Source-linked AI summary
Projection-based low-rank assembly in IgA
Tom-Christian Riemer, Martin Stoll
TL;DR
IgA’s shared spline representation gives accurate geometries but makes three-dimensional mass and stiffness assembly expensive. The paper develops a projection-based TT low-rank approach for orientation-preserving tensor-product B-spline geometries, and reports competitive performance with interpolation-based assembly while handling singular interpolation systems and nearly singular geometries.
Problem
IgA mass and stiffness matrices are expensive to assemble and store, especially in three dimensions, because geometry-dependent weights make conventional multivariate integration costly.
Method
The method represents polynomial geometry-dependent factors in reduced spline product spaces and projects the reciprocal determinant onto a tensor-product spline space, with computations carried out in TT format.
Results
The method is competitive with interpolation-based assembly, remains applicable when the interpolation system is singular, and is slightly more accurate for nearly singular geometries.
Takeaways & Limitations
Projection-based assembly provides a low-rank alternative that extends applicability to geometry and interpolation cases problematic for interpolation-based assembly.
Takeaways & Limitations
The approach is restricted to orientation-preserving tensor-product B-spline geometries and can become costly when control-point counts or TT ranks are large.
Abstract
from arXiv · showhide
Isogeometric Analysis (IgA) uses the same spline functions to represent the computational domain and to approximate the solution. This allows exact geometry descriptions, but the resulting mass and stiffness matrices are expensive to assemble and to store, especially in three dimensions. We present a projection-based low-rank approach for assembling the mass and stiffness tensors of orientation-preserving tensor-product B-spline geometries. For the mass tensor, we exploit the polynomial structure of the determinant of the Jacobian of the geometry map and represent it in reduced spline product spaces by univariate coefficient transfer operators; with exact quadrature and without truncation, the resulting low-rank tensor is an exact reformulation of the standard Galerkin mass tensor. For the stiffness tensor, we split the rational weight function into a polynomial numerator, again represented in reduced spline product spaces, and the reciprocal determinant, which is in general not a spline function and is therefore approximated by an $L^2$-projection onto a tensor-product spline space. Both constructions are carried out entirely in the tensor-train (TT) format, with the projection system solved by the alternating minimal energy (AMEn) method, so that full high-order coefficient tensors are never formed and the multidimensional integrals reduce to univariate integrals and contracted products. The method is implemented in MATLAB using GeoPDEs and the TT-Toolbox. Numerical experiments show that it is competitive with full assembly and with the interpolation-based low-rank method, and that it applies in two situations in which interpolation is problematic: a singular interpolation system and nearly singular geometries. The construction is restricted to orientation-preserving tensor-product B-spline geometries and does not cover NURBS.
1 Introduction
IgA combines geometry representation and solution approximation through shared spline functions, but tensor-product structure and high-degree bases make three-dimensional matrix assembly and storage costly. The paper introduces a projection-based TT low-rank construction for mass and stiffness tensors that remains useful when interpolation fails or becomes inaccurate.
- Motivation: IgA uses shared spline functions for the computational domain and solution, enabling exact or highly accurate geometry descriptions but making mass and stiffness matrices expensive in three dimensions.The cost arises from tensor-product structure, high-degree bases, costly multivariate quadrature, and large matrix structures.
- Motivation: Low-rank tensor methods exploit separable geometry-dependent weights so multidimensional integrals become sums of products of univariate integrals.Earlier approaches combined separated representations with TT coefficient tensors computed without explicitly forming full tensors.
- Contribution: The proposed method assembles mass and stiffness tensors for orientation-preserving tensor-product B-spline geometries using projection-based low-rank constructions in TT format.Mass assembly uses the Jacobian determinant’s polynomial structure; stiffness assembly separates a polynomial numerator from a projected reciprocal determinant.
- Contribution: With exact quadrature and no truncation, the mass construction is an exact reformulation of the standard Galerkin mass tensor, while stiffness projection approximates the non-spline reciprocal determinant.AMEn solves the projection system, and full high-order coefficient tensors are never formed.
- Results: The method is competitive with full and interpolation-based assembly and applies when interpolation systems are singular or geometries are nearly singular.The numerical implementation uses MATLAB, GeoPDEs, and the TT-Toolbox.
