arXiv subjects

Pablo Antolin

Publications and source records attributed to Pablo Antolin.

At least 19 recordsLinked to original sources

arXiv2026

Deflation-based preconditioning for immersed finite element methods and immersogeometric analysis

Trimming is a ubiquitous operation in computer-aided-design whereby parts of a geometry are merged, intersected, or simply discarded. While it grants virtually unlimited flexibility in geometric design, it introduces a plethora of other difficulties when such geometries are used within immersed finite element methods. In particular, small cut elements lead to severely ill-conditioned system matrices requiring dedicated penalization, stabilization, or preconditioning techniques. In this work, we highlight the limitations of existing preconditioning strategies by first carefully examining the condition number of the diagonally scaled matrix and later providing realistic counter-examples for some well-established preconditioning strategies. Building on those insights, we propose a robust deflation-based preconditioning technique tailored to immersed finite element methods.

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arXiv2026

A ROM-based BDDC solver for unfitted p-FEM level-set-based two-dimensional lattice structures

We present a domain decomposition method for the fast simulation of large two-dimensional lattice structures described by level set functions. The method does not rely on homogenization or multiscale techniques, and therefore avoids their underlying assumptions such as scale separation and periodicity. Individual cells are defined through level set functions and mapped into physical space using arbitrary order mappings, which allows the creation of complex graded designs with varying geometries and topologies. The discretization is based on unfitted p-FEM, where each cell is approximated by a single high order element. This choice naturally handles the implicit geometric description and provides high accuracy with a moderate number of degrees of freedom. The solver is built on the Balanced Domain Decomposition by Constraints method, where each cell corresponds to one subdomain. To accelerate the assembly of the cell stiffness matrices, we combine a fast assembly technique that separates the contributions of the geometric mapping from the trimmed domain with a reduced order model based on the matrix discrete empirical interpolation method. The ROM surrogate is trained offline and can be reused for any geometric mapping, restricting the expensive quadrature on cut elements to the training stage. A stabilization term is introduced to ensure the scalability of the solver when using the ROM approximation, at the cost of a small and controllable error. We validate the method through a series of numerical experiments and demonstrate its performance on a 2D problem with more than 17,000 cells of varying geometry, which is solved in approximately 30 seconds on a standard laptop. The number of solver iterations grows only mildly as the number of subdomains increases, provided the ratio between subdomain and mesh sizes is kept constant, consistent with the scalability properties of BDDC methods.

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arXiv2026

Efficient Fine-Scale Simulation of Nonlinear Hyperelastic Lattice Structures

With the growing maturity of additive manufacturing, the fabrication of architected or lattice-based metamaterials has become a reality for industrial applications. These materials combine lightweight design with tailored mechanical properties, most of which exhibit pronounced nonlinear, especially large-deformation, behaviors. The main numerical challenge therefore lies in performing nonlinear simulations of such lattice structures, which may contain thousands of geometrically intricate unit cells, while lacking sufficient scale separation for multiscale homogenization schemes to be applicable straightforwardly. In this work, we propose a dedicated solver for the full volumetric fine-scale simulation of nonlinear hyperelastic lattice structures that drastically reduces both memory and computational costs. The key idea is to exploit the intrinsic self-similarity of the cells through a reduced-order modeling strategy applied within a domain-decomposition framework. At each Newton iteration, a limited set of principal cells is identified through a dedicated, weakly intrusive, EIM-like approach, allowing all local tangent operators to be expressed as linear combinations of a few principal ones. This enables fast and memory-efficient operator assembly, and then feeds an efficient inexact FETI-DP based preconditioner at the solution stage, resulting in a quasi matrix-free algorithm for the nonlinear analysis. Numerical experiments in two and three dimensions demonstrate significant computational gains, with runtime reductions from several hours to a few tens of minutes and memory savings by factors of about three, while maintaining full fine-scale accuracy. Notably, the proposed strategy enables the computation of problems involving thousands of cells (i.e., millions of degrees of freedom) within a few minutes on an off-the-shelf laptop.

