Consistent Parametric Model Order Reduction by Matrix Interpolation for Varying Underlying Meshes

Parametric model order reduction (pMOR) is a powerful tool for accelerating finite element (FE) simulations while maintaining parametric dependencies. For geometric parameters, pMOR by matrix interpolation is a well-suited approach because it does not require an affine representation of the parametric dependency, which is often not available for geometric parameters. However, the method requires that the underlying FE mesh has the same number of degrees of freedom and the same topology for all parameter configurations. This requirement can be difficult or even impossible to achieve for large parameter ranges or when automatic meshing is used. In this work, we propose a novel framework for pMOR by matrix interpolation for varying underlying meshes. The key idea is to understand the sampled reduced bases as continuous displacement fields that can be represented in different discretizations. By using mesh morphing and basis interpolation, the sampled reduced bases described in varying meshes can all be represented in terms of one reference mesh. This not only allows for performing pMOR by matrix interpolation, but also enables comparing the subspaces that the reduced bases span, which is important to detect strong changes that could lead to inconsistencies in the reduced operators. For mesh morphing, two strategies, namely morphing by spring analogy with elastic hardening and radial basis function morphing, were implemented and tested. Numerical experiments on a beam-shaped plate and a plate with a hole for one- and two-dimensional parameter spaces show that the proposed framework achieves high accuracy for both morphing methods and performs significantly better than two existing approaches for pMOR by matrix interpolation for varying underlying meshes.

💡 Research Summary

The paper addresses a fundamental limitation of parametric model order reduction (pMOR) by matrix interpolation: the requirement that all finite‑element (FE) models share the same number of degrees of freedom and identical mesh topology. When geometric parameters change, meshes often become distorted or are regenerated automatically, breaking this assumption and leading to inconsistencies in the reduced bases, which in turn corrupt the interpolation of reduced operators.
To overcome this, the authors propose a novel framework that treats each sampled reduced basis as a continuous displacement field. By selecting a reference mesh, they morph this mesh to each sampled geometry and evaluate the displacement fields on the morphed reference mesh, thereby expressing all reduced bases in a common high‑dimensional space. Two mesh‑morphing strategies are implemented: (1) a spring‑analogy method with elastic hardening, which models the mesh as a network of springs and minimizes deformation energy, and (2) radial‑basis‑function (RBF) morphing, which directly interpolates node coordinates. Both approaches preserve characteristic geometric features (boundaries, corners) while smoothly deforming the remaining interior points.
Once all reduced bases are represented on the reference mesh, the standard matrix‑interpolation pipeline can be applied. The concatenated bases are processed by a singular‑value decomposition to obtain a common coordinate system R; a transformation matrix T_k = (Rᵀ V_k)⁻¹ aligns each local reduced model with R. The transformed reduced operators (mass, damping, stiffness, input, output) are then interpolated entry‑wise or via manifold‑aware techniques (e.g., interpolation on the tangent space of SPD matrices).
A key contribution is the quantitative detection of “inconsistencies” between reduced bases. By computing subspace angles θ_l from the singular values of V_iᵀ V_j, the method identifies large deviations caused by mode switching, truncation, or severe mesh changes. When angles exceed a prescribed threshold, adaptive sampling or clustering can be employed to partition the parameter space into regions of consistent bases, ensuring reliable interpolation within each region.
The framework is validated on two benchmark structures: a beam‑shaped plate and a plate with a central hole. Both one‑dimensional (single geometric parameter) and two‑dimensional (two geometric parameters) parameter spaces are examined. Results show that both morphing techniques achieve high accuracy, with RBF morphing slightly outperforming the spring‑analogy method for highly non‑linear deformations. Compared to two existing approaches—operator‑based optimization and zero‑padding of reduced bases—the proposed method yields significantly lower errors and maintains positive‑definite reduced operators. Moreover, the ability to compute subspace angles enables early detection of problematic regions, which the prior methods lack.
In summary, the authors deliver a comprehensive solution that (i) removes the equal‑mesh restriction, (ii) provides a rigorous way to compare and align reduced bases across varying meshes, and (iii) improves the robustness and accuracy of matrix‑interpolation‑based pMOR. This advancement opens the door for efficient parametric analyses in contexts where automatic meshing or large geometric variations are unavoidable, such as design optimization, real‑time simulation, and digital twins.