Project Details
Abstract
Theoretical framework. The theoretical framework of this project dwells in the analysis of new virtual element methods (VEMs) tailored for the approximation of certain classes of differential problems.
A clever definition of trial and test spaces allow for the construction of methods with an accuracy superior to that achieved with standard methods.
Objectives. A first objective of the project is the efficient approximation of solutions to wave propagation problems, such as 2D and 3D Helmholtz problems and time-harmonic Maxwell problems with (piecewise) smooth wave number. A second class of problems is provided by 2D and 3D linear and nonlinear elasticity problems. The leading aim behind the project consists in exploiting the flexibility of the definition of VE spaces. On the one hand, the VEM is a generalization of the finite element method to polygonal/polyhedral grids: it is possible to use adaptive methods, with no problems arising from the presence of hanging nodes.
On the other hand, VE spaces are defined locally as spaces of functions that are solutions to local problems. This allows us to define special VE spaces incorporating properties of the continuous problem. In particular, this project lays at the border of mathematics, scientific computing, physics, and mechanical engineering.
Methodology. The project is split into three distinct work packages. Extensions of the nonconforming Trefftz VEM to the 3D Helmholtz equation, the time-harmonic Maxwell equation, the Helmholtz equation with nonconstant (piecewise smooth) wave number, as well as techniques to reduce the ill-conditioning of wave based methods, are the target of the first work package. In the second workpackage, I aim to introduce the hypercircle, or two energy principle, in the context of VEM, initially for the Poisson problem, and subsequently to the Helmholtz problem. Finally, the third work package focuses on an enriched version of the virtual element method for linear and nonlinear elasticity problems.
Innovation. The innovation of this project consists in providing a structured approach, in order to fully exploit the flexibility of VEM in the construction of ``problem-oriented'' approximation spaces, in the sense that local spaces contain information of the problem to approximate. Virtual elements are used not only because they allow for polygonal/polyhedral meshes, but also because they provide a certain accuracy with a computational cost that is considerably lower than that of standard methods, such as finite elements. The VEMs that will be developed within this project can be regarded as a ``second generation'' VEM.
Primary researchers involved. The group I intend to form should consist of myself, two postdocs, and three external collaborators: E. Artioli, J. Gedicke, G. Manzini.
A clever definition of trial and test spaces allow for the construction of methods with an accuracy superior to that achieved with standard methods.
Objectives. A first objective of the project is the efficient approximation of solutions to wave propagation problems, such as 2D and 3D Helmholtz problems and time-harmonic Maxwell problems with (piecewise) smooth wave number. A second class of problems is provided by 2D and 3D linear and nonlinear elasticity problems. The leading aim behind the project consists in exploiting the flexibility of the definition of VE spaces. On the one hand, the VEM is a generalization of the finite element method to polygonal/polyhedral grids: it is possible to use adaptive methods, with no problems arising from the presence of hanging nodes.
On the other hand, VE spaces are defined locally as spaces of functions that are solutions to local problems. This allows us to define special VE spaces incorporating properties of the continuous problem. In particular, this project lays at the border of mathematics, scientific computing, physics, and mechanical engineering.
Methodology. The project is split into three distinct work packages. Extensions of the nonconforming Trefftz VEM to the 3D Helmholtz equation, the time-harmonic Maxwell equation, the Helmholtz equation with nonconstant (piecewise smooth) wave number, as well as techniques to reduce the ill-conditioning of wave based methods, are the target of the first work package. In the second workpackage, I aim to introduce the hypercircle, or two energy principle, in the context of VEM, initially for the Poisson problem, and subsequently to the Helmholtz problem. Finally, the third work package focuses on an enriched version of the virtual element method for linear and nonlinear elasticity problems.
Innovation. The innovation of this project consists in providing a structured approach, in order to fully exploit the flexibility of VEM in the construction of ``problem-oriented'' approximation spaces, in the sense that local spaces contain information of the problem to approximate. Virtual elements are used not only because they allow for polygonal/polyhedral meshes, but also because they provide a certain accuracy with a computational cost that is considerably lower than that of standard methods, such as finite elements. The VEMs that will be developed within this project can be regarded as a ``second generation'' VEM.
Primary researchers involved. The group I intend to form should consist of myself, two postdocs, and three external collaborators: E. Artioli, J. Gedicke, G. Manzini.
| Status | Active |
|---|---|
| Effective start/end date | 1/03/21 → 28/02/25 |
Keywords
- virtual element method
- wave propagation
- enriched Galerkin method
- plane wave method
- adaptivity and hypercircle
- approximation of singularities