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Namespace ae108::elements::shape

Namespace List > ae108 > elements > shape

Compute the values, gradients, and support points of a set of shape functions. More...

Classes

Type Name
struct AutomaticGradientTrait <class Shape_>
struct GradientTrait <class Shape>
struct Hexa8
struct PointTrait <class Shape>
Contains a member value that defines the points of the shape.
struct Quad4
struct Seg2
class ShapeBase <class SizeType_, class ValueType_, Dimension_, Size_>
struct Tet10
struct Tet4
struct Tri3
struct Tri6
struct ValueTrait <class Shape>

Public Functions

Type Name
AE108_ELEMENTS_SHAPE_DEFINE_GRADIENTS (Hexa8, {{ {{-.125 *(xi[1] - 1.) *(xi[2] - 1.), -.125 *(xi[0] - 1.) *(xi[2] - 1.), -.125 *(xi[0] - 1.) *(xi[1] - 1.)}}, {{.125 *(xi[1] - 1.) *(xi[2] - 1.),.125 *(xi[0]+1.) *(xi[2] - 1.),.125 *(xi[0]+1.) *(xi[1] - 1.)}}, {{-.125 *(xi[1]+1.) *(xi[2] - 1.), -.125 *(xi[0]+1.) *(xi[2] - 1.), -.125 *(xi[0]+1.) *(xi[1]+1.)}}, {{.125 *(xi[1]+1.) *(xi[2] - 1.),.125 *(xi[0] - 1.) *(xi[2] - 1.),.125 *(xi[0] - 1.) *(xi[1]+1.)}}, {{.125 *(xi[1] - 1.) *(xi[2]+1.),.125 *(xi[0] - 1.) *(xi[2]+1.),.125 *(xi[0] - 1.) *(xi[1] - 1.)}}, {{-.125 *(xi[1] - 1.) *(xi[2]+1.), -.125 *(xi[0]+1.) *(xi[2]+1.), -.125 *(xi[0]+1.) *(xi[1] - 1.)}}, {{.125 *(xi[1]+1.) *(xi[2]+1.),.125 *(xi[0]+1.) *(xi[2]+1.),.125 *(xi[0]+1.) *(xi[1]+1.)}}, {{-.125 *(xi[1]+1.) *(xi[2]+1.), -.125 *(xi[0] - 1.) *(xi[2]+1.), -.125 *(xi[0] - 1.) *(xi[1]+1.)}}, }})
AE108_ELEMENTS_SHAPE_DEFINE_GRADIENTS (Quad4, {{ {{.25 *(xi[1] - 1.),.25 *(xi[0] - 1.)}}, {{-.25 *(xi[1] - 1.), -.25 *(xi[0]+1.)}}, {{.25 *(xi[1]+1.),.25 *(xi[0]+1.)}}, {{-.25 *(xi[1]+1.), -.25 *(xi[0] - 1.)}}, }})
AE108_ELEMENTS_SHAPE_DEFINE_GRADIENTS (Seg2, {{ {{-.5}}, {{.5}}, }})
