arXiv subjects

Alberto Paganini

Publications and source records attributed to Alberto Paganini.

9 recordsLinked to original sources

arXiv2026

Shape optimisation of nonlinear Naghdi shells on discrete geometries

A thin shell carries load through its shape and, at finite deflections, its stiffness changes with the load itself, so a shape optimum found with a linear model can be far from optimal. We present an automated framework that embeds the fully geometrically nonlinear shell response in the shape-optimisation loop. The forward model, a five-parameter nonlinear Naghdi shell stabilised against locking by partial selective reduced integration, operates directly on a discrete (faceted) triangulation with a numerically recovered director field -- dispensing with the exact mid-surface parameterisation of isogeometric approaches, a chart that ceases to exist once the geometry itself is the design variable. Implemented in Firedrake, the model generates its residual, consistent tangent and adjoint automatically; shape derivatives, computed by algorithmic differentiation through the full load-continuation solve, drive the Fireshape/ROL trust-region optimiser. The forward solver reproduces the Sze/Abaqus benchmark for a clamped semi-cylindrical shell under a point load, capturing the progressive stiffening that a geometrically linear model cannot reproduce. The optimisation is validated against the COMSOL benchmark, a sheet-metal bracket under bending: the framework develops the same off-mid-plane corrugation mechanism and attains an 87% reduction of elastic strain energy within the prescribed displacement budget, matching the benchmark's magnitude and area change. Applied to the curved semi-cylinder, it forms a smooth stiffening crease that reduces the shell's average deflection under load by 78%.

math.NA↗

arXiv2026

A Cloud-Accessible Open-Source Framework for the Electromagnetic Modelling of Applied Superconductors

We present the H-cloud formalism, a cloud-accessible and open-source finite-element framework for electromagnetic modelling of applied superconductors. The proposed method expresses the nonlinear electromagnetic response of type-II superconductors in a curl-conforming discretisation based on N\'ed\'elec finite elements, where the tangential applied-field boundary condition, nonlinear E-J power law, and fully implicit time-discrete residual are stated explicitly at the variational level, all within a scripted Python finite-element workflow. The weak form is used as the basis for forward simulation and for extension to adjoint differentiation and PDE-constrained optimisation, while keeping the governing assumptions, boundary conditions, and solver structure fully visible to the user. The implementation is realised in Firedrake with UFL and PETSc-backed nonlinear solvers, allowing the identical script to run on local machines and in browser-accessible environments such as Google Colab without reformulating the problem. The method is verified on the canonical magnetisation benchmark of a cylindrical superconductor under Bean-like penetration conditions and then benchmarked against an independently constructed COMSOL model for a practical high temperature superconducting Bi2212 wire. Across matched mesh studies, the open-source workflow reproduces the commercial-reference magnetisation loops to within approximately \(1\%\) , with relative peak errors below 1.5%, while cloud execution preserves the same numerical solution at rather modest additional runtime considering the use of (freely available) reduced hardware resources. The proposed framework provides a rigorous, reproducible, and portable route for superconducting simulation, benchmarking, and future optimisation-led modelling of applied and functional superconductors, shareable and executable into open cloud environments.

cond-mat.supr-con↗

arXiv2024

Level-set shape optimization via polytopic discontinuous Galerkin methods

We introduce a new level-set shape optimization approach based on polytopic (i.e., polygonal in two and polyhedral in three spatial dimensions) discontinuous Galerkin methods. The approach benefits from the geometric mesh flexibility of polytopic discontinuous Galerkin methods to resolve the zero-level set accurately and efficiently. Additionally, we employ suitable Runge-Kutta discontinuous Galerkin methods to update the level-set function on a fine underlying simplicial mesh. We discuss the construction and implementation of the approach, explaining how to modify shape derivate formulas to compute consistent shape gradient approximations using discontinuous Galerkin methods, and how to recover dG functions into smoother ones. Numerical experiments on unconstrained and PDE-constrained test cases evidence the good properties of the proposed methodology.

