Carlos Jerez-Hanckes

• Ph.D. Applied Mathematics
• Full Professor at Universidad Adolfo Ibanez

We consider the three-dimensional time-harmonic electromagnetic (EM) wave scattering transmission problem involving heterogeneous scatterers. The fields are approximated using the local multiple traces formulation (MTF), originally introduced for acoustic scattering. This scheme assigns independent boundary unknowns to each subdomain and weakly enf...

Quantifying the effects on electromagnetic waves scattered by objects of uncertain shape is key for robust design, particularly in high-precision applications. Assuming small random perturbations departing from a nominal domain, the first-order sparse boundary element method (FOSB) has been proven to directly compute statistical moments with poly-l...

We establish shape holomorphy results for general weakly- and hyper-singular boundary integral operators arising from second-order partial differential equations in unbounded two-dimensional domains with multiple finite-length open arcs. After recasting the corresponding boundary value problems as boundary integral equations, we prove that their so...

We present a novel computational scheme to solve acoustic wave transmission problems over two-dimensional composite scatterers, i.e. penetrable obstacles possessing junctions or triple points. The continuous problem is cast as a local multiple traces time-domain boundary integral formulation. For discretization in time and space, we resort to convo...

Quantifying the effects on electromagnetic waves scattered by objects of uncertain shape is key for robust design, particularly in high precision applications. Assuming small random perturbations departing from a nominal domain, the first-order sparse boundary element method (FOSB) has been proven to directly compute statistical moments with poly-l...

We establish shape holomorphy results for general weakly- and hyper-singular boundary integral operators arising from second-order partial differential equations in unbounded two-dimensional domains with multiple finite-length open arcs. After recasting the corresponding boundary value problems as boundary integral equations, we prove that their so...

The work concerns the multiscale modeling of a nerve fascicle of myelinated axons. We present a rigorous derivation of a macroscopic bidomain model describing the behavior of the electric potential in the fascicle based on the FitzHugh–Nagumo membrane dynamics. The approach is based on the two-scale convergence machinery combined with the method of...

We continue our study [R. Aylwin, C. Jerez-Hanckes, C. Schwab and J. Zech, Domain uncertainty quantification in computational electromagnetics, SIAM/ASA J. Uncertain. Quant. 8 (2020) 301–341] of the numerical approximation of time-harmonic electromagnetic fields for the Maxwell lossy cavity problem for uncertain geometries. We adopt the same affine...

We continue our study [Domain Uncertainty Quantification in Computational Electromagnetics, JUQ (2020), 8:301--341] of the numerical approximation of time-harmonic electromagnetic fields for the Maxwell lossy cavity problem for uncertain geometries. We adopt the same affine-parametric shape parametrization framework, mapping the physical domains to...

The work concerns the multiscale modeling of a nerve fascicle of myelinated axons. We present a rigorous derivation of a macroscopic bidomain model describing the behavior of the electric potential in the fascicle based on the FitzHugh-Nagumo membrane dynamics. The approach is based on the two-scale convergence machinery combined with the method of...

We study the elastic time-harmonic wave scattering problems on unbounded domains with boundaries composed of finite collections of disjoints finite open arcs (or cracks) in two dimensions. Specifically, we present a fast spectral Galerkin method for solving the associated weakly- and hyper-singular boundary integral equations (BIEs) arising from Di...

We consider the time-harmonic electromagnetic transmission problem for the unit sphere. Appealing to a vector spherical harmonics analysis, we prove the first stability result of the local multiple traces formulation (MTF) for electromagnetics, originally introduced by Hiptmair and Jerez-Hanckes (2012) for the acoustic case, paving the way towards...

We investigate a range of techniques for the acceleration of Calderón (operator) preconditioning in the context of boundary integral equation methods for electromagnetic transmission problems. Our objective is to mitigate as far as possible the high computational cost of the barycentrically-refined meshes necessary for the stable discretisation of...