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arXiv2026

A General and Robust 3D Finite Element Dynamics Framework for Railway Vehicle-Bridge Interaction with Nonlinear Wheel-Rail Contact Modeling

A key challenge in 3D finite element models of coupled railway vehicle-bridge dynamics is the rigorous definition of kinematic constraints and the development of an efficient, robust solution. This paper presents a novel approach that can be implemented in general finite element software using constraint equations tailored to wheel-rail contact behavior, essential for analyzing lateral vehicle-bridge interactions. The method employs absolute coordinates to describe the motion of nodes defining the track position and orientation for each wheelset, without assuming infinitesimal displacements or rotations. This general formulation enables realistic simulations of extreme scenarios involving large lateral movements caused by strong winds or earthquakes. The proposed wheel-rail contact model is first validated against published results, and a 3D numerical example demonstrates the method's performance and capabilities.

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arXiv2025

On design, analysis, and hybrid manufacturing of microstructured blade-like geometries

With the evolution of new manufacturing technologies such as multi-material 3D printing, one can think of new type of objects that consist of considerably less, yet heterogeneous, material, consequently being porous, lighter and cheaper, while having the very same functionality as the original object when manufactured from one single solid material. We aim at questioning five decades of traditional paradigms in geometric CAD and focus at new generation of CAD objects that are not solid, but contain heterogeneous free-form internal microstructures. We propose a unified manufacturing pipeline that involves all stages, namely design, optimization, manufacturing, and inspection of microstructured free-form geometries. We demonstrate our pipeline on an industrial test case of a blisk blade that sustains the desired pressure limits, yet requires significantly less material when compared to the solid counterpart.

cs.GR↗

arXiv2025

Stabilization techniques for immersogeometric analysis of plate and shell problems in explicit dynamics

Finite element plate and shell formulations are ubiquitous in structural analysis for modeling all kinds of slender structures, both for static and dynamic analyses. The latter are particularly challenging as the high order nature of the underlying partial differential equations and the slenderness of the structures all impose a stringent constraint on the critical time step in explicit dynamics. Unfortunately, badly cut elements in immersed finite element discretizations further aggravate the issue. While lumping the mass matrix often increases the critical time step, it might also trigger spurious oscillations in the approximate solution thereby compromising the numerical solution. In this article, we extend our previous work in \cite{voet2025stabilization} to allow stable immersogeometric analysis of plate and shell problems with lumped mass matrices. This technique is based on polynomial extensions and restores a level of accuracy comparable to boundary-fitted discretizations.

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arXiv2024

The Immersed Boundary Conformal Method for Kirchhoff-Love and Reissner-Mindlin shells

This work utilizes the Immersed Boundary Conformal Method (IBCM) to analyze Kirchhoff-Love and Reissner-Mindlin shell structures within an immersed domain framework. Immersed boundary methods involve embedding complex geometries within a background grid, which allows for great flexibility in modeling intricate shapes and features despite the simplicity of the approach. The IBCM method introduces additional layers conformal to the boundaries, allowing for the strong imposition of Dirichlet boundary conditions and facilitating local refinement. In this study, the construction of boundary layers is combined with high-degree spline-based approximation spaces to further increase efficiency. The Nitsche method, employing non-symmetric average operators, is used to couple the boundary layers with the inner patch, while stabilizing the formulation with minimal penalty parameters. High-order quadrature rules are applied for integration over cut elements and patch interfaces. Numerical experiments demonstrate the efficiency and accuracy of the proposed formulation, highlighting its potential for complex shell structures modeled through Kirchhoff-Love and Reissner-Mindlin theories. These tests include the generation of conformal interfaces, the coupling of Kirchhoff-Love and Reissner-Mindlin theories, and the simulation of a damaged shell.

