Banach space
This is a complete space equipped with a norm.
Divergence (of a tensor)
Let \(\sigma: \Omega \to \R^{d\times d}\) be a second-order tensor, i.e. a matrix-valued mapping on \(\Omega\).
The divergence of \(\sigma\) is the vector field \(\dv\sigma : \Omega \to \R^d\) with entries:
\[(\dv\sigma(\x))_i = \sum\limits_{j=1}^d \frac{\partial \sigma_{ij}}{\partial x_j}(\x), \quad i=1,\ldots,d.\]
Eulerian (viewpoint)
The Eulerian description of a medium…
Frobenius inner product
The Frobenius inner product between two square matrices \(M, N \in \mathbb{R}^{d\times d}\) is the real value \(M:N\) defined by:
\[M:N = \text{tr}(M^TN) = \sum\limits_{i,j=1}^d m_{ij} n_{ij}.\]
Hilbert space
A Hilbert space is, in particular a Banach space.
Lagrangian (viewpoint)
The Lagrangian description of a medium…
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Level set method
The level set method generally consists in representing a domain \(\Omega \subset \R^d\) in an implicit way when dealing with its evolution, i.e.
This method is more extensively described in Section 2.5.
Mean curvature
We refer to Section 2.2 for further details.
Strain
Strain is synonymous to deformation.
Variational inequality
A variational inequality is
Prototypes are obstacle problems, or contact problems.
Variational problem
A variational problem is defined by
Vorticity
The vorticity of a fluid is the trend to create eddies.
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