Pith. sign in

REVIEW 3 major objections 6 minor 52 references

Gradient-based shape optimization for the reduction of particle erosion in bended pipes

T0 review · 3 major / 6 minor · reviewed 2026-08-14 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The paper derives a closed-form shape derivative for an Eulerian erosion model and shows that gradient-based reshaping of a pipe bend cuts predicted peak erosion by 76% at the design particle size and at least 20% across a range of sizes.

desk verdict A solid formal contribution to shape optimization for erosion, but the 76% reduction claim rests on the same model that overpredicts at the target Stokes number. read the letter →

arxiv 1908.04712 v1 pith:I2F2RKMA submitted 2019-08-13 math.OC

Raphael Hohmann , Christian Leithäuser This is my paper · ORCID
classification math.OC MSC 49Q1049K2076D55
keywords shapeoptimizationparticleerosionEulerianmodelcontinuousadjointderivativepipebendStokesnumberDeanvortices
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Erosion in bent pipes is usually studied by tracking individual particles; this paper instead treats the particle cloud as a continuous field and derives, through continuous adjoint calculus, the shape derivative of a wall-erosion cost functional. That derivative tells how predicted erosion responds to small deformations of the bend surface, which turns erosion reduction into a PDE-constrained shape optimization problem solved by gradient descent. Applied to a 90° pipe bend, the method produces a deformed geometry whose predicted peak erosion is 76% lower for the design particle size (Stk = 0.33) and whose total erosion is at least 20% lower for every tested Stokes number above 0.2. The point of the exercise is that the optimized shape is not hand-tuned; it emerges from a systematic, gradient-based procedure, and the same adjoint formulas extend to other erosion models by replacing one function.

What carries the argument

The load-bearing object is the continuous adjoint shape derivative $dJ(\Omega)[V]$ in Eq. (26): a formula for how the reduced cost changes when the bend is deformed by a vector field $V$. It is assembled from the forward solution $(u,p,v,\alpha)$ and adjoint variables $(\lambda_u,\lambda_p,\lambda_v,\lambda_\alpha)$, with volume terms for each PDE constraint and surface terms for erosion and curvature regularization; the derivative is then projected into a smooth mesh deformation via the linear-elasticity saddle-point system (Eqs. (30)–(33)) so each gradient step gives a legitimate geometry change.

What would settle it

Measure or simulate wall erosion on the initial and optimized bends using Lagrangian particle tracking or an experimental aerosol flow with the design particle size (Stk ≈ 0.33). If the optimized geometry does not show substantially lower erosion than the initial bend—or if the gap is much smaller than the model's 76%—then the reported reduction is an artifact of the Eulerian approximation rather than a physical property of the shape.

Watch

Extended reading notes

Core claim

The paper's central claim is that the Eulerian semi-derivative of the reduced erosion cost functional exists in closed form (Theorem 2, Eq. (26)) and can be evaluated from the forward fluid/particle state and a decoupled set of adjoint equations (Eqs. (22)–(25)). The derivative consists of boundary integrals over the deformable wall—capturing the erosion model, the normal convection of impact rate, and the Willmore curvature regularization—plus volume integrals encoding how the Navier-Stokes, particle-velocity, and volume-fraction equations respond to domain changes. Embedded in Algorithm 1, with the derivative projected onto mesh deformations through linear elasticity and a restricted-gradient correction, it drives a gradient descent that stops at a locally optimal shape. For the benchmark 90° bend, the optimized geometry lowers the integrated erosion by at least 20% for all tested Stokes numbers above 0.2 and by 76% for the design species, and the improvement is attributed not to fewer impacts but to more favorable impact angles and lower impact speeds.

Load-bearing premise

The entire 76% reduction is computed with a simplified particle model that, by the authors' own validation, overpredicts impact rates for the very small particles around the design condition; if that error grows after the geometry changes, the optimized bend may not be physically better.

