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A Stress-Based Estimator for Pressure and Stress Recovery from Velocity Measurements

Nicolás Barnafi, Benjamín Cisternas, Ernesto Castillo, Felipe Galarce

arXiv:2609.09484v1math.NAcs.NA

TL;DR

Pressure estimation from velocity measurements remains a longstanding engineering challenge, especially when direct pressure acquisition is expensive or invasive. SVE reconstructs the full stress field from velocity data and recovers pressure afterward; the paper reports robustness to measurement degradation and accurate recovery in a clean transient Venturi benchmark.

  • Problem

    Pressure estimation from velocity measurements is a longstanding challenge, while direct pressure acquisition can be expensive, invasive, or destructive.

    Source: §1 · p. 1

  • Method

    SVE reformulates pressure recovery as full stress reconstruction, with pressure recovered afterward from the reconstructed stress.

    Source: §1 · p. 2, §2 · p. 4, §6 · p. 20

  • Results

    SVE is reported well-posed and robust against measurement degradation, and accurately recovers pressure in a clean transient Venturi benchmark.

    Source: §1 · p. 2, §6 · p. 20

  • Takeaways & Limitations

    The authors conclude that SVE provides a stable and accurate framework for pressure recovery.

    Source: §6 · p. 20

    A stated comparator caveat is that PPE underestimates field peaks with noisy data, largely because of imposed artificial pressure smoothness requirements.

    Source: §1 · p. 2

Abstract

from arXiv · show

Non-invasive pressure field estimation from velocity measurements is a longstanding engineering problem. We propose, analyze, and test a pressure-recovery method that computes a full stress field from velocity measurements, and leaves the pressure estimation as a cheap post-processing step. The method relies on a stress-velocity first order formulation of the Navier-Stokes equations, and we show that the formulation accounts for deviations from incompressibility in the measured velocity field by construction. In addition, we theoretically establish the convergence of the finite element (FE) approximation scheme, the stability of the stress recovery with respect to finite-resolution velocity measurements, and then validate this theory numerically. Our results show that the proposed estimator is robust in convective flow regimes and remains accurate at reduced spatial resolution, improving upon state-of-the-art pressure-recovery strategies.

1 Introduction

The section frames pressure recovery from velocity data as a longstanding engineering challenge and introduces SVE, which reconstructs stress before recovering pressure. It presents the formulation's numerical advantages and theoretical properties, then outlines tests of convergence, resolution sensitivity, noise robustness, and convective-flow performance.

1 Introduction

  • The introduction identifies noisy-data peak underestimation as a limitation of PPE and names vWERP and STE as comparison baselines.It states that vWERP and STE outperform PPE.Source: §1 · p. 2
  • SVE recasts pressure recovery as full Cauchy stress reconstruction from velocity measurements, with pressure recovered afterward from the stress tensor's isotropic component.This makes stress reconstruction, rather than direct pressure estimation, the central problem.Source: §1 · p. 2
  • Stress reconstruction also provides access to shear-related quantities, including wall shear stress and oscillatory shear index.Source: §1 · p. 2
  • The elliptic, symmetric formulation admits H(div)-conforming finite elements and avoids auxiliary velocity-pressure saddle-point problems and artificial boundary conditions.The introduction contrasts these avoided requirements with STE, vWERP, and PPE.Source: §1 · p. 2
  • The method is reported well-posed and robust against measurement degradation, with an observability result coupling the stress and velocity fields.The introduction notes this coupling is expected for a stress-velocity formulation.Source: §1 · p. 2
  • The numerical tests assess convergence, sensitivity to measurement resolution and noise, and robustness in a moderately convective Reynolds-number regime mimicking blood flow.Raviart-Thomas elements are used for the finite-element implementation.Source: §1 · pp. 2–3

2 The pressure recovery problem

This section formulates pressure recovery from velocity data as stress reconstruction, with pressure computed afterward. It also treats measured velocity as potentially degraded and not exactly divergence-free.

