Finite Element Methods (FEM) are powerful numerical techniques used in engineering and scientific applications to solve Partial Differential Equations (PDEs). These methods have been widely utilized in the field of electromagnetics for solving a range of electromagnetic problems, including Maxwell’s equations.
The main idea behind FEM is to divide the solution domain into small elements, such as triangular or tetrahedral shapes, and then solve a set of equations at the nodes of these elements to determine the solution across the entire domain. The solution at each node is approximated using interpolation functions, and the solution for the entire domain is obtained by summing the solutions at all nodes.
One of the key benefits of using FEM in electromagnetics is its ability to handle complex geometries and boundary conditions. The solution domain can be discretized into elements of any shape, size, or position, and boundary conditions can be easily imposed at the nodes of the elements. This feature makes FEM an ideal tool for solving electromagnetic problems that involve complex or irregular geometries.
FEM can handle both static and dynamic electromagnetic problems. For static problems, the time-invariant Maxwell’s equations are solved, while for dynamic problems, the time-dependent Maxwell’s equations are solved using time-stepping methods. This versatility makes FEM a suitable tool for solving a wide range of electromagnetic problems, including those that involve coupled electromagnetic-thermal interactions, nonlinear material properties, and time-varying magnetic permeability.
FEM can also be used to solve electromagnetic problems across a wide range of frequency regimes, from low to high frequencies. For low-frequency problems, a frequency-domain FEM method can be used, while for high-frequency problems, a time-domain FEM method can be utilized. This versatility makes FEM a suitable tool for solving a wide range of electromagnetic problems, regardless of the frequency regime involved.
In terms of accuracy, FEM can provide high-resolution solutions for electromagnetic problems. The accuracy can be improved by refining the mesh, increasing the order of the interpolation functions, or using adaptive mesh refinement techniques. These features make FEM an ideal tool for solving electromagnetic problems that require high accuracy solutions.
Despite its many benefits, FEM does have some limitations. One of the main limitations is the computation time required to solve the problem, as the solution requires the solution of a large system of equations. Additionally, the solution may be sensitive to the choice of mesh, interpolation functions, and boundary conditions, making it important to choose these carefully.
In conclusion, FEM is a widely used and powerful numerical method in the field of electromagnetics. Its ability to handle complex geometries and boundary conditions, handle multiple physics, and provide high-resolution solutions make it a valuable tool for solving a wide range of electromagnetic problems. Despite its limitations, FEM continues to be a popular choice for solving electromagnetic problems due to its accuracy and versatility. The FEM method has proven to be a reliable and effective tool for solving complex electromagnetic problems and has contributed significantly to the advancement of the field of electromagnetics.
FEM steps
To solve the electromagnetic field using the finite element method, these steps are required:
Define the geometry of the transformer, including the locations and shapes of the dielectrics.
Discretize the geometry into a finite element mesh. This involves dividing the domain into smaller elements, typically triangles or quadrilaterals, and defining the nodal points at the vertices of these elements.
Define the finite element shape functions for the elements. These are polynomial functions that are used to interpolate the field quantities (e.g. electric potential) at the nodal points from the element-averaged values.
Define the weak form of the equation. This involves expressing the differential equation in terms of integral equations over the element domains, using the shape functions to interpolate the field quantities and their gradients.
Assemble the global stiffness matrix and load vector from the elemental contributions. This involves evaluating the integrals in the weak form and summing the contributions from each element.
Apply any necessary boundary conditions to the global stiffness matrix and load vector.
Solve the linear system of equations represented by the global stiffness matrix and load vector to obtain the nodal field values.
Interpolate the element-averaged field values from the nodal values using the shape functions
FEM Packages
Elmer is an open-source finite element method (FEM) software that is well-known for numerical simulations and modeling of a wide range of physics problems. It allows for multi-physics simulations, meaning it can handle multiple physical processes at once, such as heat transfer, fluid dynamics, and electromagnetics. The software is highly customizable, and users can create their models by writing subroutines.
