Add comments (work in progress): DO NOT MERGE #1
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jorgensd
reviewed
Oct 21, 2024
| # 1) The triangulation covers $\Omega$: $\cup_{j=1}^{M}K_j=\bar{\Omega}$ | ||
| # 2) No overlapping polyons: $\mathrm{int} K_i \cap \mathrm{int} K_j=\emptyset$ for $i\neq j$. | ||
| # 3) No vertex lines in the interior of a facet or edge of another element | ||
| # "No vertex lies"? |
jorgensd
reviewed
Oct 21, 2024
| # "No vertex lies"? | ||
| # | ||
| # We will call our polygonal domain $\mathcal{K}={K_j}_{j=1}^{M}$. | ||
| # Maybe add some brackets around K_j? |
jorgensd
reviewed
Oct 21, 2024
| # using the mapping $F_j:K_{ref}\mapsto K_j$. | ||
| # | ||
| # We define the Jacobian of this mapping as $\mathbf{J_j}$. | ||
| # I would add a definition of the Jacobian here |
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You mean writing out the complete definition with basis functions and derivatives?
jorgensd
reviewed
Oct 21, 2024
| basix.CellType.triangle, 8, basix.LatticeType.gll, exterior=False, method=basix.LatticeSimplexMethod.warp | ||
| ) | ||
| # - | ||
| # Is this code snippet suppose to produce some output? |
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No, not sure why I made it a dropdown. will revert
jorgensd
reviewed
Oct 21, 2024
| # as the linear combination of the three basis functions on the reference element $X$. | ||
| # | ||
| # $$x = F_j(X)= \sum_{i=0}^3 \mathbf{p}_i \phi_i(X).$$ | ||
| # I guess i = 0 to 2? |
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Fixed first it here.
Can we discuss point 2 when Im back?
jorgensd
reviewed
Oct 21, 2024
| # ```{admonition} Can we use a similar kind of mapping on a quadrilateral/tetrahedral/hexahedral element? | ||
| # :class: dropdown tip | ||
| # Yes, for any polytope that we can describe with a Lagrange finite element, we can define a mapping from a reference to a physical element. | ||
| # I had to google the word polytope, maybe add a definition here or add it to the glossary |
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I've simplified the definition.
jorgensd
reviewed
Oct 21, 2024
| # As we have already seen, we can describe any cell in our subdivided domain with a mapping from the reference element. | ||
| # However, as we want to integrate over each element individually, we need to map the basis functions to and from the reference element. | ||
| # We call this map: $(\mathcal{F}_j\phi)(x)$. | ||
| # Can we add domain and codomain to the map? |
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Would need to think about this one, as it should hold for an arbitrary basis function (in any space),
jorgensd
reviewed
Oct 21, 2024
| # $$ | ||
| # (\mathcal{F}_j\phi)(x)= \phi(F_j^{-1}(x))=\phi(X). | ||
| # $$ | ||
| # I didn't really ge this. What does `j` represent? |
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There is a unique mapping for each cell in your physical mesh. It ties back to the definition of
jorgensd
reviewed
Oct 21, 2024
| # :class: dropdown, tip | ||
| # Their tangential component is only non-zero on the edge they are associated with, but the normal component can be non-zero there too. | ||
| # This implies that they are normal to the other edges. | ||
| # I would maybe add some equations here to make it more clear |
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It's not easy to see from equations (at least not for a simple-minded person like myself).
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