Paul Louis George is Director of Research at the French Institute for Research in Computer Science and Automation (Inria) and one of the most internationally recognized experts in meshing.Houman Borouchaki is Professor at the University of Technology of Troyes (UTT) in France. He is an expert on meshing problems, geometric modeling and applications in solid mechanics.Frédéric Alauzet is a researcher at Inria, both with particular expertise in meshing adaptation, error estimators, resolution methods (advanced solvers in fluid mechanics) and remeshing methods.Adrien Loseille is a researcher at Inria, both with particular expertise in meshing adaptation, error estimators, resolution methods (advanced solvers in fluid mechanics) and remeshing methods.Patrick Laug is a researcher at Inria with particular expertise in geometric modeling and the generation of curve and surface meshes.Loïc Maréchal, a long-time collaborator at Inria as an engineer, is an essential reference on hexahedra.

• Foreword ixIntroduction xiChapter 1. Metrics, Definitions and Properties 11.1. Definitions and properties 21.2. Metric interpolation and intersection 61.2.1. Metric interpolation 71.2.2. Metric intersection 131.3. Geometric metrics 141.3.1. Geometric metric for a curve 161.3.2. Geometric metric for a surface 171.3.3. Turning any metric into a geometric metric 231.4. Meshing metrics 231.5. Metrics gradation 241.6. Element metric 311.6.1. Metric of a simplicial element 311.6.2. Metric of a non-simplicial element 371.6.3. Metric of an element of arbitrary degree 381.7. Element shape and metric quality 381.8. Practical computations in the presence of a metric 461.8.1. Calculation of the length 461.8.2. The calculation of an angle, area or volume 49Chapter 2. Interpolation Errors and Metrics 532.1. Some properties 542.2. Interpolation error of a quadratic function 552.3. Bézier formulation and interpolation error 622.3.1. For a quadratic function 632.3.2. For a cubic function 662.3.3. For a polynomial function of arbitrary degree 802.3.4. Error threshold or mesh density 852.4. Computations of discrete derivatives 862.4.1. The L2double projection method 862.4.2. Green formula 882.4.3. Least square and Taylor 89Chapter 3. Curve Meshing 933.1. Parametric curve meshing 953.1.1. Curve in R3 953.1.2. About metrics used and computations of lengths 993.1.3. Curve plotted on a patch 1033.2. Discrete curve meshing 1043.3. Remeshing a meshed curve 104Chapter 4. Simplicial Meshing 1074.1. Definitions 1084.2. Variety (surface) meshing 1094.2.1. Patch-based meshing 1104.2.2. Discrete surface remeshing 1194.2.3. Meshing using a volume mesher 1204.3. The meshing of a plane or of a volume domain 1224.3.1. Tree-based method 1234.3.2. Front-based method 1264.3.3. Delaunay-based method 1294.3.4. Remeshing of a meshed domain 1344.4. Other generation methods? 136Chapter 5. Non-simplicial Meshing 1415.1. Definitions 1425.2. Variety meshing 1435.3. Construction methods for meshing a planar or volume domain 1455.3.1. Cylindrical geometry and extrusion method 1475.3.2. Algebraic methods and block-based methods 1485.3.3. Tree-based method 1725.3.4. Pairing method 1745.3.5. Polygonal or polyhedral cell meshing 1765.3.6. Construction of boundary layers 1775.4. Other generation methods 1825.4.1. “Q-morphism” or “H-morphism” meshing 1825.4.2. Meshing using a reference frame field 1835.5. Topological invariants (quadrilaterals and hexahedra) 185Chapter 6. High-order Mesh Construction 1956.1. Straight meshes 1966.1.1. Local node numbering 1966.1.2. Overall node numeration 2016.1.3. Node positions 2046.1.4. On filling up matrices according to element degrees 2076.2. Construction of curved meshes 2086.2.1. First-degree mesh 2096.2.2. Node creation 2096.2.3. Deformation and validation 2106.2.4. General scheme 2116.3. Curved meshes on a variety, curve or surface 215Chapter 7. Mesh Optimization 2257.1. Toward a definition of quality 2267.2. Optimization process 2337.2.1. Global methods 2337.2.1.1. Optimization of a cost function 2337.2.1.2. Iterative relaxation of the position of vertices by duality (simplices) 2347.2.1.3. Global optimization of the position of vertices (quadrilaterals and hexahedra) 2357.2.2. Local operators and local methods 2367.2.2.1. Vertex moves by barycentering 2367.2.2.2. Vertex moves and Laplacian operator 2377.2.2.3. Moving or removing vertices and flips by insertion or reinsertion 2417.2.2.4. Edge flips 2417.2.2.5. Cluster of edge flips 2437.2.2.6. Edge or face