A geometrically exact isogeometric Kirchhoff plate: Feature‐preserving automatic meshing and C1 rational triangular Bézier spline discretizations

Peer Reviewedhttps://deepblue.lib.umich.edu/bitstream/2027.42/144603/1/nme5809.pdfhttps://deepblue.lib.umich.edu/bitstream/2027.42/144603/2/nme5809_am.pd

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

Isogeometric treatment of large deformation contact and debonding problems with T-splines: a review

AbstractWithin a setting where the isogeometric analysis (IGA) has been successful at bringing two different research fields together, i.e. Computer Aided Design (CAD) and numerical analysis, T-spline IGA is applied in this work to frictionless contact and mode-I debonding problems between deformable bodies in the context of large deformations. Based on the concept of IGA, the smooth basis functions are adopted to describe surface geometries and approximate the numerical solutions, leading to higher accuracy in the contact integral evaluation. The isogeometric discretizations are here incorporated into an existing finite element framework by using Bézier extraction, i.e. a linear operator which maps the Bernstein polynomial basis on Bézier elements to the global isogeometric basis. A recently released commercial T-spline plugin for Rhino is herein used to build the analysis models adopted in this study.In such context, the continuum is discretized with cubic T-splines, as well as with Non Uniform Rational B-Splines (NURBS) and Lagrange polynomial elements for comparison purposes, and a Gauss-point-to-surface (GPTS) formulation is combined with the penalty method to treat the contact constraints. The purely geometric enforcement of the non-penetration condition in compression is generalized to encompass both contact and mode-I debonding of interfaces which is approached by means of cohesive zone (CZ) modeling, as commonly done by the scientific community to analyse the progressive damage of materials and interfaces. Based on these models, non-linear relationships between tractions and relative displacements are assumed. These relationships dictate both the work of separation per unit fracture surface and the peak stress that has to be reached for the crack formation. In the generalized GPTS formulation an automatic switching procedure is used to choose between cohesive and contact models, depending on the contact status. Some numerical results are first presented and compared in 2D for varying resolutions of the contact and/or cohesive zone, including frictionless sliding and cohesive debonding, all featuring the competitive accuracy and performance of T-spline IGA. The superior accuracy of T-spline interpolations with respect to NURBS and Lagrange interpolations for a given number of degrees of freedom (Dofs) is always verified. The isogeometric formulation is also extended to 3D bodies, where some examples in large deformations based on T-spline discretizations show an high smoothness of the reaction history curves

Advanced Computational Engineering

The finite element method is the established simulation tool for the numerical solution of partial differential equations in many engineering problems with many mathematical developments such as mixed finite element methods (FEMs) and other nonstandard FEMs like least-squares, nonconforming, and discontinuous Galerkin (dG) FEMs. Various aspects on this plus related topics ranging from order-reduction methods to isogeometric analysis has been discussed amongst the pariticpants form mathematics and engineering for a large range of applications

Isogeometric analysis: an overview and computer implementation aspects

Isogeometric analysis (IGA) represents a recently developed technology in computational mechanics that offers the possibility of integrating methods for analysis and Computer Aided Design (CAD) into a single, unified process. The implications to practical engineering design scenarios are profound, since the time taken from design to analysis is greatly reduced, leading to dramatic gains in efficiency. The tight coupling of CAD and analysis within IGA requires knowledge from both fields and it is one of the goals of the present paper to outline much of the commonly used notation. In this manuscript, through a clear and simple Matlab implementation, we present an introduction to IGA applied to the Finite Element (FE) method and related computer implementation aspects. Furthermore, implemen- tation of the extended IGA which incorporates enrichment functions through the partition of unity method (PUM) is also presented, where several examples for both two-dimensional and three-dimensional fracture are illustrated. The open source Matlab code which accompanies the present paper can be applied to one, two and three-dimensional problems for linear elasticity, linear elastic fracture mechanics, structural mechanics (beams/plates/shells including large displacements and rotations) and Poisson problems with or without enrichment. The Bezier extraction concept that allows FE analysis to be performed efficiently on T-spline geometries is also incorporated. The article includes a summary of recent trends and developments within the field of IGA

An interactive geometry modeling and parametric design platform for isogeometric analysis

In this paper an interactive parametric design-through-analysis platform is proposed to help design engineers and analysts make more effective use of Isogeometric Analysis (IGA) to improve their product design and performance. We develop several Rhinoceros (Rhino) plug-ins to take input design parameters through a user-friendly interface, generate appropriate surface and/or volumetric models, perform mechanical analysis, and visualize the solution fields, all within the same Computer-Aided Design (CAD) program. As part of this effort we propose and implement graphical generative algorithms for IGA model creation and visualization based on Grasshopper, a visual programming interface to Rhino. The developed platform is demonstrated on two structural mechanics examples—an actual wind turbine blade and a model of an integrally bladed rotor (IBR). In the latter example we demonstrate how the Rhino functionality may be utilized to create conforming volumetric models for IGA