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arXiv2024

An Interior Penalty coupling strategy for Isogeometric non-conformal Kirchhoff-Love shell patches

This work focuses on the coupling of trimmed shell patches using Isogeometric Analysis, based on higher continuity splines that seamlessly meet the $C^1$ requirement of Kirchhoff-Love-based discretizations. Weak enforcement of coupling conditions is achieved through the symmetric interior penalty method, where the fluxes are computed using their correct variationally consistent expression that was only recently proposed and is unprecedentedly adopted herein in the context of coupling conditions. The constitutive relationships account for generically laminated materials, although the proposed tests are conducted under the assumption of uniform thickness and lamination sequence. Numerical experiments assess the method for an isotropic and a laminated plate, as well as an isotropic hyperbolic paraboloid shell from the new shell obstacle course. The boundary conditions and domain force are chosen to reproduce manufactured analytical solutions, which are taken as reference to compute rigorous convergence curves in the $L^2$, $H^1$, and $H^2$ norms, that closely approach optimal ones predicted by theory. Additionally, we conduct a final test on a complex structure comprising five intersecting laminated cylindrical shells, whose geometry is directly imported from a STEP file. The results exhibit excellent agreement with those obtained through commercial software, showcasing the method's potential for real-world industrial applications.

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arXiv2023

Reduced Order Modeling based Inexact FETI-DP solver for lattice structures

This paper addresses the overwhelming computational resources needed with standard numerical approaches to simulate architected materials. Those multiscale heterogeneous lattice structures gain intensive interest in conjunction with the improvement of additive manufacturing as they offer, among many others, excellent stiffness-to-weight ratios. We develop here a dedicated HPC solver that benefits from the specific nature of the underlying problem in order to drastically reduce the computational costs (memory and time) for the full fine-scale analysis of lattice structures. Our purpose is to take advantage of the natural domain decomposition into cells and, even more importantly, of the geometrical and mechanical similarities among cells. Our solver consists in a so-called inexact FETI-DP method where the local, cell-wise operators and solutions are approximated with reduced order modeling techniques. Instead of considering independently every cell, we end up with only few principal local problems to solve and make use of the corresponding principal cell-wise operators to approximate all the others. It results in a scalable algorithm that saves numerous local factorizations. Our solver is applied for the isogeometric analysis of lattices built by spline composition, which offers the opportunity to compute the reduced basis with macro-scale data, thereby making our method also multiscale and matrix-free. The solver is tested against various 2D and 3D analyses. It shows major gains with respect to black-box solvers; in particular, problems of several millions of degrees of freedom can be solved with a simple computer within few minutes.

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arXiv2023

Fast parametric analysis of trimmed multi-patch isogeometric Kirchhoff-Love shells using a local reduced basis method

This contribution presents a model order reduction framework for real-time efficient solution of trimmed, multi-patch isogeometric Kirchhoff-Love shells. In several scenarios, such as design and shape optimization, multiple simulations need to be performed for a given set of physical or geometrical parameters. This step can be computationally expensive in particular for real world, practical applications. We are interested in geometrical parameters and take advantage of the flexibility of splines in representing complex geometries. In this case, the operators are geometry-dependent and generally depend on the parameters in a non-affine way. Moreover, the solutions obtained from trimmed domains may vary highly with respect to different values of the parameters. Therefore, we employ a local reduced basis method based on clustering techniques and the Discrete Empirical Interpolation Method to construct affine approximations and efficient reduced order models. In addition, we discuss the application of the reduction strategy to parametric shape optimization. Finally, we demonstrate the performance of the proposed framework to parameterized Kirchhoff-Love shells through benchmark tests on trimmed, multi-patch meshes including a complex geometry. The proposed approach is accurate and achieves a significant reduction of the online computational cost in comparison to the standard reduced basis method.

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arXiv2023

Stabilized isogeometric formulation of the Stokes problem on overlapping patches

We present a novel stabilized isogeometric formulation for the Stokes problem, where the geometry of interest is obtained via overlapping NURBS (non-uniform rational B-spline) patches, i.e., one patch on top of another in an arbitrary but predefined hierarchical order. All the visible regions constitute the computational domain, whereas independent patches are coupled through visible interfaces using Nitsche's formulation. Such a geometric representation inevitably involves trimming, which may yield trimmed elements of extremely small measures (referred to as bad elements) and thus lead to the instability issue. Motivated by the minimal stabilization method that rigorously guarantees stability for trimmed geometries [1], in this work we generalize it to the Stokes problem on overlapping patches. Central to our method is the distinct treatments for the pressure and velocity spaces: Stabilization for velocity is carried out for the flux terms on interfaces, whereas pressure is stabilized in all the bad elements. We provide a priori error estimates with a comprehensive theoretical study. Through a suite of numerical tests, we first show that optimal convergence rates are achieved, which consistently agrees with our theoretical findings. Second, we show that the accuracy of pressure is significantly improved by several orders using the proposed stabilization method, compared to the results without stabilization. Finally, we also demonstrate the flexibility and efficiency of the proposed method in capturing local features in the solution field.

