This work is concerned with the use of isogeometric analysis based on Non- Uniform Rational B-Splines (NURBS) to develop efficient and robust numerical techniques to deal with the problems of incompressibility in the fields of solid and fluid mechanics. Towards this, two types of formulations, mixed Galerkin and least-squares, are studied. During the first phase of this work, mixed Galerkin formulations, in the context of isogeometric analysis, are presented. Two-field and three-field mixed variational formulations - in both small and large strains - are presented to obtain accurate numerical solutions for the problems modelled with nearly incompressible and elasto-plastic materials. The equivalence of the two mixed formulations, for the considered material models, is derived; and the computational advantages of using two-field formulations are illustrated. Performance of these formulations is assessed by studying several benchmark examples. The ability of the mixed methods, to accurately compute limit loads for problems involving elastoplastic material models; and to deal with volumetric locking, shear locking and severe mesh distortions in finite strains, is illustrated. Later, finite element formulations are developed by combining least-squares and isogeometric analysis in order to extract the best of both. Least-squares finite element methods (LSFEMs) based on the use of governing differential equations directly - without the need to reduce them to equivalent lower-order systems - are developed for compressible and nearly incompressible elasticity in both the small and finite strain regimes; and incompressible Navier-Stokes. The merits of using Gauss-Newton scheme instead of Newton-Raphson method to solve the underlying nonlinear equations are presented. The performance of the proposed LSFEMs is demonstrated with several benchmark examples from the literature. Advantages of using higher-order NURBS in obtaining optimal convergence rates for non-norm-equivalent LSFEMs; and the robustness of LSFEMs, for Navier-Stokes, in obtaining accurate numerical solutions without the need to incorporate any artificial stabilisation techniques, are demonstrated.

U-splines are a novel approach to the construction of a spline basis for representing smooth objects in Computer-Aided Design (CAD) and Computer-Aided Engineering (CAE). A spline is a piecewise-defined function that satisfies continuity constraints between adjacent cells in a mesh. U-splines differ from existing spline constructions, such as Non-Uniform Rational B-splines (NURBS), subdivision surfaces, T-splines, and hierarchical B-splines, in that they can accommodate local variation in cell size, polynomial degree, and smoothness simultaneously over more varied mesh configurations. Mixed cell types (e.g., triangle and tetrahedron and quadrilateral and hexahedral cells in the same mesh) and T-junctions are also supported. The U-spline construction is presented for curves, surfaces, and volumes with higher dimensional generalizations possible. A set of requirements are given to ensure that the U-spline basis is positive, forms a partition of unity, is complete, and is locally linearly independent.

This thesis aims to develop a design-oriented simulation approach for cloth analysis. Our approach is built on the framework of NURBS-based isogeometric analysis, which utilizes NURBS as the basis functions of analysis. NURBS is a class of parametric geometry to represent curves and surfaces in computer-aided design (CAD) programs. Recently, NURBS geometry has been used directly in analysis. The overall goal of this thesis is to develop a computation infrastructure that enables cloth analysis directly on NURBS geometry. The advantage of NURBS in the context of cloth modeling lies in the geometric smoothness. Using NURBS, it's easy to construct surfaces with C1 or higher order of continuity. Compared to C0 finite element geometry, the NURBS geometry is more effective in capturing wrinkles and folders of cloth, which are characteristics of cloth motion. The NURBS geometry enables the use of rotation-free Kirchhoff-Love shell. The rotation-free shell model not only saves freedoms, but also makes the contact/impact treatment much easier. The major contribution of this work is the development of a NURBS cloth modeling approach. The mechanical model of cloth and its implementation with NURBS geometry will be presented in detail. Proper constitutive laws are employed for fabric materials. Since NURBS geometry from CAD typically contains multiple patches and trimmed patches, a certain treatment is proposed so that geometry can be used directly in analysis. Another contribution of this work relates to a contact/impact algorithm. Contact problems in cloth simulation have been a bottleneck of continuum-based approach. Since the general contact method doesn't work well for cloth simulation, a special contact treatment is developed. The present contact model distinguishes three types of contact interactions. The first is the persistent contact force. This force in essence is the traditional penalty force, but is applied when a contact pair is within a separation tolerance instead of being penetrated. This essentially smears the abrupt contact reaction into a relatively smooth force defined only a small thickness. The second is trajectory impact, which deals with the reaction when impact occurs in a time step. The treatment ensures that a point stays on a same side of the surface it impacts on. The third is self-intersection. Intersection resorting force is introduced when the initial configuration has self-intersections, or when the trajectory impact force fails to eliminate all the collisions. We proposed a new method, the method of area minimization, to handle intersections. The contact models have been integrated into an operator-split integration algorithm. A notable feature of this integration is that the contact/impact response is singled out from the momentum equation. This work also proposes a continuum-based strain limiting scheme. Because the in-plane stiffness of cloth is much higher than the bending stiffness, numerical difficulty is encountered in either implicit or explicit time integration. The strain limiting is a numerical technique that formulates the in-plane response as a constraint problem to allow the use of lower in-plane stiffness. A number of examples are presented to show the performance of the proposed approach. In the wrinkling study, the simulated wrinkle pattern looks similar with the experimental results. In the contact study, it is found that the current method can accurately recover a constant contact pressure field (press patch test), can handle contacts of multi-layer folds and produce realistic draping effect. The intersection resolution method is illustrated to be robust to various kinds of intersections. The fast projection method can enlarge time steps while limiting the in-plane strain. The current method is also applied to the analysis of a soft armor. Beginning from CAD models the armor was put on the human body by a try-on simulation. In multi-layer models, the intersection resolution method is used to resolve the intersections between layers. Subsequently, cloth dynamics are simulated for different human motions. Mechanical indexes such as the extra torque caused by the armors, pressure force on the body, and stress in the armor are predicted. Parametric studies are performed to investigate the change in mechanical metrics under altered design parameters.