flip by reinsertion 2447.2.2.7. Edge slicing 2447.2.2.8. Removal of an edge by merging 2457.2.2.9. Metric field update 2467.2.2.10. Topological and metric criteria 2467.2.2.11. Strategies 2467.3. Planar mesh 2487.4. Surface mesh 2507.5. Volume meshing 2517.6. High-degree meshing 254Chapter 8. Mesh Adaptation 2658.1. Generic framework for adaptive computation, the continuous mesh 2668.1.1. Duality between discrete and continuous geometric entities 2678.1.2. Duality between discrete and continuous interpolation error 2698.1.3. Discrete–continuous duality in one diagram 2728.2. Optimal control of the interpolation error in Lp-norm 2728.3. Generic scheme of stationary adaptation 2798.3.1. Error estimators 2828.3.2. Interpolation of solution fields 2878.4. Unsteady adaptation 2898.4.1. Space–time error estimators based on the characteristics of the solution 2908.4.2. Extension of the error analysis for the fixed-point algorithm for unsteady mesh adaptation 2918.4.3. Mesh adaptation for unsteady problems 2928.4.4. Unsteady mesh adaptation targeted at a function of interest 2948.4.5. Conservative interpolation of solution fields 2958.5. Mobile geometry with or without deformation 2978.5.1. General context of the adaptation for mobile and/or deformable geometries 2978.5.2. ALE continuous optimal mesh minimizing the interpolation error in Lp-norm 2988.5.3. Space–time error estimator for moving geometry problems 300Chapter 9. Meshing and Parallelism 3039.1. Renumbering via a filling curve 3049.2. Parallelism: two memory paradigms and different strategies 3079.3. Algorithm parallelization for mesh construction 3129.4. Parallelization of a mesh construction process, partition then meshing 3249.5. Mesh parallelization, meshing then partition 326Chapter 10. Applications 33110.1. Surface meshing 33210.2. In computational fluid dynamics 33410.3. Computational solid mechanics 34110.4. Computational electromagnetism 34510.5. Renumbering and parallelism 34610.6. Other more exotic applications 349Chapter 11. Some Algorithms and Formulas 35311.1. Local numbering of nodes of high-order elements 35411.2. Length computations etc., in the presence of a metric field 36411.3. Quality 369Conclusions and Perspectives 373Bibliography 375Index 387

Paul Louis George is Director of Research at the French Institute for Research in Computer Science and Automation (Inria) and one of the most internationally recognized experts in meshing.Houman Borouchaki is Professor at the University of Technology of Troyes (UTT) in France. He is an expert on meshing problems, geometric modeling and applications in solid mechanics.Frédéric Alauzet is a researcher at Inria, both with particular expertise in meshing adaptation, error estimators, resolution methods (advanced solvers in fluid mechanics) and remeshing methods.Adrien Loseille is a researcher at Inria, both with particular expertise in meshing adaptation, error estimators, resolution methods (advanced solvers in fluid mechanics) and remeshing methods.Patrick Laug is a researcher at Inria with particular expertise in geometric modeling and the generation of curve and surface meshes.Loïc Maréchal, a long-time collaborator at Inria as an engineer, is an essential reference on hexahedra.

• Foreword ixIntroduction xiChapter 1. Metrics, Definitions and Properties 11.1. Definitions and properties 21.2. Metric interpolation and intersection 61.2.1. Metric interpolation 71.2.2. Metric intersection 131.3. Geometric metrics 141.3.1. Geometric metric for a curve 161.3.2. Geometric metric for a surface 171.3.3. Turning any metric into a geometric metric 231.4. Meshing metrics 231.5. Metrics gradation 241.6. Element metric 311.6.1. Metric of a simplicial element 311.6.2. Metric of a non-simplicial element 371.6.3. Metric of an element of arbitrary degree 381.7. Element shape and metric quality 381.8. Practical computations in the presence of a metric 461.8.1. Calculation of the length 461.8.2. The calculation of an angle, area or volume 49Chapter 2. Interpolation Errors and Metrics 532.1. Some properties 542.2. Interpolation error of a quadratic function 552.3. Bézier formulation and interpolation error 622.3.1. For a quadratic function 632.3.2. For a cubic function 662.3.3. For a polynomial function of arbitrary degree 802.3.4. Error threshold or mesh density 852.4. Computations of discrete derivatives 862.4.1. The L2double projection method 862.4.2. Green formula 882.4.3. Least square and Taylor 89Chapter 3. Curve Meshing 933.1. Parametric curve meshing 953.1.1. Curve in R3 953.1.2. About metrics used and computations of lengths 993.1.3. Curve plotted on a patch 1033.2. Discrete curve meshing 1043.3. Remeshing a meshed curve 104Chapter 4. Simplicial Meshing 1074.1. Definitions 1084.2. Variety (surface) meshing 1094.2.1. Patch-based meshing 1104.2.2. Discrete surface remeshing 1194.2.3. Meshing using a volume mesher 1204.3. The meshing of a plane or of a volume domain 1224.3.1. Tree-based method 1234.3.2. Front-based method 1264.3.3. Delaunay-based method 1294.3.4. Remeshing of a meshed domain 1344.4. Other generation methods? 