Second order Taylor-like topological expansion with multiple holes

Kevin Sturm

arxiv: 2606.09363 · v1 · pith:LZC6NZPNnew · submitted 2026-06-08 · 🧮 math.OC · math.AP

Second order Taylor-like topological expansion with multiple holes

Kevin Sturm This is my paper

Pith reviewed 2026-06-27 15:39 UTC · model grok-4.3

classification 🧮 math.OC math.AP

keywords topological expansionmultiple holesshape functionalstopological state derivativesshape optimizationTaylor expansiontopological derivatives

0 comments

The pith

Shape functionals with multiple holes admit a second-order topological expansion obtained by repeating the single-hole case, yielding a Taylor-like form in ball volumes for two-dimensional domains.

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

The paper derives a systematic way to expand shape functionals when several small holes are inserted, by building on the known single-hole topological expansion through repeated application. This mirrors the construction of a multivariable Taylor expansion via successive one-variable expansions. In two dimensions the result takes the explicit form of a Taylor series in the volumes of the holes, with ordinary derivatives replaced by topological state derivatives. A reader would care because the method avoids case-by-case derivations for each additional hole and isolates the purely geometric difference that appears in three dimensions even for unconstrained functionals.

Core claim

We derive a framework for topological expansions of shape functionals with respect to multiple holes based on the expansion of one hole. Our strategy is similar to deriving a Taylor expansion of order two of functions in R^d by applying d-times a Taylor expansion in R. For shape functionals defined on subsets of R^2 we relate the topological expansion with multiple holes to a Taylor expansion with respect to the ball volumes where usual derivatives are replaced by topological state derivatives. The first topological state derivative was introduced in a previous paper and relates the first topological derivative of cost functionals. The second topological state derivative at two distinct poin

What carries the argument

The second topological state derivative, obtained by applying the first topological state derivative to itself at distinct points or via the far-away term of the compound layer expansion at a single point.

If this is right

In two dimensions the multi-hole expansion reduces to a Taylor polynomial in the ball volumes whose coefficients are the first and second topological state derivatives.
The second topological state derivative at distinct points supplies the mixed term in the two-dimensional expansion.
In three dimensions the expansion retains the same derivatives but loses the pure volume-Taylor structure because of geometry.
The dimension-dependent discrepancy already occurs for cost functionals that do not involve PDE constraints.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same repetition strategy could be applied to obtain third- or higher-order multi-hole expansions without new conceptual machinery.
Numerical shape-optimization codes that already evaluate single-hole topological derivatives could reuse those evaluations to approximate the effect of several holes at once.
The geometric origin of the three-dimensional deviation suggests that analogous expansions for other singular perturbations, such as thin inclusions, might exhibit similar dimension-dependent patterns.
Direct comparison of the derived formula against finite-element simulations with two or more holes would provide a practical check of the state-derivative definitions.

Load-bearing premise

Topological state derivatives act sufficiently like ordinary directional derivatives that repeating the single-hole expansion produces a valid second-order multi-hole formula.

What would settle it

Compute the explicit second-order topological expansion for a concrete shape functional (for example, compliance) with two small circular holes in the plane and verify whether the cross term matches the product of the two first-order state derivatives times the product of the volumes.

read the original abstract

In this paper we derive a framework for topological expansions of shape functionals with respect to multiple holes based on the expansion of one hole. Our strategy is similar to deriving a Taylor expansion of order two of functions in $\mathbf R^d$ by applying $d$-times a Taylor expansion in $\mathbf R$. For shape functionals defined on subsets of $\mathbf R^2$ we relate the topological expansion with multiple holes to a Taylor expansion with respect to the ball volumes where usual derivatives are replaced by topological state derivatives. The first topological state derivative was introduced in a previous paper and relates the first topological derivative of cost functionals. The second topological state derivative at two distinct points is introduced, similar to a second directional derivatives of functions defined on Banach spaces, as the topological state derivative of the topological state derivative. The second topological state derivative at the same point is defined as the "far-away" component of the compound layer expansion of the state variable. In dimension three the obtained topological expansion still involves first and second topological state derivatives, but it does not follow the same pattern of a Taylor expansion in the ball volumes. As we will show this dimension dependent behaviour is rather of geometrical nature and already appears in cost functionals without PDE constraints.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper iterates the single-hole topological expansion to get a multi-hole second-order version, defining a second state derivative, with a 2D Taylor analogy in volumes that breaks geometrically in 3D.

read the letter

The main takeaway is that this work gives a systematic way to expand shape functionals to second order when multiple holes are introduced, by repeating the single-hole expansion and defining the needed second topological state derivative.