Editorial extensions

If this is right

  • If the central claim is right, pipe bends can be erosion-optimized with a gradient method rather than by geometrical intuition or trial-and-error parameter studies.
  • The optimized bend's benefit is not limited to the design particle size: predicted erosion falls by at least 20% for every Stokes number above 0.2 in the tested range.
  • Because only the partial derivatives of the erosion function enter the adjoint equations, the same derivative machinery applies to other erosion models (e.g., Finnie or E/CRC) with minor substitutions.
  • The improvement mechanism is identifiable: high impact rates, steep angles, and high speeds are spatially separated on the optimized wall, so erosion hot spots are dispersed.
  • The optimized geometry is locally optimal within the chosen deformation class, as indicated by the decrease of the cost and projected gradient norms over iterations.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A natural testable extension is to build or simulate the optimized bend with Lagrangian particle tracking or experiments, since the 76% figure is a prediction of an Eulerian model that itself overpredicts small-particle impacts at the validation stage.
  • The adjoint-derivative structure suggests the same shape optimization could be run for other target Stokes numbers or for a multi-species cost; the reported persistence of improvement across Stokes numbers hints that a single robust shape may serve a whole particle-size distribution.
  • Because the cost functional can be modified without re-deriving the full system—only the erosion model's derivatives change—this framework could be coupled with surrogate or space-mapping strategies for turbulent erosion, which the paper names as future work.
  • The same shape calculus could be applied to multi-velocity or moment-based particle models once their adjoints are derived, potentially addressing particle trajectory crossing that the single-velocity model cannot represent.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 6 minor

Summary. The manuscript derives a continuous adjoint shape calculus for minimizing particle erosion in a 3D pipe bend using a one-way coupled, single-velocity Eulerian particle model. The main theoretical result is Theorem 2, Eq. (26), which gives the Eulerian semi-derivative of the reduced cost functional, together with the adjoint equations (22)-(25). The authors implement the derivative via a linear-elasticity Riesz projection and a gradient descent method (Algorithm 1), validate the forward model's impact rates against experimental and numerical references for Stokes numbers above 0.43, and optimize a 90-degree bend for Stk = 0.33. They report a 76% reduction in maximal erosion at the target Stokes number and at least a 20% reduction in integrated erosion for every tested Stokes number above 0.2.

Significance. If the central claims hold, this is a useful first demonstration of gradient-based shape optimization for an Eulerian erosion model on a 3D pipe bend, and Eq. (26) is a nontrivial theoretical contribution that can be adapted to other erosion models. The authors deserve credit for deriving the adjoint system explicitly, for validating the forward impact rates against independent experimental and numerical references, and for using externally calibrated erosion constants from the literature. The strength of the quantitative conclusions is, however, limited by the fact that the optimized shape is evaluated only with the same model that overpredicts in the target regime; this is a correctness-risk concern rather than an internal inconsistency.

major comments (3)
  1. [Sections 5.1 and 5.2, Figures 3, 5, and 8] The optimization target Stk = 0.33 lies in the range Stk <= 0.43 where the authors state that the Eulerian model overpredicts impact rates, and the optimized geometry is evaluated only with the same Eulerian model. Because a gradient-based method can exploit systematic model bias (for example in impact-rate magnitude, impact-angle distribution, or artificial diffusion) to reduce the predicted objective without improving the physical erosion profile, the reported 76% and greater-than-20% reductions are not established as physical improvements. The authors should validate the optimized geometry with an independent method, such as Lagrangian particle tracking or experiments, or at minimum quantify the sensitivity of the optimized shape to the model discrepancy at Stk = 0.33.
  2. [Section 2.3 and Theorem 2] The reduced cost functional J(Omega) is defined under an unproved assumption that Eq. (7) has a unique solution, and Eq. (28) additionally requires the forward state to be shape differentiable. The nonlinear Schiller-Naumann drag term and the sign-dependent boundary conditions for the volume fraction alpha make these assumptions nontrivial. Since Theorem 2 and Algorithm 1 rest on this formal calculus, the paper should either prove the needed well-posedness and differentiability in a suitable setting or explicitly mark the derivation as formal and state how this limitation affects the validity of the optimality conditions.
  3. [Sections 2.1 and 5.2] The authors acknowledge in Section 2.1 that a single particle velocity cannot represent crossing particle trajectories, a limitation known to be relevant in pipe bends. The optimized geometry is a strongly deformed bend, and no evidence is given that this limitation does not become worse on the deformed geometry. This is not a circularity, but it is a concrete correctness risk for the reported reductions; the paper should address it, for example by comparing the optimized geometry against a Lagrangian particle model or by demonstrating that trajectory crossing is negligible in the deformed bend.
minor comments (6)
  1. [Equations (8)-(10) and Section 2.1] The symbol g is used both for the gravity vector in the momentum equations and for the erosion integrand in the cost functional (10); this notational conflict should be resolved for clarity.
  2. [Adjoint equations (22)-(24)] The derivation of the adjoint equations omits a detailed discussion of the boundary terms on the sign-dependent set Gamma^-(qv); because the function space Z(Omega) depends on the forward solution, the treatment of these terms should be clarified.
  3. [Algorithm 1] The stopping criterion 'until converged' is vague; the authors should specify the tolerance or the precise test used on the relative decrease of J and the gradient norm shown in Figure 4a.
  4. [Throughout] There are several language issues, including 'bended' for 'bent' and 'the later of which' for 'the latter of which'; a careful proofreading pass is recommended.
  5. [Figure 3 and Section 5.1] The statement that the deviation at low Stokes numbers is 'within the range of deviations among the reference studies' is qualitative; reporting the numerical values of the deviations would make the validation more informative.
  6. [Section 5.1 and Figure 2] The paper models only half of the bend by symmetry; it should state explicitly whether the deformable boundary Gamma_f and the optimized geometry are constrained to respect that symmetry.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the shape derivative is derived from the state and adjoint equations, the erosion model constants come from external literature, and the forward model is benchmarked against independent experimental and numerical references.

