Geometric Partial Differential Equations - Part 2
- 1st Edition, Volume 22 - January 26, 2021
- Imprint: North Holland
- Editors: Andrea Bonito, Ricardo Horacio Nochetto
- Language: English
- Hardback ISBN:9 7 8 - 0 - 4 4 4 - 6 4 3 0 5 - 6
- eBook ISBN:9 7 8 - 0 - 4 4 4 - 6 4 3 0 6 - 3
Besides their intrinsic mathematical interest, geometric partial differential equations (PDEs) are ubiquitous in many scientific, engineering and industrial applications. They re… Read more
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Request a sales quoteBesides their intrinsic mathematical interest, geometric partial differential equations (PDEs) are ubiquitous in many scientific, engineering and industrial applications. They represent an intellectual challenge and have received a great deal of attention recently. The purpose of this volume is to provide a missing reference consisting of self-contained and comprehensive presentations. It includes basic ideas, analysis and applications of state-of-the-art fundamental algorithms for the approximation of geometric PDEs together with their impacts in a variety of fields within mathematics, science, and engineering.
- About every aspect of computational geometric PDEs is discussed in this and a companion volume. Topics in this volume include stationary and time-dependent surface PDEs for geometric flows, large deformations of nonlinearly geometric plates and rods, level set and phase field methods and applications, free boundary problems, discrete Riemannian calculus and morphing, fully nonlinear PDEs including Monge-Ampere equations, and PDE constrained optimization
- Each chapter is a complete essay at the research level but accessible to junior researchers and students. The intent is to provide a comprehensive description of algorithms and their analysis for a specific geometric PDE class, starting from basic concepts and concluding with interesting applications. Each chapter is thus useful as an introduction to a research area as well as a teaching resource, and provides numerous pointers to the literature for further reading
- The authors of each chapter are world leaders in their field of expertise and skillful writers. This book is thus meant to provide an invaluable, readable and enjoyable account of computational geometric PDEs
The targeted audience is mathematically trained research scientists and engineers with basic knowledge in partial differential equations and their numerical approximations.
- Cover image
- Title page
- Table of Contents
- Copyright
- Contributors
- Preface
- Chapter 1: Shape and topology optimization
- Abstract
- 1. Introduction
- 2. Structural optimization problems: models and examples
- 3. Existence and non-existence of optimal shapes: the homogenization effect
- 4. Differentiation with respect to the domain: the boundary variation method of Hadamard
- 5. Shape optimization using Hadamard's method: theoretical framework
- 6. “Body-fitted” geometric optimization
- 7. Eulerian shape capturing topology optimization via the level set method
- 8. A Lagrangian, level set based mesh evolution method for shape and topology optimization
- 9. Shape optimization in different physical contexts and a few recent challenges
- 10. A glimpse to other popular shape and topology optimization frameworks
- 11. Perspectives for shape and topology optimization
- Acknowledgements
- References
- Chapter 2: Optimal transport: discretization and algorithms
- Abstract
- 1. Introduction
- 2. Optimal transport theory
- 3. Discrete optimal transport
- 4. Semi-discrete optimal transport
- 5. Appendix
- Acknowledgements
- References
- Chapter 3: Optimal control of geometric partial differential equations
- Abstract
- 1. Introduction
- 2. The sharp interface approach
- 3. The phase field approach
- References
- Chapter 4: Lagrangian schemes for Wasserstein gradient flows
- Abstract
- 1. Introduction
- 2. Benefit from the gradient flow structures
- 3. 1D Wasserstein using the inverse distribution functions
- 4. Multi-D: first discretize, then optimize
- 5. Optimize then discretize
- 6. Other variational approaches
- Appendix A.
- References
- Chapter 5: The Q-tensor model with uniaxial constraint
- Abstract
- 1. Physics of liquid crystals
- 2. Modeling of nematic liquid crystals
- 3. Mathematical formulation
- 4. The uniaxially constrained Q-model
- 5. Colloidal effects
- 6. Electric fields
- 7. The Landau-de Gennes model with and without the uniaxial constraint
- 8. Conclusion
- References
- Chapter 6: Approximating the total variation with finite differences or finite elements
- Abstract
- 1. Introduction
- 2. Finite differences discretizations of the total variation
- 3. Finite elements based approximations
- 4. Dual discretizations
- 5. Conclusion and perspectives
- Acknowledgements
- References
- Chapter 7: Numerical simulation and benchmarking of drops and bubbles
- Abstract
- 1. Introduction
- 2. Mathematical model
- 3. Discrete projection methods for the incompressible Navier–Stokes equations
- 4. Treatment of surface tension effects
- 5. The Q2/P1/Q1-FEM-interface tracking approach
- 6. The Q2/P1/Q2-FEM-interface capturing approach
- 7. Numerical benchmarking and simulations
- 8. Conclusions
- Acknowledgements
- References
- Chapter 8: Smooth multi-patch discretizations in Isogeometric Analysis
- Abstract
- 1. Introduction
- 2. Multi-patch parametrizations
- 3. Isogeometric spaces
- 4. The AS-G1 construction
- 5. The D-patch construction
- 6. The polar construction
- 7. Numerical tests
- Acknowledgements
- References
- Index
- Edition: 1
- Volume: 22
- Published: January 26, 2021
- No. of pages (Hardback): 570
- No. of pages (eBook): 570
- Imprint: North Holland
- Language: English
- Hardback ISBN: 9780444643056
- eBook ISBN: 9780444643063
AB
Andrea Bonito
Andrea Bonito is professor in the Department of Mathematics at Texas A&M University.
Together with Ricardo H. Nochetto they have more than forty years of experience in the variational formulation and approximation of a wide range of geometric partial differential equations (PDEs). Their work encompass fundamental studies of numerical PDEs: the design, analysis and implementation of efficient numerical algorithms for the approximation of PDEs; and their applications in modern engineering, science, and bio-medical problems.
Affiliations and expertise
Professor, Department of Mathematics, Texas A&M University, USARN
Ricardo Horacio Nochetto
Ricardo H. Nochetto is professor in the Department of Mathematics and the Institute for Physical Science and Technology at the University of Maryland, College Park.
Together with Andrea Bonito they have more than forty years of experience in the variational formulation and approximation of a wide range of geometric partial differential equations (PDEs). Their work encompass fundamental studies of numerical PDEs: the design, analysis and implementation of efficient numerical algorithms for the approximation of PDEs; and their applications in modern engineering, science, and bio-medical problems.
Affiliations and expertise
Professor, Department of Mathematics and the Institute for Physical Science and Technology, Institute for Physical Science and Technology, University of Maryland, USARead Geometric Partial Differential Equations - Part 2 on ScienceDirect