Discrete equations of piezoelectric structures

Contents Functions

1.3 Discrete equations of piezoelectric structures

Hamilton's principle is used to derive the dynamic variational principle [1]:

(87)

where V is the volume of the piezoelectric structure, ρ is the mass density, {u} is the displacement field and {δ u } its variation, {φ} is the elecctric pontential and {δ φ } its variation. {f } is the volumic force, {ρ_e } the volumic charge density, {t } the vector of applied surface forces on Ω₁ and {σ } the charge density applied on Ω₂. The variational principle is the starting point for all discrete finite element formulations. 3D and shell approximations are detailed below.

1.3.1 Piezoelectric solid finite elements

For 3D solids, the discretized strain and electric fields are linked to the discretized displacement vector (u,v,w) and electric potential φ by:

(88)

where N,x u is a short notation for

(89)

and N_i(x,y,z) are the finite element shape functions. Plugging (88) in (87) leads to the discrete set of equations which are written in the matrix form:

(90)

where {q_mech } contains the mechanical degrees of freedom (3 per node related to u,v,w), and {V } contains the electrical degrees of freedom (1 per node, the electric potential φ). {F_mech } is the vector of applied external mechanical forces, and {Q } is the vector of applied external charges.

1.3.2 Piezoelectric shell finite elements

Shell strain is defined by the membrane, curvature and transverse shear as well as the electric field components. In the piezoelectric multi-layer shell elements implemented in SDT, it is assumed that in each piezoelectric layer i=1...n, the electric field takes the form E^→= (0 0 E_zi). E_zi is assumed to be constant over the thickness h_i of the layer and is therefore given by E_zi=−Δ φ_i/h_i where Δ φ_i is the difference of potential between the electrodes at the top and bottom of the piezoelectric layer i. It is also assumed that the piezoelectric principal axes are parallel to the structural orthotropy axes.

Figure 1.10: Multi-layer shell piezoelectric element

The discretized strain and electric fields of a piezoelectric shell take the form

(91)

There are thus n additional degrees of freedom Δ φ_i, n being the number of piezoelectric layers in the laminate shell. The constitutive laws are obtained by using the "piezoelectric plates" hypothesis (16) and the definitions of the generalized forces N,M,Q and strains є,κ,γ for shells:

(92)

D_zi is the electric displacement in piezoelectric layer , z_mi is the distance between the midplane of the shell and the midplane of piezoelectric layer i (Figure 1.10), G_i is given by

(93)

where ^* refers to the piezoelectric properties under the piezoelectric plate assumption as detailed in section 1.2.2 and [R_s]_i are rotation matrices associated to the angle θ of the principal axes 1,2 of the piezoelectric layer given by:

(94)

Plugging (91) into (87) leads again to:

(95)

where {q_mech } contains the mechanical degrees of freedom (5 per node corresponding to the displacements u,v,w and rotations rx,ry), and {V } contains the electrical degrees of freedom. The electrical dofs are defined at the element level, and there are as many as there are active layers in the laminate. Note that the electrical degree of freedom is the difference of the electric potential between the top and bottom electrodes Δ φ.

1.3.3 Full order model

Piezoelectric models are described using both mechanical q_mech and electric potential DOF V. As detailed in sections section 1.3.1 and section 1.3.2, one obtains models of the form

(96)

for both piezoelectric solids and shells, where Z_CC(s) is the dynamic stiffness expressed as a function of the Laplace variable s.

For piezoelectric shell elements, electric DOF correspond to the difference of potential on the electrodes of one layer, while the corresponding load is the charge Q. In SDT, the electric DOFs for shells are unique for a single shell property and are thus giving an implicit definition of electrodes (see p_piezo Shell). Note that a common error is to fix all DOF when seeking to fix mechanical DOFs, calls of the form 'x==0 -DOF 1:6' avoid this error.
For volume elements, each volume node is associated with an electric potential DOF and one defines multiple point constraints to enforce equal potential on nodes linked by a single electrode and sets one of the electrodes to zero potential (see p_piezo ElectrodeMPC and section 2.5 for a tutorial on how to set these contraints). During assembly the constraints are eliminated and the resulting model has electrical DOFs that correspond to differences of potential and loads to charge.

Figure 1.11: Short circuit: voltage actuator, charge sensor

Short circuit (SC), charge sensors configurations correspond to cases where the potential is forced to zero (the electrical circuit is shorted). In (96), this corresponds to a case where the potential (electrical DOF) is fixed and the charge corresponds to the resulting force associated with this boundary condition.