full rationale

The paper's central derivation is self-contained. Theorem 2 computes the Eulerian semi-derivative of the reduced cost functional from the shape Lagrangian, the forward PDE system, and the adjoint equations; the derivative is not fitted to the target reduction and no fitted parameter is renamed as a prediction. The erosion constants (m, Hv, n1, n2) are taken from Oka et al. [27], and the Eulerian particle model is cited from external sources [4,5,25,41]. The only self-citation, reference [24], is not load-bearing for the main claims. The forward model's impact rates are validated against independent experimental and numerical data in Figure 3, with the authors explicitly noting that the model 'seems to slightly over-predict η for Stk≤ 0.43' and that crossing particle trajectories and rebound 'pose severe difficulties' for the single-velocity model. These are acknowledged limitations that create correctness risk for the quantitative 76% reduction at Stk=0.33, but they do not make the derivation circular: the optimized geometry is evaluated with the same Eulerian model used to generate it, which is a standard model-based optimization workflow rather than a reduction of the prediction to the input. No equation is equivalent to another by construction, no parameter is fitted to the quantity it is used to predict, and no uniqueness theorem or ansatz is imported from the authors' prior work to force the result. The reported reduction is a consequence of the gradient-descent minimization, not an independent physical confirmation, but that is a limitation of the validation strategy, not circularity in the derivation.

Assumptions & free parameters 4 free parameters · 6 assumptions · 0 invented entities

The core derivation does not postulate new physical entities. It relies on empirical erosion constants, artificial regularization parameters, and standard shape-calculus regularity assumptions. The most consequential external inputs are the Oka erosion constants and the single-velocity Eulerian particle model, both taken from prior literature.

free parameters (4)
  • Willmore regularization weight c1 = c1 = 0.01 * (integral of g on initial Gamma_w) / (integral of 0.5 h^2 on initial Gamma_f)
    Chosen by hand in Section 5.2 to keep the curvature regularization small relative to the erosion cost.
  • Artificial viscosity K = K = 10^4
    Added in Eq. (3) for regularity only; value set in Table 1 and used in all simulations.
  • Artificial Peclet number Pe = Pe = 10^8
    Added in Eq. (5) for regularization only; value set in Table 1.
  • Oka erosion constants m, Hv, n1, n2 = m = 2.36, Hv = 2, n1 = 0.78, n2 = 1.25
    Empirical material constants for stainless steel taken from Oka et al. [27]. They are not fitted in this paper but define the objective, so the reported erosion reductions inherit their calibration.
assumptions (6)
  • domain assumption The weak forward problem Eq. (7) has a unique solution on every admissible domain.
    Stated as an assumption in Section 2.3; no existence or regularity proof is provided.
  • domain assumption The particle phase is dilute and one-way coupled, so it does not affect the carrier fluid.
    Used throughout Section 2 to decouple the fluid and particle equations; invalid for high particle loading.
  • domain assumption A single locally averaged particle velocity field with artificial diffusion captures erosion-relevant impacts despite known issues with rebound and trajectory crossing.
    Acknowledged as a limitation in Section 2.1; validation in Section 5.1 shows deviations at low Stokes numbers.
  • domain assumption The flow is stationary, laminar, incompressible, and the Schiller-Naumann drag correlation holds for particle Reynolds numbers below 1000.
    Used in Eq. (3) and Table 1; restricts the method to laminar regimes.
  • domain assumption The Oka erosion model, Eqs. (8) and (9), describes the erosion rate for the wall material.
    Basis of the cost functional Eq. (10); the functional form and constants are taken from external empirical studies [27, 28].
  • standard math Boundaries and deformations are smooth enough (C^2 and C^{2,1}) for the shape calculus identities in Lemma 1 and the Willmore curvature term.
    Standard regularity assumptions in shape calculus, invoked in Section 3.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Gradient-based shape optimization for the reduction of particle erosion in bended pipes." pith.science (2026). https://pith.science/paper/I2F2RKMA

