Geometry#

Struphy models are implemented in curvilinear coordinates and can be run on a variaty of mapped domains. Besides analytical mappings, there are discrete spline mappings available (IGA approach).

The (physical) domain \(\Omega \subset \mathbb R^3\) is an open subset of \(\mathbb R^3\), defined by a diffeomorphism

\[F:(0, 1)^3 \to \Omega\,,\qquad \boldsymbol{\eta} \mapsto F(\boldsymbol \eta) = \mathbf x\,,\]

mapping points \(\boldsymbol{\eta} \in (0, 1)^3 = \hat\Omega\) of the (logical) unit cube to physical points \(\mathbf x \in \Omega\). The corresponding Jacobain matrix \(DF:\hat\Omega \to \mathbb R^{3\times 3}\), its volume element \(\sqrt g: \hat\Omega \to \mathbb R\) and the metric tensor \(G:\hat\Omega \to \mathbb R^{3\times 3}\) are defined by

\[DF_{i,j} = \frac{\partial F_i}{\partial \eta_j}\,,\qquad \sqrt g = |\textnormal{det}(DF)|\,,\qquad G = DF^\top DF\,.\]

Only right-handed mappings (\(\textnormal{det}(DF) > 0\)) are admitted.

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