Elmer was developed in the mid-90s at Helsinki University of Technology and has since gained popularity among engineers, scientists, and researchers worldwide. The latest version, Elmer 8.5, was released in 2021.
Elmer has a user-friendly interface, making it easy to set up and run simulations. The software includes a graphical user interface (GUI) for model setup, data input, and results visualization, as well as tools for mesh generation, post-processing, and data analysis.
The software supports a broad range of finite element methods, including linear and non-linear static and dynamic analysis, and eigenvalue problems. It also includes multi-grid techniques for efficient solution of large-scale problems and solvers for different types of problems like linear and non-linear systems of equations, and partial differential equations.
Elmer is versatile and can be used for various applications, including structural mechanics, fluid dynamics, heat transfer, electromagnetics, and more. It can be applied to fields such as engineering, physics, biology, and others. The software is also highly extensible, allowing users to write their own subroutines to extend its capabilities and incorporate custom models, boundary conditions, and features. There is a large community of users and developers who share models and extensions on the Elmer website and forums.
Elmer is optimized for high performance computing, running on a wide range of platforms, including desktop computers, laptops, and high-performance computing clusters. The software is designed for parallel processing, allowing users to speed up their simulations using multi-core processors and parallel processing.
However, Elmer also has limitations. It is a complex software and requires a good understanding of finite element method and numerical simulation to effectively use it. Furthermore, Elmer requires a significant amount of computational resources, and users need access to a high-performance computer to run large-scale simulations.
FreeFEM++ is an open-source software tool that solves partial differential equations (PDEs) in two and three dimensions. It is a versatile solution for complex PDEs in various fields, including mathematics, engineering, physics, and biology. The software uses finite element methods to approximate the solution to PDEs.
FreeFEM++ allows users to define the geometry, mesh, and boundary conditions of the problem through a high-level scripting language. It supports triangular and quadrilateral meshes and includes adaptive meshing options for refining the mesh where accuracy is needed. The software comes with a comprehensive set of built-in functions for solving a wide range of PDEs, including linear and nonlinear systems, and it supports user-defined functions.
FreeFEM++ has visualization and exploration tools, such as shading, lighting, and texture mapping options, to interact with the solution. The software has a graphical user interface and supports multiple file formats for importing and exporting data. It is highly customizable, allowing users to write their own scripts and plugins and to access the source code to modify it to fit specific needs.
FreeFEM++ is designed to be efficient and fast, even for large and complex models, and it supports parallel processing. The software is user-friendly, with a comprehensive set of tutorials and documentation, making it easy to get started and become proficient in using it.
deal.II is a flexible and powerful open-source C++ software library designed to solve partial differential equations (PDEs) with finite element methods. It can be applied in various fields, such as engineering, physics, and mathematics. The library provides tools for defining the geometry, mesh, and boundary conditions of the problem and supports both triangular and quadrilateral meshes. It also offers adaptive meshing, enabling refinement of the mesh in areas where higher accuracy is needed.
deal.II comes with a comprehensive set of built-in functions for solving various types of PDEs, including linear and nonlinear systems, and allows users to create their own solvers with user-defined functions. The high-level scripting language makes it easy for users to write and modify code. The library includes visualization tools and a graphical user interface that allow users to interact with the solution and manipulate its properties, with support for multiple file formats for import and export of data.
deal.II is highly customizable and users can write their own scripts and plugins to enhance its functionality. It features a comprehensive API and supports scripting in C++, which enables automation of complex processing tasks. As an open-source software, users can access the source code and modify it to meet their specific needs.
deal.II is fast and efficient, making it ideal for use with large and complex models. It also supports parallel processing, allowing for quick generation and processing of large solutions. The library is user-friendly, with a comprehensive set of tutorials and documentation to help users get started and become proficient in using it. deal.II is widely used and well-documented, with a large and active community of users contributing to its development and sharing their knowledge and experiences.
