Day3 1D to 2D - macOS Serverjupiter.ethz.ch/.../2018/docs/Day3_1D_to_2D.pdf · Numerical Quadrature...

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Introduction to Finite Element Modelling in Geosciences: From 1D to 2D (See Chapter 5 in the Course Notes) Day 3 Patrick Sanan

Transcript of Day3 1D to 2D - macOS Serverjupiter.ethz.ch/.../2018/docs/Day3_1D_to_2D.pdf · Numerical Quadrature...

Page 1: Day3 1D to 2D - macOS Serverjupiter.ethz.ch/.../2018/docs/Day3_1D_to_2D.pdf · Numerical Quadrature in 2D Change of Coordinates - we've seen it before! 2D Gauss-Legendre Integration

Introduction to Finite Element Modelling in Geosciences:

From 1D to 2D

(See Chapter 5 in the Course Notes)

Day 3Patrick Sanan

Page 2: Day3 1D to 2D - macOS Serverjupiter.ethz.ch/.../2018/docs/Day3_1D_to_2D.pdf · Numerical Quadrature in 2D Change of Coordinates - we've seen it before! 2D Gauss-Legendre Integration

From 1D to 2DThe good news (and a powerful feature of the FEM) is that most things remain the same, regardless of dimension (or physics!)

Main conceptual complication : Quadrature in 2D (but if you understood the "change of variables" section from the last lecture, you are most of the way there)

• Choose physics (PDE)• Mesh domain• Choose basis/interpolation functions• Define element matrices and vectors• Assemble elementwise matrices/vectors into global matrices/vectors• Solve system• Plot, postprocess, analyze, ...

Main practical complication: indexing

Page 3: Day3 1D to 2D - macOS Serverjupiter.ethz.ch/.../2018/docs/Day3_1D_to_2D.pdf · Numerical Quadrature in 2D Change of Coordinates - we've seen it before! 2D Gauss-Legendre Integration

Numerical Quadrature in 2D

Change of Coordinates - we've seen it before!

2D Gauss-Legendre Integration - product of 1D rules!

Warning! Make sure to correctly compute derivatives of shape functions (see 5.24)

Z 1

�1

Z 1

�1f(⇠, ⌘)d~⇠ ⇡

nxX

i=1

nyX

j=1

f(⇠i, ⌘j)wiwj

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Convenient to introduce linear numbering

=

NipX

k=1

f(⇠k, ⌘k)Wk

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Nip.= nxny

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Page 4: Day3 1D to 2D - macOS Serverjupiter.ethz.ch/.../2018/docs/Day3_1D_to_2D.pdf · Numerical Quadrature in 2D Change of Coordinates - we've seen it before! 2D Gauss-Legendre Integration

Isoparametric ElementsImportant point: you could choose any set of basis functions ( ) that you like,

and if you use a reference element, you can choose any way you like to map it to the spatial/deformed element.

We'd like to make choices that give, simple, efficient methods, though

Terminology Warning! Another place where FEM literature can be confusing is the term "element". Usually, we use this term to mean the subdomains into which we slice our domain. It can also be used to refer to the basis functions. Here, it refers to the latter!

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⌦eref

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⌦e<latexit sha1_base64="s0lGEjbJ7f3TKgBbRYwDbQnABDc=">AAAC0XichVFLS8NAEJ7GV1tfVY9egkXwVBIR9Fh84UWsaB/QVtmk2xiaF5u0UIsgXr151T+mv8WD366poEW6YTOz33zz7cyOFXlunBjGe0abmZ2bX8jm8otLyyurhbX1Whz2hc2rduiFomGxmHtuwKuJm3i8EQnOfMvjdat3JOP1ARexGwbXyTDibZ85gdt1bZYAarQufO6wG35bKBolQy190jFTp0jpqoSFD2pRh0KyqU8+cQooge8Roxhfk0wyKALWphEwAc9VcU4PlEduHywOBgPaw9/BqZmiAc5SM1bZNm7xsAUyddrGPlWKFtjyVg4/hv3EvleY8+8NI6UsKxzCWlDMKcVz4AndgTEt00+Z41qmZ8quEurSgerGRX2RQmSf9o/OMSICWE9FdDpRTAcaljoP8AIBbBUVyFceK+iq4w4sU5YrlSBVZNATsPL1UQ/GbP4d6qRT2y2ZRsm83CuWD9OBZ2mTtmgHU92nMp1RBXXIab7QK71pV9pQe9SevqlaJs3ZoF9Le/4CRciR3w==</latexit><latexit sha1_base64="s0lGEjbJ7f3TKgBbRYwDbQnABDc=">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</latexit><latexit sha1_base64="s0lGEjbJ7f3TKgBbRYwDbQnABDc=">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</latexit><latexit sha1_base64="s0lGEjbJ7f3TKgBbRYwDbQnABDc=">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</latexit>

Page 5: Day3 1D to 2D - macOS Serverjupiter.ethz.ch/.../2018/docs/Day3_1D_to_2D.pdf · Numerical Quadrature in 2D Change of Coordinates - we've seen it before! 2D Gauss-Legendre Integration

Isoparametric Elements

Isoparametric elements use the same basis functions for the mapping from the reference element and for the interpolation functions!