136Chapter 5. Non-simplicial Meshing 1415.1. Definitions 1425.2. Variety meshing 1435.3. Construction methods for meshing a planar or volume domain 1455.3.1. Cylindrical geometry and extrusion method 1475.3.2. Algebraic methods and block-based methods 1485.3.3. Tree-based method 1725.3.4. Pairing method 1745.3.5. Polygonal or polyhedral cell meshing 1765.3.6. Construction of boundary layers 1775.4. Other generation methods 1825.4.1. “Q-morphism” or “H-morphism” meshing 1825.4.2. Meshing using a reference frame field 1835.5. Topological invariants (quadrilaterals and hexahedra) 185Chapter 6. High-order Mesh Construction 1956.1. Straight meshes 1966.1.1. Local node numbering 1966.1.2. Overall node numeration 2016.1.3. Node positions 2046.1.4. On filling up matrices according to element degrees 2076.2. Construction of curved meshes 2086.2.1. First-degree mesh 2096.2.2. Node creation 2096.2.3. Deformation and validation 2106.2.4. General scheme 2116.3. Curved meshes on a variety, curve or surface 215Chapter 7. Mesh Optimization 2257.1. Toward a definition of quality 2267.2. Optimization process 2337.2.1. Global methods 2337.2.1.1. Optimization of a cost function 2337.2.1.2. Iterative relaxation of the position of vertices by duality (simplices) 2347.2.1.3. Global optimization of the position of vertices (quadrilaterals and hexahedra) 2357.2.2. Local operators and local methods 2367.2.2.1. Vertex moves by barycentering 2367.2.2.2. Vertex moves and Laplacian operator 2377.2.2.3. Moving or removing vertices and flips by insertion or reinsertion 2417.2.2.4. Edge flips 2417.2.2.5. Cluster of edge flips 2437.2.2.6. Edge or face flip by reinsertion 2447.2.2.7. Edge slicing 2447.2.2.8. Removal of an edge by merging 2457.2.2.9. Metric field update 2467.2.2.10. Topological and metric criteria 2467.2.2.11. Strategies 2467.3. Planar mesh 2487.4. Surface mesh 2507.5. Volume meshing 2517.6. High-degree meshing 254Chapter 8. Mesh Adaptation 2658.1. Generic framework for adaptive computation, the continuous mesh 2668.1.1. Duality between discrete and continuous geometric entities 2678.1.2. Duality between discrete and continuous interpolation error 2698.1.3. Discrete–continuous duality in one diagram 2728.2. Optimal control of the interpolation error in Lp-norm 2728.3. Generic scheme of stationary adaptation 2798.3.1. Error estimators 2828.3.2. Interpolation of solution fields 2878.4. Unsteady adaptation 2898.4.1. Space–time error estimators based on the characteristics of the solution 2908.4.2. Extension of the error analysis for the fixed-point algorithm for unsteady mesh adaptation 2918.4.3. Mesh adaptation for unsteady problems 2928.4.4. Unsteady mesh adaptation targeted at a function of interest 2948.4.5. Conservative interpolation of solution fields 2958.5. Mobile geometry with or without deformation 2978.5.1. General context of the adaptation for mobile and/or deformable geometries 2978.5.2. ALE continuous optimal mesh minimizing the interpolation error in Lp-norm 2988.5.3. Space–time error estimator for moving geometry problems 300Chapter 9. Meshing and Parallelism 3039.1. Renumbering via a filling curve 3049.2. Parallelism: two memory paradigms and different strategies 3079.3. Algorithm parallelization for mesh construction 3129.4. Parallelization of a mesh construction process, partition then meshing 3249.5. Mesh parallelization, meshing then partition 326Chapter 10. Applications 33110.1. Surface meshing 33210.2. In computational fluid dynamics 33410.3. Computational solid mechanics 34110.4. Computational electromagnetism 34510.5. Renumbering and parallelism 34610.6. Other more exotic applications 349Chapter 11. Some Algorithms and Formulas 35311.1. Local numbering of nodes of high-order elements 35411.2. Length computations etc., in the presence of a metric field 36411.3. Quality 369Conclusions and Perspectives 373Bibliography 375Index 387