From 27b4b2d914ca0be2a821582393c46378391af5af Mon Sep 17 00:00:00 2001 From: apple Date: Fri, 18 Apr 2025 01:06:38 -0700 Subject: [PATCH 01/27] test --- .DS_Store | Bin 0 -> 8196 bytes YiyueTest/yiyueTest.jl | 0 2 files changed, 0 insertions(+), 0 deletions(-) create mode 100644 .DS_Store create mode 100644 YiyueTest/yiyueTest.jl diff --git a/.DS_Store b/.DS_Store new file mode 100644 index 0000000000000000000000000000000000000000..955a089d59c9ed403053e3391c9ae3eea955cf20 GIT binary patch literal 8196 zcmeHML2eT<6n$OcCy3M=OiO}P1f?QKh#h1yB!w9!N>W5OD0|KT5+~pm zT!FK&g7<9~jXe{ASRg9CCHs%;_rAZMXX1VrfH^#y>;roMdaPn^TxHWzWL#>aZ22A` z8}fky6THGHR>&J@?N9|&0aZX1Pz6+he?bAB*;0BN&V4tktty}j{Fe&I??b{WHuZMx zto`X=V^aXyF1u~vGxGq;NxV(HT{~;3^y#t(*#7qr*5ljGx)~ z48_>l$xAF8CUsU@RX`PJDj>Q0F8UbbjCp%4e?Ka!Y+O`hj>{)?z}o+A3!mhJ?=#oW z@g~B3Zoy6EMj*tQTM=%X{wxkNMlz1j?<2HC?9ivQuTf!vJmP$dI5)|;N8bgbd5*i5 zQ}StLZz~oYU9qo35&GGhMt*jwRn9pxMk@RTZ#V;J@MC?Dom04`08?cif7c{3dcfYs}0eS;M1& literal 0 HcmV?d00001 diff --git a/YiyueTest/yiyueTest.jl b/YiyueTest/yiyueTest.jl new file mode 100644 index 0000000000..e69de29bb2 From a3fc7e5a8e57738f970dedaa8f7cb80cc52011aa Mon Sep 17 00:00:00 2001 From: Yiyue Feng <79966731+yiyuef@users.noreply.github.com> Date: Fri, 18 Apr 2025 23:18:36 -0700 Subject: [PATCH 02/27] Update math_framework.md --- docs/src/math_framework.md | 55 ++++++++++++++++++++++++++++++++++++++ 1 file changed, 55 insertions(+) diff --git a/docs/src/math_framework.md b/docs/src/math_framework.md index dbfd6dd04e..46798c6639 100644 --- a/docs/src/math_framework.md +++ b/docs/src/math_framework.md @@ -74,6 +74,58 @@ while the _contravariant basis_ is the opposite: gradient in ``x`` of the coordi \mathbf{e}^i = \nabla_x \xi^i ``` +Here is an intuitive example of how the _contravariant basis_ and _covariant basis_ changes when a coordinate is being transformed. + +![Different bases supported in ClimaCore.jl](controva_cova.png) + +Start with the _contravariant basis_, which is the case that most of us is more familier. As it can be seen in the figure below, we have 2D vector $\vec{a}$, +and two unit vectors $\vec{e_{1}}$ and $\vec{e_{2}}$, served as two basis. Thus we can represent vector $\vec{a}$ as: +```math +\vec{a} = a^{1} \vec{e_{1}} + a^{2} \vec{e_{2}} +``` + +And now we set amplify one of the basis $\vec{a}$ by 2 times, that is $\vec{e_{1}}^{'} = 2\vec{e_{1}}$. If we want to maintain the vector $\vec{a}$ to stay still, +obviously we need to set $a^{1}$ to its half: +```math +a^{1'} = \frac{1}{2} a^{1} +``` + +With this being said, the components we use to describe a certain vector are changing in the opposite manner of the the basis, and we are calling these +components _contravariant components_. The _contravariant vector_ is a vector with _contravariant components_. + +Now consider the _covariant basis_. We still have a 2D vector, but calling it $\vec{b}$. So if with the unit vectors $\vec{e_{1}}$ and $\vec{e_{2}}$, we can still +represent vector $\vec{b}$ in a "controvariant component" manner: +```math +\vec{b} = b^{1} \vec{e_{1}} + b^{2} \vec{e_{2}} +``` + +Then we try to project vector $\vec{b}$ onto the two unit vectors, and we have the following: +```math +b_{1} = \vec{b}\cdot \vec{e_{1}} +``` +```math +b_{2} = \vec{b}\cdot \vec{e_{2}} +``` + +If we plug $\vec{b} = b^{1} \vec{e_{1}} + b^{2} \vec{e_{2}}$ into the above two equations, we can have: +```math +b_{1} = \vec{b}\cdot \vec{e_{1}} = b^{1} \vec{e_{1}} \cdot \vec{e_{1}}+ b^{2} \vec{e_{2}}\cdot \vec{e_{1}} +``` +```math +b_{2} = \vec{b}\cdot \vec{e_{2}} = b^{2} \vec{e_{2}} \cdot \vec{e_{1}}+ b^{2} \vec{e_{2}}\cdot \vec{e_{2}} +``` + +And in this case, if we still amplify one