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volumetric parameterization and trivariate B-spline fitting using harmonic functions

COMPUTER AIDED GEOMETRIC DESIGN 2009年第6期26卷 648-664页

作者： Martin, T. Cohen, E. Kirby, R. M. Univ Utah Sch Comp Salt Lake City UT 84112 USA Univ Utah Sci Comp & Imaging Inst Salt Lake City UT USA

We present a methodology based oil discrete volumetric harmonic functions to parameterize a volumetric model in a way that it can be used to fit a single trivariate B-spline to data so that simulation attributes can also be modeled. The resulting model representation is suitable for isogeometric analysis [Hughes, T.J., Cottrell, J.A., B., Y., 2005. Isogeometric analysis: Cad, finite elements. nurbs, exact geometry, and mesh refinement. Computer Methods in Applied Mechanics and Engineering 194, 4135-4195]. Input data consists of both a closed triangle mesh representing the exterior geometric shape of the object and interior triangle meshes that can represent material attributes or other interior features. The trivariate B-spline geometric and attribute representations are generated from the resulting parameterization, creating trivariate B-spline material property representations over the same parameterization in a way that is related to [Martin, W., Cohen, E., 2001. Representation and extraction of volumetric attributes using trivariate splines. In: Symposium on Solid and Physical Modeling, pp. 234-240] but is suitable for application to a much larger family of shapes and attributes. The technique constructs a B-spline representation with guaranteed quality of approximation to the original data. Then we focus attention oil a model Of Simulation interest. a femur, consisting of hard outer cortical bone and inner trabecular bone. The femur is a reasonably complex object to model with a single trivariate B-spline since the shape overhangs make it impossible to model by sweeping planar slices. The representation is used in an elastostatic isogeometric analysis, demonstrating its ability to suitably represent objects for isogeometric analysis. (c) 2008 Elsevier B.V. All rights reserved.

关键词： Trivariate b-spline modeling and generation volumetric parameterization Model acquisition for simulation

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volumetric parameterization of complex objects by respecting multiple materials

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COMPUTERS & GRAPHICS-UK 2010年第3期34卷 187-197页

作者： Martin, Tobias Cohen, Elaine Univ Utah Sch Comp Salt Lake City UT 84112 USA

In this paper we present a methodology to create higher order parametric trivariate representations such as B-splines or T-splines, from closed triangle meshes with higher genus or bifurcations. The input can consist of multiple interior boundaries which represent inner object material attributes. Fundamental to our approach is the use of a midsurface in combination with harmonic functions to decompose the object into a small number of trivariate tensor-product patches that respect material attributes. The methodology is applicable to thin solid models which we extend using the flexibility of harmonic functions and demonstrate our technique, among other objects, on a genus-1 pelvis data set containing an interior triangle mesh separating the cortical part of the bone from the trabecular part. Finally, a B-spline representation is generated from the parameterization. (C) 2010 Elsevier Ltd. All rights reserved.

关键词： Trivariate b-spline modeling and generation volumetric parameterization Model acquisition for simulation

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volumetric parameterization and trivariate B-spline fitting using harmonic functions 08

Volumetric parameterization and trivariate B-spline fitting ...

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ACM Solid and Physical Modeling Symposium (ACM SPM 2008)

作者： Martin, T. Cohen, E. Kirby, R. M. Univ Utah Sch Comp Salt Lake City UT 84112 USA Univ Utah Sci Comp & Imaging Inst Salt Lake City UT USA

ISBN: (纸本)9781605581064

关键词： Trivariate b-spline modeling and generation volumetric parameterization Model acquisition for simulation

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Stretch-Minimizing volumetric parameterization

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Journal of Computer Science & Technology 2015年第3期30卷 553-564页

作者：金耀钱归平赵杰伊常建童若锋张建军 Department of Computer Science and Engineering Zhejiang University Hangzhou 310027 China College of New Media Zhejiang University of Media and Communications Hangzhou 310018 China National Centre for Computer Animation Bournemouth University Bournemouth BH12 5BB U.K.

Not many methods for parameterization guarantee bijectivity or local injectivity, which is essential for foldover- free mappings. Stretch-minimizing parameterization which is widely used for surface parameterization, provides foldover-free mappings and is capable of trading off between angle and area distortions. We extend its usage to volumetric parameterization in this paper by deriving a 3D version of stretch-distortion energy and incorporating fixed boundary conditions. Our energy definition includes a naturM barrier term which effectively prevents elements from collapsing and folding over. It saves the effort in other methods of formulating additional energy or constrains to ensure the local injectivity. We propose to minimize the overall energy integrated over the whole mesh with a relaxation-enhanced solver and optimize the energy globally. This is different from the conventional approach of surface parameterization where mesh nodes are optimized individually. Compared with other volumetric parameterizations, method, being foldover-free and offering a good trade-off between our approach bears the advantages of stretch-minimizing angle and volume distortions.