@misc{pith2026190804712,
  author       = {Pith},
  title        = {Pith review of: Gradient-based shape optimization for the reduction of particle erosion in bended pipes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/I2F2RKMA}},
  note         = {Machine review of arXiv:1908.04712}
}
read the original abstract

In this paper we consider a shape optimization problem for the minimization of the erosion, that is caused by the impact of inert particles onto the walls of a bended pipe. Using the continuous adjoint approach, we formally compute the shape derivative of the optimization problem, which is based on a one-way coupled, fully Eulerian description of a monodisperse particle jet, that is transported in a carrier fluid. We validate our approach by numerically optimizing a three-dimensional pipe segment with respect to a single particle species using a gradient descent method, and show, that the erosion rates on the optimized geometry are reduced with respect to the initial bend for a broader range of particle Stokes numbers.

Figures

Figures reproduced from arXiv: 1908.04712 by the authors.

Figure 1
Figure 1. Two-dimensional description of the bended pipe [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 3
Figure 3. Comparison of calculated impact rates with numerical and experimental [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figure 4
Figure 4. Application of Algorithm 1 to the initial geometry from Figure 2 [PITH_FULL_IMAGE:figures/full_fig_p014_4.png] view at source ↗
Figures from the paper (4 more)
Figure 5
Figure 5. Figure 5: Erosion rates on Ω1 and Ω18 In order to investigate this decrease of the objective more closely, we recall that e is defined in eq. (8) 14 [PITH_FULL_IMAGE:figures/full_fig_p014_5.png]
Figure 6
Figure 6. Figure 6: Impact rates, velocities and angles for the initial geometry [PITH_FULL_IMAGE:figures/full_fig_p015_6.png]
Figure 7
Figure 7. Figure 7: Particle impact rates, angles and velocities for the optimized geometry [PITH_FULL_IMAGE:figures/full_fig_p015_7.png]
Figure 8
Figure 8. Figure 8: Erosion and impact rates on the initial and optimized geometry for different Stokes numbers [PITH_FULL_IMAGE:figures/full_fig_p016_8.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

52 extracted references · 52 canonical work pages

  1. [1]

    An introduction to the space mapping technique

    Bakr, M.H., Bandler, J.W., Madsen, K., Søndergaard, J., 2001. An introduction to the space mapping technique. Opti- mization and Engineering 2, 369–384

    work page 2001

  2. [2]

    Space mapping: the state of the art

    Bandler, J.W., Cheng, Q.S., Dakroury, S.A., Mohamed, A.S., Bakr, M.H., Madsen, K., Sondergaard, J., 2004. Space mapping: the state of the art. IEEE Transactions on Microwave theory and techniques 52, 337–361

    work page 2004

  3. [3]

    Parametric FEM for geometric biomembranes

    Bonito, A., Nochetto, R.H., Pauletti, M.S., 2010. Parametric FEM for geometric biomembranes. Journal of Computational Physics 229, 3171–3188. 16

    work page 2010

  4. [4]

    Three-dimensional Eulerian approach to droplet impingement simu- lation using fensap-ice, part 1: model, algorithm, and validation

    Bourgault, Y., Boutanios, Z., Habashi, W.G., 2000. Three-dimensional Eulerian approach to droplet impingement simu- lation using fensap-ice, part 1: model, algorithm, and validation. Journal of Aircraft 37, 95–103

    work page 2000

  5. [5]

