This example shows how to calculate the internal heating in a composite pillar due to losses in the constituent materials. The unit cell (figure 1) is fired in a pulsed mode meaning the method for calculating thermal response is slightly different to that of a continuous wave (cw) excitation. With pulsed excitation a short broadband pulse of ultrasound is transmitted, and the model is run until all significant waves have left the model or have been absorbed at the boundaries. Data is gathered over the entire simulation for loss (energy deposition).
As with any of our models it is imperative that the attenuation properties of the material are accurate to ensure accurate calculation of energy loss in each material. OnScale has highly versatile damping models that allow not only for separation of the shear and longitudinal damping, but for frequency dependence of the damping. The load in the initial mechanical simulation (i.e., the acoustic wave propagation model) uses an electrical load, two electrodes are defined, the top electrode is excited with a broadband pulse and the bottom electrode is grounded.
The wave propagates from the front face of the unit cell and moves towards the end of the water load where it is absorbed at the boundaries and the wave exits the model. The ‘loss’ array is saved at the end of the simulation for use as in input function for the thermal model.
Use the acoustic model as a base to begin building the thermal model; the files are similar up to the point of the “site” commands. Take the ThermalUnitCell.flxinp file and discard all commands after “site” has been completed both models will be identical. The material properties for the thermal model are the same as those for the acoustic model. The only alteration is in the mesh density. As thermal gradients are typically much smaller than acoustic gradients, the mesh can be less dense for the thermal model. However, since this is a simple example, we have kept the mesh density the same for both models.
The first significant difference between acoustic and thermal models is in the “calc” command. Because this is a thermal model, temperature (“tmpr”) rather than stresses, strain, or pressures is calculated. As you want to see the hottest points throughout the simulation, use the “calc max” option to store the highest temperatures. You must tell OnScale that this is a thermal rather than acoustic calculation. Using the “heat” command, tell OnScale to use the conjugate gradient solver (“slvr cgds”), which is ideal for large thermal models. Also tell OnScale that, due to the difference in time step (“cupl off”), the thermal and mechanical fields are not coupled. Boundary conditions (“boun”) are defined differently in thermal models than in acoustical ones. By default, any unspecified boundary in a thermal model is symmetrical; that is, you assume that an identical structure and load exist on the other side of the surface plane. Any other surface, either in contact with a heat source or acting as a radiator to infinity, must be specified. Use the “boun” commands to import the acoustic absorption loss from OnScale for use as a thermal drive in the 2nd model.
Unlike with most acoustic simulations in OnScale, you must specify a time step with the “time” command. As this is a thermal simulation, there are no wavespeeds with which to set the appropriate time step. Using even the true material velocities would result in a time step suited to a microseconds-long acoustical simulation rather than a seconds-long thermal one. Fortunately, the implicit thermal solution is unconditionally stable; that is, the model will be stable and run to completion no matter what time step is specified. You must, however, specify a time step that is small enough to capture the temperature variation over time anywhere in the model. Finally, unlike with acoustic models, with thermal models the base settings must be determined—in this case, the starting temperature. Use the “set tmpr” commands to set the temperature to 20°C.
The temperature at the top of the composite is plotted (figure 3) alongside a plot of temperature vs time showing how the temperature on the front face rises over time. Although this energy deposition and temperature rise could be calculated simultaneously, the significant difference in time scale between thermal and mechanical effects makes such an approach inefficient. Because the mechanical wave propagation occurs on a scale of microseconds and the thermal effects on a scale of seconds (a factor of 1,000,000 difference), the mechanical and thermal models may be decoupled without loss of simulation accuracy. The usual sequence is to run the mechanical ultrasound propagation model to completion and then use the results in a second, thermal model that is run at a different time step. Thus, why we needed two OnScale input files for this example: one for the mechanical model and another for the thermal model. Using the implicit thermal solver and manually set our timestep we were able to accurately simulate the temperature rise in the model over a time period of 1200s (20 minutes).
Discussion of Thermal Example
The thermal example shows how to determine heat losses created in a piezoelectric model by splitting the mechanical and thermal simulations into two sections. First you model the acoustic wave interactions in the model and then you use the resulting acoustic loss data to model the thermal source in the tissue. The decoupling of the two models allows for efficient modelling, as they occur on vastly different time scales. The example also shows how to calculate the “pmax” or pressure maximum developed in the model. Using the implicit thermal solver and manually set our timestep we were able to accurately simulate the temperature rise in the model over a time period of 1200s (20 minutes).