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Person# Riccardo Puppi

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Related research domains (5)

Geometry

Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician who works in the field of geometry is called a geometer. Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry, which includes the notions of point, line, plane, distance, angle, surface, and curve, as fundamental concepts.

Finite element method

The finite element method (FEM) is a popular method for numerically solving differential equations arising in engineering and mathematical modeling. Typical problem areas of interest include the traditional fields of structural analysis, heat transfer, fluid flow, mass transport, and electromagnetic potential. The FEM is a general numerical method for solving partial differential equations in two or three space variables (i.e., some boundary value problems).

Graphical user interface

The graphical user interface, or GUI (ˌdʒi:juːˈaɪ or ˈɡu:i ), is a form of user interface that allows users to interact with electronic devices through graphical icons and audio indicators such as primary notation, instead of text-based UIs, typed command labels or text navigation. GUIs were introduced in reaction to the perceived steep learning curve of command-line interfaces (CLIs), which require commands to be typed on a computer keyboard. The actions in a GUI are usually performed through direct manipulation of the graphical elements.

Related publications (8)

We propose two different discrete formulations for the weak imposition of the Neumann boundary conditions of the Darcy flow. The Raviart-Thomas mixed finite element on both triangular and quadrilateral meshes is considered for both methods. One is a consistent discretization depending on a weighting parameter scaling as $\mathcal O(h^{-1})$, while the other is a penalty-type formulation obtained as the discretization of a perturbation of the original problem and relies on a parameter scaling as $\mathcal O(h^{-k-1})$, $k$ being the order of the Raviart-Thomas space. We rigorously prove that both methods are stable and result in optimal convergent numerical schemes with respect to appropriate mesh-dependent norms, although the chosen norms do not scale as the usual $L^2$-norm. However, we are still able to recover the optimal a priori $L^2$-error estimates for the velocity field, respectively, for high-order and the lowest-order Raviart-Thomas discretizations, for the first and second numerical schemes. Finally, some numerical examples validating the theory are exhibited.

2021Annalisa Buffa, Pablo Antolin Sanchez, Xiaodong Wei, Riccardo Puppi

We present a novel method for isogeometric analysis (IGA) to directly work on geometries constructed by Boolean operations including difference (i.e., trimming), union, and intersection. Particularly, this work focuses on the union operation, which involves multiple independent, generally nonconforming and trimmed, spline patches. Given a series of patches, we overlay them one on top of one another in a certain order. While the invisible part of each patch is trimmed away, the visible parts of all the patches constitute the entire computational domain. We employ Nitsche's method to weakly couple independent patches through visible interfaces. Moreover, we propose a minimal stabilization method to address the instability issue that arises on the interfaces shared by small trimmed elements. We show, in theory, that our proposed method recovers stability and guarantees well-posedness of the problem as well as optimal error estimates. To conclude, we numerically verify the theory by solving the Poisson's equation on various geometries that are obtained by the union operation.

Modern manufacturing engineering is based on a

`design-through-analysis'' workflow. According to this paradigm, a prototype is first designed with Computer-aided-design (CAD) software and then finalized by simulating its physical behavior, which usually involves the simulation of Partial Differential Equations (PDEs) on the designed product. The simulation of PDEs is often performed via finite element discretization techniques.A severe bottleneck in the entire process is undoubtedly the interaction between the design and analysis phases. The prototyped geometries must undergo the time-consuming and human-involved meshing and feature removal processes to become `

analysis-suitable''. This dissertation aims to develop and study numerical solvers for PDEs to improve the integration between numerical simulation and geometric modeling. The thesis is made of two parts. In the first one, we focus our attention on the analysis of isogeometric methods which are robust in geometries constructed using Boolean operations. We consider geometries obtained via trimming (or set difference) and union of multiple overlapping spline patches. As differential model problems, we consider both elliptic (the Poisson problem, in particular) and saddle point problems (the Stokes problem, in particular). As it is standard, the Nitsche method is used for the weak imposition of the essential boundary conditions and to weakly enforce the transmission conditions at the interfaces between the patches. After proving through well-constructed examples that the Nitsche method is not uniformly stable, we design a minimal stabilization technique based on a stabilized computation of normal fluxes (and on a simple modification of the pressure space in the case of the Stokes problem). The main core of this thesis is devoted to the derivation and rigorous mathematical analysis of a stabilization procedure to recover the well-posedness of the discretized problems independently of the geometric configuration in which the domain has been constructed. In the second part of the thesis, we consider a different approach. Instead of considering the underlying spline parameterization of the geometrical object, we immerse it in a much simpler and readily meshed domain. From the mathematical point of view, this approach is closely related to the isogeometric discretizations in trimmed domains treated in the first part. In this case, we consider the Raviart-Thomas finite element discretization of the Darcy flow. First, we analyze a Nitsche and a penalty method for the weak imposition of the essential boundary conditions on a boundary fitted mesh, a problem that was not studied before, not needed for our final goal, but still interesting by itself. Then, we consider the case of a general domain immersed in an underlying mesh unfitted with the boundary. We focus on the Nitsche method presented for the boundary fitted case and study its extension to the unfitted setting. We show that the so-called ghost penalty stabilization provides an effective solution to recover the well-posedness of the formulation and the well-conditioning of the resulting linear system.