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A POD-selective inverse distance weighting method for fast parametrized shape morphing

Francesco Ballarin, Alessandro D'Amario, Simona Perotto, Gianluigi Rozza

arXiv:1710.09243v2math.NAcs.NA

TL;DR

Shape morphing for parametrized PDEs needs to balance computational cost and accuracy, particularly when standard IDW uses many control points. The paper combines geometric selection of a smaller control-point set with POD of admissible displacements; the reported tests show substantial CPU-time savings, with practical timing caveats.

  • Problem

    Standard IDW can require many control points, making assembly and storage of its dense matrix costly.

    Source: §2.1.1 · p. 4

  • Method

    The method geometrically selects a control-point subset for IDW and combines selective IDW with POD reduction of admissible displacements.

    Source: §1 · p. 2, §2.2 · p. 5, §2.2.1 · p. 6

  • Results

    The combined POD-IDW and POD-ESIDW techniques save up to two orders of magnitude in CPU time, while SIDW/ESIDW have small relative errors of a few percentage points against standard IDW.

    Source: §4 · p. 26

  • Takeaways & Limitations

    SIDW/ESIDW reduce control-point counts without excessively compromising approximation accuracy, and POD can further reduce computational burden.

    Source: §4 · p. 25

    Actual speedup may be lower than the ideal speedup because timing includes mesh updating as well as deformation.

    Source: §2.3.1 · p. 8

Abstract

from arXiv · show

Efficient shape morphing techniques play a crucial role in the approximation of partial differential equations defined in parametrized domains, such as for fluid-structure interaction or shape optimization problems. In this paper, we focus on Inverse Distance Weighting (IDW) interpolation techniques, where a reference domain is morphed into a deformed one via the displacement of a set of control points. We aim at reducing the computational burden characterizing a standard IDW approach without significantly compromising the accuracy. To this aim, first we propose an improvement of IDW based on a geometric criterion which automatically selects a subset of the original set of control points. Then, we combine this new approach with a dimensionality reduction technique based on a Proper Orthogonal Decomposition of the set of admissible displacements. This choice further reduces computational costs. We verify the performances of the new IDW techniques on several tests by investigating the trade-off reached in terms of accuracy and efficiency.

1 Introduction

Shape morphing supports applications involving changing domains and repeated geometric configurations, but standard IDW can become costly when many control points are used. The paper proposes geometric control-point selection and POD-based dimensionality reduction to reduce that burden.

  • Changing domains and repeated geometric configurations arise in applications such as fluid-structure interaction and shape optimization.Source: §1 · p. 1
  • Morphing a reference mesh avoids generating a new mesh for every deformed configuration, a potentially time-consuming step not always integrated into the solver.Source: §1 · p. 1
  • IDW morphs a reference domain by displacing control points and applying an interpolation map to the discretized domain.Source: §1 · pp. 1–2
  • Because morphing-map evaluation grows more complex with more control points, their selection should keep the set small; existing choices can depend on prior knowledge of the phenomenon.Source: §1 · p. 2
  • The paper proposes geometrically based control-point selection independent of the underlying PDE, then applies POD to admissible displacements to further reduce computational cost.Source: §1 · p. 2
  • The paper compares standard and selective IDW on structural configurations, then investigates POD combined with selective IDW for shape morphing.Source: §1 · p. 2

2 Inverse Distance Weighting techniques

This section introduces IDW for shape morphing in an FSI context and presents a variant intended to address computational limits of the original method.

  • The section considers shape-morphing techniques for FSI mesh motion, extending structural displacement into the fluid domain.Source: §2 · p. 2
  • It introduces standard IDW interpolation before proposing a variant to address computational limits.Source: §2 · p. 2
  • Unlike original FFD, later NURBS-FFD variants allow greater freedom in choosing control points.Source: §2 · p. 2

2.1 The Inverse Distance Weighting (IDW) approach

IDW interpolates values using control-point weights that diminish with distance. In shape morphing, control-point displacements drive a deformation map, but the dense IDW matrix can make large control-point sets costly to assemble and store.

2.1 The Inverse Distance Weighting (IDW) approach

  • IDW was extended beyond its original two-dimensional interpolation setting to higher-dimensional and applied problems.Source: §2.1 · p. 3
  • The local IDW weights depend inversely on distance, with power p tuning that dependence; in FSI, p is important for avoiding compenetration effects.Source: §2.1 · p. 3

2.1.1 IDW for shape morphing

  • For shape morphing, the reference domain is the initial physical configuration, and the IDW interpolant defines the shape-parametrization map from control points to mesh deformation.Source: §2.1.1 · p. 3
  • Control-point displacements are inputs to the deformation, while the computed deformation represents the displaced mesh nodes.Source: §2.1.1 · pp. 3–4
  • The IDW matrix is dense, and a large control-point set can make its assembly and storage computationally burdensome.Source: §2.1.1 · p. 4

2.2 The Selective Inverse Distance Weighting (SIDW) approach

SIDW reduces the computational effort of standard IDW by geometrically selecting a subset of control points to describe the initial and deformed domains. For shape morphing, it uses the selected points to interpolate internal-node displacements, with optional enrichment for problem-specific constraints.

