Sensitivity Analysis Techniques for the Quantification of Uncertainty in Electromagnetic Simulations

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Nov 16, 2013 (3 years and 11 months ago)

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17. November 2013

| TU Darmstadt | Fachbereich 18 | Institut Theorie Elektromagnetischer Felder | Ulrich Römer |
1


Sensitivity Analysis Techniques for the
Quantification of Uncertainty in
Electromagnetic Simulations


Ulrich
Römer
*, Stephan Koch, Thomas
Weiland


Institut

für

Theorie

Elektromagnetischer

Felder, TU Darmstadt

ESCO 2012 (European Seminar on Computing)

Pilsen, Czech Republic, June 25
-

29, 2012

*Supported through Deutsche
Forschungsgemeinschaft, SFB 634

17. November 2013

| TU Darmstadt | Fachbereich 18 | Institut Theorie Elektromagnetischer Felder | Ulrich Römer |
2


Outline


Magnetostatics Model Problem


Uncertainties


Worst
-
Case
-
Scenario Method


Shape
Calculus


Numerical Examples






Magnetostatic Problem


Magnetostatics






Weak formulation




Space of divergence free functions


Unique solution, even in nonlinear case
*



17. November 2013

| TU Darmstadt | Fachbereich 18 | Institut Theorie Elektromagnetischer Felder | Ulrich Römer |
3


*

F.Bachinger, U. Langer, J. Schöberl; Numer
. Math. (2005
)

Magnetostatics Problem (cont‘d)


Physical Quantities of Interest (QOI
)

-
Multipole coefficients
(2
-
D)



-
Point evaluation (magnetic field)




Abstract form (divergence theorem)



17. November 2013

| TU Darmstadt | Fachbereich 18 | Institut Theorie Elektromagnetischer Felder | Ulrich Römer |
4


Beam pipe

small


Types of uncertainties

-
Shape (fabrication tolerances)

-
Material (measurement)

-
Coil current variation



Problem dependent on (stochastic) domain


Worst
-
Case
-
Scenario Method

-
Little information on input uncertainties

-
Restrictive design specifications

-
Important ingredient: Sensitivity analysis





17. November 2013

| TU Darmstadt | Fachbereich 18 | Institut Theorie Elektromagnetischer Felder | Ulrich Römer |
5


Uncertainties

I
. Babuška
, J. Chleboun; Numer
. Math. (
2003)

E
. Rosseel, H. De Gersem, S. Vandewalle; Commun. Comput. Phys.
(
2010)

H. Harbrecht, R. Schneider, C. Schwab; Numer. Math.
(
2008)


Velocity method
*






Stochastic domains


Admissible domains


Approximate admissible domain

Uncertain domains

17. November 2013

| TU Darmstadt | Fachbereich 18 | Institut Theorie Elektromagnetischer Felder | Ulrich Römer |
6


*M. C. Delfour, J.
-
P. Zolésio; Shapes and Geometries (2011)

B
-
spline boundary

Worst
-
Case
-
Scenario Method


Taylor‘s expansion


(Anti)
-
optimization problem
*




Linear
programming
problem for shape uncertainty

-
Control points wih box
constraints


But: Contribution from remainder hard to evaluate


Solve directly

**


Constraint optimization with gradient




17. November 2013

| TU Darmstadt | Fachbereich 18 | Institut Theorie Elektromagnetischer Felder | Ulrich Römer |
7


* I. Babuška, F. Nobile, R. Tempone; Num. Math.
(
2004)

Remainder

** I. Hlaváček, J. Chleboun, I. Babuška; Uncertain Input Data Problems... (2004)

Shape Calculus


Shape derivative



Distribution free approach (optimization)


Special transformation behaviour for electromagnetic fields


Structure theorem for differential forms
*




For each QOI we require the solution of the adjoint
problem

17. November 2013

| TU Darmstadt | Fachbereich 18 | Institut Theorie Elektromagnetischer Felder | Ulrich Römer |
8


*

R. Hiptmair, J. Li; Research Report ETH (2011)

Shape Calculus (cont‘d)


Solutions obtained through numerical simulation




Boundary
integral equations appealing


Not feasible here in the nonlinear case


Finite Element Method employed


Neglecting „mesh sensitivities“ in the discrete case, e.g.,




17. November 2013

| TU Darmstadt | Fachbereich 18 | Institut Theorie Elektromagnetischer Felder | Ulrich Römer |
9






2D and 3D linear/nonlinear
magnetostatic solver

-
Spline „edge elements“



Tensor product splines on macro patches


Sensitivity analysis


Spline boundary representation

-
Appealing for shape optimization

Numerical Results

17. November 2013

| TU Darmstadt | Fachbereich 18 | Institut Theorie Elektromagnetischer Felder | Ulrich Römer |
10






GeoPDES

C. de Falco, A. Reali, R.
Vazquez

Advances in Engineering Software (2011)





*

A. Buffa, G. Sangalli, J. Rivas, R. Vazquez; SIAM J. Numer. Anal. (2011)

Numerical Examples (cont‘d)


Surface with uncertainty

-
Analytical adjoint and state solution







17. November 2013

| TU Darmstadt | Fachbereich 18 | Institut Theorie Elektromagnetischer Felder | Ulrich Römer |
11


1680

0.0018

5220

1.311

e
-
4

11856

3.032

e
-
5

rel. error (shape derivative)

10 %

7,07 %

Material

20 %

6.74 %

Source

20 %

7.22 %

Numerical Examples (cont‘d)


2
-
D magnet design

-
Nonlinear material

17. November 2013

| TU Darmstadt | Fachbereich 18 | Institut Theorie Elektromagnetischer Felder | Ulrich Römer |
12


Summary


Worst
-
Case
-
Scenario Method


Shape calculus

-
Approximate WCS, problems to bound the remainder

-
Gradient based optimization problem


Application to linear/nonlinear magnetostatics

-
Uncertain
domains, material, sources


17. November 2013

| TU Darmstadt | Fachbereich 18 | Institut Theorie Elektromagnetischer Felder | Ulrich Römer |
13


Thank you for your attention!!!