Optimal piezoelectric energy harvesting strategy Joint work with B. - - PowerPoint PPT Presentation

Optimal piezoelectric energy harvesting strategy

Joint work with B. Kaltenbacher

Pavel Krejčí

Matematický ústav AV ČR Žitná 25, Praha 1

Padova September 26, 2017

Pavel Krejčí (Matematický ústav AV ČR) Piezoelectric energy harvesting September 26, 2017 1 / 24

Plan of the talk

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Plan of the talk Experimental observations

Pavel Krejčí (Matematický ústav AV ČR) Piezoelectric energy harvesting September 26, 2017 2 / 24

Plan of the talk Experimental observations Problems in constitutive modeling, Principles of Thermodynamics

Pavel Krejčí (Matematický ústav AV ČR) Piezoelectric energy harvesting September 26, 2017 2 / 24

Plan of the talk Experimental observations Problems in constitutive modeling, Principles of Thermodynamics Preisach hysteresis model and Preisach free energy

Pavel Krejčí (Matematický ústav AV ČR) Piezoelectric energy harvesting September 26, 2017 2 / 24

Plan of the talk Experimental observations Problems in constitutive modeling, Principles of Thermodynamics Preisach hysteresis model and Preisach free energy Magnetostrictive and piezoelectric energy exchange

Pavel Krejčí (Matematický ústav AV ČR) Piezoelectric energy harvesting September 26, 2017 2 / 24

Plan of the talk Experimental observations Problems in constitutive modeling, Principles of Thermodynamics Preisach hysteresis model and Preisach free energy Magnetostrictive and piezoelectric energy exchange Feedback effects

Pavel Krejčí (Matematický ústav AV ČR) Piezoelectric energy harvesting September 26, 2017 2 / 24

Plan of the talk Experimental observations Problems in constitutive modeling, Principles of Thermodynamics Preisach hysteresis model and Preisach free energy Magnetostrictive and piezoelectric energy exchange Feedback effects Optimal energy harvesting process - necessary optimality conditions

Pavel Krejčí (Matematický ústav AV ČR) Piezoelectric energy harvesting September 26, 2017 2 / 24

Plan of the talk Experimental observations Problems in constitutive modeling, Principles of Thermodynamics Preisach hysteresis model and Preisach free energy Magnetostrictive and piezoelectric energy exchange Feedback effects Optimal energy harvesting process - necessary optimality conditions Other applications and conclusions

Pavel Krejčí (Matematický ústav AV ČR) Piezoelectric energy harvesting September 26, 2017 2 / 24

Plan of the talk Experimental observations Problems in constitutive modeling, Principles of Thermodynamics Preisach hysteresis model and Preisach free energy Magnetostrictive and piezoelectric energy exchange Feedback effects Optimal energy harvesting process - necessary optimality conditions Other applications and conclusions References

Pavel Krejčí (Matematický ústav AV ČR) Piezoelectric energy harvesting September 26, 2017 2 / 24

Magnetostrictive and piezoelectric materials Magnetostrictive and piezoelectric materials exhibit mechanical deformation under the influence of electric or magnetic field and, vice versa, produce electric or magnetic field under mechanical loading.

✂ ☛ ☞ ✌

Applications: Actuators, sensors, harvesters, active or passive damping A 2 input (e.g., strain ε and electric field E ) – 2 output (dielectric displacement D and stress σ ) model is necessary for describing these phenomena.

Pavel Krejčí (Matematický ústav AV ČR) Piezoelectric energy harvesting September 26, 2017 3 / 24

Magnetostrictive and piezoelectric materials Magnetostrictive and piezoelectric materials exhibit mechanical deformation under the influence of electric or magnetic field and, vice versa, produce electric or magnetic field under mechanical loading.

☎ ✕ ✖ ✗ ✆

H Applications: Actuators, sensors, harvesters, active or passive damping A 2 input (e.g., strain ε and electric field E ) – 2 output (dielectric displacement D and stress σ ) model is necessary for describing these phenomena.

Pavel Krejčí (Matematický ústav AV ČR) Piezoelectric energy harvesting September 26, 2017 4 / 24

Magnetostrictive and piezoelectric materials Magnetostrictive and piezoelectric materials exhibit mechanical deformation under the influence of electric or magnetic field and, vice versa, produce electric or magnetic field under mechanical loading.

✭ ✮ ✯ ✰

Hysteresis! A 2 input (e.g., strain ε and electric field E ) – 2 output (dielectric displacement D and stress σ ) model is necessary for describing these phenomena.

Pavel Krejčí (Matematický ústav AV ČR) Piezoelectric energy harvesting September 26, 2017 5 / 24

Magnetostrictive and piezoelectric materials Magnetostrictive and piezoelectric materials exhibit mechanical deformation under the influence of electric or magnetic field and, vice versa, produce electric or magnetic field under mechanical loading.

☎ ✤ ✥ ✦ ✝

H Applications: Actuators, sensors, harvesters, active or passive damping

Pavel Krejčí (Matematický ústav AV ČR) Piezoelectric energy harvesting September 26, 2017 6 / 24

Magnetostrictive and piezoelectric materials Magnetostrictive and piezoelectric materials exhibit mechanical deformation under the influence of electric or magnetic field and, vice versa, produce electric or magnetic field under mechanical loading.

☎ ✤ ✥ ✦ ✝

H Applications: Actuators, sensors, harvesters, active or passive damping A 2 input (e.g., strain ε and electric field E ) – 2 output (dielectric displacement D and stress σ ) model is necessary for describing these phenomena.

Pavel Krejčí (Matematický ústav AV ČR) Piezoelectric energy harvesting September 26, 2017 6 / 24

Magnetic and magnetoelastic curves of Galfenol at various preloads

Measured by Daniele Davino, Università del Sannio, Benevento

Pavel Krejčí (Matematický ústav AV ČR) Piezoelectric energy harvesting September 26, 2017 7 / 24

Terfenol D, commercial presentation by Etrema Products Inc. Strain (ppm) Applied field (Oe ≈ 80 A/m)

Pavel Krejčí (Matematický ústav AV ČR) Piezoelectric energy harvesting September 26, 2017 8 / 24

Problems in constitutive modeling

Pavel Krejčí (Matematický ústav AV ČR) Piezoelectric energy harvesting September 26, 2017 9 / 24

Problems in constitutive modeling

A constitutive relation (D, σ) = F[E, ε] is compatible with the First and the Second Principle of Thermodynamics only if there exists a free energy operator W = W[E, ε] such that for all isothermal processes we have ˙ DE + ˙ εσ − ˙ W = ∆ ≥ 0 , where ∆ is the dissipation rate.

