The Flex Representation Method (FRM) leverages unique computational advantages of splines to address limitations in the process of building CAE simulation models from CAD geometric models. Central to the approach is the envelope CAD domain that encapsulates a CAD model. An envelope CAD domain can be of arbitrary topological and geometric complexity. Envelope domains are constructed from spline representations, like U-splines, that are analysis-suitable. The envelope CAD domain can be used to approximate none, some, or all of the features in a CAD model. This yields additional simulation modeling options that simplify the model-building process while leveraging the properties of splines to control the accuracy and robustness of computed solutions. Modern integration techniques are adapted to envelope domains to maintain accurate solutions regardless of the CAD envelope chosen. The potential of the method is illustrated through several carefully selected benchmark problems.

Constructing spline models for isogeometric analysis is important in integrating design and analysis. Converting designed CAD (Computer Aided Design) models with B-reps to analysis-suitable volumetric T-spline is fundamental for the integration. In this thesis, we work on two directions to achieve this: (a) using Boolean operations and skeletons to build polycubes for feature-preserving high-genus volumetric T-spline construction; and (b) developing weighted T-splines with arbitrary degree for T-spline surface and volume modeling which can be used for analysis. In this thesis, we first develop novel algorithms to build feature-preserving polycubes for volumetric T-spline construction. Then a new type of T-spline named the weighted T-spline with arbitrary degree is defined. It is further used in converting CAD models to analysis-suitable volumetric T-splines. An algorithm is first developed to use Boolean operations in CSG (Constructive Solid Geometry) to generate polycubes robustly, then the polycubes are used to generate volumetric rational solid T-splines. By solving a harmonic field with proper boundary conditions, the input surface is automatically decomposed into regions that are classified into topologically either a cube or a torus. Two Boolean operations, union and difference, are performed with the primitives and polycubes are generated by parametric mapping. With polycubes, octree subdivision is carried out to obtain a volumetric T-mesh. The obtained T-spline surface is C2-continuous everywhere except the local region surrounding irregular nodes, where the surface continuity is elevated from C0 to G1. B´ezier elements are extracted from the constructed solid T-spline models, which are further used in isogeometric analysis. The Boolean operations preserve the topology of the models inherited from design and can generate volumetric T-spline models with better quality. Furthermore, another algorithm is developed which uses skeleton as a guidance to the polycube construction. From the skeleton of the input model, initial cubes in the interior are first constructed. By projecting corners of interior cubes onto the surface and generating a new layer of boundary cubes, the entire interior domain is split into different cubic regions. With the splitting result, octree subdivision is performed to obtain T-spline control mesh or T-mesh. Surface features are classified into three groups: open curves, closed curves and singularity features. For features without introducing new singularities like open or closed curves, we preserve them by aligning to the parametric lines during subdivision, performing volumetric parameterization from frame field, or modifying the skeleton. For features introducing new singularities, we design templates to handle them. With a valid T-mesh, we calculate rational trivariate T-splines and extract B´ezier elements for isogeometric analysis. Weighted T-spline basis functions are designed to satisfy partition of unity and linear independence. The weighted T-spline is proved to be analysis-suitable. Compared to standard T-splines, weighted T-splines have less geometrical constraint and can decrease the number of control points significantly. Trimmed NURBS surfaces of CAD models are reparameterized with weighted T-splines by a new edge interval extension algorithm, with bounded surface error introduced. With knot interval duplication, weighted T-splines are used to deal with extraordinary nodes. With B´ezier coefficient optimization, the surface continuity is elevated from C0 to G1 for the one-ring neighborhood elements. Parametric mapping and sweeping methods are developed to construct volumetric weighted T-splines for isogeometric analysis. Finally, we develop an algorithm to construct arbitrary degree T-splines. The difference between odd degree and even degree T-splines are studied in detail. The methods to extract knot intervals, calculate new weights to handle extraordinary nodes, and extract B´ezier elements for analysis are investigated with arbitrary degrees. Hybrid degree weighted Tspline is generated at designated region with basis functions of different degrees, for the purpose of performing local p-refinement. We also study the convergence rate for T-spline models of different degrees, showing that hybrid degree weighted T-splines have better performance after p-refinement. In summary, we develop novel methods to construct volumetric T-splines based on polycube and sweeping methods. Arbitrary degree weighted T-spline is proposed, with proved analysis-suitable properties. Weighted T-spline basis functions are used to reparameterize trimmed NURBS surfaces, handling extraordinary nodes, based on which surface and volumetric weighted T-spline models are constructed for isogeometric analysis.

