# 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.

### 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 $ar{B}$ 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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