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arXiv2022

A localized reduced basis approach for unfitted domain methods on parameterized geometries

This work introduces a reduced order modeling (ROM) framework for the solution of parameterized second-order linear elliptic partial differential equations formulated on unfitted geometries. The goal is to construct efficient projection-based ROMs, which rely on techniques such as the reduced basis method and discrete empirical interpolation. The presence of geometrical parameters in unfitted domain discretizations entails challenges for the application of standard ROMs. Therefore, in this work we propose a methodology based on i) extension of snapshots on the background mesh and ii) localization strategies to decrease the number of reduced basis functions. The method we obtain is computationally efficient and accurate, while it is agnostic with respect to the underlying discretization choice. We test the applicability of the proposed framework with numerical experiments on two model problems, namely the Poisson and linear elasticity problems. In particular, we study several benchmarks formulated on two-dimensional, trimmed domains discretized with splines and we observe a significant reduction of the online computational cost compared to standard ROMs for the same level of accuracy. Moreover, we show the applicability of our methodology to a three-dimensional geometry of a linear elastic problem.

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arXiv2022

Analysis-aware defeaturing of complex geometries with Neumann features

Local modifications of a computational domain are often performed in order to simplify the meshing process and to reduce computational costs and memory requirements. However, removing geometrical features of a domain often introduces a non-negligible error in the solution of a differential problem in which it is defined. In this work, we extend the results from [1] by studying the case of domains containing an arbitrary number of distinct Neumann features, and by performing an analysis on Poisson's, linear elasticity, and Stokes' equations. We introduce a simple, computationally cheap, reliable, and efficient a posteriori estimator of the geometrical defeaturing error. Moreover, we also introduce a geometric refinement strategy that accounts for the defeaturing error: Starting from a fully defeatured geometry, the algorithm determines at each iteration step which features need to be added to the geometrical model to reduce the defeaturing error. These important features are then added to the (partially) defeatured geometrical model at the next iteration, until the solution attains a prescribed accuracy. A wide range of two- and three-dimensional numerical experiments are finally reported to illustrate this work.

math.NA↗

arXiv2022

Reduced order modelling of nonaffine problems on parameterized NURBS multipatch geometries

This contribution explores the combined capabilities of reduced basis methods and IsoGeometric Analysis (IGA) in the context of parameterized partial differential equations. The introduction of IGA enables a unified simulation framework based on a single geometry representation for both design and analysis. The coupling of reduced basis methods with IGA has been motivated in particular by their combined capabilities for geometric design and solution of parameterized geometries. In most IGA applications, the geometry is modelled by multiple patches with different physical or geometrical parameters. In particular, we are interested in nonaffine problems characterized by a high-dimensional parameter space. We consider the Empirical Interpolation Method (EIM) to recover an affine parametric dependence and combine domain decomposition to reduce the dimensionality. We couple spline patches in a parameterized setting, where multiple evaluations are performed for a given set of geometrical parameters, and employ the Static Condensation Reduced Basis Element (SCRBE) method. At the common interface between adjacent patches a static condensation procedure is employed, whereas in the interior a reduced basis approximation enables an efficient offline/online decomposition. The full order model over which we setup the RB formulation is based on NURBS approximation, whereas the reduced basis construction relies on techniques such as the Greedy algorithm or proper orthogonal decomposition (POD). We demonstrate the developed procedure using an illustrative model problem on a three-dimensional geometry featuring a multi-dimensional geometrical parameterization.

