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

Construction of smooth maps with mean value coordinates

Bernstein polynomials are a classical tool in Computer Aided Design to create smooth maps with a high degree of local control. They are used for the construction of B\'ezier surfaces, free-form deformations, and many other applications. However, classical Bernstein polynomials are only defined for simplices and parallelepipeds. These can in general not directly capture the shape of arbitrary objects. Instead, a tessellation of the desired domain has to be done first. We construct smooth maps on arbitrary sets of polytopes such that the restriction to each of the polytopes is a Bernstein polynomial in mean value coordinates (or any other generalized barycentric coordinates). In particular, we show how smooth transitions between different domain polytopes can be ensured

Methods for constraint-based conceptual free-form surface design

Zusammenfassung Der constraint-basierte Entwurf von Freiformfl„chen ist eine m„chtige Methode im Computer gest�tzten Entwurf. Bekannte Realisierungen beschr„nken sich jedoch meist auf Interpolation von Rand- und isoparametrischen Kurven. In diesem Zusammenhang sind die sog. "Multi-patch" Methoden die am weitesten verbreitete Vorgehensweise. Hier versucht man Fl„chenverb„nde aus einem Netz von dreidimensionalen Kurven (oft gemischt mit unstrukturierten Punktewolken) derart zu generieren, dass die Kurven und Punkte von den Fl„chen interpoliert werden. Die Kurven werden als R„nder von rechteckigen oder dreieckigen bi-polynomialen oder polynomialen Fl„chen betrachtet. Unter dieser Einschr„nkung leidet die Flexibilit„t des Verfahrens. In dieser Dissertation schlagen wir vor, beliebige, d.h. auch nicht iso-parametrische, Kurven zu verwenden. Dadurch ergeben sich folgende Vorteile: Erstens kann so beispielsweise eine B-spline Fl„che entlang einer benutzerdefinierten Kurve verformt werden w„hrend andere Kurven oder Punkte fixiert sind. Zweitens, kann eine B-spline Fl„che Kurven interpolieren, die sich nicht auf iso-parametrische Linien der Fl„che abbilden lassen. Wir behandeln drei Arten von Constraints: Inzidenz einer beliebigen Kurve auf einer B-spline Fl„che, Fixieren von Fl„chennormalen entlang einer beliebigen Kurve (dieser Constraint dient zur Herstellung von tangentialen šberg„ngen zwischen zwei Fl„chen) und die sog. Variational Constrains. Letztere dienen unter anderem zur Optimierung der physikalischen und optischen Eigenschaften der Fl„chen. Es handelt sich hierbei um die Gausschen Normalgleichungen der Fl„chenfunktionale zweiter Ordnung, wie sie in der Literatur bekannt sind. Die Dissertation gliedert sich in zwei Teile. Der erste Teil befasst sich mit der Aufstellung der linearen Gleichungssysteme, welche die oben erw„hnten Constraints repr„sentieren. Der zweite Teil behandelt Methoden zum L”sen dieser Gleichungssysteme. Der Kern des ersten Teiles ist die Erweiterung und Generalisierung des auf Polarformen (Blossoms) basierenden Algorithmus f�r Verkettung von Polynomen auf Bezier und B-spline Basis: Gegeben sei eine B-spline Fl„che und eine B-spline Kurve im Parameterraum der Fl„che. Wir zeigen, dass die Kontrollpunkte der dreidimensionalen Fl„chenkurve, welche als polynomiale Verkettung der beiden definiert ist, durch eine im Voraus berechenbare lineare Tranformation (eine Matrix) der Fl„chenkontrollpunkte ausgedr�ckt werden k”nnen. Dadurch k”nnen Inzidenzbeziehungen zwischen Kurven und Fl„chen exakt und auf eine sehr elegante und kompakte Art definiert werden. Im Vergleich zu den bekannten Methoden ist diese Vorgehensweise effizienter, numerisch stabiler und erh”ht nicht die Konditionszahl der zu l”senden linearen Gleichungen. Die Effizienz wird erreicht durch Verwendung von eigens daf�r entwickelten Datenstrukturen und sorgf„ltige Analyse von kombinatorischen Eigenschaften von Polarformen. Die Gleichungen zur Definition von Tangentialit„ts- und Variational Constraints werden als Anwendung und Erweiterung dieses Algorithmus implementiert. Beschrieben werden auch symbolische und