# FEValues

## CellValues

JuAFEM.CellValuesType
CellScalarValues([::Type{T}], quad_rule::QuadratureRule, func_interpol::Interpolation, [geom_interpol::Interpolation])
CellVectorValues([::Type{T}], quad_rule::QuadratureRule, func_interpol::Interpolation, [geom_interpol::Interpolation])

A CellValues object facilitates the process of evaluating values of shape functions, gradients of shape functions, values of nodal functions, gradients and divergences of nodal functions etc. in the finite element cell. There are two different types of CellValues: CellScalarValues and CellVectorValues. As the names suggest, CellScalarValues utilizes scalar shape functions and CellVectorValues utilizes vectorial shape functions. For a scalar field, the CellScalarValues type should be used. For vector field, both subtypes can be used.

Arguments:

• T: an optional argument (default to Float64) to determine the type the internal data is stored as.
• quad_rule: an instance of a QuadratureRule
• func_interpol: an instance of an Interpolation used to interpolate the approximated function
• geom_interpol: an optional instance of a Interpolation which is used to interpolate the geometry

Common methods:

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JuAFEM.reinit!Function
reinit!(cv::CellValues, x::Vector)
reinit!(bv::FaceValues, x::Vector, face::Int)

Update the CellValues/FaceValues object for a cell or face with coordinates x. The derivatives of the shape functions, and the new integration weights are computed.

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JuAFEM.getdetJdVFunction
getdetJdV(fe_v::Values, q_point::Int)

Return the product between the determinant of the Jacobian and the quadrature point weight for the given quadrature point: $\det(J(\mathbf{x})) w_q$

This value is typically used when one integrates a function on a finite element cell or face as

$\int\limits_\Omega f(\mathbf{x}) d \Omega \approx \sum\limits_{q = 1}^{n_q} f(\mathbf{x}_q) \det(J(\mathbf{x})) w_q$ $\int\limits_\Gamma f(\mathbf{x}) d \Gamma \approx \sum\limits_{q = 1}^{n_q} f(\mathbf{x}_q) \det(J(\mathbf{x})) w_q$

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JuAFEM.shape_valueFunction
shape_value(fe_v::Values, q_point::Int, base_function::Int)

Return the value of shape function base_function evaluated in quadrature point q_point.

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JuAFEM.shape_gradientFunction
shape_gradient(fe_v::Values, q_point::Int, base_function::Int)

Return the gradient of shape function base_function evaluated in quadrature point q_point.

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JuAFEM.shape_symmetric_gradientFunction
shape_symmetric_gradient(fe_v::Values, q_point::Int, base_function::Int)

Return the symmetric gradient of shape function base_function evaluated in quadrature point q_point.

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JuAFEM.shape_divergenceFunction
shape_divergence(fe_v::Values, q_point::Int, base_function::Int)

Return the divergence of shape function base_function evaluated in quadrature point q_point.

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JuAFEM.function_valueFunction
function_value(fe_v::Values, q_point::Int, u::AbstractVector)

Compute the value of the function in a quadrature point. u is a vector with values for the degrees of freedom. For a scalar valued function, u contains scalars. For a vector valued function, u can be a vector of scalars (for use of VectorValues) or u can be a vector of Vecs (for use with ScalarValues).

The value of a scalar valued function is computed as $u(\mathbf{x}) = \sum\limits_{i = 1}^n N_i (\mathbf{x}) u_i$ where $u_i$ are the value of $u$ in the nodes. For a vector valued function the value is calculated as $\mathbf{u}(\mathbf{x}) = \sum\limits_{i = 1}^n N_i (\mathbf{x}) \mathbf{u}_i$ where $\mathbf{u}_i$ are the nodal values of $\mathbf{u}$.

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JuAFEM.function_gradientFunction
function_scalar_gradient(fe_v::Values{dim}, q_point::Int, u::AbstractVector)

Compute the gradient of the function in a quadrature point. u is a vector with values for the degrees of freedom. For a scalar valued function, u contains scalars. For a vector valued function, u can be a vector of scalars (for use of VectorValues) or u can be a vector of Vecs (for use with ScalarValues).

The gradient of a scalar function is computed as $\mathbf{\nabla} u(\mathbf{x}) = \sum\limits_{i = 1}^n \mathbf{\nabla} N_i (\mathbf{x}) u_i$ where $u_i$ are the nodal values of the function. For a vector valued function the gradient is computed as $\mathbf{\nabla} \mathbf{u}(\mathbf{x}) = \sum\limits_{i = 1}^n \mathbf{\nabla} N_i (\mathbf{x}) \otimes \mathbf{u}_i$ where $\mathbf{u}_i$ are the nodal values of $\mathbf{u}$.

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JuAFEM.function_symmetric_gradientFunction
function_symmetric_gradient(fe_v::Values, q_point::Int, u::AbstractVector)

Compute the symmetric gradient of the function, see function_gradient. Return a SymmetricTensor.

The symmetric gradient of a scalar function is computed as $\left[ \mathbf{\nabla} \mathbf{u}(\mathbf{x_q}) \right]^\text{sym} = \sum\limits_{i = 1}^n \frac{1}{2} \left[ \mathbf{\nabla} N_i (\mathbf{x}_q) \otimes \mathbf{u}_i + \mathbf{u}_i \otimes \mathbf{\nabla} N_i (\mathbf{x}_q) \right]$ where $\mathbf{u}_i$ are the nodal values of the function.

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JuAFEM.function_divergenceFunction
function_divergence(fe_v::Values, q_point::Int, u::AbstractVector)

Compute the divergence of the vector valued function in a quadrature point.

The divergence of a vector valued functions in the quadrature point $\mathbf{x}_q)$ is computed as $\mathbf{\nabla} \cdot \mathbf{u}(\mathbf{x_q}) = \sum\limits_{i = 1}^n \mathbf{\nabla} N_i (\mathbf{x_q}) \cdot \mathbf{u}_i$ where $\mathbf{u}_i$ are the nodal values of the function.

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JuAFEM.spatial_coordinateFunction
spatial_coordinate(fe_v::Values{dim}, q_point::Int, x::AbstractVector)

Compute the spatial coordinate in a quadrature point. x contains the nodal coordinates of the cell.

The coordinate is computed, using the geometric interpolation, as $\mathbf{x} = \sum\limits_{i = 1}^n M_i (\mathbf{x}) \mathbf{\hat{x}}_i$

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## FaceValues

All of the methods for CellValues apply for FaceValues as well. In addition, there are some methods that are unique for FaecValues:

JuAFEM.FaceValuesType
FaceScalarValues([::Type{T}], quad_rule::QuadratureRule, func_interpol::Interpolation, [geom_interpol::Interpolation])
FaceVectorValues([::Type{T}], quad_rule::QuadratureRule, func_interpol::Interpolation, [geom_interpol::Interpolation])

A FaceValues object facilitates the process of evaluating values of shape functions, gradients of shape functions, values of nodal functions, gradients and divergences of nodal functions etc. on the faces of finite elements. There are two different types of FaceValues: FaceScalarValues and FaceVectorValues. As the names suggest, FaceScalarValues utilizes scalar shape functions and FaceVectorValues utilizes vectorial shape functions. For a scalar field, the FaceScalarValues type should be used. For vector field, both subtypes can be used.

Note

The quadrature rule for the face should be given with one dimension lower. I.e. for a 3D case, the quadrature rule should be in 2D.

Arguments:

Common methods:

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JuAFEM.getcurrentfaceFunction
getcurrentface(fv::FaceValues)

Return the current active face of the FaceValues object (from last reinit!).

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