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arxiv: 2604.09113 · v1 · submitted 2026-04-10 · 🧮 math.NA · cs.NA

A ROM-based BDDC solver for unfitted p-FEM level-set-based lattice structures

Gonzalo Bonilla Moreno , Giuliano Guarino , Pablo Antolin This is my paper

Pith reviewed 2026-05-10 17:06 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords BDDCreduced order modelunfitted finite elementslevel setlattice structuresdomain decompositionp-FEMMDEIM
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The pith

A BDDC solver accelerated by reduced-order models computes large level-set lattice structures with bounded iterations on standard hardware.

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

The paper develops a domain-decomposition approach for simulating lattice structures whose cells are defined by level-set functions and arbitrary mappings. Each cell is treated as a single high-order unfitted finite element, and the Balanced Domain Decomposition by Constraints method assigns one subdomain per cell. Assembly of the cell stiffness matrices is accelerated by a reduced-order model based on matrix discrete empirical interpolation that is trained once offline and reused across different geometries. A stabilization term is added to preserve the solver's theoretical scalability properties despite the approximation. Tests on a graded 2D lattice exceeding 17,000 cells confirm that iteration counts remain bounded when the subdomain-to-mesh-size ratio is held fixed and that the full solve completes in roughly 30 seconds on a laptop.

Core claim

The central claim is that the combination of unfitted p-FEM discretization, BDDC domain decomposition with one subdomain per cell, and an MDEIM-based reduced-order model for stiffness-matrix assembly, together with a stabilization term, produces an accurate and scalable solver for arbitrarily graded and topologically varying lattice structures described by level sets, without invoking homogenization or periodicity assumptions.

What carries the argument

The MDEIM reduced-order surrogate for cell stiffness matrices, trained offline on geometric-mapping contributions and combined with a stabilization term inside the BDDC preconditioner for unfitted elements.

If this is right

  • Large graded lattices with thousands of distinct cell geometries can be solved directly at full resolution rather than through homogenization.
  • Solver iterations stay bounded when the ratio of subdomain size to mesh size is fixed, matching classical BDDC theory.
  • Expensive quadrature on cut elements is confined to an offline training stage and does not recur during repeated solves or design iterations.
  • The method accommodates arbitrary mappings and topology changes between cells without requiring scale separation.

Where Pith is reading between the lines

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

  • The same offline-training strategy could be applied to three-dimensional lattices once the ROM basis is extended to higher-dimensional mappings.
  • Because the reduced-order model is independent of the specific physical problem, the framework might be reused across different constitutive laws inside the same lattice geometry.
  • The controllable error introduced by stabilization opens a route to trading a modest accuracy loss for further speed gains in early-stage design exploration.

Load-bearing premise

The stabilization term added to the reduced-order approximation keeps the introduced error small enough that it does not degrade solution quality or destroy BDDC scalability in lattices with varying cell topologies.

What would settle it

If the number of BDDC iterations grows without bound as the number of subdomains increases while the ratio of subdomain diameter to local mesh size is kept constant, the claimed scalability would be refuted.

Figures

Figures reproduced from arXiv: 2604.09113 by Giuliano Guarino, Gonzalo Bonilla Moreno, Pablo Antolin.

Figure 1
Figure 1. Figure 1: Construction of a single cell of the lattice structure. The untrimmed parametric domain [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: 2D TPMS corresponding to the level-set functions in Table [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: A lattice structure is constructed by juxtaposing six cells in a compatible way. The compatibility [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Position of the Gauss-Lobatto-Legendre nodes of the Lagrangian basis superimposed to the trimmed [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Classification of the DoFs associated with a single cell (a) and with a two-by-two cells structure (b). [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Classification of the coarse DoFs associated with a single cell (a) and with a two-by-two cells [PITH_FULL_IMAGE:figures/full_fig_p012_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Comparison between unfitted p-FEM element method and CutFEM for the manufactured problem [PITH_FULL_IMAGE:figures/full_fig_p019_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Benchmark geometry for the tests in Sections [PITH_FULL_IMAGE:figures/full_fig_p020_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: (a) Accuracy of the fast assembly interpolation measured in the [PITH_FULL_IMAGE:figures/full_fig_p021_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Error of the fast assembly tensor depending on the basis size [PITH_FULL_IMAGE:figures/full_fig_p022_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Effect of the stabilization in the solver iterations and the solution error. (a) Iterations with and [PITH_FULL_IMAGE:figures/full_fig_p023_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Performances of three different solvers are compared: BDDC, Cholesky decomposition, and SOR. [PITH_FULL_IMAGE:figures/full_fig_p024_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Iterations (a) and computational time (b) associated with the solution of the problem using a [PITH_FULL_IMAGE:figures/full_fig_p025_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Iterations (a) and computational time (b) as a function of number of cells. In (b), the computa [PITH_FULL_IMAGE:figures/full_fig_p026_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: (a) Geometry of the sandwich wing example in Section [PITH_FULL_IMAGE:figures/full_fig_p026_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: (a) Geometry of lattice structure representing a wrench as described in Section [PITH_FULL_IMAGE:figures/full_fig_p027_16.png] view at source ↗
read the original abstract