of the basis $\vec{a}$ by 2 times, it is not difficult to find that $b_{1}$ would also be amplified by 2 times! +```math +b_{1}^{'} = b^{1} \vec{e_{1}}^{'} \cdot \vec{e_{1}}+ b^{2} \vec{e_{2}}\cdot \vec{e_{1}}^{'} = 2 (b^{1} \vec{e_{1}} \cdot \vec{e_{1}}+ b^{2} \vec{e_{2}}\cdot \vec{e_{1}} ) +``` + +In this case, the components of this vector are changing in the same manner of the the basis, and we are calling these +components _covariant components_. The _covariant vector_ is a vector with _covariant components_. + +To summarize, from what we can see from the figure as well, parallel projections would lead to controvariant components, while perpendicular projection would lead to covariant components. + + **Note**: @@ -85,6 +137,9 @@ while the _contravariant basis_ is the opposite: gradient in ``x`` of the coordi * things get a little more complicated in the presence of terrain, but ``\xi^3`` is radially aligned - the 3rd covariant component is aligned with W, but the 3rd contravariant component may not be (e.g. at the surface it is normal to the boundary). + + + ### Cartesian bases Analogously to `CartesianPoint`s, in ClimaCore, there are also `CartesianVector`s: From 2167cb30c11ea7d5b353dca5c47c708465bfc8ba Mon Sep 17 00:00:00 2001 From: Yiyue Feng <79966731+yiyuef@users.noreply.github.com> Date: Fri, 18 Apr 2025 23:19:13 -0700 Subject: [PATCH 03/27] Add files via upload --- docs/src/controva_cova.png | Bin 0 -> 92060 bytes 1 file 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correction --- docs/src/math_framework.md | 26 ++++++++++++++++++++------ 1 file changed, 20 insertions(+), 6 deletions(-) diff --git a/docs/src/math_framework.md b/docs/src/math_framework.md index 46798c6639..5f27d9eba8 100644 --- a/docs/src/math_framework.md +++ b/docs/src/math_framework.md @@ -62,6 +62,19 @@ then may want to consider different bases, as not all operators accept all bases _Covariance_ and _contravariance_ describe how the quantitative description of certain geometric or physical entities changes with a change of basis. +More specifically, + +__Covariant objects__ —whether you mean covariant components (the coefficients that sit in front of the basis) +or the covariant basis vectors themselves—co-vary with the coordinate grid. In other words, when you change coordinates, +these quantities transform in the same way as the coordinate differentials. By convention they carry lower indices (subscripts). + +__Contravariant objects__—whether you mean contravariant components or the contravariant basis vectors—vary contra +to the coordinate grid. That is, they transform in the opposite way to the coordinate differentials so as to keep +tensorial expressions invariant. By convention they carry upper indices (superscripts). + +In ClimaCore.jl, `CovariantVector`s are aligned with the _contravariant basis vectors_, but have _covariant components_. +Conversely, `ContravariantVector`s are aligned with the _covariant basis vectors_, but have _contravariant components_. + In ClimaCore.jl, the _covariant basis_ is specified by the partial derivative of the transformation from the reference element ``\xi \in [-1,1]^d`` (where ``d`` is the dimensionality of the domain ``\Omega``) to ``x`` in the physical space: @@ -74,11 +87,12 @@ while the _contravariant basis_ is the opposite: gradient in ``x`` of the coordi \mathbf{e}^i = \nabla_x \xi^i ``` -Here is an intuitive example of how the _contravariant basis_ and _covariant basis_ changes when a coordinate is being transformed. + +Here is a visual representation of how vectors can be represented in _contravariant_ and _covariant_ components. ![Different bases supported in