Analytically Benchmarked Spatial and Temporal Discretization Guidelines for Finite Element Simulation of Thermal Waves

Abstract

Finite element simulation is widely used to design and interpret thermal wave-based measurement systems. The accuracy of the results obtained from such simulations depends on how finely the model is resolved in space and time. The discretization resolution is often chosen case by case rather than following a well-defined guideline. This work recommends a discretization protocol for finite element simulation of thermal waves where the spatial discretization is scaled to the thermal diffusion length and the temporal discretization to the number of time steps per cycle. The associated discretization errors are quantified using three benchmark problems of increasing complexity. The spatial phase error converges at second order in the element size, reaching 0.14 degrees at ten elements per diffusion length, whereas the temporal error decreases only at first order in the number of time steps per cycle; thus, the time step can be set from a target phase tolerance $ε$. Each error is assessed against a reference that isolates it. The benchmark case with subsurface defects shows the consequence of violating the recommended discretization protocol.