    A finite element method study of Eulerian droplets impingement models

    Bourgault, Y., Habashi, W.G., Dompierre, J., Baruzzi, G.S., 1999. A finite element method study of Eulerian droplets impingement models. International Journal for Numerical Methods in Fluids 29, 429–449

    work page 1999

  6. [6]

    Prediction of aerosol deposition in 90 ◦ bends using LES and an efficient Lagrangian tracking method

    Breuer, M., Baytekin, H., Matida, E., 2006. Prediction of aerosol deposition in 90 ◦ bends using LES and an efficient Lagrangian tracking method. Journal of Aerosol Science 37, 1407–1428

    work page 2006

  7. [7]

    Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations

    Brooks, A.N., Hughes, T.J., 1982. Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations. Computer Methods in Applied Mechanics and Engi- neering 32, 199–259

    work page 1982

  8. [8]

    COMSOL Multiphysics

    COMSOL AB, 2017. COMSOL Multiphysics. Stockholm, Sweden. URL: https://comsol.com

    work page 2017

Show all 52 references
  1. [9]

    Note on the motion of fluid in a curved pipe

    Dean, W.R., 1927. Note on the motion of fluid in a curved pipe. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 4, 208–223

    1927

  2. [10]

    Shapes and geometries: metrics, analysis, differential calculus, and optimization

    Delfour, M.C., Zol´ esio, J.P., 2011. Shapes and geometries: metrics, analysis, differential calculus, and optimization. volume 22 of Advances in Design and Control . second ed., SIAM, Philadelphia

    2011

  3. [11]

    A quadrature-based moment method for dilute fluid-particle flows

    Desjardins, O., Fox, R.O., Villedieu, P., 2008. A quadrature-based moment method for dilute fluid-particle flows. Journal of Computational Physics 227, 2514–2539

    2008

  4. [12]

    Feedback Petrov-Galerkin methods for convection-dominated problems

    Do Carmo, E.G.D., Gale˜ ao, A.C., 1991. Feedback Petrov-Galerkin methods for convection-dominated problems. Computer Methods in Applied Mechanics and Engineering 88, 1–16

    1991

  5. [13]

    Innovative pipe wall design to mitigate elbow erosion: A CFD analysis

    Duarte, C.A.R., de Souza, F.J., 2017. Innovative pipe wall design to mitigate elbow erosion: A CFD analysis. Wear 380, 176–190

    2017

  6. [14]

    Robust mesh deformation using the linear elasticity equations, in: Proceedings of the Fourth Inter- national Conference on Computational Fluid Dynamics 2006, Springer, Berlin

    Dwight, R.P., 2009. Robust mesh deformation using the linear elasticity equations, in: Proceedings of the Fourth Inter- national Conference on Computational Fluid Dynamics 2006, Springer, Berlin. pp. 401–406

    2009

  7. [15]

    A regularized Newton method in electrical impedance tomography using shape Hessian information

    Eppler, K., Harbrecht, H., 2005. A regularized Newton method in electrical impedance tomography using shape Hessian information. Control and Cybernetics 34, 203–225

    2005

  8. [16]

    Theory and Practice of Finite Elements

    Ern, A., Guermond, J.L., 2004. Theory and Practice of Finite Elements. volume 159 of Applied Mathematical Sciences . Springer, New York

    2004

  9. [17]

    First and second order shape optimization based on restricted mesh deformations

    Etling, T., Herzog, R., Loayza, E., Wachsmuth, G., 2018. First and second order shape optimization based on restricted mesh deformations. arXiv:1810.10313

    2018 arXiv

  10. [18]

    Antierosion in a 90 bend by particle impaction

    Fan, J., Yao, J., Cen, K., 2002. Antierosion in a 90 bend by particle impaction. AIChE Journal 48, 1401–1412

    2002

  11. [19]

    Some observations on the erosion of ductile metals

    Finnie, I., 1972. Some observations on the erosion of ductile metals. Wear 19, 81–90

    1972

  12. [20]

    Higher-order moment models for laminar multiphase flows with accurate particle- stream crossing

    Forgues, F., McDonald, J.G., 2019. Higher-order moment models for laminar multiphase flows with accurate particle- stream crossing. International Journal of Multiphase Flow 114, 28–38