Pavel Krejčí (Matematický ústav AV ČR) Piezoelectric energy harvesting September 26, 2017 9 / 24

Problems in constitutive modeling

A constitutive relation (D, σ) = F[E, ε] is compatible with the First and the Second Principle of Thermodynamics only if there exists a free energy operator W = W[E, ε] such that for all isothermal processes we have ˙ DE + ˙ εσ − ˙ W = ∆ ≥ 0 , where ∆ is the dissipation rate. Scalar counterparts of this energy balance are known, e.g., for the Preisach model for ferromagnetism: If m = P[h] is the constitutive relation between the magnetic field h and the magnetization m with a Preisach operator P and with the associated Preisach free energy operator W = W[h], then the inequality ˙ mh − ˙ W = ∆ ≥ 0 holds for all processes.

Pavel Krejčí (Matematický ústav AV ČR) Piezoelectric energy harvesting September 26, 2017 9 / 24

Problems in constitutive modeling

A constitutive relation (D, σ) = F[E, ε] is compatible with the First and the Second Principle of Thermodynamics only if there exists a free energy operator W = W[E, ε] such that for all isothermal processes we have ˙ DE + ˙ εσ − ˙ W = ∆ ≥ 0 , where ∆ is the dissipation rate. Scalar counterparts of this energy balance are known, e.g., for the Preisach model for ferromagnetism: If m = P[h] is the constitutive relation between the magnetic field h and the magnetization m with a Preisach operator P and with the associated Preisach free energy operator W = W[h], then the inequality ˙ mh − ˙ W = ∆ ≥ 0 holds for all processes. !!! Dissipated energy is manifested by heat production which can damage the device or reduce its accuracy;

Pavel Krejčí (Matematický ústav AV ČR) Piezoelectric energy harvesting September 26, 2017 9 / 24

Problems in constitutive modeling

A constitutive relation (D, σ) = F[E, ε] is compatible with the First and the Second Principle of Thermodynamics only if there exists a free energy operator W = W[E, ε] such that for all isothermal processes we have ˙ DE + ˙ εσ − ˙ W = ∆ ≥ 0 , where ∆ is the dissipation rate. Scalar counterparts of this energy balance are known, e.g., for the Preisach model for ferromagnetism: If m = P[h] is the constitutive relation between the magnetic field h and the magnetization m with a Preisach operator P and with the associated Preisach free energy operator W = W[h], then the inequality ˙ mh − ˙ W = ∆ ≥ 0 holds for all processes. !!! Dissipated energy is manifested by heat production which can damage the device or reduce its accuracy; !!! Hysteresis losses can influence the harvester efficiency.

Pavel Krejčí (Matematický ústav AV ČR) Piezoelectric energy harvesting September 26, 2017 9 / 24

Preisach operator

☎

Pavel Krejčí (Matematický ústav AV ČR) Piezoelectric energy harvesting September 26, 2017 10 / 24

Preisach operator

Let pr be the mapping which with a parameter r > 0 and with a function h ∈ W 1,1(0, T) associates the solution ξr ∈ W 1,1(0, T) of the constrained rate independent equation |h(t) − ξr(t)| ≤ r , ˙ ξr(t)(h(t) − ξr(t)) = r| ˙ ξr(t)| , ξr(0) = min{h(0) + r, max{0, h(0) − r}}.

☎

r −r ξr h

Pavel Krejčí (Matematický ústav AV ČR) Piezoelectric energy harvesting September 26, 2017 10 / 24

Preisach operator

Let pr be the mapping which with a parameter r > 0 and with a function h ∈ W 1,1(0, T) associates the solution ξr ∈ W 1,1(0, T) of the constrained rate independent equation |h(t) − ξr(t)| ≤ r , ˙ ξr(t)(h(t) − ξr(t)) = r| ˙ ξr(t)| , ξr(0) = min{h(0) + r, max{0, h(0) − r}}.

☎

r −r ξr h For each fixed value of the memory depth r , the function t → ξr(t) describes the mechanical play with threshold r .

Pavel Krejčí (Matematický ústav AV ČR) Piezoelectric energy harvesting September 26, 2017 10 / 24

Preisach operator

Let pr be the mapping which with a parameter r > 0 and with a function h ∈ W 1,1(0, T) associates the solution ξr ∈ W 1,1(0, T) of the constrained rate independent equation |h(t) − ξr(t)| ≤ r , ˙ ξr(t)(h(t) − ξr(t)) = r| ˙ ξr(t)| , ξr(0) = min{h(0) + r, max{0, h(0) − r}}.

☎

r −r ξr h For each fixed value of the memory depth r , the function t → ξr(t) describes the mechanical play with threshold r . At each fixed time t > 0, the function r → ξr(t) describes the memory state at time t .

Pavel Krejčí (Matematický ústav AV ČR) Piezoelectric energy harvesting September 26, 2017 10 / 24

Preisach operator

Let pr be the mapping which with a parameter r > 0 and with a function h ∈ W 1,1(0, T) associates the solution ξr ∈ W 1,1(0, T) of the constrained rate independent equation |h(t) − ξr(t)| ≤ r , ˙ ξr(t)(h(t) − ξr(t)) = r| ˙ ξr(t)| , ξr(0) = min{h(0) + r, max{0, h(0) − r}}.

☎

r −r ξr h For each fixed value of the memory depth r , the function t → ξr(t) describes the mechanical play with threshold r . At each fixed time t > 0, the function r → ξr(t) describes the memory state at time t . The play ξr = pr[h] satisfies the energy balance equation ˙ ξrh − ˙ W = ∆ with W = 1

2ξ2 r , ∆ = r| ˙

ξr|.

Pavel Krejčí (Matematický ústav AV ČR) Piezoelectric energy harvesting September 26, 2017 10 / 24

Preisach operator II

Pavel Krejčí (Matematický ústav AV ČR) Piezoelectric energy harvesting September 26, 2017 11 / 24

Preisach operator II

Given a nonnegative function ψ ∈ L1((0, ∞) × R) (the Preisach density), the Preisach operator is defined by the integral formula m(t) = P[h](t) = ∞ pr [h](t) ψ(r, v) dv dr .