numerische Operationen auf B-spline Polynomen (Multiplikation, Differenzierung, Integration). Dabei wird konsistent die Matrixdarstellung von B-spline Polynomen verwendet. Das L”sen dieser Art von Constraintproblemen bedeutet das Finden der Kontrollpunkte einer B-spline Fl„che derart, dass die definierten Bedingungen erf�llt werden. Dies wird durch L”sen von, im Allgemeinen, unterbestimmten und schlecht konditionierten linearen Gleichungssystemen bewerkstelligt. Da in solchen F„llen keine eindeutige, numerisch stabile L”sung existiert, f�hren die �blichen Methoden zum L”sen von linearen Gleichungssystemen nicht zum Erfolg. Wir greifen auf die Anwendung von sog. Regularisierungsmethoden zur�ck, die auf der Singul„rwertzerlegung (SVD) der Systemmatrix beruhen. Insbesondere wird die L-curve eingesetzt, ein "numerischer Hochfrequenzfilter", der uns in die Lage versetzt eine stabile L”sung zu berechnen. Allerdings reichen auch diese Methoden im Allgemeinen nicht aus, eine Fl„che zu generieren, welche die erw�nschten „sthetischen und physikalischen Eigenschaften besitzt. Verformt man eine Tensorproduktfl„che entlang einer nicht isoparametrischen Kurve, entstehen unerw�nschte Oszillationen und Verformungen. Dieser Effekt wird "Surface-Aliasing" genannt. Wir stellen zwei Methoden vor um diese Aliasing-Effekte zu beseitigen: Die erste Methode wird vorzugsweise f�r Deformationen einer existierenden B-spline Fl„che entlang einer nicht isoparametrischen Kurve angewendet. Es erfogt eine Umparametrisierung der zu verformenden Fl„che derart, dass die Kurve in der neuen Fl„che auf eine isoparametrische Linie abgebildet wird. Die Umparametrisierung einer B- spline Fl„che ist keine abgeschlossene Operation; die resultierende Fl„che besitzt i.A. keine B-spline Darstellung. Wir berechnen eine beliebig genaue Approximation der resultierenden Fl„che mittels Interpolation von Kurvennetzen, die von der umzuparametrisierenden Fl„che gewonnen werden. Die zweite Methode ist rein algebraisch: Es werden zus„tzliche Bedingungen an die L”sung des Gleichungssystems gestellt, die die Aliasing-Effekte unterdr�cken oder ganz beseitigen. Es wird ein restriktionsgebundenes Minimum einer Zielfunktion gesucht, deren globales Minimum bei "optimaler" Form der Fl„che eingenommen wird. Als Zielfunktionen werden Gl„ttungsfunktionale zweiter Ordnung eingesetzt. Die stabile L”sung eines solchen Optimierungsproblems kann aufgrund der nahezu linearen Abh„ngigkeit des Gleichungen nur mit Hilfe von Regularisierungsmethoden gewonnen werden, welche die vorgegebene Zielfunktion ber�cksichtigen. Wir wenden die sog. Modifizierte Singul„rwertzerlegung in Verbindung mit dem L-curve Filter an. Dieser Algorithmus minimiert den Fehler f�r die geometrischen Constraints so, dass die L”sung gleichzeitig m”glichst nah dem Optimum der Zielfunktion ist.The constrained-based design of free-form surfaces is currently limited to tensor-product interpolation of orthogonal curve networks or equally spaced grids of points. The, so- called, multi-patch methods applied mainly in the context of scattered data interpolation construct surfaces from given boundary curves and derivatives along them. The limitation to boundary curves or iso-parametric curves considerably lowers the flexibility of this approach. In this thesis, we propose to compute surfaces from arbitrary (that is, not only iso-parametric) curves. This allows us to deform a B-spline surface along an arbitrary user-defined curve, or, to interpolate a B-spline surface through a set of curves which cannot be mapped to iso-parametric lines of the surface. We consider three kinds of constraints: the incidence of a curve on a B-spline surface, prescribed surface normals along an arbitrary curve incident on a surface and the, so-called, variational constraints which enforce a physically and optically advantageous shape of the computed surfaces. The thesis is divided into two parts: in the first part, we describe efficient methods to set up the equations for above mentioned linear constraints between curves and surfaces. In the second part, we