2 Preliminaries
The preliminaries define tensor-product B-spline geometries and the geometry-induced weights used in IgA assembly. They motivate low-rank representations by showing that determinant-based weights can separate, whereas stiffness weights are generally rational and nonseparable.
- B-spline preliminaries: B-spline bases are defined from knot vectors, with derivatives represented in reduced-degree spaces associated with reduced knot vectors.The paper also distinguishes h-refinement by knot insertion from p-refinement by increasing spline degree.
- Tensor-product geometries: Tensor-product B-spline spaces form multivariate bases from univariate factors, allowing geometry and solution coefficients to be organized as multidimensional tensors.The geometry map contracts coordinate control-point tensors with the multivariate basis tensor.
- Geometry assumptions: The proposed construction assumes orientation-preserving tensor-product B-spline geometries, excluding rational NURBS parameterizations because their direct tensor-product separability is lost.The orientation-preserving assumption makes the Jacobian determinant positive.
- Geometry-induced weights: The mass weight ω is a spline with a finite separated representation, whereas stiffness weights qkl are generally rational, nonseparable, and outside polynomial tensor-product spline spaces.Consequently, conventional treatment of these weights leads to computationally expensive multivariate quadrature.
- Low-rank representations: TT format is used instead of CP format because CP approximation can be ill-posed and numerically unstable, while interpolation-based assembly reduces multivariate matrix assembly to small univariate matrices when its system is nonsingular.Interpolation recovers the target function exactly within the interpolation space; remaining error comes from truncation, numerical solution, or auxiliary-space choices.
3 Projection-based low-rank method
The method assembles mass and stiffness tensors in low-rank form by exploiting tensor-product B-spline structure and projection-based coefficient representations. Mass assembly is exact under exact quadrature and no truncation, whereas stiffness assembly approximates the reciprocal determinant through projection.
- Overview: The mass and stiffness assembly methods rewrite multivariate Galerkin integrals as products of univariate integrals compatible with the TT format.Both methods exploit tensor-product B-spline structure and coefficient transfer operators obtained by univariate L2-projections.
- Mass tensor: Compressing geometry-induced modes preserves the solution-space modes while reducing the representation of the mass tensor.The last three modes associated with geometry are compressed, while the two modes associated with the solution space remain unchanged.
- Mass tensor: The determinant weight lies in a reduced tensor-product spline space whose basis size grows linearly with n(d), rather than cubically with the triple-product representation.This reduced representation contains the required products of geometry basis functions.
- Mass tensor: Without tensor truncation and with exact univariate spline integrals, the low-rank mass tensor is an exact reformulation of standard Galerkin assembly.Approximation errors arise only from tensor truncation or inexact quadrature.
- Stiffness tensor: For stiffness assembly, numerator functions are represented exactly, while the common reciprocal determinant is projected onto a tensor-product spline space and introduces approximation error.The reciprocal is generally rational rather than a spline function, so projection makes it separable but not exact.
- Stiffness tensor: The reciprocal determinant is approximated once and then combined with all numerator functions, requiring one projection system instead of one system per independent stiffness entry.This concentrates approximation on the common non-polynomial factor shared by the stiffness entries.
4 Limitations and implementation details
The method’s efficiency depends on tensor-product B-spline geometries with manageable control-point tensor ranks and univariate sizes, while projection accuracy can increase computational costs. Its implementation preserves the low-rank structure by applying transfers and solves directly to TT representations.
- Structural scope: The construction requires orientation-preserving tensor-product B-spline geometry maps and excludes non-tensor-product or rational parameterizations such as NURBS.
- Geometry complexity: Efficiency depends on moderate TT ranks of the geometry’s control-point tensors because determinant and numerator ranks grow through tensor products and mode permutations.
- Geometry complexity: For fixed degree, univariate integrals and coefficient transfers scale linearly in n(d), but n(d) = 104 control points per direction can still make operators and integration costly.
- Possible remedies: A low-rank multi-patch formulation may reduce control points and TT ranks per patch, but introduces additional coupling and interface treatment, with no guaranteed rank reduction from geometry refinement.