MFEM is a flexible, efficient and open-source C++ library that solves partial differential equations (PDEs) using finite element methods. It caters to various fields such as mathematics, physics, engineering, and biology, in addressing complex PDEs.
The library provides tools to define the problem’s geometry, mesh, and boundary conditions, supporting both triangular and quadrilateral meshes. MFEM also offers adaptive meshing, enabling users to refine the mesh for greater accuracy in specific areas. Its parallel processing feature allows for fast processing of large solutions.
MFEM provides built-in functions for solving different PDEs, including linear and nonlinear systems, and allows for user-defined functions to create custom solvers. The library includes a high-level scripting language for defining problems, making code modification and writing easy.
The library has tools to visualize and examine the solution, including shading, lighting, and texture mapping options. MFEM also supports multiple file formats for data import and export, making integration with other software seamless.
MFEM is highly customizable and offers a comprehensive API, making it possible to automate complex tasks through scripting in C++. As an open-source software, users have access to the source code and can make modifications to suit their needs.
Designed to be fast and efficient, MFEM is suitable for use with large and complex models. The library has a user-friendly interface with tutorials and documentation to make it easy for users to start and become proficient. MFEM is well-documented and widely used, with a large and active community that contributes to its development and shares their knowledge.
MOFEM is a finite element software solution developed by researchers at the University of Strathclyde in Scotland. It is an open-source C++ library designed for solving complex partial differential equations (PDEs) using the finite element method. The software is applicable in a variety of fields including mathematics, engineering, physics, and biology.
MOFEM offers a range of tools for defining the problem’s geometry, mesh, and boundary conditions, and it supports both triangular and quadrilateral meshes. It includes options for adaptive meshing, allowing refinement in specific areas where a higher degree of accuracy is needed. The software also supports parallel processing, making it capable of solving large problems quickly.
MOFEM has built-in functions for solving various PDEs, including linear and nonlinear systems, and it allows users to create custom solvers by defining their own functions. It includes a high-level scripting language for defining problems, making it simple to write and modify code.
The software provides tools for visualizing and analyzing solutions, including shading, lighting, and texture mapping options. It supports multiple file formats for importing and exporting data and can be integrated with other software tools.
MOFEM is highly customizable, and users can write their own scripts and plugins to extend its features. It includes a comprehensive API and supports scripting in C++, making it possible to automate complex processing tasks. As an open-source solution, the source code is available for modification to meet specific needs.
MOFEM is both fast and efficient, making it suitable for use with large and complex models. The software comes with comprehensive tutorials and documentation, making it user-friendly and easy for users to get started and become proficient. MOFEM has a large and active user community who contribute to its development and share their experiences and knowledge.
FEM meshers
GMSH is a flexible and powerful open-source 3D mesh generator. It is used for computational simulations and analysis in a range of engineering, scientific, and industrial applications. The software enables the creation of discretized representations of 3D objects known as meshes for use in simulations.
GMSH offers tools for defining the geometry of objects, including a graphical interface for creating and manipulating shapes and curves, as well as support for various file formats for importing and exporting geometry. The software employs a combination of algorithms, such as advancing front and Delaunay triangulation, to create high-quality triangular and tetrahedral meshes. GMSH also supports adaptive meshing, allowing for refinement in specific areas where higher accuracy is required.
For post-processing and visualization, GMSH offers shading, lighting, and texture mapping options. It features a graphical user interface that enables interaction with the mesh and manipulation of its properties, and supports multiple file formats for importing and exporting meshes. The software is highly customizable, with a comprehensive API and support for scripting in Python for automating complex processing tasks.
GMSH is designed for efficiency, making it suitable for handling large and complex models. It supports parallel processing, allowing for fast generation and processing of large meshes. The software is also user-friendly, with a comprehensive set of tutorials and documentation for ease of use.
