Green's Function

compose_stress_greens_func(ee::AbstractArray, ev::NTuple, ve::NTuple, vv::NTuple{N, <:Tuple},
    ϵcomp::NTuple, σcomp::NTuple) where N

Concatenate tuple of matrix or tuple of tuple of matrix which arrange them in a way to update traction/stress rate using only one BLAS call. It does nothing to the elastic Green's function and specifically used for the outputs from stress_greens_func and stress_greens_func.

Arguments

• ee::AbstractArray: traction Green's function within the elastic fault
• ev::NTuple: stress Green's function from elastic fault to inelastic asthenosphere
• ve::NTuple: traction Green's function inelastic asthenosphere to elastic fault
• vv::NTuple: stress Green's function within inelastic asthenosphere
• ϵcomp: strain component to be considered, must be a subset of σcomp
• σcomp: stress component to be considered

compose_stress_greens_func(mf::OkadaMesh, me::SBarbotMeshEntity,
    λ::T, μ::T, ft::PlaneFault, comp::NTuple{N, <:Symbol}) where {T, N}

Shortcut function for computing all 4 Green's function for viscoelastic relaxation. Arguments stays the same as stress_greens_func.

stress_greens_func(mf::AbstractMesh{2}, ma::SBarbotMeshEntity,
    λ::T, μ::T, ft::FlatPlaneFault,
    σcomp::NTuple{N, Symbol}; kwargs...) where {T<:Real, I<:Integer, N}

Compute stress Green's function from fault mesh to asthenosphere mesh.

Arguments

• mf::AbstractMesh{2}: mesh of fault
• ma::SBarbotMeshEntity{3}: mesh of asthenosphere
• λ::T: Lamé's first parameter
• μ::T: shear modulus
• ft::FlatPlaneFault: fault type, either DIPPING() or STRIKING()
• σcomp::NTuple{N, Symbol}: stress components to consider

KWARGS Arguments

The same as previously mentioned:

• nrept::Integer
• buffer_ratio::Real

Output

The output is a tuple of length(σcomp) matrix, each corresponds $σ_{ij}$ in the same order as given by σcomp.

stress_greens_func(mesh::LineOkadaMesh, λ::T, μ::T, ft::FlatPlaneFault; kwargs...) where T

Compute traction Green function in 1-D elastic fault in LineOkadaMesh.

Arguments

• mesh::LineOkadaMesh: the line mesh coupled with dc3d
• λ::T: Lamé's first parameter
• μ::T: shear modulus
• ft::FlatPlaneFault: fault type, either DIPPING() or STRIKING()

KWARGS Arguments

• ax_ratio::Real: ratio of along-strike to along-downdip, default is 12.5

stress_greens_func(mesh::RectOkadaMesh, λ::T, μ::T, ft::FlatPlaneFault;
    fourier_domain=true, kwargs...) where T

Compute traction Green's function in 2-D elastic fault in RectOkadaMesh. Translational symmetry is considered.

Arguments

• mesh::RectOkadaMesh: the rectangular mesh coupled with dc3d
• λ::T: Lamé's first parameter
• μ::T: shear modulus
• ft::FlatPlaneFault: fault type, either DIPPING() or STRIKING()

KWARGS Arguments

• fourier_domain::Bool: whether or not transform the tensor to fourier domain
• nrept::Integer: number of periodic summation is performed
• buffer_ratio::Real: ratio of length of buffer zone (along-strike) to that of fault (along-strike) It is recommended to set at least 1 for strike-slip fault for mimicing zero-dislocation at ridge on both sides.

stress_greens_func(mesh::TDTri3MeshEntity, λ::T, μ::T, ft::FlatPlaneFault; kwargs...) where T

Compute traction Green's function in 2-D elastic fault in TDTri3MeshEntity.

Arguments

• mesh::TDTri3MeshEntity: the triangular mesh coupled with td_stress_hs
• λ::T: Lamé's first parameter
• μ::T: shear modulus
• ft::FlatPlaneFault: fault type, either DIPPING() or STRIKING()

KWARGS Arguments

• nrept::Integer: number of periodic summation is performed
• buffer_ratio::Real: ratio of length of buffer zone (along-strike) to that of fault (along-strike). Notice the direction of strike is the average value of mesh.ss and the length is the strike projected maximum horizontal expansion.

stress_greens_func(ma::SBarbotMeshEntity{3}, λ::T, μ::T,
    ϵcomp::NTuple{N1, Symbol}, σcomp::NTuple{N2, Symbol}; kwargs...) where {T, N1, N2}

