Hysteresis modelling by Preisach Operators in Piezoelectricity

Transcription

1 Hysteresis modelling by Preisach Operators in Piezoelectricity Barbara Kaltenbacher 1 Thomas Hegewald Manfred Kaltenbacher Department of Sensor Technology University of Erlangen 1 DFG research group Inverse Problems in Piezoelectricity Piezo-Workshop, Söllerhaus, July

2 Hysteresis modelling by Preisach Operators Barbara Kaltenbacher 2 Thomas Hegewald Manfred Kaltenbacher Department of Sensor Technology University of Erlangen 2 DFG research group Inverse Problems in Piezoelectricity Piezo-Workshop, Söllerhaus, July

3 Hysteresis modelling by Preisach Operators in Piezoelectricity? Barbara Kaltenbacher 3 Thomas Hegewald Manfred Kaltenbacher Department of Sensor Technology University of Erlangen 3 DFG research group Inverse Problems in Piezoelectricity Piezo-Workshop, Söllerhaus, July

4 Material Parameter Identification What we are can do... Identification of material tensors from indirect measurements (PDE based) Optimal choice of measurement frequencies Identification of reversible nonlinearities What we are working on... Identification of irreversible nonlinearities Piezo-Workshop, Söllerhaus, July

5 Overview Preisach Operators: examples, definition, properties Modelling of irreversible strain and polarization: Comparison of different approaches Piezo-Workshop, Söllerhaus, July

6 Hysteresis in Piezoelectricity Measured polarization and strain at large electric field excitation (E 2M V/m): Piezo-Workshop, Söllerhaus, July

7 input: Hysteresis t output: t Piezo-Workshop, Söllerhaus, July

8 input: Hysteresis output t output: input t Piezo-Workshop, Söllerhaus, July

9 input: Hysteresis output magnetics t piezoelectricity output: input plasticity... memory t Volterra property rate independence Krasnoselksii-Pokrovskii (1983), Mayergoyz (1991), Visintin (1994), Krejčí (1996), Brokate-Sprekels (1996) Piezo-Workshop, Söllerhaus, July

10 Rate independence input: output output: input Piezo-Workshop, Söllerhaus, July

11 A Simple Example I: The Relay R β,α [v](t) = w(t) { +1 if v(t) > α or (w(ti ) = +1 v(t) > β) = 1 if v(t) < β or (w(t i ) = 1 v(t) < α) t [t i, t i+1 ] Piezo-Workshop, Söllerhaus, July

12 A Simple Example II: The Mechanical Play F r [v](t) = w(t) = max{v(t) r, min{v(t) + r, w(t i )}} t [t i, t i+1 ] w t (t) = { vt (t) if w(t) v(t) r 0 else Piezo-Workshop, Söllerhaus, July

13 A Simple Example III: The Elastic-Plastic Element G r [v](t) = w(t) = min{r, max{ r, v(t) v(t i ) + w(t i )}} t [t i, t i+1 ] w t (t) = { vt (t) if w(t) r 0 else G r = id F r Piezo-Workshop, Söllerhaus, July

14 A General Hysteresis Model: The Preisach Operator weighted superposition of relays with Preisach weight function defined on the Preisach plane S = S + S : S P[v](t) = = α,β S α,β S + (t) (β, α)r β,α [v](t) d(α, β) (β, α) d(α, β) α,β S (t) S + (β, α) d(α, β) Piezo-Workshop, Söllerhaus, July

15 The Preisach Memory Piezo-Workshop, Söllerhaus, July

16 The Preisach Memory Piezo-Workshop, Söllerhaus, July

17 The Preisach Memory Piezo-Workshop, Söllerhaus, July

18 The Preisach Memory Piezo-Workshop, Söllerhaus, July

19 The Preisach Memory Piezo-Workshop, Söllerhaus, July

20 A General Hysteresis Model: The Preisach Operator weighted superposition of relays with Preisach weight function defined on the Preisach plane S = S + S : S P[v](t) = = = w α,β S α,β S + (t) Fr [v](t) 0 0 (β, α)r β,α [v](t) d(α, β) (β, α) d(α, β) α,β S (t) (s r, s + r) ds dr S + (β, α) d(α, β) Piezo-Workshop, Söllerhaus, July

