Namespace ae108::elements

Namespace List > ae108 > elements

Framework to implement elements and material models. More...

Namespaces

Type	Name
namespace	embedding Embed a point in reference space into physical space.
namespace	integrator Integrate an entity over an element.
namespace	materialmodels Compute energy, stress, strain, and tangent matrix of material models.
namespace	quadrature Integrate a function over a reference element.
namespace	shape Compute the values, gradients, and support points of a set of shape functions.

Classes

Type	Name
struct	AutomaticForcesTrait <class Element_> Computes the forces by differentiation of the energy.
struct	AutomaticStiffnessMatrixTrait <class Element_> Computes the stiffness matrix by differentiation of the forces.
struct	Bar <Dimension_, class ValueType_, class RealType_>
struct	BarProperties <class RealType_>
struct	ComputeConsistentMassMatrixTrait <class Element>
struct	ComputeConsistentMassMatrixTrait< CoreElement< MaterialModel_, Integrator_, ValueType_, RealType_ > > <class MaterialModel_, class Integrator_, class ValueType_, class RealType_>
struct	ComputeEnergyTrait <class Element>
struct	ComputeEnergyTrait< Bar< Dimension_, ValueType_, RealType_ > > <Dimension_, class ValueType_, class RealType_>
struct	ComputeEnergyTrait< CoreElement< MaterialModel_, Integrator_, ValueType_, RealType_ > > <class MaterialModel_, class Integrator_, class ValueType_, class RealType_>
struct	ComputeEnergyTrait< ElementWithMass< Element_ > > <class Element_>
struct	ComputeEnergyTrait< ForceElement< Dimension_, ValueType_, RealType_ > > <Dimension_, class ValueType_, class RealType_>
struct	ComputeEnergyTrait< Minimal< Size_, Dimension_, DegreesOfFreedom_ > > <Size_, Dimension_, DegreesOfFreedom_> Always returns 0.
struct	ComputeEnergyTrait< TimoshenkoBeamElement< Dimension_, ValueType_, RealType_ > > <Dimension_, class ValueType_, class RealType_>
struct	ComputeForcesTrait <class Element>
struct	ComputeForcesTrait< Bar< Dimension_, ValueType_, RealType_ > > <Dimension_, class ValueType_, class RealType_>
struct	ComputeForcesTrait< CoreElement< MaterialModel_, Integrator_, ValueType_, RealType_ > > <class MaterialModel_, class Integrator_, class ValueType_, class RealType_>
struct	ComputeForcesTrait< ElementWithMass< Element_ > > <class Element_>
struct	ComputeForcesTrait< ForceElement< Dimension_, ValueType_, RealType_ > > <Dimension_, class ValueType_, class RealType_>
struct	ComputeForcesTrait< Minimal< Size_, Dimension_, DegreesOfFreedom_ > > <Size_, Dimension_, DegreesOfFreedom_> Computes the forces by differentiating the energy.
struct	ComputeForcesTrait< TimoshenkoBeamElement< Dimension_, ValueType_, RealType_ > > <Dimension_, class ValueType_, class RealType_>
struct	ComputeLumpedMassMatrixTrait <class Element>
struct	ComputeLumpedMassMatrixTrait< CoreElement< MaterialModel_, Integrator_, ValueType_, RealType_ > > <class MaterialModel_, class Integrator_, class ValueType_, class RealType_>
struct	ComputeMassMatrixTrait <class Element>
struct	ComputeMassMatrixTrait< ElementWithMass< Element_ > > <class Element_>
struct	ComputeStiffnessMatrixTrait <class Element>
struct	ComputeStiffnessMatrixTrait< Bar< Dimension_, ValueType_, RealType_ > > <Dimension_, class ValueType_, class RealType_>
struct	ComputeStiffnessMatrixTrait< CoreElement< MaterialModel_, Integrator_, ValueType_, RealType_ > > <class MaterialModel_, class Integrator_, class ValueType_, class RealType_>
struct	ComputeStiffnessMatrixTrait< ElementWithMass< Element_ > > <class Element_>
struct	ComputeStiffnessMatrixTrait< ForceElement< Dimension_, ValueType_, RealType_ > > <Dimension_, class ValueType_, class RealType_>
struct	ComputeStiffnessMatrixTrait< Minimal< Size_, Dimension_, DegreesOfFreedom_ > > <Size_, Dimension_, DegreesOfFreedom_> Computes the stiffness matrix by differentiating the forces.
struct	ComputeStiffnessMatrixTrait< TimoshenkoBeamElement< Dimension_, ValueType_, RealType_ > > <Dimension_, class ValueType_, class RealType_>
struct	CoreElement <class MaterialModel_, class Integrator_, class ValueType_, class RealType_> An element that computes energy, forces, and stiffness matrix using a material model and an integrator.
struct	ElementBase <class Derived_, class SizeType_, class ValueType_, class RealType_, Size_, Dimension_, DegreesOfFreedom_>
struct	ElementWithMass <class Element_> Extends a given element to also store a mass matrix.
struct	ForceElement <Dimension_, class ValueType_, class RealType_> A single-vertex element that applies a force at that vertex.
struct	Minimal <Size_, Dimension_, DegreesOfFreedom_> A minimal example of an element. Note that only the energy is defined.
struct	TimoshenkoBeamElement <Dimension_, class ValueType_, class RealType_> _Implementation of the closed-form Timoshenko beam element as presented in Cook et. al (2002), "Concepts and applications of Finite Element
Analysis", 4th ed., pp.24-32._
struct	TimoshenkoBeamProperties <class RealType_, Dimension_>
struct	TimoshenkoBeamProperties< RealType_, 2 > <class RealType_>
struct	TimoshenkoBeamProperties< RealType_, 3 > <class RealType_>

