Injective hierarchical free-form deformations using THB-splines

The free-form deformation (FFD) method deforms geometry in n-dimensional space by employing an n-variate function to deform (parts of) the ambient space. The original method pioneered by Sederberg and Parry in 1986 uses trivariate tensor-product Bernstein polynomials in R3 and is controlled as a Bézier volume. We propose an extension based on truncated hierarchical B-splines (THB-splines). This offers hierarchical and local refinability, an efficient implementation due to reduced supports of THB-splines, and intuitive control point hiding during FFD interaction. Additionally, we address the issue of fold-overs by efficiently checking the injectivity of the hierarchical deformation in real-time

Optimal analysis-aware parameterization of computational domain in 3D isogeometric analysis

International audienceIn isogeometric analysis framework, computational domain is exactly described using the same representation as that employed in the CAD process. For a CAD object, we can construct various computational domain with same shape but with different parameterization. One basic requirement is that the resulting parameterization should have no self-intersections. In this paper, a linear and easy-to-check sufficient condition for injectivity of trivariate B-spline parameterization is proposed. By an example of 3D thermal conduction problem, we show that different parameterization of computational domain has different impact on the simulation result and efficiency in isogeometric analysis. For problems with exact solutions, we propose a shape optimization method to obtain optimal parameterization of computational domain. The proposed injective condition is used to check the injectivity of initial trivariate B-spline parameterization constructed by discrete Coons volume method, which is the generalization of discrete Coons patch method. Several examples and comparisons are presented to show the effectiveness of the proposed method. Compared with the initial parameterization during refinement, the optimal parameterization can achieve the same accuracy but with less degrees of freedom

Parametrization of computational domain in isogeometric analysis: methods and comparison

International audienceParameterization of computational domain plays an important role in isogeometric analysis as mesh generation in finite element analysis. In this paper, we investigate this problem in the 2D case, i.e, how to parametrize the computational domains by planar B-spline surface from the given CAD objects (four boundary planar B-spline curves). Firstly, two kinds of sufficient conditions for injective B-spline parameterization are derived with respect to the control points. Then we show how to find good parameterization of computational domain by solving a constraint optimization problem, in which the constraint condition is the injectivity sufficient conditions of planar B-spline parametrization, and the optimization term is the minimization of quadratic energy functions related to the first and second derivatives of planar B-spline parameterization. By using this method, the resulted parameterization has no self-intersections, and the isoparametric net has good uniformity and orthogonality. After introducing a posteriori error estimation for isogeometric analysis, we propose $r$-refinement method to optimize the parameterization by repositioning the inner control points such that the estimated error is minimized. Several examples are tested on isogeometric heat conduction problem to show the effectiveness of the proposed methods and the impact of the parameterization on the quality of the approximation solution. Comparison examples with known exact solutions are also presented

Axial deformation with controllable local coordinate frames.

