2D Meshless Analysis of SPM Machines Without Mesh Reconstruction
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Solution Overview
Problem
Traditional mesh-based numerical methods for analyzing surface-mounted permanent magnet (SPM) machines face challenges with motion and deformation, leading to inaccurate solutions due to mesh deformation, especially in complex areas, necessitating mesh reconstruction which is difficult to perform effectively.
Innovation Solution
A 2D meshless method is employed, where nodes are arranged in each solution region, and a support region is formed around each node to approximate derivative values using Taylor expansion and weighted least squares, converting partial differential equations into algebraic equations, allowing for the calculation of vector potential and electromagnetic parameters without the need for mesh generation.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Reliability
If mesh-based numerical methods (FDM, FEM, BEM) are used for electromagnetic analysis, then the method is sophisticated and can solve many engineering problems, but the mesh deforms severely during motion and deformation, significantly affecting solution accuracy and requiring complex mesh reconstruction
Solution Approach 1:
The patent extracts the essential discretization function from mesh-based methods and separates it from the mesh structure itself. By using only nodes without connecting elements, the method removes the mesh deformation problem while retaining the ability to discretize the domain for numerical analysis.
Solution Approach 2:
The patent introduces a weight function as an intermediary to establish relationships between nodes. This weight function serves as a mediator that allows calculation of field values at any point based on nodal values, replacing the need for explicit mesh connections while maintaining computational accuracy.
2Measurement precision
If mesh reconstruction is performed in complex areas to maintain accuracy, then solution accuracy is improved, but the reconstruction process becomes relatively difficult and time-consuming
Solution Approach 1:
The patent performs preliminary discretization by placing nodes directly in the solution domain without requiring mesh generation. This preliminary action of node placement eliminates the need for subsequent mesh reconstruction, saving time while maintaining accuracy through adaptive node distribution.
3Ease of manufacture
If traditional mesh-based methods are used, then the analysis can handle complex geometries, but the preprocessing work including mesh generation is complex and time-consuming
Solution Approach 1:
The patent extracts the discretization functionality from the mesh structure, using only nodes without elements. This extraction simplifies preprocessing by eliminating mesh generation while maintaining the ability to handle complex geometries through flexible node placement.
Solution Approach 2:
The patent changes the fundamental parameter from mesh density to node density. This parameter change allows direct control over computational resolution without the constraints of mesh topology, improving both preprocessing simplicity and computational efficiency.
Applied Scientific Principles
This section explains which scientific principles are used to turn an abstract innovation direction into a practical engineering solution.
Function Achieved in This Case
This approach simplifies preprocessing, improves computational efficiency, and allows for flexible node density adjustment, enhancing accuracy and speed by eliminating the need for mesh generation and handling complex geometries effectively.
Implementation Method 1
Construct a residual function based on Taylor expansion and weighted least squares method, and approximate the derivate values of each node as a linear combination of the function values of each node in the support region
Implementation Method 2
Construct a residual function based on Taylor expansion and weighted least squares method
Implementation Method 3
Convert the partial differential equation satisfied by each node in the solution region into algebraic equations
Implementation Method 4
Based on the vector potential of each node solved in Step 6, the flux density distribution and flux lines can be obtained; According to the electromagnetic calculation formulas, the electromagnetic parameters such as electromotive force and torque can be obtained
Data Source
AI summary
An analysis of a surface mounted permanent magnet (SPM) machine by a 2D meshless method is provided. The 2D meshless method includes steps: discrete nodes are arranged in the region to be solved; based on Taylor expansion and weighted least squares principle, the derivative value of vector potential can be approximated as a linear combination of vector potential values of each node in the support region; partial differential equations are converted into algebraic equations; by solving the algebraic equations, the vector potential of each node can be calculated, and then the distribution of the flux line and the flux density can be obtained. According to the electromagnetic calculation constraints of the machine, parameters such as the back electromotive force and electromagnetic torque of the machine can be obtained.