Identification of Nonlinear Coefficients in Hyperbolic PDEs, with Application to Piezoelectricity
In this paper we consider the problem of determining parameters in nonlinear partial differential equations of hyperbolic type from boundary measurements. In order to investigate the qualitative behavior of this class of identification problems, we analyz
- PDF / 450,935 Bytes
- 23 Pages / 467.724 x 666.139 pts Page_size
- 99 Downloads / 182 Views
dentification of Nonlinear Coefficients in Hyperbolic PDEs, with Application to Piezoelectricity Barbara Kaltenbacher Abstract. In this paper we consider the problem of determining parameters in nonlinear partial differential equations of hyperbolic type from boundary measurements. In order to investigate the qualitative behavior of this class of identification problems, we analyze the model problem of identifying c in the nonlinear wave equation dtt − (c(dx )dx )x = 0 and discuss stability and identifiability for this problem. Moreover, we derive applicability of these results to material parameter identification in piezoelectricity and provide numerical reconstruction results. Mathematics Subject Classification (2000). Primary 35R30; Secondary 35L70. Keywords. Parameter identification, nonlinear wave equation, piezoelectricity.
1. Introduction Consider the model problem of identifying the function c in the nonlinear hyperbolic PDE (1.1) ρdtt − (c(dx )dx )x = 0 x ∈ (0, L), t ∈ (0, T ) , with boundary conditions d(0, t) = 0 c(dx (L, t))dx (L, t) = g(t)
t ∈ (0, T ) ,
(1.2)
and initial conditions d(x, 0) = d0 (x),
dt (x, 0) = d1 (x)
x ∈ (0, L) ,
(1.3)
for given g : [0, T ] → R, d0 : [0, L] → R, d1 : [0, L] → R, from additional boundary measurements y(t) = d(L, t) . Supported by the German Science Foundation DFG under grant Ka 1778/1.
194
B. Kaltenbacher
This can, e.g., be seen as a model of a vibrating string of length L, with the elasticity coefficient c depending on the strain dx . At the left boundary x = 0, the string is clamped, at the right boundary x = L, it is excited in longitudinal direction by a surface load (mechanical stress) g, and measurements of the displacement d are made. Our motivation for studying (1.1) comes from the problem of determining material parameter curves for piezoelectric materials (see Section 4), in the piezoelectric PDEs , 2 = 0 in Ω ρ ∂∂t2d − BT cE Bd + eT gradφ , (1.4) −div eBd − εS gradφ = 0 in Ω . Here, d denotes the vector of mechanical displacements, φ the electric potential, ρ the mass density, and B a first order differential operator with respect to the space variables, that reflects the relation between displacements and strain (see, e.g., [17] and Section 4 below). The system (1.4) models the piezoelectric effect, i.e., a coupling between the electrical and the mechanical behavior of certain materials. The material tensors cE , εS , and e, appearing in (1.4) are the elasticity coefficients, the dielectric constants, and the piezoelectric coupling coefficients, respectively. When large excitations are applied, the material parameters will not be constants any more but depend on the field quantities, i.e., in (1.4), the entries of the material tensors cE , εS , and e are functions of the amplitude of the electric ˆ An interesting task = |gradφ| and/or the mechanical strain |S| = |Bd|. field |E| is to reconstruct these parameter functions from overdetermined measurements at the boundary, e.g., voltage-current measurements at an electrode or displacement measurements at a surfac
Data Loading...
