[Paper Review] Patch-wise Quadrature of Trimmed Surfaces in Isogeometric Analysis

Michael Loibl, Leonardo Leonetti+2 more|arXiv (Cornell University)|Jun 26, 2023

Advanced Numerical Analysis TechniquesEngineering3 citations

TL;DR

This paper proposes a patch-wise quadrature rule for trimmed NURBS surfaces in Isogeometric Analysis, extending existing 1D optimal quadrature techniques to 2D trimmed geometries by adapting integration points to the actual parametrization and trimming boundaries. The method significantly reduces quadrature points while maintaining accuracy in linear and nonlinear shell analyses, outperforming standard trimming procedures.

ABSTRACT

This work presents an efficient quadrature rule for shell analysis fully integrated in CAD by means of Isogeometric Analysis (IGA). General CAD-models may consist of trimmed parts such as holes, intersections, cut-offs etc. Therefore, IGA should be able to deal with these models in order to fulfil its promise of closing the gap between design and analysis. Trimming operations violate the tensor-product structure of the used Non-Uniform Rational B-spline (NURBS) basis functions and of typical quadrature rules. Existing efficient patch-wise quadrature rules consider actual knot vectors and are determined in 1D. They are extended to further dimensions by means of a tensor-product. Therefore, they are not directly applicable to trimmed structures. The herein proposed method extends patch-wise quadrature rules to trimmed surfaces. Thereby, the number of quadrature points can be signifficantly reduced. Geometrically linear and non-linear benchmarks of plane, plate and shell structures are investigated. The results are compared to a standard trimming procedure and a good performance is observed.

Motivation & Objective

• To address the challenge of efficient and accurate integration in Isogeometric Analysis for trimmed surfaces that violate tensor-product structures.
• To extend existing 1D patch-wise quadrature rules—designed for untrimmed patches—to 2D trimmed surfaces with complex boundaries.
• To reduce the number of quadrature points required for integration in trimmed shell structures without sacrificing accuracy.
• To enable seamless integration of CAD models with holes, cuts, and intersections into IGA workflows by maintaining high computational efficiency.

Proposed method

• Adapts 1D generalized Gaussian quadrature rules to 2D by extending the concept of optimal quadrature points to trimmed surface patches.
• Uses the actual knot vectors and geometric parameters of trimmed NURBS surfaces to determine quadrature points and weights in a patch-wise manner.
• Applies a tensor-product extension strategy that respects the trimmed domain boundaries, avoiding the inefficiencies of standard Gauss quadrature on trimmed elements.
• Employs a recursive algorithm to compute quadrature points and weights, ensuring stability and convergence for arbitrary polynomial degrees and knot distributions.
• Implements the method in a Galerkin-based isogeometric formulation for geometrically linear and nonlinear shell problems.
• Validates the approach through benchmark examples with convergence studies and comparison to standard trimming and full Gauss quadrature.

Experimental results

Research questions

• RQ1 How can patch-wise quadrature rules be extended from untrimmed to trimmed NURBS surfaces in isogeometric analysis?
• RQ2 What is the optimal number of quadrature points required for accurate integration on trimmed surface patches, and how can they be computed efficiently?
• RQ3 How does the proposed method compare in accuracy and efficiency to standard Gauss quadrature and conventional trimming techniques in shell analysis?
• RQ4 Can the method maintain optimal convergence rates and reduce computational cost in geometrically nonlinear problems involving trimmed shells?
• RQ5 What are the implications of varying quadrature point distribution on parallelization efficiency and matrix symmetry in IGA?

Key findings

• The proposed patch-wise quadrature rule reduces the number of quadrature points significantly compared to standard Gauss quadrature, especially in trimmed regions.
• For the punched plate benchmark, the method achieved a displacement result of -0.002401 at the plate center, matching high-accuracy reference solutions.
• In the curved shell example, the method yielded a displacement of -0.3610789 at point A, demonstrating robustness in nonlinear analysis.
• The method maintained optimal convergence rates across all benchmarks, confirming its reliability for both linear and nonlinear problems.
• Computational time was substantially reduced compared to standard trimming procedures, particularly in nonlinear analyses where integration is repeated multiple times.
• The approach enables efficient integration in CAD-integrated shell analysis by preserving the benefits of patch-wise quadrature even in complex trimmed geometries.