Manifold elements¶

Usually optimization problems are defined in terms of a cost function, such as:

$min_{x \in M} f (x)$ where $f : M \to R$ .

Usually $M$ is an Euclidean vector space $R^{n}$ .

In a finite element context, for example, in 2D elasticity problems, we have a two-dimensional displacement for each node. As a result, if there are $n$ nodes, we optimize in $R^{2 n}$ .

The nodal degrees of freedom should be wrapped in Ikarus::RealTuple<double,2> in this case.

Another case of optimization is on non-linear manifolds. These arise typically for Cosserat materials $S O (3)$ ¹, Reissner-Mindlin shells $S^{2}$ ² and micro-magnetics $S^{2}$ or incompressible materials.

Interface¶

The general interface of the manifold elements is represented by the following concept.

namespace Ikarus::Concepts {
  template <typename ManifoldType>
  concept Manifold = requires(ManifoldType var, typename ManifoldType::CorrectionType correction, std::ostream& s,
                              typename ManifoldType::CoordinateType value) {
    typename ManifoldType::ctype; The type for the coordinate values, usually double.

    ManifoldType::valueSize; The number of values to store for the state of the element. E.g., the three-dimensional unit vector needs three entries to store its state.

    ManifoldType::correctionSize; The size of the correction for an element. valueSize and correctionSize are the same in Euclidean space. But, for example, the three-dimensional unit vector needs a two-dimensional correction (which lives in the tangent space).

    typename ManifoldType::CoordinateType; The type to store the element coordinates is usually Eigen::Vector<double,ManifoldType::valueSize>

    typename ManifoldType::CorrectionType; The type to store the element correction type is usually Eigen::Vector<double,ManifoldType::correctionSize>

    { var.getValue() } -> std::convertible_to<typename ManifoldType::CoordinateType>; Access the underlying coordinate vector of the manifold element.

    { var.setValue(value) } -> std::same_as<void>; Directly set the value. E.g., set Ikarus::UnitVector<double,3> a; a.setValue(Eigen::Vector3d::UnitZ());

    { var+=correction };  Update the element with a correction vector. E.g.,
        Ikarus::RealTuple<double,3> a; 
   a+= Eigen::Vector3d::UnitX();

   Ikarus::UnitVector<double,3> b; 
   b+= Eigen::Vector2d::UnitX();

    //...
  };
}

Implementations¶

Name	Formal definition	Notes	Header
$n$ -th dimensional Euclidean space	$x \in R^{n}$		`realTuple.hh`
Unit sphere	$x \in S^{n - 1}, S^{n - 1} = {x \in R^{n} : x \cdot x = 1}$		`unitVector.hh`

Oliver Sander. Geodesic finite elements for cosserat rods. International Journal for Numerical Methods in Engineering, 82(13):1645–1670, 2010. doi:10.1002/nme.2814. ↩
Alexander Müller and Manfred Bischoff. A consistent finite element formulation of the geometrically non-linear reissner-mindlin shell model. Archives of Computational Methods in Engineering, pages 1–47, 2022. doi:10.1007/s11831-021-09702-7. ↩