Compute stress Green's function within SBarbotTet4MeshEntity or SBarbotHex8MeshEntity

Arguments

• ma::SBarbotMeshEntity{3}: asthenosphere mesh
• λ::T: Lamé's first parameter
• μ::T: shear modulus
• ϵcomp: the strain ϵcomponent(s) to be considered
• σcomp::NTuple{N, Symbol}: stress components to consider

Output

A tuple of $n$ tuple of matrix, each tuple represents interaction from one strain component, w.r.t. ϵcomp to the stress ϵcomponents, whose order is $σ_{ij}$ whose order is the same as σcomp, each of which is a matrix.

stress_greens_func(ma::SBarbotMeshEntity{3}, mf::AbstractMesh{2}, λ::T, μ::T, ft::PlaneFault,
    ϵcomp::NTuple{N, Symbol}; kwargs...) where {T, N}

Compute traction Green's function from SBarbotTet4MeshEntity or SBarbotHex8MeshEntity to RectOkadaMesh

Arguments

• ma::SBarbotMeshEntity{3}: asthenosphere mesh
• mf::AbstractMesh{2}: fault mesh
• λ::T: Lamé's first parameter
• μ::T: shear modulus
• ft::FlatPlaneFault: fault type, either DIPPING() or STRIKING()
• ϵcomp: the strain ϵcomponent(s) to be considered

Output

A tuple of $n$ matrix, each represents interaction from one strain to the traction on fault.

dc3d(x::T, y::T, z::T, α::T, dep::T, dip::T,
    al::Union{A, SubArray}, aw::Union{A, SubArray}, disl::A
    ) where {T <: Number, A <: AbstractVecOrMat{T}}

Calculate displacements and gradient of displacements due to a rectangular dislocation in an elastic isotropic halfspace.

Please see dc3d for details. Also this fault coordinate system is widely used in this package.

Arguments

• x, y, z: observational position, where $z ≤ 0$
• α: elastic constant
• dep: depth of fault origin
• dip: dip angle in degree
• al: a vector of 2 numbers, indicating along strike (x-axis) spanning
• aw: a vector of 2 numbers, indicating along downdip (y-z plane) spanning
• disl: a vector of 3 numbers, indicating dislocation in along-strike, along-downdip and tensile respectively.

Output

A vector of 12 numbers, each is $u_{x}$, $u_{y}$, $u_{z}$, $u_{x,x}$, $u_{y,x}$, $u_{z,x}$, $u_{x,y}$, $u_{y,y}$, $u_{z,y}$, $u_{x,z}$ $u_{y,z}$, $u_{z,z}$.

sbarbot_disp_hex8(
    x1::R, x2::R, x3::R, q1::R, q2::R, q3::R,
    L::R, T::R, W::R, theta::R,
    epsv11p::R, epsv12p::R, epsv13p::R, epsv22p::R, epsv23p::R, epsv33p::R,
    G::R, nu::R,
    ) where R

Compute displacement arisen from inelastic strain in Hex8 elements. Please see original version for complete details, especially the coordinate system used here.

Arguments

• x1, x2, x3: observational position, where $x_{3} ≥ 0$
• q1, q2, q3: Hex8 element position, where $q_{3} ≥ 0$
• L, T, W: Hex8 element length, thickness and width
• theta: strike angle
• epsv**: strain components, each is $ϵ_{11}$, $ϵ_{12}$, $ϵ_{13}$, $ϵ_{22}$, $ϵ_{23}$, $ϵ_{33}$
• G: shear modulus
• nu: poisson ratio

Output

A vector of 3 numbers, each represents $u_{1}$, $u_{2}$, $u_{3}$

Notice

• Inplace version: sbarbot_disp_hex8!(u, args...) where u is a vector of 3 numbers.

sbarbot_stress_hex8(
    x1::R, x2::R, x3::R, q1::R, q2::R, q3::R,
    L::R, T::R, W::R, theta::R,
    epsv11p::R, epsv12p::R, epsv13p::R, epsv22p::R, epsv23p::R, epsv33p::R,
    G::R, nu::R,
    ) where R

Compute stress arisen from inelastic strain in Hex8 elements. Please see original version for complete details, especially the coordinate system used here.

