ROM methodology: We aim at consolidating the development of computational reduction techniques for problems described by parametrized mathematical models, governed by partial differential equations (PDEs). Parameters might be both physical (material properties, nondimensional coefficients such as Reynolds number, boundary conditions, forcing terms) and geometric (i.e., quantities which characterize the shape of the domain). The focus of this project concerns scientific computing and modelling, with a special interest in computational mechanics, electromagnetism, optimization, and control. In all these cases, iterative minimization procedures entailing several numerical resolutions of PDEs (each time with different values of control or design variables or different physical/geometric scenarios) are involved, thus requiring high computational efficiency. For this reason, suitable reduced order model (ROM) techniques, such as reduced basis methods (RBM) and hierarchical model reduction (HiMod), provide an effective strategy to contain the overall computational cost. HiMod, for instance, provides surrogate models for problems characterized by an intrinsic directionality, with transverse components relevant locally. The idea of HiMod is to discretize the full model via a combination of a 1D finite element approximation along the leading direction with a modal expansion for the transverse dynamics. This leads to solving a system of 1D models. Research activities have led to a significant development of ROM for many problems, and to real-life applications in several scenarios. However, to make these techniques more efficient and viable in a more applied and technological context, several methodological issues are still to be investigated and developed. For instance, in order to perform efficient numerical simulations in complex and variable geometric configurations, as required for instance in engineering or medical applications, ROMs need to be coupled with efficient adaptive and/or parameterization techniques. In order to treat complex geometric properties, the recent developments of the Isogeometric Analysis for curves and surfaces, based on B-Spline or NURBS (Non-Uniform Rational B-Spline), can help ROM to become more performing. This guarantees an efficient and highly integrated treatment and processing of geometries developed within CAD systems, as well as the generation of high-quality computational grids, without resorting to any geometric approximation. To make ROM even more efficient, the combination with BEM (Boundary Element Method) is advisable in linear problems instead of the classical finite elements. Another possible tool to save on the computational cost is represented by the employment of anisotropic adaptive meshes, whose elements are automatically tuned in terms of size, shape, and orientation.
Our research project is largely of fundamental and methodological nature, hence we expect to contribute to the advance of knowledge in the framework of numerical analysis for the approximation of PDEs. Nevertheless, we expect that our results will provide new insights for the design of numerical schemes in several application areas. The novelties of our research will concern new analysis tools for the adopted methodologies and new software for scientific computing involving problems ranging from academic examples to real life applications in the fields of fluid-dynamics, electromagnetism, structural mechanics, cardiology and in general multiphysics problems in computational science and engineering.
Models and methods developed within the research program will contribute to open new scenarios in the numerical approximation of PDEs, making it possible the solution of more complex problems with significantly reduced computational times (order of seconds). In addition to industry, the techniques developed within this project have a great potential impact in biomedical and biological fields, ensuring the possibility to increase the complexity of the problems to be addressed, for example by including also uncertainty quantification of some parameters of interest. A remarkable strength of the proposed methodology is the possibility to offer a clear separation between computational resources, together with computational stages. Clusters and supercomputers are useful in the offline stage to generate a basis of solutions, by means of repetitive and expensive computations. Less powerful devices, such as laptops but even tablets or smartphones, enable during the online stage to obtain real-time calculations. The development of the proposed methods is complementary to important investigations dealing with high-performance, large scale, parallel computing environments. By easily accessing a database of pre-computed solutions, reduced-order methods make it promptly available both large computing platforms and increasingly complex mathematical models, such as for example in healthcare or shipyard facilities.