2 The pressure recovery problem

  • The paper's route to a full pressure field is to reconstruct the stress tensor and recover pressure from its trace in a post-processing step.Source: §2 · p. 3
  • Velocity measurements are modeled as projections onto an observation space, allowing the data to represent under-resolution, voxel-wise sampling, or observations over only part of the domain.The numerical tests use observation spaces induced by voxel grids at different resolutions.Source: §2 · p. 4
  • The measured velocity field may violate incompressibility because of noise, partial measurements, or modeling errors, and the formulation accounts for deviations in incompressibility and momentum balance.A bias term associates recovered pressure with a corrected divergence-free velocity field.Source: §2 · p. 4
  • The stress tensor is the unknown field reconstructed from known velocity-dependent terms, and pressure is then recovered independently in post-processing.The section describes this as reducing pressure recovery to a stress-reconstruction problem driven by velocity data.Source: §2 · p. 4

3 Stress-Velocity Estimator

This section presents a stress-velocity estimator that reconstructs the full stress field from velocity measurements and recovers pressure from the stress trace, while accounting for non-solenoidal measurements. It establishes well-posedness and convergence, including for finite measurements approaching the measured velocity in L2(0, T; H1(Ω)), and describes incompressible-flow recovery and explicit data mismatch.

3 Stress-Velocity Estimator

  • The estimator replaces direct pressure recovery with reconstruction of a stress field from known velocity, followed by pressure recovery from the stress trace.The stress field satisfies momentum conservation and a constitutive relation.Source: §3 · p. 4

3.1 Continuous formulation

  • The continuous formulation reconstructs the full Cauchy stress using momentum balance and a deviatoric constitutive constraint, then recovers pressure a posteriori from its trace.The pressure recovery does not require additional assumptions about the explicit rheological law.Source: §3.1 · p. 5

3.2 Correction for non-solenoidal measurements

  • The recovered stress corresponds to a divergence-free velocity even when measurements are not; divergence-free measurements need no correction.The correction field z represents incompatibility with the strong formulation and is zero when div u = 0.Source: §3.2 · pp. 6–7

3.3 Discrete formulation

  • The discrete method uses Raviart–Thomas finite elements for stress and BDF2 to approximate transient velocity time derivatives; pressure is post-processed at each time step.At each step, the fully discrete SVE solves a stress-reconstruction problem driven by the known velocity field.Source: §3.3 · pp. 7–8

3.4 Well-posedness analysis

  • The continuous problem has a unique solution by the Lax–Milgram lemma, and conformity of the discrete space allows the same analysis for the discrete problem.The proof establishes boundedness and coercivity of the bilinear form.Source: §3.4 · p. 8
  • Recovery converges when the measurement operator converges in L2(0, T; H1(Ω)).The section also states that the infinite-measurement scenario can be approximated through partial measurements.Source: §3.4 · p. 9
  • The SVE formulation corresponds to recovery of an incompressible flow, and its data mismatch can be computed explicitly.Source: §3.4 · p. 9
  • The SVE formulation is well-posed, and its proposed finite-element scheme is convergent, including for Raviart–Thomas discretizations.Standard finite-element theory also gives a well-posed, convergent discrete problem for conforming approximation spaces.Source: §3.4 · p. 9

4 Baseline methods

This section reviews the Pressure Poisson Estimator (PPE), Virtual Work-Energy Relative Pressure Estimator (vWERP), and Stokes Estimator (STE) as pressure-recovery baselines. The PPE is the standard approach, while vWERP and STE have reported better accuracy and robustness against noisy and noise-free measurements.

4 Baseline methods

  • Prior work reports that vWERP and STE improve accuracy and robustness over the classical PPE, including with noisy velocity measurements.The section also recalls PPE as the standard baseline for pressure recovery from velocity data.Source: §4 · p. 9

4.1 Pressure Poisson Estimator

  • The PPE obtains pressure by solving a Poisson equation driven by inertial terms, then computes the pressure drop from the recovered field.For an incompressible Newtonian fluid, its continuous viscous contribution vanishes; the section specifies piecewise linear spatial elements and first- or second-order backward differences for time derivatives.Source: §4.1 · p. 10