To date, efforts to model fracture and crack propagation have focused on two broad approaches: discrete and continuum damage descriptions. The discrete approach incorporates a discontinuity into the displacement field that must be tracked and updated. Examples of this approach include XFEM, element deletion, and cohesive zone models. The continuum damage, or smeared crack, approach incorporates a damage parameter into the model that controls the strength of the material. An advantage of this approach is that it does not require interface tracking since the damage parameter varies continuously over the domain. An alternative approach is to use a phase-field to describe crack propagation. In the phase-field approach to modeling fracture the problem is reformulated in terms of a coupled system of partial differential equations. A continuous scalar-valued phase-field is introduced into the model to indicate whether the material is in the unfractured or fractured ''phase''. The evolution of the phase-field is governed by a partial differential equation that includes a driving force that is a function of the strain energy of the body in question. This leads to a coupling between the momentum equation and the phase-field equation. The phase-field model also includes a length scale parameter that controls the width of the smooth approximation to the discrete crack. This allows discrete cracks to be modeled down to any desired length scale. Thus, this approach incorporates the strengths of both the discrete and continuum damage models, i.e., accurate modeling of individual cracks with no interface tracking. The research presented in this dissertation focuses on developing phase-field models for dynamic fracture. A general formulation in terms of the usual balance laws supplemented by a microforce balance law governing the evolution of the phase-field is derived. From this formulation, small-strain brittle and large-deformation ductile models are then derived. Additionally, a fourth-order theory for the phase-field approximation of the crack path is postulated. Convergence and approximation results are obtained for the proposed theories. In this work, isogeometric analysis, and particularly T-splines, plays an important role by providing a smooth basis that allows local refinement. Several numerical simulations have been performed to evaluate the proposed theories. These results show that phase-field models are a powerful tool for predicting fracture. / text

The Reproducing Kernel Element Method (RKEM) implementation of the Fleck-Hutchinson phenomenological strain gradient theory in 1D, 2D and 3D is implemented in this research. Fleck-Hutchinson theory fits within the framework of Touplin- Mindlin theories and deals with first order strain gradients and associated work conjugate higher-order stress. Theories at the intrinsic or material length scales find applications in size dependent phenomena. In elasticity, length scale enters the constitutive equation through the elastic strain energy function which depends on both strain as well as the gradient of rotation and stress. The displacement formulation of the Touplin Mindlin theory involve diffrential equations of the fourth order, in conventional finite element method C1 elements are required to solve such equations, however C1 elements are cumbersome in 2D and unknown in 3D. The high computational cost and large number of degrees of freedom soon place such a formulation beyond the realm of practicality. Recently, some mixed and hybrid formulations have developed which require only C0 continuity but none of these elements solve complicated geometry problems in 2D and there is no problem yet solved in 3D. The large number of degrees of freedom is still inevitable for these formulation. As has been demonstrated earlier RKEM has the potential to solve higher-order problems, due to its global smoothness and interpolation properties. This method has the promise to solve important problems formulated with higher order derivatives, such as the strain gradient theory.

