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BEAST.jl documentation

BEAST provides a number of types modelling concepts and a number of algorithms for the efficient and simple implementation of boundary and finite element solvers. It provides full implementations of these concepts for the LU based solution of boundary integral equations for the Maxwell and Helmholtz systems.

Because Julia only compiles code at execution time, users of this library can hook into the code provided in this package at any level. In the extreme case it suffices to provide overwrites of the assemble functions. In that case, only the LU solution will be performed by the code here.

At the other end it suffices that users only supply integration kernels that act on the element-element interaction level. This package will manage all required steps for matrix assembly.

For the Helmholtz 2D and Maxwell 3D systems, complete implementations are supplied. These models will be discussed in detail to give a more concrete idea of the APIs provides and how to extend them.

Central to the solution of boundary integral equations is the assembly of the system matrix. The system matrix is fully determined by specifying a kernel G, a set of trial functions, and a set of test functions.

Basis

Sets of both trial and testing functions are implemented by models following the basis concept. The term basis is somewhat misleading as it is nowhere required nor enforced that these functions are linearly independent. Models implementing the Basis concept need to comply to the following semantics.

  • numfunctions(basis): number of functions in the Basis.
  • coordtype(basis): type of (the components of) the values taken on by the functions in the Basis.
  • scalartype(d): the scalar field underlying the vector space the basis functions take value in.
  • refspace(basis): returns the ReferenceSpace of local shape functions on which the Basis is built.
  • assemblydata(basis): assemblydata returns an iterable collection elements of geometric elements and a look table ad for use in assembly of interaction matrices. In particular, for an index element_idx into elements and an index local_shape_idx in basis of local shape functions refspace(basis), ad[element_idx, local_shape_idx] returns the iterable collection of (global_idx, weight) tuples such that the local shape function at local_shape_idx defined on the element at element_idx contributes to the basis function at global_idx with a weight of weight.
  • geometry(basis): returns an iterable collection of Elements. The order in which these Elements are encountered corresponds to the indices used in the assembly data structure.

Reference Space

The reference space concept defines an API for working with spaces of local shape functions. The main role of objects implementing this concept is to allow specialization of the functions that depend on the precise reference space used.

The functions that depend on the type and value of arguments modeling reference space are:

Kernel

A kernel is a fairly simple concept that mainly exists as part of the definition of a Discrete Operator. A kernel should obey the following semantics:

In many function definitions the kernel object is referenced by operator or something similar. This is a misleading name as an operator definition should always be accompanied by the domain and range space.

Discrete Operator

Informally speaking, a Discrete Operator is a concept that allows for the computation of an interaction matrix. It is a kernel together with a test and trial basis. A Discrete Operator can be passed to assemble and friends to compute its matrix representation.

A discrete operator is a triple (kernel, test_basis, trial_basis), where kernel is a Kernel, and test_basis and trial_basis are Bases. In addition, the following expressions should be implemented and behave according to the correct semantics:

In the context of fast methods such as the Fast Multipole Method other algorithms on Discrete Operators will typically be defined to compute matrix vector products. These algorithms do not explicitly compute and store the interaction matrix (this would lead to unacceptable computational and memory complexity).

BEAST.elements — Function

elements(geo)

Create an iterable collection of the elements stored in geo. The order in which this collection produces the elements determines the index used for lookup in the data structures returned by assemblydata and quaddata.

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BEAST.scalartype — Function
scalartype(x)

The scalar field over which the values of a global or local basis function, or an operator are defined. This should always be a scalar type, even if the basis or operator takes on values in a vector or tensor space. This data type is used to determine the eltype of assembled discrete operators.

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BEAST.assemblydata — Function
charts, admap, act_to_global = assemblydata(basis; onlyactives=true)

Given a basis this function returns a data structure containing the information required for matrix assemble, that is, the vector charts containing Simplex elements, a variable admap of type AssemblyData, and a mapping from indices of actively used simplices to global simplices.

When onlyactives is true, another layer of indices is introduced to filter out all cells of the mesh that are not in the union of the support of the basis functions (i.e., when the basis functions are defined only on a part of the mesh).

admap is, in essence, a three-dimensional array of named tuples, which, by wrapping it in the struct AssemblyData, allows the definition of iterators. The tuple consists of the two entries

admap[i,r,c].globalindex
admap[i,r,c].coefficient

Here, c and r are indices in the iterable set of (active) simplices and the set of shape functions on each cell/simplex: r ranges from 1 to the number of shape functions on a cell/simplex, c ranges from 1 to the number of active simplices, and i ranges from 1 to the number of maximal number of basis functions, where any of the shape functions contributes to.

For example, for continuous piecewise linear lagrange functions (c0d1), each of the three shape functions on a triangle are associated with exactly one Lagrange function, and therefore i is limited to 1.

Note: When onlyactives=false, the indices c correspond to the position of the corresponding cell/simplex whilst iterating over geometry(basis). When onlyactives=true, then act_to_global(c) correspond to the position of the corresponding cell/simplex whilst iterating over geometry(basis).

For a triplet (i,r,c), globalindex is the index in the basis of the ith basis function that has a contribution from shape function r on (active) cell/simplex c. coefficient is the coefficient of that contribution in the linear combination defining that basis function in terms of shape function.

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BEAST.geometry — Function
geometry(basis)

Returns an iterable collection of geometric elements on which the functions in basis are defined. The order the elements are encountered needs correspond to the element indices used in the data structure returned by assemblydata.

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BEAST.refspace — Function
refspace(basis)

Returns the ReferenceSpace of local shape functions on which the basis is built.

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BEAST.quaddata — Function
quaddata(operator, test_refspace, trial_refspace, test_elements, trial_elements)

Returns an object cashing data required for the computation of boundary element interactions. It is up to the client programmer to decide what (if any) data is cached. For double numberical quadrature, storing the integration points for example can significantly speed up matrix assembly.

  • operator is an integration kernel.
  • test_refspace and trial_refspace are reference space objects. quadata

is typically overloaded on the type of these local spaces of shape functions. (See the implementation in maxwell.jl for an example).

  • test_elements and trial_elements are iterable collections of the geometric

elements on which the finite element space are defined. These are provided to allow computation of the actual integrations points - as opposed to only their coordinates.

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BEAST.quadrule — Function
quadrule(operator, test_refspace, trial_refspace, test_index, test_chart, trial_index, trial_chart, quad_data)

Based on the operator kernel and the test and trial elements, this function builds an object whose type and data fields specify the quadrature rule that needs to be used to accurately compute the interaction integrals. The quad_data object created by quaddata is passed to allow reuse of any precomputed data such as quadrature points and weights, geometric quantities, etc.

The type of the returned quadrature rule will help in deciding which method of momintegrals to dispatch to.

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BEAST.momintegrals! — Function

momintegrals!(biop, tshs, bshs, tcell, bcell, interactions, strat)

Function for the computation of moment integrals using simple double quadrature.

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