get_Neumann_dΓ(...)::Vector{Gridap.CellData.GenericMeasure}

Return a collection of boundary triangulations at the specified Neumann boundaries.

residual_Neumann(...)::Function

Return the Neumann residual as a FUNCTION.

A bundle of helper tools to work with discrete models in space/time.

Create a new tag from a geometry and a coordinate-based filter function. The filter function takes in vertex coordinates and returns a boolean values. A geometrical entity is tagged if all its vertices pass the filter.

See also

• Gridap.Geometry.face_labeling_from_vertex_filter
• Gridap.Geometry.merge!

Return the aspect ratio of the underlying cartesian elements as a string. This function is only available for an underlying CartesianGrid.

Example

aspect_ratio(Ω)             # "51:51:5"
aspect_ratio(Ω, tol=0.05)   # "10:10:1"
aspect_ratio(uh⁺, tol=0.1)  # "10:10:1"

Return the element size for a cartesian mesh. This function is only available for an underlying CartesianGrid.

Example

element_size(model)   # Compute the diagonal
element_size(uh, :x)  # Get the x-size of the underlying grid

Shortcuts for the tags of cartesian discrete models.

Example

geometry = CartesianDiscreteModel(domain, partition)
labels = get_face_labeling(geometry)
add_tag_from_tags!(labels, "top",    CartesianTags.faceXY1)  # Edges and vertices are excluded
add_tag_from_tags!(labels, "bottom", CartesianTags.faceXY0⁺) # Edges and vertices are included
add_tag_from_tags!(labels, "x_sym", [CartesianTags.face0YZ; CartesianTags.edge0Y0; CartesianTags.edge0Y1])
add_tag_from_tags!(labels, "center_axis", CartesianTags.edge00Z⁺)

The evolution functions have been designed to apply variable boundary conditions.

Return a constant function which is always evaluated to 1.

\[f(x) = 1\]

Return the Heaviside function.

\[f(t) = H(t,T)\]

Return a bounded ramp function from 0 to 1. By default, the slope is the identity. Otherwise, the scaling factor is 1/T.

\[f(t) = \begin{cases} 0 &, t < 0 \\ t/T &, 0 \leq t < T \\ 1 &, t \geq T \end{cases}\]

Return a sigmoid-like function centered at T and edges at ±ϵ. by default, T=ϵ.

\[u(t) = \frac{t - T + \epsilon}{2\epsilon} \\[10pt] f(t) = \begin{cases} 0 &, u < 0 \\ 3u^2 - 2u^3 &, 0 \leq u < 1 \\ 1 &, u \geq 1 \end{cases}\]

Return a triangular evolution function ranging from 0 to 1, centered at Tmax, having edges at T0 and 2Tmax-T0. By default, T0=0 and Tmax=T.

\[f(t) = \begin{cases} 0 &, t < T_0 \\ \frac{t-T_0}{T_{max}-T_0} &, T_0 \leq t < T_{max} \\ 1-\frac{t-T_{max}}{T_{max}-T_0} &, T_{max} \leq t < 2T_{max}-T_0 \\ 0 &, t \geq 2T_{max}-T_0 \end{cases}\]

Initialize the state variables for the given constitutive model and discretization.

Simplified eight-chain model by Arruda and Boyce. the implementation uses the first five terms of the inverse Langevin function.

\[\Psi = C_1 \sum_{i=1}^{3} \alpha_i \beta^{i-1} (I_1^i - 3^i)\]

Neo-Hooke hyperelastic model

\[\Psi = \frac{1}{2}\mu (I_1 - 3)\]

Coercive volumetric energy term of the form:

\[\Psi = \frac{1}{\kappa} (J-1)^2\]

Yeoh constitutive model.

\[\Psi = \sum_{i=1}^3 C_i (I_1 - 3)^i\]

Update the state variables. The state variables must be initialized using the function 'CellState' with the constitutive model.

Right Cauchy-Green deformation tensor.

Return the dissipation and its derivatives if any.

Elastic right Cauchy-Green deformation tensor.

Residual of the return mapping algorithm and its Jacobian with respect to {Ce,λα} for incompressible case

Arguments

Return

• res
• ∂res

First Piola-Kirchhoff for the incompressible case

Arguments

• obj::ViscousIncompressible: The visous model
• Se_: Elastic 2nd Piola (function of C)
• ∂Se∂Ce_: 2nd Piola Derivatives (function of C)
• F: Current deformation gradient
• Fn: Previous deformation gradient
• A: State variables (Uvα and λα)

Return

• Pα::Gridap.TensorValues.TensorValue

Return mapping for the incompressible case

# Arguments
- `::ViscousIncompressible`
- `Se_::Function`: Elastic Piola (function of C)
- `∂Se∂Ce_::Function`: Piola Derivatives (function of C)
- `∇u_::TensorValue`
- `∇un_::TensorValue`
- `A::VectorValue`: State variables (10-component vector gathering Uvα and λα)

# Return
- `::bool`: indicates whether the state variables should be updated
- `::VectorValue`: State variables at new time step

Return the energy density and its derivatives as functions of C instead of F.

