arXiv subjects

Rafael Vázquez

Publications and source records attributed to Rafael Vázquez.

At least 19 recordsLinked to original sources

arXiv2026

Isogeometric multi-patch preconditioner for time-dependent electromagnetic problems

We build upon recent developments in high-order spline-based geometric methods for the three-dimensional initial boundary value problem of Maxwell's equations. Previous schemes were limited to single-patch geometries. To address the computational challenges associated with inverting mass matrices in multi-patch settings, we extend existing techniques for 0-forms mass and stiffness matrices to 1-forms mass matrices. Our approach leverages the tensor-product structure of spline spaces to construct a spectrally equivalent preconditioner for the 1-forms mass matrices. Specifically, we design an efficient preconditioner for single-patch domains that is robust with respect to both mesh size and spline degree. In the multi-patch case, this methodology is extended by integrating the single-patch preconditioner with an additive Schwarz approach, ensuring robustness and scalability with respect to the number of patches. We analyze the computational complexity of the proposed preconditioner and validate its robustness and efficiency through comprehensive numerical experiments.

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arXiv2026

Tree-cotree gauging for two-dimensional hierarchical splines

In magnetostatics and eddy current problems, formulated in terms of the magnetic vector potential, the solution is not unique, because the addition of an irrotational function to the solution remains a valid solution. The tree-cotree decomposition is a gauging technique to recover uniqueness when using finite elements, which consists in considering the mesh as a graph, and building a spanning tree on that graph. The idea has been recently extended to isogeometric analysis, applying the construction of the spanning tree on the control mesh, or equivalently, on the Greville grid. In the present paper we extend the construction to hierarchical splines, a set of splines with multi-level structure for adaptive refinement, by constructing a spanning tree for each single level. Since for degree $p=1$ the spaces of finite elements and hierarchical splines coincide, the presented construction is also valid for quadrilateral finite element meshes with hanging nodes. To assess the correctness of the method, we present numerical results for Maxwell eigenvalue problem.

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arXiv2026

Construction of exact refinements for the two-dimensional hierarchical B-spline de Rham complex

The de Rham complex arises naturally when studying problems in electromagnetism and fluid mechanics. Stable numerical methods to solve these problems can be obtained by using a discrete de Rham complex that preserves the structure of the continuous one. This property is not necessarily guaranteed when the discrete function spaces are hierarchical B-splines, and research shows that an arbitrary choice of refinement domains may give rise to spurious harmonic fields that ruin the accuracy of the solution. We will focus on the two-dimensional de Rham complex over the unit square $Ω\subseteq \mathbb{R}^2$, and provide theoretical results and a constructive algorithm to ensure that the structure of the complex is preserved: when a pair of functions are in conflict some additional functions, forming an L-chain between the pair, are also refined. Another crucial aspect to consider in the hierarchical setting is the notion of admissibility, as it is possible to obtain optimal convergence rates of numerical solutions and improved stability by limiting the multi-level interaction of basis functions. We show that, under a common restriction, the admissibility class of the first space of the discrete complex persists throughout the remaining spaces. As such, admissible refinement can be combined with our new algorithm to obtain admissible meshes that also respect the structure of the de Rham complex. Moreover, we detail how our algorithm can be easily included in standard adaptive mesh refinement schemes. Finally, we include numerical results that motivate the importance of the previous concerns for the vector Laplace and Maxwell eigenvalue problems.

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arXiv2026

Adaptive refinement in defeaturing problems via an equilibrated flux a posteriori error estimator

An adaptive refinement strategy, based on an equilibrated flux a posteriori error estimator, is proposed in the context of defeaturing problems. Defeaturing consists of removing features from complex domains to simplify mesh generation and reduce the computational cost of simulations. It is a common procedure, for example, in computer aided design for simulation-based manufacturing. However, depending on the problem at hand, geometrical simplification may significantly deteriorate the accuracy of the solution. The proposed adaptive strategy is hence twofold: starting from a defeatured geometry, it performs both standard mesh refinement and geometrical refinement by selecting, at each step, which features must be reintroduced to significantly improve accuracy. Similar adaptive strategies have been previously developed using residual-based error estimators within an IGA framework. Here, instead, we extend a previously developed equilibrated flux a posteriori error analysis, designed for standard finite element discretizations, to make it fully applicable within the adaptive procedure. In particular, we address the assembly of the equilibrated flux estimator in presence of elements trimmed by the boundary of included features, adopting a CutFEM strategy to handle feature inclusion. The resulting estimator allows us to bound both the defeaturing and the numerical sources of error, with additional contributions accounting for the weak imposition of boundary conditions.