To simulate increasingly complex physical phenomena and systems, tightly integrated design-through-analysis (DTA) tools are essential. In this dissertation, the complementary strengths of isogeometric analysis and T-splines are coupled and enhanced to create a seamless DTA framework. In all cases, the technology de- veloped meets the demands of both design and analysis. In isogeometric analysis, the smooth geometric basis is used as the basis for analysis. It has been demonstrated that smoothness offers important computational advantages over standard finite elements. T-splines are a superior alternative to NURBS, the current geometry standard in computer-aided design systems. T-splines can be locally refined and can represent complicated designs as a single watertight geometry. These properties make T-splines an ideal discretization technology for isogeometric analysis and, on a higher level, a foundation upon which unified DTA technologies can be built. We characterize analysis-suitable T-splines and develop corresponding finite element technology, including the appropriate treatment of extraordinary points (i.e., unstructured meshing). Analysis-suitable T-splines form a practically useful subset of T-splines. They maintain the design flexibility of T-splines, including an efficient and highly localized refinement capability, while preserving the important analysis-suitable mathematical properties of the NURBS basis. We identify Bézier extraction as a unifying paradigm underlying all isogeometric element technology. Bézier extraction provides a finite element representation of NURBS or T-splines, and facilitates the incorporation of T-splines into existing finite element programs. Only the shape function subroutine needs to be modified. Additionally, Bézier extraction is automatic and can be applied to any T-spline regardless of topological complexity or polynomial degree. In particular, it represents an elegant treatment of T-junctions, referred to as "hanging nodes" in finite element analysis We then detail a highly localized analysis-suitable h-refinement algorithm. This algorithm introduces a minimal number of superfluous control points and preserves the properties of an analysis-suitable space. Importantly, our local refinement algorithm does not introduce a complex hierarchy of meshes. In other words, all local refinement is done on one control mesh on a single hierarchical “level” and all control points have similar influence on the shape of the surface. This feature is critical for its adoption and usefulness as a design tool. Finally, we explore the behavior of T-splines in finite element analysis. It is demonstrated that T-splines possess similar convergence properties to NURBS with far fewer degrees of freedom. We develop an adaptive isogeometric analysis framework which couples analysis-suitable T-splines, local refinement, and Bézier extraction and apply it to the modeling of damage and fracture processes. These examples demonstrate the feasibility of applying T-spline element technology to very large problems in two and three dimensions and parallel implementations. / text

This paper was presented in WCSMO-9, Shizuoka.

La méthode isogéométrique (IGA) récemment proposée en tant que méthode numérique générique offre de réelles perspectives dans l’unification des modèles géométriques et computationnel. La méthode isogéométrique est intiment liée à la méthode des éléments finis (FEM) étant donné que la méthode est basée sur le même cadre variationnel. Cette méthode a montré dans de nombreuses circonstances de très bonne qualités numériques notamment avec des maillages grossiers (précision numérique, capacité à supporter de grandes déformations…). Notre objectif final dans ce travail est de fournir un environnement de base, numérique et logiciel, pour la simulation de problèmes à champs et physiques multiples pour des pièces élastomériques de type industriel. Dans ce contexte, les points numériques à développer pour l’IGA sont le traitement de l’incompressibilité et le caractère multi-champs du problème thermique dans la formulation de Galerkin. Ainsi dans ce travail nous proposons en premier, un paradigme objet de l’IGA intégré au sein d’une architecture orientée objet en Java, initialement con?ue pour résoudre des problèmes multi-champs couplés en transformations finies. L’approche proposée s’appuie pleinement sur le contexte variationnel existant dans le code dans le cadre des éléments finis pour réduire les développements pour MEF et IGA (une formulation développée en IGA tourne en MEF et vice versa). Dans un second temps, nous avons étudié le problème de l’incompressibilité pour notamment réduire le verrouillage numérique existant toujours sur l’IGA standard. Par un souci de simplicité, nous adoptons des formulations mixtes à 2 champs (déplacement/pression). Afin d’essayer de satisfaire la condition inf-sup en relachant la contrainte sur le déplacement, nous avons développé deux idées de la littérature (naturelle en NURBS) qui consiste à soit dupliquer une fois les n?uds intérieurs du patch des déplacements ou subdiviser les éléments du patch des déplacements. Nous avons étendu ce type d’éléments aux transformations finies. Enfin, et de manière originale, nous avons adopté la même stratégie pour les problèmes à 2-champs pour la thermomécanique. Différentes simulations à petites et grandes déformations confirment le potentiel de l’approche. Enfin, nous évaluons l’ensemble sur un modèle quasi-incompressible thermo-visco-élastique de type Zener sur des éprouvettes classiques dans un contexte physique complexe. / Recently proposed as a general purpose numerical method, the Isogeometric Analysis (IGA) offers great perspective to bridge the gap between CAD and CAE. The IGA is closely related to the finite element method (FEM) as the method is based on the same variational framework. Moreover, this method has shown in many circumstances to be have a better accuracy than the FEM (large mesh distortions…). Our final aim in this work is to simulate complex multiphysics problems for elastomers industrial parts. As matter of fact, the two main numerical issues in this context is the incompressibility/quasi-incompressibility of the material and the thermochemical coupling in Galerkin formulations. First, we propose, a programming paradigm of the IGA in an existing Java object-oriented hierarchy initially designed for solving multi-fields coupled problems at finite strains. We develop an approach that fully take benefit of the original architecture to reduce developments for both FEM and IGA (one problem developed in FEM can be run in IGA and vice versa). Second, we investigate volumetric locking issues persisting for low order NURBS element observed with standard displacement formulation as finite elements. To cure the problem, we adopt two-fields mixed formulation (displacement/pressure) for the sake of simplicity and target at assessing different discretizations in stability (inf-sup condition). The basic idea is to first to increase the internal knot’s multiplicity or to subdivide the patch for displacements. These ideas that are directly inspired from patches properties, have been found in the literature for the Stokes problem and extended to large strain in solid mechanics. The comparison between the two-fields mixed formulation and a strain projection method is lead at small and large strains. At last, we originally adopt a similar strategy for thermomechanical problem at small and large strains. In the context two-fields formulation, displacement/temperature, the LBB stability condition must be fulfilled to guaranty stability. Thus, we investigate the choices of patches for two-fields formulation displacement/temperature fields for IGA applied to thermoelasticity. Several numerical results for thermomechanical problems at small and finite strains, linear and nonlinear have been presented. At last, an incompressible viscous thermo-hyperelastic model is evaluated in the IGA framework with the proposed approach.

