Thesis Summary:

The main goal of this doctoral thesis is to develop, analyze, implement and apply several virtual element methods (VEM) conforming in H² on general polygonal meshes for solving diverse fourth order problems that arise in solid mechanics, with the purpose of establishing an original contribution in the virtual element theory. We firstly present two linear eigenvalue problems for thin plates, namely vibration and buckling problems of Kirchhoff plates. In relation to the vibration problem of Kirchhoff plates, the study is focused on developing a C¹-virtual element method for the analysis of the numerical approximation of the vibration frequencies and vibration modes of thin plates. A variational formulation based only on the transverse displacement of the plate is proposed. A conforming discretization of H² by means of VEM which is simple in terms of degrees of freedom and coding aspects is introduced. Under standard assumptions on the computational domain, it is established that the resulting scheme provides a correct approximation of the spectrum. In addition, optimal order error estimates for the eigenfunctions and a double order for the eigenvalues are proved. On the other hand, a virtual element method of high order on polygonal meshes for solving the buckling problem governed by Kirchhoff equations is developed as second work of this thesis. Then, a C¹ conforming virtual element discretization of arbitrary order ≥ 2 is introduced. In addition, the standard spectral theory for compact operators is applied to prove that the resulting scheme provides a correct approximation of the spectrum. Moreover, optimal order error estimates for the buckling modes and a double order for the buckling coefficients are proved. Subsequently, we introduce and analyze a C¹-virtual element method for solving a nonlinear problem of plates modelled by von Kármán equations. A variational formulation based on C¹-conforming discretization by means of VEM is proposed. The method has the advantages of supporting general polygonal meshes and is simple in terms of coding aspects. We prove that the discrete problem is well posed for h (letter usually chosen to denote the discretization parameter) small enough. Moreover, optimal error estimates are derived. We finally propose and analyze a virtual element method for numerically solving a non-linear nonself- adjoint eigenvalue problem called the transmission eigenvalue problem. By introducing a new unknown (which belongs to H_0^1 ) in the system of equations the eigenvalue problem is linearized. Next, a variational formulation on H_0^2 × H_0^1 is defined. Then, by defining a solution operator, which result to be non-self-adjoint with respect to usual seminorm of H_0^2 × H_0^1 and compact, the spectrum of the variational formulation is characterized (which is the spectrum of the transmission eigenvalue problem). A C¹×C⁰ conforming discretization by means of the VEM is proposed. The classical spectral theory for non-sel-fadjoint compact operators is employed in order to analyze the correct spectral approximation. The optimal order of convergence for the eigenvalues and the eigenfunctions is derived. For all the situations described above, several numerical experiments illustrating good performance of the proposed methods, and confirming the theoretical analysis, are presented.

ISI Publications from the Thesis

David MORA, Iván VELáSQUEZ: Virtual elements for the transmission eigenvalue problem on polytopal meshes. SIAM Journal on Scientific Computing, vol. 43, 4, pp. A2425-A2447, (2021).

Carlo LOVADINA, David MORA, Iván VELáSQUEZ: A virtual element method for the von Kármán equations. ESAIM: Mathematical Modelling and Numerical Analysis, vol. 55, 2, pp. 533-560, (2021).

David MORA, Iván VELáSQUEZ: Virtual element for the buckling problem of Kirchhoff-Love plates. Computer Methods in Applied Mechanics and Engineering, vol. 360, Art. Num. 112687, (2020).

David MORA, Iván VELáSQUEZ: A virtual element method for the transmission eigenvalue problem. Mathematical Models and Methods in Applied Sciences, vol. 28, 14, pp. 2803-2831, (2018).

David MORA, Gonzalo RIVERA, Iván VELáSQUEZ: A virtual element method for the vibration problem of Kirchhoff plates. ESAIM: Mathematical Modelling and Numerical Analysis, vol. 52, 4, pp. 1437-1456, (2018).