Geometry

Mantis' Geometry module contains all functionality related to geometry.

Geometry

An $(n, m)$ geometry $Φ$ is a collection of $L$ mappings ${Φ_{i}}_{i = 1}^{L}$ that map the canonical $n$ -dimensional domain, $Ω^{0} := [0, 1]^{n}$ into $L$ $n$ -dimensional simply connected subdomains, $Ω_{i}^{1}$ with $i = 1, \dots, L$ , of $R^{m}$ . Moreover, $⋂_{i = 1}^{L} Ω_{i}^{1} = \emptyset$ and using Core: Argument ${\overset{―}{Ω}}_{i}^{1} \cap {\overset{―}{Ω}}_{j}^{1} \subset \partial Ω_{i}^{1} \cup \partial Ω_{j}^{1}$ with $i, j = 1, \dots, L$ .

Note that

Φ_{i} (ξ_{1}, \dots, ξ_{n}) = (x_{1}, \dots, x_{m}),

and we use $Φ_{i, j} = x_{j}$ to mean the $j$ -th component of the mapping $Φ$ of element $i$ .

Tensor Product Geometry

Given an $(n_{1}, m_{1})$ geometry $Φ^{1}$ of $L_{1}$ mappings and an $(n_{2}, m_{2})$ geometry $Φ^{2}$ of $L_{2}$ mappings, i.e.,

Φ_{i}^{1} : [0, 1]^{n_{1}} \mapsto Ω_{i}^{1} \subset R^{m_{1}}, i = 1, \dots, L_{1}

and

Φ_{i}^{2} : [0, 1]^{n_{2}} \mapsto Ω_{i}^{2} \subset R^{m_{2}}, i = 1, \dots, L_{2}

the tensor product geometry $Φ := Φ^{1} \otimes Φ^{2}$ is an $(n_{1} + n_{2}, m_{1} + m_{2})$ geometry made up of a collection of $L_{1} L_{2}$ mappings $Φ_{k}$

Φ_{k = L_{1} (j - 1) + I} : [0, 1]^{n_{1}} \times [0, 1]^{n_{2}} \mapsto Ω_{k} = Ω_{i}^{1} \times Ω_{j}^{2} \subset R^{m_{1} + m_{2}}, i = 1, \dots, L_{1}, and j = 1, \dots, L 2 .

Specifically, we have

Φ_{L_{1} (j - 1) + i, l} (ξ_{1}, \dots, ξ_{n}) := {\begin{cases} Φ_{i, l}^{1} (ξ_{1}, \dots, ξ_{n_{1}}), & if l \leq n_{1} \\ Φ_{j, l - n_{1}}^{2} (ξ_{n_{1} + 1}, \dots, ξ_{n_{1} + n_{2}}), & if n_{1} < l \leq n_{1} + n_{2} \end{cases}, i = 1, \dots, L_{1}, j = 1, \dots, L_{2}, and l = 1, \dots, m_{1} + m_{2} .

The Jacobian of this geometry

J_{l, v}^{k} := \frac{\partial Φ_{k, l}}{\partial ξ_{v}}

is given by

\frac{\partial Φ_{L_{1} (j - 1) + i, l}}{\partial ξ_{v}} (ξ_{1}, \dots, ξ_{n}) := {\begin{cases} \frac{\partial Φ_{i, l}^{1}}{\partial ξ_{v}} (ξ_{1}, \dots, ξ_{n_{1}}), & if l \leq n_{1}, and v \leq m_{1} \\ \frac{\partial Φ_{j, l - n_{1}}^{2}}{\partial ξ_{u - m_{1}}} (ξ_{n_{1} + 1}, \dots, ξ_{n_{1} + n_{2}}), & if n_{1} < l \leq n_{1} + n_{2}, and m_{1} < v \leq m_{1} + m_{2} \\ 0 & otherwise \end{cases}, i = 1, \dots, L_{1}, j = 1, \dots, L_{2}, and l = 1, \dots, m_{1} + m_{2} .

Evaluation

Given the NTuple ξ of $n$ Vectors, $ξ^{i}$ , $i = 1, \dots, n$ , each containing $m_{i}$ unidimensional coordinates $ξ_{j}^{i}$ , $i = 1, \dots, n$ and $j = 1, \dots m_{i}$ , the tensor product geometry is evaluated at the element element_idx and at the $\prod_{i = 1}^{n} m_{i}$ tensor product points $V_{k = j_{1} + \sum_{i = 2}^{n} (j_{i} - 1) \prod_{l = 1}^{i - 1} m_{l}} = (ξ_{j_{1}}^{1}, \dots, ξ_{j_{n}}^{n})$ , with $j_{i} = 1, \dots, m_{i}$ .

The output is a matrix, $X$ of dimensions $(\prod_{i = 1}^{n} m_{i}) \times m$ (the number of tensor product points where the geometry is evaluated in element element_idx, and the dimension of the embedding space to where the canonical element is mapped into. Specifically:

X_{k, l} = Φ_{r, l} (ξ_{j_{1}}^{1}, \dots, ξ_{j_{n}}^{n}),

where $r =$ element_idx, and $k = j_{1} + \sum_{i = 2}^{n} (j_{i} - 1) \prod_{l = 1}^{i - 1} m_{l}$ , as before.

Jacobian

Given the NTuple ξ of $n$ Vectors, $ξ^{i}$ , $i = 1, \dots, n$ , each containing $m_{i}$ unidimensional coordinates $ξ_{j}^{i}$ , $i = 1, \dots, n$ and $j = 1, \dots m_{i}$ , evaluates the Jacobian of the tensor product geometry at the element element_idx and at the $\prod_{i = 1}^{n} m_{i}$ tensor product points $V_{k = j_{1} + \sum_{i = 2}^{n} (j_{i} - 1) \prod_{l = 1}^{i - 1} m_{l}} = (ξ_{j_{1}}^{1}, \dots, ξ_{j_{n}}^{n})$ , with $j_{i} = 1, \dots, m_{i}$ .

The output is a matrix, $J$ of dimensions $(\prod_{i = 1}^{n} m_{i}) \times m \times n$ (the number of tensor product points where the geometry is evaluated in element element_idx, the dimension of the embedding space to where the canonical element is mapped into, and the dimension of the canonical element, which is the same as the dimension of the element's manifold). Specifically:

J_{k, l, s} = \frac{\partial Φ_{r, l}}{\partial ξ_{s}} (ξ_{j_{1}}^{1}, \dots, ξ_{j_{n}}^{n}),

where $r = element_idx$ , and $k = j_{1} + \sum_{i = 2}^{n} (j_{i} - 1) \prod_{l = 1}^{i - 1} m_{l}$ , as before.

All docstrings from Mantis.Geometry

Mantis.Geometry Module

julia

module Geometry

Contains all geometry structure definitions and related methods.

Geometry ​

Geometry ​

Tensor Product Geometry ​

Evaluation ​

Jacobian ​

All docstrings from Mantis.Geometry ​

Geometry

Geometry

Tensor Product Geometry

Evaluation

Jacobian

All docstrings from Mantis.Geometry