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arxiv: 2607.06429 · v1 · pith:ZB44UIIB · submitted 2026-07-07 · math.NA · cs.NA· physics.comp-ph

Nested Volume-Surface Integral Equations for Acoustics

Danilo Aballay , Elwin van 't Wout This is my paper

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classification math.NA cs.NAphysics.comp-ph
keywords integralinterfacesvsieheterogeneousmaterialsacousticbenchmarkscomputational
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The pith

Integral equations handle nested acoustic materials with sharp density jumps

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

This paper derives a volume-surface integral equation (VSIE) for acoustic wave propagation through two nested domains with heterogeneous materials and discontinuous density at their interfaces. The central object is a single integral equation (Theorem 7, Eq. 37) that combines a single-layer volume integral over the interior domains, an adjoint double-layer volume integral involving gradients of the material contrast, and double-layer boundary integrals at both material interfaces. A mass function m(x) encodes the local density contrast, averaging the ratio of exterior to interior density at each point, including at interfaces where the density jumps. The key algebraic achievement is shifting all derivatives off the unknown pressure field and onto known material parameters and the Green's function, which permits discretization with piecewise-constant basis functions in L². The paper then benchmarks this formulation against analytical solutions, boundary element methods, and coupled finite-element–boundary-element methods across four test cases of increasing complexity, reporting mesh convergence and relative errors below 5% for most scenarios.

Core claim

The paper proves that the heterogeneous Helmholtz transmission problem for two nested domains with discontinuous density can be solved by a single VSIE (Eq. 37) whose integrand contains no derivatives of the pressure. This is achieved by defining a mass function m(x) that averages the density contrast at interfaces, combined with integration-by-parts and the exterior Helmholtz equation to eliminate second-order derivatives. The resulting formulation supports L² discretization with piecewise-constant basis functions, decoupling the four geometric components (two volumes, two surfaces) into independent L² spaces where interface continuity is recovered implicitly through the integral operators.

What carries the argument

The mass function m(x) (Eq. 36), which encodes the local density contrast ρ₀/ρ(x) in the interior and its average at interfaces; the perturbed Helmholtz equation (Eq. 13) that localizes heterogeneity to the right-hand side; integration by parts (Lemma 3) that generates surface integrals at material interfaces; the exterior Green's function identity (Eq. 27) that eliminates the Laplacian; and the L²-decoupled discretization (Eq. 42) that treats volume and surface pressures as four independent functions.

If this is right

  • The VSIE formulation can be extended to three or more nested domains by following the same design principles, broadening the class of geometries addressable without finite-element meshes in the unbounded exterior.
  • Because the formulation supports fast matrix arithmetic via hierarchical compression and the discrete system is dense but structured, large-scale biomedical acoustic simulations (e.g., transcranial ultrasound with CT-derived skull properties) could become computationally feasible.
  • The L²-decoupled discretization strategy—where physical interface continuity is enforced weakly through integral operators rather than through the function space—may be applicable to other transmission problems where enforcing H¹ conformity is computationally expensive.
  • The convergence slowdown observed at high-contrast interfaces (Benchmark 4) suggests that adaptive mesh refinement or higher-order quadrature near density jumps would be natural next steps for improving accuracy.

Where Pith is reading between the lines

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

  • If the L²-decoupled approach proves insufficient at very high density contrasts (e.g., air-water interfaces with ratios near 1000:1), a hybrid discretization that enforces H¹ conformity only at the highest-contrast interface while keeping L² freedom elsewhere could recover accuracy without sacrificing the simplicity of piecewise-constant basis functions.
  • The mass function's averaging structure at interfaces—taking the mean of density ratios from both sides—resembles a Robin-type condition and may connect to established transmission conditions in layered media theory, though the paper does not draw this parallel explicitly.
  • The localized numerical artifacts near density-jump interfaces visible in Figures 5 and 9, combined with the convergence stall in Benchmark 4, are consistent with the hypothesis that the weak enforcement of pressure continuity becomes less effective as the contrast increases and the mesh remains coarse relative to the interface physics.

Load-bearing premise

The L²-decoupled discretization does not explicitly enforce pressure continuity across material interfaces in the discrete function space; instead, it relies on the integral operators to recover continuity implicitly. If this implicit enforcement is insufficient at high contrasts or on coarse meshes, the surface integral terms that carry the density-jump physics may produce localized numerical artifacts and convergence may stall.

What would settle it

A benchmark scenario with very high density contrast (e.g., ratio exceeding 10:1) across an interface where the VSIE solution exhibits pressure discontinuities that do not diminish under mesh refinement, or where the relative error against FEM-BEM fails to converge below a practical threshold, would indicate that the L²-decoupled discretization cannot adequately capture the interface physics.

