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The proposal concerns the development of numerical methods suitable for simulations for safety assessments in several fields of civil and environmental engineering. Civil structures are subject to aging and the evolution of the surrounding environment increases the possibility of dramatic events. Moreover, most natural catastrophic events are hardly predictable and local or global human activities can strongly impact the stability of the environment surrounding civil structures, for example by changing the hydrogeological properties of the soil. This project aims to develop new numerical tools suitable to tackle all the engineering models and the physical phenomena needed to describe such catastrophic events overcoming usual difficulties such as, for example, mesh generation problems, mesh adaptation problems, mesh quality problems, and incompressibility conditions. The Virtual Element Method (VEM) is a polygonal/polyhedral Finite Element Method that was developed for this aim. Although its great success in managing many of the overmentioned problems, some issues still affect its applicability in fields like non-linear elasticity, elastoplasticity, viscoplasticity and in many engineering fields that deal with strongly non-linear problems, strongly anisotropic problems or problems governed by uncommon differential operators. The main issue is related to the non-consistent stabilization term needed by these methods to get well-posed discrete problems for the simulation. This project will focus on stabilization-free VEMs with the aim of retaining the key features of VEM but without the side-effect of the pollution introduced by the stabilizing terms. Stabilization-free VEMs can be developed for general discretizations, but for some specific engineering applications, “ad hoc” methods can be more effective and will be developed. Stabilization-free methods are preferable for low-order methods, whereas for high-order methods can be less effective. For high-order methods, optimal stabilizations requiring a low computational cost will be investigated in the project. Stabilization-free VEMs will be derived and applied to elasticity, elastoplasticity and Richards equations for modelling unsaturated soils. Moreover, optimal cheap stabilization methods, mainly for high-order VEMs and optimal preconditioner and solvers will be investigated from a mathematical viewpoint. These mathematical tools will be applied to understand slope and soil stability and to forecast natural catastrophic events like landslides and to water resources conservation and preservation simulating the roots-soil interaction in unsaturated and fractured porous media. The developed methods will be useful for simulations in the field of geothermal energy exploitation and storage that can help reducing the human impact on the earth's environment.