unfitted discontinuous galerkin in brain research and the

Unfitted Discontinuous Galerkin inBrain Research and theTopology Preserving Marching CubesDUNE User Meeting 2015

Andreas Nüßing September 28th, 2015living knowledgeWWU Münster

WESTFÄLISCHEWILHELMS-UNIVERSITÄTMÜNSTER

livin

gkn

owle

dge

WW

UM

ünst

er

WESTFÄLISCHEWILHELMS-UNIVERSITÄTMÜNSTER UDG in Brain Research and the TPMC 2 /14

EEG forward problem

I inhomogeneous Poisson equationI homogeneous Neumann boundary

conditionI strongly irregular right hand sideI domain of complex shapeI discontinuous, anisotropic

conductivity tensor

Andreas Nüßing, [email protected]

livin

gkn

owle

dge

WW

UM

ünst

er


UDG = Unfitted Discontinous GalerkinI domain given by piecewise multilinear level setI separate computational grid from domain geometryI use DG on structured grid and restrict basis functions to

domain geometry

outside

inside

Level set grid combined


livin

gkn

owle

dge

WW

UM

ünst

er


TPMC = Topology Preserving Marching Cubes

I method for evaluating volume and surface integral overimplicitely defined domains

I based on marching cubes algorithm from computer graphicsI construct element wise reconstruction of level setI keep guarantees on topology of the reconstruction

I library available on GitHub:http://www.github.com/tpmc

I DUNE wrapper dune-mc

[Engwer & Nüßing, In preparation for submission]


livin

gkn

owle

dge

WW

UM

ünst

er


TPMC = Topology Preserving Marching Cubes

I multilinear level set defined by corner valuesI each set of corner values can be reduced to a base caseI value independent triangulations stored in a look up table

[Engwer & Nüßing, In preparation for submission]


livin

gkn

owle

dge

WW

UM

ünst

er


Results

I Evaluation in a multi layer sphere modelI Discontinuous tissue conductivities:

Radius (mm) 92 86 80 78Conductivity (S/m) 0.33 0.0042 1.79 0.33

I comparison of potential at electrodes with quasi analyticalsolution

re(Unum,Uana) =‖Unum − Uana‖2

‖Uana‖2

I sample over different dipolesI 500 unit dipoles with random direction on each ofI 15 eccentricities in the inner compartment

[Nüßing, Wolters, Brinck & Engwer, In preparation for submission]


livin

gkn

owle

dge

WW

UM

ünst

er


convergence of UDG



livin

gkn

owle

dge

WW

UM

ünst

er


convergence of UDG

0.14

89

0.38

35

0.55

35

0.67

66

0.76

57

0.83

03

0.87

71

0.91

09

0.93

55

0.95

33

0.96

61

0.97

55

0.98

22

0.98

71

0.99

06

eccentricity

10-4

10-3

10-2

10-1

100

re

Evaluation of UDG for different grid sizes

UDG 40k

UDG 218k

UDG 1335k



livin

gkn

owle

dge

WW

UM

ünst

er


comparison to DG on conforming mesh



livin

gkn

owle

dge

WW

UM

ünst

er


comparison to DG on conforming mesh

0.14

89

0.38

35

0.55

35

0.67

66

0.76

57

0.83

03

0.87

71

0.91

09

0.93

55

0.95

33

0.96

61

0.97

55

0.98

22

0.98

71

0.99

06

eccentricity

10-4

10-3

10-2

10-1

re

Comparison with the DG method on a conforming grid

DG 1766k

UDG 1335k



livin

gkn

owle

dge

WW

UM

ünst

er


trancranial direct current stimulation

[Neuroconn]

applications:I understanding current flowI optimized stimulation protocols


livin

gkn

owle

dge

WW

UM

ünst

er


tDCS on realistic data

MRI image Segmentation UDG level sets


livin

gkn

owle

dge

WW

UM

ünst

er


tDCS on realistic data

u u |σ∇u|


livin

gkn

owle

dge

WW

UM

ünst

er


Outlook

I investigate source model for UDGI smoothing using constrained mean curvature flowI integrate DUNE into software for brain research

e.g. FieldTrip, BesaI application to data from children

Thank you for your attention!


unfitted discontinuous galerkin in brain research and the

Documents