Chow, Yuk Pui.Thesis (M.Phil.)--Chinese University of Hong Kong, 2010.Includes bibliographical references (leaves 83-87).Abstracts in English and Chinese.Chapter 1. --- Introduction --- p.13-16Chapter 1.1. --- Motivation --- p.13Chapter 1.2 --- Objectives --- p.14-15Chapter 1.3 --- Thesis Organization --- p.16Chapter 2. --- Related Works --- p.17-24Chapter 2.1 --- Axial and the Free Form Deformation --- p.17Chapter 2.1.1 --- The Free-Form Deformation --- p.18Chapter 2.1.2 --- The Lattice-based Representation --- p.18Chapter 2.1.3 --- The Axial Deformation --- p.19-20Chapter 2.1.4 --- Curve Pair-based Representation --- p.21-22Chapter 2.2 --- Self Intersection Detection --- p.23-24Chapter 3. --- Axial Deformation with Controllable LCFs --- p.25-46Chapter 3.1 --- Related Methods --- p.25Chapter 3.2 --- Axial Space --- p.26-27Chapter 3.3 --- Definition of Local Coordinate Frame --- p.28-29Chapter 3.4 --- Constructing Axial Curve with LCFs --- p.30Chapter 3.5 --- Point Projection Method --- p.31-32Chapter 3.5.1 --- Optimum Reference Axial Curve Point --- p.33Chapter 3.6 --- Advantages using LCFs in Axial Deformation --- p.34Chapter 3.6.1 --- Deformation with Smooth Interpolated LCFs --- p.34-37Chapter 3.6.2 --- Used in Closed-curve Deformation --- p.38-39Chapter 3.6.3 --- Hierarchy of Axial Curve --- p.40Chapter 3.6.4 --- Applications in Soft Object Deformation --- p.41Chapter 3.7 --- Experiments and Results --- p.42-46Chapter 4. --- Self Intersection Detection of Axial Curve with LCFs --- p.47-76Chapter 4.1 --- Related Works --- p.48-49Chapter 4.2 --- Algorithms for Solving Self-intersection Problem with a set of LCFs --- p.50-51Chapter 4.2.1 --- The Intersection of Two Plane --- p.52Chapter 4.2.1.1 --- Constructing the Normal Plane --- p.53-54Chapter 4.2.1.2 --- A Line Formed by Two Planes Intersection --- p.55-57Chapter 4.2.1.3 --- Problems --- p.58Chapter 4.2.1.4 --- Sphere as Constraint --- p.59-60Chapter 4.2.1.5 --- Intersecting Line between Two Circular Discs --- p.61Chapter 4.2.2 --- Distance between a Mesh Vertex and a Curve Point --- p.62-63Chapter 4.2.2.1 --- Possible Cases of a Line and a Circle --- p.64-66Chapter 4.3 --- Definition Proof --- p.67Chapter 4.3.1 --- Define the Meaning of Self-intersection --- p.67Chapter 4.3.2 --- Cross Product of Two Vectors --- p.68Chapter 4.4 --- Factors Affecting the Accuracy of the Algorithm --- p.69Chapter 4.3.1 --- High Curvature of the Axial Curve --- p.69-70Chapter 4.3.2 --- Mesh Density of an Object. --- p.71-73Chapter 4.5 --- Architecture of the Self Intersection Algorithm --- p.74Chapter 4.6 --- Experimental Results --- p.75- 79Chapter 5. --- Conclusions and Future Development --- p.80-82Chapter 5.1 --- Contribution and Conclusions --- p.80-81Chapter 5.2 --- Limitations and Future Developments --- p.82References --- p.83-8

A Survey of Spatial Deformation from a User-Centered Perspective

The spatial deformation methods are a family of modeling and animation techniques for indirectly reshaping an object by warping the surrounding space, with results that are similar to molding a highly malleable substance. They have the virtue of being computationally efficient (and hence interactive) and applicable to a variety of object representations. In this paper we survey the state of the art in spatial deformation. Since manipulating ambient space directly is infeasible, deformations are controlled by tools of varying dimension - points, curves, surfaces and volumes - and it is on this basis that we classify them. Unlike previous surveys that concentrate on providing a single underlying mathematical formalism, we use the user-centered criteria of versatility, ease of use, efficiency and correctness to compare techniques

Virtual prototyping with surface reconstruction and freeform geometric modeling using level-set method

More and more products with complex geometries are being designed and manufactured by computer aided design (CAD) and rapid prototyping (RP) technologies. Freeform surface is a geometrical feature widely used in modern products like car bodies, airfoils and turbine blades as well as in aesthetic artifacts. How to efficiently design and generate digital prototypes with freeform surfaces is an important issue in CAD. This paper presents the development of a Virtual Sculpting system and addresses the issues of surface reconstruction from dexel data structures and freeform geometric modeling using the level-set method from distance field structure. Our virtual sculpting method is based on the metaphor of carving a solid block into a 3D freeform object using a 3D haptic input device integrated with the computer visualization. This dissertation presents the result of the study and consists primarily of four papers --Abstract, page iv

