Fast Operators

partial_assemble_parallel!(mesh::Mesh3d,k,in_sources,shape_function::Triangular;
                                fOn=true,gOn=true,n=3,progress=false,depth=1)

Assembles only local contributions in the BEM computations. This is used for singularity extraction when using the Fast Mulitpole Method (FMM) for BEM.

unroll_interpolations(interpolations)

Merges the element interpolations from interpolate_elements into vectors.

create_coefficient_map(weights,physics_topology,physics_function,n_gauss)

Returns the mapping from nodal values to coefficients for the FMM/H-matrix.

This mapping is represented by a (sparse) matrix $C$ with rows given as

\[ \underbrace{\text{jacobian}(\mathbf{u}_j)w_j\mathbf{T}^{e(j)}(\mathbf{u}_j)\mathbf{L}^{e(j)}}_{j\text{th row of } \mathbf{C}},\]

where each of the $j$ rows corresponds to the Gaussian points from all elements and $e(j)$ is a function that returns the element number that Gaussian point $j$ is located on.

scale_columns!(dipvecs,normals,scalings)

Saves the $i$th column of normals scaled by the $i$th value of scalings in dipvecs.

Used when creating the "dipole-vectors" required for using the FMM.

setup_fast_operator(mesh,zk,n_gauss,nearfield,offset,depth;single_layer=true)

Computes sources, normals, C_map and the nearfield_correction required when setting up a fast operator.

create_single_layer_matrix(X, Y, k)

Creates a KernelMatrix representing the single layer potential.

evaluate_targets(A::FMMFOperator,x,targets)

Evaluates the G-operator with coefficients x at the targets positions. The wavenumber used is defined by the FMMGOperaetor.

evaluate_targets(A::FMMFOperator,k,x,targets)

Evaluates the F-operator with coefficients x at the targets positions. The wavenumber used, k, is supplied by the user.

evaluate_targets(A::FMMGOperator,x,targets)

Evaluates the G-operator with coefficients x at the targets positions. The wavenumber used is defined by the FMMGOperaetor.

evaluate_targets(A::FMMGOperator,k,x,targets)

Evaluates the G-operator with coefficients x at the targets positions. The wavenumber used, k, is supplied by the user.

FMMGOperator(k,tol,targets,sources,C,coefficients,nearfield_correction)
FMMGOperator(mesh,k;tol=1e-6,n_gauss=3,nearfield=true,offset=0.2,depth=1)

A LinearMap that represents the BEM $\mathbf{G}$ matrix through the FMM. This matrix has $k$th row given by $\mathbf{z}=\mathbf{z}_k$ in the following

\[\begin{aligned} \left(\int_{\Gamma} G(\mathbf{x},\mathbf{z})\mathbf{T}(\mathbf{x}) \mathrm{d}S_\mathbf{x}\right)\mathbf{y} &\approx \left(\sum_{e=1}^{N}\left(\sum_{i=1}^{Q}G(\mathbf{x}^e(\mathbf{u}_i),\mathbf{z})\text{jacobian}(\mathbf{u}_i)w_i\mathbf{T}^e(\mathbf{u}_i)\right)\mathbf{L}^e\right)\mathbf{y} \newline &= \left(\sum_{j=1}^{NQ}G(\mathbf{x}_j,\mathbf{z})\underbrace{\text{jacobian}(\mathbf{u}_j)w_j\mathbf{T}^{e(j)}(\mathbf{u}_j)\mathbf{L}^{e(j)}}_{j\text{th row of } \mathbf{C}}\right)\mathbf{y} \newline &= \begin{bmatrix} G(\mathbf{x}_1,\mathbf{z}) & G(\mathbf{x}_2,\mathbf{z}) & \dots & G(\mathbf{x}_{NQ},\mathbf{z}) \end{bmatrix} \mathbf{C}\mathbf{y}, \end{aligned}\]

where the subscript $j$ refers to an ordering of the collection of Gaussian points from all elements and $e(j)$ is a function that returns the element number that Gaussian point $j$ is located on. Note that $\mathbf{C}$ is the same same for all $k$'s.

The remaining multiplication with the Green's functions is done utilizing the Flatiron Institute Fast Multipole libraries.

FMMFOperator(k,tol,targets,sources,normals,C,coefficients,dipvecs,nearfield_correction)
FMMFOperator(mesh,k;n_gauss=3,tol=1e-6,nearfield=true,offset=0.2,depth=1,)