We consider the problem of domain approximation in finite element methods for Maxwell equations on curved domains, i.e., when affine or polynomial meshes fail to exactly cover the domain of interest. In such cases, one is forced to approximate the domain by a sequence of polyhedral domains arising from inexact meshes. We deduce conditions on the qu...

We study the numerical solution of forward and inverse time-harmonic acoustic scattering problems by randomly shaped obstacles in three-dimensional space using a fast isogeometric boundary element method. Within the isogeometric framework, realizations of the random scatterer can efficiently be computed by simply updating the NURBS mappings which r...

We solve first-kind Fredholm boundary integral equations arising from Helmholtz and Laplace problems on bounded, smooth screens in three dimensions with either Dirichlet or Neumann conditions. The proposed Galerkin–Bubnov methods take as discretization elements pushed-forward weighted azimuthal projections of standard spherical harmonics onto the u...

We extend the operator preconditioning framework Hiptmair (2006) [10] to Petrov-Galerkin methods while accounting for parameter-dependent perturbations of both variational forms and their preconditioners, as occurs when performing numerical approximations. By considering different perturbation parameters for the original form and its preconditioner...

We present a fast spectral Galerkin scheme for the discretization of boundary integral equations arising from two-dimensional Helmholtz transmission problems in multi-layered periodic structures or gratings. Employing suitably parametrized Fourier basis and excluding cut-off frequen- cies (also known as Rayleigh-Wood frequencies), we rigorously est...

Multilayered diffraction gratings are an essential component in many optical devices due to their ability to engineer light. We propose a first-order optimization strategy to maximize diffraction efficiencies of such structures by a fast approximation of the underlying boundary integral equations for polarized electromagnetic fields. A parametric r...

We study the effects of numerical quadrature rules on error convergence rates when solving Maxwell-type variational problems via the curl-conforming or edge finite element method. A complete a priori error analysis for the case of bounded polygonal and curved domains with non-homogeneous coefficients is provided. We detail sufficient conditions wit...

We consider the time-harmonic scalar wave scattering problems with Dirichlet, Neumann, impedance and transmission boundary conditions. Under this setting, we analyze how sensitive diffracted fields and Cauchy data are to small perturbations of a given nominal shape. To this end, we follow [K. Eppler, Int.~J.~Appl. Math. Comput. Sci. 10(3) (2000), p...

We present a novel computational scheme to solve acoustic wave transmission problems over composite scatterers, i.e. penetrable obstacles possessing junctions or triple points. Our continuous problem is cast as a multiple traces time-domain boundary integral formulation valid in two and three dimensions. Numerically, our two-dimensional non-conform...

We extend the general operator preconditioning framework [R. Hiptmair, Comput. Math. with Appl. 52 (2006), pp. 699-706] to account for parameter-dependent perturbations of variational forms and their preconditioning. These perturbations can arise, for example, from quadrature approximation or machine precision, when solving variational formulations...

We solve first-kind Fredholm boundary integral equations arising from Helmholtz and Laplace problems on bounded, smooth screens in three-dimensions with either Dirichlet or Neumann conditions. The proposed Galerkin-Bubnov method takes as discretization elements pushed-forward weighted azimuthal projections of standard spherical harmonics onto the c...

We present a fast spectral Galerkin scheme for the discretization of boundary integral equations arising from two-dimensional Helmholtz transmission problems in multi-layered periodic structures or gratings. Employing suitably parametrized Fourier basis and excluding Rayleigh-Wood anomalies, we rigorously establish the well-posedness of both contin...

We study the effects of numerical quadrature rules on error convergence rates when solving Maxwell-type variational problems via the curl-conforming or edge finite element method. A complete {\em a priori} error analysis for the case of bounded polygonal and curved domains with non-homogeneous coefficients is provided. We detail sufficient conditio...

We study the numerical solution of forward and inverse acoustic scattering problems by randomly shaped obstacles in three-dimensional space using a fast isogeometric boundary element method. Within the isogeometric framework, realizations of the random scatterer can efficiently be computed by simply updating the NURBS mappings which represent the s...