We present a domain decomposition method for the fast simulation of large lattice structures described by level set functions. The method does not rely on homogenization or multiscale techniques, and therefore avoids their underlying assumptions such as scale separation and periodicity. Individual cells are defined through level set functions and mapped into physical space using arbitrary order mappings, allowing the creation of complex graded designs with varying geometries and topologies. The discretization is based on unfitted p-FEM, where each cell is approximated by a single high order element. This choice naturally handles the implicit geometric description and provides high accuracy with a moderate number of degrees of freedom. The solver is built on the Balanced Domain Decomposition by Constraints (BDDC) method, where each cell corresponds to one subdomain. To accelerate the assembly of the cell stiffness matrices, we combine a fast assembly technique that separates the contributions of the geometric mapping from the trimmed domain with a reduced order model (ROM) based on the matrix discrete empirical interpolation method (MDEIM). The ROM surrogate is trained offline and reused for any geometric mapping, restricting the expensive quadrature on cut elements to the training stage. A stabilization term ensures the scalability of the solver when using the ROM approximation, at the cost of a small and controllable error. We validate the method through numerical experiments and demonstrate its performance on a complex 2D problem with more than 17,000 cells of varying geometry, solved in approximately 30 seconds on a standard laptop. The number of solver iterations remains bounded as the number of subdomains grows, provided the ratio between subdomain and mesh sizes is kept constant, in agreement with classical BDDC scalability properties.

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 / 3 minor

Summary. The paper proposes a BDDC domain decomposition solver for unfitted p-FEM discretizations of level-set-based lattice structures. Each cell is treated as a subdomain and discretized with a single high-order element; cell stiffness matrices are assembled via a combination of geometric-mapping separation and an MDEIM-based reduced-order model trained offline. A stabilization term is added to restore the stability properties required for BDDC scalability when the ROM surrogate is used. The method is demonstrated on a single large 2D graded lattice containing more than 17,000 cells of varying geometry and topology, solved in roughly 30 seconds on a laptop, with iteration counts remaining bounded when the subdomain-to-mesh-size ratio is held constant.

Significance. If the stabilization term can be shown to preserve both BDDC iteration bounds and solution accuracy across topologically varying cells, the approach would provide a practical route to direct high-order simulation of complex, non-periodic lattices without homogenization assumptions. The offline ROM training and reuse for arbitrary mappings addresses a key assembly bottleneck in unfitted p-FEM, and the reported 30-second solve time on 17k cells indicates engineering relevance. However, the current evidence rests on one numerical example and the assumption that the ROM-induced perturbation remains controllable.

major comments (3)
  1. [Numerical experiments] Numerical experiments section: the single 17,000-cell demonstration reports bounded iterations and acceptable runtime, yet contains no systematic convergence study, no comparison against full quadrature assembly, and no quantification of the L2 or energy error introduced by the ROM-plus-stabilization approximation across cells with differing cut configurations. This is load-bearing for the central claim that the error remains “small and controllable.”
  2. [Stabilization term] Stabilization term description (likely §3): the manuscript states that the added stabilization restores scalability “at the cost of a small and controllable error,” but provides neither an a-priori bound on the perturbation to the local stiffness matrices nor an analysis of how the stabilization coefficient interacts with the varying Jacobians and cut topologies present in graded lattices. Without such analysis the extension of classical BDDC theory to the ROM setting remains empirical.
  3. [ROM training] ROM training and reuse (Section on MDEIM): the offline training set is described as sufficient for “any geometric mapping,” yet no coverage study or sensitivity test is reported for the range of mappings and topology changes encountered in the target graded structures. If the training set under-samples certain cut configurations, the local matrix error can propagate through the BDDC coarse space, undermining the observed iteration bound.
minor comments (3)
  1. [Abstract] The abstract claims agreement with “classical BDDC scalability properties,” but the manuscript does not restate the precise assumptions (e.g., quasi-uniform subdomain partitioning, sufficient overlap) under which the ROM perturbation is expected to preserve those properties.
  2. [Numerical experiments] Hardware and implementation details for the 30-second timing are not provided; a brief statement of processor, memory, and whether the code is serial or parallel would allow readers to assess reproducibility.
  3. [Notation] Notation for the stabilization coefficient and the MDEIM interpolation points should be introduced once and used consistently; occasional re-definition of symbols across sections reduces readability.