This dissertation discusses the thermomechanical analyses performed on threaded fasteners and curvilinearly stiffened composite panels with internal cutouts. The former problem was analyzed using a global/local approach using the commercial finite element software ANSYS while a fully functional code using isogeometric analysis was developed from scratch for the latter. For the threaded fasteners, a global simplified 3D model is built to evaluate the deformation of the structure. A second local model reproducing accurately the threads of the fasteners is used for the accurate assessment of the stresses in the vicinity of the fasteners. The isogeometric analysis code, capable of performing static, buckling and vibration analysis on stiffened composite plates with cutouts using single patch, multiple patches and level set methods is then discussed. A novel way to achieve displacement compatibility between the panel and stiffeners interfaces is introduced. An easy way of modeling plates with complicated cutouts by using edge curves and generating a ruled NURBS surface between them is described. Influence on the critical thermal buckling load and the fundamental mode of vibration due to the presence of circular, elliptical and complicated cutouts is also investigated. Results of parametric studies are presented which show the influence of ply orientation, size and orientation of the cutout, and the position and profile of the curvilinear stiffener. The numerical examples show high reliability and efficiency when compared with other published solutions and those obtained using ABAQUS, a commercial software. / PHD / Aircraft in flight are subjected to different loads due to maneuvers and gust; there external forces cause internal loads and depend on the location of the panel in the aircraft. The internal loads, may result in the buckling of the panel. Hence, there is a need for studying structural efficiency and develop strong and stiff lightweight structures. Stiffened composite panels is a technology capable of addressing these needs. However, when used in space vehicles moving at hypersonic speeds, such structures experience significant temperature rise in a very short time resulting from the aerodynamic heating due to friction between the vehicle surface and the atmosphere. Such phenomena is more prominent during reentry and launch processes. Hence, it is really important to consider thermal effects while designing and analyzing such structures. Composite stiffened panels have many advantages like small manufacturing cost, high stability, great energy absorption, superior damage tolerance etc. One of the main failure modes for stiffened composite panels is thermal buckling. An extensive literature review on thermal buckling of stiffened composite panels was conducted in this dissertation. Thermal buckling and vibration analysis as well as a parametric study of a stiffened composite panel with internal cutouts was conducted, and verified using ABAQUS, a Finite Element Software.