21 Memory deletion and Shape Functions I Deletion rules: Monotone deletion: only local extrema of the input are relevant for the output Madelung rule: inner minor loops are forgotten Wipe out: previous local extrema are erased from memory by subsequent local extrema with larger modulus (v(τ)) τ [0,t] deletion v(t i ) i {1,...,N} Piezo-Workshop, Söllerhaus, July

22 Memory deletion and Shape Functions II Everett (or shape) function: E(v, v ) = 2 (β, α) d(α, β) if v < v E( v, v ) = E(v, v ) v β α v N P[v](t) = (β, α)r β,α [v](t) d(α, β) = P[v 0 ] + E(v(t i ), v(t i+1 )) α,β S essential for efficient computations: i=1 avoid evaluation of integrals store only a few past time instances Hysteresis in PDEs: each point in space has its own memory! Piezo-Workshop, Söllerhaus, July

23 Regularity and Monotonicity Properties of Preisach Operators Lipschitz continuity on C[0, T ] Coercivity: P[v] t (t)v t (t) µ v t (t) 2 if 0 sup s IR (s r, s + r) dr < if 0 < µ 2 E Convexity: t (v tp[v] t ) (t) 2v tt (t)p[v] t (t) if sign(v v ) 2 2E(v, v ) 0 However... P is not differentiable in a classical sense, even for smooth P is not a monotone operator well-posedness of PDEs with hysteresis: [Visintin 1994]: parabolic, [Krejčí 1996]: hyperbolic Piezo-Workshop, Söllerhaus, July

24 Hysteresis Identification from Input-Output model Given input (v(t)) t [0,T ] measure output (w(t)) t [0,T ] = (P[v](t)) t [0,T ]. Identify P (i.e., ) from P[v](t) = α,β S (β, α)r β,α[v](t) d(α, β) = w(t) linear integral equation. Nonuniqueness, since v : [0, T ] IR but : }{{}}{{} S IR! IR 1 IR 2 α 4 α 3 α 2 α 1 β 1 β 2 β 3 β 4 β 5 v α β yields E on α β 5 β 4 β 3 β 2 β 1 α 1 α 2 α 3 α 4 β identifiability of E from Λ P : {v n } n IN {P[v n ]} n IN with v n α = v n β n and α n = β n = ( 1 n, 2 n,..., n n ) [Hoffmann&Meyer 89] Piezo-Workshop, Söllerhaus, July

25 Hysteresis Identification: Instability... singular values σ j of weightfunction - to - measurement map... j 2 Identification of is as ill-posed as twice differentiation: E(v, v ) = 2 (β, α) d(α, β) = 1 2 E v β α v Piezo-Workshop, Söllerhaus, July

26 Hysteresis in Piezoelectricity, First Approach: Preisach Operator for P i and Functional Dependence of S i on P i S = S r + S i D = D r + D i = ɛ 0 E + P r + P i S r = st + d t E D r = dt + ɛ T E 1-D model: S S 33 =: S, T T 33 =: T, D D 3 =: D, E E 33 =: E P i = P sat P[e] d = f(p) S i = g( p ) or S i = g(p) e = p = 1 E E max 1 P i P sat Piezo-Workshop, Söllerhaus, July

27 Identification of f, g, and P by an Alternating Iteration P[e] = λ Λ a λ P λ [e], f(p) = J d j=0 α d jp j, g( p ) = J S j=0 α S j p j or g(p) = J S j=0 α S j p j Set f 0 : d 0 d lin For k = 1, 2, 3,... α d 0 = d lin D meas = f k (p k )T + ɛ T E + P i P i k+1 p k+1 = p = 1 P sat P i k+1 S meas = st + f(p k+1 )E + g( p k+1 ) f k+1, g k+1 Endfor p = P[e] P Piezo-Workshop, Söllerhaus, July