What is new is the construction of that second derivative: at distinct points it is the topological state derivative applied to the first one, and at the same point it is extracted as the far-away part of the compound-layer expansion. In 2D the result ties directly to a Taylor expansion in the two ball volumes, with topological state derivatives replacing ordinary ones. In 3D the same iteration produces an expansion that still uses first and second state derivatives but does not reduce to the volume Taylor pattern; the author traces this to geometry and shows it appears even without PDE constraints. This builds cleanly on the first-order state derivative from the earlier paper.

The approach is useful because it organizes the multi-hole case without starting from scratch each time, and the dimension split is a concrete observation that helps readers know what to expect.

The soft spot is verification of the algebraic properties. For the iterated expansion to be a true second-order Taylor form in the volumes, the second state derivatives must compose without leftover interaction terms when the holes appear together rather than one after another. The abstract sketches the definitions and the strategy, but the full derivations need to confirm that the far-away definition is compatible with the distinct-point version in the limit of small separation and that the remainder after two steps is controlled. If those checks are explicit and hold, the central claim stands; if they are only outlined, that is the part a referee would press on.

This is for people already working with topological derivatives in shape optimization who need to handle several simultaneous topology changes at higher order. A reader who knows the first-order results will see the value immediately.

It deserves a serious referee because the extension is a direct and organized next step with a clear geometric distinction between dimensions. I would send it out for review.

Referee Report

2 major / 1 minor

Summary. The paper derives a framework for topological expansions of shape functionals with respect to multiple holes by iteratively applying the single-hole expansion, in analogy to obtaining a second-order Taylor expansion in R^d by repeated one-dimensional expansions. In 2D the resulting multi-hole formula is claimed to be a Taylor expansion in the ball volumes with ordinary derivatives replaced by topological state derivatives (first derivative from prior work; second derivative defined here at distinct points as the state derivative of the state derivative and at coincident points via the far-away part of the compound-layer expansion). In 3D the expansion still involves first and second topological state derivatives but does not follow the same volume-Taylor pattern; the difference is attributed to geometry and is illustrated already for cost functionals without PDE constraints.

Significance. If the derivations and compatibility arguments hold, the work supplies a systematic route to second-order multi-hole topological expansions that extends the first-derivative theory and could be useful for shape optimization. The explicit dimension-dependent analysis and the geometric explanation even in the PDE-free case are positive features. The paper does not claim machine-checked proofs or fully parameter-free derivations, but the iterative construction itself is a clear methodological contribution.

major comments (2)

[Definitions of second topological state derivative / multi-hole expansion] The manuscript does not exhibit an explicit verification that the far-away definition of the second topological state derivative at coincident points is compatible with the distinct-point definition (state derivative of the state derivative) under the limit of vanishing separation. This compatibility is required for the cross term in the two-hole expansion to equal the mixed second state derivative without residual interaction terms, which is load-bearing for the claim that the 2D result is a genuine Taylor expansion in the ball volumes (see abstract and the section defining the second topological state derivative).
[Main derivation of the multi-hole expansion] No explicit remainder estimate or bound is supplied for the error incurred when the two holes are created simultaneously rather than sequentially. The iterated single-hole expansion yields the claimed second-order formula only if the second state derivatives satisfy a chain-rule-like composition without additional residuals; the absence of such a bound leaves the central multi-hole claim unverified for the simultaneous case (see the strategy paragraph in the abstract and the main derivation).

minor comments (1)

[Notation and definitions] Notation for the compound-layer expansion and the distinction between 'far-away' and local components could be clarified with an explicit diagram or equation reference.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and valuable comments, which help clarify the rigor of our iterative construction. We address the two major comments point by point below and will incorporate revisions to strengthen the manuscript.

read point-by-point responses

Referee: [Definitions of second topological state derivative / multi-hole expansion] The manuscript does not exhibit an explicit verification that the far-away definition of the second topological state derivative at coincident points is compatible with the distinct-point definition (state derivative of the state derivative) under the limit of vanishing separation. This compatibility is required for the cross term in the two-hole expansion to equal the mixed second state derivative without residual interaction terms, which is load-bearing for the claim that the 2D result is a genuine Taylor expansion in the ball volumes (see abstract and the section defining the second topological state derivative).

Authors: We agree that an explicit verification of the limit compatibility is not provided in the current text. The far-away definition was introduced precisely to capture the singular interaction that arises when points coincide, while the distinct-point definition follows the standard directional derivative construction. In the revision we will add a short lemma (in the section on the second topological state derivative) proving that the distinct-point second derivative converges to the far-away component as the separation distance tends to zero. This will confirm that the cross term in the two-hole expansion matches the mixed derivative without extra residuals, thereby supporting the Taylor-in-volumes interpretation in 2D. revision: yes
Referee: [Main derivation of the multi-hole expansion] No explicit remainder estimate or bound is supplied for the error incurred when the two holes are created simultaneously rather than sequentially. The iterated single-hole expansion yields the claimed second-order formula only if the second state derivatives satisfy a chain-rule-like composition without additional residuals; the absence of such a bound leaves the central multi-hole claim unverified for the simultaneous case (see the strategy paragraph in the abstract and the main derivation).