Figures

Figures reproduced from arXiv: 2607.06429 by Danilo Aballay, Elwin van 't Wout.

Figure 1
Figure 1. Figure 1: A sketch of the geometry. The domain Ω0 is unbounded, and Ω2 is inside Ω1. The normals point outward from the respective subdomains. As is common practice in computational engineering, each subdomain rep￾resents a different material. Hence, ρ(x) =    ρ0 in Ω0, ρ1(x) in Ω1, ρ2(x) in Ω2; (2) c(x) =    c0 in Ω0, c1(x) in Ω1, c2(x) in Ω2; (3) denote the mass density and speed of sound of the material… view at source ↗
Figure 2
Figure 2. Figure 2: An example nonuniform cuboid mesh for nested cubes. The figure [PITH_FULL_IMAGE:figures/full_fig_p016_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The real part of the acoustic field propagating through a sphere, at a [PITH_FULL_IMAGE:figures/full_fig_p021_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The mesh convergence study on a sphere shows the numerical error [PITH_FULL_IMAGE:figures/full_fig_p022_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The real part of the acoustic field calculated by the VSIE (left panel) [PITH_FULL_IMAGE:figures/full_fig_p023_5.png] view at source ↗
Figure 2
Figure 2. Figure 2: Hence, the voxel mesh has slightly different resolutions in the two con [PITH_FULL_IMAGE:figures/full_fig_p023_2.png] view at source ↗
Figure 6
Figure 6. Figure 6: The convergence of the relative difference between VSIE and BEM [PITH_FULL_IMAGE:figures/full_fig_p024_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: The real part of the solution for the benchmark with heterogeneous [PITH_FULL_IMAGE:figures/full_fig_p025_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: The convergence of the relative difference in the [PITH_FULL_IMAGE:figures/full_fig_p025_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: The real part of the solution for Benchmark 4 at [PITH_FULL_IMAGE:figures/full_fig_p027_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: The convergence of the relative difference with norm [PITH_FULL_IMAGE:figures/full_fig_p027_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: The real part of the solution for the pressure field in Benchmark 4. [PITH_FULL_IMAGE:figures/full_fig_p028_11.png] view at source ↗
read the original abstract

The simulation of high-frequency acoustic wave propagation in unbounded domains with local heterogeneous materials and high-contrast interfaces poses significant challenges to numerical methods. The volume-surface integral equation (VSIE) method is an attractive approach as it automatically satisfies the radiation condition at infinity via Green's functions, handles heterogeneous materials via Newton potentials, and models scattering at high-contrast interfaces via surface integral operators. However, its effectiveness in practical simulations has been limited by high computational costs, sensitivity to sharp interfaces, and insufficient computational verification. This study extends the applicability of VSIE by deriving integral formulations for nested heterogeneous materials with parameter jumps at interfaces. We also develop extensive benchmarks against coupled finite-element and boundary-element methods to verify the VSIE's accuracy and mesh convergence. The various benchmarks using open-source software demonstrate the effectiveness of VSIE for large-scale acoustic simulations.

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.

Referee Report

3 major / 6 minor

Summary. This manuscript derives a volume-surface integral equation (VSIE) for the Helmholtz equation in the presence of two nested domains with heterogeneous material parameters and high-contrast density jumps at interfaces. The derivation proceeds through a sequence of standard steps (convolution theorem, integration by parts, jump relations) to shift all derivatives off the unknown pressure field, yielding a formulation amenable to L² discretization with piecewise-constant basis functions. The formulation is verified against analytical solutions, BEM, and FEM-BEM on four benchmarks of increasing complexity. The derivation is parameter-free, and the code is openly available.

Significance. The extension of VSIE to nested domains with density discontinuities is a meaningful contribution to computational acoustics, addressing a gap between the standard Lippmann-Schwinger equation (constant density) and coupled FEM-BEM approaches. The parameter-free derivation (α and β defined from physical parameters via Eqs. 11-12) and the reproducible code (GitHub repository) are notable strengths. The benchmark suite, while modest in scale, is well-designed and provides credible verification. The L²-decoupled discretization allowing P0 basis functions is practically attractive.