BAMG (Bidimensional Anisotrope Mesh Generator) is a cutting-edge software tool that is designed to generate two-dimensional meshes for use in a wide range of scientific and engineering applications. BAMG creates triangular or quadrilateral meshes that are ideal for finite element or finite volume methods. These types of methods are commonly used in fields such as fluid dynamics, elasticity, and electromagnetic simulations.
One of the key features of BAMG is its ability to generate anisotropic refinements. This means that the mesh elements can be stretched or shrunk in different directions to achieve a desired level of accuracy in specific regions of the mesh. This capability is particularly useful for problems that have complex geometries and varying material properties, as it allows for a more customized mesh refinement process.
The mesh generation process in BAMG starts with a coarse initial mesh that covers the domain of interest. This mesh is then refined using a series of iterative algorithms that are designed to improve the mesh quality and accuracy. BAMG uses a combination of edge splitting, vertex insertion, and local mesh smoothing algorithms to generate high-quality meshes. The mesh quality is evaluated using a set of mesh quality metrics, such as aspect ratio, skew angle, and element size, and the mesh refinement algorithms are designed to improve these metrics.
Another important feature of BAMG is its ability to handle complex geometries and material properties. BAMG can be used to generate meshes for domains with arbitrary shapes, such as those defined by curved boundaries or multiple disjoint subdomains. BAMG also supports the use of curved edges and curved boundaries, which is crucial for accurately resolving complex geometries.
BAMG is also capable of handling heterogeneous material properties, such as varying conductivity or permeability, by allowing for anisotropic refinements in specific regions of the mesh. This is particularly useful for electromagnetic simulations, where the material properties can have a significant impact on the simulation results. BAMG can generate anisotropic meshes that accurately resolve these material properties, resulting in more accurate simulation results.
Triangle Mesher is a versatile software tool that specializes in generating triangular meshes, a discretized representation of two-dimensional shapes. This two-dimensional Delaunay triangulation software produces high-quality triangular meshes that are ideal for finite element analysis, computer graphics, and geographic information systems (GIS).
The main objective of Triangle Mesher is to create triangular meshes that have a high degree of conformity, meaning that the triangles are as equilateral as possible and have a small aspect ratio. This results in more accurate numerical solutions for finite element analysis and improved display quality for computer graphics. The software employs various algorithms to achieve this, such as Delaunay triangulation, constrained Delaunay triangulation, and conforming Delaunay triangulation.
Triangle Mesher can accept a wide range of input file formats, including plain text files, AutoCAD files, and others. It also offers features that allow users to control the mesh generation process, such as setting the maximum triangle area, minimum angle, and maximum aspect ratio. The software can also incorporate elements like holes, Steiner points, and constraints to generate meshes that meet specific requirements.
In addition to mesh generation, Triangle Mesher provides tools for post-processing the generated meshes, including refining the mesh, checking mesh quality, and visualization. The refined mesh can then be used for further analysis or computer graphics rendering.
Triangle Mesher is highly customizable and provides a C language library that can be utilized to create custom mesh generators, post-processors, or visualization tools. As it is open-source software, users can access the source code and make modifications to meet their specific needs.
Triangle Mesher is widely utilized by professionals in various fields, including engineers, scientists, and researchers for applications such as finite element analysis, computer graphics, and geographic information systems. The software is also used in industries like civil engineering, mechanical engineering, and geology to generate triangular meshes for simulations and analysis. It is included in many FEM applications.
Matlab’s distmesh is a two-dimensional mesh generator that creates triangular meshes fit for finite element or finite volume methods. It is a user-friendly tool designed to help explore the behavior of partial differential equations (PDEs) and other problems with simple geometries and uniform mesh refinements.
The process of generating meshes with distmesh starts with a coarse initial mesh that covers the desired area. This mesh is then refined using iterative algorithms that work to improve its quality and accuracy. Distmesh uses a combination of edge splitting, vertex insertion, and local mesh smoothing algorithms to generate high-quality meshes, which are evaluated using mesh quality metrics such as aspect ratio, skew angle, and element size. The refinement algorithms aim to improve these metrics and result in an optimal mesh.