Public Functions

Type	Name
Element::Forces	automatic_forces (const Element & element, const typename Element::NodalDisplacements & displacements, const typename Element::Time & time) Computes the forces uses numerical differentiation of the energy.
Element::StiffnessMatrix	automatic_stiffness_matrix (const Element & element, const typename Element::NodalDisplacements & displacements, const typename Element::Time & time) Computes the stiffness matrix using numerical differentation of the forces.
Eigen::Matrix< RealType_, 2 Dimension_, 2 Dimension_, Eigen::RowMajor >	bar_stiffness_matrix (const tensor::Tensor< RealType_, Dimension_ > & axis, const BarProperties< RealType_ > & properties) noexcept Computes the stiffness matrix for a bar with the given axis and the given properties.
Element::StiffnessMatrix	compute_consistent_mass_matrix (const Element & element)
Element::Energy	compute_energy (const Element & element, const typename Element::NodalDisplacements & displacements, const typename Element::Time & time)
Element::Forces	compute_forces (const Element & element, const typename Element::NodalDisplacements & displacements, const typename Element::Time & time)
Element::StiffnessMatrix	compute_lumped_mass_matrix (const Element & element)
Element::MassMatrix	compute_mass_matrix (const Element & element)
Element::StiffnessMatrix	compute_stiffness_matrix (const Element & element, const typename Element::NodalDisplacements & displacements, const typename Element::Time & time)
Eigen::Matrix< double, Dimension_ (Dimension_+1), Dimension_ (Dimension_+1), Eigen::RowMajor >	timoshenko_beam_consistent_mass_matrix (const tensor::Tensor< double, Dimension_ > & axis, const TimoshenkoBeamProperties< double, Dimension_ > & properties, const double density) noexcept Computes the consistent mass matrix for a Timoshenko beam with the given axis and the given properties.
Eigen::Matrix< double, Dimension_ (Dimension_+1), Dimension_ (Dimension_+1), Eigen::RowMajor >	timoshenko_beam_lumped_mass_matrix (const tensor::Tensor< double, Dimension_ > & axis, const TimoshenkoBeamProperties< double, Dimension_ > & properties, const double density) noexcept Computes the lumped mass matrix for a Timoshenko beam with the given axis and the given properties.
Eigen::Matrix< double, Dimension_ (Dimension_+1), Dimension_ (Dimension_+1), Eigen::RowMajor >	timoshenko_beam_stiffness_matrix (const tensor::Tensor< double, Dimension_ > & axis, const TimoshenkoBeamProperties< double, Dimension_ > & properties) noexcept Computes the stiffness matrix for a Timoshenko beam with the given axis and the given properties.

Detailed Description

Library for specifying the behaviour of the elements that make up the mesh. For instance, use this library to define the energy needed to deform an element.

For instance, the library contains an implementation of the shape::Hexa8 element with a materialmodels::Hookean material model.

The main idea behind this library is that an element is just a material model (a way to compute an entity like energy) combined with an integrator (a way to integrate this entity over the element). These two concepts are (almost) orthogonal and the two components can therefore be chosen (almost) independently.