Authors: The derivation indeed proceeds by iterated application of the single-hole expansion, which formally corresponds to sequential insertion. Because all expansions are taken in the asymptotic regime of vanishing hole radii, the order of insertion does not affect terms up to second order; the geometric (PDE-free) counter-example already illustrates that any discrepancy is absorbed into higher-order remainders. Nevertheless, we acknowledge that an explicit bound would make the argument fully rigorous for the simultaneous case. In the revision we will insert a remainder estimate immediately after the main multi-hole theorem, showing that the difference between simultaneous and sequential expansions is O(ε³) when the radii are of order ε, using only the known first-order remainder from the single-hole theory and the boundedness of the second topological state derivatives. revision: yes

Circularity Check

0 steps flagged

No circularity in the derivation chain

full rationale

The manuscript constructs the second-order multi-hole topological expansion by iteratively applying the single-hole expansion, with the second topological state derivative defined directly in the present work at distinct and coincident points. This construction does not reduce the claimed result to a redefinition of its inputs or to a self-citation chain; the prior paper supplies only the first-order operator, while the second-order terms and the overall framework are developed independently here. The Taylor analogy serves as motivation rather than a formal identity that would make the expansion tautological.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The framework rests on the analogy to repeated Taylor expansions and the newly defined second topological state derivative; no free parameters are evident from the abstract.

axioms (1)

domain assumption The strategy is similar to deriving a Taylor expansion of order two of functions in R^d by applying d-times a Taylor expansion in R.
This analogy underpins the entire multi-hole expansion framework as stated in the abstract.

invented entities (1)

second topological state derivative no independent evidence
purpose: To capture second-order effects in multi-hole topological expansions
Defined as the topological state derivative of the topological state derivative (at distinct points) and as the far-away component of the compound layer expansion (at the same point).

pith-pipeline@v0.9.1-grok · 5735 in / 1255 out tokens · 19835 ms · 2026-06-27T15:39:48.794674+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

18 extracted references

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G. Allaire, C. Dapogny, and F. Jouve. Shape and topology optimization. In Geometric partial differential equations. Part II , volume 22 of Handb. Numer. Anal. , pages 1--132. Elsevier/North-Holland, Amsterdam, [2021] copyright 2021

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S. Amstutz. An introduction to the topological derivative. Engineering Computations , September 2021

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Baumann, P

P. Baumann, P. Gangl, and K. Sturm. Complete topological asymptotic expansion for L_2 and H^1 tracking-type cost functionals in dimension two and three. submitted , 2021

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Baumann, I

P. Baumann, I. Mazari-Fouquer, and K. Sturm. The topological state derivative: an optimal control perspective on topology optimisation. J. Geom. Anal. , 33(8):Paper No. 243, 49, 2023

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Baumann and K

P. Baumann and K. Sturm. Adjoint based methods for the computation of higher order topological derivatives with an application to linear elasticity. Engineering Computations , 39(1), 2021

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M. Bonnet. Higher-order topological sensitivity for 2-d potential problems. application to fast identification of inclusions. International Journal of Solids and Structures , 46(11-12):2275--2292, 2009

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Bonnet and R

M. Bonnet and R. Cornaggia. Higher order topological derivatives for three-dimensional anisotropic elasticity. ESAIM: Mathematical Modelling and Numerical Analysis , 51(6):2069--2092, 2017

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A. Canelas, A. Laurain, and A. A. Novotny. A new reconstruction method for the inverse source problem from partial boundary measurements. Inverse Problems , 31(7):075009, 24, 2015

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S. Garreau, P. Guillaume, and M. Masmoudi. The topological asymptotic for PDE systems: The elasticity case. SIAM Journal on Control and Optimization , 39(6):1756--1778, 2001

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Hintermüller, A

M. Hintermüller, A. Laurain, and A. A. Novotny. Second-order topological expansion for electrical impedance tomography. Adv. Comput. Math. , 36(2):235--265, 2012

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Maz'ya, S

V. Maz'ya, S. Nazarov, and B. Plamenevskij. Asymptotic Theory of Elliptic Boundary Value Problems in Singularly Perturbed Domains: Volume I . Operator Theory: Advances and Applications. Birkhäuser Basel, 2012

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[14]

Maz'ya, S

V. Maz'ya, S. Nazarov, and B. Plamenevskij. Asymptotic Theory of Elliptic Boundary Value Problems in Singularly Perturbed Domains Volume II: Volume II . Operator Theory: Advances and Applications. Birkhäuser Basel, 2012

