![]() Open a new Gmsh document, use OpenCASCADE if it asks (Figure 1).Please get in touch if you’d like more detail, or see my previous posts. All of the steps will be included but I will just list them without much explanation. I am going to keep this section minimal as I have given detailed instructions for constructing geometries in my previous posts. 4.1 – The Geometry, Domains and Mesh 4.1.1 – Geometry ![]() Both the gmsh files and the mesh are available in the repository. This section will make extensive use of the software package GMsh, you can obtain it from here. ![]() The external field E_0 is of course given by (phi_a – phi_b)/h. Which is the same result you get if you solve the Laplace equation. If a long cylinder made from a linear dielectric is uniformly polarised, then the field inside the material is given by I’ll only solve the field inside the material, since the result turns out to be uniform, making it simple to compare with the numerical results later on. I have chosen to solve this problem by means of calculating the polarisation resulting from the external field and the subsequent resulting fields as an infinite series. Fig 1- Geometry of 2D problem, a circular linear dielectric region surrounded by free space, positioned between two parallel plates held at a constant potential difference. An electric potential of ϕₐ is imposed on the upper boundary and ϕᵦ on the lower these will generate the uniform electric field. Free space surrounds the dielectric circle, forming a rectangular outer geometry of width w and height h. Located at the geometric centre of the geometry is a circular region of radius a, which is filled with a dielectric material of relative permittivity ϵᵣ. 2 – GeometryĪ schematic layout of the geometry for this problem is presented in Figure 1. I’ve also included one additional 2D example which has a charge density as the source. The technique is general and can also be used for 3D meshes, although I won’t demonstrate this in the post, I have tested it and a 3D example is available in the repository along with the input and output files for this example. The uniform field will be formed by positioning the circle between two parallel plates of which the plate separation is much larger than the radius of the circle. This will then be compared to the solution of the equivalent 3D problem, an infinite cylindrical dielectric in a uniform field. I will begin by demonstrating the technique for the 2D problem of a dielectric circle in a uniform field. The boundary conditions at the interface between materials are automatically satisfied and need not be specified manually. The technique itself is simple, and requires only that a subdomain be specified for the electric permittivity. In this post I will introduce a technique to include linear dielectrics of arbitrary shape, and thus to do away with this simplification. Articles thus far relating to the topic of electrostatics, have made the simplification that all space within the problem was homogeneous and isotropic.
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