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

December 2nd, 2019

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. The starting point of the formulation is the Hellinger-Reissner variational principle with independent displacement, membrane, and shear strains as the unknown fields. To overcome locking, the strain variables are  interpolated with lower-order spline bases while the variations of the strain variables are interpolated with the corresponding Bézier dual bases. Leveraging the biorthogonality property of the Bézier dual basis, the strain variables are condensed out of the system with only a slight increase in the bandwidth of the resulting linear system. The condensed approach preserves the accuracy of the non-condensed mixed approach but with fewer degrees of freedom. From a practical point of view, since the Bézier dual basis is completely specified through Bézier extraction, any spline space that admits Bézier extraction can utilize the proposed approach directly.

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

November 2nd, 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. In the context of fourth-order problems, both $C^0$ and $C^1$ continuity constraints must be applied at each intersection. We develop a novel geometry-independent $C^1$ continuity constraint that also preserves the sparsity of the coupled problem. The performance of the proposed formulation is verified through several challenging second- and fourth-order problems.

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Christopher Whetten, Matthew Sederberg, Michael Scott

May 14th, 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. *IGA_INCLUDE_BEZIER is a new keyword that was recently implemented in LS-DYNA to enable the use  of  unstructured  spline  models  such  as  U-splines  and  T-splines.  This  is  significant  because  it  allows for more complex models to be used for IGA in LS-DYNA.

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

February 20th, 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. Bézier projection is a local projection technique with optimal approximation properties, which in many cases produces solutions that are comparable to global $L^2$ projection. In the context of B̄ methods, the use of Bézier projection produces sparse stiffness matrices with only a slight increase in bandwidth when compared to standard displacement-based methods. Of particular importance is that the approach is applicable to any spline representation that can be written in Bézier form like NURBS, T-splines, LR-splines, etc. We discuss in detail how to integrate this approach into an existing finite element framework with minimal disruption through the use of Bézier extraction operators and a newly introduced dual Bézier extraction operator. We then demonstrate the behavior of the approach through several challenging benchmark problems.

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arXiv: 1708.07743 ;   DOI: 10.1016/j.cma.2018.02.019

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

November 13th, 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. We also develop weakly continuous geometry as a particular application of isogeometric Bézier dual mortaring. Weakly continuous geometry is a geometry description where the weak continuity constraints are built into properly modified Bézier extraction operators. As a result, multi-patch models can be processed in a solver directly without having to employ a mortaring solution strategy. We demonstrate the utility of the approach on several challenging benchmark problems.

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arXiv: 1711.01009 ;   DOI: 10.1016/j.cma.2018.01.023

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

April 28th, 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. Bézier projection is used to develop a unified framework for spline operations including cell subdivision and merging, degree elevation and reduction, basis roughening and smoothing, and spline reparameterization. In fact, Bézier projection provides a quadrature-free approach to refinement and coarsening of splines. In this sense, Bézier projection provides the fundamental building block for $hpkr$-adaptivity in isogeometric analysis.

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arXiv: 1404.7155;   DOI: 10.1016/j.cma.2014.07.014