math.NA↗

arXiv2024

G-Adaptivity: optimised graph-based mesh relocation for finite element methods

We present a novel, and effective, approach to achieve optimal mesh relocation in finite element methods (FEMs). The cost and accuracy of FEMs is critically dependent on the choice of mesh points. Mesh relocation (r-adaptivity) seeks to optimise the mesh geometry to obtain the best solution accuracy at given computational budget. Classical r-adaptivity relies on the solution of a separate nonlinear "meshing" PDE to determine mesh point locations. This incurs significant cost at remeshing, and relies on estimates that relate interpolation- and FEM-error. Recent machine learning approaches have focused on the construction of fast surrogates for such classical methods. Instead, our new approach trains a graph neural network (GNN) to determine mesh point locations by directly minimising the FE solution error from the PDE system Firedrake to achieve higher solution accuracy. Our GNN architecture closely aligns the mesh solution space to that of classical meshing methodologies, thus replacing classical estimates for optimality with a learnable strategy. This allows for rapid and robust training and results in an extremely efficient and effective GNN approach to online r-adaptivity. Our method outperforms both classical, and prior ML, approaches to r-adaptive meshing. In particular, it achieves lower FE solution error, whilst retaining the significant speed-up over classical methods observed in prior ML work.

cs.LG↗

arXiv2021

Control of bifurcation structures using shape optimization

Many problems in engineering can be understood as controlling the bifurcation structure of a given device. For example, one may wish to delay the onset of instability, or bring forward a bifurcation to enable rapid switching between states. We propose a numerical technique for controlling the bifurcation diagram of a nonlinear partial differential equation by varying the shape of the domain. Specifically, we are able to delay or advance a given branch point to a target parameter value. The algorithm consists of solving a shape optimization problem constrained by an augmented system of equations, the Moore--Spence system, that characterize the location of the branch points. Numerical experiments on the Allen--Cahn, Navier--Stokes, and hyperelasticity equations demonstrate the effectiveness of this technique in a wide range of settings.

math.NA↗

arXiv2020

Fireshape: a shape optimization toolbox for Firedrake

We introduce Fireshape, an open-source and automated shape optimization toolbox for the finite element software Firedrake. Fireshape is based on the moving mesh method and allows users with minimal shape optimization knowledge to tackle with ease challenging shape optimization problems constrained to partial differential equations (PDEs).

math.OC↗

arXiv2018

Automated shape differentiation in the Unified Form Language

We discuss automating the calculation of weak shape derivatives in the Unified Form Language (Aln{\ae}s et al., ACM Trans. Math. Softw., 2014) by introducing an appropriate additional step in the pullback from physical to reference space that computes G\^ateaux derivatives with respect to the coordinate field. We illustrate the ease of use with several examples.

math.NA↗

arXiv2018

Weakly-normal basis vector fields in RKHS with an application to shape Newton methods

We construct a space of vector fields that are normal to differentiable curves in the plane. Its basis functions are defined via saddle point variational problems in reproducing kernel Hilbert spaces (RKHSs). First, we study the properties of these basis vector fields and show how to approximate them. Then, we employ this basis to discretise shape Newton methods and investigate the impact of this discretisation on convergence rates.

math.OC↗

arXiv2018

Formal solutions for polarized radiative transfer. III. Stiffness and instability

Efficient numerical approximation of the polarized radiative transfer equation is challenging because this system of ordinary differential equations exhibits stiff behavior, which potentially results in numerical instability. This negatively impacts the accuracy of formal solvers, and small step-sizes are often necessary to retrieve physical solutions. This work presents stability analyses of formal solvers for the radiative transfer equation of polarized light, identifies instability issues, and suggests practical remedies. In particular, the assumptions and the limitations of the stability analysis of Runge-Kutta methods play a crucial role. On this basis, a suitable and pragmatic formal solver is outlined and tested. An insightful comparison to the scalar radiative transfer equation is also presented.

astro-ph.SR↗