- Projection-system cost: Projection accuracy for the reciprocal determinant depends on the richness of Sρ; refinement or degree elevation enlarges systems, raises integrand degrees, and can make univariate computations dominant.
- Implementation: Implementation avoids full high-order tensors by applying derivative, product, mixed-transfer, and integration operators directly to TT cores, with rounding and AMEn-based projection solves controlling computation.
5 Numerical experiments
The experiments compare projection-based low-rank assembly with full and interpolation-based assembly across several B-spline geometries, solution degrees, refinements, tolerances, and quadrature rules. Results show strong accuracy and efficiency in regular cases, while projection remains applicable when interpolation is singular or nearly singular geometries challenge approximation.
- Experimental setup: The experiments compare assembly times, memory requirements, and low-rank errors against fully assembled reference operators.Peak assembly memory is measured separately from final operator storage, and low-rank errors are computed relative to GeoPDEs reference matrices.
- Experimental setup: Three mass-tensor geometries and one additional almost singular stiffness-tensor geometry cover increasing approximation difficulty.The mass cases are a thick flag, rotor blade, and twisted beam; the additional stiffness case has a nearly vanishing Jacobian determinant.
- Mass tensor: Projection-based mass assembly is consistently faster than interpolation-based assembly for the thick flag, with slightly higher memory use and errors near machine precision.For the thick flag, tighter tolerances do not materially improve the already near-machine-precision projection errors.
- Mass tensor: For the rotor and twisted beam, projection-based mass assembly reaches high accuracy, while interpolation fails to approximate the twisted-beam mass tensor accurately.The twisted-beam interpolation error remains approximately 10^-1 independently of the prescribed tolerance, whereas projection follows the tolerance with small TT-rounding deviations.
- Stiffness tensor: Stiffness accuracy depends on the projection space: refining it with the solution space improves convergence, but increases assembly time and peak memory.The fixed default space can stagnate or miss tighter tolerances, while the refined space approaches prescribed thresholds for the thick flag and rotor.
- Stiffness tensor: For nearly singular geometries and singular interpolation systems, refined projection remains accurate and resource-stable, whereas interpolation is unavailable or substantially more costly.The twisted-beam interpolation system is singular; for the almost singular geometry, projection error approaches tolerance while time and memory remain nearly constant under refinement.
6 Conclusion
The paper presents a projection-based low-rank assembly method for mass and stiffness tensors in orientation-preserving tensor-product B-spline geometries. It is competitive with interpolation-based assembly, handles singular interpolation systems and nearly singular geometries, and has scope limitations.
- The mass tensor construction is an exact reformulation of Galerkin assembly with exact quadrature and no tensor truncation.It exploits the polynomial determinant of the Jacobian and represents it in a reduced spline product space.
- The stiffness tensor splits polynomial numerators from the reciprocal determinant, approximating the latter by an L2-projection onto a tensor-product spline space.The construction is performed in TT format, avoiding explicit formation of high-order coefficient tensors.
- The proposed method is competitive with interpolation-based assembly, with nearly singular geometries showing improved stiffness accuracy because projection does not enforce large values near det(∇G) = 0.For the mass tensor, the method is almost consistently faster and more accurate; for stiffness, it also has a slight memory advantage.
- The method remains applicable when the interpolation system is singular, including the twisted beam example where interpolation-based assembly fails to approximate the mass tensor accurately.The projection-based mass assembly follows the prescribed tolerance, with small deviations attributed to TT-rounding errors.
- The stiffness approximation is governed by the richness of Sρ: the default projection space can be insufficient, while refinement with the solution space follows prescribed tolerances.The refined projection space improves approximation quality and can keep assembly time and memory nearly constant under refinement.
- The construction is restricted to orientation-preserving tensor-product B-spline geometries and does not cover NURBS parameterizations.The orientation assumption can be relaxed to geometries whose Jacobian determinant has constant negative sign by negating the coefficient tensor.
Conflict of interest
The paper reports no conflicts of interest, no organizational support, and availability of the computer codes and algorithms.
- The authors declare that they have no conflict of interest.
- The authors report no organizational support for the submitted work.
- The authors have no relevant competing interests to declare.
- The computer codes and algorithms are available in a public GitHub repository.