Numerical Analysis and Modeling

Luca Bergamaschi,
Massimiliano Ferronato,
Annamaria Mazzia,
Pietro Teatini,
Carlo Janna,
Claudia Zoccarato,
Matteo Frigo,
Laura Gazzola,

Research areas:

Numerical Linear Algebra
The research focuses on the development of fast solution techniques for linear systems or eigenproblems typically arising from real world problems. This research is of paramount importance since in several applications the most time-consuming stage of a simulation stems from the solution of linear algebra problems. In this field, iterative solution methods based on Krylov subspaces are very attractive due to their potential parallelism that makes them viable on large supercomputers. A key point to make iterative methods effective is the availability of a suitable preconditioner, that is an operator capable of transforming the original problem in an easier one. In particular, the group studies and develops preconditioners for systems arising from continuous optimization or multiphysics problems where the resulting matrix has typically a block form and the preconditioner mimics the operator properties by taking the same block structure and properly approximating the Schur complement. The research also concerns Scalable preconditioners based on approximate inverses or Algebraic multigrid. These preconditioners are purely algebraic as they do not need any information other than the system matrix and are amenable of extremely efficient parallel implementation.

Discretization Techniques
Several natural processes can be described through Partial Differential Equations (PDEs) or system of PDEs which need to be solved numerically to allow for prediction or design. The numerical solution of such problems is typically based on the discretization of the continuum through finite elements (FE) or finite differences (FD). FE are usually the method of choice for instance in mechanical problems, however, in other contexts different techniques are preferred and represent the main focus of this research. Interface Elements (IE) are developed to model mechanical discontinuities in solid bodies, such as fault and fractures in geological formations. Finite Volume (FV) and mixed techniques are studied in problems where mass conservation is a key property that the numerical model has to preserve, such as in multiphase flow in porous media. Finally, in problems where the continuum geometry is very complex and difficult to subdivide in regular pieces, Meshless (MLPG) or Virtual Elements (VEM) are studied in order to reduce the discretization troubles.

Numerical Modeling
The research focuses on the simulation of the mechanics of fractures in porous media in saturated and unsaturated conditions by developing and implementing numerical models based on finite element, mixed finite elements, finite volumes and meshless analysis. The applications concern the simulations of subsurface hydrology and geomechanical consequences related to the exploitation of natural resources in aquifers or deep hydrocarbon reservoirs. Specifically, the research is aimed at predicting anthropogenic subsidence and potential induced seismicity triggered by fluid extraction, e.g. gas/oil and/or water, from the subsurface and by underground gas storage in depleted reservoirs.
Moreover, in geological structures characterized by system of faults, the research is aimed at simulating the possible fault re-activation with numerical models based on interface finite elements able to reproduce the mechanical behavior of faults and fractures in the subsurface. Specific applications are aimed at simulating the unexpected consequences due to the exploitation of subsurface resources such as water or hydrocarbons. Finally, Data Assimilation techniques are studied and developed for a more robust and efficient calibration of numerical models and estimation of the associate uncertainties arising from mathematical approximations of physical processes.

Keywords: Numerical Linear Algebra, Preconditioning, Scientific Computing, Discretization Techniques, Numerical Modeling