discuss methods for solving such constraints. The core of the first part is the extension and generalization of the blossom-based polynomial composition algorithm for B-splines: let be given a B-spline surface and a B-spline curve in the domain of that surface. We compute a matrix which represents a linear transformation of the surface control points such that after the transformation we obtain the control points of the curve representing the polynomial composition of the domain curve and the surface. The result is a 3D B-spline curve always exactly incident on the surface. This, so-called, composition matrix represents a set of linear curve-surface incidence constraints. Compared to methods used previously our approach is more efficient, numerically more stable and does not unnecessarily increase the condition number of the matrix. The thesis includes a careful analysis of the complexity and combinatorial properties of the algorithm. We also discuss topics regarding algebraic operations on B-spline polynomials (multiplication, differentiation, integration). The matrix representation of B-spline polynomials is used throughout the thesis. We show that the equations for tangency and variational constraints are easily obtained re-using the methods elaborated for incidence constraints. The solving of generalized curve-surface constraints means to find the control points of the unknown surface given one or several curves incident on that surface. This is accomplished by solving of large and, generally, under-determined and badly conditioned linear systems of equations. In such cases, no unique and numerically stable solution exists. Hence, the usual methods such as Gaussian elimination or QR-decomposition cannot be applied in straightforward manner. We propose to use regularization methods based on Singular Value Decomposition (SVD). We apply the so-called L-curve, which can be seen as an numerical high-frequency filter. The filter automatically singles out a stable solution such that best possible satisfaction of defined constraints is achieved. However, even the SVD along with the L-curve filter cannot be applied blindly: it turns out that it is not sufficient to require only algebraic stability of the solution. Tensor-product surfaces deformed along arbitrary incident curves exhibit unwanted deformations due to the rectangular structure of the model space. We discuss a geometric and an algebraic method to remove this, so-called, Surface aliasing effect. The first method reparametrizes the surface such that a general curve constraint is converted to iso-parametric curve constraint which can be easily solved by standard linear algebra methods without aliasing. The reparametrized surface is computed by means of the approximated surface-surface composition algorithm, which is also introduced in this thesis. While this is not possible symbolically, an arbitrary accurate approximation of the resulting surface is obtained using constrained curve network interpolation. The second method states additional constraints which suppress or completely remove the aliasing. Formally we solve a constrained least square approximation problem: we minimize an surface objective function subject to defined curve constraints. The objective function is chosen such that it takes in the minimal value if the surface has optimal shape; we use a linear combination of second order surface smoothing functionals. When solving such problems we have to deal with nearly linearly dependent equations. Problems of this type are called ill-posed. Therefore sophisticated numerical methods have to be applied in order to obtain a set of degrees of freedom (control points of the surface) which are sufficient to satisfy given constraints. The remaining unused degrees of freedom are used to enforce an optically pleasing shape of the surface. We apply the Modified Truncated SVD (MTSVD) algorithm in connection with the L-curve filter which determines a compromise between an optically pleasant shape of the surface and constraint satisfaction in a particularly efficient manner