4.2 Virtual Work-Energy Relative Pressure Estimator

  • vWERP uses a precomputed divergence-free test function to estimate a global pressure drop, assuming pressure is nearly constant over inlet and outlet sections.Its velocity-dependent terms can be evaluated directly, but the method does not reconstruct the complete pressure field.Source: §4.2 · pp. 10–11

4.3 Stokes Estimator

  • STE solves an auxiliary Stokes-type system from the known velocity field to recover the full pressure field, from which the pressure drop is evaluated.Integrating by parts reduces direct differentiation of noisy data and leads to a more stable right-hand-side evaluation.Source: §4.3 · pp. 11–12

5 Numerical tests

Two-dimensional numerical tests examine finite-element convergence, sensitivity to velocity-measurement resolution, and pressure recovery in a Venturi channel. They also compare SVE with STE and vWERP in noise-free and noisy measurement cases.

5 Numerical tests

  • The numerical-test program uses two-dimensional flows to assess FE convergence, velocity-measurement quality, and Venturi pressure recovery with clean and perturbed velocity data.The Venturi benchmarks are designed to compare SVE with vWERP and STE; the supplied passages describe the noisy-data test as an assessment of robustness, not its result.Source: §5 · p. 12

5.1 Convergence test

  • The stress-velocity recovery operator shows first-order error decay, consistent with lowest-order Raviart–Thomas approximation and supporting the operator’s consistency.The convergence test compares the finite-element approximation against a manufactured analytical stress solution; the results are summarized in Table 1.Source: §5.1 · p. 13, Table 1 · p. 13

5.2 Velocity-measurement convergence test

  • Both velocity H1 and stress H(div) errors converge with a least-squares slope close to 0.72, indicating stress error follows measurement error up to a multiplicative constant.This behavior supports the steady-case stability estimate that bounds SVE reconstruction error by velocity-measurement error.Source: §5.2 · p. 15

5.3 Venturi channel

  • In the noise-free Venturi case, SVE and STE recover pressure fields visually indistinguishable from ground truth, and all three estimators follow the reference pressure-drop curve.The pressure-drop comparison includes vWERP, which directly estimates pressure drop, and STE and SVE, which derive it from reconstructed pressure fields.Source: §5.3 · p. 18
  • In the noise-free Venturi case, SVE predicted pressure drop mildly but persistently more accurately than STE and vWERP throughout the cycle.Source: Figure 7 · p. 18, §5.3 · p. 18

5.4 Venturi channel with noisy measurements

  • Noisy inputs were generated from a ground-truth velocity field using independent Gaussian noise at VENC = 0.55 m/s and SNR = 10.These settings were described as representative of typical 4D Flow MRI data.Source: §5.4 · pp. 18–19
  • With noisy velocity measurements, SVE showed the closest agreement with the reference pressure-drop curve among SVE, STE, and vWERP.The comparison used the ensemble-averaged curve from 30 independent noisy simulations.Source: Figure 8 · p. 19, §5.4 · p. 19

6 Conclusions and perspectives

The SVE reconstructs the full stress tensor from velocity measurements and recovers pressure from its trace. Its theory and benchmarks support accurate pressure recovery, including under noisy measurements.

  • SVE reconstructs stress as a single-field H(div; Ω) problem, avoids auxiliary fields and saddle-point structure, and obtains pressure from the stress trace.Source: §6 · p. 20
  • The recovery operator is well posed, and its FE discretization shows first-order error decay in both L2(Ω) and H(div; Ω) norms.Source: §6 · p. 20
  • The measurement-driven reconstruction works with partial velocity information, and its stress error converges at the velocity-measurement error rate, as bounded by the stability estimate.Source: §6 · p. 20
  • For clean Navier–Stokes Venturi data, SVE recovered pressure nearly indistinguishable from ground truth and gave the closest pressure-drop agreement.Source: §6 · p. 20
  • Under noisy velocity measurements, SVE's pressure-drop curve stayed closer to ground truth than STE's and vWERP's.Source: §6 · p. 20
  • Future work is directed toward non-Newtonian rheologies and complex three-dimensional geometries.Source: §6 · p. 20
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