The main objective of this research is to present a robust numerical framework based upon Isogeometric analysis (IGA) for simulation of thermo-hydro-mechanical (THM) processes in variably saturated soils. The proposed platform employs the Bézier extraction operator to connect IGA to the conventional finite element analysis (FEA), allowing to take advantage of features offered by the two methods. In the first part, the formulation and numerical implementation for fully coupled numerical simulation of THM problems in saturated porous media are presented. The results are compared against analytical solutions and experimental tests available in the literature. In the second part, the proposed method is used to study the temperature effect on the hydro-mechanical response of sd supporting hydrocarbon pipelines, an aspect that has been overlooked in the majority of previous studies. The results highlight the need for considering nonisothermal behavior in different analysis and design stages of sd-buried pipelines. In the third part, the proposed IGA-FEA framework is extended to evaluate the nonisothermal elasto-plastic behavior of unsaturated soils. Drucker-Prager yield surface is used as criterion to limit the modified effective stress where the model follows small strain, quasi-static loading conditions. The framework is used to simulate strain localization of unsaturated dense sand subjected to undrained compression loading. In comparison with FEA, the present method smoothly distributes plastic strain over the adjacent elements. The parametric study highlights the importance of considering temperature effects in elasto-plastic analysis of unsaturated soils.

In contrast to the structural modes, which describe the physical motion of vibrating structures, acoustic radiation modes describe the radiated sound power. Radiation modes are beneficial in active noise control because reducing an efficiently radiating radiation mode guarantees the reduction of radiated sound power. Much work has been done to calculate the radiation modes for simple geometries, where analytic solutions are available. In this work, isogeometric analysis (IGA) is used to provide a tool capable of analyzing the radiation modes of arbitrarily complex geometries. IGA offers increased accuracy and efficiency by using basis functions generated from Non-Uniform Rational B-Splines (NURBS) or T-Splines, which can represent geometries exactly. Results showing this increased accuracy and efficiency with IGA using T-Splines are shown for a sphere to validate the method, comparing with the exact analytical solution as well as results from a traditional boundary element method. A free cylindrical shell is also analyzed to show the usefulness of this method. Expected similarities, as well as expected differences, are observed between this free shell and a baffled cylindrical shell.

The fundamental advantages of applying Isogeometric Analysis (IGA) to shell analysis have been extensively demonstrated across a wide range of problems and formulations. However, a phenomenon called numerical locking is still a major challenge in IGA shell analysis, which can lead to dramatically deteriorated analysis accuracy. Additionally, for complex thin-walled structures, a simple and robust coupling technique is desired to sew together models composed of multiple patches. This dissertation focuses on addressing these challenges of IGA shell analysis. First, an isogeometric dual mortar method is developed for multi-patch coupling. This method is based on Be ?zier extraction and projection and can be employed during the creation and editing of geometry through properly modified extraction operators. It is applicable to any spline space which has a representation in Be ?zier form. The error in the method can be adaptively controlled, in some cases recovering optimal higher-order rates of convergence, by leveraging the exact refineability of the proposed dual spline basis without introducing any additional degrees-of-freedom into the linear system. This method can be used not only for shell elements but also for heat transfer and solid elements, etc. Next, a mixed formulation for IGA shell analysis is proposed that addresses both shear and membrane locking and improves the quality of computed stresses. The starting point of the formulation is the modified 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 proposed dual spline bases. As a result, the strain variables can be condensed out of the system with only a slight increase in the bandwidth of the resulting linear system and the condensed approach preserves the accuracy of the non-condensed mixed approach but with fewer degrees-of-freedom. Finally, as an alternative, new quadrature rules are developed to release membrane and shear locking. These quadrature rules asymptotically only require one point for Reissner-Mindlin (RM) shell elements and two points for Kirchhoff-Love (KL) shell elements in B-spline and NURBS-based isogeometric shell analysis, independent of the polynomial order p of the elements. The quadrature points are Greville abscissae and the quadrature weights are calculated by solving a linear moment fitting problem in each parametric direction. These quadrature rules are free of spurious zero-energy modes and any spurious finite-energy modes in membrane stiffness can be easily stabilized by using a higher-order Greville rule.

The use of the finite element method within an optimization workflow is fraught with challenges that limit the automation of such workflows. These challenges are inherent to the traditional finite element formulations which are heavily dependent on a manual meshing process that introduces variability that is challenging to account for within an automated workflow. The recently developed flex representation method (FRM) provides a salient solution to the manual meshing process without sacrificing solution accuracy. In response to the development of FRM a global automotive company requested a study to explore the applicability of FRM to one of their sizing-optimization problems: the constrained optimization of a wheel undergoing a rigidity test. In this study we develop an optimization framework based on the DAKOTA optimization framework, the open-source FreeCAD computer-aided design software, and an implementation of FRM within the Coreform IGA solver. Within this framework we demonstrate in the affirmative that FRM enables a highly robust sizing-optimization workflow while requiring minimal effort to prepare the FRM model.

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