AE108_ELEMENTS_SHAPE_DEFINE_GRADIENTS (Tet10, {{ {{1. - 4. *(1 - xi[0] - xi[1] - xi[2]), 1. - 4. *(1 - xi[0] - xi[1] - xi[2]), 1. - 4. *(1 - xi[0] - xi[1] - xi[2])}}, {{-1.+4. *xi[0], 0., 0.}}, {{0., -1.+4. *xi[1], 0.}}, {{0., 0., -1.+4. *xi[2]}}, {{4. *((1 - xi[0] - xi[1] - xi[2]) - xi[0]), -4. *xi[0], -4. *xi[0]}}, {{4. *xi[1], 4. *xi[0], 0.}}, {{-4. *xi[1], 4. *((1 - xi[0] - xi[1] - xi[2]) - xi[1]), -4 *xi[1]}}, {{-4. *xi[2], -4. *xi[2], 4. *((1 - xi[0] - xi[1] - xi[2]) - xi[2])}}, {{4. *xi[2], 0., 4. *xi[0]}}, {{0., 4. *xi[2], 4. *xi[1]}}, }})
AE108_ELEMENTS_SHAPE_DEFINE_GRADIENTS (Tet4, {{ {{-1., -1., -1.}}, {{1., 0., 0.}}, {{0., 1., 0.}}, {{0., 0., 1.}}, }})
AE108_ELEMENTS_SHAPE_DEFINE_GRADIENTS (Tri3, {{ {{-1., -1.}}, {{1., 0.}}, {{0., 1.}}, }})
AE108_ELEMENTS_SHAPE_DEFINE_GRADIENTS (Tri6, {{ {{4. *(xi[0]+xi[1]) - 3., 4. *(xi[0]+xi[1]) - 3.}}, {{4. *xi[0] - 1., 0.}}, {{0., 4 *xi[1] - 1}}, {{4. *(1 - 2 *xi[0] - xi[1]), -4. *xi[0]}}, {{4. *xi[1], 4. *xi[0]}}, {{-4. *xi[1], 4. *(1 - xi[0] - 2. *xi[1])}}, }})
AE108_ELEMENTS_SHAPE_DEFINE_POINTS (Hexa8, {{ {{-1., -1., -1.}}, {{1., -1., -1.}}, {{1., 1., -1.}}, {{-1., 1., -1.}}, {{-1., -1., 1.}}, {{1., -1., 1.}}, {{1., 1., 1.}}, {{-1., 1., 1.}}, }})
AE108_ELEMENTS_SHAPE_DEFINE_POINTS (Quad4, {{ {{-1., -1.}}, {{1., -1.}}, {{1., 1.}}, {{-1., 1.}}, }})
AE108_ELEMENTS_SHAPE_DEFINE_POINTS (Seg2, {{ {{-1.}}, {{1.}}, }})
AE108_ELEMENTS_SHAPE_DEFINE_POINTS (Tet10, {{ {{0., 0., 0.}}, {{1., 0., 0.}}, {{0., 1., 0.}}, {{0., 0., 1.}}, {{.5, 0., 0.}}, {{.5,.5, 0.}}, {{0.,.5, 0.}}, {{0., 0.,.5}}, {{.5, 0.,.5}}, {{0.,.5,.5}}, }})
AE108_ELEMENTS_SHAPE_DEFINE_POINTS (Tet4, {{ {{0., 0., 0.}}, {{1., 0., 0.}}, {{0., 1., 0.}}, {{0., 0., 1.}}, }})
AE108_ELEMENTS_SHAPE_DEFINE_POINTS (Tri3, {{ {{0., 0.}}, {{1., 0.}}, {{0., 1.}}, }})
AE108_ELEMENTS_SHAPE_DEFINE_POINTS (Tri6, {{ {{0., 0.}}, {{1., 0.}}, {{0., 1.}}, {{0.5, 0.}}, {{0.5, 0.5}}, {{0., 0.5}}, }})
