Modeling and processing of physically based dynamic t-splines - Ma, Yuewen (2014)
This research explores how to incorporate physical quantities such as mass distributions, internal deformation energies, damping, and forces into the T-spline geometric substrate and its related geometric processing algorithms. The research is inspired by the successful physics-based design paradigm of dynamic non-uniform rational B-splines (D-NURBS) and T-spline technology. D-NURBS introduce physical quantities into the NURBS geometric formulation through the application of Lagrangian mechanics and provide a new design paradigm that integrates standard geometric design and physics-based design. However, there exist serious weaknesses within the D-NURBS scheme. D-NURBS do not support local refinement which is very important in geometric design and engineering computation, D-NURBS generally do not have closed-form solutions and need numerical methods, and currently D-NURBS lack very effective surface editing tools. T-splines are a free-form surface technology that solve most of the limitations inherent in NURBS representation. In particular, T-splines allow local refinement and are well-suited for geometric modeling, and engineering computation and analysis. Therefore, it is tempting to develop physically-based dynamic T-splines (or PD-T-splines) and related algorithms to overcome the weaknesses of D-NURBS. In this thesis, we first present the formulation of PD-T-splines. The dynamic equation that governs the motion of the PD-T-splines is derived based on the work-energy version of Lagrangian Dynamics. The highly non-linear equation motivates us to further propose Efficient PD-T-splines (or E-PD-T-splines) by freezing the weights, whose dynamic equation becomes a linear ordinary differential equations (ODE). A closed form solution is derived based on the physical point of view that a vibration system can be viewed as a coupling of individual vibrant modes. Based on this physical insight, a modal reduction technique is developed to reduce computational complexity. We also consider incorporating the local refinement property of T-splines into PD-T-splines, which means the change of the generalized coordinates as well as the dynamic equation. We present an efficient method to achieve local update of these physical matrices along with local refinement. Second, we present a new modeling tool based on general curve handles. Curve based modeling techniques become prevalent and are shown more effective than point based. We propose to deform the T-spline surface by adjusting the sketched curve handles on the surface. The curve handle is defined by the composition between a 2D domain curve and the incident T-spline surface. To facilitate the adjustment of the composite curve, physically based dynamic composite (PBDC) curve is developed directly based on the composite representation, which supports arbitrary type of 2D domain curves. To avoid the possible rank deficiency of its dynamic equation, we carefully examine the relation between the dimension of the composite blending functions and the rank of their corresponding construction matrix. We present an approach to identify a basis for the vector space spanned by those blending functions, and compute their dependency. After filtering out the deficient DOFs, we obtain a regularized PBDC curve, which serves as the main tool to adjust the general curve handle on the surface. Finally, the surface deformation is achieved through introducing the curve force into PD-T-splines, which connects the handle curve and the 3D target curve in space. Third, to explore geometric processing of PD-T-Splines such as morphing and rendering, we consider the problem of constrained mapping. We reveal the relation between a smooth mapping and its induced transformation and propose a refinement strategy based on the longest edge bisection to guarantee that the induced transformation is locally foldover free if the smooth mapping is locally bijective. To make the RBF-based mapping locally bijective, we propose to find non-intersecting trajectories to guide the warping and derive a bound for the warping stepsize. Furthermore, to make the warping process more effective, we derive the truly foldover free condition for the induced transformation. Based on this condition and the bound for the local bijective smooth mapping, we determine whether we perform local mesh refinement or warping and the actual warping stepsize as well. Integrating all these technical components provides a constrained mesh warping algorithm that is effective, provably foldover free, able to handle a large number of constraints, and able to output a visually pleasing result without extra smoothing optimization. Fourth, we present a novel solution to the T-spline morphing problem. It is well known that morphing involves two basic steps: vertex correspondence and vertex path. We solve the vertex correspondence problem by consistent approximation technique, which aligns important feature points using our proposed constrained mapping technique and approximates the two input surfaces using the same preimage of control T-mesh. Observing that PD-T-splines naturally produce the dynamic evolution from one pose to another, the vertex path problem can be readily solved by introducing PD-T-splines into morphing. In this case, we impose the boundary conditions into the ODE, which is also analytically solvable. Furthermore, we also introduce extra DOFs into the physical model to enable speed control. Finally, we consider the rendering of the dynamic evolution in both modeling and morphing. We present an efficient method to map a static image onto the evolution sequence. Here we would like to achieve dynamic rendering in realtime along with dynamic evolution. Rather than applying the technique on every frame separately, we propose to establish the constrained map making use of the existing map. We first compute a satisfactory map for the initial frame. Then we generate the subsequent maps from the first map, which only involves the calculation of the barycentric coordinates. In this way, the proposed method can render the dynamic evolution in realtime.