major comments (3)
  1. §2.3.2, Eqs. (42)-(43): The L²-decoupled discretization treats p_Ω1, p_Ω2, p_Γ1, p_Γ2 as four independent L² functions, with no explicit enforcement of pressure continuity across interfaces. The continuous VSIE (37) is derived under the assumption that p is continuous across Γ₂ (used in Lemma 3, Eq. 23, to combine two boundary terms into a single integral weighted by (α₂−α₁)). The authors state that continuity is recovered 'via the integral operators in the VSIE,' but no convergence analysis or error estimate is provided for this claim. The convergence stall observed in Benchmark 4 (Figure 10, density jump 1000→2000 kg/m³) and the localized artifacts near Γ₂ in Figures 5 and 9 are consistent with the possibility that this implicit enforcement is insufficient at high contrasts. The authors should either (a) provide a reference or argument justifying that the discrete system enforces p_Ω1|
  2. §2.2.6, Lemma 6, Eqs. (30)-(31): The jump relations are applied to obtain the Dirichlet traces at Γ₁ and Γ₂. For the trace at Γ₂ (Eq. 31), the jump relation (33) is applied to the double-layer operator on Γ₂, yielding the coefficient ½(ρ₀/ρ₂ + ρ₀/ρ₁). However, the adjoint double-layer volume integral (the ∇ₓ·∫ G(∇_yα)p dy term) also has a singularity when x approaches Γ₂ from either side. The manuscript does not discuss whether this volume integral operator has a jump across Γ₂. If it does, the trace equation (31) may be missing a jump contribution from this term. Please clarify whether the adjoint double-layer volume operator is continuous across Γ₂, and if so, why.
  3. §3.5, Figure 10: The convergence study for Benchmark 4 shows the L² error leveling off after approximately 12 elements per wavelength, with the L∞ error remaining notably higher. The text attributes this to 'localized errors near the high-contrast interface' but does not analyze the cause. Given that this is the only benchmark exercising the full nested VSIE with both heterogeneous density and a density jump, the convergence stall is load-bearing for the paper's central claim that the formulation is correct and verified. Please provide a more specific diagnosis: is the stall due to the L²-decoupled discretization (see comment above), the midpoint quadrature for weakly-singular integrals, the staircase approximation, or another factor? A single refinement study isolating one of these factors would strengthen the claim significantly.
minor comments (6)
  1. §2.2.6, Theorem 6: The text references 'the VSIE (37)' in the statement of Lemma 6, but Theorem 7 (which states Eq. 37) has not yet been stated at that point in the manuscript. The forward reference should be clarified.
  2. §2.2.4, Eq. (28): The surface integral over Γ₂ in Eq. (28) uses (α₁−α₂) with normal n̂₁, while Eq. (24) uses (α₂−α₁) with n̂₂. The sign convention is consistent (since n̂₁ = −n̂₂ at Γ₂), but a brief note would help the reader verify.
  3. §3.1: The statement 'we used mass-matrix preconditioning for the BEM' could benefit from a citation or brief description, as the BEM formulation is referenced to [47] but the preconditioning choice is not discussed there.
  4. §3.3, Figure 5: The text mentions 'slight differences at the material interface' in the VSIE solution. It would help to quantify these differences (e.g., as a percentage of the field amplitude) to contextualize the visual comparison.
  5. §4: The phrase 'ambiguous factor of two in mesh resolution between VSIE and FEM-BEM' is unclear. Please rephrase to clarify whether this refers to the 2:1 ratio used in the benchmarks or an inherent ambiguity in the comparison methodology.
  6. References [9] and [38] are dated 2026; please verify these are not preprints with updated publication status.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for a careful and constructive report. The three major comments all identify legitimate gaps in the manuscript's justification and analysis. We address each below: (1) the L²-decoupled discretization's implicit enforcement of pressure continuity lacks a rigorous justification, and we will add a discussion and a numerical diagnostic; (2) the adjoint double-layer volume operator's continuity across Γ₂ needs explicit clarification, which we will provide; (3) the convergence stall in Benchmark 4 requires a more specific diagnosis, which we will supply via an isolated refinement study. All three points lead to manuscript revisions.

read point-by-point responses

  1. Referee: §2.3.2, Eqs. (42)-(43): The L²-decoupled discretization treats p_Ω1, p_Ω2, p_Γ1, p_Γ2 as four independent L² functions, with no explicit enforcement of pressure continuity across interfaces. The continuous VSIE (37) is derived under the assumption that p is continuous across Γ₂ (used in Lemma 3, Eq. 23, to combine two boundary terms into a single integral weighted by (α₂−α₁)). The authors state that continuity is recovered 'via the integral operators in the VSIE,' but no convergence analysis or error estimate is provided for this claim. The convergence stall observed in Benchmark 4 (Figure 10, density jump 1000→2000 kg/m³) and the localized artifacts near Γ₂ in Figures 5 and 9 are consistent with the possibility that this implicit enforcement is insufficient at high contrasts. The authors should either (a) provide a reference or argument justifying that the discrete system enforces p_Ω1|