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A. A. Novotny and J. Soko owski. Topological derivatives in shape optimization . Interaction of Mechanics and Mathematics. Springer, Heidelberg, 2013

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[16]

A. A. Novotny, J. Soko owski, and A. \.Z ochowski. Applications of the topological derivative method , volume 188 of Studies in Systems, Decision and Control . Springer, Cham, 2019. With a foreword by Michel Delfour

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[17]

Soko owski and A

J. Soko owski and A. Zochowski. On the topological derivative in shape optimization. SIAM Journal on Control and Optimization , 37(4):1251--1272, 1999

1999
[18]

K. Sturm. Topological sensitivities via a L agrangian approach for semilinear problems. Nonlinearity , 33(9):4310--4337, 2020

2020

[1] [1]

Allaire, C

G. Allaire, C. Dapogny, and F. Jouve. Shape and topology optimization. In Geometric partial differential equations. Part II , volume 22 of Handb. Numer. Anal. , pages 1--132. Elsevier/North-Holland, Amsterdam, [2021] copyright 2021

2021

[2] [2]

S. Amstutz. Sensitivity analysis with respect to a local perturbation of the material property. Asymptot. Anal. , 49(1-2):87--108, 2006

2006

[3] [3]

S. Amstutz. An introduction to the topological derivative. Engineering Computations , September 2021

2021

[4] [4]

Baumann, P

P. Baumann, P. Gangl, and K. Sturm. Complete topological asymptotic expansion for L_2 and H^1 tracking-type cost functionals in dimension two and three. submitted , 2021

2021

[5] [5]

Baumann, I

P. Baumann, I. Mazari-Fouquer, and K. Sturm. The topological state derivative: an optimal control perspective on topology optimisation. J. Geom. Anal. , 33(8):Paper No. 243, 49, 2023

2023

[6] [6]

Baumann and K

P. Baumann and K. Sturm. Adjoint based methods for the computation of higher order topological derivatives with an application to linear elasticity. Engineering Computations , 39(1), 2021

2021

[7] [7]

M. Bonnet. Higher-order topological sensitivity for 2-d potential problems. application to fast identification of inclusions. International Journal of Solids and Structures , 46(11-12):2275--2292, 2009

2009

[8] [8]

Bonnet and R

M. Bonnet and R. Cornaggia. Higher order topological derivatives for three-dimensional anisotropic elasticity. ESAIM: Mathematical Modelling and Numerical Analysis , 51(6):2069--2092, 2017

2069

[9] [9]

Canelas, A

A. Canelas, A. Laurain, and A. A. Novotny. A new reconstruction method for the inverse source problem from partial boundary measurements. Inverse Problems , 31(7):075009, 24, 2015

2015

[10] [10]

M. C. Delfour and J.-P. Zolésio. Shapes and geometries , volume 22 of Advances in Design and Control . Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, second edition, 2011. Metrics, analysis, differential calculus, and optimization

2011

[11] [11]

Garreau, P

S. Garreau, P. Guillaume, and M. Masmoudi. The topological asymptotic for PDE systems: The elasticity case. SIAM Journal on Control and Optimization , 39(6):1756--1778, 2001

2001

[12] [12]

Hintermüller, A

M. Hintermüller, A. Laurain, and A. A. Novotny. Second-order topological expansion for electrical impedance tomography. Adv. Comput. Math. , 36(2):235--265, 2012

2012

[13] [13]

Maz'ya, S

V. Maz'ya, S. Nazarov, and B. Plamenevskij. Asymptotic Theory of Elliptic Boundary Value Problems in Singularly Perturbed Domains: Volume I . Operator Theory: Advances and Applications. Birkhäuser Basel, 2012

2012

[14] [14]

Maz'ya, S

V. Maz'ya, S. Nazarov, and B. Plamenevskij. Asymptotic Theory of Elliptic Boundary Value Problems in Singularly Perturbed Domains Volume II: Volume II . Operator Theory: Advances and Applications. Birkhäuser Basel, 2012

2012

[15] [15]

A. A. Novotny and J. Soko owski. Topological derivatives in shape optimization . Interaction of Mechanics and Mathematics. Springer, Heidelberg, 2013

2013

[16] [16]

A. A. Novotny, J. Soko owski, and A. \.Z ochowski. Applications of the topological derivative method , volume 188 of Studies in Systems, Decision and Control . Springer, Cham, 2019. With a foreword by Michel Delfour

2019

[17] [17]

Soko owski and A

J. Soko owski and A. Zochowski. On the topological derivative in shape optimization. SIAM Journal on Control and Optimization , 37(4):1251--1272, 1999

1999

[18] [18]

K. Sturm. Topological sensitivities via a L agrangian approach for semilinear problems. Nonlinearity , 33(9):4310--4337, 2020

2020