~x = (x, y, . . .) = �(~⇠)<latexit sha1_base64="QjLuVU5Y20MjHZSk/hw3Bb8aBr8=">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</latexit><latexit sha1_base64="QjLuVU5Y20MjHZSk/hw3Bb8aBr8=">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</latexit><latexit sha1_base64="QjLuVU5Y20MjHZSk/hw3Bb8aBr8=">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</latexit><latexit sha1_base64="QjLuVU5Y20MjHZSk/hw3Bb8aBr8=">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</latexit>

2

64xy...

3

75 =hN1(~⇠) N2(~⇠) N3(~⇠) . . .

i

2

6664

x1 y1 . . .x2 y2 . . .x3 y3 . . ....

.... . .

3

7775

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⌦eref

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⌦e<latexit sha1_base64="s0lGEjbJ7f3TKgBbRYwDbQnABDc=">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</latexit><latexit sha1_base64="s0lGEjbJ7f3TKgBbRYwDbQnABDc=">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</latexit><latexit sha1_base64="s0lGEjbJ7f3TKgBbRYwDbQnABDc=">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</latexit><latexit sha1_base64="s0lGEjbJ7f3TKgBbRYwDbQnABDc=">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</latexit>

Page 6: Day3 1D to 2D - macOS Serverjupiter.ethz.ch/.../2018/docs/Day3_1D_to_2D.pdf · Numerical Quadrature in 2D Change of Coordinates - we've seen it before! 2D Gauss-Legendre Integration

Jacobian Computation

Can precompute this!

J =

"@x@⇠

@y@⇠

@x@⌘

@y@⌘

#

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x = N1(⇠, ⌘)x1 +N2(⇠, ⌘)x2 + . . .

y = N1(⇠, ⌘)y1 +N2(⇠, ⌘)y2 + . . .<latexit sha1_base64="HQ39r43A7Q/8TLk1ppBQzsJ71Rg=">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</latexit><latexit sha1_base64="HQ39r43A7Q/8TLk1ppBQzsJ71Rg=">AAADLnichVHRStxAFD1Ga3W1urWPfQkugtKyJItQXwSprfSlRaGrgpFlkh3XYbNJSGaXjeI/+Rv9gUKF0lfffG3BB0/GWNRFnDC5d86958y9c/0kVJl2nF9j1vjEi8mXU9OVmdlXc/PV1wu7WdxPA9kM4jBO932RyVBFsqmVDuV+kkrR80O553c3i/jeQKaZiqPvOk/kYU90InWkAqEJtarB0F63v7XcZW+o3ntSi5Vhy7XfEWrchxqEvLAd68zzKvljSj5Kye9RWtWaU3fMskcdt3RqKNd2XL2AhzZiBOijB4kImn4IgYzfAVw4SIgd4pRYSk+ZuMQZKuT2mSWZIYh2+e/wdFCiEc+FZmbYAW8JuVMybSxxbxlFn9nFrZJ+RnvNfWKwzpM3nBrlosKc1qfitFH8SlzjmBnPMXtl5l0tzzOLrjSOsGa6UawvMUjRZ/Bf5xMjKbGuidj4bDI71PDNecAXiGibrKB45TsF23TcphXGSqMSlYqCeilt8fqsh2N2Hw911Nlt1F2n7u6s1jY+lgOfwlssYplT/YANfME26wjwA1f4i3/WufXT+m39uU21xkrOGzxY1uUNW+Gw0Q==</latexit><latexit sha1_base64="HQ39r43A7Q/8TLk1ppBQzsJ71Rg=">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</latexit><latexit sha1_base64="HQ39r43A7Q/8TLk1ppBQzsJ71Rg=">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</latexit>

N1(⌘, ⇠)<latexit sha1_base64="CL3E1ISnqA0NJ56Fh6IMQXL7pxw=">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</latexit><latexit sha1_base64="CL3E1ISnqA0NJ56Fh6IMQXL7pxw=">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</latexit><latexit sha1_base64="CL3E1ISnqA0NJ56Fh6IMQXL7pxw=">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</latexit><latexit sha1_base64="CL3E1ISnqA0NJ56Fh6IMQXL7pxw=">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</latexit>

J =

"@N1@⇠

@N2@⇠

@N3@⇠ . . . @N9

@⇠@N1@⌘

@N2@⌘

@N3@⌘ . . . @N9

@⌘

#

2

666664

x1 y1x2 y2x3 y3...

