arXiv subjects

Francesc Verdugo

Publications and source records attributed to Francesc Verdugo.

At least 19 recordsLinked to original sources

arXiv2026

Topology-optimized distributed 3d anisotropic Raman emission

Topology optimization (TO) of 3D surface-enhanced Raman scattering (SERS) substrates faces challenges in managing field singularities and modeling orientation-averaged anisotropic molecules. We present 3D TO for manufacturable SERS substrates that maximize spatially averaged signals from randomly oriented, anisotropic molecules in both elastic and inelastic scattering. A new trace formulation provides a closed-form rotational average of anisotropic Raman tensors, which are not equivalent to isotropic molecules because of tensor nonlinearity. Optimized silver and Si3N4 devices show that lengthscale constraints are sufficient to suppress designs that rely on unphysical mathematical field divergences at sharp corners. Metallic designs deliver broadband enhancement and remain robust to typical Raman shifts, whereas dielectric designs yield narrower, quality-factor-limited gains that are inferior to metallic designs for quality factors below about 500. Our approach readily incorporates additional physics, such as a nonlinear damage model. Together, these results provide a practical route to improved manufacturable SERS substrates and extend naturally to other distributed-emitter design problems.

physics.optics↗

arXiv2024

Space-time unfitted finite elements on moving explicit geometry representations

This work proposes a novel variational approximation of partial differential equations on moving geometries determined by explicit boundary representations. The benefits of the proposed formulation are the ability to handle large displacements of explicitly represented domain boundaries without generating body-fitted meshes and remeshing techniques. For the space discretization, we use a background mesh and an unfitted method that relies on integration on cut cells only. We perform this intersection by using clipping algorithms. To deal with the mesh movement, we pullback the equations to a reference configuration (the spatial mesh at the initial time slab times the time interval) that is constant in time. This way, the geometrical intersection algorithm is only required in 3D, another key property of the proposed scheme. At the end of the time slab, we compute the deformed mesh, intersect the deformed boundary with the background mesh, and consider an exact transfer operator between meshes to compute jump terms in the time discontinuous Galerkin integration. The transfer is also computed using geometrical intersection algorithms. We demonstrate the applicability of the method to fluid problems around rotating (2D and 3D) geometries described by oriented boundary meshes. We also provide a set of numerical experiments that show the optimal convergence of the method.

cs.CE↗

arXiv2023

Space-time unfitted finite element methods for time-dependent problems on moving domains

We propose a space-time scheme that combines an unfitted finite element method in space with a discontinuous Galerkin time discretisation for the accurate numerical approximation of parabolic problems with moving domains or interfaces. We make use of an aggregated finite element space to attain robustness with respect to the cut locations. The aggregation is performed slab-wise to have a tensor product structure of the space-time discrete space, which is required in the numerical analysis. We analyse the proposed algorithm, providing stability, condition number bounds and anisotropic \emph{a priori} error estimates. A set of numerical experiments confirm the theoretical results for a parabolic problem on a moving domain. The method is applied for a mass transfer problem with changing topology.

math.NA↗

arXiv2022

Trace formulation for photonic inverse design with incoherent sources

Spatially incoherent light sources, such as spontaneously emitting atoms, naively require Maxwell's equations to be solved many times to obtain the total emission, which becomes computationally intractable in conjunction with large-scale optimization (inverse design). We present a trace formulation of incoherent emission that can be efficiently combined with inverse design, even for topology optimization over thousands of design degrees of freedom. Our formulation includes previous reciprocity-based approaches, limited to a few output channels (e.g. normal emission), as special cases, but generalizes to a continuum of emission directions by exploiting the low-rank structure of emission problems. We present several examples of incoherent-emission topology optimization, including tailoring the geometry of fluorescent particles, a periodically emitting surface, and a structure emitting into a waveguide mode, as well as discussing future applications to problems such as Raman sensing and cathodoluminescence.

physics.optics↗

arXiv2022

Robust high-order unfitted finite elements by interpolation-based discrete extension

In this work, we propose a novel formulation for the solution of partial differential equations using finite element methods on unfitted meshes. The proposed formulation relies on the discrete extension operator proposed in the aggregated finite element method. This formulation is robust with respect to the location of the boundary/interface within the cell. One can prove enhanced stability results, not only on the physical domain, but on the whole active mesh. However, the stability constants grow exponentially with the polynomial order being used, since the underlying extension operators are defined via extrapolation. To address this issue, we introduce a new variant of aggregated finite elements, in which the extension in the physical domain is an interpolation for polynomials of order higher than two. As a result, the stability constants only grow at a polynomial rate with the order of approximation. We demonstrate that this approach enables robust high-order approximations with the aggregated finite element method. The proposed method is consistent, optimally convergent, and with a condition number that scales optimally for high order approximation.