A LinearMap that represents the BEM $\mathbf{H}$ matrix through the FMM. This matrix has $k$th row given by $\mathbf{z}=\mathbf{z}_k$ in the following

\[\begin{aligned} \left(\int_{\Gamma}\frac{\partial G(\mathbf{x}, \mathbf{z})}{\partial\mathbf{n}(\mathbf{x})}\mathbf{T}(\mathbf{x}) \mathrm{d}S_{\mathbf{x}}\right)\mathbf{y} &\approx \left(\sum_{e=1}^{N}\left(\sum_{i=1}^{Q}\frac{\partial G(\mathbf{x}^e(\mathbf{u}_i), \mathbf{z})}{\partial\mathbf{n}(\mathbf{x})}\text{jacobian}(\mathbf{u}_i)w_i\mathbf{T}^e(\mathbf{u}_i)\right)\mathbf{L}^e\right)\mathbf{y}\newline &= \left(\sum_{j=1}^{NQ}\nabla G(\mathbf{x}_j, \mathbf{z})\cdot \mathbf{n}(\mathbf{x}_j)\underbrace{\text{jacobian}(\mathbf{u}_j)w_j\mathbf{T}^{e(j)}(\mathbf{u}_j)\mathbf{L}^{e(j)}}_{j\text{th row of }\mathbf{C}}\right)\mathbf{y}\newline &= \begin{bmatrix} \nabla G(\mathbf{x}_1, \mathbf{z}) \cdot \mathbf{n}(\mathbf{x}_1) & \dots & \nabla G(\mathbf{x}_{NQ}, \mathbf{z})\cdot \mathbf{n}(\mathbf{x}_{NQ}) \end{bmatrix} \mathbf{C}\mathbf{y}, \end{aligned}\]

where $\mathbf{C}$ is coefficient mapping (the same for all $k$).

The remaining multiplication with the Green's functions is performed utilizing the Flatiron Institute Fast Multipole libraries.

HGOperator(k,G,C,nearfield_correction,coefficients)
HGOperator(mesh,k;tol=1e-4,n_gauss=3,nearfield=true,offset=0.2,depth=1)

A LinearMap that represents the BEM $\mathbf{G}$ matrix through the H-matrix approach. This matrix has $k$th row given by $\mathbf{z}=\mathbf{z}_k$ in the following

\[\begin{aligned} \left(\int_{\Gamma} G(\mathbf{x},\mathbf{z})\mathbf{T}(\mathbf{x}) \mathrm{d}S_\mathbf{x}\right)\mathbf{y} &\approx \left(\sum_{e=1}^{N}\left(\sum_{i=1}^{Q}G(\mathbf{x}^e(\mathbf{u}_i),\mathbf{z})\text{jacobian}(\mathbf{u}_i)w_i\mathbf{T}^e(\mathbf{u}_i)\right)\mathbf{L}^e\right)\mathbf{y} \newline &= \left(\sum_{j=1}^{NQ}G(\mathbf{x}_j,\mathbf{z})\underbrace{\text{jacobian}(\mathbf{u}_j)w_j\mathbf{T}^{e(j)}(\mathbf{u}_j)\mathbf{L}^{e(j)}}_{j\text{th row of } \mathbf{C}}\right)\mathbf{y} \newline &= \begin{bmatrix} G(\mathbf{x}_1,\mathbf{z}) & G(\mathbf{x}_2,\mathbf{z}) & \dots & G(\mathbf{x}_{NQ},\mathbf{z}) \end{bmatrix} \mathbf{C}\mathbf{y}, \end{aligned}\]

where the subscript $j$ refers to an ordering of the collection of Gaussian points from all elements and $e(j)$ is a function that returns the element number that Gaussian point $j$ is located on. Note that $\mathbf{C}$ is the same same for all $k$'s.

The remaining multiplication with the Green's functions is done utilizing the HMatrices.jl package.

HFOperator(k,H,C,nearfield_correction,coefficients)
HFOperator(mesh,k;tol=1e-4,n_gauss=3,nearfield=true,offset=0.2,depth=1)

A LinearMap that represents the BEM $\mathbf{H}$ matrix through the H-matrix approach. This matrix has $k$th row given by $\mathbf{z}=\mathbf{z}_k$ in the following

\[\begin{aligned} \left(\int_{\Gamma}\frac{\partial G(\mathbf{x}, \mathbf{z})}{\partial\mathbf{n}(\mathbf{x})}\mathbf{T}(\mathbf{x}) \mathrm{d}S_{\mathbf{x}}\right)\mathbf{y} &\approx \left(\sum_{e=1}^{N}\left(\sum_{i=1}^{Q}\frac{\partial G(\mathbf{x}^e(\mathbf{u}_i), \mathbf{z})}{\partial\mathbf{n}(\mathbf{x})}\text{jacobian}(\mathbf{u}_i)w_i\mathbf{T}^e(\mathbf{u}_i)\right)\mathbf{L}^e\right)\mathbf{y}\newline &= \left(\sum_{j=1}^{NQ}\nabla G(\mathbf{x}_j, \mathbf{z})\cdot \mathbf{n}(\mathbf{x}_j)\underbrace{\text{jacobian}(\mathbf{u}_j)w_j\mathbf{T}^{e(j)}(\mathbf{u}_j)\mathbf{L}^{e(j)}}_{j\text{th row of }\mathbf{C}}\right)\mathbf{y}\newline &= \begin{bmatrix} \nabla G(\mathbf{x}_1, \mathbf{z}) \cdot \mathbf{n}(\mathbf{x}_1) & \dots & \nabla G(\mathbf{x}_{NQ}, \mathbf{z})\cdot \mathbf{n}(\mathbf{x}_{NQ}) \end{bmatrix} \mathbf{C}\mathbf{y}, \end{aligned}\]

where $\mathbf{C}$ is coefficient mapping (the same for all $k$). The remaining multiplication with the Green's functions is performed utilizing the HMatrices.jl package.