SHAPE DEFORMATION FOR OBJECTS OF GREATLY DISSIMILAR SHAPES WITH SMOOTH MANIFOLD

Ph.DDOCTOR OF PHILOSOPH

Shape deformations based on vector fields

This thesis explores applications of vector field processing to shape deformations. We present a novel method to construct divergence-free vector fields which are used to deform shapes by vector field integration (Chapter 2). The resulting deformation is volume-preserving and no self-intersections occur. We add more controllability to this approach by introducing implicit boundaries (Chapter 3), a shape editing method which resembles the well-known boundary constraint modeling metaphor. While the vector fields are originally defined in space, we also present a surface-based version of this approach which allows for more exact boundary selection and deformation control (Chapter 4). We show that vectorfield- based shape deformations can be used to animate elastic motions without complex physical simulations (Chapter 5). We also introduce an alternative approach to exactly preserve the volume of skinned triangle meshes (Chapter 6). This is accomplished by constructing a displacement field on the mesh surface which restores the original volume after deformation. Finally, we demonstrate that shape deformation by vector field integration can also be used to visualize smoke-like streak surfaces in dynamic flow fields (Chapter 7).In dieser Dissertation werden verschiedene Anwendungen der Vektorfeldverarbeitung im Bereich Objektdeformation untersucht. Wir präsentieren eine neuartige Methode zur Konstruktion von divergenzfreien Vektorfeldern, welche mittels Integration zum Deformieren von Objekten verwendet werden (Kapitel 2). Die so entstehende Deformation ist volumenerhaltend und keine Selbstüberschneidungen treten auf. Inspiriert von etablierten, auf Randbedingungen beruhenden Methoden, erweitern wir diese Idee hinsichtlich Kontrollierbarkeit mittels impliziten Abgrenzungen (Kapitel 3). Während die ursprüngliche Konstruktion im Raum definiert ist, präsentieren wir auch eine oberflächenbasierte Version, welche ein genaueres Festlegen der Abgrenzungen und bessere Kontrolle ermöglicht (Kapitel 4). Wir zeigen, dass vektorfeldbasierte Deformationen auch zur Animation von elastischen Bewegungen benutzt werden können, ohne dass komplexe Simulationen nötig sind (Kapitel 5). Des weiteren zeigen wir eine alternative Möglichkeit, mit der man das Volumen von Dreiecksnetzen erhalten kann, welche mittels Skelett-Animation deformiert werden (Kapitel 6). Dies erreichen wir durch ein Deformationsfeld auf der Oberfläche, das das ursprüngliche Volumen wieder hergestellt. Wir zeigen außerdem, dass Deformierungen mittels Vektorfeld-Integration auch zur Visualisierung von Rauch in dynamischen Flüssen genutzt werden können(Kapitel 7)

Reduced Models for Optimal Control, Shape Optimization and Inverse Problems in Haemodynamics

The objective of this thesis is to develop reduced models for the numerical solution of optimal control, shape optimization and inverse problems. In all these cases suitable functionals of state variables have to be minimized. State variables are solutions of a partial differential equation (PDE), representing a constraint for the minimization problem. The solution of these problems induce large computational costs due to the numerical discretization of PDEs and to iterative procedures usually required by numerical optimization (many-query context). In order to reduce the computational complexity, we take advantage of the reduced basis (RB) approximation for parametrized PDEs, once the state problem has been reformulated in parametrized form. This method enables a rapid and reliable approximation of parametrized PDEs by constructing low-dimensional, problem-specific approximation spaces. In case of PDEs defined over domains of variable shapes (e.g. in shape optimization problems) we need to introduce suitable, low-dimensional shape parametrization techniques in order to tackle the geometrical complexity. Free-Form Deformations and Radial-Basis Functions techniques have been analyzed and successfully applied with this aim. We analyze the reduced framework built by coupling these tools and apply it to the solution of optimal control and shape optimization problems. Robust optimization problems under uncertain conditions are also taken into consideration. Moreover, both deterministic and Bayesian frameworks are set in order to tackle inverse identification problems. As state equations, we consider steady viscous flow problems described by Stokes or Navier-Stokes equations, for which we provide a detailed analysis and construction of RB approximation and a posteriori error estimation. Several numerical test cases are also illustrated to show efficacy and reliability of RB approximations. We exploit this general reduced framework to solve some optimization and inverse problems arising in haemodynamics. More specifically, we focus on the optimal design of cardiovascular prostheses, such as bypass grafts, and on inverse identification of pathological conditions or flow/shape features in realistic parametrized geometries, such as carotid artery bifurcations