Pavel Krejčí (Matematický ústav AV ČR) Piezoelectric energy harvesting September 26, 2017 11 / 24

Preisach operator II

Given a nonnegative function ψ ∈ L1((0, ∞) × R) (the Preisach density), the Preisach operator is defined by the integral formula m(t) = P[h](t) = ∞ pr [h](t) ψ(r, v) dv dr . The free energy W associated with P has the form W (t) = W[h](t) = ∞ pr [h](t) vψ(r, v) dv dr ,

Pavel Krejčí (Matematický ústav AV ČR) Piezoelectric energy harvesting September 26, 2017 11 / 24

Preisach operator II

Given a nonnegative function ψ ∈ L1((0, ∞) × R) (the Preisach density), the Preisach operator is defined by the integral formula m(t) = P[h](t) = ∞ pr [h](t) ψ(r, v) dv dr . The free energy W associated with P has the form W (t) = W[h](t) = ∞ pr [h](t) vψ(r, v) dv dr , and the energy balance equation (we denote ξr = pr[h]) ˙ mh − ˙ W = ∞ ˙ ξr(h − ξr)ψ(r, ξr) dr = ∞ r| ˙ ξr|ψ(r, ξr) dr = | ˙ D| ≥ 0

Pavel Krejčí (Matematický ústav AV ČR) Piezoelectric energy harvesting September 26, 2017 11 / 24

Preisach operator II

Given a nonnegative function ψ ∈ L1((0, ∞) × R) (the Preisach density), the Preisach operator is defined by the integral formula m(t) = P[h](t) = ∞ pr [h](t) ψ(r, v) dv dr . The free energy W associated with P has the form W (t) = W[h](t) = ∞ pr [h](t) vψ(r, v) dv dr , and the energy balance equation (we denote ξr = pr[h]) ˙ mh − ˙ W = ∞ ˙ ξr(h − ξr)ψ(r, ξr) dr = ∞ r| ˙ ξr|ψ(r, ξr) dr = | ˙ D| ≥ 0 holds with the dissipation operator D(t) = D[h](t) = ∞ pr [h](t) rψ(r, v) dv dr .

Pavel Krejčí (Matematický ústav AV ČR) Piezoelectric energy harvesting September 26, 2017 11 / 24

Preisach operator and Preisach free energy

2 1 1 2 0.5 0.5

• Fig. 1: The Preisach constitutive

relation m = P[h].

2 1 1 2 0.1 0.2 0.3 0.4 0.5

• Fig. 2: The Preisach free energy

W = W[h].

Pavel Krejčí (Matematický ústav AV ČR) Piezoelectric energy harvesting September 26, 2017 12 / 24

Preisach operator and Preisach free energy

2 1 1 2 0.5 0.5

• Fig. 1: The Preisach constitutive

relation m = P[h].

2 1 1 2 0.1 0.2 0.3 0.4 0.5

• Fig. 2: The Preisach free energy

W = W[h].

• Theorem. Both operators P and W admit a locally Lipschitz continuous

extension to a mapping C[0, T] → C[0, T].

Pavel Krejčí (Matematický ústav AV ČR) Piezoelectric energy harvesting September 26, 2017 12 / 24

Preisach operator and Preisach free energy

2 1 1 2 0.5 0.5

• Fig. 1: The Preisach constitutive

relation m = P[h].

2 1 1 2 0.1 0.2 0.3 0.4 0.5

• Fig. 2: The Preisach free energy

W = W[h].

• Theorem. Both operators P and W admit a locally Lipschitz continuous

extension to a mapping C[0, T] → C[0, T]. Conjecture: The Preisach free energy operator describes the electro-mechanical or magneto-mechanical interaction.

Pavel Krejčí (Matematický ústav AV ČR) Piezoelectric energy harvesting September 26, 2017 12 / 24

Piezoelectricity

Pavel Krejčí (Matematický ústav AV ČR) Piezoelectric energy harvesting September 26, 2017 13 / 24

Piezoelectricity

In order to model the self-similar behavior in the input- (strain ε and the electric field E ) output (dielectric displacement D and stress σ ) hysteresis diagram, the simplest choice is D = ωε + κE + P, σ = Kε − ωE + S, W = K 2 ε2 + κ 2 E 2 + V , P = P[u], S = f ′(ε)W[u], V = f (ε)W[u], u = E f (ε) with a Preisach operator P and Preisach free energy W , a positive self-similarity function f (ε), and physical constants K, ω, κ.

Pavel Krejčí (Matematický ústav AV ČR) Piezoelectric energy harvesting September 26, 2017 13 / 24

Piezoelectricity

In order to model the self-similar behavior in the input- (strain ε and the electric field E ) output (dielectric displacement D and stress σ ) hysteresis diagram, the simplest choice is D = ωε + κE + P, σ = Kε − ωE + S, W = K 2 ε2 + κ 2 E 2 + V , P = P[u], S = f ′(ε)W[u], V = f (ε)W[u], u = E f (ε) with a Preisach operator P and Preisach free energy W , a positive self-similarity function f (ε), and physical constants K, ω, κ. We then have the correct energy balance ˙ DE + ˙ εσ − ˙ W

Pavel Krejčí (Matematický ústav AV ČR) Piezoelectric energy harvesting September 26, 2017 13 / 24

Piezoelectricity

In order to model the self-similar behavior in the input- (strain ε and the electric field E ) output (dielectric displacement D and stress σ ) hysteresis diagram, the simplest choice is D = ωε + κE + P, σ = Kε − ωE + S, W = K 2 ε2 + κ 2 E 2 + V , P = P[u], S = f ′(ε)W[u], V = f (ε)W[u], u = E f (ε) with a Preisach operator P and Preisach free energy W , a positive self-similarity function f (ε), and physical constants K, ω, κ. We then have the correct energy balance ˙ DE + ˙ εσ − ˙ W = ˙ PE + ˙ εS − ˙ V

Pavel Krejčí (Matematický ústav AV ČR) Piezoelectric energy harvesting September 26, 2017 13 / 24

Piezoelectricity

In order to model the self-similar behavior in the input- (strain ε and the electric field E ) output (dielectric displacement D and stress σ ) hysteresis diagram, the simplest choice is D = ωε + κE + P, σ = Kε − ωE + S, W = K 2 ε2 + κ 2 E 2 + V , P = P[u], S = f ′(ε)W[u], V = f (ε)W[u], u = E f (ε) with a Preisach operator P and Preisach free energy W , a positive self-similarity function f (ε), and physical constants K, ω, κ. We then have the correct energy balance ˙ DE + ˙ εσ − ˙ W = ˙ PE + ˙ εS − ˙ V = f (ε)(u∂tP[u] − ∂tW[u]) ≥ 0.

Pavel Krejčí (Matematický ústav AV ČR) Piezoelectric energy harvesting September 26, 2017 13 / 24

Mechanical depolarization

Pavel Krejčí (Matematický ústav AV ČR) Piezoelectric energy harvesting September 26, 2017 14 / 24

Mechanical depolarization

The model presented above is not able to describe the phenomena of demagnetization and depolarization at zero field and exhibits other bigger or smaller discrepancies with experiments at low field values.