A voltage actuator corresponds to the same problem with V=V_In (built in SDT using fe_load DofSet entries). The closed circuit charge is associated with the contraint on the enforced voltage and can be computed by extracting the second row of (96)

(97)

p_piezo ElectrodeSensQ provides utilities to build the charge sensors, including sensor combinations.

SC is the only possibly boundary condition in a FEM model where voltage is the unknown. The alternative is to leave the potential free which corresponds to not specifying any boundary condition. When computing modes under voltage actuation, the proper boundary condition is a short circuit.

Figure 1.12: Open circuit (voltage sensor, charge actuator)

Open circuit (OC), voltage sensor , configurations correspond to cases where the charge remains zero and a potential is created on the electrodes due to mechanical deformations. A piezoelectric actuator driven using a charge source also would correspond to this configuration (but the usual is voltage driving).

The voltage DOF {V} associated to open-circuits are left free in (96). Since electrostatics are normally considered, Z_vv is actually frequency independent and the voltage DOFs could be condensed exactly

(98)

Since voltage is an explicit DOF, it can be observed using fe_case SensDOF sensor entries. Similarly charge is dual to the voltage, so a charge input would be a simple point load on the active DOF associated to an electrode. Note that specifying a charge distribution does not make sense since you cannot both enforce the equipotential condition and specify a charge distribution that results from this constraint.

It is possible to observe charge in an OC condition, but this is of little interest since this charge will remain at 0.

1.3.4 Using the Electrode stack entry

SDT 6.6 underwent significant revisions to get rid of solver strategies that were specific to piezo applications. The info,Electrodes of earlier releases is thus no longer necessary. To avoid disruption of user procedures, you can still use the old format with a .ver=0 field.
p_piezo ElectrodeInit is used to build/verify a data structure describing master electric DOFs associated with electrodes defined in your model. The info,Electrode stack entry is a structure with fields

.data rows NodeId IsOpen gives the electrode nodes and for each one 1 if the circuit is open (voltage free), and 0 if it is closed (voltage enforced or fixed, actuator).
.ver=1 is used to specify that the more general piezoelectric strategies of SDT >= 6.6 are used. This is the combined with the p_piezo Electrode2Case command which builds piezo loads and sensors. For SDT 6.5 strategies, use .ver=0.
.def .DOF, .lab_in only needed when combining multiple electrodes into a single input. The .lab_in is a cell array of strings, you should end the string with V so that it shows Q for associated charge sensors.
Each column gives the weighting coefficients associated with each electrode. Thus def=[1;0;1] corresponds to a single equal input on electrodes 1 and 3. Note that it does not make sense to combine electrical DOFs that are of mixed nature (actuator/sensor).
The .DOF field should contain NodeId+.21 since the potential corresponds to DOF .21.
The .lab_in field can be used to provide labels associated with each actuator/sensor defined as a column of def. You should end the label with V so that the collocated sensor ends with a Q label.
.cta .lab (optional) can be used to combine electrodes into sensors / actuators. Each row of .cta defines a sensor (with matching .lab). Each column corresponds to an electrode declared in the .data field. You cannot combine open and closed circuit electrodes. It is possible to use both a .cta and a .def field.

[model,data]=p_piezo('ElectrodeInit',model); generates a default value for the electrode stack entry. Combination of actuators and sensors (both charge and voltage) is illustrated in section 2.1.3.

1.3.5 Model reduction

When building reduced or state-space models to allow faster simulation, the validity of the reduction is based on assumptions on bandwidth, which drive modal truncation, and considered loads which lead to static correction vectors.

Modes of interest are associated with boundary conditions in the absence of excitation. For the electric part, these are given by potential set to zero (grounded or shorted electrodes) and enforced by actuators (defined as DofSet in SDT) which in the absence of excitation is the same as shorting.

Excitation can be mechanical F_mech, charge on free electric potential DOF Q_In and enforced voltage V_In. One thus seeks to solve a problem of the form

(99)

Using the classical modal synthesis approach (implemented as fe2ss('free')), one builds a Ritz basis combining modes with grounded electrodes (V_In=0), static responses to mechanical and charge loads and static response to enforced potential

(100)

In this basis, one notes that the static response associated with enforced potential V_In does not verify the boundary condition of interest for the state-space model where V_In=0. Since it is desirable to retain the modes with this boundary condition as the first vectors of basis (100) and to include static correction as additional vectors, the strategy used here is to rewrite reduction as

(101)

where the response associated with reduced DOFs q_R verifies V_In=0 and the total response is found by adding the enforced potential on the voltage DOF only. The presence of this contribution corresponds to a D term in state-space models. The usual SDT default is to include it as a residual vector as shown in (100), but to retain the shorted boundary conditions, form (101) is prefered.