ClimaCore.jl](controva_cova.png) -Start with the _contravariant basis_, which is the case that most of us is more familier. As it can be seen in the figure below, we have 2D vector $\vec{a}$, +Start with the _contravariant components_, which is exactly the definition of vectors we usually meet. As it can be seen in the figure above, we have a 2D vector $\vec{a}$, and two unit vectors $\vec{e_{1}}$ and $\vec{e_{2}}$, served as two basis. Thus we can represent vector $\vec{a}$ as: ```math \vec{a} = a^{1} \vec{e_{1}} + a^{2} \vec{e_{2}} @@ -91,9 +105,9 @@ a^{1'} = \frac{1}{2} a^{1} ``` With this being said, the components we use to describe a certain vector are changing in the opposite manner of the the basis, and we are calling these -components _contravariant components_. The _contravariant vector_ is a vector with _contravariant components_. +components _contravariant components_. This case corresponds to the definition of `ContravariantVector`s in ClimaCore.jl. -Now consider the _covariant basis_. We still have a 2D vector, but calling it $\vec{b}$. So if with the unit vectors $\vec{e_{1}}$ and $\vec{e_{2}}$, we can still +Now consider the _covariant components_. We still have a 2D vector, but calling it $\vec{b}$. So if with the unit vectors $\vec{e_{1}}$ and $\vec{e_{2}}$, we can still represent vector $\vec{b}$ in a "controvariant component" manner: ```math \vec{b} = b^{1} \vec{e_{1}} + b^{2} \vec{e_{2}} @@ -121,9 +135,9 @@ b_{1}^{'} = b^{1} \vec{e_{1}}^{'} \cdot \vec{e_{1}}+ b^{2} \vec{e_{2}}\cdot \vec ``` In this case, the components of this vector are changing in the same manner of the the basis, and we are calling these -components _covariant components_. The _covariant vector_ is a vector with _covariant components_. +components _covariant components_. This case corresponds to the definition of `CovariantVector`s in ClimaCore.jl. -To summarize, from what we can see from the figure as well, parallel projections would lead to controvariant components, while perpendicular projection would lead to covariant components. +From these two illustrative examples, we can see that parallel projections would lead to controvariant components, while perpendicular projection would lead to covariant components. From 1cd67669e84b007059ced433ac82ebb88e0c8a65 Mon Sep 17 00:00:00 2001 From: apple Date: Sun, 4 May 2025 00:12:19 -0700 Subject: [PATCH 06/27] adding polar coordinates --- docs/src/math_framework.md | 60 ++++++++++++++++++++++++++++++++------ 1 file changed, 51 insertions(+), 9 deletions(-) diff --git a/docs/src/math_framework.md b/docs/src/math_framework.md index 5f27d9eba8..038915a6b4 100644 --- a/docs/src/math_framework.md +++ b/docs/src/math_framework.md @@ -87,27 +87,34 @@ while the _contravariant basis_ is the opposite: gradient in ``x`` of the coordi \mathbf{e}^i = \nabla_x \xi^i ``` - +If we plot these basis in a curvilinear space, _Covariant basis_ “ride along” the coordinate surfaces (parallel), +while _contravariant basis_ “stick out” of those surfaces (perpendicular). See the plot below: + +![normal and tangent](normal_tangent.png) + Here is a visual representation of how vectors can be represented in _contravariant_ and _covariant_ components. -![Different bases supported in ClimaCore.jl](controva_cova.png) +![Different bases](controva_cova.png) -Start with the _contravariant components_, which is exactly the definition of vectors we usually meet. As it can be seen in the figure above, we have a 2D vector $\vec{a}$, +Start with the _contravariant components_, which is exactly the definition of vectors we usually meet. +As it can be seen in the figure above, we have a 2D vector $\vec{a}$, and two