[Paper Review] A POD-selective inverse distance weighting method for fast parametrized shape morphing

Francesco Ballarin, Alessandro D'Amario+2 more|arXiv (Cornell University)|Jan 1, 2019

Model Reduction and Neural Networks48 references50 citations

TL;DR

This paper proposes a selective Inverse Distance Weighting (SIDW) method combined with Proper Orthogonal Decomposition (POD) to accelerate parametrized shape morphing in PDEs. By geometrically selecting control points and reducing deformation modes via POD, the method cuts CPU time by up to two orders of magnitude while maintaining accuracy within 10−5 tolerance, achieving near-optimal efficiency in fluid-structure interaction and shape optimization contexts.

ABSTRACT

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 that 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.

Motivation & Objective

• Address the high computational cost of standard Inverse Distance Weighting (IDW) in parametrized shape morphing for PDEs.
• Reduce the number of control points in IDW using a geometric criterion to lower computational complexity.
• Integrate Proper Orthogonal Decomposition (POD) to further compress deformation modes and accelerate online evaluation.
• Achieve a balance between computational efficiency and morphing accuracy in engineering applications such as fluid-structure interaction and shape optimization.
• Enable real-time or multi-query simulations by minimizing online computational cost via offline-online decomposition.

Proposed method

• Introduce Selective IDW (SIDW) that automatically selects a minimal subset of control points based on geometric proximity and distribution in the domain.
• Apply a geometric criterion to exclude less influential control points, reducing the number of points used in IDW interpolation without significant accuracy loss.
• Use Proper Orthogonal Decomposition (POD) to extract dominant modes from a snapshot matrix of deformations, enabling dimensionality reduction.
• Construct a reduced-order model where the deformation is approximated as a linear combination of a few POD modes.
• Combine SIDW with POD to form POD-ESIDW, reducing both control point count and deformation space dimension.
• Use a least-squares regression in the online phase to compute morphing maps using only a few POD basis functions, drastically cutting online cost.

Experimental results

Research questions

• RQ1 Can a geometric criterion for control point selection reduce computational cost in IDW-based shape morphing without degrading accuracy?
• RQ2 To what extent can POD-based dimensionality reduction further accelerate the morphing process in parametrized domains?
• RQ3 How does the combination of selective control points and POD reduction affect the trade-off between accuracy and computational efficiency?
• RQ4 What is the minimal number of POD modes required to maintain a prescribed tolerance (e.g., 10−5) in the energy of discarded modes?
• RQ5 How does the error in the morphing map vary with rotation angle and control point selection in complex geometries like hulls or airfoils?

Key findings

• The POD-ESIDW method reduced CPU time by approximately 90% compared to standard IDW, with only two POD modes needed to achieve a 10−5 tolerance in energy content.
• The selective control point selection (ESIDW) alone halved the CPU time compared to standard IDW, demonstrating significant efficiency gains.
• The relative error in morphing remained below 1% for ESIDW across all test cases, indicating minimal accuracy loss despite reduced control points.
• For the rotating hull case, two POD modes were sufficient to capture the deformation with energy tolerance of 10−5, confirming the method’s low-dimensional representation capability.
• The error in POD-ESIDW remained consistently at around 10−2 regardless of rotation angle, showing robustness to geometric changes.
• The offline phase for POD-ESIDW took 150.26 seconds, which is acceptable given the online speedup, especially in multi-query or real-time applications.