Pavel Krejčí (Matematický ústav AV ČR) Piezoelectric energy harvesting September 26, 2017 14 / 24

Mechanical depolarization

The model presented above is not able to describe the phenomena of demagnetization and depolarization at zero field and exhibits other bigger or smaller discrepancies with experiments at low field values. As a correction of the model, a modification including a mean field feedback correction term has recently been proposed in the form D = ωε + κE + P, σ = Kε − ωE + S, W = K 2 ε2 + κ 2 E 2 + V , P = P[u], S = f ′(ε)W[u] + 1 2a′(ε)P2, V = f (ε)W[u] + 1 2a(ε)P2, u = 1 f (ε)(E − a(ε)P), with an empirical feedback function a(ε). The energy balance then reads as before ˙ DE + ˙ εσ − ˙ W

Pavel Krejčí (Matematický ústav AV ČR) Piezoelectric energy harvesting September 26, 2017 14 / 24

Mechanical depolarization

The model presented above is not able to describe the phenomena of demagnetization and depolarization at zero field and exhibits other bigger or smaller discrepancies with experiments at low field values. As a correction of the model, a modification including a mean field feedback correction term has recently been proposed in the form D = ωε + κE + P, σ = Kε − ωE + S, W = K 2 ε2 + κ 2 E 2 + V , P = P[u], S = f ′(ε)W[u] + 1 2a′(ε)P2, V = f (ε)W[u] + 1 2a(ε)P2, u = 1 f (ε)(E − a(ε)P), with an empirical feedback function a(ε). The energy balance then reads as before ˙ DE + ˙ εσ − ˙ W = ˙ PE + ˙ εS − ˙ V

Pavel Krejčí (Matematický ústav AV ČR) Piezoelectric energy harvesting September 26, 2017 14 / 24

Mechanical depolarization

The model presented above is not able to describe the phenomena of demagnetization and depolarization at zero field and exhibits other bigger or smaller discrepancies with experiments at low field values. As a correction of the model, a modification including a mean field feedback correction term has recently been proposed in the form D = ωε + κE + P, σ = Kε − ωE + S, W = K 2 ε2 + κ 2 E 2 + V , P = P[u], S = f ′(ε)W[u] + 1 2a′(ε)P2, V = f (ε)W[u] + 1 2a(ε)P2, u = 1 f (ε)(E − a(ε)P), with an empirical feedback function a(ε). The energy balance then reads as before ˙ DE + ˙ εσ − ˙ W = ˙ PE + ˙ εS − ˙ V = f (ε)(u∂tP[u] − ∂tW[u]) ≥ 0.

Pavel Krejčí (Matematický ústav AV ČR) Piezoelectric energy harvesting September 26, 2017 14 / 24

Piezoelectric harvester: Case without inductance

Pavel Krejčí (Matematický ústav AV ČR) Piezoelectric energy harvesting September 26, 2017 15 / 24

Piezoelectric harvester: Case without inductance

The dynamics of a piezoelectric harvester subject to an impressed time-dependent mechanical force σimp(t) can be described by the system d dt ρ¨ ε+ν ˙ ε+σ = σimp, d dt ˙ D + αE = 0, where σ and D satisfy the piezoelectric constitutive equations.

Pavel Krejčí (Matematický ústav AV ČR) Piezoelectric energy harvesting September 26, 2017 15 / 24

Piezoelectric harvester: Case without inductance

The dynamics of a piezoelectric harvester subject to an impressed time-dependent mechanical force σimp(t) can be described by the system d dt ρ¨ ε+ν ˙ ε+σ = σimp, d dt ˙ D + αE = 0, where σ and D satisfy the piezoelectric constitutive equations.

Pavel Krejčí (Matematický ústav AV ČR) Piezoelectric energy harvesting September 26, 2017 15 / 24

Piezoelectric harvester: Case without inductance

The dynamics of a piezoelectric harvester subject to an impressed time-dependent mechanical force σimp(t) can be described by the system d dt ρ¨ ε+ν ˙ ε+Kε − ωE + f ′(ε)W[u] + 1

2a′(ε)P2[u] = σimp,

d dt d dt (ωε + κE + P[u]) + αE = 0, where ρ, ν, K, ω, κ, α are physical constants.

Pavel Krejčí (Matematický ústav AV ČR) Piezoelectric energy harvesting September 26, 2017 15 / 24

Piezoelectric harvester: Case without inductance

The dynamics of a piezoelectric harvester subject to an impressed time-dependent mechanical force σimp(t) can be described by the system d dt ρ¨ ε+ν ˙ ε+Kε − ωE + f ′(ε)W[u] + 1

2a′(ε)P2[u] = σimp,

d dt d dt (ωε + κE + P[u]) + αE = 0, where ρ, ν, K, ω, κ, α are physical constants. We have D = ωε + κE + P[u] and E = f (ε)u + a(ε)P[u], hence, u + 1 + κa(ε) κf (ε) P[u] = D − ωε κf (ε) . (1)

Pavel Krejčí (Matematický ústav AV ČR) Piezoelectric energy harvesting September 26, 2017 15 / 24

Piezoelectric harvester: Case without inductance

The dynamics of a piezoelectric harvester subject to an impressed time-dependent mechanical force σimp(t) can be described by the system d dt ρ¨ ε+ν ˙ ε+Kε − ωE + f ′(ε)W[u] + 1

2a′(ε)P2[u] = σimp,

d dt d dt (ωε + κE + P[u]) + αE = 0, where ρ, ν, K, ω, κ, α are physical constants. We have D = ωε + κE + P[u] and E = f (ε)u + a(ε)P[u], hence, u + 1 + κa(ε) κf (ε) P[u] = D − ωε κf (ε) . (1) Theorem (K+K 2016). The operator R which with D, ε ∈ C[0, T] associates the solution u = R[D, ε] ∈ C[0, T] of equation (??) is Lipschitz continuous.

Pavel Krejčí (Matematický ústav AV ČR) Piezoelectric energy harvesting September 26, 2017 15 / 24

Piezoelectric harvester: Case without inductance

The dynamics of a piezoelectric harvester subject to an impressed time-dependent mechanical force σimp(t) can be described by the system d dt ρ¨ ε+ν ˙ ε+Kε−ω κ (D−ωε−P[R[D, ε]])+f ′(ε)W[R[D, ε]]+ 1

2a′(ε)P2[R[D, ε]] = σimp,

d dt ˙ D + α κ (D−ωε−P[R[D, ε]]) = 0, where ρ, ν, K, ω, κ, α are physical constants. We have D = ωε + κE + P[u] and E = f (ε)u + a(ε)P[u], hence, u + 1 + κa(ε) κf (ε) P[u] = D − ωε κf (ε) . (1) Theorem (K+K 2016). The operator R which with D, ε ∈ C[0, T] associates the solution u = R[D, ε] ∈ C[0, T] of equation (??) is Lipschitz continuous. Our equations thus can be reduced to a simple ODE system with a locally Lipschitz continuous right-hand side, for which all results about local existence, uniqueness, and continuous data dependence are available.

