# Coreform reports

Coreform reports are authored by Coreform technical staff often in collaboration with other researchers and students. They are meant to provide interested parties with an in-depth treatment of the technical foundations of spline-based simulation.

### Efficient and Robust Quadratures for Isogeometric Analysis: Reduced Gauss and Gauss-Greville rules

Z. Zou, T.J.R Hughes, M.A. Scott, Di Miao, and R.A. Sauer

Published: 16 August 2021

This work proposes two efficient quadrature rules, reduced Gauss quadrature and Gauss- Greville quadrature, for B-spline and NURBS based isogeometric analysis. The rules are con- structed to exactly integrate one-dimensional B-spline basis functions of degree $p$, and continuity class $C^{p−k}$, where $k$ is the highest order of derivatives appearing in the Galerkin formulation of the problem under consideration. This is the same idea we utilized in [1], but the rules therein produced negative weights for certain non-uniform meshes. The present work improves upon [1] in that the weights are guaranteed to be positive for all meshes. The reduced Gauss quadrature rule is built element-wise according to the element basis degree and smoothness. The Gauss-Greville quadrature rule combines the proposed reduced Gauss quadrature and Gre- ville quadrature [1]. Both quadrature rules involve many fewer quadrature points than the full Gauss quadrature rule and avoid negative quadrature weights for arbitrary knot vectors. The proposed quadrature rules are stable and accurate, and they can be constructed without solv- ing nonlinear equations, therefore providing efficient and easy-to-use alternatives to full Gauss quadrature. Various numerical examples, including curved shells, demonstrate that they achieve good accuracy, and for $p = 5$ and $6$ eliminate locking.

### Isogeometric Bézier dual mortaring: Kirchhoff-Love shell problem

Di Miao, Zhihui Zou, Michael A. Scott, Michael J. Borden, and Derek C. Thomas

Published: 13 April 2021

In this paper we develop an isogeometric Bézier dual mortar method for coupling multi-patch Kirchhoff-Love shell structures. The proposed approach weakly enforces the continuity of the solution at patch interfaces through a dual mortar method and can be applied to both conforming and non-conforming discretizations. As the employed dual basis functions have local supports and satisfy the biorthogonality property, the resulting stiffness matrix is sparse. In addition, the coupling accuracy is optimal because the dual basis possesses the polynomial reproduction property. We also formulate the continuity constraints through the Rodrigues' rotation operator which gives an unified framework for coupling patches that are intersected with $G^1$ continuity as well as patches that meet at a kink. Several linear and nonlinear examples demonstrated the performance and robustness of the proposed coupling techniques.

### Galerkin formulations of isogeometric shell analysis: Alleviating locking with Greville quadratures and higher-order elements

Z. Zou, T.J.R Hughes, M.A. Scott, R.A. Sauer, and E.J. Savitha

Published: 6 January 2021

We propose new quadrature schemes that asymptotically require only four in-plane points for Reissner-Mindlin shell elements and nine in-plane points for Kirchhoff-Love shell elements in B-spline and NURBS-based isogeometric shell analysis, independent of the polynomial degree $p$ of the elements. The quadrature points are Greville abscissae associated with $p$th-order B-spline basis functions whose continuities depends on the specific Galerkin formulations, and the quadrature weights are calculated by solving a linear moment fitting problem in each parametric direction. The proposed shell element formulations are shown through numerical studies to be rank sufficient and to be free of spurious modes. The studies reveal comparable accuracy, in terms of both displacement and stress, compared with fully integrated spline-based shell elements, while at the same time reducing storage and computational cost associated with forming element stiffness and mass matrices and force vectors. The high accuracy with low computational cost makes the proposed quadratures along with higher-order spline bases, in particular polynomial orders, $p=5$ and 6, good choices for alleviating membrane and shear locking in shells.