math.NA↗

arXiv2022

Designing Structures that Maximize Spatially Averaged Surface-Enhanced Raman Spectra

We present a general framework for inverse design of nanopatterned surfaces that maximize spatially averaged surface-enhanced Raman (SERS) spectra from molecules distributed randomly throughout a material or fluid, building upon a recently proposed trace formulation for optimizing incoherent emission. This leads to radically different designs than optimizing SERS emission at a single known location, as we illustrate using several 2D design problems addressing effects of hot-spot density, angular selectivity, and nonlinear damage. We obtain optimized structures that perform about 4 times better than coating with optimized spheres or bowtie structures and about 20 times better when the nonlinear damage effects are included.

physics.optics↗

arXiv2022

A monolithic Finite Element formulation for the hydroelastic analysis of Very Large Floating Structures

In this work we present a novel monolithic Finite Element Method (FEM) for the hydroelastic analysis of Very Large Floating Structures (VLFS) with arbitrary shapes that is stable, energy conserving and overcomes the need of an iterative algorithm. The new formulation enables a fully monolithic solution of the linear free-surface flow, described by linear potential flow, coupled with floating thin structures, described by the Euler-Bernoulli beam or Poisson-Kirchhoff plate equations. The formulation presented in this work is general in the sense that solutions can be found in the frequency and time domains, it overcomes the need of using elements with C1 continuity by employing a continuous/discontinuous Galerkin (C/DG) approach, and it is suitable for Finite Elements of arbitrary order. We show that the proposed approach can accurately describe the hydroelastic phenomena of VLFS with a variety of tests, including structures with elastic joints, variable bathymetry and arbitrary structural shapes.

cs.CE↗

arXiv2022

Geometrical discretisations for unfitted finite elements on explicit boundary representations

Unfitted (also known as embedded or immersed) finite element approximations of partial differential equations are very attractive because they have much lower geometrical requirements than standard body-fitted formulations. These schemes do not require body-fitted unstructured mesh generation. In turn, the numerical integration becomes more involved, because one has to compute integrals on portions of cells (only the interior part). In practice, these methods are restricted to level-set (implicit) geometrical representations, which drastically limit their application. Complex geometries in industrial and scientific problems are usually determined by (explicit) boundary representations. In this work, we propose an automatic computational framework for the discretisation of partial differential equations on domains defined by oriented boundary meshes. The geometrical kernel that connects functional and geometry representations generates a two-level integration mesh and a refinement of the boundary mesh that enables the straightforward numerical integration of all the terms in unfitted finite elements. The proposed framework has been applied with success on all analysis-suitable oriented boundary meshes (almost 5,000) in the Thingi10K database and combined with an unfitted finite element formulation to discretise partial differential equations on the corresponding domains.

math.NA↗

arXiv2021

The software design of Gridap: a Finite Element package based on the Julia JIT compiler

We present the software design of Gridap, a novel finite element library written exclusively in the Julia programming language, which is being used by several research groups world-wide to simulate complex physical phenomena such as magnetohydrodynamics, photonics, weather modeling, non-linear solid mechanics, and fluid-structure interaction problems. The library provides a feature-rich set of discretization techniques for the numerical approximation of a wide range of PDEs, including linear, nonlinear, single-field, and multi-field equations. An expressive API allows users to define PDEs in weak form by a syntax close to the mathematical notation. While this is also available in previous codes, the main novelty of Gridap is that it implements this API without introducing a DSL plus a compiler of variational forms. Instead, it leverages the Julia just-in-time compiler to build efficient code, specialized for the concrete problem at hand. As a result, there is no need to use different languages for the computational back-end and the user front-end anymore, thus eliminating the so-called two-language problem. Gridap also provides a low-level API that is modular and extensible via the multiple-dispatch paradigm of Julia and provides easy access to the main building blocks of the library. The main contribution of this paper is the detailed presentation of the novel software abstractions behind the Gridap design that leverages the new software possibilities provided by the Julia language. The second main contribution of the article is a performance comparison against FEniCS. We measure CPU times needed to assemble discrete systems of linear equations for different problem types and show that the performance of Gridap is comparable to FEniCS, demonstrating that the new software design does not compromise performance. Gridap is freely available at Github and distributed under an MIT license.

cs.MS↗

arXiv2021

Linking ghost penalty and aggregated unfitted methods

In this work, we analyse the links between ghost penalty stabilisation and aggregation-based discrete extension operators for the numerical approximation of elliptic partial differential equations on unfitted meshes. We explore the behavior of ghost penalty methods in the limit as the penalty parameter goes to infinity, which returns a strong version of these methods. We observe that these methods suffer locking in that limit. On the contrary, aggregated finite element spaces are locking-free because they can be expressed as an extension operator from well-posed to ill-posed degrees of freedom. Next, we propose novel ghost penalty methods that penalise the distance between the solution and its aggregation-based discrete extension. These methods are locking-free and converge to aggregated finite element methods in the infinite penalty parameter limit. We include an exhaustive set of numerical experiments in which we compare weak (ghost penalty) and strong (aggregated finite elements) schemes in terms of error quantities, condition numbers and sensitivity with respect to penalty coefficients on different geometries, intersection locations and mesh topologies.