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arXiv2026

Tree-Cotree-Based IETI-DP for Eddy Current Problems in Time-Domain

For low-frequency electromagnetic problems, where wave-propagation effects can be neglected, eddy current formulations are commonly used as a simplification of the full Maxwell's equations. In this setup, time-domain simulations, needed to capture transient startup responses or nonlinear behavior, are often computationally expensive. We propose a novel tearing and interconnecting approach for eddy currents in time-domain and investigate its scalability.

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arXiv2025

Tree-Cotree-Based Tearing and Interconnecting for 3D Magnetostatics: A Dual-Primal Approach

The simulation of electromagnetic devices with complex geometries and large-scale discrete systems benefits from advanced computational methods like IsoGeometric Analysis and Domain Decomposition. In this paper, we employ both concepts in an Isogeometric Tearing and Interconnecting method to enable the use of parallel computations for magnetostatic problems. We address the underlying non-uniqueness by using a graph-theoretic approach, the tree-cotree decomposition. The classical tree-cotree gauging is adapted to be feasible for parallelization, which requires that all local subsystems are uniquely solvable. Our contribution consists of an explicit algorithm for constructing compatible trees and combining it with a dual-primal approach to enable parallelization. The correctness of the proposed approach is proved and verified by numerical experiments, showing its accuracy, scalability and optimal convergence.

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arXiv2024

Adaptive analysis-aware defeaturing: the case of Neumann boundary conditions

Removing geometrical details from a complex domain is a classical operation in computer aided design. This procedure simplifies the meshing process, and it enables faster simulations with less memory requirements. However, depending on the partial differential equation that one wants to solve, removing some important geometrical features may greatly impact the solution accuracy. Unfortunately, the effect of geometrical simplification on the accuracy of the problem solution is often neglected or its evaluation is based on engineering expertise, only due to the lack of reliable tools. It is therefore important to have a better understanding of the effect of geometrical model simplification, also called defeaturing, to improve our control on the simulation accuracy along the design and analysis phases. In this work, we consider as a model problem the Poisson equation on a geometry with Neumann features, we consider some finite element discretization of it, and we build an adaptive strategy that is twofold. Firstly, it is able to perform geometrical refinements, that is, to choose at each iteration step which geometrical feature is important to obtain an accurate solution. Secondly, it performs standard mesh refinements; since the geometry changes at each iteration, the algorithm is designed to be used with an immersed method. To drive this adaptive strategy, we introduce an a posteriori estimator of the energy error between the exact solution defined in the exact fully-featured geometry, and the numerical approximation of the solution defined in the defeatured geometry. The reliability of the estimator is proven for very general (potentially trimmed multipatch) geometric configurations, and in particular for IGA with hierarchical B-splines. Finally, numerical experiments are performed to validate the presented theory and to illustrate the capabilities of the proposed adaptive strategy.

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arXiv2023

An equilibrated flux a posteriori error estimator for defeaturing problems

An a posteriori error estimator based on an equilibrated flux reconstruction is proposed for defeaturing problems in the context of finite element discretizations. Defeaturing consists in the simplification of a geometry by removing features that are considered not relevant for the approximation of the solution of a given PDE. In this work, the focus is on Poisson equation with Neumann boundary conditions on the feature boundary. The estimator accounts both for the so-called defeaturing error and for the numerical error committed by approximating the solution on the defeatured domain. Unlike other estimators that were previously proposed for defeaturing problems, the use of the equilibrated flux reconstruction allows to obtain a sharp bound for the numerical component of the error. Furthermore, it does not require the evaluation of the normal trace of the numerical flux on the feature boundary: this makes the estimator well-suited for finite element discretizations, in which the normal trace of the numerical flux is typically discontinuous across elements. The reliability of the estimator is proven and verified on several numerical examples. Its capability to identify the most relevant features is also shown, in anticipation of a future application to an adaptive strategy.