28 Results: J d = J S = 4, S i = g( p ) (I) Preisach Weight Function: Polynomial Coefficients: α d = 1.0e 09 ( ) α S = 1.0e 02 ( ) Piezo-Workshop, Söllerhaus, July

29 Results: J d = J S = 4, S i = g( p ) (I) Measurement versus Fit Piezo-Workshop, Söllerhaus, July

30 Results: J d = J S = 4, S i = g( p ) (II) Preisach Weight Function: Polynomial Coefficients: α d = 1.0e 09 ( ) α S = 1.0e 02 ( ) Piezo-Workshop, Söllerhaus, July

31 Results: J d = J S = 4, S i = g( p ) (II) Measurement versus Fit Piezo-Workshop, Söllerhaus, July

32 Results: J d = J S = 4, S i = g(p) (I) Preisach Weight Function: Polynomial Coefficients: α d = 1.0e 09 ( ) α S = 1.0e 03 ( ) Piezo-Workshop, Söllerhaus, July

33 Results: J d = J S = 4, S i = g(p) (I) Measurement versus Fit Piezo-Workshop, Söllerhaus, July

34 Results: J d = J S = 4, S i = g(p) (II) Preisach Weight Function: Polynomial Coefficients: α d = 1.0e 09 ( ) α S = 1.0e 03 ( ) Piezo-Workshop, Söllerhaus, July

35 Results: J d = J S = 4, S i = g(p) (II) Measurement versus Fit Piezo-Workshop, Söllerhaus, July

36 Hysteresis in Piezoelectricity, Second Approach: Preisach operators for P i and for S i S = S r + S i D = D r + D i = ɛ 0 E + P r + P i S r = st + d t E D r = dt + ɛ T E 1-D model: S S 33 =: S, T T 33 =: T, D D 3 =: D, E E 33 =: E P i = P sat P 1 [e] d = d lin p S i = S sat P 2 [e] e = p = 1 E E max 1 P i P sat Piezo-Workshop, Söllerhaus, July

37 Identification of P 1 and P 2 by Linear Least Squares P 1 [e] = λ Λ a λ P λ [e], P 2 [e] = λ Λ b λ P λ [e] D meas = d lin P 1 [e]t + ɛ T E + P sat P 1 [e] P 1 [e] = 1 P sat +d lin T (Dmeas ɛ T E) P 1 S meas = st + d lin P 1 [e]e + S sat P 2 [e] P 2 [e] = sign(p 2 [e]) }{{} =:sign(p 1 [e]) P 2 1 S sat (S meas st d lin P 1 [e]e) Piezo-Workshop, Söllerhaus, July

38 Results: P i = P sat P 1 [e], S i = S sat P 2 [e], d = d lin P sat P i (II) Preisach Weight Functions: Piezo-Workshop, Söllerhaus, July

39 Results: P i = P sat P 1 [e], S i = S sat P 2 [e], d = d lin P sat P i (I) Measurement versus Fit Piezo-Workshop, Söllerhaus, July

40 Results: P i = P sat P 1 [e], S i = S sat P 2 [e], d = d lin P sat P i (II) Preisach Weight Functions: Piezo-Workshop, Söllerhaus, July

41 Results: P i = P sat P 1 [e], S i = S sat P 2 [e], d = d lin P sat P i (II) Measurement versus Fit Piezo-Workshop, Söllerhaus, July

42 Comparison to 2nd Set of Measurements Piezo-Workshop, Söllerhaus, July

43 Fit to 2nd Set of Measurements Preisach Weight Functions: Piezo-Workshop, Söllerhaus, July

44 Fit to 2nd Set of Measurements Measurement versus Fit Piezo-Workshop, Söllerhaus, July

45 Summary and Outlook Preisach operators for hysteresis modelling first attempts to application in piezoelectricity hysteresis identification within piezoelectric PDEs formulate thermodynamic laws as constrains on Preisach weight function advanced models: microscopically motivated internal variables, generalizations to 3-d [Kamlah et al.] Piezo-Workshop, Söllerhaus, July