math.NA↗

arXiv2021

The aggregated unfitted finite element method on parallel tree-based adaptive meshes

In this work, we present an adaptive unfitted finite element scheme that combines the aggregated finite element method with parallel adaptive mesh refinement. We introduce a novel scalable distributed-memory implementation of the resulting scheme on locally-adapted Cartesian forest-of-trees meshes. We propose a two-step algorithm to construct the finite element space at hand by means of a discrete extension operator that carefully mixes aggregation constraints of problematic degrees of freedom, which get rid of the small cut cell problem, and standard hanging degree of freedom constraints, which ensure trace continuity on non-conforming meshes. Following this approach, we derive a finite element space that can be expressed as the original one plus well-defined linear constraints. Moreover, it requires minimum parallelization effort, using standard functionality available in existing large-scale finite element codes. Numerical experiments demonstrate its optimal mesh adaptation capability, robustness to cut location and parallel efficiency, on classical Poisson $hp$-adaptivity benchmarks. Our work opens the path to functional and geometrical error-driven dynamic mesh adaptation with the aggregated finite element method in large-scale realistic scenarios. Likewise, it can offer guidance for bridging other scalable unfitted methods and parallel adaptive mesh refinement.

math.NA↗

arXiv2021

Physics-informed neural networks with hard constraints for inverse design

Inverse design arises in a variety of areas in engineering such as acoustic, mechanics, thermal/electronic transport, electromagnetism, and optics. Topology optimization is a major form of inverse design, where we optimize a designed geometry to achieve targeted properties and the geometry is parameterized by a density function. This optimization is challenging, because it has a very high dimensionality and is usually constrained by partial differential equations (PDEs) and additional inequalities. Here, we propose a new deep learning method -- physics-informed neural networks with hard constraints (hPINNs) -- for solving topology optimization. hPINN leverages the recent development of PINNs for solving PDEs, and thus does not rely on any numerical PDE solver. However, all the constraints in PINNs are soft constraints, and hence we impose hard constraints by using the penalty method and the augmented Lagrangian method. We demonstrate the effectiveness of hPINN for a holography problem in optics and a fluid problem of Stokes flow. We achieve the same objective as conventional PDE-constrained optimization methods based on adjoint methods and numerical PDE solvers, but find that the design obtained from hPINN is often simpler and smoother for problems whose solution is not unique. Moreover, the implementation of inverse design with hPINN can be easier than that of conventional methods.

physics.comp-ph↗

arXiv2020

A user-guide to Gridap -- grid-based approximation of partial differential equations in Julia

We present Gridap, a new scientific software library for the numerical approximation of partial differential equations (PDEs) using grid-based approximations. Gridap is an open-source software project exclusively written in the Julia programming language. The main motivation behind the development of this library is to provide an easy-to-use framework for the development of complex PDE solvers in a dynamically typed style without sacrificing the performance of statically typed languages. This work is a tutorial-driven user guide to the library. It covers some popular linear and nonlinear PDE systems for scalar and vector fields, single and multi-field problems, conforming and nonconforming finite element discretizations, on structured and unstructured meshes of simplices and hexahedra.

cs.MS↗

arXiv2020

A generic finite element framework on parallel tree-based adaptive meshes

In this work we formally derive and prove the correctness of the algorithms and data structures in a parallel, distributed-memory, generic finite element framework that supports h-adaptivity on computational domains represented as forest-of-trees. The framework is grounded on a rich representation of the adaptive mesh suitable for generic finite elements that is built on top of a low-level, light-weight forest-of-trees data structure handled by a specialized, highly parallel adaptive meshing engine, for which we have identified the requirements it must fulfill to be coupled into our framework. Atop this two-layered mesh representation, we build the rest of data structures required for the numerical integration and assembly of the discrete system of linear equations. We consider algorithms that are suitable for both subassembled and fully-assembled distributed data layouts of linear system matrices. The proposed framework has been implemented within the FEMPAR scientific software library, using p4est as a practical forest-of-octrees demonstrator. A strong scaling study of this implementation when applied to Poisson and Maxwell problems reveals remarkable scalability up to 32.2K CPU cores and 482.2M degrees of freedom. Besides, a comparative performance study of FEMPAR and the state-of-the-art deal.ii finite element software shows at least comparative performance, and at most factor 2-3 improvements in the h-adaptive approximation of a Poisson problem with first- and second-order Lagrangian finite elements, respectively.

