Taiga.FastDiagonalization(model::Model{Dim, T}; method::Type{<:DataApproximationMethod} = Wachspress, kwargs)

Computes a block diagonal fast diagonalization preconditioner for Model.

Arguments:

• model: Model model definition
• method: method for data approximation
• kwargs: optional keyword arguments

BMatrix{Dim, F}

Struct holding B-matrices (strain-displacement matrices). Supports linear indexing.

BenchmarkSolution

Struct holding displacement and stress mappings for benchmark problems.

Fields:

• displacement::Function: displacement vector as a function of physical coordinates
• stress::Function: stress tensor as a function of physical coordinates

CauchyStress{Dim, T <: Strain{Dim}} <: AbstractMapping{Dim,Dim,Dim}

Mapping for evaluation of Cauchy stress:

$\boldsymbol \sigma = \mathbf C : \boldsymbol \varepsilon$.

Fields:

• strain::T: Strain
• material::Material{Dim}: Material

Isotropic <: Material{3,6}

Isotropic material.

Fields:

• E::Function: Young's modulus
• ν::Function: Poisson's ratio

LinearOperator{Dim, T} <: Taiga.LinearOperator{Dim,T}

Linear operator for linear elastic models.

Fields:

• C::SparseMatrixCSC{T}: sparse extraction operator
• K::SparseMatrixCSC{T}: sparse stiffness matrix without constraints
• L::SparseMatrixCSC{T}: sparse stiffness matrix including constraints
• b::Vector{T}: load vector

abstract type Material{Dim,S} <: AbstractMapping{Dim,S,S} end

All linear elastic materials derive from this. Materials are mappings in Dim × S × S.

Model{Dim, T, M <: GeometricMapping{Dim}, V <: VectorSplineSpace{Dim, T}, W <: Field{Dim}, U <: Material{Dim}} <: Taiga.Model{Dim,T}

Linear elastic model.

Fields:

• F::M: geometric mapping
• S::V: test and trial space
• Δ::Partition{Dim, T}: partition
• uʰ::W: solution field
• ūʰ::W: field compatible with boundary conditions
• t::Function: traction
• material::U: material

Orthotropic <: Material{3,6}

Orthotropic material.

Fields:

• E₁::Function: Young's modulus
• E₂::Function: Young's modulus
• E₃::Function: Young's modulus
• ν₂₃::Function: Poisson's ratio
• ν₃₂::Function: Poisson's ratio
• ν₁₃::Function: Poisson's ratio
• ν₃₁::Function: Poisson's ratio
• ν₁₂::Function: Poisson's ratio
• ν₂₁::Function: Poisson's ratio
• G₂₃::Function: Shear modulus
• G₁₃::Function: Shear modulus
• G₁₂::Function: Shear modulus

PlaneStrain <: Material{2,3}

Plane strain material.

Fields:

• E::Function: Young's modulus
• ν::Function: Poisson's ratio

PlaneStress <: Material{2,3}

Plane stress material.

Fields:

• E::Function: Young's modulus
• ν::Function: Poisson's ratio

PullbackBilinearForm{Dim, T1 <: GeometricMapping{Dim}, T2 <: Material{Dim}}

Pullback of the bilinear form.

Fields:

• mapping::T1: geometric mapping
• material::T2: material
• B::BMatrix{Dim}: B-matrices.

PullbackBoundaryLinearForm{Dim, S}

Pullback of the linear form on the boundary. The parameter 1 ≤ S ≤ 2Dim denotes the coressponding boundary face.

Fields:

• mapping::GeometricMapping{Dim}: geometric mapping
• traction::Function: traction tensor as a function of physical coordinates

Strain{Dim, T1 <: GeometricMapping{Dim}, T2 <: Field{Dim, Dim}} <: AbstractMapping{Dim,Dim,Dim}

Mapping for evaluation of strain:

$\boldsymbol \varepsilon = \frac 1 2 (\nabla \mathbf u + \nabla \mathbf u^T)$.