Pavel Krejčí (Matematický ústav AV ČR) Piezoelectric energy harvesting September 26, 2017 15 / 24

Piezoelectric harvester: Case with inductance

Pavel Krejčí (Matematický ústav AV ČR) Piezoelectric energy harvesting September 26, 2017 16 / 24

Piezoelectric harvester: Case with inductance

The dynamics of a piezoelectric harvester subject to an impressed time-dependent mechanical force σimp(t) can be described by the system d dt ρ¨ ε+ν ˙ ε+σ = σimp, d2 dt2 ¨ D + α ˙ E + βE = 0, where σ and D satisfy the piezoelectric constitutive equations.

Pavel Krejčí (Matematický ústav AV ČR) Piezoelectric energy harvesting September 26, 2017 16 / 24

Piezoelectric harvester: Case with inductance

The dynamics of a piezoelectric harvester subject to an impressed time-dependent mechanical force σimp(t) can be described by the system d dt ρ¨ ε+ν ˙ ε+σ = σimp, d2 dt2 ¨ D + α ˙ E + βE = 0, where σ and D satisfy the piezoelectric constitutive equations.

Pavel Krejčí (Matematický ústav AV ČR) Piezoelectric energy harvesting September 26, 2017 16 / 24

Piezoelectric harvester: Case with inductance

The dynamics of a piezoelectric harvester subject to an impressed time-dependent mechanical force σimp(t) can be described by the system d dt ρ¨ ε+ν ˙ ε+Kε − ωE + f ′(ε)W[u] + 1

2a′(ε)P2[u] = σimp,

d2 dt2 d2 dt2 (ωε + κE + P[u]) + α ˙ E + βE = 0, where ρ, ν, K, ω, κ, α, β are physical constants.

Pavel Krejčí (Matematický ústav AV ČR) Piezoelectric energy harvesting September 26, 2017 16 / 24

Piezoelectric harvester: Case with inductance

The dynamics of a piezoelectric harvester subject to an impressed time-dependent mechanical force σimp(t) can be described by the system d dt ρ¨ ε+ν ˙ ε+Kε − ωE + f ′(ε)W[u] + 1

2a′(ε)P2[u] = σimp,

d2 dt2 d2 dt2 (ωε + κE + P[u]) + α ˙ E + βE = 0, where ρ, ν, K, ω, κ, α, β are physical constants. We rewrite the system in the form ρ¨ ε+ν ˙ ε+Kε − ωE + f ′(ε)W[u] + 1

2a′(ε)P2[u] = σimp,

d dt (ωε + κE + P[u]) + αE + βΦ = 0, ˙ Φ − E = 0.

Pavel Krejčí (Matematický ústav AV ČR) Piezoelectric energy harvesting September 26, 2017 16 / 24

Piezoelectric harvester: Case with inductance

The dynamics of a piezoelectric harvester subject to an impressed time-dependent mechanical force σimp(t) can be described by the system d dt ρ¨ ε+ν ˙ ε+Kε − ωE + f ′(ε)W[u] + 1

2a′(ε)P2[u] = σimp,

d2 dt2 d2 dt2 (ωε + κE + P[u]) + α ˙ E + βE = 0, where ρ, ν, K, ω, κ, α, β are physical constants. We rewrite the system in the form ρ¨ ε+ν ˙ ε+Kε − ωE + f ′(ε)W[u] + 1

2a′(ε)P2[u] = σimp,

d dt (ωε + κE + P[u]) + αE + βΦ = 0, ˙ Φ − E = 0. Passing to the new variable D = ωε + κE + P[u] and substituting u = R[D, ε], we obtain as before an ODE system for the unknowns D, ε, Φ with a locally Lipschitz continuous right-hand side.

Pavel Krejčí (Matematický ústav AV ČR) Piezoelectric energy harvesting September 26, 2017 16 / 24

Energy balance

Pavel Krejčí (Matematický ústav AV ČR) Piezoelectric energy harvesting September 26, 2017 17 / 24

Energy balance

In the system ρ¨ ε+ν ˙ ε+Kε − ωE + f ′(ε)W[u] + 1

2a′(ε)P2[u] = σimp,

d dt (ωε + κE + P[u]) + αE + βΦ = 0, ˙ Φ − E = 0,

Pavel Krejčí (Matematický ústav AV ČR) Piezoelectric energy harvesting September 26, 2017 17 / 24

Energy balance

In the system ρ¨ ε+ν ˙ ε+Kε − ωE + f ′(ε)W[u] + 1

2a′(ε)P2[u] = σimp,

d dt (ωε + κE + P[u]) + αE + βΦ = 0, ˙ Φ − E = 0, we multiply the first equation by ˙ ε, the second equation by E , the third equation by βΦ, and sum them up to obtain d dt ρ 2 ˙ ε2 + c 2ε2 + κ 2 E 2 + β 2 Φ2 + f (ε)W[u] + 1 2b(ε)P2[u]

• +ν ˙

ε2 + αE 2 + f (ε)

• u d

dt P[u] − d dt W[u]

• = ˙

ε σimp.

Pavel Krejčí (Matematický ústav AV ČR) Piezoelectric energy harvesting September 26, 2017 17 / 24

Energy balance

In the system ρ¨ ε+ν ˙ ε+Kε − ωE + f ′(ε)W[u] + 1

2a′(ε)P2[u] = σimp,

d dt (ωε + κE + P[u]) + αE + βΦ = 0, ˙ Φ − E = 0, we multiply the first equation by ˙ ε, the second equation by E , the third equation by βΦ, and sum them up to obtain d dt ρ 2 ˙ ε2 + c 2ε2 + κ 2 E 2 + β 2 Φ2 + f (ε)W[u] + 1 2b(ε)P2[u]

• +ν ˙

ε2 + αE 2 + f (ε)

• u d

dt P[u] − d dt W[u]

• = ˙

ε σimp. The solution thus remains bounded in the whole existence range. This implies in turn that the solution exists globally and depends continuously on the data and

• n the physical parameters.

Pavel Krejčí (Matematický ústav AV ČR) Piezoelectric energy harvesting September 26, 2017 17 / 24

Special case: The play operator

Pavel Krejčí (Matematický ústav AV ČR) Piezoelectric energy harvesting September 26, 2017 18 / 24

Special case: The play operator

Consider the special case P = λpr , with some λ > 0.

Pavel Krejčí (Matematický ústav AV ČR) Piezoelectric energy harvesting September 26, 2017 18 / 24

Special case: The play operator

Consider the special case P = λpr , with some λ > 0. The energy inequality holds for the choice W[u] = λ

2 p2 r [u].

Pavel Krejčí (Matematický ústav AV ČR) Piezoelectric energy harvesting September 26, 2017 18 / 24

Special case: The play operator

Consider the special case P = λpr , with some λ > 0. The energy inequality holds for the choice W[u] = λ

2 p2 r [u].

