In this book we gather recent mathematical developments and engineering applications of Trefftz methods, with particular emphasis on the Method of Fundamental Solutions (MFS). These are true meshless methods that have the advantage of avoiding the need to set up a mesh altogether, and therefore going beyond the reduction of the mesh to a boundary. These Trefftz methods have advantages in several engineering applications, for instance in inverse problems where the domain is unknown and some numerical methods would require a remeshing approach.
Trefftz methods are also known to perform very well with regular domains and regular data in boundary value problems, achieving exponential convergence. On the other hand, they may also under certain conditions, exhibit instabilities and lead to ill-conditioned systems.
This book is divided into ten chapters that illustrate recent advances in Trefftz methods and their application to engineering problems. The first eight chapters are devoted to the MFS and variants whereas the last two chapters are devoted to related meshless engineering applications. Part of these selected contributions were presented in the 9th International Conference on Trefftz Methods and 5th International Conference on the MFS, held in 2019, July 29-31, in Lisbon, Portugal.
Carlos Alves graduated from the University of Lisbon and received his PhD in Applied Mathematics from Ecole Polytechnique, France. He is a Professor in the department of Mathematics at Instituto Superior Técnico, University of Lisbon. He has published more than 50 papers on numerical analysis, meshless methods and inverse problems in partial differential equations. He has also published two books with Springer as an editor, and he is an editorial board member of Appl. Math. Comp., Inv. Probl. Sc. Eng., and Eng. Anal. Bound. Elements.
Andreas Karageorghis completed both his undergraduate and graduate studies at the University of Oxford and had held positions at the University of Kentucky, the University of Wales and Southern Methodist University. Currently, he is a Professor at the University of Cyprus in the department of Mathematics and Statistics. He has published over 150 research papers in international journals and his research interests include numerical algorithms and scientific computing.
Vitor Leitão graduated from Instituto Superior Técnico of the Technical University of Lisbon, Portugal, and received his Ph.D from the University of Portsmouth, UK. He is currently an Associate Professor of Civil Engineering at Instituto Superior Técnico of the Technical University of Lisbon. He has been the author or co-author of around 100 research works, co-editor of 8 conference proceedings books and special issues, and has been a member in the editorial board of Eng. Anal. Bound. Elements. His current research interests include meshless methods, in particular radial basis functions and fundamental solutions for structural mechanics applications.
Svilen S. Valtchev graduated from the Instituto Superior Técnico of the University of Lisbon where he was also a teaching assistant, and received his Ph.D. in Mathematics from the same university. He is currently an Assistant Professor at the Polytechnic of Leiria and a Researcher at Center for Computational and Stochastic Mathematics of the University of Lisbon. He has authored and co-authored more than 20 research papers and book chapters on meshfree methods for partial differential equations, with applications in acoustic and elastic wave propagation.
1 Chen, M. et al., Solving Partial Di?erential Equations on Surfaces with Fundamental Solutions.- 2 Akhmouch, L. et al., Solving magneto-hydrodynamic (MHD) channel ?ows at large Hartmann numbers by using the method of fundamental solutions.- 3 Gáspár, C. et al., Application of Quadtrees in the Method of Fundamental Solutions using Multi-Level Tools.- 4 Liu, Q., Method of Fundamental Solutions without Fictitious Boundary for Anisotropic Elasticity Problems Based on Mechanical Equilibrium Desingularization.- 5 Barbeiro, S. and Serranho, P., The method of fundamental solutions for the direct elastography problem in the human retina.- 6 Martins, Nuno F. M., Identi?cation and reconstruction of body forces in a Stokes system using shear waves.- 7 Marin, L., MFS-Fading Regularization Method for Inverse BVPs in Anisotropic Heat Conduction.- 8. Mocerino, A., et al., Non-intrusive Estimate of Spatially Varying Internal Heat Flux in Coiled Ducts: Method of Fundamental Solutions Applied to the Reciprocity Functional Approach.- 9 Moldovan, D.I., et al., Uni?ed hybrid-Tre?tz ?nite element formulation for dynamic problems.- 10. Fu, Z.-J. et al., Acoustic bandgap calculation of liquid phononic crystals via the meshless generalized ?nite di?erence method