### U-splines: splines over unstructured meshes

Derek C. Thomas, Luke Engvall, Steven K. Schmidt, Kevin Tew, Michael A. Scott

Published: 11 November 2020

U-splines are a novel approach to the construction of a spline basis for representing smooth objects in Computer- Aided Design (CAD) and Computer-Aided Engineering (CAE). A spline is a piecewise-defined function that satisfies continuity constraints between adjacent elements in a mesh. U-splines differ from existing spline constructions, such as Non-Uniform Rational B- splines (NURBS), subdivision surfaces, T-splines, and hierarchical B-splines, in that they can accommodate local geometrically exact adaptivity in h (element size), p (polynomial degree), and k (smoothness) simultaneously over more varied mesh topology. U-splines have no restrictions on the placement of T-junctions in the mesh. Mixed element meshes (e.g., triangle and quadrilateral elements in the same surface mesh) are also supported. We conjecture that the U-spline basis is positive, forms a partition of unity, is linearly independent, and provides optimal approximation when used in analysis.

### An isogeometric Reissner-Mindlin shell element based on Bézier dual basis functions: overcoming locking and improved coarse mesh accuracy

Zhihui Zou, Michael A. Scott, Di Miao, Manfred Bischoff, and Bastian Oesterle

Published: 2 June 2020

We develop a mixed nonlinear isogeometric Reissner-Mindlin shell element for the analysis of thin-walled structures that leverages Bézier dual basis functions to address both shear and membrane locking and to improve the quality of computed stresses. The accuracy of computed solutions over coarse meshes, that have highly non-interpolatory control meshes, is achieved through the application of a continuous rotational approach.

### Isogeometric Bézier dual mortaring: the enriched Bézier dual basis with application to second- and fourth-order problems

Di Miao, Zhihui Zou, Michael A. Scott, Michael J. Borden, and Derek C. Thomas

Published: 2 November 2019

In this paper, we present an algorithm to construct enriched Bézier dual basis functions that can reproduce higher-order polynomials. Our construction is unique in that it is based on Bézier extraction and projection, allowing it to be used for tensor product and unstructured polynomial spline spaces, is well-conditioned, and is quadrature-free. When used as a basis for dual mortar methods, optimal approximations are achieved for both second- and fourth-order problems.

### Isogeometric analysis using the *IGA_INCLUDE_BEZIER keyword in LS-DYNA

Christopher Whetten, Matthew Sederberg, Michael Scott

Published: 14 May 2019

In contrast to the laborious and error-prone process of translating computer-aided design (CAD) into computer-aided engineering (CAE) models, isogeometric analysis (IGA) performs the finite element analysis (FEA) simulation directly on CAD geometry, using smooth spline basis functions. LS-DYNA is a leader in the industrial adoption of IGA, and has recently made a significant enhancement to broaden the possible use of IGA within LS-DYNA.

### Bézier $\bar{B}$ projection

Di Miao, Michael J. Borden, Michael A. Scott, Derek C. Thomas

Published: 20 February 2018

We demonstrate the use of Bézier projection to alleviate locking phenomena in structural mechanics applications of isogeometric analysis. We call this method Bézier $\bar{B}$ projection. To demonstrate the utility of the approach for both geometry and material locking phenomena we focus on transverse shear locking in Timoshenko beams and volumetric locking in nearly compressible linear elasticity although the approach can be applied generally to other types of locking phenemona as well.

### Isogeometric Bézier dual mortaring: Refineable higher-order spline dual bases and weakly continuous geometry

Z. Zou, M. A. Scott, M. J. Borden, D. C. Thomas, W. Dornisch, E. Brivadis

Published: 13 November 2017

In this paper we develop the isogeometric Bézier dual mortar method. It is based on Bézier extraction and projection and is applicable to any spline space which can be represented in Bézier form (i.e., NURBS, T-splines, LR-splines, etc.). The approach weakly enforces the continuity of the solution at patch interfaces and the error can be adaptively controlled by leveraging the refineability of the underlying dual spline basis without introducing any additional degrees of freedom.

### Bézier projection: a unified approach for local projection and quadrature-free refinement and coarsening of NURBS and T-splines with particular application to isogeometric design and analysis

Derek C. Thomas, Michael A. Scott, John A. Evans, Kevin Tew, Emily J. Evans

Published: 28 April 2014

We introduce Bézier projection as an element-based local projection methodology for B-splines, NURBS, and T-splines. This new approach relies on the concept of Bézier extraction and an associated operation introduced here, spline reconstruction, enabling the use of Bézier projection in standard finite element codes. Bézier projection exhibits provably optimal convergence and yields projections that are virtually indistinguishable from global $L^{2}$ projection.

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