unit vectors $\vec{e_{1}}$ and $\vec{e_{2}}$, served as two basis. Thus we can represent vector $\vec{a}$ as: ```math \vec{a} = a^{1} \vec{e_{1}} + a^{2} \vec{e_{2}} ``` -And now we set amplify one of the basis $\vec{a}$ by 2 times, that is $\vec{e_{1}}^{'} = 2\vec{e_{1}}$. If we want to maintain the vector $\vec{a}$ to stay still, +And now we set amplify one of the basis $\vec{a}$ by 2 times, that is $\vec{e_{1}}^{'} = 2\vec{e_{1}}$. +If we want to maintain the vector $\vec{a}$ to stay still, obviously we need to set $a^{1}$ to its half: ```math a^{1'} = \frac{1}{2} a^{1} ``` -With this being said, the components we use to describe a certain vector are changing in the opposite manner of the the basis, and we are calling these -components _contravariant components_. This case corresponds to the definition of `ContravariantVector`s in ClimaCore.jl. +With this being said, the components we use to describe a certain vector are changing in the opposite manner of the the basis, +and we are calling these components _contravariant components_. This case corresponds to the definition of `ContravariantVector`s in ClimaCore.jl. -Now consider the _covariant components_. We still have a 2D vector, but calling it $\vec{b}$. So if with the unit vectors $\vec{e_{1}}$ and $\vec{e_{2}}$, we can still +Now consider the _covariant components_. We still have a 2D vector, but calling it $\vec{b}$. +So if with the unit vectors $\vec{e_{1}}$ and $\vec{e_{2}}$, we can still represent vector $\vec{b}$ in a "controvariant component" manner: ```math \vec{b} = b^{1} \vec{e_{1}} + b^{2} \vec{e_{2}} @@ -129,7 +136,8 @@ b_{1} = \vec{b}\cdot \vec{e_{1}} = b^{1} \vec{e_{1}} \cdot \vec{e_{1}}+ b^{2} \ b_{2} = \vec{b}\cdot \vec{e_{2}} = b^{2} \vec{e_{2}} \cdot \vec{e_{1}}+ b^{2} \vec{e_{2}}\cdot \vec{e_{2}} ``` -And in this case, if we still amplify one of the basis $\vec{a}$ by 2 times, it is not difficult to find that $b_{1}$ would also be amplified by 2 times! +And in this case, if we still amplify one of the basis $\vec{a}$ by 2 times, +it is not difficult to find that $b_{1}$ would also be amplified by 2 times! ```math b_{1}^{'} = b^{1} \vec{e_{1}}^{'} \cdot \vec{e_{1}}+ b^{2} \vec{e_{2}}\cdot \vec{e_{1}}^{'} = 2 (b^{1} \vec{e_{1}} \cdot \vec{e_{1}}+ b^{2} \vec{e_{2}}\cdot \vec{e_{1}} ) ``` @@ -137,7 +145,41 @@ b_{1}^{'} = b^{1} \vec{e_{1}}^{'} \cdot \vec{e_{1}}+ b^{2} \vec{e_{2}}\cdot \vec In this case, the components of this vector are changing in the same manner of the the basis, and we are calling these components _covariant components_. This case corresponds to the definition of `CovariantVector`s in ClimaCore.jl. -From these two illustrative examples, we can see that parallel projections would lead to controvariant components, while perpendicular projection would lead to covariant components. +From these two illustrative examples, we can see that parallel projections would lead to controvariant components, +while perpendicular projection would lead to covariant components. + +As to better connect the original idea of _covariant components_ / _contravariant components_ with the real application in ClimaCore.jl, +we bring the case of __polar coordinates__ -- a classic example of a __curvilinear coordinate system__. + +First of all, we have the Polar‐to‐Cartesian mapping as following: +```math +\vec{r} (r,\theta ) = (r \cos \theta, r \sin \theta) +``` + +Then we have the _covariant basis_: +```math +\vec{e_{r} } = \frac{\partial \vec{r} }{\partial r} = \frac{\partial (r \cos \theta, r \sin \theta) }{\partial r} = (\cos \theta, \sin \theta) +``` +```math +\vec{e_{\theta} } = \frac{\partial \vec{r} }{\partial \theta } = \frac{\partial (r \cos \theta, r \sin \theta) }{\partial \theta } = (-r\sin \theta,r \cos \theta) +``` + +$\vec{e_{r} }$ represents the direction that is tangent to the "radius direction", and $\vec{e_{\theta} }$ is tangent to the circle curve itself. + +And the _contravariant basis_: + +since we have $r(x,y) = \sqrt{x^{2} +y^{2} }$ and $\theta (x,y) = \arctan (\frac{y}{x} )$. Thus, + +```math +\vec{e^{r} } = \nabla r(x,y) = (\frac{x}{\sqrt{x^{2} +y^{2} }}, \frac{y}{\sqrt{x^{2} +y^{2} }}) = (\frac{x}{r},\frac{y}{r} ) = (\cos \theta, \sin \theta) +``` +```math +\vec{e^{\theta } } = \nabla \theta(x,y) = (\frac{-y}{x^{2} +y^{2}}, \frac{x}{x^{2} +y^{2}}) = (\frac{-\sin \theta}{r} , \frac{\cos \theta}{r}) +``` + +$\vec{e^{r} }$ represents the direction that is perpendicular to the circle curve, and $\vec{e^{\theta} }$ is perpendicular to the "radius direction". + + From 477caa3f454196cb935956db124a4655e1e1e265 Mon Sep 17 00:00:00 2001 From: apple Date: Sun, 4 May 2025 00:23:09 -0700 Subject: [PATCH 07/27] adding images --- .DS_Store | Bin 8196 -> 8196 bytes .gitignore | 2 +- docs/.DS_Store | Bin 0 -> 6148 bytes docs/src/normal_tangent.png | Bin 0 -> 250864 bytes 4 files changed, 1 insertion(+), 1 deletion(-) create mode 100644 docs/.DS_Store create mode 100644 docs/src/normal_tangent.png diff --git a/.DS_Store b/.DS_Store index 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See the plot below: +If we plot these bases in a curvilinear space, _covariant basis_ vectors “ride along” the coordinate surface liness (parallel), while _contravariant basis_ vectors “stick out” of those surface lines (perpendicular). See the plot from [Yatunin2025](@cite) below: ![normal and tangent](normal_tangent.png) -Here is a visual representation of how vectors can be represented in _contravariant_ and _covariant_ components (reference [Yatunin2025](@cite).). +Here is a visual representation of how vectors can be represented in _contravariant_ and _covariant_ components. ![Different bases](controva_cova.png) From d6c9bf8c19d798cc55deff75a41c270f817894bd Mon Sep 17 00:00:00 2001 From: apple Date: Wed, 7 May 2025 21:12:25 -0700 Subject: [PATCH 15/27] remove DS_Store3 --- docs/.DS_Store | Bin 6148 -> 0 bytes 1 file changed, 0 insertions(+), 0 deletions(-) delete mode 100644 docs/.DS_Store diff --git a/docs/.DS_Store b/docs/.DS_Store deleted file mode 100644 index f8a9739bb03897686ad86357bc9ecf39d90a6939..0000000000000000000000000000000000000000 GIT binary patch literal 0 HcmV?d00001 literal 6148 zcmeHKPfrs;6n_KNZUJTaQvsvN#$HSy2&5PjW2gnA#26~ASOl!Q?X<3JcbeTT2!y0p z_2dWe19ymq5d1rs)bLQDNJ3oU$G 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zSWj=?snh-Xz~Iop@IX9%W@I!zc6RJsK_9Zr;%3F;uQ|^g=B1rFU8{I@DO)p(wp+;u z^l6Z_cWFT%Y;#?vrH2Poxt$vN3?EF0bT>UpU6F68T=r>@Z_!r^dcR-vnB!!aZwEGW zSJ!KKq56ORZSoB*1Gu%6dMN=1r<6$Y7;DF!up?^cK*0RIlaf^Cwt@pSShxtMVLI(bM2KfG9VB?rl z*pMi;4(!Ag05O8oLQv;_8hDJMFr}~|5iKZ0h9b&PqOKT3hU2`U{89=V5@k3Lb@?Fb z$wb{yhiqn_@%#J#;h=C7G7vKGUok);3)zJ Date: Fri, 16 May 2025 21:37:11 -0700 Subject: [PATCH 26/27] phrasing modification --- .DS_Store | Bin 0 -> 8196 bytes docs/.DS_Store | Bin 0 -> 6148 bytes docs/src/math_framework.md | 7 +++---- 3 files changed, 3 insertions(+), 4 deletions(-) create mode 100644 .DS_Store create mode 100644 docs/.DS_Store diff --git a/.DS_Store b/.DS_Store new file mode 100644 index 0000000000000000000000000000000000000000..4a208048eaa58271c1a44f92dfc47c7353c28bdb GIT binary patch literal 8196 zcmeHML2uJA6n;PyqO}2%lN!6q)QZ>{o z@E`aK9Qh~w7f$d!+l)HxjxZ!_$^KmXd-lucmt-#^BGDN5HKG+Fa!?s7i)dyPp65DM z=Jd!iP{5ujBA-qvq>O?owjHnvSOu&CRspMkRp4Jx0Pk!r&L!`CHEUa|fK}kXRDhok 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