We first rewrite the operator equation u + 1 + κa(ε) κf (ε) P[u] = D − ωε κf (ε)

Pavel Krejčí (Matematický ústav AV ČR) Piezoelectric energy harvesting September 26, 2017 18 / 24

Special case: The play operator

Consider the special case P = λpr , with some λ > 0. The energy inequality holds for the choice W[u] = λ

2 p2 r [u].

We first rewrite the operator equation u + λB(ε)pr[u]() () = () ()A(D, ε). The inversion formula is explicit in terms of the play operator with moving threshold: ξ = pr[u] = pR(ε) A(D, ε) 1 + λB(ε)

• ,

R(ε) = r 1 + λB(ε).

Pavel Krejčí (Matematický ústav AV ČR) Piezoelectric energy harvesting September 26, 2017 18 / 24

Special case: The play operator

Consider the special case P = λpr , with some λ > 0. The energy inequality holds for the choice W[u] = λ

2 p2 r [u].

We first rewrite the operator equation u + λB(ε)pr[u]() () = () ()A(D, ε). The inversion formula is explicit in terms of the play operator with moving threshold: ξ = pr[u] = pR(ε) A(D, ε) 1 + λB(ε)

• ,

R(ε) = r 1 + λB(ε). All operators in the balance equation thus admit a representation in terms of ξ R[D, ε] = A(D, ε) − λB(ε)ξ, (P ◦ R)[D, ε] = λξ, (W ◦ R)[D, ε] = λ 2 ξ2.

Pavel Krejčí (Matematický ústav AV ČR) Piezoelectric energy harvesting September 26, 2017 18 / 24

Special case: The play operator II

Pavel Krejčí (Matematický ústav AV ČR) Piezoelectric energy harvesting September 26, 2017 19 / 24

Special case: The play operator II

The system of balance equations has the form ˙ y(t) = F(t, y(t), ξ(t); θ), where y = (ε, ˙ ε, D, Φ) is the unknown vector function, and θ ∈ Θ is the constant vector of physical parameters.

Pavel Krejčí (Matematický ústav AV ČR) Piezoelectric energy harvesting September 26, 2017 19 / 24

Special case: The play operator II

The system of balance equations has the form ˙ y(t) = F(t, y(t), ξ(t); θ), where y = (ε, ˙ ε, D, Φ) is the unknown vector function, and θ ∈ Θ is the constant vector of physical parameters. The moving play operator ξ = pR(ε)

• A(D,ε)

1+λB(ε)

• admits a representation in terms of differential inclusion

˙ ξ(t) ∈ ∂I[−1,1](a(t)), a = 1 r (A(D, ε) − (1 + λB(ε))ξ), where I[−1,1] is the indicator function of the interval [−1, 1] and ∂I[−1,1] is its subdifferential.

Pavel Krejčí (Matematický ústav AV ČR) Piezoelectric energy harvesting September 26, 2017 19 / 24

Special case: The play operator II

The system of balance equations has the form ˙ y(t) = F(t, y(t), ξ(t); θ), where y = (ε, ˙ ε, D, Φ) is the unknown vector function, and θ ∈ Θ is the constant vector of physical parameters. The moving play operator ξ = pR(ε)

• A(D,ε)

1+λB(ε)

• admits a representation in terms of differential inclusion

˙ ξ(t) ∈ ∂I[−1,1](a(t)), a = 1 r (A(D, ε) − (1 + λB(ε))ξ), where I[−1,1] is the indicator function of the interval [−1, 1] and ∂I[−1,1] is its

• subdifferential. This inclusion can be in turn rewritten in the form

˙ a(t) + ∂I[−1,1](a(t)) ∋ g(t, y(t), a(t); θ).

Pavel Krejčí (Matematický ústav AV ČR) Piezoelectric energy harvesting September 26, 2017 19 / 24

Special case: The play operator II

The system of balance equations has the form ˙ y(t) = ˆ F(t, y(t), a(t); θ), where y = (ε, ˙ ε, D, Φ) is the unknown vector function, and θ ∈ Θ is the constant vector of physical parameters. The moving play operator ξ = pR(ε)

• A(D,ε)

1+λB(ε)

• admits a representation in terms of differential inclusion

˙ ξ(t) ∈ ∂I[−1,1](a(t)), a = 1 r (A(D, ε) − (1 + λB(ε))ξ), where I[−1,1] is the indicator function of the interval [−1, 1] and ∂I[−1,1] is its

• subdifferential. This inclusion can be in turn rewritten in the form

˙ a(t) + ∂I[−1,1](a(t)) ∋ g(t, y(t), a(t); θ). The next goal is to maximize the harvested energy T J(t, y(t), a(t); θ)(t) dt − → min with respect to the physical parameter vector θ ∈ Θ if y(0), a(0) are given.

Pavel Krejčí (Matematický ústav AV ČR) Piezoelectric energy harvesting September 26, 2017 19 / 24

Approximation

Pavel Krejčí (Matematický ústav AV ČR) Piezoelectric energy harvesting September 26, 2017 20 / 24

Approximation

Complement the cost functional with the term |θ − θ∗|2 , where θ∗ is a value where the minimum is achieved, and replace the system ˙ y(t) = ˆ F(t, y(t), a(t); θ) ˙ a(t) + ∂I[−1,1](a(t)) ∋ g(t, y(t), a(t); θ) with ˙ yγ(t) = ˆ F(t, yγ(t), aγ(t); θγ) ˙ aγ(t) + 1 γ Ψ′(aγ(t)) = g(t, yγ(t), aγ(t); θγ), for γ > 0, where Ψ(a) = 1 6((a2 − 1)+)3;

Pavel Krejčí (Matematický ústav AV ČR) Piezoelectric energy harvesting September 26, 2017 20 / 24

Approximation

Complement the cost functional with the term |θ − θ∗|2 , where θ∗ is a value where the minimum is achieved, and replace the system ˙ y(t) = ˆ F(t, y(t), a(t); θ) ˙ a(t) + ∂I[−1,1](a(t)) ∋ g(t, y(t), a(t); θ) with ˙ yγ(t) = ˆ F(t, yγ(t), aγ(t); θγ) ˙ aγ(t) + 1 γ Ψ′(aγ(t)) = g(t, yγ(t), aγ(t); θγ), for γ > 0, where Ψ(a) = 1 6((a2 − 1)+)3; For γ → 0, (yγ, aγ) converge strongly to solutions (y∗, a∗) in W 1,2(0, T) of the system ˙ y∗(t) = ˆ F(t, y∗(t), a∗(t); θ∗) ˙ a∗(t) + ∂I[−1,1](a∗(t)) ∋ g(t, y∗(t), a∗(t); θ∗).