One of the advantages of using distmesh is its simplicity and ease of use. The software is specifically designed for exploring PDEs and other problems with simple geometries, and it is easy to customize. Users can specify parameters such as the number of elements, element size, and boundary conditions.
Additionally, distmesh is able to handle arbitrary shapes and curved boundaries. This makes it suitable for generating meshes for domains with complex shapes, such as those defined by curved edges and boundaries.
Distmesh has a wide range of applications, including fluid dynamics, elasticity, and electromagnetic simulations, as well as solving PDEs with complex geometries and material properties. It is also highly customizable and can be easily integrated into other simulation software.
However, distmesh has some limitations. One is the lack of support for anisotropic refinements, meaning the mesh elements cannot be stretched or shrunk in different directions to achieve desired accuracy in specific regions. This can limit the accuracy of simulation results in problems with complex geometries and varying material properties.
Another limitation is the limited mesh quality control capabilities. While distmesh can generate high-quality meshes, it does not have advanced mesh quality control algorithms like other mesh generators, which may result in lower quality meshes in complex problems.
OneMesh is a cloud-based software designed for generating 3D meshes, a discretized representation of an object. It offers an easy way to produce high-quality meshes for simulations, analysis, and visualization. The software utilizes a combination of algorithms including Delaunay triangulation, advancing front and primal-dual optimization to produce uniform and conformal meshes with minimal elements, improved accuracy, and faster computation.
OneMesh has a user-friendly interface and supports multiple input file formats such as STL and OBJ. Users can control mesh parameters such as size, element quality and more to get the desired mesh quality for their simulation. The software also provides mesh refinement capabilities through mesh sizing functions specified by the user and a graphical user interface for mesh visualization and editing.
OneMesh supports parallel computing, making it suitable for large datasets and models. It is highly scalable and flexible to meet the needs of various industries and applications, with constant updates and new features added regularly. Additionally, the software is easy to use and accessible with comprehensive tutorials and documentation, making mesh generation accessible to non-technical users.
In summary, OneMesh is a cloud-based, user-friendly software solution for generating high-quality 3D meshes for simulations, analysis, and visualization. With its advanced features, support for parallel computing and ease of use, OneMesh is a valuable tool for professionals in different industries.
Netgen is a well known, open-source mesh generator that can handle both 2D and 3D geometries. The purpose of Netgen is to create high-quality and efficient meshes for use in finite element and finite volume simulations. This software is used in scientific computing and engineering, and it has been applied in areas such as fluid dynamics, solid mechanics, and electromagnetics.
Netgen is based on the Delaunay triangulation algorithm, which creates meshes by connecting a set of points in a way that leads to triangles with maximum circumradius. This results in well-formed meshes with a minimum number of triangles, and thus, faster and more efficient simulations.
Netgen also offers advanced mesh refinement features, allowing for fine meshes to be created in areas where a higher level of accuracy is required. This is made possible through mesh sizing functions, which can be set by the user. Additionally, Netgen can handle curved boundaries, making it a suitable option for complex geometry mesh generation.
Netgen is customizable, with users able to specify element numbers, size, and boundary conditions, among other things. It also has a graphical user interface, making it easy to use, even for those without much programming experience. The software includes advanced visualization features, enabling users to view the mesh and its properties, such as element size and shape quality, to verify its quality and ensure it meets simulation requirements.
Netgen’s versatility is one of its strengths, as it can generate meshes for a wide range of geometries, including curved boundaries and multiple subdomains. The software also supports various element shapes, including triangles and quadrilaterals, making it suitable for various applications.
However, Netgen has limited support for parallel computing, which can limit its usefulness for large-scale simulations.
