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Manifold elements

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

$$ \min_{\boldsymbol{x} \in \mathcal{M}} f(\boldsymbol{x} ) $$ where \( f: \mathcal{M} \rightarrow \mathbb{R} \).

Usually \( \mathcal{M} \) is an Euclidean vector space \( \mathbb{R}^n \).

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 \( {\mathbb{R}^{2n}} \).

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 \( \mathcal{S}\mathcal{O}(3) \)1, Reissner-Mindlin shells \( \mathcal{S}^2 \)2 and micro-magnetics \( \mathcal{S}^{2} \) 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; // (1)!
    ManifoldType::valueSize; // (2)!
    ManifoldType::correctionSize; // (3)!
    typename ManifoldType::CoordinateType; // (4)!
    typename ManifoldType::CorrectionType; // (5)!
    { var.getValue() } -> std::convertible_to<typename ManifoldType::CoordinateType>; // (6)!
    { var.setValue(value) } -> std::same_as<void>; // (7)!
    { var+=correction };  // (8)!
    //...
  };
}
  1. The type for the coordinate values, usually double.
  2. 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.
  3. 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).
  4. The type to store the element coordinates is usually Eigen::Vector<double,ManifoldType::valueSize>
  5. The type to store the element correction type is usually Eigen::Vector<double,ManifoldType::correctionSize>
  6. Access the underlying coordinate vector of the manifold element.
  7. Directly set the value. E.g., set Ikarus::UnitVector<double,3> a; a.setValue(Eigen::Vector3d::UnitZ());

  8. 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 $$ \boldsymbol{x} \in \mathbb{R}^n $$ realTuple.hh
Unit sphere $$ \boldsymbol{x} \in \mathcal{S}^{n-1}, \quad \mathcal{S}^{n-1} = \left\{ \boldsymbol{x} \in \mathbb{R}^n : \boldsymbol{x}\cdot \boldsymbol{x} = 1 \right\} $$ unitVector.hh

  1. 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. ↩

  2. 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. ↩