API reference

⊗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

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})

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.

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

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.