We investigate a range of techniques for the acceleration of Calder\'on (operator) preconditioning in the context of boundary integral equation methods for electromagnetic transmission problems. Our objective is to mitigate as far as possible the high computational cost of the barycentrically-refined meshes necessary for the stable discretisation o...

This work presents the implementation, numerical examples, and experimental convergence study of first- and second-order optimization methods applied to one-dimensional periodic gratings. Through boundary integral equations and shape derivatives, the profile of a grating is optimized such that it maximizes the diffraction efficiency for given diffr...

This work presents the implementation, analysis, and convergence study of first- and second-order optimization methods applied to one-dimensional periodic gratings. Through boundary integral equations and shape derivatives, the profile of a grating (taken to be a perfect electric conductor) is optimized such that it maximizes the diffraction effici...

We present a spectral Galerkin numerical scheme for solving Helmholtz and Laplace problems with Dirichlet boundary conditions on a finite collection of open arcs in two-dimensional space. A boundary integral method is employed, giving rise to a first kind Fredholm equation whose variational form is discretized using weighted Chebyshev polynomials....

We study the mapping properties of boundary integral operators arising when solving two-dimensional, time-harmonic waves scattered by periodic domains. For domains assumed to be at least Lipschitz regular, we propose a novel explicit representation of Sobolev spaces for quasi-periodic functions that allows for an analysis analogous to that of Helmh...

We consider the time-harmonic electromagnetic transmission problem for the unit sphere. Appealing to a vector spherical harmonics analysis, we prove the first stability result of the local multiple trace formulation (MTF) for electromagnetics, originally introduced by Hiptmair and Jerez-Hanckes [Adv. Comp. Math. 37 (2012), 37-91] for the acoustic c...

Calderón multiplicative preconditioners are an effective way to improve the condition number of first kind boundary integral equations yielding provable mesh independent bounds. However, when discretizing by local low-order basis functions as in standard Galerkin boundary element methods, their computational performance worsens as meshes are refine...

We present a spectral numerical scheme for solving Helmholtz and Laplace problems with Dirichlet boundary conditions on an unbounded non-Lipschitz domain $$\mathbb {R}^2 \backslash \overline {\Gamma }$$ ℝ 2 ∖ Γ ¯ , where Γ is a finite collection of open arcs. Through an indirect method, a first kind formulation is derived whose variational form is...

We consider the numerical solution of time-harmonic acoustic scattering by obstacles with uncertain geometries for Dirichlet, Neumann, impedance and transmission boundary conditions. In particular, we aim to quantify diffracted fields originated by small stochastic perturbations of a given relatively smooth nominal shape. Using first-order shape Ta...

Despite its solid mathematical background, standard Calderón preconditioning for the Electric Field Integral Equation scales poorly with respect to mesh refinement due to its construction over barycentric meshes. Based on hierarchical matrices, our proposed algorithm optimally splits solution and preconditioner accuracies, significantly reducing co...

Fog water is a valuable resource in places were fresh water is scarce and fog events occur frequently. Fog collectors (FCs) are the technology currently used for harvesting it. In this work we present an electrostatic method for achieving this. A radial electric field is generated between two electrodes which exerts electric forces over the fog dro...

This paper explores the performance potential of gratings based on tungsten/hafnia (W/HfO2) stacks for thermophotovoltaic thermal emitters via numerical simulations. Structures consisting of a W grating over a HfO2 spacer layer and a W substrate are analyzed over a range of geometries. For shallow gratings (W grating thickness much smaller than the...

We show existence and uniqueness of the outgoing solution for the Maxwell problem with an impedance boundary condition of Leontovitch type in a half-space. Due to the presence of surface waves guided by an infinite surface, the established radiation condition differs from the classical one when approaching the boundary of the half-space. This speci...

We present a fast Galerkin spectral method to solve logarithmic singular equations on segments. The proposed method uses weighted first-kind Chebyshev polynomials. Convergence rates of several orders are obtained for fractional Sobolev spaces $H^{-1/2}$ (or $H^{-1/2}_{00}$). Main tools are the approximation properties of the discretization basis, t...