    2019

  13. [21]

    Higher-order quadrature-based moment methods for kinetic equations

    Fox, R.O., 2009. Higher-order quadrature-based moment methods for kinetic equations. Journal of Computational Physics 228, 7771–7791

    2009

  14. [22]

    Shape optimization of an electric motor subject to nonlinear magnetostatics

    Gangl, P., Langer, U., Laurain, A., Meftahi, H., Sturm, K., 2015. Shape optimization of an electric motor subject to nonlinear magnetostatics. SIAM Journal on Scientific Computing 37, B1002–B1025

    2015

  15. [23]

    Optimization with PDE constraints

    Hinze, M., Pinnau, R., Ulbrich, M., Ulbrich, S., 2008. Optimization with PDE constraints. volume 23. Springer Science & Business Media

    2008

  16. [24]

    Shape optimization of a polymer distributor using an eulerian residence time model

    Hohmann, R., Leith¨ auser, C., 2019. Shape optimization of a polymer distributor using an eulerian residence time model. SIAM Journal on Scientific Computing 41, B625–B648

    2019

  17. [25]

    Eulerian modeling of in-flight icing due to supercooled large droplets

    Honsek, R., Habashi, W.G., Aub´ e, M.S., 2008. Eulerian modeling of in-flight icing due to supercooled large droplets. Journal of aircraft 45, 1290–1296

    2008

  18. [26]

    Pneumatic conveying design guide

    Mills, D., 2004. Pneumatic conveying design guide. second ed., Elsevier Butterworth-Heinemann

    2004

  19. [27]

    The impact angle dependence of erosion damage caused by solid particle impact

    Oka, Y., Ohnogi, H., Hosokawa, T., Matsumura, M., 1997. The impact angle dependence of erosion damage caused by solid particle impact. Wear 203, 573–579

    1997

  20. [28]

    Practical estimation of erosion damage caused by solid particle impact: Part 1: Effects of impact parameters on a predictive equation

    Oka, Y.I., Okamura, K., Yoshida, T., 2005. Practical estimation of erosion damage caused by solid particle impact: Part 1: Effects of impact parameters on a predictive equation. Wear 259, 95–101

    2005

  21. [29]

    Overview of the incompressible Navier–Stokes simulation capabilities in the MOOSE framework

    Peterson, J.W., Lindsay, A.D., Kong, F., 2018. Overview of the incompressible Navier–Stokes simulation capabilities in the MOOSE framework. Advances in Engineering Software 119, 68–92

    2018

  22. [30]

    Inertial particle deposition in a 90 ◦ laminar flow bend: an Eulerian fluid particle approach

    Pilou, M., Tsangaris, S., Neofytou, P., Housiadas, C., Drossinos, Y., 2011. Inertial particle deposition in a 90 ◦ laminar flow bend: an Eulerian fluid particle approach. Aerosol Science and Technology 45, 1376–1387

    2011

  23. [31]

    Experimental study of particle deposition in bends of circular cross section

    Pui, D.Y., Romay-Novas, F., Liu, B.Y., 1987. Experimental study of particle deposition in bends of circular cross section. Aerosol Science and Technology 7, 301–315

    1987

  24. [32]

    Numerical solution scheme for inert, disperse, and dilute gas-particle flows

    Sachdev, J., Groth, C., Gottlieb, J., 2007. Numerical solution scheme for inert, disperse, and dilute gas-particle flows. International Journal of Multiphase Flow 33, 282–299

    2007

  25. [33]

    Reducing bend erosion with a twisted tape insert

    dos Santos, V.F., de Souza, F.J., Duarte, C.A.R., 2016. Reducing bend erosion with a twisted tape insert. Powder Technology 301, 889–910

    2016

  26. [34]

    Two-phase flows - second-order schemes and boundary conditions

    Saurel, R., Daniel, E., Loraud, J.C., 1994. Two-phase flows - second-order schemes and boundary conditions. AIAA Journal 32, 1214–1221

    1994

  27. [35]

    A drag coefficient correlation

    Schiller, L., N.A., 1935. A drag coefficient correlation. Zeitschrift des Vereins Deutscher Ingenieure 77, 318–320

    1935

  28. [36]

    Impulse response approximations of discrete shape Hessians with application in CFD