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arXiv2021

Region extraction in mesh intersection

Region extraction is a very common task in both Computer Science and Engineering with several applications in object recognition and motion analysis, among others. Most of the literature focuses on regions delimited by straight lines, often in the special case of intersection detection among two unstructured meshes. While classical region extraction algorithms for line drawings and mesh intersection algorithms have proved to be able to deal with many applications, the advances in Isogeometric Analysis require a generalization of such problem to the case in which the regions to be extracted are bounded by an arbitrary number of curved segments. In this work we present a novel region extraction algorithm that allows a precise numerical integration of functions defined in different spline spaces. The presented algorithm has several interesting applications in contact problems, mortar methods, and quasi-interpolation problems.

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arXiv2021

DeepBND: a Machine Learning approach to enhance Multiscale Solid Mechanics

Effective properties of materials with random heterogeneous structures are typically determined by homogenising the mechanical quantity of interest in a window of observation. The entire problem setting encompasses the solution of a local PDE and some averaging formula for the quantity of interest in such domain. There are relatively standard methods in the literature to completely determine the formulation except for two choices: i) the local domain itself and the ii) boundary conditions. Hence, the modelling errors are governed by the quality of these two choices. The choice i) relates to the degree of representativeness of a microscale sample, i.e., it is essentially a statistical characteristic. Naturally, its reliability is higher as the size of the observation window becomes larger and/or the number of samples increases. On the other hand, excepting few special cases there is no automatic guideline to handle ii). Although it is known that the overall effect of boundary condition becomes less important with the size of the microscale domain, the computational cost to simulate such large problem several times might be prohibitive even for relatively small accuracy requirements. Here we introduce a machine learning procedure to select most suitable boundary conditions for multiscale problems, particularly those arising in solid mechanics. We propose the combination Reduced-Order Models and Deep Neural Networks in an offline phase, whilst the online phase consists in the very same homogenisation procedure plus one (cheap) evaluation of the trained model for boundary conditions. Hence, the method allows an implementation with minimal changes in existing codes and the use of relatively small domains without losing accuracy, which reduces the computational cost by several orders of magnitude.

math.NA↗

arXiv2021

Robust Numerical Integration on Curved Polyhedra Based on Folded Decompositions

We present a novel method to perform numerical integration over curved polyhedra enclosed by high-order parametric surfaces. Such a polyhedron is first decomposed into a set of triangular and/or rectangular pyramids, whose certain faces correspond to the given parametric surfaces. Each pyramid serves as an integration cell with a geometric mapping from a standard parent domain (e.g., a unit cube), where the tensor-product Gauss quadrature is adopted. As no constraint is imposed on the decomposition, certain resulting pyramids may intersect with themselves, and thus their geometric mappings may present negative Jacobian values. We call such cells the folded cells and refer to the corresponding decomposition as a folded decomposition. We show that folded cells do not cause any issues in practice as they are only used to numerically compute certain integrals of interest. The same idea can be applied to planar curved polygons as well. We demonstrate both theoretically and numerically that folded cells can retain the same accuracy as the cells with strictly positive Jacobians. On the other hand, folded cells allow for a much easier and much more flexible decomposition for general curved polyhedra, on which one can robustly compute integrals. In the end, we show that folded cells can flexibly and robustly accommodate real-world complex geometries by presenting several examples in the context of immersed isogeometric analysis, where involved sharp features can be well respected in generating integration cells.

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arXiv2021

Fast and Multiscale Formation of Isogeometric matrices of Microstructured Geometric Models

The matrix formation associated to high-order discretizations is known to be numerically demanding. Based on the existing procedure of interpolation and lookup, we design a multiscale assembly procedure to reduce the exorbitant assembly time in the context of isogeometric linear elasticity of complex microstructured geometries modeled via spline compositions. The developed isogeometric approach involves a polynomial approximation occurring at the macro-scale and the use of lookup tables with pre-computed integrals incorporating the micro-scale information. We provide theoretical insights and numerical examples to investigate the performance of the procedure. The strategy turns out to be of great interest not only to form finite element operators but also to compute other quantities in a fast manner as for instance sensitivity analyses commonly used in design optimization.

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