In conclusion, Netgen is a customizable and versatile mesh generator for 2D and 3D geometries. Its advanced mesh refinement features and support for curved boundaries make it a great choice for creating high-quality meshes for finite element and finite volume simulations
Hermes Library is an open-source numerical library designed for the simulation of complex physical processes in the fields of electromagnetics and heat transfer. Developed at the Czech Technical University in Prague, it is widely used by researchers and engineers for the solution of partial differential equations (PDEs). The library offers a collection of high-performance solvers that utilize advanced numerical methods such as the finite element method, finite volume method, and boundary element method.
Hermes Library has been crafted to be flexible and user-friendly, allowing users to easily modify and extend the solvers to their specific requirements. Written in C++ and with a comprehensive API, it is simple for users to integrate the library into their own software systems. The library also includes built-in support for parallel computing, which greatly enhances the speed and efficiency of simulations, especially for large and complex problems.
In addition, Hermes Library includes advanced visualization capabilities that enable users to visualize the results of simulations, including electric and magnetic fields, heat transfer, and fluid flow. This helps in verifying the accuracy of the simulations and ensuring that the results meet the requirements of the application.
Hermes Library is well-suited for the simulation of complex physical processes in the fields of electromagnetics and heat transfer, such as the interaction of electromagnetic fields with materials and structures, and the heat transfer in electronic devices. Its support for parallel computing, advanced visualization capabilities, and flexible and user-friendly API make it a valuable tool for researchers and engineers in these fields and a suitable choice for custom simulation software development.
MeshLab is an open-source and cross-platform 3D mesh processing and editing tool that is designed to handle large and complex 3D models in fields like computer graphics, cultural heritage, engineering, and design. It has a range of features for editing and processing 3D meshes, including cleaning, repairing, simplifying, texturing, and visualizing.
MeshLab also has tools to remove unwanted features from the mesh such as noise and spikes, and tools to repair holes or missing parts. The software has various algorithms for reducing the complexity of 3D meshes, making it efficient to work with large models.
The software also has options for visualizing and exploring 3D meshes through shading, lighting, and texture mapping, with a 3D viewer that enables users to rotate, zoom and pan the mesh. It supports multiple file formats like STL, PLY, OBJ, and VRML.
MeshLab is highly customizable and users can write their own scripts and plugins, or even modify the source code, to extend its capabilities. It includes a comprehensive API and supports scripting in Python, which enables automation of complex processing tasks.
The software has a user-friendly interface and includes a set of tutorials and documentation to make it easy to use. It is also designed to be fast and efficient, supporting parallel processing for quick generation and processing of large meshes.
TetGen is a flexible, open-source software for generating high-quality tetrahedral meshes from 3D polyhedral domains. This software package was created to provide an efficient solution for finite element simulations and computational geometry applications. Written in C++, TetGen is user-friendly, making it a widely used option for various fields.
TetGen supports domains defined with both triangular and quadrilateral faces and can generate both Delaunay and conforming Delaunay tetrahedral meshes. The software allows for size constraint mesh generation, giving users control over the size of tetrahedra to improve solution accuracy. It also provides options for mesh refinement in specific areas where a higher degree of accuracy is necessary.
TetGen offers a variety of tools for defining geometry, mesh, and boundary conditions and supports multiple file formats for data import and export, making it easily integrable with other software tools. The software provides visualization and exploration options such as shading, lighting, and texture mapping.
TetGen is designed to be quick and efficient, making it suitable for even the largest and most complex models. The software is highly customizable and users can write their own scripts and plugins to enhance its capabilities. The source code is accessible and can be adjusted to meet specific requirements.
The software comes with comprehensive tutorials and documentation, making it simple for users to get started and become proficient with the software. TetGen also has a large, active user community who contribute to its development and share their knowledge and experiences.
In conclusion, TetGen is a powerful, versatile software for generating high-quality tetrahedral meshes. Its ease of use, combined with its efficient design and support for parallel processing, makes it a favored choice for a wide range of applications in areas such as mathematics, engineering, physics, and biology.