Arguments

The same as sbarbot_disp_hex8

Output

A vector of 6 numbers, each represents $σ_{11}$, $σ_{12}$, $σ_{13}$, $σ_{22}$, $σ_{23}$, $σ_{33}$

Notice

• Inplace version: sbarbot_stress_hex8!(u, args...) where u is a vector of 6 numbers.

sbarbot_disp_tet4(quadrature::Q,
    x1::R, x2::R, x3::R, A::U, B::U, C::U, D::U,
    e11::R, e12::R, e13::R, e22::R, e23::R, e33::R, nu::R
    ) where {R, U, Q}

Compute displacement arisen from inelastic strain in Tet4 elements. Please see original version for complete details, especially the coordinate system used here.

Arguments

• quadrature: quadrature rule for integration, see FastGaussQuadrature.jl
• x1, x2, x3: observational position, where $x_{3} ≥ 0$
• A, B, C, D: a list of 3 numbers for each, each of which represents coordinates of the vertex. All depth coordinates must be greater or equal to 0 (no checking is performed here)
• e**: strain components, each is $ϵ_{11}$, $ϵ_{12}$, $ϵ_{13}$, $ϵ_{22}$, $ϵ_{23}$, $ϵ_{33}$
• nu: poisson ratio

Output

A vector of 3 numbers, each represents $u_{1}$, $u_{2}$, $u_{3}$

Notice

• Inplace version: sbarbot_disp_tet4!(u, args...) where u is a vector of 3 numbers.

sbarbot_stress_tet4(quadrature::Q,
    x1::R, x2::R, x3::R, A::U, B::U, C::U, D::U,
    e11::R, e12::R, e13::R, e22::R, e23::R, e33::R, G::R, nu::R
    ) where {R, U, Q}

Compute stress arisen from inelastic strain in Tet4 elements. Please see original version for complete details, especially the coordinate system used here.

Arguments

• quadrature: quadrature rule for integration, see FastGaussQuadrature.jl
• x1, x2, x3: observational position
• A, B, C, D: a list of 3 numbers for each, each of which represents coordinates of the vertex
• e**: strain components, each is $ϵ_{11}$, $ϵ_{12}$, $ϵ_{13}$, $ϵ_{22}$, $ϵ_{23}$, $ϵ_{33}$
• G: shear modulus
• nu: poisson ratio

Output

A vector of 6 numbers, each represents $σ_{11}$, $σ_{12}$, $σ_{13}$, $σ_{22}$, $σ_{23}$, $σ_{33}$

Notice

• Inplace version: sbarbot_stress_tet4!(u, args...) where u is a vector of 6 numbers.

td_disp_fs(X::T, Y::T, Z::T, P1::V, P2::V, P3::V, Ss::T, Ds::T, Ts::T, nu::T) where {T, V}

Compute displacement risen from triangular dislocation in elastic fullspace. Please see original version (in supporting information) for details, especially the coordinate system used here.

Arguments

The same as td_disp_hs

td_disp_hs(X::T, Y::T, Z::T, P1::V, P2::V, P3::V, Ss::T, Ds::T, Ts::T, nu::T) where {T, V}

Compute displacement risen from triangular dislocation in elastic halfspace. Please see original version (in supporting information) for details, especially the coordinate system used here.

Arguments

• X, Y, Z: observational coordinates
• P1, P2, P3: three triangular vertices coordinates respectively
• Ss, Ds, Ts: triangular dislocation vector, Strike-slip, Dip-slip, Tensile-slip respectively
• nu: poisson ratio

Output

By order: $u_x$, $u_y$, $u_z$

td_stress_fs(X::T, Y::T, Z::T, P1::V, P2::V, P3::V, Ss::T, Ds::T, Ts::T, λ::T, μ::T) where {T, V}

Compute stress risen from triangular dislocation in elastic fullspace. Please see original version (in supporting information) for details, especially the coordinate system used here.

Arguments

The same as td_stress_hs

td_stress_hs(X::T, Y::T, Z::T, P1::V, P2::V, P3::V, Ss::T, Ds::T, Ts::T, λ::T, μ::T) where {T, V}

Compute stress risen from triangular dislocation in elastic halfspace. Please see original version (in supporting information) for details, especially the coordinate system used here.

Arguments

• X, Y, Z: observational coordinates
• P1, P2, P3: three triangular vertices coordinates respectively
• Ss, Ds, Ts: triangular dislocation vector, Strike-slip, Dip-slip, Tensile-slip respectively
• λ: Lamé's first parameter
• μ: shear modulus

Output

By order: $σ_{xx}$, $σ_{yy}$, $σ_{zz}$, $σ_{xy}$, $σ_{xz}$, $σ_{yz}$. Please be aware of its different order, where principle components come first, against sbarbot_stress_hex8 and sbarbot_stress_tet4 aside from coordinate system difference.