...x9 y9

3

777775

<latexit sha1_base64="w+s5o8mZ3RGTCiZKawJbAidM5Nw=">AAAEqHicjVFbi9NAFD6NUXfXy3b10ZfBoogPJWkF7YOwuLqIoFSw25VOKZN0kg7Njcm0tIb8UMWfsg+emaaK7ZbuhMm5fec7l/GySOTKcX7WrFv27Tt3Dw6P7t1/8PC4fvLoIk9n0uc9P41SeemxnEci4T0lVMQvM8lZ7EW8703PdLw/5zIXafJNLTM+jFmYiED4TKFrdFL7TWOmJl5QfCrJW0IjHqgB9XgokoJJyZZl4etTEhpI5hc0Y1IJFpEvI7f8Z9GFKMnzbUzrBpj2BgYh0ThV+XXYzgaW0j19ccX2N7YL1N4E3bg1DaY8GVc7JFSKcKKGO/ZbksXIRc4l/nGixahljNbKaBujrQ06X1dfKSbeMfEOuabeqN5wmo45ZFtxK6UB1emm9V9AYQwp+DCDGDgkoFCPgEGO3wBccCBD3xAK9EnUhIlzKOEIc2eI4ohg6J3iP0RrUHkTtDVnbrJ9rBLhlZhJ4Bnec8PoIVpX5ajnKK/w/jC+cGeFwjDrDpcoPWQ8NIyf0a9ggoh9mXGFXPeyP1NPpSCAN2Yagf1lxqPn9P/yvMeIRN/URAh8MMgQOTxjz3EDCcoedqC3vGYgZuIxSmYkNyxJxciQT6LU28d+8JndzUfdVi5aTddpul9fNU7fVQ9+AE/gKbzAV30Np/ARutiHb51ZwpJWbr+0u3bf/r6CWrUq5zH8d2zvD3XUO2I=</latexit><latexit sha1_base64="w+s5o8mZ3RGTCiZKawJbAidM5Nw=">AAAEqHicjVFbi9NAFD6NUXfXy3b10ZfBoogPJWkF7YOwuLqIoFSw25VOKZN0kg7Njcm0tIb8UMWfsg+emaaK7ZbuhMm5fec7l/GySOTKcX7WrFv27Tt3Dw6P7t1/8PC4fvLoIk9n0uc9P41SeemxnEci4T0lVMQvM8lZ7EW8703PdLw/5zIXafJNLTM+jFmYiED4TKFrdFL7TWOmJl5QfCrJW0IjHqgB9XgokoJJyZZl4etTEhpI5hc0Y1IJFpEvI7f8Z9GFKMnzbUzrBpj2BgYh0ThV+XXYzgaW0j19ccX2N7YL1N4E3bg1DaY8GVc7JFSKcKKGO/ZbksXIRc4l/nGixahljNbKaBujrQ06X1dfKSbeMfEOuabeqN5wmo45ZFtxK6UB1emm9V9AYQwp+DCDGDgkoFCPgEGO3wBccCBD3xAK9EnUhIlzKOEIc2eI4ohg6J3iP0RrUHkTtDVnbrJ9rBLhlZhJ4Bnec8PoIVpX5ajnKK/w/jC+cGeFwjDrDpcoPWQ8NIyf0a9ggoh9mXGFXPeyP1NPpSCAN2Yagf1lxqPn9P/yvMeIRN/URAh8MMgQOTxjz3EDCcoedqC3vGYgZuIxSmYkNyxJxciQT6LU28d+8JndzUfdVi5aTddpul9fNU7fVQ9+AE/gKbzAV30Np/ARutiHb51ZwpJWbr+0u3bf/r6CWrUq5zH8d2zvD3XUO2I=</latexit><latexit sha1_base64="w+s5o8mZ3RGTCiZKawJbAidM5Nw=">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</latexit><latexit sha1_base64="w+s5o8mZ3RGTCiZKawJbAidM5Nw=">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</latexit>

Compare to (5.23) in the notes

⌦eref

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⌦e<latexit sha1_base64="s0lGEjbJ7f3TKgBbRYwDbQnABDc=">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</latexit><latexit sha1_base64="s0lGEjbJ7f3TKgBbRYwDbQnABDc=">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</latexit><latexit sha1_base64="s0lGEjbJ7f3TKgBbRYwDbQnABDc=">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</latexit><latexit sha1_base64="s0lGEjbJ7f3TKgBbRYwDbQnABDc=">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</latexit>

Page 7: Day3 1D to 2D - macOS Serverjupiter.ethz.ch/.../2018/docs/Day3_1D_to_2D.pdf · Numerical Quadrature in 2D Change of Coordinates - we've seen it before! 2D Gauss-Legendre Integration

What would a non-isoparametric element look like?

Defined by 4 green points, even though there are 9 basis functions

4 < 9 , so subparametric

Yellow points defined by midpoints

⌦eref

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⌦eisoparametric

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⌦esubparametric

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Page 8: Day3 1D to 2D - macOS Serverjupiter.ethz.ch/.../2018/docs/Day3_1D_to_2D.pdf · Numerical Quadrature in 2D Change of Coordinates - we've seen it before! 2D Gauss-Legendre Integration

Boundary ConditionsDirichlet boundary conditions (prescribed value)

Neumann boundary conditions (prescribed normal derivative)

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@u

@n= g(~x)

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Modify system

Modify righthand side