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5.1.4.1. Basis Set

The Analyst solvers are based on the finite-element method, in which a solution to the Maxwell equations is expanded on each mesh element in terms of basis functions. Basis functions may be scalar or vector depending on the nature of the problem, they have local support (i.e. they are zero outside the element), and they have unknown amplitude inside the element. The problem of determining the amplitudes, or coefficients, is reduced to a matrix equation by applying the weighted-residual method to the underlying partial differential equations. For linear problems (those for which material properties are not functions of field strength), this approach yields a sparse linear system that is solved for the coefficients using standard methods.

Basis functions on simplexes (triangles and tetrahedrons) are most simply constructed from so-called scalar nodal functions. There is one nodal function for each corner node in the simplex, and it is unity on this node and falls linearly to zero at the remaining corner nodes. The lowest order scalar basis functions are just these nodal functions, with higher-order functions constructed as polynomials of these functions. Scalar functions are used when you solve for scalar potentials or fields, such as in electrostatics and magnetostatics problems.

Basis functions are typically arranged in groups or sets that each encompass sufficient flexibility to represent the underlying solution to a particular order of accuracy. The basis sets in these solvers are also hierarchical so that sets for higher order representations contain, as a subset, functions for all lower-order representations.

For electromagnetic problems involving coupled electric and magnetic fields, Analyst typically uses an electric field formulation on hierarchical vector basis functions on triangles and tetrahedrons with flat sides. Vector basis functions are defined in terms of the nodal basis functions and also gradients of these functions. Included within Analyst are basis sets which can represent the electromagnetic field with up to fifth-order accuracy.

Based on historical and academic precedent, the vector basis set of order n+1 accuracy is designated hn.5 within Analyst, with the interpolation order for the electric field given by n+1, and the interpolation order for the magnetic field given by n. In general, as you increase n, you obtain higher accuracy in the field solution at the cost of an increased computational burden in the calculation. The first order, linear basis set consists of six functions, one associated with each edge in the tetrahedron. This basis set is designated h0.5; it represents the electric field to first-order, and the magnetic field (obtained by taking the curl of the electric field) to zero-th order. The second order basis set (h1.5, representing the electric field to second-order and the magnetic field to first-order) contains the six linear functions of the h0.5 set, as well as 14 additional functions (one for each edge and two for each face on the tetrahedron). This composition is a characteristic of hierarchical basis sets. Similarly, there are 45 functions in the cubic set h2.5, 84 in the quartic set h3.5, and 145 in the quintic set h4.5.

Because the number of functions and associated unknowns depends on the basis set, so too do the computer resources required to solve a problem using the basis set. Generally, the time needed to solve a problem increases in multiple for each increment in the basis set order. For example, if it takes X seconds to solve a problem using the h0.5 basis set on a given mesh, it generally takes 5X to 10X to solve on the mesh using h1.5, and 10X to 20X for h2.5. Direct timing comparisons are difficult as there are other factors that affect run times, such as how fast the AMR process converges, so the choice of basis set is often made on other grounds. Often the choice is determined by experimentation to discover what works best on a given class of problem. When in doubt, start with the quadratic h1.5 basis set.

Analyst includes capacity for hybrid basis sets, in which the accuracy of the basis set may vary within the mesh depending on the system geometry. These hybrid sets are designated as cn.5, and they are hierarchical vector basis sets. They are equivalent to the h0.5 basis set in regions of the model away from conducting corners and edges, but use higher orders (up to hn.5) basis sets on corners and edges. Hierarchical basis sets generally give solution accuracies similar to the corresponding hn.5 basis sets, but at a lower computational cost.

Note that in the driven frequency solver RF3p, the basis set for the port solve is automatically one order higher than the chosen full solve basis set. Since the default basis set for the full solve is h1.5 (quadratic), the default for the port solve is h2.5 (cubic).

The three families of basis sets are described in the following table.

Basis set

Description

Solvers

nx.0 where x = 1, 2

This is the simplest basis set, in that it describes only scalar field unknowns. As such, it is appropriate for use in problems that can be described by a scalar potential, as in the electrostatic and magnetostatic cases. The interpolation order of the n basis set is uniform across the mesh. n1.0 uses linear interpolation functions, and n2.0 uses quadratic interpolation functions.

hx.5 where x = 0, 1, 2, 3, 4

This basis set consists of vector functions over each element, making it appropriate for non-static electromagnetic field calculations. The interpolation order of the h basis set is uniform across the mesh. An hx.5 basis set represents the electric field with an interpolation order of x+1, and it represents the magnetic field with an interpolation order of x. For example, h1.5 is considered the quadratic basis set. It represents the electric field with quadratic interpolating functions, and the magnetic field with linear interpolating functions.

cx.5 where x = 1, 2, 3, 4

This basis set is a hybridization of the hx.5 basis set. The cx.5 basis set uses an interpolation level that varies across the mesh: cx.5 basis sets use hx.5 on conducting corners and edges, and h0.5 in regions of the model away from corners and edges. The cx.5 basis sets generally give solution accuracies similar to the corresponding h basis sets, but for a lower computational cost.

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