AE108_ELEMENTS_SHAPE_DEFINE_VALUE (Hexa8, {{ -.125 *(xi[0] - 1.) *(xi[1] - 1.) *(xi[2] - 1.),.125 *(xi[0]+1.) *(xi[1] - 1.) *(xi[2] - 1.), -.125 *(xi[0]+1.) *(xi[1]+1.) *(xi[2] - 1.),.125 *(xi[0] - 1.) *(xi[1]+1.) *(xi[2] - 1.),.125 *(xi[0] - 1.) *(xi[1] - 1.) *(xi[2]+1.), -.125 *(xi[0]+1.) *(xi[1] - 1.) *(xi[2]+1.),.125 *(xi[0]+1.) *(xi[1]+1.) *(xi[2]+1.), -.125 *(xi[0] - 1.) *(xi[1]+1.) *(xi[2]+1.), }})
AE108_ELEMENTS_SHAPE_DEFINE_VALUE (Quad4, {{ .25 *(xi[0] - 1.) *(xi[1] - 1.), -.25 *(xi[0]+1.) *(xi[1] - 1.),.25 *(xi[0]+1.) *(xi[1]+1.), -.25 *(xi[0] - 1.) *(xi[1]+1.), }})
AE108_ELEMENTS_SHAPE_DEFINE_VALUE (Seg2, {{ -.5 *(xi[0] - 1.),.5 *(xi[0]+1.), }})
AE108_ELEMENTS_SHAPE_DEFINE_VALUE (Tet10, {{ -(1. - xi[0] - xi[1] - xi[2]) *(1. - 2. *(1 - xi[0] - xi[1] - xi[2])), -xi[0] *(1 - 2. *xi[0]), -xi[1] *(1. - 2. *xi[1]), -xi[2] *(1. - 2. *xi[2]), 4. *xi[0] *(1 - xi[0] - xi[1] - xi[2]), 4. *xi[0] *xi[1], 4. *xi[1] *(1 - xi[0] - xi[1] - xi[2]), 4. *xi[2] *(1 - xi[0] - xi[1] - xi[2]), 4. *xi[0] *xi[2], 4. *xi[1] *xi[2], }})
AE108_ELEMENTS_SHAPE_DEFINE_VALUE (Tet4, {{ 1 - xi[0] - xi[1] - xi[2], xi[0], xi[1], xi[2], }})
AE108_ELEMENTS_SHAPE_DEFINE_VALUE (Tri3, {{ 1 - xi[0] - xi[1], xi[0], xi[1], }})
AE108_ELEMENTS_SHAPE_DEFINE_VALUE (Tri6, {{(1 - xi[0] - xi[1]) *(1 - 2 *xi[0] - 2 *xi[1]), xi[0] *(2 *xi[0] - 1), xi[1] *(2 *xi[1] - 1), 4 *xi[0] *(1 - xi[0] - xi[1]), 4 *xi[0] *xi[1], 4 *xi[1] *(1 - xi[0] - xi[1]), }})
Shape::template Collection< typename Shape::Gradient > automatic_gradients (const typename Shape::Point & xi)
Computes the gradients using numerical differentiation.
Shape::template Collection< typename Shape::Gradient > compute_gradients (const typename Shape::Point & xi) noexcept
Shape::template Collection< typename Shape::value_type > compute_values (const typename Shape::Point & xi) noexcept
Shape::template Collection< typename Shape::Point > get_points () noexcept

