Ultrasound Simulation Convergence: Test the Design Decision

By ALEVA · Orange County, California, USA · November 7, 2026

Engineering question: Before selecting a probe geometry or beamforming configuration from an acoustic simulation, how can an OEM team establish that numerical settings are not determining the apparent winner?

A detailed pressure map is an output, not an accuracy certificate. The practical acceptance question is whether the quantities that drive the design decision remain sufficiently stable when numerical resolution changes. This article proposes a verification workflow; it reports no ALEVA hardware measurement or completed simulation.

Define the output before refining the grid

Choose the decision variable first: focal location, pressure at a specified physical coordinate, an explicitly defined beam width, or the received waveform at a sensor. Keep the source waveform, physical geometry, material properties and observation interval fixed while testing numerical settings. Record how source and receiver positions are represented when the grid changes; otherwise a refinement experiment can silently become a different physical model.

The official k-Wave documentation explains that grid spacing and sound speed determine the highest frequency the spatial grid can represent. Its time-varying-source example also warns that a temporally sampled source can contain frequencies the spatial grid cannot support. A sufficiently fine time step therefore does not, by itself, establish adequate spatial resolution. Review the relevant signal bandwidth, not just the nominal center frequency. A grid-support limit is not an accuracy guarantee. If automatic source filtering changes with the grid, document the effective source bandwidth and use a common supported band for the comparison. [1, 2]

Separate three numerical checks

Spatial resolution: refine the grid and compare outputs at the same physical locations. If interpolation is needed, record it as part of the comparison. Do not hide focal displacement by aligning every waveform or recentering every image before measuring the error.

Temporal resolution: reduce the time step while holding the spatial model fixed. Compare arrival time, phase and the agreed amplitude measure. A stable solver can still produce an output that is insufficiently accurate for the intended decision.

Domain and boundary treatment: vary the domain extent and absorbing-boundary settings separately. k-Wave uses a perfectly matched layer to suppress the return of waves associated with its Fourier-based spatial derivatives. Boundary treatment is therefore a numerical setting to document, rather than an invisible default. [3]

Compare alternatives under the same test

For two candidate designs, retain the individual results at each refinement level as well as their difference. If the design ordering changes, or the remaining numerical variation is comparable to the proposed advantage, the simulation has not yet established a robust preference. Set the acceptable variation from the engineering requirement before inspecting the final plots; no universal percentage is proposed here.

A 2026 peer-reviewed study by Almuna-Morales and colleagues describes a volume–surface integral-equation approach for heterogeneous skull models and compares it with a high-resolution boundary-element reference. Its solver and physical scope differ from a general imaging-probe model. The useful lesson for an OEM evaluation is to document the reference and model assumptions, rather than transplant one method’s grid rule into another solver. [4]

Budget the cost and state the limit

For an idealized three-dimensional uniform grid covering a fixed volume, halving spacing along all axes multiplies the cell count by eight. If the time step is also halved for the same simulated duration, the number of time steps doubles. That is sixteen times as many cell–time-step combinations, by arithmetic—not a measured runtime prediction. FFT costs, boundary layers, stored sensor data, solver implementation and available memory change the actual resource requirement.

Numerical convergence tests sensitivity to the discretization. It does not establish that material properties, source calibration or the physical model describe the real probe correctly. Independent experimental validation remains a separate task.

For OEM/ODM handover, request the solver version, input files, geometry and source definitions, grid and time-step settings, boundary configuration, comparison procedure and acceptance record. Preserve unresolved differences. Release a simulation-backed design decision only with an explicit statement of what was tested and which physical uncertainties remain.

References

1. k-Wave: Initial Value Problem in a Homogeneous Propagation Medium. Official documentation; publication date not stated, consulted November 7, 2026.

2. k-Wave: time-varying pressure source in a homogeneous medium. Official documentation; publication date not stated, consulted November 7, 2026.

3. k-Wave: kspaceFirstOrder3D. Official function documentation; publication date not stated, consulted November 7, 2026.

4. Almuna-Morales et al. Full-wave modeling of transcranial ultrasound using volume-surface integral equations and CT-derived heterogeneous skull data. Ultrasonics 161, 107954 (2026). Published online January 6, 2026; May 2026 issue. Indexed abstract; UCL-hosted manuscript. The manuscript is distinct from the journal’s final formatted version.