Free-form-deformation parameterization for multilevel 3D shape optimization in aerodynamics

A versatile parameterization technique is developed for 3D shape optimization in aerodynamics. Special attention is paid to construct a hierarchical parameterization by progressive enrichment of the parametric space. After a brief review of possible approaches, the free-form deformation framework is elected for a 3D tensorial Bézier parameterization. The classical degree-elevation algorithm applicable to Bézier curves is still valid for tensor products, and its application yields a hierarchy of embedded parameterizations. A drag-reduction optimization of a 3D wing in transonic regime is carried out by applying the Nelder-Mead simplex algorithm and a genetic algorithm. The new parameterization including degree-elevation is validated by numerical experimentation and its performance assessed

Isogeometric Approximation of Variational Problems for Shells

The interaction of applied geometry and numerical simulation is a growing field in the interplay of com- puter graphics, computational mechanics and applied mathematics known as isogeometric analysis. In this thesis we apply and analyze Loop subdivision surfaces as isogeometric tool because they provide great flexibility in handling surfaces of arbitrary topology combined with higher order smoothness. Compared with finite element methods, isogeometric methods are known to require far less degrees of freedom for the modeling of complex surfaces but at the same time the assembly of the isogeo- metric matrices is much more time-consuming. Therefore, we implement the isogeometric subdivision method and analyze the experimental convergence behavior for different quadrature schemes. The mid-edge quadrature combines robustness and efficiency, where efficiency is additionally increased via lookup tables. For the first time, the lookup tables allow the simulation with control meshes of arbitrary closed connectivity without an initial subdivision step, i.e. triangles can have more than one vertex with valence different from six. Geometric evolution problems have many applications in material sciences, surface processing and modeling, bio-mechanics, elasticity and physical simulations. These evolution problems are often based on the gradient flow of a geometric energy depending on first and second fundamental forms of the surface. The isogeometric approach allows a conforming higher order spatial discretization of these geometric evolutions. To overcome a time-error dominated scheme, we combine higher order space and time discretizations, where the time discretization based on implicit Runge-Kutta methods. We prove that the energy diminishes in every time-step in the fully discrete setting under mild time-step restrictions which is the crucial characteristic of a gradient flow. The overall setup allows for a general type of fourth-order energies. Among others, we perform experiments for Willmore flow with respect to different metrics. In the last chapter of this thesis we apply the time-discrete geodesic calculus in shape space to the space of subdivision shells. By approximating the squared Riemannian distance by a suitable energy, this approach defines a discrete path energy for a consistent computation of geodesics, logarithm and exponential maps and parallel transport. As approximation we pick up an elastic shell energy, which measures the deformation of a shell by membrane and bending contributions of its mid-surface. Bézier curves are a fundamental tool in computer-aided geometric design. We extend these to the subdivision shell space by generalizing the de Casteljau algorithm. The evaluation of Bézier curves depends on all input data. To solve this problem, we introduce B-splines and cardinal splines in shape space by gluing together piecewise Bézier curves in a smooth way. We show examples of quadratic and cubic Bézier curves, quadratic and cubic B-splines as well as cardinal splines in subdivision shell space

An isogeometric finite element formulation for phase transitions on deforming surfaces

This paper presents a general theory and isogeometric finite element implementation for studying mass conserving phase transitions on deforming surfaces. The mathematical problem is governed by two coupled fourth-order nonlinear partial differential equations (PDEs) that live on an evolving two-dimensional manifold. For the phase transitions, the PDE is the Cahn-Hilliard equation for curved surfaces, which can be derived from surface mass balance in the framework of irreversible thermodynamics. For the surface deformation, the PDE is the (vector-valued) Kirchhoff-Love thin shell equation. Both PDEs can be efficiently discretized using $C^1$-continuous interpolations without derivative degrees-of-freedom (dofs). Structured NURBS and unstructured spline spaces with pointwise $C^1$-continuity are utilized for these interpolations. The resulting finite element formulation is discretized in time by the generalized-$\alpha$ scheme with adaptive time-stepping, and it is fully linearized within a monolithic Newton-Raphson approach. A curvilinear surface parameterization is used throughout the formulation to admit general surface shapes and deformations. The behavior of the coupled system is illustrated by several numerical examples exhibiting phase transitions on deforming spheres, tori and double-tori.Comment: fixed typos, extended literature review, added clarifying notes to the text, added supplementary movie file