Detailed Description

Provides traits to compute the values, gradients, and support points of a set of shape functions. Here, “support point” refers to a point where a shape function is equal to one whereas all other shape functions are equal to zero. There is also a trait to compute the gradients numerically.

The part includes three sets of shape functions for which these traits are implemented: @refHexa8, Quad4, and Seg2.

Public Functions Documentation

function AE108_ELEMENTS_SHAPE_DEFINE_GRADIENTS

ae108::elements::shape::AE108_ELEMENTS_SHAPE_DEFINE_GRADIENTS (
    Hexa8,
    {{ {{-.125 *(xi[1] - 1.) *(xi[2] - 1.), -.125 *(xi[0] - 1.) *(xi[2] - 1.), -.125 *(xi[0] - 1.) *(xi[1] - 1.)}}, {{.125 *(xi[1] - 1.) *(xi[2] - 1.),.125 *(xi[0]+1.) *(xi[2] - 1.),.125 *(xi[0]+1.) *(xi[1] - 1.)}}, {{-.125 *(xi[1]+1.) *(xi[2] - 1.), -.125 *(xi[0]+1.) *(xi[2] - 1.), -.125 *(xi[0]+1.) *(xi[1]+1.)}}, {{.125 *(xi[1]+1.) *(xi[2] - 1.),.125 *(xi[0] - 1.) *(xi[2] - 1.),.125 *(xi[0] - 1.) *(xi[1]+1.)}}, {{.125 *(xi[1] - 1.) *(xi[2]+1.),.125 *(xi[0] - 1.) *(xi[2]+1.),.125 *(xi[0] - 1.) *(xi[1] - 1.)}}, {{-.125 *(xi[1] - 1.) *(xi[2]+1.), -.125 *(xi[0]+1.) *(xi[2]+1.), -.125 *(xi[0]+1.) *(xi[1] - 1.)}}, {{.125 *(xi[1]+1.) *(xi[2]+1.),.125 *(xi[0]+1.) *(xi[2]+1.),.125 *(xi[0]+1.) *(xi[1]+1.)}}, {{-.125 *(xi[1]+1.) *(xi[2]+1.), -.125 *(xi[0] - 1.) *(xi[2]+1.), -.125 *(xi[0] - 1.) *(xi[1]+1.)}}, }}
)

function AE108_ELEMENTS_SHAPE_DEFINE_GRADIENTS

ae108::elements::shape::AE108_ELEMENTS_SHAPE_DEFINE_GRADIENTS (
    Quad4,
    {{ {{.25 *(xi[1] - 1.),.25 *(xi[0] - 1.)}}, {{-.25 *(xi[1] - 1.), -.25 *(xi[0]+1.)}}, {{.25 *(xi[1]+1.),.25 *(xi[0]+1.)}}, {{-.25 *(xi[1]+1.), -.25 *(xi[0] - 1.)}}, }}
)

function AE108_ELEMENTS_SHAPE_DEFINE_GRADIENTS

ae108::elements::shape::AE108_ELEMENTS_SHAPE_DEFINE_GRADIENTS (
    Seg2,
    {{ {{-.5}}, {{.5}}, }}
)

function AE108_ELEMENTS_SHAPE_DEFINE_GRADIENTS

ae108::elements::shape::AE108_ELEMENTS_SHAPE_DEFINE_GRADIENTS (
    Tet10,
    {{ {{1. - 4. *(1 - xi[0] - xi[1] - xi[2]), 1. - 4. *(1 - xi[0] - xi[1] - xi[2]), 1. - 4. *(1 - xi[0] - xi[1] - xi[2])}}, {{-1.+4. *xi[0], 0., 0.}}, {{0., -1.+4. *xi[1], 0.}}, {{0., 0., -1.+4. *xi[2]}}, {{4. *((1 - xi[0] - xi[1] - xi[2]) - xi[0]), -4. *xi[0], -4. *xi[0]}}, {{4. *xi[1], 4. *xi[0], 0.}}, {{-4. *xi[1], 4. *((1 - xi[0] - xi[1] - xi[2]) - xi[1]), -4 *xi[1]}}, {{-4. *xi[2], -4. *xi[2], 4. *((1 - xi[0] - xi[1] - xi[2]) - xi[2])}}, {{4. *xi[2], 0., 4. *xi[0]}}, {{0., 4. *xi[2], 4. *xi[1]}}, }}
)

function AE108_ELEMENTS_SHAPE_DEFINE_GRADIENTS

ae108::elements::shape::AE108_ELEMENTS_SHAPE_DEFINE_GRADIENTS (
    Tet4,
    {{ {{-1., -1., -1.}}, {{1., 0., 0.}}, {{0., 1., 0.}}, {{0., 0., 1.}}, }}
)

function AE108_ELEMENTS_SHAPE_DEFINE_GRADIENTS

ae108::elements::shape::AE108_ELEMENTS_SHAPE_DEFINE_GRADIENTS (
    Tri3,
    {{ {{-1., -1.}}, {{1., 0.}}, {{0., 1.}}, }}
)