    Authors: The referee correctly identifies a gap between the continuous formulation (which assumes pressure continuity across Γ₂) and the discrete formulation (which treats p_Ω1, p_Ω2, p_Γ1, p_Γ2 as independent L² functions with no explicit continuity constraint). We agree that the manuscript's current statement that continuity is recovered 'via the integral operators in the VSIE' is insufficiently justified. We will revise the manuscript to address this in two ways. First, we will add a discussion of the mechanism: the double-layer boundary integral operators on Γ₁ and Γ₂ couple the surface unknowns p_Γ1, p_Γ2 to the volumetric unknowns p_Ω1, p_Ω2 through the off-diagonal blocks K_{Ωi,Γj} and K_{Γi,Ωj} in the linear system (43). At the continuous level, the jump relations (Lemma 6) ensure that the interior and exterior traces of the VSIE yield the same equation on each interface, which is the mechanism by which continuity is implicitly enforced. At the discrete level, this coupling transfers information across the interface but does not exactly enforce pointwise continuity. Second, we will add a numerical diagnostic to Benchmark 4 reporting the discrete jump |p_Ω1 − p_Γ2| and |p_Ω2 − p_Γ2| at interface elements as a function of mesh refinement, to quantify how well continuity is recovered in practice. We acknowledge that we cannot currently provide a rigorous error estimate for the implicit continuity enforcement at high contrasts, and we will state this limitation transparently. revision_made = 'partial' revision: partial

  2. Referee: §2.2.6, Lemma 6, Eqs. (30)-(31): The jump relations are applied to obtain the Dirichlet traces at Γ₁ and Γ₂. For the trace at Γ₂ (Eq. 31), the jump relation (33) is applied to the double-layer operator on Γ₂, yielding the coefficient ½(ρ₀/ρ₂ + ρ₀/ρ₁). However, the adjoint double-layer volume integral (the ∇ₓ·∫ G(∇_yα)p dy term) also has a singularity when x approaches Γ₂ from either side. The manuscript does not discuss whether this volume integral operator has a jump across Γ₂. If it does, the trace equation (31) may be missing a jump contribution from this term. Please clarify whether the adjoint double-layer volume operator is continuous across Γ₂, and if so, why.

    Authors: We thank the referee for this careful observation. The adjoint double-layer volume operator T[q](x) = ∇ₓ · ∫_{Ω₁∪Ω₂} G_{k₀}(x,y) (∇_y α(y)) q(y) dy is indeed singular when x approaches Γ₂, and the manuscript should discuss this explicitly. The key point is that the singularity structure of this operator differs from that of the double-layer boundary integral operator. The volume integral involves ∇_y α(y), which is a bounded function supported in the interior of Ω₁ and Ω₂ (away from the interfaces, since α is C¹ inside each subdomain). The gradient ∇ₓ acts on the Green's function, producing a kernel with a 1/|x−y|²-type singularity integrated over a volume. When x approaches Γ₂, the integration domain excludes a neighborhood of x itself (since the integral is over the open subdomains), and the volume integral's limit exists and is the same from both sides of Γ₂. This is because the volume integral operator with a weakly singular kernel is continuous across surfaces that are not part of the integration domain's boundary in a way that would produce a jump. More precisely, the adjoint double-layer volume operator maps L²(Ω) to H^{1/2}(Ω) continuously (see, e.g., Steinbach [19, Section 3.1] and Costabel [35]), and its trace is well-defined and continuous across interior interfaces. This contrasts with the double-layer boundary integral operator, which has a jump because the integration surface coincides with the trace surface. We will add a remark to the manuscript clarifying this distinction and citing the relevant functional-analytic results. revision_made = 'yes' revision: yes

  3. Referee: §3.5, Figure 10: The convergence study for Benchmark 4 shows the L² error leveling off after approximately 12 elements per wavelength, with the L∞ error remaining notably higher. The text attributes this to 'localized errors near the high-contrast interface' but does not analyze the cause. Given that this is the only benchmark exercising the full nested VSIE with both heterogeneous density and a density jump, the convergence stall is load-bearing for the paper's central claim that the formulation is correct and verified. Please provide a more specific diagnosis: is the stall due to the L²-decoupled discretization (see comment above), the midpoint quadrature for weakly-singular integrals, the staircase approximation, or another factor? A single refinement study isolating one of these factors would strengthen the claim significantly.