2.2 The Selective Inverse Distance Weighting (SIDW) approach

  • SIDW selects a subset of control points to reduce standard IDW's computational effort while describing the initial and deformed domains.The selection is geometry-based and independent of the underlying partial differential equations; the approach is intended to support large displacements and a uniform reduced-point distribution.Source: §2.2 · p. 4
  • The algorithm tessellates the region outside an initial control point's radius into annuli, then iteratively selects points and removes nearby candidates from consideration.When a selection area is exhausted, it moves to the next annulus or resets within the current one if that annulus is disconnected from the selection area.Source: §2.2 · pp. 5–6, Figure 2 · p. 7
  • Smaller R selects more control points, while larger b selects fewer; smaller a generally creates more annuli and a finer point distribution.The interplay among R, a, and b is problem-dependent, and the authors note that choosing them a priori is difficult.Source: §2.2 · p. 6

2.2.1 SIDW for shape morphing

  • For shape morphing, SIDW filters control points for internal-node updates, while boundary nodes use displacements assigned at the original control points.The selection and matrix assembly are performed once; optional enrichment adds points for constraints such as fixed nodes, forming the ESIDW variant.Source: §2.2.1 · p. 6

2.3 IDW versus SIDW interpolation

The section compares standard IDW with selective variants on structural and fluid-mesh deformations, using geometric control-point selection to reduce computational effort. The tests examine accuracy, runtime, mesh quality, and sensitivity to selection parameters across wing and hull configurations.

2.3.1 Structural deformation of a wing

  • For sufficiently large R, relative L2(Ωh) error against IDW deformation was approximately 4% for SIDW versus about 1% for ESIDW; both converge to zero as R decreases, and ESIDW preserves initial maximum mesh quality as R increases.Source: §2.3.1 · p. 12
  • For R = 0.2 or 0.3, choosing a ≈ 1 reduced SIDW’s control points by about an order of magnitude for only slightly higher relative error; random selection had similar average error but greater variability.At small R, decreasing a improved accuracy because SIDW selected nearly all available control points; random-selection errors could improve by about one standard deviation or worsen by up to three.Source: §2.3.1 · pp. 13–14

2.3.2 Fluid mesh motion around a wing

  • 57.07 s: ESIDW reduced the fluid-wing control-point set to 9,339 and produced a 5.86% relative error; for R > 2, control-point cardinality fell by about an order of magnitude.For R ≤ 1, maximum and mean mesh-quality values remained 4.31 and 1.48, with negligible variation for ESIDW.Source: §2.3.2 · pp. 15, 17

2.3.3 Fluid mesh motion around a rotating hull

  • 2.56 s: ESIDW reduced the rotating-hull deformation time from IDW’s 4.51 s, with a 2.42% relative error and mesh-quality values largely independent of interpolation and R.For the hull, maximum Q changed from 2.47 to 2.56 and mean Q from 1.62 to 1.63 after rotation.Source: §2.3.3 · pp. 17, 19

3 POD for SIDW shape morphing

POD is combined with selective IDW in an offline-online workflow to reduce the cost of repeated shape morphing. Across the tests, the savings and accuracy vary by configuration, while reported mesh-quality effects remain limited.

3 POD for SIDW shape morphing

  • The POD basis is built offline from deformation snapshots by singular value decomposition, retaining modes according to an energy tolerance.For a new parameter, the method fits POD coefficients to the control-point displacement and reconstructs the internal-node deformation; the reduced solve assumes full column rank of cW+Z.Source: §3 · pp. 19–20
  • When the retained mode count is much smaller than the control-point count, the reduced method is expected to save computation relative to the full SIDW problem.The online stage uses precomputed matrices, solves a system of size N, and forms the reduced linear combination.Source: §3 · p. 21

3.1.1 Structural deformation of a wing

  • For structural wing deformation, POD morphing takes 0.013 s, compared with 0.25 s and 0.034 s for the corresponding unreduced approaches.Relative error versus standard IDW is less than 10^-8 for POD-IDW and rises from 1.06% to 1.10% for POD-ESIDW.Source: §3.1.1 · pp. 22–23, Table 5 · p. 22
  • For large wing deformations, the normalized mesh-quality index stays near 0.28 for µ > 0.3, with little influence from POD or ESIDW.The considered displacement reaches more than 800% of the wing’s longitudinal dimension.Source: §3.1.1 · p. 23, Figure 15 · p. 23

3.1.2 Fluid mesh motion around a wing

  • For fluid mesh motion around a wing, selection plus POD reduces CPU time per morphing by two orders of magnitude versus standard IDW, using a one-mode POD basis.The offline phases take 2479.51 s for POD-IDW and 1613.65 s for POD-ESIDW.Source: §3.1.2 · pp. 23–24, Table 6 · p. 23

3.1.3 Fluid mesh motion around a rotating hull

  • For the rotating hull, two POD modes give relative error below 10^-8 for POD-IDW, while POD-ESIDW error stays of order 10^-2 and is weakly sensitive to rotation angle.POD reduction essentially preserves ESIDW mesh quality, with contained dependence on rotation.Source: §3.1.3 · p. 25, Figure 17 · p. 25

4 Conclusions and perspectives

The proposed IDW variants reduce computational costs while retaining accuracy close to standard IDW. POD reduction adds further savings, with one or two basis functions sufficient for a 10^-5 tolerance on discarded-mode energy in the tested cases.

4 Conclusions and perspectives

  • Geometric control-point selection is general across dimensions and arbitrary meshes, and the tested SIDW/ESIDW variants substantially reduce control points without excessively compromising accuracy.POD and least-squares regression further reduce the task to a linear system sized by the number of selected basis functions; one or two basis functions sufficed for a 10^-5 tolerance on discarded-mode energy.Source: §4 · p. 25
  • Combined POD-IDW and POD-ESIDW save up to two orders of magnitude in CPU time, while SIDW/ESIDW have few-percentage-point relative errors versus standard IDW.The offline phase remains the most time-consuming, and POD reduction does not significantly improve accuracy further.Source: §4 · p. 26
  • Future work could integrate the methods into an FSI solver or optimization contexts, adapt control-point selection to quantities of interest, or combine reduction with active subspaces.The authors also identify combination with reduction procedures for parametrized problems as a prospective direction.Source: §4 · p. 26
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