The Serendipity Pyramid Finite Element

Transcription

1 The Seendipity Pyamid Finite Element Andew Gillette - Univesity of Aizona Andew Gillette - U. Aizona Seendipity Pyamid Element MAFELAP June / 24

2 Table of Contents 1 A bief pictoal oveview 2 Context and backgound 3 Shape functions and degees of feedom 4 Dimension compaison and minimality Andew Gillette - U. Aizona Seendipity Pyamid Element MAFELAP June / 24

3 Outline 1 A bief pictoal oveview 2 Context and backgound 3 Shape functions and degees of feedom 4 Dimension compaison and minimality Andew Gillette - U. Aizona Seendipity Pyamid Element MAFELAP June / 24

4 Regula pyamid + = Andew Gillette - U. Aizona Seendipity Pyamid Element MAFELAP June / 24

5 Seendipity pyamid + = Andew Gillette - U. Aizona Seendipity Pyamid Element MAFELAP June / 24

6 What lies inside the pyamids? Classical pyamids have mysteious inteios. Andew Gillette - U. Aizona Moden pyamids have tiny pyamids inside! Seendipity Pyamid Element MAFELAP June / 24

7 Outline 1 A bief pictoal oveview 2 Context and backgound 3 Shape functions and degees of feedom 4 Dimension compaison and minimality Andew Gillette - U. Aizona Seendipity Pyamid Element MAFELAP June / 24

8 Pyamids and the Peiodic Table of Finite Elements The peiodic table of finite elements (pepaed by Doug Anold & Andes Logg): Squae-based pyamids povide a link between tet and hex meshing egimes. A confoming pyamid element should match and elements on its faces. Seendipity elements offe substantial eduction in local basis size. Tenso-poduct-based pyamids ae known; seendipity-based elements ae not! Andew Gillette - U. Aizona Seendipity Pyamid Element MAFELAP June / 24

9 A histoy of pyamid elements Fist implementations in computational electomagnetics BEDROSIAN 1992, ZGAINSKI ET AL 1996, COULOMB ET AL 1997 Genealizations to highe ode elements and exact sequences BERGOT, COHEN, AND DURUFLÉ 2010 NIGAM, PHILLIPS 2012 (two papes) Moden implementation and additional analysis FUENTES, KEITH, DEMKOWICZ, NAGARAJ 2015 (implementation) WITHERDEN, VINCENT 2015 (quadatue) CHAN, WARBURTON 2015 (tace inequalities) Seentipity elements and pyamids ARNOLD, AWANOU 2011 (seendipity elements on cubes; supelinea degee) LIU ET AL 2004 and 2011 (piecewise-defined quadatic seendipity pyamid) Andew Gillette - U. Aizona Seendipity Pyamid Element MAFELAP June / 24

10 Refeence geomety and mappings We follow the geomety conventions of Nigam and Phillips. The infinite pyamid geomety is K := { (x, y, z) R 3 : 0 x, y 1, 0 z }. The efeence pyamid geomety is ˆK := { (ˆx, ŷ, ẑ) R 3 : 0 ˆx, ŷ, ẑ, ˆx 1 ẑ, ŷ 1 ẑ }. K Define a bijective change of coodinates φ : K ˆK by ( ) ˆx φ(x, y, z) = 1 + ẑ, ŷ 1 + ẑ, ẑ, 0 z < 1 + ẑ (0, 0, 1), z = Given u : ˆK R, the pullback of u to K by φ is φ u : K R whee (φ u)(x, y, z) := u(φ(x, y, z)) ˆK Andew Gillette - U. Aizona Seendipity Pyamid Element MAFELAP June / 24

11 Pullback example Conside the (possibly ational) function u : ˆK R u = ˆx a ŷ b (1 ẑ) c a b with c a b 0. The pullback of u to K by φ fo 0 z < is K ˆK (φ u)(x, y, z) = u(φ(x, y, z)) ( ) ˆx = u 1 + ẑ, ŷ 1 + ẑ, ẑ 1 + ẑ ( ˆx a ŷ b = 1 ẑ (1 + ẑ) a+b 1 + ẑ = ˆx a ŷ b (1 + ẑ) c and (φ u)(x, y, ) = u(0, 0, 1) = 0. ) c a b Note that u pulls back to a nice ational function that vanishes as z Andew Gillette - U. Aizona Seendipity Pyamid Element MAFELAP June / 24