cs.MS↗

arXiv2019

Distributed-memory parallelization of the aggregated unfitted finite element method

The aggregated unfitted finite element method (AgFEM) is a methodology recently introduced in order to address conditioning and stability problems associated with embedded, unfitted, or extended finite element methods. The method is based on removal of basis functions associated with badly cut cells by introducing carefully designed constraints, which results in well-posed systems of linear algebraic equations, while preserving the optimal approximation order of the underlying finite element spaces. The specific goal of this work is to present the implementation and performance of the method on distributed-memory platforms aiming at the efficient solution of large-scale problems. In particular, we show that, by considering AgFEM, the resulting systems of linear algebraic equations can be effectively solved using standard algebraic multigrid preconditioners. This is in contrast with previous works that consider highly customized preconditioners in order to allow one the usage of iterative solvers in combination with unfitted techniques. Another novelty with respect to the methods available in the literature is the problem sizes that can be handled with the proposed approach. While most of previous references discussing linear solvers for unfitted methods are based on serial non-scalable algorithms, we propose a parallel distributed-memory method able to efficiently solve problems at large scales. This is demonstrated by means of a weak scaling test defined on complex 3D domains up to 300M degrees of freedom and one billion cells on 16K CPU cores in the Marenostrum-IV platform. The parallel implementation of the AgFEM method is available in the large-scale finite element package FEMPAR.

math.NA↗

arXiv2018

Mixed aggregated finite element methods for the unfitted discretization of the Stokes problem

In this work, we consider unfitted finite element methods for the numerical approximation of the Stokes problem. It is well-known that this kind of methods lead to arbitrarily ill-conditioned systems. In order to solve this issue, we consider the recently proposed aggregated finite element method, originally motivated for coercive problems. However, the well-posedness of the Stokes problem is far more subtle and relies on a discrete inf-sup condition. We consider mixed finite element methods that satisfy the discrete version of the inf-sup condition for body-fitted meshes, and analyze how the discrete inf-sup is affected when considering the unfitted case. We propose different aggregated mixed finite element spaces combined with simple stabilization terms, which can include pressure jumps and/or cell residuals, to fix the potential deficiencies of the aggregated inf-sup. We carry out a complete numerical analysis, which includes stability, optimal a priori error estimates, and condition number bounds that are not affected by the small cut cell problem. For the sake of conciseness, we have restricted the analysis to hexahedral meshes and discontinuous pressure spaces. A thorough numerical experimentation bears out the numerical analysis. The aggregated mixed finite element method is ultimately applied to two problems with non-trivial geometries.

math.NA↗

arXiv2017

The aggregated unfitted finite element method for elliptic problems

Unfitted finite element techniques are valuable tools in different applications where the generation of body-fitted meshes is difficult. However, these techniques are prone to severe ill conditioning problems that obstruct the efficient use of iterative Krylov methods and, in consequence, hinders the practical usage of unfitted methods for realistic large scale applications. In this work, we present a technique that addresses such conditioning problems by constructing enhanced finite element spaces based on a cell aggregation technique. The presented method, called aggregated unfitted finite element method, is easy to implement, and can be used, in contrast to previous works, in Galerkin approximations of coercive problems with conforming Lagrangian finite element spaces. The mathematical analysis of the new method states that the condition number of the resulting linear system matrix scales as in standard finite elements for body-fitted meshes, without being affected by small cut cells, and that the method leads to the optimal finite element convergence order. These theoretical results are confirmed with 2D and 3D numerical experiments.

cs.CE↗

arXiv2017

Robust and scalable domain decomposition solvers for unfitted finite element methods

Unfitted finite element methods, e.g., extended finite element techniques or the so-called finite cell method, have a great potential for large scale simulations, since they avoid the generation of body-fitted meshes and the use of graph partitioning techniques, two main bottlenecks for problems with non-trivial geometries. However, the linear systems that arise from these discretizations can be much more ill-conditioned, due to the so-called small cut cell problem. The state-of-the-art approach is to rely on sparse direct methods, which have quadratic complexity and are thus not well suited for large scale simulations. In order to solve this situation, in this work we investigate the use of domain decomposition preconditioners (balancing domain decomposition by constraints) for unfitted methods. We observe that a straightforward application of these preconditioners to the unfitted case has a very poor behavior. As a result, we propose a {customization} of the classical BDDC methods based on the stiffness weighting operator and an improved definition of the coarse degrees of freedom in the definition of the preconditioner. These changes lead to a robust and algorithmically scalable solver able to deal with unfitted grids. A complete set of complex 3D numerical experiments show the good performance of the proposed preconditioners.

math.NA↗