Duality based error estimation for electrostatic force computation - PowerPoint PPT Presentation

Duality based error estimation for electrostatic force computation 4. November 2010 1 / 33 Duality based error estimation for electrostatic force computation Author: Simon Pintarelli Supervisor: Prof. Ralf Hiptmair 4. November 2010

Duality based error estimation for electrostatic force computation 4. November 2010 2 / 33 Outline Outline

Duality based error estimation for electrostatic force computation 4. November 2010 3 / 33 Outline Duality based error estimation 1 Application to electrostatic force computation 2 Conclusion 3

Duality based error estimation for electrostatic force computation 4. November 2010 4 / 33 Duality based error estimation Duality based error estimation

Duality based error estimation for electrostatic force computation 4. November 2010 5 / 33 Duality based error estimation Duality based error estimation Primal problem a ( u h , v h ) = � f , v h � ∀ v h ∈ V h Quantity of interest We are not interested in the solution u directly but in a (linear) functional F ( u ). Dual problem a ( v , z ) = � F , v � ∀ v ∈ V

Duality based error estimation for electrostatic force computation 4. November 2010 6 / 33 Duality based error estimation Primal problem + dual problem + Galerkin orthogonality F ( e ) = a ( e , z ) = a ( e , z − v h ) = � f , z − v h � − a ( u h , z − v h ) =: ρ ( u h )( z − v h ) v h ∈ V h after cell-wise integration by parts (for the Poisson problem − ∆ u = f ) � � f + ∆ u h , z − v h � K + � 1 � � ρ ( u h )( z − v h ) = 2[ ∂ n u h ] , z − v h � ∂ K K ∈T h Dual weighted residual method � | F ( e ) | ≤ ρ K ω K K ∈T h ρ K : cell residuals “smoothness indicators” ω K : weights “influence factors”

Duality based error estimation for electrostatic force computation 4. November 2010 7 / 33 Duality based error estimation Practical error estimators Practical error estimators

Duality based error estimation for electrostatic force computation 4. November 2010 8 / 33 Duality based error estimation Practical error estimators Practical error estimators Starting point � F ( e ) = {� R h , z − v h � K + � r h , z − v h � ∂ K } K ∈T h � | F ( e ) | ≤ η := η K K ∈T h The previous error representations contained the exact solution z of the dual problem which is unkown and cannot be computed. derive approximate error representations ˜ E ( u h ).

Duality based error estimation for electrostatic force computation 4. November 2010 9 / 33 Duality based error estimation Practical error estimators Important properties of an error estimator Sharpness η should be a sharp upper bound for the error in the quantity of ˜ interest. Effectivity The approximate local error indicators ˜ η K should be effective for mesh refinement

Duality based error estimation for electrostatic force computation 4. November 2010 10 / 33 Duality based error estimation Practical error estimators Approximation by a higher-order method EST1 � � � R h , z (2) − I h z (2) h � K + � r h , z (2) − I h z (2) � F ( e ) ≈ h � ∂ K h h K ∈T h � � � � R h , z (2) − I h z (2) h � K + � r h , z (2) − I h z (2) η K = h � ∂ K � � h h � Expensive : dual problem is solved with a higher-order method (biquadratic FE) estimated error turned out to be close to the true error in most cases. not reliable : under-estimation can occur.

Duality based error estimation for electrostatic force computation 4. November 2010 11 / 33 Duality based error estimation Practical error estimators Approximation by higher-order interpolation EST2 � � � � R h , I (2) h z h − z h � K + � r h , I (2) F ( e ) ≈ h z h − z h � ∂ K K ∈T h dual problem is solved with bilinear FE and interpolated to biquadratic FE on each element. less computational cost compared to the previous estimator error estimate not as accurate as from the the higher-order method.

Duality based error estimation for electrostatic force computation 4. November 2010 12 / 33 Duality based error estimation Practical error estimators Approximation by difference quotients EST3 ω 2 K = � z − I h z � 2 K + h K � z − I h z � 2 ∂ K ≤ c 2 I h 2 K �∇ 2 z � 2 K The second derivatives ∇ 2 z can be replaced by suitable second-order difference quotients. � h 3 / 2 F ( e ) ≤ c I K ρ K � [ ∂ n z h ] � ∂ K K ∈T h usually strong overestimation

Duality based error estimation for electrostatic force computation 4. November 2010 13 / 33 Duality based error estimation Practical error estimators Gradient recovery EST4 The second derivatives ∇ 2 z can be obtained by patchwise gradient recovery.