关键词： stretch-minimizing volumetric parameterization foldover-free

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Constructing volumetric parameterization based on directed graph simplification of l1 polycube structure from complex shapes

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COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING 2019年 351卷 422-440页

作者： Chen, Long Xu, Gang Wang, Shiyi Shi, Zeyun Huang, Jin Univ Shanghai Sci & Technol Sch Mech Engn Shanghai 200093 Peoples R China Hangzhou Dianzi Univ Sch Comp Sci & Technol Hangzhou 310018 Zhejiang Peoples R China Zhejiang Univ State Key Lab CAD & CG Hangzhou 310058 Zhejiang Peoples R China Hunan Normal Univ Sch Math & Stat MOE LCSM Changsha 410081 Hunan Peoples R China

volumetric spline parameterization of complex geometry plays a key role in isogeometric analysis (IGA). In this paper, we propose a general framework to construct volumetric parameterization from complex shapes based on directed graph simplification of the l(1) polycube structure. By minimizing the l(1)-norm of the normals on the input triangular meshes, a polycube structure can be generated with robustness, efficiency, and controllability. Then an algorithm is proposed for l(1) polycube structure simplification with a directed graph, which is an abstract structure generated from the polycube. After simplification, the vertex number of the directed graph will decrease, and the polycube structure will become more simple. From the simplified polycube structure, we construct a segmented surface by spline fitting techniques, and finally we fill each block with a trivariate B-spline volume with C-0-constraints. The proposed method can generate volumetric parameterization without an interior extraordinary vertex, and it has very high potential for constructing volumetric parameterization in IGA simulations. Some volume parameterization examples from complex shapes are presented to show the robustness and efficiency of the proposed method. (C) 2019 Elsevier B.V. All rights reserved.

关键词： volumetric parameterization l(1) optimization Polycube structure Directed graph Isogeometric analysis

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DiagPara: volumetric parameterization with energy-minimizing pair of diagonal surfaces from given boundaries

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JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2024年 449卷

作者： Li, Xiaowei Hu, Qinghua Gu, Renshu Xu, Jinlan Wu, Haiyan Xu, Gang Hangzhou Dianzi Univ Sch Comp Sci & Technol Hangzhou 310018 Peoples R China CSSC SEAGO Syst Technol Co Ltd Shanghai 200010 Peoples R China

An efficient approach for constructing volumetric parameterization with internal energyminimizing pair of diagonal surfaces from given boundaries is presented. The expression for the internal energy of one diagonal surface pair is formulated. Then necessary and sufficient conditions for trivariate tensor -product B & eacute;zier volume with minimal internal energy of the diagonal surface pair are derived. From the derived conditions, an efficient construction algorithm for single- and multi -block B & eacute;zier volumes with minimal diagonal internal energy by the Lagrange multiplier method is presented. The auxiliary linear system only needs to be solved once. As soon the explicit computation masks are obtained, the interior control points of the volume are then directly calculated. Experimental results are given to illustrate the volumetric parameterization effects of the presented approach.

关键词： volumetric parameterization Diagonal surface pair Energy minimization Isogeometric analysis

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volumetric parameterization with truncated hierarchical B-splines for isogeometric analysis

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COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING 2022年 401卷

作者： Zheng, Ye Chen, Falai Univ Sci & Technol China Sch Math Sci Hefei 230026 Anhui Peoples R China

Constructing spline parameterizations for computational domains is one of the fundamental problems in isogeometric analysis. In this paper, we present an efficient method for parameterizing volumetric computational domains using truncated hierarchical B-splines (THB). To ensure the injectivity of the parameterization, we utilize a collocation-based framework where the foldovers are eliminated on an increasing collocation set until the injectivity can be verified. A local-global solver is employed to eliminate any foldover effectively by projecting the mapping onto a bounded conformal distortion space. Local refinement of THB splines is also performed to enlarge the injectivity space. Various sufficient conditions are proposed to check the injectivity of the mapping efficiently. The quality of the bijective mapping is further improved by optimizing a geometrical energy, which is also embedded in the collocation framework to ensure injectivity. A coarse-to-fine variant of our method, which is more robust, is also presented. THB splines not only reduce the number of degrees of freedom in the parametric representation, but also increase the possibility of producing a bijective parameterization by local refinement. Some examples are provided to demonstrate the efficiency, effectiveness and robustness of our method.(c) 2022 Elsevier B.V. All rights reserved.

