N. Warncke, Ioana Ciotir, A. Tonnoir, Zoé Lambert, C. Gout
{"title":"Analytical approach to Galerkin BEMs on polyhedral surfaces","authors":"N. Warncke, Ioana Ciotir, A. Tonnoir, Zoé Lambert, C. Gout","doi":"10.5802/smai-jcm.50","DOIUrl":"https://doi.org/10.5802/smai-jcm.50","url":null,"abstract":"","PeriodicalId":376888,"journal":{"name":"The SMAI journal of computational mathematics","volume":"20 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116322920","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In the context of isogeometric analysis, globally C1 isogeometric spaces over unstructured quadrilateral meshes allow the direct solution of fourth order partial differential equations on complex geometries via their Galerkin discretization. The design of such smooth spaces has been intensively studied in the last five years, in particular for the case of planar domains, and is still task of current research. In this paper, we first give a short survey of the developed methods and especially focus on the approach [28]. There, the construction of a specific C1 isogeometric spline space for the class of so-called analysis-suitable G1 multi-patch parametrizations is presented. This particular class of parameterizations comprises exactly those multi-patch geometries, which ensure the design of C1 spaces with optimal approximation properties, and allows the representation of complex planar multi-patch domains. We present known results in a coherent framework, and also extend the construction to parametrizations that are not analysis-suitable G1 by allowing higher-degree splines in the neighborhood of the extraordinary vertices and edges. Finally, we present numerical tests that illustrate the behavior of the proposed method on representative examples. 2010 Mathematics Subject Classification. 65N30.
{"title":"Isogeometric analysis with $C^1$ functions on planar, unstructured quadrilateral meshes","authors":"M. Kapl, G. Sangalli, T. Takacs","doi":"10.5802/smai-jcm.52","DOIUrl":"https://doi.org/10.5802/smai-jcm.52","url":null,"abstract":"In the context of isogeometric analysis, globally C1 isogeometric spaces over unstructured quadrilateral meshes allow the direct solution of fourth order partial differential equations on complex geometries via their Galerkin discretization. The design of such smooth spaces has been intensively studied in the last five years, in particular for the case of planar domains, and is still task of current research. In this paper, we first give a short survey of the developed methods and especially focus on the approach [28]. There, the construction of a specific C1 isogeometric spline space for the class of so-called analysis-suitable G1 multi-patch parametrizations is presented. This particular class of parameterizations comprises exactly those multi-patch geometries, which ensure the design of C1 spaces with optimal approximation properties, and allows the representation of complex planar multi-patch domains. We present known results in a coherent framework, and also extend the construction to parametrizations that are not analysis-suitable G1 by allowing higher-degree splines in the neighborhood of the extraordinary vertices and edges. Finally, we present numerical tests that illustrate the behavior of the proposed method on representative examples. 2010 Mathematics Subject Classification. 65N30.","PeriodicalId":376888,"journal":{"name":"The SMAI journal of computational mathematics","volume":"70 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126763923","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Blanche Buet, J. Mirebeau, Y. Gennip, François Desquilbet, Johann Dréo, G. P. Leonardi, S. Masnou, Carola-Bibian Schönlieb
This paper arose from a minisymposium held in 2018 at the 9th International Conference on Curves and Surface in Arcachon, France, and organized by Simon Masnou and Carola-Bibiane Schonlieb. This minisymposium featured a variety of recent developments of geometric partial differential equations and variational models which are directly or indirectly related to several problems in image and data processing. The current paper gathers three contributions which are in connection with the talks of three minisymposium speakers: Blanche Buet, Jean-Marie Mirebeau, and Yves van Gennip. The first contribution (Section 1) by Yves van Gennip provides a short overview of recent activity in the field of PDEs on graphs, without aiming to be exhaustive. The main focus is on techniques related to the graph Ginzburg–Landau variational model, but some other research in the field is also mentioned at the end of the section. The second contribution (Section 2), written by Jean-Marie Mirebeau, Francois Desquilbet, Johann Dreo, and Frederic Barbaresco presents a recent numerical method devoted to computing curves that globally minimize an energy featuring both a data driven term, and a second order curvature penalizing term. Applications to image segmentation are discussed, and recent progress on radar network configuration, in which the optimal curves represent an opponent’s trajectories, is described in detail. Lastly, Section 3 is devoted to a work by Blanche Buet, Gian Paolo Leonardi, and Simon Masnou on the definition and the approximation of weak curvatures for a large class of generalized surfaces, and in particular for point clouds, based on the geometric measure theoretic notion of varifolds.