Isogeometric analysis (IGA) is a numerical method for solving partial differential equations (PDE). In this master thesis we explain a concept of IGA with special emphasis on problems on closed domains created by a single NURBS patch. For them we show a process how to modify the NURBS basis to ensure the highest possible continuity of the function space. Then we solve the minimal surface problem using two different Newton type methods. The first one is based on the classical approach using PDE, in the second one we use unique advantages of IGA to directly minimize the area functional.

The objective of this work is to develop a novel 3-D biological particulate dynamics framework to simulate blood flow in the micro circulation. This entails the amalgamation of concepts from various fields namely blood flow dynamics, solid mechanics, fluid-structure interaction and computational data structures. It is envisioned that this project will serve as a harbinger for implementing a multi-scale simulation model with applications in a vast array of situations from blood flows in heart valves to studying cancer metastasis. The primary motivation for this work stems from the need for establishing a simple, effective and holistic framework for performing blood flow simulations, taking into account the extremely 3-D nature of flow, the particle interactions and fluid structure interaction between blood and its constituent elements. Many current models to simulate blood cells rely on finite element methods which render large scale simulations extremely computationally intensive. The development of a framework for simulating blood flow is tied together with achieving a framework for performing an investigation of cancer metastasis. Cancer initially develops at a primary site and spreads through the body to secondary sites using the circulatory systems of the body - the blood circulatory system and the lymphatic system. It is known that all the cancer cells that enter into the circulation do not survive the harsh environment, though the exact cause of this is still undetermined. Moreover, the mechanical properties of cancer cells are not well documented and appropriate computational models require that experiments be conducted to determine the same. Thus the end goal of this work is to establish a system to analyze and simulate 3-D blood particulate dynamics, including cancer cells, from a holistic standpoint in order to understand more about the phenomenon of blood flow as a whole, and cancer metastasis in particular.

The discretization of time-dependent wave propagation is plagued with dispersion in which the wavefield is perceived to travel with an erroneous velocity. To remediate the problem, simulations are run on dense and computationally expensive grids yielding plausible approximate solutions. This work introduces an error analysis tool which can be used to obtain optimal simulation parameters that account for mesh size, orders of spatial and temporal discretizations, angles of propagation, temporal stability conditions (usually referred to as CFL conditions), and time of propagation. The classical criteria of 10-15 nodes per wavelength for second-order finite differences, and 4-5 nodes per wavelength for fourth-order spectral elements are shown to be unrealistic and overly-optimistic simulation parameters for different propagation times. This work analyzes finite differences, spectral elements, optimally-blended spectral elements, and isogeometric analysis.