Fields:

• mapping::T1: geometric mapping
• displacement::T2: displacement field

VonMisesStress{Dim, T <: CauchyStress{Dim}}

Mapping for evaluation of von Mises stress.

\[\begin{aligned} \mathbf{s} &= \boldsymbol{\sigma} - \frac{1}{2} \mathrm{tr}(\boldsymbol{\sigma}) \mathbf{I} \\ J_2 &= \mathbf{s} : \mathbf{s}) \\ \boldsymbol{\sigma}_{\mathrm{vM}} &= \sqrt{3 J_2} \end{aligned}\]

Fields:

• stress::T: CauchyStress

SparseArrays.sparse(L::LinearOperator)

Get sparse linear operator.

assemble_matrix(acc::ElementAccessor{Dim}, partition::Partition{Dim}, pullback::PullbackBilinearForm{Dim}; show_progress::Bool = true)

Assemble stiffness matrix.

Arguments:

• acc: element accessor
• partition: partition
• pullback: pullback of bilinear form
• show_progress: boolean flag for formation progress bar

assemble_vector(acc::ElementAccessor{Dim}, partition::Partition{Dim}, pullbacks::NTuple{N, PullbackBoundaryLinearForm})

Assemble forcing vector.

Arguments:

• acc: element accessor
• partition: partition
• pullbacks: tuple of pullbacks of the linear form on each of the boundaries

benchmark_hole_in_plate_2d(; E, ν)

Hole in a plate benchmark in two dimensions.

benchmark_hole_in_plate_3d(; E, ν)

Hole in a plate benchmark in three dimensions.

benchmark_spherical_cavity(; E, ν)

Spherical cavity benchmark in three dimensions.

material_matrix(::Type{Isotropic}, E, ν)

Returns the elasticity tensor in Voigt notation.

material_matrix(::Type{Orthotropic}, E₁, E₂, E₃, ν₂₃, ν₃₂, ν₁₃, ν₃₁, ν₁₂, ν₂₁, G₂₃, G₁₃, G₁₂)

Returns the elasticity tensor in Voigt notation.

material_matrix(::Type{PlaneStrain}, E, ν)

Returns the elasticity tensor in Voigt notation.

material_matrix(::Type{PlaneStress}, E, ν)

Returns the elasticity tensor in Voigt notation.

σ_voigt(A::SMatrix) -> SVector
σ_voigt(v::SVector) -> SMatrix

Convert between standard and Voigt notation for stress.

• σ_voigt(A::SMatrix{2,2}): Converts a 2×2 stress tensor to a 3-component Voigt vector.
• σ_voigt(A::SMatrix{3,3}): Converts a 3×3 stress tensor to a 6-component Voigt vector.
• σ_voigt(v::SVector{3}): Converts a 3-component Voigt stress vector to a 2×2 stress tensor.
• σ_voigt(v::SVector{6}): Converts a 6-component Voigt stress vector to a 3×3 stress tensor.

Shear components remain unchanged in Voigt notation.

ϵ_voigt(A::SMatrix) -> SVector
ϵ_voigt(v::SVector) -> SMatrix

Convert between standard and Voigt notation for strain.

• ϵ_voigt(A::SMatrix{2,2}): Converts a 2×2 strain tensor to a 3-component Voigt vector.
• ϵ_voigt(A::SMatrix{3,3}): Converts a 3×3 strain tensor to a 6-component Voigt vector.
• ϵ_voigt(v::SVector{3}): Converts a 3-component Voigt strain vector to a 2×2 strain tensor.
• ϵ_voigt(v::SVector{6}): Converts a 6-component Voigt strain vector to a 3×3 strain tensor.

Shear components are multiplied by 2 when converting to Voigt notation.

Taiga.apply_particular_solution(L::LinearOperator, model::Model, x₀::V)

Return particular solution.

Arguments:

• L: linear operator
• model: model
• x₀: homogeneous solution

Taiga.forcing!(b::AbstractVector, L::LinearOperator, model::Model)

Update forcing vector.

Arguments:

• b: cache vector
• L: linear operator
• model: model

Taiga.forcing(L::LinearOperator, model::Model)

Return forcing vector.

Taiga.linear_operator(model::Model; show_progress::Bool = true)

Construct linear operator.