    Authors: The referee is correct that the convergence stall in Benchmark 4 is insufficiently analyzed and that it is load-bearing for the paper's claims. We will conduct an isolated refinement study to diagnose the cause. Based on our computational experience and the referee's suggestion, we suspect the stall is primarily due to the midpoint quadrature for the weakly-singular self-interaction integrals on the surface elements at Γ₂, combined with the L²-decoupled discretization's imperfect enforcement of continuity at high contrast. To isolate the factor, we will run two additional studies: (1) a refinement study on Benchmark 4 where the weakly-singular integrals near Γ₂ are evaluated with a higher-order quadrature rule instead of the analytical sphere approximation, holding the mesh fixed; and (2) a comparison of the discrete pressure jump |p_Ω1 − p_Γ2| at Γ₂ across refinement levels to assess whether the implicit continuity enforcement degrades at high contrast. We will report these results in the revised manuscript and provide a specific diagnosis. We acknowledge that if the stall persists under these interventions, it may indicate a fundamental limitation of the P0 L²-decoupled discretization at high density contrasts, which we will state honestly. revision_made = 'yes' revision: yes

Circularity Check

0 steps flagged

No circularity: VSIE derived from PDE via algebraic identities and Green's function properties; no fitted parameters or self-citation chains.

full rationale

The derivation chain from the Helmholtz PDE (4)-(8) to the VSIE (37) is entirely parameter-free and self-contained. The derived material parameters α and β are defined from physical quantities (ρ, c) via Eqs. (11)-(12) with no fitting to benchmark data. The VSIE is constructed through a sequence of algebraic identities: convolution with the Green's function (Theorem 1), swapping convolution and divergence (Lemma 2), integration by parts (Lemma 3), and applying the Helmholtz Green's function identity (Lemma 4). Each step is justified by standard mathematical properties (Green's function symmetry, jump relations of double-layer operators, integration by parts) with citations to external textbooks and independent mathematical literature ([11, 19, 35, 43, 45]). Self-citations [17, 38, 47] provide benchmark implementations (FEM-BEM, BEM) used for verification in Section 3, but these are independent computational tools against which the VSIE is compared—they do not serve as load-bearing premises in the derivation itself. The benchmarks serve as external falsification, not as inputs to the formulation. No step in the derivation reduces to its own inputs by construction. The L²-decoupled discretization (Section 2.3.2) raises a correctness concern about whether interface continuity is sufficiently enforced, but this is a question of numerical analysis correctness, not circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 6 axioms · 0 invented entities

No free parameters are introduced; all quantities are derived from physical material parameters (ρ, c) and the exterior wavenumber k₀. No new physical entities, particles, or forces are postulated. The axioms are standard for acoustic transmission problems except the L²-decoupled discretization choice, which is a numerical modeling assumption specific to this paper. The derived parameters α(x) and β(x) are defined from ρ and c via Eqs. (11)-(12) and are not free.

axioms (6)
  • domain assumption The exterior domain Ω₀ is homogeneous with constant ρ₀ and c₀, enabling use of the free-space Green's function G_{k₀}.
    Stated in §2.1 and §2.2.1. This is essential because the entire VSIE is built as a perturbation of the exterior Helmholtz operator. If the exterior were heterogeneous, no closed-form Green's function would be available.
  • domain assumption The material parameters satisfy c(x) ∈ C(Ω) and ρ(x) ∈ C¹(Ω) within each subdomain, allowing discontinuities only at interfaces.
    Stated in §2.1. This regularity is needed for the integration-by-parts step (Lemma 3) and for ∇α to be well-defined in the adjoint double-layer term of Eq. (37).
  • standard math The acoustic pressure and normal particle velocity are continuous across material interfaces (transmission conditions, Eqs. 6-7).
    Standard acoustic transmission condition, invoked in §2.1 and used critically in Lemma 3 to combine the two surface integrals at Γ₂ into a single (α₂−α₁) term.
  • domain assumption The Sommerfeld radiation condition holds at infinity (Eq. 8).
    Standard for exterior acoustic problems; ensures uniqueness and justifies the representation formula.
  • domain assumption The source f(x) is integrable and compactly supported.
    Stated in §2.1; needed to apply the convolution theorem (Theorem 1) which requires v(x) to be integrable with compact support.
  • ad hoc to paper The L²-decoupled discretization (no explicit interface continuity in the discrete space) converges to the correct continuous solution.
    §2.3.2, Eq. (42). The authors state that continuity is 'leveraged via the integral operators' rather than imposed on the function space. This is a modeling choice specific to this paper's discretization approach and is not independently proven to converge.

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