function AE108_ELEMENTS_SHAPE_DEFINE_GRADIENTS

ae108::elements::shape::AE108_ELEMENTS_SHAPE_DEFINE_GRADIENTS (
    Tri6,
    {{ {{4. *(xi[0]+xi[1]) - 3., 4. *(xi[0]+xi[1]) - 3.}}, {{4. *xi[0] - 1., 0.}}, {{0., 4 *xi[1] - 1}}, {{4. *(1 - 2 *xi[0] - xi[1]), -4. *xi[0]}}, {{4. *xi[1], 4. *xi[0]}}, {{-4. *xi[1], 4. *(1 - xi[0] - 2. *xi[1])}}, }}
)

function AE108_ELEMENTS_SHAPE_DEFINE_POINTS

ae108::elements::shape::AE108_ELEMENTS_SHAPE_DEFINE_POINTS (
    Hexa8,
    {{ {{-1., -1., -1.}}, {{1., -1., -1.}}, {{1., 1., -1.}}, {{-1., 1., -1.}}, {{-1., -1., 1.}}, {{1., -1., 1.}}, {{1., 1., 1.}}, {{-1., 1., 1.}}, }}
)

function AE108_ELEMENTS_SHAPE_DEFINE_POINTS

ae108::elements::shape::AE108_ELEMENTS_SHAPE_DEFINE_POINTS (
    Quad4,
    {{ {{-1., -1.}}, {{1., -1.}}, {{1., 1.}}, {{-1., 1.}}, }}
)

function AE108_ELEMENTS_SHAPE_DEFINE_POINTS

ae108::elements::shape::AE108_ELEMENTS_SHAPE_DEFINE_POINTS (
    Seg2,
    {{ {{-1.}}, {{1.}}, }}
)

function AE108_ELEMENTS_SHAPE_DEFINE_POINTS

ae108::elements::shape::AE108_ELEMENTS_SHAPE_DEFINE_POINTS (
    Tet10,
    {{ {{0., 0., 0.}}, {{1., 0., 0.}}, {{0., 1., 0.}}, {{0., 0., 1.}}, {{.5, 0., 0.}}, {{.5,.5, 0.}}, {{0.,.5, 0.}}, {{0., 0.,.5}}, {{.5, 0.,.5}}, {{0.,.5,.5}}, }}
)

function AE108_ELEMENTS_SHAPE_DEFINE_POINTS

ae108::elements::shape::AE108_ELEMENTS_SHAPE_DEFINE_POINTS (
    Tet4,
    {{ {{0., 0., 0.}}, {{1., 0., 0.}}, {{0., 1., 0.}}, {{0., 0., 1.}}, }}
)

function AE108_ELEMENTS_SHAPE_DEFINE_POINTS

ae108::elements::shape::AE108_ELEMENTS_SHAPE_DEFINE_POINTS (
    Tri3,
    {{ {{0., 0.}}, {{1., 0.}}, {{0., 1.}}, }}
)

function AE108_ELEMENTS_SHAPE_DEFINE_POINTS

ae108::elements::shape::AE108_ELEMENTS_SHAPE_DEFINE_POINTS (
    Tri6,
    {{ {{0., 0.}}, {{1., 0.}}, {{0., 1.}}, {{0.5, 0.}}, {{0.5, 0.5}}, {{0., 0.5}}, }}
)

function AE108_ELEMENTS_SHAPE_DEFINE_VALUE

ae108::elements::shape::AE108_ELEMENTS_SHAPE_DEFINE_VALUE (
    Hexa8,
    {{ -.125 *(xi[0] - 1.) *(xi[1] - 1.) *(xi[2] - 1.),.125 *(xi[0]+1.) *(xi[1] - 1.) *(xi[2] - 1.), -.125 *(xi[0]+1.) *(xi[1]+1.) *(xi[2] - 1.),.125 *(xi[0] - 1.) *(xi[1]+1.) *(xi[2] - 1.),.125 *(xi[0] - 1.) *(xi[1] - 1.) *(xi[2]+1.), -.125 *(xi[0]+1.) *(xi[1] - 1.) *(xi[2]+1.),.125 *(xi[0]+1.) *(xi[1]+1.) *(xi[2]+1.), -.125 *(xi[0] - 1.) *(xi[1]+1.) *(xi[2]+1.), }}
)