关键词： Isogeometric analysis volumetric parameterization Truncated hierarchical splines Bounded distortion Geometrical energy

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Penalty function-based volumetric parameterization method for isogeometric analysis

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COMPUTER AIDED GEOMETRIC DESIGN 2022年 94卷

作者： Ji, Ye Wang, Meng-Yun Pan, Mao-Dong Zhang, Yi Zhu, Chun-Gang Dalian Univ Technol Sch Math Sci Dalian 116024 Peoples R China Key Lab Computat Math & Data Intelligence Liaonin Dalian 116024 Peoples R China Nanjing Univ Aeronaut & Astronaut Dept Math Nanjing 210016 Peoples R China MIIT Key Lab Math Modelling & High Performance Comp Ai Nanjing 211106 Peoples R China

In isogeometric analysis, constructing bijective and low-distorted parameterizations is a fundamental task. Compared with the planar problem, the volumetric case is more challenging in both robustness and efficiency. In this paper, we present a robust and efficient volumetric parameterization method based on the idea of penalty functions and the Jacobian regularization technique. The proposed method does not require an already bijective initialization and thus avoids an extra foldover elimination step. The main contributions of this work lie in three aspects. First, a new objective function that characterizes the volume distortion is established using the Divergence Theorem. Second, we employ a novel penalty function for the Jacobian regularization. The full analytical gradient of the objective function is also deduced to enhance the numerical stability in gradient-based optimization. Third, we develop a reduced numerical integration strategy to accelerate the new algorithm. Several numerical examples demonstrate that our method significantly outperforms the current competitive approaches both in terms of robustness and efficiency. (C) 2022 Elsevier B.V. All rights reserved.

关键词： Isogeometric analysis volumetric parameterization Penalty function Jacobian regularization

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Efficient construction of multi-block volumetric spline parameterization by discrete mask method

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JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2015年 290卷 589-597页

作者： Xu, Gang Mourrain, Bernard Wu, Xiangyang Chen, Long Hui, Kin-chuen Hangzhou Dianzi Univ Hangzhou 310018 Zhejiang Peoples R China INRIA Sophia Antipolis Galaad F-06902 Valbonne France Univ Shanghai Sci & Technol Shanghai 200093 Peoples R China Chinese Univ Hong Kong Dept Mech & Automat Engn Shatin Hong Kong Peoples R China

Isogeometric analysis attempts to unify the mathematical languages in design and analysis to realize the seamless integration of CAD and CAE. In three-dimensional isogeometric analysis, parametric volume is employed as the computational domain for a given set of boundary information. In this paper, we propose a discrete mask method for the efficient construction of multi-block volumetric parameterization based on a set of given boundary spline surfaces. Given the block-partition information of a model, the interior control points can be obtained efficiently by solving a sparse linear system. The existence and uniqueness of the solution for the linear system are also proved. After performing a preprocess on the non-compatible boundary surfaces, the discrete Coons mask is generalized to a unified form, which provides more choices for the construction of inner control points. The proposed method is not only suitable for the multi-block case with C-1 continuity, but can also be used for the case in which the boundary surfaces are of different degrees, and with different number of control points and knot vectors. Several examples are presented to illustrate the effectiveness of the proposed method. (C) 2015 Elsevier B.V. All rights reserved.

关键词： Isogeometric analysis volumetric parameterization Discrete mask Multi-block

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volumetric spline parameterization for isogeometric analysis

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COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING 2020年 359卷

作者： Pan, Maodong Chen, Falai Tong, Weihua Univ Sci & Technol China Sch Math Sci Hefei 230026 Anhui Peoples R China

Given the spline representation of the boundary of a three dimensional domain, constructing a volumetric spline parameterization of the domain (i.e., a map from a unit cube to the domain) with the given boundary is a fundamental problem in isogeometric analysis, and the quality of parameterization has a great influence on the accuracy and efficiency in subsequent analysis. In this paper, we propose a collocation-based approach for constructing volumetric parameterization with good quality. Firstly, a harmonic map is computed between a unit cube and the computational domain. Then the task of finding a good parameterization is modeled as a max-min optimization problem with a collocation-based formulation of the bijectivity condition of a map, and an algorithm based on the coarse-to-fine and divide-and-conquer strategy is proposed to solve the optimization problem efficiently. Finally, to further improve the quality of the parameterization, the MIPS (Most Isometric parameterizations) method is adopted to reduce the conformal distortion of the solved map. We provide several examples to demonstrate the feasibility of our approach and to compare our approach with several state-of-the-art methods. The experimental results show that our algorithm can produce parameterizations with higher quality than previous approaches especially for complex domains. (C) 2019 Elsevier B.V. All rights reserved.

关键词： volumetric parameterization Isogeometric analysis Max-min optimization Collocation MIPS

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