Visco-Elastic model for incompressible case

Arguments

• obj::ViscousIncompressible: The visous model
• Se_: Elastic 2nd Piola (function of C)
• ∂Se∂Ce_: 2nd Piola Derivatives (function of C)
• ∇u_: Current deformation gradient
• ∇un_: Previous deformation gradient
• A: State variables (Uvα and λα)

Return

• Cα::Gridap.TensorValues.TensorValue

ViscousPiola(Se::Function, Ce::SMatrix, invUv::SMatrix, F::SMatrix)::SMatrix

Viscous 1st Piola-Kirchhoff stress

Arguments

• Se Elastic Piola (function of C)
• Ce Elastic right Green-Cauchy deformation tensor
• invUv Inverse of viscous strain
• F Deformation gradient

Return

• Pα::SMatrix

Multiplicative decomposition of visous strain.

Return

• Ue::TensorValue
• Uv::TensorValue
• Uv⁻¹::TensorValue

ViscousTangentOperator::TensorValue

Tangent operator for the incompressible case

Arguments

• obj::ViscousIncompressible
• Se_::Function: Function of C
• ∂Se∂Ce_::Function: Function of C
• F::TensorValue: Deformation tensor
• Ce_trial: Right Green-Cauchy deformation tensor at intermediate step
• Ce: Right Green-Cauchy deformation tensor at curent step
• invUv
• invUvn
• λα

Return

• Cv::TensorValue{9,9}: A fourth-order tensor in flattened notation

returnmappingalgorithm!

Compute the elastic Cauchy deformation tensor and the incompressibility condition.

Arguments

• obj::ViscousIncompressible: The viscous model
• Se_::Function: Elastic 2nd Piola-Kirchhoff stress (function of C)
• ∂Se_∂Ce_::Function: Derivatives of elastic 2nd Piola-Kirchhoff stress (function of C)
• F: Deformation gradient
• Ce_trial: Elastic right Green-Cauchy at intermediate statep
• Ce: Elastic right Green-Cauchy deformation tensor
• λα: incompressibility constraint (Lagrange multiplier)

Return

• Ce
• λα

Set the time step to be used internally by the constitutive model.

∂Ce∂C(::ViscousIncompressible, γα, ∂Se∂Ce, invUvn, Ce, Cetrial, λα, F)

Tangent operator of Ce for the incompressible case

Arguments

• ::ViscousIncompressible The viscous model
• γα: Characteristic time τα / (τα + Δt)
• ∂Se∂Ce_: Function of C
• ...

Return

• ∂Ce∂C

Tangent operator of Ce at fixed Uv

∂Ce∂(Uv^{-1})

⊗₁²(A::VectorValue{D}, B::VectorValue{D})::TensorValue{D,D}

Outer product of two first-order tensors (vectors), returning a second-order tensor (matrix).

⊗₁²³(A::VectorValue{D}, B::TensorValue{D})::TensorValue{D,D*D}

Outer product of a first-order and second-order tensors (vector and matrix), returning a third-order tensor represented in a D x D² flattened matrix using combined indices.

⊗₁²³(A::TensorValue{D}, B::VectorValue{D})::TensorValue{D,D*D}

Outer product of a second-order and first-order tensors (matrix and vector), returning a third-order tensor represented in a D x D² flattened matrix using combined indices.

⊗₁₃²⁴(A::TensorValue{D}, B::TensorValue{D})::TensorValue{D*D}

Outer product of two second-order tensors (matrices), returning a fourth-order tensor represented in a D² x D² flattened matrix using combined indices.

⊗₁²³(A::TensorValue{D}, B::TensorValue{D})::TensorValue{D,D*D}

Outer product of a second-order and first-order tensors (matrix and vector), returning a third-order tensor represented in a D x D² flattened matrix using combined indices.

⊗₁₃²⁴(A::TensorValue{D}, B::TensorValue{D})::TensorValue{D*D}

Outer product of two second-order tensors (matrices), returning a fourth-order tensor represented in a D² x D² flattened matrix using combined indices.

⊗₁₄²³(A::TensorValue{D}, B::TensorValue{D})::TensorValue{D*D}

Outer product of two second-order tensors (matrices), returning a fourth-order tensor represented in a D² x D² flattened matrix using combined indices.

The scaling N×N matrix

_Kroneckerδδ(δδ::Function, N::Int)::TensorValue{N*N,N*N,Float64}

Delta Kronecker outer product according to the δδ(i,j,k,l) function

Return the linear index of a N-dimensional tensor

Return the cartesian indices of an N-dimensional second-order tensor

Return the cartesian indices of an N-dimensional fourth-order tensor

cof(A::TensorValue)::TensorValue

Calculate the cofactor of a matrix.

contraction_IP_JPKL(A::TensorValue{D}, H::TensorValue{D*D})::TensorValue{D*D}

Performs a tensor contraction between a second-order tensor (of size D × D) and a fourth-order tensor (represented as a D² × D² matrix in flattened index notation). The operation follows the index contraction pattern, where addition is performed for repeated indices.

contraction_IP_PJKL(A::TensorValue{D}, H::TensorValue{D*D})::TensorValue{D*D}

Performs a tensor contraction between a second-order tensor (of size D × D) and a fourth-order tensor (represented as a D² × D² matrix in flattened index notation). The operation follows the index contraction pattern, where addition is performed for repeated indices.

Jacobian regularization

jacobian(...)::Gridap.CellData.Integrand

Calculate the jacobian using the given constitutive model and finite element functions.

residual(...)::Gridap.CellData.Integrand

Calculate the residual using the given constitutive model and finite element functions.