12 Outline 1 A bief pictoal oveview 2 Context and backgound 3 Shape functions and degees of feedom 4 Dimension compaison and minimality Andew Gillette - U. Aizona Seendipity Pyamid Element MAFELAP June / 24

13 Shape functions on K and ˆK The supelinea degee of a monomial is defined by: ( n ) sldeg := α i. αi 1 i=1 x α i i Define shape functions on K as follows: Q [,] := S [,] := { } x a y b : 0 a, b j (1 + z) j j=0 { } x a y b : 0 a, b j, sldeg(x a y b ) j (1 + z) j j=0 Define shape functions on ˆK as those whose pullback is a shape function on K. ( ) { φ Q [,] := u : ˆK } R : φ u Q [,]. ( φ S [,] ) := { u : ˆK } R : φ u S [,]. Andew Gillette - U. Aizona Seendipity Pyamid Element MAFELAP June / 24

14 Counting the shape functions dim Q [,] dim S [,] = dim = { } x a y b : 0 a, b j (1 + z) j j=0 dim Q j Λ 0 (I 2 ) 2d tenso poduct, ode j j=0 = 1 6 ( ). = dim = 1 + Since φ is an isomophism, ( ) dim φ { } x a y b : 0 a, b j, sldeg(x a y b ) j (1 + z) j j=0 dim S j Λ 0 (I 2 ) j=1 = 1 6 ( ) S [,] = dim S [,] and dim φ 2d seendipity, ode j ( S [,] ) = dim S [,]. Andew Gillette - U. Aizona Seendipity Pyamid Element MAFELAP June / 24

15 The lowest ode bubble function on K and ˆK Define the lowest ode bubble function b : K R by b(x, y, z) := x(1 x)y(1 y)z (1 + z) 3 b vanishes on K i.e. on all of x = 0, x = 1, y = 0, y = 1, z = 0 and as z. We can ewite b(x, y, z) = x(1 x)y(1 y) x(1 x)y(1 y). (1 + z) 2 (1 + z) 3 The tem x 2 y 2 appeas afte expanding numeatos, so b Q[3,3] (1 + z) 3 3. To allow x 2 y 2 as a numeato in a seendipity shape function we need o a highe exponent in the denominato since sldeg(x 2 y 2 ) = 4. In all, we have b Q [,] 3 and b S [,] 5. But whee ae these mysteious numbes 3 and 5 in the geomety? b(x, y, z) (1 + z) 2, Andew Gillette - U. Aizona Seendipity Pyamid Element MAFELAP June / 24

16 Visualizing the dimension Y 2 Λ0 Y 3 Λ0 Y 4 Λ0 Y 5 Λ0 Y 2 Λ 0 Y 3 Λ 0 Y 4 Λ 0 Y 5Λ 0 2D tenso poduct Q and seendipity elements S stack to make pyamids Fist inteio degees of feedom appea at ode 3 fo Q and ode 5 fo S Andew Gillette - U. Aizona Seendipity Pyamid Element MAFELAP June / 24

17 Visualizing the inteio dimension Y 5 Λ0 inteio Y 5Λ 0 inteio Note: Q (esp. S ) pyamids have tiny Q 3 (esp. S 5) pyamids inside! Andew Gillette - U. Aizona Seendipity Pyamid Element MAFELAP June / 24

18 Classical degees of feedom To each vetex v, To each edge e, To each tiangula face, To the paallelogam face, To the thee-dimensional inteio int, u u(v) u (t e u) q, e u (t u) q, u (t u) q, u int (t int u) q, q P e q P q P q R int P v P e P P R int ( ) Y Λ 0 R P 2 Λ 1 (e) P 3 Λ 2 ( ) Q 1 Λ2 ( ) φ b Q [ 3, 3] 3 Λ 3 (int) ( ) Y Λ 0 R P 1 Λ1 (e) P 2 Λ2 ( ) P 4 Λ 2 ( ) φ b S [ 5, 5] 5 Λ 3 (int) ( whee φ b Q [ 3, 3] 3 ) { } Λ 3 (int) := span u dv : φ u = bq with q Q [ 3, 3] 3 Andew Gillette - U. Aizona Seendipity Pyamid Element MAFELAP June / 24