    Schmidt, S., Schulz, V., 2009. Impulse response approximations of discrete shape Hessians with application in CFD. SIAM Journal on Control and Optimization 48, 2562–2580

    2009

  29. [37]

    Shape derivatives for general objective functions and the incompressible Navier-Stokes equations

    Schmidt, S., Schulz, V., 2010. Shape derivatives for general objective functions and the incompressible Navier-Stokes equations. Control and Cybernetics 39, 677–713

    2010

  30. [38]

    Computational comparison of surface metrics for PDE constrained shape optimization

    Schulz, V., Siebenborn, M., 2016. Computational comparison of surface metrics for PDE constrained shape optimization. 17 Computational Methods in Applied Mathematics 16, 485–496

    2016

  31. [39]

    Structured inverse modeling in parabolic diffusion problems

    Schulz, V.H., Siebenborn, M., Welker, K., 2015. Structured inverse modeling in parabolic diffusion problems. SIAM Journal on Control and Optimization 53, 3319–3338

    2015

  32. [40]

    Differentiation with respect to the domain in boundary value problems

    Simon, J., 1980. Differentiation with respect to the domain in boundary value problems. Numerical Functional Analysis and Optimization 2, 649–687

    1980

  33. [41]

    The calculation of inertial particle transport in dilute gas-particle flows

    Slater, S.A., Young, J.B., 2001. The calculation of inertial particle transport in dilute gas-particle flows. International Journal of Multiphase Flow 27, 61–87

    2001

  34. [42]

    Introduction to shape optimization: shape sensitivity analysis

    Soko lowski, J., Zol´ esio, J.P., 1992. Introduction to shape optimization: shape sensitivity analysis. volume 16 of Springer Series in Computational Mathematics . Springer, Berlin, Heidelberg

    1992

  35. [43]

    Minimax Lagrangian approach to the differentiability of nonlinear PDE constrained shape functions without saddle point assumption

    Sturm, K., 2015. Minimax Lagrangian approach to the differentiability of nonlinear PDE constrained shape functions without saddle point assumption. SIAM Journal on Control and Optimization 53, 2017–2039

    2015

  36. [44]

    Optimale Steuerung partieller Differentialgleichungen

    Tr¨ oltzsch, F., 2005. Optimale Steuerung partieller Differentialgleichungen. Springer, Wiesbaden

    2005

  37. [45]

    Numerical study of particle deposition in bends of a circular cross-section-laminar flow regime

    Tsai, C.J., Pui, D.Y., 1990. Numerical study of particle deposition in bends of a circular cross-section-laminar flow regime. Aerosol Science and Technology 12, 813–831

    1990

  38. [46]

    Analysis of particle transport and deposition of micron-sized particles in a 90 ◦ bend using a two-fluid Eulerian–Eulerian approach

    Vasquez, E.S., Walters, K.B., Walters, D.K., 2015. Analysis of particle transport and deposition of micron-sized particles in a 90 ◦ bend using a two-fluid Eulerian–Eulerian approach. Aerosol Science and Technology 49, 692–704

    2015

  39. [47]

    Riemannian geometry

    Willmore, T.J., 1996. Riemannian geometry. Oxford University Press

    1996

  40. [48]

    Numerical optimization

    Wright, S., Nocedal, J., 1999. Numerical optimization. Springer, New York

    1999

  41. [49]

    The Schur complement and its applications

    Zhang, F., 2006. The Schur complement and its applications. volume 4. Springer Science & Business Media

    2006

  42. [50]

    Comparison of computed and measured particle velocities and erosion in water and air flows

    Zhang, Y., Reuterfors, E., McLaury, B.S., Shirazi, S., Rybicki, E., 2007. Comparison of computed and measured particle velocities and erosion in water and air flows. Wear 263, 330–338

    2007

  43. [51]

    Numerical analysis of mitigating elbow erosion with a rib

    Zhu, H., Li, S., 2018. Numerical analysis of mitigating elbow erosion with a rib. Powder Technology 330, 445–460

    2018

  44. [52]

    Weakly differentiable functions: Sobolev spaces and functions of bounded variation

    Ziemer, W.P., 2012. Weakly differentiable functions: Sobolev spaces and functions of bounded variation. volume 120. Springer Science & Business Media. 18

    2012

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.