function AE108_ELEMENTS_SHAPE_DEFINE_VALUE

ae108::elements::shape::AE108_ELEMENTS_SHAPE_DEFINE_VALUE (
    Quad4,
    {{ .25 *(xi[0] - 1.) *(xi[1] - 1.), -.25 *(xi[0]+1.) *(xi[1] - 1.),.25 *(xi[0]+1.) *(xi[1]+1.), -.25 *(xi[0] - 1.) *(xi[1]+1.), }}
)

function AE108_ELEMENTS_SHAPE_DEFINE_VALUE

ae108::elements::shape::AE108_ELEMENTS_SHAPE_DEFINE_VALUE (
    Seg2,
    {{ -.5 *(xi[0] - 1.),.5 *(xi[0]+1.), }}
)

function AE108_ELEMENTS_SHAPE_DEFINE_VALUE

ae108::elements::shape::AE108_ELEMENTS_SHAPE_DEFINE_VALUE (
    Tet10,
    {{ -(1. - xi[0] - xi[1] - xi[2]) *(1. - 2. *(1 - xi[0] - xi[1] - xi[2])), -xi[0] *(1 - 2. *xi[0]), -xi[1] *(1. - 2. *xi[1]), -xi[2] *(1. - 2. *xi[2]), 4. *xi[0] *(1 - xi[0] - xi[1] - xi[2]), 4. *xi[0] *xi[1], 4. *xi[1] *(1 - xi[0] - xi[1] - xi[2]), 4. *xi[2] *(1 - xi[0] - xi[1] - xi[2]), 4. *xi[0] *xi[2], 4. *xi[1] *xi[2], }}
)

function AE108_ELEMENTS_SHAPE_DEFINE_VALUE

ae108::elements::shape::AE108_ELEMENTS_SHAPE_DEFINE_VALUE (
    Tet4,
    {{ 1 - xi[0] - xi[1] - xi[2], xi[0], xi[1], xi[2], }}
)

function AE108_ELEMENTS_SHAPE_DEFINE_VALUE

ae108::elements::shape::AE108_ELEMENTS_SHAPE_DEFINE_VALUE (
    Tri3,
    {{ 1 - xi[0] - xi[1], xi[0], xi[1], }}
)

function AE108_ELEMENTS_SHAPE_DEFINE_VALUE

ae108::elements::shape::AE108_ELEMENTS_SHAPE_DEFINE_VALUE (
    Tri6,
    {{(1 - xi[0] - xi[1]) *(1 - 2 *xi[0] - 2 *xi[1]), xi[0] *(2 *xi[0] - 1), xi[1] *(2 *xi[1] - 1), 4 *xi[0] *(1 - xi[0] - xi[1]), 4 *xi[0] *xi[1], 4 *xi[1] *(1 - xi[0] - xi[1]), }}
)

function automatic_gradients

Computes the gradients using numerical differentiation. 

template<class Shape>
Shape::template Collection< typename Shape::Gradient > ae108::elements::shape::automatic_gradients (
    const typename Shape::Point & xi
)

function compute_gradients

template<class Shape>
Shape::template Collection< typename Shape::Gradient > ae108::elements::shape::compute_gradients (
    const typename Shape::Point & xi
) noexcept

function compute_values

template<class Shape>
Shape::template Collection< typename Shape::value_type > ae108::elements::shape::compute_values (
    const typename Shape::Point & xi
) noexcept

function get_points

template<class Shape>
Shape::template Collection< typename Shape::Point > ae108::elements::shape::get_points () noexcept

The documentation for this class was generated from the following file elements/src/include/ae108/elements/shape/automatic_gradients.h