Construction of analysis-suitable G1G^1 planar multi-patch parameterizations

Isogeometric analysis allows to define shape functions of global $C^{1}$ continuity (or of higher continuity) over multi-patch geometries. The construction of such $C^{1}$-smooth isogeometric functions is a non-trivial task and requires particular multi-patch parameterizations, so-called analysis-suitable $G^{1}$ (in short, AS-$G^{1}$) parameterizations, to ensure that the resulting $C^{1}$ isogeometric spaces possess optimal approximation properties, cf. [7]. In this work, we show through examples that it is possible to construct AS-$G^{1}$ multi-patch parameterizations of planar domains, given their boundary. More precisely, given a generic multi-patch geometry, we generate an AS-$G^{1}$ multi-patch parameterization possessing the same boundary, the same vertices and the same first derivatives at the vertices, and which is as close as possible to this initial geometry. Our algorithm is based on a quadratic optimization problem with linear side constraints. Numerical tests also confirm that $C^{1}$ isogeometric spaces over AS-$G^{1}$ multi-patch parameterized domains converge optimally under mesh refinement, while for generic parameterizations the convergence order is severely reduced

T-spline based unifying registration procedure for free-form surface workpieces in intelligent CMM

With the development of the modern manufacturing industry, the free-form surface is widely used in various fields, and the automatic detection of a free-form surface is an important function of future intelligent three-coordinate measuring machines (CMMs). To improve the intelligence of CMMs, a new visual system is designed based on the characteristics of CMMs. A unified model of the free-form surface is proposed based on T-splines. A discretization method of the T-spline surface formula model is proposed. Under this discretization, the position and orientation of the workpiece would be recognized by point cloud registration. A high accuracy evaluation method is proposed between the measured point cloud and the T-spline surface formula. The experimental results demonstrate that the proposed method has the potential to realize the automatic detection of different free-form surfaces and improve the intelligence of CMMs

Fitting Terrestrial Laser Scanner Point Clouds with T-Splines: Local Refinement Strategy for Rigid Body Motion

T-splines have recently been introduced to represent objects of arbitrary shapes using a smaller number of control points than the conventional non-uniform rational B-splines (NURBS) or B-spline representatizons in computer-aided design, computer graphics and reverse engineering. They are flexible in representing complex surface shapes and economic in terms of parameters as they enable local refinement. This property is a great advantage when dense, scattered and noisy point clouds are approximated using least squares fitting, such as those from a terrestrial laser scanner (TLS). Unfortunately, when it comes to assessing the goodness of fit of the surface approximation with a real dataset, only a noisy point cloud can be approximated: (i) a low root mean squared error (RMSE) can be linked with an overfitting, i.e., a fitting of the noise, and should be correspondingly avoided, and (ii) a high RMSE is synonymous with a lack of details. To address the challenge of judging the approximation, the reference surface should be entirely known: this can be solved by printing a mathematically defined T-splines reference surface in three dimensions (3D) and modeling the artefacts induced by the 3D printing. Once scanned under different configurations, it is possible to assess the goodness of fit of the approximation for a noisy and potentially gappy point cloud and compare it with the traditional but less flexible NURBS. The advantages of T-splines local refinement open the door for further applications within a geodetic context such as rigorous statistical testing of deformation. Two different scans from a slightly deformed object were approximated; we found that more than 40% of the computational time could be saved without affecting the goodness of fit of the surface approximation by using the same mesh for the two epochs

Automatic and Interactive Mesh to T-Spline Conversion

In Geometry Processing, and more specifically in surface approximation, one of the most important issues is the automatic generation of a quad-dominant control mesh from an arbitrary shape (e.g. a scanned mesh). One of the first fully automatic solutions was proposed by Eck and Hoppe in 1996. However, in the industry, designers still use manual tools (see e.g. cyslice). The main difference between a control mesh constructed by an automatic method and the one designed by a human user is that in the second case, the control mesh follows the features of the model. More precisely, it is well known from approximation theory that aligning the edges with the principal directions of curvature improves the smoothness of the reconstructed surface, and this is what designers intuitively do. In this paper, our goal is to automatically construct a control mesh driven by the anisotropy of the shape, mimicking the mesh that a designer would create manually. The control mesh generated by our method can be used by a wide variety of representations (splines, subdivision surfaces...). We demonstrate our method applied to the automatic conversion from a mesh of arbitrary topology into a T-Spline surface. Our method first extracts an initial mesh from a PGP (Periodic Global Parameterization). To facilitate user-interaction, we extend the PGP method to take into account optional user-defined information. This makes it possible to locally tune the orientation and the density of the control mesh. The user can also interactively remove edges or sketch additional ones. Then, from this initial control mesh, our algorithm generates a valid T-Spline control mesh by enforcing some validity constraints. The valid T-Spline control mesh is finally fitted to the original surface, using a classic regularized optimization procedure. To reduce the L-infinity approximation error below a user-defined threshold, we iteratively use the T-Spline adaptive local refinement