这篇论文源于2018年在法国阿卡松举行的第九届曲线与曲面国际会议上的一个小型研讨会，由Simon Masnou和Carola-Bibiane Schonlieb组织。这次小型研讨会的特色是几何偏微分方程和变分模型的各种最新发展，这些模型直接或间接地与图像和数据处理中的几个问题有关。本文收集了三个贡献，这些贡献与三位迷你研讨会发言人的谈话有关:Blanche Buet, Jean-Marie Mirebeau和Yves van Gennip。Yves van Gennip的第一篇文章(第1节)简要概述了图上偏微分方程领域的最新活动，但并不打算详尽。主要重点是与图金兹堡-朗道变分模型相关的技术，但在本节末尾也提到了该领域的其他一些研究。第二个贡献(第2节)，由Jean-Marie Mirebeau, Francois Desquilbet, Johann Dreo和Frederic Barbaresco撰写，提出了一种最新的数值方法，用于计算具有数据驱动项和二阶曲率惩罚项的能量全局最小化曲线。讨论了图像分割的应用，并详细描述了雷达网络配置的最新进展，其中最优曲线代表对手的轨迹。最后，第3节专门介绍了Blanche Buet, Gian Paolo Leonardi和Simon Masnou的工作，该工作基于几何测量理论的变分概念，对一类广义曲面，特别是点云的弱曲率的定义和近似。
{"title":"Partial differential equations and variational methods for geometric processing of images","authors":"Blanche Buet, J. Mirebeau, Y. Gennip, François Desquilbet, Johann Dréo, G. P. Leonardi, S. Masnou, Carola-Bibian Schönlieb","doi":"10.5802/smai-jcm.55","DOIUrl":"https://doi.org/10.5802/smai-jcm.55","url":null,"abstract":"This paper arose from a minisymposium held in 2018 at the 9th International Conference on Curves and Surface in Arcachon, France, and organized by Simon Masnou and Carola-Bibiane Schonlieb. This minisymposium featured a variety of recent developments of geometric partial differential equations and variational models which are directly or indirectly related to several problems in image and data processing. The current paper gathers three contributions which are in connection with the talks of three minisymposium speakers: Blanche Buet, Jean-Marie Mirebeau, and Yves van Gennip. The first contribution (Section 1) by Yves van Gennip provides a short overview of recent activity in the field of PDEs on graphs, without aiming to be exhaustive. The main focus is on techniques related to the graph Ginzburg–Landau variational model, but some other research in the field is also mentioned at the end of the section. The second contribution (Section 2), written by Jean-Marie Mirebeau, Francois Desquilbet, Johann Dreo, and Frederic Barbaresco presents a recent numerical method devoted to computing curves that globally minimize an energy featuring both a data driven term, and a second order curvature penalizing term. Applications to image segmentation are discussed, and recent progress on radar network configuration, in which the optimal curves represent an opponent’s trajectories, is described in detail. Lastly, Section 3 is devoted to a work by Blanche Buet, Gian Paolo Leonardi, and Simon Masnou on the definition and the approximation of weak curvatures for a large class of generalized surfaces, and in particular for point clouds, based on the geometric measure theoretic notion of varifolds.","PeriodicalId":376888,"journal":{"name":"The SMAI journal of computational mathematics","volume":"69 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133569357","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
H. Calandra, Zoé Lambert, C. Gout, A. Atle, Marie Bonnasse-Gahot, Julien Diaz, Simon Ettouati
In this paper, we present the recent advances in using discontinuous Galerkin method for solving wave equation in the context of seismic depth imaging and full wave inversion. We show some examples and the way forward to some advanced schemes coupling different numerical approximations we believe will provide the necessary tools for building the next seismic depth imaging generation codes for TOTAL Exploration&Production. This contribution is linked to the mini symposium (MS) Mathematical tools in energy industry (organized at Arcachon during the 9th International conference Curves and Surfaces).
