Finite element modeling of transient ultrasonic waves in linear viscoelastic media
Linear viscoelasticity offers a minimal framework within which to construct a causal model for wave propagation in absorptive media. Viscoelastic media are often described as media with memory, that is, the present state of stress is dependent on the present strain and the complete time history of strain weighted by time convolution with an appropriate time-dependent stress relaxation modulus. An axisymmetric, displacement based finite element method for modeling pulsed ultrasonic waves in linear, homogeneous viscoelastic media is developed that does not require storage of the complete time history of displacement at every node. This is accomplished by modeling the stress relaxation moduli as discrete or continuous spectra of decaying exponentials. The viscoelastic finite element method serves as a test bed for studying three inverse methods for recovering time dependent longitudinal moduli from pulsed ultrasonic waves transmitted through a slab of viscoelastic material with properties known a priori. Specifically, two existing inverse methods called propagator methods, denoted here as the two-slab method and slab-substitution method, are modeled and compared to show relative advantages and disadvantages of both. Both methods require attenuation and wave speed as a function of frequency derived from transmitted wave data for inversion and recovery of modulus data. Several different variables such as measurement location and source radius are varied to discern those variables that have greatest influence on accuracy of reconstructed moduli. It is found that an increase in source aperture radius causes the greatest improvement in modulus accuracy. Another novel inverse method known as wave splitting is applied to numerical data generated by the finite element test bed. Wave splitting requires a time-dependent transmission kernel for recovery of a viscoelastic modulus rather than frequency-dependent attenuation and wave speed. It is shown that in principle wave splitting can recover the material modulus with a data derived from a simulated ultrasonic experiment, but it is not as robust as the other two frequency-domain inverse methods studied. Its main drawback is that transmission kernel data required for inversion must be known for the same thickness of viscoelastic slab implying that pulses with relatively high center frequencies must be propagated through slabs whose thickness is only appropriate for low frequency measurement. Material attenuation quickly reduces transmitted waves at high frequencies to unacceptable low levels when propagated through thick slabs appropriate for pulses centered at lower frequencies. In general, the finite element method has been utilized as an effective tool for comparing alternative inverse methods.