Pavel Krejčí (Matematický ústav AV ČR) Piezoelectric energy harvesting September 26, 2017 20 / 24

Necessary optimality conditions

• Theorem. Let ˆ

F, g, J be continuously differentiable, and let (y∗, a∗, θ∗) be a local maximizer of the problem. Then there exist adjoint states p∗ ∈ W 1,2(0, T; Rn), q∗ ∈ BV (0, T) such that −˙ p∗(t) = ∂y ˆ F(t, y∗(t), a∗(t); θ∗) · p∗(t) + ∂yg(t, y∗(t), a∗(t); θ∗) q∗(t) − ∂yJ(t, y∗(t), a∗(t); θ∗) for t ∈ (0, T), p∗(T) = 0, q∗(t) g(t, y∗(t), a∗(t); θ∗) = 0 for a. e. t ∈ {s ∈ (0, T) : |a∗(s)| = 1}, −˙ q∗(t) = ∂ag(t, y∗(t), a∗(t); θ∗) q∗(t) + ∂a ˆ F(t, y∗(t), a∗(t); θ∗) · p∗(t) − ∂aJ(t, y∗(t), a∗(t); θ∗) for a. e. t ∈ {s ∈ (0, T) : |a∗(s)| < 1} , q∗(T) = 0 0 ∈ T

• ∂θJ(t, y∗(t), a∗(t); θ∗) − ∂θ ˆ

F(t, y∗(t), a∗(t); θ∗) · p∗(t) − ∂θg(t, y∗(t), a∗(t); θ∗) q∗(t)

• dt + ∂IΘ(θ∗).

Pavel Krejčí (Matematický ústav AV ČR) Piezoelectric energy harvesting September 26, 2017 21 / 24

Necessary optimality conditions

• Theorem. Let ˆ

F, g, J be continuously differentiable, and let (y∗, a∗, θ∗) be a local maximizer of the problem. Then there exist adjoint states p∗ ∈ W 1,2(0, T; Rn), q∗ ∈ BV (0, T) such that −˙ p∗(t) = ∂y ˆ F(t, y∗(t), a∗(t); θ∗) · p∗(t) + ∂yg(t, y∗(t), a∗(t); θ∗) q∗(t) − ∂yJ(t, y∗(t), a∗(t); θ∗) for t ∈ (0, T), p∗(T) = 0, q∗(t) g(t, y∗(t), a∗(t); θ∗) = 0 for a. e. t ∈ {s ∈ (0, T) : |a∗(s)| = 1}, −˙ q∗(t) = ∂ag(t, y∗(t), a∗(t); θ∗) q∗(t) + ∂a ˆ F(t, y∗(t), a∗(t); θ∗) · p∗(t) − ∂aJ(t, y∗(t), a∗(t); θ∗) for a. e. t ∈ {s ∈ (0, T) : |a∗(s)| < 1} , q∗(T) = 0 0 ∈ T

• ∂θJ(t, y∗(t), a∗(t); θ∗) − ∂θ ˆ

F(t, y∗(t), a∗(t); θ∗) · p∗(t) − ∂θg(t, y∗(t), a∗(t); θ∗) q∗(t)

• dt + ∂IΘ(θ∗).

Pavel Krejčí (Matematický ústav AV ČR) Piezoelectric energy harvesting September 26, 2017 21 / 24

Necessary optimality conditions

• Theorem. Let ˆ

F, g, J be continuously differentiable, and let (y∗, a∗, θ∗) be a local maximizer of the problem. Then there exist adjoint states p∗ ∈ W 1,2(0, T; Rn), q∗ ∈ BV (0, T) such that −˙ p∗(t) = ∂y ˆ F(t, y∗(t), a∗(t); θ∗) · p∗(t) + ∂yg(t, y∗(t), a∗(t); θ∗) q∗(t) − ∂yJ(t, y∗(t), a∗(t); θ∗) for t ∈ (0, T), p∗(T) = 0, q∗(t) g(t, y∗(t), a∗(t); θ∗) = 0 for a. e. t ∈ {s ∈ (0, T) : |a∗(s)| = 1}, −˙ q∗(t) = ∂ag(t, y∗(t), a∗(t); θ∗) q∗(t) + ∂a ˆ F(t, y∗(t), a∗(t); θ∗) · p∗(t) − ∂aJ(t, y∗(t), a∗(t); θ∗) for a. e. t ∈ {s ∈ (0, T) : |a∗(s)| < 1} , q∗(T) = 0 0 ∈ T

• ∂θJ(t, y∗(t), a∗(t); θ∗) − ∂θ ˆ

F(t, y∗(t), a∗(t); θ∗) · p∗(t) − ∂θg(t, y∗(t), a∗(t); θ∗) q∗(t)

• dt + ∂IΘ(θ∗).

Pavel Krejčí (Matematický ústav AV ČR) Piezoelectric energy harvesting September 26, 2017 21 / 24

Necessary optimality conditions

• Theorem. Let ˆ

F, g, J be continuously differentiable, and let (y∗, a∗, θ∗) be a local maximizer of the problem. Then there exist adjoint states p∗ ∈ W 1,2(0, T; Rn), q∗ ∈ BV (0, T) such that −˙ p∗(t) = ∂y ˆ F(t, y∗(t), a∗(t); θ∗) · p∗(t) + ∂yg(t, y∗(t), a∗(t); θ∗) q∗(t) − ∂yJ(t, y∗(t), a∗(t); θ∗) for t ∈ (0, T), p∗(T) = 0, q∗(t) g(t, y∗(t), a∗(t); θ∗) = 0 for a. e. t ∈ {s ∈ (0, T) : |a∗(s)| = 1}, −˙ q∗(t) = ∂ag(t, y∗(t), a∗(t); θ∗) q∗(t) + ∂a ˆ F(t, y∗(t), a∗(t); θ∗) · p∗(t) − ∂aJ(t, y∗(t), a∗(t); θ∗) for a. e. t ∈ {s ∈ (0, T) : |a∗(s)| < 1} , q∗(T) = 0 0 ∈ T

• ∂θJ(t, y∗(t), a∗(t); θ∗) − ∂θ ˆ

F(t, y∗(t), a∗(t); θ∗) · p∗(t) − ∂θg(t, y∗(t), a∗(t); θ∗) q∗(t)

• dt + ∂IΘ(θ∗).

