Non-Uniform Rational B-Splines and Rational Bezier Triangles for Isogeometric Analysis of Structural Applications

Isogeometric Analysis (IGA) is a major advancement in computational analysis that bridges the gap between a computer-aided design (CAD) model, which is typically constructed using Non-Uniform Rational B-splines (NURBS), and a computational model that traditionally uses Lagrange polynomials to represent the geometry and solution variables. In IGA, the same shape functions that are used in CAD are employed for analysis. The direct manipulation of CAD data eliminates approximation errors that emanate from the process of converting the geometry from CAD to Finite Element Analysis (FEA). As a result, IGA allows the exact geometry to be represented at the coarsest level and maintained throughout the analysis process. While IGA was initially introduced to streamline the design and analysis process, this dissertation shows that IGA can also provide improved computational results for complex and highly nonlinear problems in structural mechanics. This dissertation addresses various problems in structural mechanics in the context of IGA, with the use of NURBS and rational Bézier triangles for the description of the parametric and physical spaces. The approaches considered here show that a number of important properties (e.g., high-order smoothness, geometric exactness, reduced number of degrees of freedom, and increased flexibility in discretization) can be achieved, leading to improved numerical solutions. Specifically, using B-splines and a layer-based discretization, a distributed plasticity isogeometric frame model is formulated to capture the spread of plasticity in large-deformation frames. The modeling approach includes an adaptive analysis where the structure of interest is initially modeled with coarse mesh and knots are inserted based on the yielding information at the quadrature points. It is demonstrated that improvement on efficiency and convergence rates is attained. With NURBS, an isogeometric rotation-free multi-layered plate formulation is developed based on a layerwise deformation theory. The derivation assumes a separate displacement field expansion within each layer, and considers transverse displacement component as C0-continuous at dissimilar material interfaces, which is enforced via knot repetition. The separate integration of the in-plane and through-thickness directions allows to capture the complete 3D stresses in a 2D setting. The proposed method is used to predict the behavior of advanced materials such as laminated composites, and the results show advantages in efficiency and accuracy. To increase the flexibility in discretizing complex geometries, rational Bézier triangles for domain triangulation is studied. They are further coupled with a Delaunay-based feature-preserving discretization algorithm for static bending and free vibration analysis of Kirchhoff plates. Lagrange multipliers are employed to explicitly impose high-order continuity constraints and the augmented system is solved iteratively without increasing the matrix size. The resulting discretization is geometrically exact, admits small geometric features, and constitutes C1-continuity. The feature-preserving rational Bézier triangles are further applied to smeared damage modeling of quasi-brittle materials. Due to the ability of Lagrange multipliers to raise global continuity to any desired order, the implicit fourth- and sixth-order gradient damage models are analyzed. The inclusion of higher-order terms in the nonlocal Taylor expansion improves solution accuracy. A local refinement algorithm that resolves marked regions with high resolution while keeping the resulting mesh conforming and well-conditioned is also utilized to improve efficiency. The outcome is a unified modeling framework where the feature-preserving discretization is able to capture the damage initiation and early-stage propagation, and the local refinement technique can then be applied to adaptively refine the mesh in the direction of damage propagation.PHDCivil EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/147668/1/ningliu_1.pd

An immersed boundary hierarchical B-spline method for flexoelectricity

© 2019. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/This paper develops a computational framework with unfitted meshes to solve linear piezoelectricity and flexoelectricity electromechanical boundary value problems including strain gradient elasticity at infinitesimal strains. The high-order nature of the coupled PDE system is addressed by a sufficiently smooth hierarchical B-spline approximation on a background Cartesian mesh. The domain of interest is embedded into the background mesh and discretized in an unfitted fashion. The immersed boundary approach allows us to use B-splines on arbitrary domain shapes, regardless of their geometrical complexity, and could be directly extended, for instance, to shape and topology optimization. The domain boundary is represented by NURBS, and exactly integrated by means of the NEFEM mapping. Local adaptivity is achieved by hierarchical refinement of B-spline basis, which are efficiently evaluated and integrated thanks to their piecewise polynomial definition. Nitsche's formulation is derived to weakly enforce essential boundary conditions, accounting also for the non-local conditions on the non-smooth portions of the domain boundary (i.e. edges in 3D or corners in 2D) arising from Mindlin's strain gradient elasticity theory. Boundary conditions modeling sensing electrodes are formulated and enforced following the same approach. Optimal error convergence rates are reported using high-order B-spline approximations. The method is verified against available analytical solutions and well-known benchmarks from the literature.Peer ReviewedPostprint (author's final draft

Numerical Testing of a New Positivity-Preserving Interpolation Algorithm

An important component of a number of computational modeling algorithms is an interpolation method that preserves the positivity of the function being interpolated. This report describes the numerical testing of a new positivity-preserving algorithm that is designed to be used when interpolating from a solution defined on one grid to different spatial grid. The motivating application is a numerical weather prediction (NWP) code that uses spectral elements as the discretization choice for its dynamics core and Cartesian product meshes for the evaluation of its physics routines. This combination of spectral elements, which use nonuniformly spaced quadrature/collocation points, and uniformly-spaced Cartesian meshes combined with the desire to maintain positivity when moving between these necessitates our work. This new approach is evaluated against several typical algorithms in use on a range of test problems in one or more space dimensions. The results obtained show that the new method is competitive in terms of observed accuracy while at the same time preserving the underlying positivity of the functions being interpolated.Comment: 58 pages, 17 figure

On the formulation of a BEM in the Bézier–Bernstein space for the solution of Helmholtz equation

This paper proposes a novel boundary element approach formulated on the Bézier-Bernstein basis to yield a geometry-independent field approximation. The proposed method is geometrically based on both computer aid design (CAD) and isogeometric analysis (IGA), but field variables are independently approximated from the geometry. This approach allows the appropriate approximation functions for the geometry and variable field to be chosen. We use the Bézier–Bernstein form of a polynomial as an approximation basis to represent both geometry and field variables. The solution of the element interpolation problem in the Bézier–Bernstein space defines generalised Lagrange interpolation functions that are used as element shape functions. The resulting Bernstein–Vandermonde matrix related to the Bézier–Bernstein interpolation problem is inverted using the Newton-Bernstein algorithm. The applicability of the proposed method is demonstrated solving the Helmholtz equation over an unbounded region in a two-and-a-half dimensional (2.5D) domainMinisterio de Economía y Competitividad BIA2016-75042-C2-1-RFondos FEDER POCI-01-0247-FEDER-01775