Adaptive Knot Placement in Non-uniform B-spline Surface Fitting

针对非均匀b样条的节点设置问题,提出一种利用非均匀b样条曲面拟合离散数据的迭代算法,通过优化节点分布来改进拟合曲面的质量.该算法以带参数化的三角网格曲面为输入,在首次迭代中根据输入曲面的几何特征将其对应的参数域划分成若干个子区域,并使得每个子区域上累积的几何特征信息量近似相等,子区域的重心坐标将取为首次迭代的节点;在随后的迭代中,保证前次迭代生成的重心位置固定不变,并根据前次迭代得到的曲面拟合误差再次将区域划分成累积误差接近相等的子区域,新增加的子区域重心的坐标选为拟加入的节点.文中算法自适应地在曲面形状复杂或拟合误差大的区域引入更多的控制顶点,使得拟合曲面的质量得以逐步改进.实验结果表明,该算法快速有效,在拟合具有明显几何特征的输入数据时具有优势.Knot placement of non-uniform B-spline is studied, and an iterative surface fitting scheme is proposed by exploring the degrees of freedom of knots to improve the fitting surface's quality.Our algorithm takes as input triangular meshes with parameterization.In the first iteration, the parametric domain is partitioned into several sub-regions with equally accumulated surface geometric information, and the coordinates of the centroids are chosen as the candidates of knots; in the following iteration steps, we partition the regions according to the fitting errors analogously while the centroids generated by previous steps remain unchanged.The fitting surface's quality is progressively improved as more control points are adaptively introduced into the region of the surface with more features or larger fitting error.Several experiments demonstrate the efficacy of our method in fitting surface with distinct geometric features.国家自然科学基金(61100105;61100107;61170324;61272300); 福建省自然科学基金(2011J05007;2012J01291

T-splinit

Pintojen mallinnus splineillä on keskeisessä asemassa CAD- ja grafiikkasovelluksissa. Niinpä spliniesitysten tehokkuus on jatkuvasti tutkimuksen alla. T-splinit (Sederberg & al) ovat NURBS-pintojen yleistyksiä, jotka sallivat harvemman ohjaushilan. Tässä tutkielmassa perehdytään T-splinien toimintaan, varmistetaan niiden toimivuutta sekä esitetään muutamia lisätarkennuksia ja -todistuksia. Työn osana on toteutettu T-splinirutiineja, jotka operoivat paikallisesti hilaverkossa ja jotka ovat omalta osaltaan lisäsivät ymmärrystä niiden toiminnasta. Algoritmeista keskeisin on lisäysalgoritmi, jonka toteutus näytti vahvistavan sen esitetyt hyvät ominaisuudet. Vaikka T-splinit ovat jo kaupallisessa käytössä, niiden teoreettinen tutkimus on vielä varsin kesken

Abstract Adaptive T-spline Surface Fitting to Z-Map Models

Surface fitting refers to the process of constructing a smooth representation for an object surface from a fairly large number of measured 3D data points. This paper presents an automatic algorithm to construct smooth parametric surfaces using T-splines from z-map data. The algorithm begins with a rough surface approximation and then progressively refines it in the regions where the approximation accuracy does not meet the requirement. The topology of the resulting T-spline surface is determined adaptively based on the local geometric character of the input data and the geometry of the control points is obtained by a least squares procedure. The advantage of the approach is that the resulting surface is C 2 continuous and the refinement is essentially local, resulting in a small number of control points for the surface. CR Categories: I.3.5 [Computer Graphics]: Computational Geometry and Object Modeling—curve, surface, solid and object representations; Keywords: T-splines, z-map models, adaptive fitting, surface reconstruction

Evolvability-guided Optimization of Linear Deformation Setups for Evolutionary Design Optimization

Richter A. Evolvability-guided Optimization of Linear Deformation Setups for Evolutionary Design Optimization. Bielefeld: Universität Bielefeld; 2019.Andreas Richter gratefully acknowledges the financial support from Honda Research Institute Europe (HRI-EU).This thesis targets efficient solutions for optimal representation setups for evolutionary design optimization problems. The representation maps the abstract parameters of an optimizer to a meaningful variation of the design model, e.g., the shape of a car. Thereby, it determines the convergence speed to and the quality of the final result. Thus, engineers are eager to employ well-tuned representations to achieve high-quality design solutions. But, setting up optimal representations is a cumbersome process because the setup procedure requires detailed knowledge about the objective functions, e.g., a fluid dynamics simulation, and the parameters of the employed representation itself. Thus, we target efficient routines to set up representations automatically to support engineers from their tedious, partly manual work. Inspired by the concept of evolvability, we present novel quality criteria for the evaluation of linear deformations commonly applied as representations. We define and analyze the criteria variability, regularity, and improvement potential which measure the expected quality and convergence speed of an evolutionary design optimization process based on the linear deformation setup. Moreover, we target the efficient optimization of deformation setups with respect to these three criteria. In dynamic design optimization scenarios a suitable compromise between exploration and exploitation is crucial for efficient solutions. We discuss the construction of optimal compromises for these dynamic scenarios with our criteria because they characterize exploration and exploitation. As a result an engineer can initialize and adjust the deformation setup for improved convergence speed of a design process and for enhanced quality of the design solutions with our methods