Simulated Author's Rebuttal

3 responses · 1 unresolved

We thank the referee for the constructive and detailed comments. We address each major comment point by point below, indicating the revisions we will make to strengthen the manuscript while remaining faithful to the existing results.

read point-by-point responses

  1. Referee: [Numerical experiments] Numerical experiments section: the single 17,000-cell demonstration reports bounded iterations and acceptable runtime, yet contains no systematic convergence study, no comparison against full quadrature assembly, and no quantification of the L2 or energy error introduced by the ROM-plus-stabilization approximation across cells with differing cut configurations. This is load-bearing for the central claim that the error remains “small and controllable.”

    Authors: We agree that additional numerical evidence would better support the claim of controllable error. In the revised manuscript we will add: (i) a systematic convergence study for representative single cells with varying cut configurations and polynomial degrees, comparing ROM-based solutions against full-quadrature references in both L2 and energy norms; (ii) a direct accuracy comparison between ROM and full assembly on a smaller multi-cell lattice; and (iii) tabulated relative matrix and solution errors across a range of cut topologies. These additions will quantify the approximation error and its propagation. revision: yes

  2. Referee: [Stabilization term] Stabilization term description (likely §3): the manuscript states that the added stabilization restores scalability “at the cost of a small and controllable error,” but provides neither an a-priori bound on the perturbation to the local stiffness matrices nor an analysis of how the stabilization coefficient interacts with the varying Jacobians and cut topologies present in graded lattices. Without such analysis the extension of classical BDDC theory to the ROM setting remains empirical.

    Authors: The stabilization term is introduced to recover the spectral equivalence properties needed for BDDC scalability when the local matrices are replaced by their ROM surrogates. While the current manuscript relies on numerical evidence that iteration counts remain bounded and comparable to the non-ROM case, we acknowledge the absence of an a-priori perturbation bound. In the revision we will expand the description of the stabilization coefficient, report its sensitivity to Jacobian variation and cut topology through additional numerical tests, and clarify that the theoretical extension remains empirical at present. revision: partial

  3. Referee: [ROM training] ROM training and reuse (Section on MDEIM): the offline training set is described as sufficient for “any geometric mapping,” yet no coverage study or sensitivity test is reported for the range of mappings and topology changes encountered in the target graded structures. If the training set under-samples certain cut configurations, the local matrix error can propagate through the BDDC coarse space, undermining the observed iteration bound.

    Authors: The MDEIM training set was constructed from a diverse collection of mappings and cut configurations representative of graded lattices. We agree that explicit coverage and sensitivity results would increase confidence in generalization. The revised manuscript will include a new subsection detailing the training-set composition together with approximation-error metrics for both in-distribution and out-of-distribution mappings and topologies, including extreme cut ratios, to assess potential impact on the coarse-space correction. revision: yes

standing simulated objections not resolved
  • A rigorous a-priori bound on the ROM-induced perturbation to the local stiffness matrices and its interaction with arbitrary Jacobians and cut topologies.

Circularity Check

0 steps flagged

No significant circularity; derivation builds on external BDDC and MDEIM theory with independent numerical validation

full rationale

The paper's core chain—unfitted p-FEM discretization of level-set lattices, BDDC domain decomposition with one cell per subdomain, MDEIM-based ROM for fast stiffness assembly, and an added stabilization term—relies on established external references for BDDC scalability and MDEIM training. The stabilization is introduced to restore properties lost under ROM approximation and is validated numerically on a 17k-cell graded lattice example, with iteration counts shown to remain bounded under standard subdomain-to-mesh ratio conditions. No step reduces a claimed prediction or uniqueness result to a self-fit, self-citation chain, or definitional renaming; the error controllability is treated as an empirical assumption supported by experiments rather than derived tautologically from the target data.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claim rests on the ROM surrogate preserving sufficient accuracy for BDDC scalability when paired with stabilization; the offline training step and the constant subdomain-to-mesh-size ratio are key external requirements.

free parameters (1)
  • stabilization coefficient
    Introduced to restore scalability under ROM approximation; its specific value is chosen for the reported experiments but not quantified in the abstract.
axioms (2)
  • domain assumption Classical BDDC iteration bounds hold when the ratio of subdomain size to mesh size is held constant
    Invoked to explain the observed bounded iteration count.
  • ad hoc to paper The ROM approximation error remains small and controllable after stabilization
    Added to justify use of the surrogate in the solver.

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