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arXiv2023

Energy conservative isogeometric techniques for the wave equation

We analyze the wave equation in mixed form, with periodic and/or Dirichlet homogeneous boundary conditions, and nonconstant coefficients that depend on the spatial variable. For the discretization, the weak form of the second equation is replaced by a strong form, written in terms of a projection operator. The system of equations is discretized with B-splines forming a De Rham complex along with suitable commutative projectors for the approximation of the second equation. The discrete scheme is energy conservative when discretized in time with a conservative method such as Crank-Nicolson. We propose a convergence analysis of the method to study the dependence with respect to the mesh size $h$, with focus on the consistency error. Numerical results show optimal convergence of the error in energy norm, and a relative error in energy conservation for long-time simulations of the order of machine precision.

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arXiv2023

Arbitrary order spline representation of cohomology generators for isogeometric analysis of eddy current problems

The eddy current problem has many relevant practical applications in science, ranging from non-destructive testing to magnetic confinement of plasma in fusion reactors. It arises when electrical conductors are immersed in an external time-varying magnetic field operating at frequencies for which electromagnetic wave propagation effects can be neglected. Popular formulations of the eddy current problem either use the magnetic vector potential or the magnetic scalar potential. They have individual advantages and disadvantages. One challenge is related to differential geometry: Scalar potential based formulations run into trouble when conductors are present in non-trivial topology, as approximation spaces must be then augmented with generators of the first cohomology group of the non-conducting domain. For all existing algorithms based on lowest order methods it is assumed that the extension of the graph-based algorithms to high-order approximations requires hierarchical bases for the curl-conforming discrete spaces. However, building on insight on de Rham complexes approximation with splines, we will show in the present submission that the hierarchical basis condition is not necessary. Algorithms based on spanning tree techniques can instead be adapted to work on an underlying hexahedral mesh arising from isomorphisms between spline spaces of differential forms and de Rham complexes on an auxiliary control mesh.

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arXiv2023

Adaptive isogeometric phase-field modeling of the Cahn-Hilliard equation: Suitably graded hierarchical refinement and coarsening on multi-patch geometries

We present an adaptive scheme for isogeometric phase-field modeling, to perform suitably graded hierarchical refinement and coarsening on both single- and multi-patch geometries by considering truncated hierarchical spline constructions which ensures $C^1$ continuity between patches. We apply the proposed algorithms to the Cahn-Hilliard equation, describing the time-evolving phase separation processes of immiscible fluids. We first verify the accuracy of the hierarchical spline scheme by comparing two classical indicators usually considered in phase-field modeling, for then demonstrating the effectiveness of the grading strategy in terms of accuracy per degree of freedom. A selection of numerical examples confirms the performance of the proposed scheme to simulate standard modes of phase separation using adaptive isogeometric analysis with smooth THB-spline constructions.

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arXiv2023

High order geometric methods with splines: fast solution with explicit time-stepping for Maxwell equations

We introduce a high-order spline geometric approach for the initial boundary value problem for Maxwell's equations. The method is geometric in the sense that it discretizes in structure preserving fashion the two de Rham sequences of differential forms involved in the formulation of the continuous system. Both the Ampere--Maxwell and the Faraday equations are required to hold strongly, while to make the system solvable two discrete Hodge star operators are used. By exploiting the properties of the chosen spline spaces and concepts from exterior calculus, a non-standard explicit in time formulation is introduced, based on the solution of linear systems with matrices presenting Kronecker product structure, rather than mass matrices as in the standard literature. These matrices arise from the application of the exterior (wedge) product in the discrete setting, and they present Kronecker product structure independently of the geometry of the domain or the material parameters. The resulting scheme preserves the desirable energy conservation properties of the known approaches. The computational advantages of the newly proposed scheme are studied both through a complexity analysis and through numerical experiments in three dimensions.

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arXiv2023

Adaptive isogeometric methods with $C^1$ (truncated) hierarchical splines on planar multi-patch domains

Isogeometric analysis is a powerful paradigm which exploits the high smoothness of splines for the numerical solution of high order partial differential equations. However, the tensor-product structure of standard multivariate B-spline models is not well suited for the representation of complex geometries, and to maintain high continuity on general domains special constructions on multi-patch geometries must be used. In this paper we focus on adaptive isogeometric methods with hierarchical splines, and extend the construction of $C^1$ isogeometric spline spaces on multi-patch planar domains to the hierarchical setting. We introduce a new abstract framework for the definition of hierarchical splines, which replaces the hypothesis of local linear independence for the basis of each level by a weaker assumption. We also develop a refinement algorithm that guarantees that the assumption is fulfilled by $C^1$ splines on certain suitably graded hierarchical multi-patch mesh configurations, and prove that it has linear complexity. The performance of the adaptive method is tested by solving the Poisson and the biharmonic problems.