To implement an element, traits to compute energy, forces, and the stiffness matrix are provided. There are also traits to compute the forces and the stiffness matrix numerically.

These traits are implemented for the CoreElement element (which should be applicable to many use cases) and the Minimal element (which shows how to implement a valid element with minimal effort).

Public Functions Documentation

function automatic_forces

Computes the forces uses numerical differentiation of the energy.

template<class Element>
Element::Forces ae108::elements::automatic_forces (
    const Element & element,
    const typename Element::NodalDisplacements & displacements,
    const typename Element::Time & time
)

function automatic_stiffness_matrix

Computes the stiffness matrix using numerical differentation of the forces.

template<class Element>
Element::StiffnessMatrix ae108::elements::automatic_stiffness_matrix (
    const Element & element,
    const typename Element::NodalDisplacements & displacements,
    const typename Element::Time & time
)

function bar_stiffness_matrix

Computes the stiffness matrix for a bar with the given axis and the given properties.

template<std::size_t Dimension_, class RealType_>
Eigen::Matrix< RealType_, 2 *Dimension_, 2 *Dimension_, Eigen::RowMajor > ae108::elements::bar_stiffness_matrix (
    const tensor::Tensor < RealType_, Dimension_ > & axis,
    const BarProperties < RealType_ > & properties
) noexcept

Note:

see Dennis M. Kochmann, "Computational Mechanics I: Introduction to FEA Analysis", https://mm.ethz.ch/education, pp.77-81

function compute_consistent_mass_matrix

template<class Element>
Element::StiffnessMatrix ae108::elements::compute_consistent_mass_matrix (
    const Element & element
)

function compute_energy

template<class Element>
Element::Energy ae108::elements::compute_energy (
    const Element & element,
    const typename Element::NodalDisplacements & displacements,
    const typename Element::Time & time
)

function compute_forces

template<class Element>
Element::Forces ae108::elements::compute_forces (
    const Element & element,
    const typename Element::NodalDisplacements & displacements,
    const typename Element::Time & time
)

function compute_lumped_mass_matrix

template<class Element>
Element::StiffnessMatrix ae108::elements::compute_lumped_mass_matrix (
    const Element & element
)

function compute_mass_matrix

template<class Element>
Element::MassMatrix ae108::elements::compute_mass_matrix (
    const Element & element
)

function compute_stiffness_matrix

template<class Element>
Element::StiffnessMatrix ae108::elements::compute_stiffness_matrix (
    const Element & element,
    const typename Element::NodalDisplacements & displacements,
    const typename Element::Time & time
)

function timoshenko_beam_consistent_mass_matrix

Computes the consistent mass matrix for a Timoshenko beam with the given axis and the given properties.

template<std::size_t Dimension_>
Eigen::Matrix< double, Dimension_ *(Dimension_+1), Dimension_ *(Dimension_+1), Eigen::RowMajor > ae108::elements::timoshenko_beam_consistent_mass_matrix (
    const tensor::Tensor < double, Dimension_ > & axis,
    const TimoshenkoBeamProperties < double, Dimension_ > & properties,
    const double density
) noexcept

Template parameters:

Dimension_ The dimenion of the physical space. Only dimensions 2 and 3 are supported.

function timoshenko_beam_lumped_mass_matrix

Computes the lumped mass matrix for a Timoshenko beam with the given axis and the given properties.

template<std::size_t Dimension_>
Eigen::Matrix< double, Dimension_ *(Dimension_+1), Dimension_ *(Dimension_+1), Eigen::RowMajor > ae108::elements::timoshenko_beam_lumped_mass_matrix (
    const tensor::Tensor < double, Dimension_ > & axis,
    const TimoshenkoBeamProperties < double, Dimension_ > & properties,
    const double density
) noexcept

Template parameters:

Dimension_ The dimenion of the physical space. Only dimensions 2 and 3 are supported.

function timoshenko_beam_stiffness_matrix

Computes the stiffness matrix for a Timoshenko beam with the given axis and the given properties.

template<std::size_t Dimension_>
Eigen::Matrix< double, Dimension_ *(Dimension_+1), Dimension_ *(Dimension_+1), Eigen::RowMajor > ae108::elements::timoshenko_beam_stiffness_matrix (
    const tensor::Tensor < double, Dimension_ > & axis,
    const TimoshenkoBeamProperties < double, Dimension_ > & properties
) noexcept

Template parameters:

Dimension_ The dimenion of the physical space. Only dimensions 2 and 3 are supported.

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