal to face: use Normal Electric Current parallel to face: use Tangential Electric 24

13 Boundary Conditions Normal Electric / Tangential Magnetic Vector Potential A n = 0; V = constant Regions Magnetic Scalar Potential Regions φ = 0 n (natural condition) Tangential Electric / Normal Magnetic Vector Potential A.n =0; = 0 ; V = 0 Regions n (natural condition for V) Magnetic Scalar φ = constant Potential Regions 25