Boundary integral methods for NIMROD resistive wall

Transcription

1 Coronado Consulting Boundary integral methods for NIMROD resistive wall D. C. Barnes, C. Akcay, J. M. Finn J. King NIMROD SUMMER MEETING 1 August, 2018

2 R-wall project With Jake (TechX), DCB (Coronado), C. Akcay, J. Finn (Tibbar) Provide alternative to 2 region Sovinec implementation Achieve higher-order convergence w. boundary-integral method 1

3 R-wall Conditions Sherwood-status Replaced old top-hat (also used by GRIN) basis Implemented Nystrom (collocation) method Young & Martinsson Tested on circular, aspect 3 torus Manufactured solution with same ring source used for solver Circular? Problem with accuracy, essentially no convergence Studied A. Becarra implementation in developer Some strange results of using different interpolations?? Suspect GRIN solver matrix for slow convergence?? 2

4 B = Φ Φ = 2 x M, M 0 continuous B External Solver = Φ Φ n n M M in BTx = ΦM ', ΦM R * [ BT] JT ET B n σ ( S ) Φ M = ds Dσ r r 1 nφ M = dsσ ( S ) n Nσ r r S = DN 1 S S 3

5 Convergence of boundary solution (tophat) Function Derivative 4

6 Resistive wall transformer solution 5

7 Improve R-wall algorithm Desired nth-order convergence (n up to 6 or so) or Round-off-ish accuracy Components of boundary integral approach Free-space Green s function Approximate integrals (logarithmic singularity) Matrix inversion and algebra Remainder of algorithm nimset time nimrod time 6

8 Vacuum solver Green s functions 7

9 Vacuum solver Green s functions n = 0 & n = 1 use elliptic integrals n > 1 by recursion 8

10 Vacuum solver Integrals 9

11 Vacuum solver Integrals Testing in stand-alone code n = 0 & n = 1 so far Manufacture solution using several random sources inside torus Give solve for φ compare nφ Issues Young & Martinsson use Nystrom method (collocation) NIMROD uses various different nodes Interpolation between various meshes 10

12 Bug in Young-Martinsson manuscript They use triage approach Far segments use GL Next door use special modified GL (Jim Bremer) Same segment use different modified GL Problem with next door numbers in doc E.g. replaced with GL and improved result Discovered that weights don t sum to 2 11

13 Hero mathematicians to the rescue! Gunnar Martinsson Jim Bremer Eternal gratitude to both for their help Jim provided code to re-compute modified GL quadratures Changed ~3 lines and voila! 12

14 Manufactured solution

15 Success! Potential Normal Derivative Tangential Derivative n = 0 14

16 Success! Potential Normal Derivative Tangential Derivative n =1 15

17 Higher-order convergence Poly_degree=4 Uniform GLL 1.00E E E E-08 Potential Convergence -- uniform nodelets 1.00E E E E E E-09 n=0 n=1 4th Potential Convergence -- GLL nodelets 1.00E n=0 n=1 4th 1.00E E E E-07 Derivative Convergence -- uniform nodelets 1.00E E E E E E-07 4th Dn=0 Dn=1 Derivative Convergence -- GLL nodelets 1.00E th Dn=0 Dn=1 16

18 Higher-order convergence Poly_degree=5 Uniform GLL 1.00E E E E E-10 Potential Convergence -- uniform nodelets (pd=5) 1.00E E E E E-10 n=0 n=1 5th Potential Convergence -- GLL nodelets (pd=5) 1.00E n=0 n=1 5th Derivative Convergence -- uniform nodelets (pd=5) 1.00E E E E E E E E th Dn=0 Dn=1 Derivative Convergence -- GLL nodelets (pd=5) 1.00E E E E E E E E th Dn=0 Dn=1 17

19 Under the rug Underlying Nystrom accurate but slow convergent Boundary Integral Convergence 3.0E E-09 y = 7E-08x E n=0 n=1 Power (n=0) Possible/probable causes Discontinuities in normal, tangent C 0 elements? 18

20 Boundary integral plans Order of priority? Higher toroidal harmonics Straightforward use recursion to get Green s Testing developer w new boundary matrices Replace GRIN w NIMbnd Use new tang derivative matrix eliminate interpolation, etc. Cihan working on this Additional verification Cihan/John working on toridal harmonics for circular torus Add 2 nd conducting wall Compare w Carl s 2 region Get Nystrom convergence Use splines to get boundary geometry May migrate to NIMROD? 19

21 Verification with toroidal harmonics Toroidal Coordinates Toroidal Harmonics ( ) Tridiagonal System Exp. of χ Exp. of nχ August 3,

22 NIMdevel items Cihan built on Tibbar machine Studied Andie s implementation of rest of algorithm Obtained Carl s notes on this R E n B t B B t n [ ] w = t µ 0 = φ = n E t 21

23 Plug in new matrices NIMdevel plans Test convergence (how?) Use toroidal harmonics, e.g. resistive wall decay problem Convergence! Clean up code and ready for merge Convergence! Improve NIMROD boundary term treatment 22

24 Other issues More complex boundaries Smooth shapes no problem expected (demo) Corners will likely kill splines, etc. Basically no problem except bookkeeping Open boundary i.e. r = 0 Worked at one time with tophat basis Fix broken branches in BI code Publication, documentation 23