Duality based error estimation for electrostatic force computation 4. November 2010 14 / 33 Duality based error estimation Practical error estimators Convergence property The error in the output functional is represented by F ( u − u h ) = a ( u − u h , z − z h ). a ( z − z h , u − u h ) ≤ | z − z h | 1 , Ω | u − u h | 1 , Ω ≤ Ch 2 | z | 2 , Ω | u | 2 , Ω Provided that the problem is sufficiently regular, i.e. z , u ∈ H 2 (Ω) the error in the output functional converges with O ( h 2 ).

Duality based error estimation for electrostatic force computation 4. November 2010 15 / 33 Application to electrostatic force computation Application to electrostatic force computation

Duality based error estimation for electrostatic force computation 4. November 2010 16 / 33 Application to electrostatic force computation Given electrostatic BVP Unkowns potential u , the force acting on the PEC, the error in the force − ∆ u = 0 x ∈ Ω u = U 0 x on Γ 1 u = 0 x on Γ 2 E ( u ) = −∇ u

Duality based error estimation for electrostatic force computation 4. November 2010 17 / 33 Application to electrostatic force computation Force computation Maxwell stress tensor T ( ∇ u ) = ∇ u · ∇ u T − 1 2 �∇ u � 2 I Force � F ( u ) = T · n d σ Γ 1 The force is given by integration of the Maxwell stress tensor over the boundary of the object. (not continuous on H 1 (Ω))

Duality based error estimation for electrostatic force computation 4. November 2010 18 / 33 Application to electrostatic force computation By applying Gauss’s theorem and inserting a cutoff function Ψ the functional F can be rewritten as an integral over the entire domain Ω. Where Ψ has to be in H 1 (Ω) and Ψ ≡ 1 on Γ 1 and Ψ ≡ 0 on Γ 2 The domain where the force is � ⇒ F ( u ) = T ( ∇ u ) · ∇ Ψ d x computed can be freely choosen Ω as long as it encloses the object of interest. “eggshell”-method

Duality based error estimation for electrostatic force computation 4. November 2010 19 / 33 Application to electrostatic force computation Linearization of F The right hand side of the dual problem must be a linear functional. ⇒ use Gateaux derivative of F . Dual problems Force in x -direction: ∀ v h ∈ V h � � a ( v h , z h D F ( u h )( v h ) x ) = x Force in y -direction: � � ∀ v h ∈ V h a ( v h , z h D F ( u h )( v h ) y ) = y

Duality based error estimation for electrostatic force computation 4. November 2010 20 / 33 Application to electrostatic force computation Adaptive mesh refinement solve coarse grid estimate linearized output η K , mark refine functional, error elements estimator no converged? yes stop

Duality based error estimation for electrostatic force computation 4. November 2010 21 / 33 Application to electrostatic force computation Results Results

Duality based error estimation for electrostatic force computation 4. November 2010 22 / 33 Application to electrostatic force computation Results Model problem 1 Error estimation Adaptive mesh refinement u | Γ 1 ≡ 0 , u | Γ 2 ≡ 1

Duality based error estimation for electrostatic force computation 4. November 2010 23 / 33 Application to electrostatic force computation Results Convergence rates 0 10 −1 10 −2 10 rel error −3 10 EST1, d = 2.45 −4 EST2, d = 2.22 10 EST3, d = 2.01 EST4, d = 2.00 RES, d = 1.78 uniform, d = 0.52 −5 10 1 2 3 4 10 10 10 10 ndofs

Duality based error estimation for electrostatic force computation 4. November 2010 24 / 33 Application to electrostatic force computation Results est1 est2 est3 est4

Duality based error estimation for electrostatic force computation 4. November 2010 25 / 33 Application to electrostatic force computation Results est1: ρ est1: ω est1: η est3: ρ est3: ω est3: η

Duality based error estimation for electrostatic force computation 4. November 2010 26 / 33 Application to electrostatic force computation Results Model problem 2 Example where mesh refinement based on an explicit residual estimator fails. expl. residual η

Duality based error estimation for electrostatic force computation 4. November 2010 27 / 33 Application to electrostatic force computation Results est2: ω est2: η

Duality based error estimation for electrostatic force computation 4. November 2010 28 / 33 Application to electrostatic force computation Results est2 explicit residual