19 Counting the degees of feedom dim Y Λ 0 = P 2 Λ 1 (e) + 4 P 3 Λ 2 ( ) + Q 1 Λ2 ( ) + = 5 + 8( 1) + 2( 2)( 1) + ( 1) 2 + = 1 6 ( ) = dim Q [,] Q [ 3, 3] 3 (2 3)( 2)( 1) 6 dim Y Λ 0 = P 1 Λ1 (e) + 4 P 2 Λ2 ( ) + P 4 Λ 2 ( ) + = 5 + 8( 1) + 2( 2)( 1) + = 1 6 ( ) = dim S [,] ( 3)( 2) 2 + S [ 5, 5] 5 ( 4)( 3)( 2) 6 Andew Gillette - U. Aizona Seendipity Pyamid Element MAFELAP June / 24

20 Unisolvence and polynomial epoduction Theoem [G., 2015] ( Λ 0 ae unisolvent fo φ The degees of feedom fo Y ( The degees of feedom fo Y Λ 0 ae unisolvent fo φ Q [,] S [,] ). ). Poof idea: Appeal to known elements fo unisolvence on bounday; use definition of shape functions fo inteio. Theoem [G., 2015] The shape functions of ode on ˆK epoduce polynomials of degee, i.e. ( ) ( ) P (R 3 ) φ and P (R 3 ) φ Q [,] S [,] Poof idea: Wite p P in powes of x, y, 1 z; show pullback is in shape function space. Andew Gillette - U. Aizona Seendipity Pyamid Element MAFELAP June / 24

21 Outline 1 A bief pictoal oveview 2 Context and backgound 3 Shape functions and degees of feedom 4 Dimension compaison and minimality Andew Gillette - U. Aizona Seendipity Pyamid Element MAFELAP June / 24

22 Compaing dimension counts key efeence dim Y Λ Gillette dim Y Λ 0 Gillette dim ˆP Begot, Cohen, Duuflé dim R (0) Nigam, Phillips (M2AN) dim U (0), Nigam, Phillips (IMA J.) Fuentes et al. Begot, Cohen, Duuflé. Jounal of Scientific Computing, Fuentes, Keith, Demkowicz, Nagaaj. Computes & Mathematics with Applications, Gillette. axiv: , 2015 Nigam, Phillips. IMA Jounal of Numeical Analysis, Nigam, Phillips. Mathematical Modelling and Numeical Analysis, Andew Gillette - U. Aizona Seendipity Pyamid Element MAFELAP June / 24

23 Minimality What is the minimum numbe of degees of feedom equied to make an ode pyamid element? On a tiangula face: P Λ 0 ( ) On the quadilateal base: S Λ 0 ( ) On the inteio: P 5Λ 3 (R 3 ) (No choice, eally) (See pape with Chistiansen below) (Loosely, b/c thee ae 5 linealy independent constaints to make a bubble function.) ( ) 2 Vitual element methods associate at least degees of feedom to the inteio 3 ( ) of a pyamid and dim P 5Λ 3 (R 3 2 ) =. 3 So, ou constuction appeas to be minimal! CHRISTIANSEN, G. Constuctions of some minimal finite element systems. Mathematical Modelling and Numeical Analysis, BEIRAO DA VEIGA, BREZZI, MARINI, RUSSO Seendipity nodal VEM spaces axiv: , Andew Gillette - U. Aizona Seendipity Pyamid Element MAFELAP June / 24

24 Acknowledgments Reseach Funding Suppoted in pat by the National Science Foundation gant DMS Slides and Pe-pints Andew Gillette - U. Aizona Seendipity Pyamid Element MAFELAP June / 24