Pavel Krejčí (Matematický ústav AV ČR) Piezoelectric energy harvesting September 26, 2017 21 / 24

Method of proof

Pavel Krejčí (Matematický ústav AV ČR) Piezoelectric energy harvesting September 26, 2017 22 / 24

Method of proof Derive necessary optimality conditions for the regularized problem with Lagrange multipliers pγ, qγ ;

Pavel Krejčí (Matematický ústav AV ČR) Piezoelectric energy harvesting September 26, 2017 22 / 24

Method of proof Derive necessary optimality conditions for the regularized problem with Lagrange multipliers pγ, qγ ; Prove estimates for pγ in W 1,2(0, T) and for qγ in W 1,1(0, T);

Pavel Krejčí (Matematický ústav AV ČR) Piezoelectric energy harvesting September 26, 2017 22 / 24

Method of proof Derive necessary optimality conditions for the regularized problem with Lagrange multipliers pγ, qγ ; Prove estimates for pγ in W 1,2(0, T) and for qγ in W 1,1(0, T); Select convergent subsequences pγ → p∗ weakly in W 1,2(0, T) and qγ → q∗ pointwise in BV (0, T) (Helly Selection Principle);

Pavel Krejčí (Matematický ústav AV ČR) Piezoelectric energy harvesting September 26, 2017 22 / 24

Method of proof Derive necessary optimality conditions for the regularized problem with Lagrange multipliers pγ, qγ ; Prove estimates for pγ in W 1,2(0, T) and for qγ in W 1,1(0, T); Select convergent subsequences pγ → p∗ weakly in W 1,2(0, T) and qγ → q∗ pointwise in BV (0, T) (Helly Selection Principle); Distinguish the cases that a∗(t) is on the boundary or in the interior

• f the admissible interval [−1, 1].

Pavel Krejčí (Matematický ústav AV ČR) Piezoelectric energy harvesting September 26, 2017 22 / 24

Applications and conclusions

Pavel Krejčí (Matematický ústav AV ČR) Piezoelectric energy harvesting September 26, 2017 23 / 24

Applications and conclusions Inversion and energy balance equations for hysteresis operators have important applications in Real time control of piezoelectric or magnetostrictive actuators and sensors, where algorithms for fast and accurate inversion of hysteresis

• perators are of central importance;

Pavel Krejčí (Matematický ústav AV ČR) Piezoelectric energy harvesting September 26, 2017 23 / 24

Applications and conclusions Inversion and energy balance equations for hysteresis operators have important applications in Real time control of piezoelectric or magnetostrictive actuators and sensors, where algorithms for fast and accurate inversion of hysteresis

• perators are of central importance;

Optimization of magnetostrictive energy harvesting processes under mechanical loading;

Pavel Krejčí (Matematický ústav AV ČR) Piezoelectric energy harvesting September 26, 2017 23 / 24

Applications and conclusions Inversion and energy balance equations for hysteresis operators have important applications in Real time control of piezoelectric or magnetostrictive actuators and sensors, where algorithms for fast and accurate inversion of hysteresis

• perators are of central importance;

Optimization of magnetostrictive energy harvesting processes under mechanical loading; Wave propagation modeling in piezoelectric solids.

Pavel Krejčí (Matematický ústav AV ČR) Piezoelectric energy harvesting September 26, 2017 23 / 24

Applications and conclusions Inversion and energy balance equations for hysteresis operators have important applications in Real time control of piezoelectric or magnetostrictive actuators and sensors, where algorithms for fast and accurate inversion of hysteresis

• perators are of central importance;

Optimization of magnetostrictive energy harvesting processes under mechanical loading; Wave propagation modeling in piezoelectric solids. The techniques based on the Preisach model Offer a tool for modeling the butterfly magnetostrictive hysteresis;

Pavel Krejčí (Matematický ústav AV ČR) Piezoelectric energy harvesting September 26, 2017 23 / 24

Applications and conclusions Inversion and energy balance equations for hysteresis operators have important applications in Real time control of piezoelectric or magnetostrictive actuators and sensors, where algorithms for fast and accurate inversion of hysteresis

• perators are of central importance;

Optimization of magnetostrictive energy harvesting processes under mechanical loading; Wave propagation modeling in piezoelectric solids. The techniques based on the Preisach model Offer a tool for modeling the butterfly magnetostrictive hysteresis; Are accessible to standard identification methods;

Pavel Krejčí (Matematický ústav AV ČR) Piezoelectric energy harvesting September 26, 2017 23 / 24

Applications and conclusions Inversion and energy balance equations for hysteresis operators have important applications in Real time control of piezoelectric or magnetostrictive actuators and sensors, where algorithms for fast and accurate inversion of hysteresis

• perators are of central importance;

Optimization of magnetostrictive energy harvesting processes under mechanical loading; Wave propagation modeling in piezoelectric solids. The techniques based on the Preisach model Offer a tool for modeling the butterfly magnetostrictive hysteresis; Are accessible to standard identification methods; Are relatively simple and robust; error estimates can easily be derived;

Pavel Krejčí (Matematický ústav AV ČR) Piezoelectric energy harvesting September 26, 2017 23 / 24

Applications and conclusions Inversion and energy balance equations for hysteresis operators have important applications in Real time control of piezoelectric or magnetostrictive actuators and sensors, where algorithms for fast and accurate inversion of hysteresis

• perators are of central importance;

Optimization of magnetostrictive energy harvesting processes under mechanical loading; Wave propagation modeling in piezoelectric solids. The techniques based on the Preisach model Offer a tool for modeling the butterfly magnetostrictive hysteresis; Are accessible to standard identification methods; Are relatively simple and robust; error estimates can easily be derived; Can be coupled with the full system of balance PDEs describing, e.g., vibrations of piezoelectric beams.

Pavel Krejčí (Matematický ústav AV ČR) Piezoelectric energy harvesting September 26, 2017 23 / 24

References

• M. Brokate: Optimale Steuerung von gewöhnlichen Differentialgleichungen

mit Nichtlinearitäten vom Hysteresis-Typ. Peter-Lang-Verlag, 1987.

• P. Krejčí: On Maxwell equations with the Preisach hysteresis operator: the
• ne-dimensional time-periodic case. Appl. Math. 34, 364–374, 1989.
• M. Brokate, P. Krejčí: Optimal control of ODE systems involving a rate

independent variational inequality. DCDS B 18 (2013), 331–348.

• D. Davino, P. Krejčí, C. Visone: Fully coupled modeling of

magnetomechanical hysteresis through ‘thermodynamic’ compatibility. Smart Materials and Structures 22, 095009-, 2013.

• B. Kaltenbacher, P. Krejčí: A thermodynamically consistent

phenomenological model for ferroelectric and ferroelastic hysteresis. ZAMM

• Z. Angew. Math. Mech. 96, 874–891, 2016.
• D. Davino, P. Krejčí, A. Pimenov, D. Rachinskii, C. Visone: Analysis of an
• perator-differential model for magnetostrictive energy harvesting. Comm. in

Nonlinear Science and Numerical Simulation 39, 504–519, 2016.

Pavel Krejčí (Matematický ústav AV ČR) Piezoelectric energy harvesting September 26, 2017 24 / 24