{"title":"Recent advances in numerical methods for solving the wave equation in the context of seismic depth imaging","authors":"H. Calandra, Zoé Lambert, C. Gout, A. Atle, Marie Bonnasse-Gahot, Julien Diaz, Simon Ettouati","doi":"10.5802/smai-jcm.51","DOIUrl":"https://doi.org/10.5802/smai-jcm.51","url":null,"abstract":"In this paper, we present the recent advances in using discontinuous Galerkin method for solving wave equation in the context of seismic depth imaging and full wave inversion. We show some examples and the way forward to some advanced schemes coupling different numerical approximations we believe will provide the necessary tools for building the next seismic depth imaging generation codes for TOTAL Exploration&Production. This contribution is linked to the mini symposium (MS) Mathematical tools in energy industry (organized at Arcachon during the 9th International conference Curves and Surfaces).","PeriodicalId":376888,"journal":{"name":"The SMAI journal of computational mathematics","volume":"3 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129419228","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
With the increasing need for volumetric B-spline representations and the lack of methodologies for creating semi-structured volumetric B-spline representations from B-spline Boundary Representations (B-Rep), hybrid approaches combining semi-structured volumetric B-splines and unstructured Bézier tetrahedra have been introduced, including one that transforms a trimmed B-spline B-Rep first to an untrimmed Hybrid B-Rep (HBRep) and then to a Hybrid Volume Representation (HV-Rep). Generally, the effect of h-refinement has not been considered over B-spline hybrid representations. Standard refinement approches to tensor product B-splines and subdivision of Bézier triangles and tetrahedra must be adapted to this representation. In this paper, we analyze possible types of h-refinement of the HV-Rep. The revised and trim basis functions for HBand HV-rep depend on a partition of knot intervals. Therefore, a naive h-refinement can change basis functions in unexpected ways. This paper analyzes the the effect of h-refinement in reducing error as well. Different h-refinement strategies are discussed. We demonstrate their differences and compare their respective consequential trade-offs. Recommendations are also made for different use cases. 2010 Mathematics Subject Classification. 65D17.
{"title":"Refinement for a Hybrid Boundary Representation and its Hybrid Volume Completion","authors":"Yang Song, E. Cohen","doi":"10.5802/smai-jcm.49","DOIUrl":"https://doi.org/10.5802/smai-jcm.49","url":null,"abstract":"With the increasing need for volumetric B-spline representations and the lack of methodologies for creating semi-structured volumetric B-spline representations from B-spline Boundary Representations (B-Rep), hybrid approaches combining semi-structured volumetric B-splines and unstructured Bézier tetrahedra have been introduced, including one that transforms a trimmed B-spline B-Rep first to an untrimmed Hybrid B-Rep (HBRep) and then to a Hybrid Volume Representation (HV-Rep). Generally, the effect of h-refinement has not been considered over B-spline hybrid representations. Standard refinement approches to tensor product B-splines and subdivision of Bézier triangles and tetrahedra must be adapted to this representation. In this paper, we analyze possible types of h-refinement of the HV-Rep. The revised and trim basis functions for HBand HV-rep depend on a partition of knot intervals. Therefore, a naive h-refinement can change basis functions in unexpected ways. This paper analyzes the the effect of h-refinement in reducing error as well. Different h-refinement strategies are discussed. We demonstrate their differences and compare their respective consequential trade-offs. Recommendations are also made for different use cases. 2010 Mathematics Subject Classification. 65D17.","PeriodicalId":376888,"journal":{"name":"The SMAI journal of computational mathematics","volume":"31 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128386444","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Splines form an elegant bridge between the continuous real world and the discrete computational world. Their tensor-product form lifts many univariate properties effortlessly to the surfaces, volumes and beyond. Irregularities, where the tensor-structure breaks down, therefore deserve attention – and provide a rich source of mathematical challenges and insights. This paper reviews and categorizes techniques for splines on meshes with irregularities. Of particular interest are quad-dominant meshes that can have n 6= 4 valent interior points and T-junctions where quad-strips end. “Generalized” splines can use quad-dominant meshes as control nets both for modeling geometry and to support engineering analysis without additional meshing. 2010 Mathematics Subject Classification. 65N35, 15A15.
{"title":"Splines for Meshes with Irregularities","authors":"J. Peters","doi":"10.5802/smai-jcm.57","DOIUrl":"https://doi.org/10.5802/smai-jcm.57","url":null,"abstract":"Splines form an elegant bridge between the continuous real world and the discrete computational world. Their tensor-product form lifts many univariate properties effortlessly to the surfaces, volumes and beyond. Irregularities, where the tensor-structure breaks down, therefore deserve attention – and provide a rich source of mathematical challenges and insights. This paper reviews and categorizes techniques for splines on meshes with irregularities. Of particular interest are quad-dominant meshes that can have n 6= 4 valent interior points and T-junctions where quad-strips end. “Generalized” splines can use quad-dominant meshes as control nets both for modeling geometry and to support engineering analysis without additional meshing. 2010 Mathematics Subject Classification. 65N35, 15A15.","PeriodicalId":376888,"journal":{"name":"The SMAI journal of computational mathematics","volume":"7 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123907296","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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