temf

1. Uncertainty Quantification
Stochastic Collocation Method
Incorporate material uncertainties into simulation of magnetic devices
2013-12-03 | TU Darmstadt | Fachbereich 18 | Institut Theorie Elektromagnetischer Felder | Ulrich Römer | 1

2. Uncertainty Quantification
Stochastic Collocation Method
Incorporate material uncertainties into simulation of magnetic devices
Stochastic magnetoquasistatics
σy ∂t Ay + × νy × Ay = J (y random realization)
2013-12-03 | TU Darmstadt | Fachbereich 18 | Institut Theorie Elektromagnetischer Felder | Ulrich Römer | 1

3. Uncertainty Quantification
Stochastic Collocation Method
Incorporate material uncertainties into simulation of magnetic devices
Stochastic magnetoquasistatics
σy ∂t Ay + × νy × Ay = J (y random realization)
+ boundary, initial and gauging conditions
2013-12-03 | TU Darmstadt | Fachbereich 18 | Institut Theorie Elektromagnetischer Felder | Ulrich Römer | 1

4. Uncertainty Quantification
Stochastic Collocation Method
Incorporate material uncertainties into simulation of magnetic devices
Stochastic magnetoquasistatics
σy ∂t Ay + × νy × Ay = J (y random realization)
+ boundary, initial and gauging conditions
Stochastic collocation approximation Ap
p + 1 nodes in each direction
tensor product grid
interpolate with Lagrange polynomials
−1 0 1
−1
−0.5
0
0.5
1
y1
y2
2013-12-03 | TU Darmstadt | Fachbereich 18 | Institut Theorie Elektromagnetischer Felder | Ulrich Römer | 1

5. Uncertainty Quantification
Stochastic Collocation Method
Incorporate material uncertainties into simulation of magnetic devices
Stochastic magnetoquasistatics
σy ∂t Ay + × νy × Ay = J (y random realization)
+ boundary, initial and gauging conditions
Stochastic collocation approximation Ap
p + 1 nodes in each direction
tensor product grid
interpolate with Lagrange polynomials
−1 0 1
−1
−0.5
0
0.5
1
y1
y2
Goal: control of stochastic error err = D×IT
f (A − Ap) dx dt
2013-12-03 | TU Darmstadt | Fachbereich 18 | Institut Theorie Elektromagnetischer Felder | Ulrich Römer | 1

6. Uncertainty Quantification
Adjoint Error Estimator
Linear adjoint problem −σy ∂t zy + × νy × zy = f,
solved again with collocation method (Butler et al., SIAM/ASA UQ 2014)
2013-12-03 | TU Darmstadt | Fachbereich 18 | Institut Theorie Elektromagnetischer Felder | Ulrich Römer | 2

7. Uncertainty Quantification
Adjoint Error Estimator
Linear adjoint problem −σy ∂t zy + × νy × zy = f,
solved again with collocation method (Butler et al., SIAM/ASA UQ 2014)
Adjoint error estimator
err =
D×IT
J − σ∂t Ap − × ν × Ap zp dx dt
2013-12-03 | TU Darmstadt | Fachbereich 18 | Institut Theorie Elektromagnetischer Felder | Ulrich Römer | 2

8. Uncertainty Quantification
Adjoint Error Estimator
Linear adjoint problem −σy ∂t zy + × νy × zy = f,
solved again with collocation method (Butler et al., SIAM/ASA UQ 2014)
Adjoint error estimator
err =
D×IT
J − σ∂t Ap − × ν × Ap zp dx dt
2013-12-03 | TU Darmstadt | Fachbereich 18 | Institut Theorie Elektromagnetischer Felder | Ulrich Römer | 2

9. Uncertainty Quantification
Adjoint Error Estimator
Linear adjoint problem −σy ∂t zy + × νy × zy = f,
solved again with collocation method (Butler et al., SIAM/ASA UQ 2014)
Adjoint error estimator
err =
D×IT
J − σ∂t Ap − × ν × Ap zp dx dt
Computational verification: 2D transformer coupled to circuit
−2
0
2
−2
0
2
−0.05
0
0.05
adjoint error estimator
y1
y2
−2
0
2
−2
0
2
−0.05
0
0.05
error reference
y1
y2
2013-12-03 | TU Darmstadt | Fachbereich 18 | Institut Theorie Elektromagnetischer Felder | Ulrich Römer | 2