Freeform curve design has existed in various forms for at least two millennia, and is important throughout computer-aided design and manufacture. With the increasing importance of animation and robotics, coupled with the increasing power of computers, there is now interest in freeform motion design, which, in part, extends techniques from curve design, as well as introducing some entirely distinct challenges. There are several approaches to freeform motion construction, and the first step in designing freeform motions is to choose a representation. Unlike for curves, there is no "standard" way of representing freeform motions, and the different tools available each have different properties. A motion can be viewed as a continuously-varying pose, where a pose is a position and an orientation. This immediately presents a problem; the dimensions of rotations and translations are different, and it is not clear how the two can be compared, such as to define distance along a motion. One solution is to treat the rotational and translational components of a motion separately, but this is inelegant and clumsy. The philosophy of this thesis is that a motion is not defined purely by rotations and translations, but that the body following a motion is a part of that motion. Specifically, the part of the body that is accounted for is its inertia tensor. The significance of the inertia tensor is that it allows the rotational and translational parts of a motion to be, in some sense, compared in a dimensionally- consistent way. Using the inertia tensor, this thesis finds the form of kinetic energy in <;1'4, and also discusses extensions of the concepts of arc length and curvature to the space of motions, allowing techniques from curve fairing to be applied to motion fairing. Two measures of motion fairness are constructed, and motion fairing is the process of minimizing the measure of a motion by adjusting degrees of freedom present in the motion's construction. This thesis uses the geometric algebra <;1'4 in the generation offreeform motions, and the fairing of such motions. <;1'4 is chosen for its particular elegance in representing rigid-body transforms, coupled with an equivalence relation between elements representing transforms more general than for ordinary homogeneous coordinates. The properties of the algebra germane to freeform motion design and improvement are given, and two distinct frameworks for freeform motion construction and modification are studied in detail.

Die Geometrie eines Druckbehälters wird unter Zuhilfenahme parametrischer Bezierkurven und durch die Integration einer externen Mathcad-Analyse in Creo hinsichtlich einer harmonischen, meridianen Spannungsverteilung optimiert.

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

[Truncated abstract] This thesis is concerned with geometrically-defined curves that can be used for interpolation in Riemannian or, more generally, semi-Riemannian manifolds. As in much of the existing literature on such curves, emphasis is placed on manifolds which are important in computer graphics and engineering applications, namely the unit 3-sphere S3 and the closely related rotation group SO(3), as well as other Lie groups and spheres of arbitrary dimension. All geometrically-defined curves investigated in the thesis are either higher order variational curves, namely critical points of cost functionals depending on (covariant) derivatives of order greater than 1, or defined by geometrical algorithms, namely generalisations to manifolds of algorithms from the field of computer aided geometric design. Such curves are needed, especially in the aforementioned applications, since interpolation methods based on applying techniques of classical approximation theory in coordinate charts often produce unnatural interpolants. However, mathematical properties of higher order variational curves and curves defined by geometrical algorithms are in need of substantial further investigation: higher order variational curves are solutions of complicated nonlinear differential equations whose properties are not well-understood; it is usually unclear how to impose endpoint derivative conditions on, or smoothly piece together, curves defined by geometrical algorithms. This thesis addresses these difficulties for several classes of curves. ... The geometrical algorithms investigated in this thesis are generalisations of the de Casteljau and Cox-de Boor algorithms, which define, respectively, polynomial B'ezier and piecewise-polynomial B-spline curves by dividing, in certain ratios and for a finite number of iterations, piecewise-linear control polygons corresponding to finite sequences of control points. We show how the control points of curves produced by the generalised de Casteljau algorithm in an (almost) arbitrary connected finite-dimensional Riemannian manifold M should be chosen in order to impose desired endpoint velocities and (covariant) accelerations and, thereby, piece the curves together in a C2 fashion. A special case of the latter construction simplifies when M is a symmetric space. For the generalised Cox-de Boor algorithm, we analyse in detail the failure of a fundamental property of B-spline curves, namely C2 continuity at (certain) knots, to carry over to M.

Free-form surfaces and implicit surfaces must be tessellated before being rendered with rasterization techniques. However ray tracing provides the means to directly render such objects without the need to first convert into polygonal meshes. Since ray tracing can handle triangle meshes as well, the question of which method is most suitable in terms of performance, quality and memory usage is addressed in this thesis. Bézier surfaces and NURBS surfaces along with basic algebraic implicit surfaces are implemented in order to test the performance relative to polygonal meshes approximating the same objects. The parametric surfaces are implemented using an iterative Newtonian method that converges on the point of intersection using a bounding volume hierarchy that stores the initial guesses. Research into intersecting rays with parametric surfaces is surveyed in order to find additional methods that speed up the computation. The implicit surfaces are implemented using common direct algebraic methods. All of the intersection tests are implemented using the Embree ray tracing API as well as a SIMD library in order to achieve interactive framerates on a CPU. The results show that both Bézier surfaces and NURBS surfaces can achieve interactive framerates on a CPU using SIMD computation, with Bézier surfaces coming close to the performance of polygonal counterparts. The implicit surfaces implemented outperform even the simplest polygonal approximations.

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Die Geometrie eines Druckbehälters wird unter Zuhilfenahme parametrischer Bezierkurven und durch die Integration einer externen Mathcad-Analyse in Creo hinsichtlich einer harmonischen, meridianen Spannungsverteilung optimiert.

Isogeometric analysis is aimed to mitigate the gap between Computer-Aided Design (CAD) and analysis by using a unified geometric representation. Thanks to the exact geometry representation and high smoothness of adopted basis functions, isogeometric analysis demonstrated excellent mathematical properties and successfully addressed a variety of problems. In particular, it allows to solve higher order Partial Differential Equations (PDEs) directly omitting the usage of mixed approaches. Unfortunately, complex CAD geometries are often constituted by multiple Non-Uniform Rational B-Splines (NURBS) patches and cannot be directly applied for finite element analysis.parIn this work, we presents a dual mortaring framework to couple adjacent patches for higher order PDEs. The development of this formulation is initiated over the simplest 4th order problem-biharmonic problem. In order to speed up the construction and preserve the sparsity of the coupled problem, we derive a dual mortar compatible C1 constraint and utilize the Bezier dual basis to discretize the Lagrange multipler spaces. We prove that this approach leads to a well-posed discrete problem and specify requirements to achieve optimal convergence. After identifying the cause of sub-optimality of Bezier dual basis, we develop an enrichment procedure to endow Bezier dual basis with adequate polynomial reproduction ability. The enrichment process is quadrature-free and independent of the mesh size. Hence, there is no need to take care of the conditioning. In addition, the built-in vertex modification yields compatible basis functions for multi-patch coupling.To extend the dual mortar approach to couple Kirchhoff-Love shell, we develop a dual mortar compatible constraint for Kirchhoff-Love shell based on the Rodrigues' rotation formula. This constraint provides a unified formulation for both smooth couplings and kinks. The enriched Bezier dual basis preserves the sparsity of the coupled Kirchhoff-Love shell formulation and yields accurate results for several benchmark problems.Like the dual mortaring formulation, locking problem can also be derived from the mixed formulation. Hence, we explore the potential of Bezier dual basis in alleviating transverse shear locking in Timoshenko beams and volumetric locking in nearly compressible linear elasticity. Interpreting the well-known B projection in two different ways we develop two formulations for locking problems in beams and nearly incompressible elastic solids. One formulation leads to a sparse symmetric symmetric system and the other leads to a sparse non-symmetric system.

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