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arXiv2022

Mathematical foundations of adaptive isogeometric analysis

This paper reviews the state of the art and discusses recent developments in the field of adaptive isogeometric analysis, with special focus on the mathematical theory. This includes an overview of available spline technologies for the local resolution of possible singularities as well as the state-of-the-art formulation of convergence and quasi-optimality of adaptive algorithms for both the finite element method (FEM) and the boundary element method (BEM) in the frame of isogeometric analysis (IGA).

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arXiv2022

An a posteriori error estimator for isogeometric analysis on trimmed geometries

Trimming consists of cutting away parts of a geometric domain, without reconstructing a global parametrization (meshing). It is a widely used operation in computer aided design, which generates meshes that are unfitted with the described physical object. This paper develops an adaptive mesh refinement strategy on trimmed geometries in the context of hierarchical B-spline based isogeometric analysis. A residual a posteriori estimator of the energy norm of the numerical approximation error is derived, in the context of the Poisson equation. The estimator is proven to be reliable, independently of the number of hierarchical levels and of the way the trimmed boundaries cut the underlying mesh. Numerical experiments are performed to validate the presented theory, and to show that the estimator's effectivity index is independent of the size of the active part of the trimmed mesh elements.

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arXiv2022

High order geometric methods with splines: an analysis of discrete Hodge--star operators

A new kind of spline geometric method approach is presented. Its main ingredient is the use of well established spline spaces forming a discrete de Rham complex to construct a primal sequence $\{X^k_h\}^n_{k=0}$, starting from splines of degree $p$, and a dual sequence $\{\tilde{X}^k_h\}_{k=0}^n$, starting from splines of degree $p-1$. By imposing homogeneous boundary conditions to the spaces of the primal sequence, the two sequences can be isomorphically mapped into one another. Within this setup, many familiar second order partial differential equations can be finally accommodated by explicitly constructing appropriate discrete versions of constitutive relations, called Hodge--star operators. Several alternatives based on both global and local projection operators between spline spaces will be proposed. The appeal of the approach with respect to similar published methods is twofold: firstly, it exhibits high order convergence. Secondly, it does not rely on the geometric realization of any (topologically) dual mesh. Several numerical examples in various space dimensions will be employed to validate the central ideas of the proposed approach and compare its features with the standard Galerkin approach in Isogeometric Analysis.

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arXiv2022

Torque Computation with the Isogeometric Mortar Method for the Simulation of Electric Machines

In this work isogeometric mortaring is used for the simulation of a six pole permanent magnet synchronous machine. Isogeometric mortaring is especially well suited for the efficient computation of rotating electric machines as it allows for an exact geometry representation for arbitrary rotation angles without the need of remeshing. The appropriate B-spline spaces needed for the solution of Maxwell's equations and the corresponding mortar spaces are introduced. Unlike in classical finite element methods their construction is straightforward in the isogeometric case. The torque in the machine is computed using two different methods, i.e., Arkkio's method and by using the Lagrange multipliers from the mortaring.

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arXiv2021

Analysis-aware defeaturing: problem setting and a posteriori estimation

Defeaturing consists in simplifying geometrical models by removing the geometrical features that are considered not relevant for a given simulation. Feature removal and simplification of computer-aided design models enables faster simulations for engineering analysis problems, and simplifies the meshing problem that is otherwise often unfeasible. The effects of defeaturing on the analysis are then neglected and, as of today, there are basically very few strategies to quantitatively evaluate such an impact. Understanding well the effects of this process is an important step for automatic integration of design and analysis. We formalize the process of defeaturing by understanding its effect on the solution of Poisson equation defined on the geometrical model of interest containing a single feature, with Neumann boundary conditions on the feature itself. We derive an a posteriori estimator of the energy error between the solutions of the exact and the defeatured geometries in $\mathbb{R}^n$, $n\in\{2,3\}$, that is simple, reliable and efficient up to oscillations. The dependence of the estimator upon the size of the features is explicit.

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