Reconstruction of flow domain boundaries from velocity data via multi-step optimization of distributed resistance
- Pártl, Ondřej
- Wilbrandt, Ulrich
- Mura, Joaquín
- Caiazzo, Alfonso
2020 Mathematics Subject Classification
- 49M41 76D55 76D07
- Brinkmann equation, gradient-based optimization, stabilized finite elements, boundary reconstruction
We reconstruct the unknown shape of a flow domain using partially available internal velocity measurements. This inverse problem is motivated by applications in cardiovascular imaging where motion-sensitive protocols, such as phase-contrast MRI, can be used to recover three-dimensional velocity fields inside blood vessels. In this context, the information about the domain shape serves to quantify the severity of pathological conditions, such as vessel obstructions. We consider a flow modeled by a linear Brinkman problem with a fictitious resistance accounting for the presence of additional boundaries. To reconstruct these boundaries, we employ a multi-step gradient-based variational method to compute a resistance that minimizes the difference between the computed flow velocity and the available data. Afterward, we apply different post-processing steps to reconstruct the shape of the internal boundaries. To limit the overall computational cost, we use a stabilized equal-order finite element method. We prove the stability and the well-posedness of the considered optimization problem. We validate our method on three-dimensional examples based on synthetic velocity data and using realistic geometries obtained from cardiovascular imaging.
- Comput. Math. Appl., 129 (2023), pp. 11--33, DOI /10.1016/j.camwa.2022.11.006 .