The paper concerns the multiscale modeling of a myelinated axon. Taking into account the microstructure with alternating myelinated parts and nodes Ranvier, we derive a nonlinear cable equation describing the potential propagation along the axon. We assume that the myelin is not a perfect insulator, and assign a low (asymptotically vanishing) condu...

We model the electrical behavior of several biological cells under external stimuli by extending and computationally improving the multiple traces formulation introduced in Henríquez et al. [Numer. Math. 136 (2016) 101-145]. Therein, the electric potential and current for a single cell are retrieved through the coupling of boundary integral operato...

We establish new explicit expressions and variational forms of boundary integral operators that provide the exact inverses of the weakly singular and hypersingular operators for -Δ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{u...

We present a novel deterministic method capable of calculating statistical moments of transverse electric polarized fields scattered by perfect electric conductor gratings with small surface random perturbations. Based on a first-order shape Taylor expansion, the resulting electric field integral equations are solved via the method of moments with...

For time-harmonic electromagnetic waves scattered by either perfectly conducting or dielectric bounded obstacles, we show that the fields depend holomorphically on the shape of the scatterer. In the presence of random geometrical perturbations, our results imply strong measurability of the fields, in weighted spaces in the exterior of the scatterer...

We present non-overlapping Domain Decomposition Methods (DDM) based on quasi-optimal transmission operators for the solution of Helmholtz transmission problems with piece-wise constant material properties. The quasi-optimal transmission boundary conditions incorporate readily available approximations of Dirichlet to Neumann operators. These approxi...

We model the electrical activity of biological cells under external stimuli via a novel boundary integral (BI) formulation together with a suitable time-space numerical discretization scheme. Ionic channels follow a non-linear dynamic behavior commonly described by systems of ordinary differential equations dependent on the electric potential jump...

We introduce new boundary integral operators which are the exact inverses of the weakly singular and hypersingular operators for the Laplacian on flat disks. Moreover, we provide explicit closed forms for them and prove the continuity and ellipticity of their corresponding bilinear forms in the natural Sobolev trace spaces. This permit us to derive...

We present Nyström discretizations of multitrace formulations and non-overlapping Domain Decomposition Methods (DDM) for the solution of Helmholtz transmission problems for bounded composite scatterers with piecewise constant material properties. We investigate the performance of DDM with both classical Robin and generalized Robin boundary conditio...

We present Nystr\"om discretizations of multitrace formulations and non-overlapping Domain Decomposition Methods (DDM) for the solution of Helmholtz transmission problems for bounded composite scatterers with piecewise constant material properties. We investigate the performance of DDM with both classical Robin and generalized Robin boundary condit...

For time-harmonic scattering of electromagnetic waves from obstacles with uncertain geometry, we perform a domain perturbation analysis. Assuming as known both the scatterers’ nominal geometry and its small-amplitude random perturbations statistics, we derive a tensorized boundary integral equation (BIE) which describes, to leading order, the secon...

We provide a novel ready-to-precondition boundary integral formulation to solve Helmholtz scattering problems by heterogenous penetrable objects in two dimensions exhibiting high-contrast ratios. By weakly imposing transmission conditions and integral representations per subdomain, we are able to devise a robust Galerkin-Petrov formulation employin...

We present an efficient method to solve high-frequency scattering problems by heterogeneous penetrable objects in two dimensions. This is achieved by extending the so-called Local Multiple Traces Formulation, introduced recently by Hiptmair and Jerez-Hanckes, to purely spectral discretizations employing weighted Chebyshev polynomials. Together with...

We consider lowest-order H − 1 2 (div Γ , Γ)-and H − 1 2 (Γ)-conforming boundary element spaces supported on part of the boundary Γ of a Lipschitz polyhedron. Assuming families of triangular meshes created by regular refinement , we prove that on these spaces the norms of the extension by zero operators with respect to (localized) trace norms incre...

We review the ideas behind and the construction of so-called local multi-trace boundary integral equations for second-order boundary value problems with piecewise constant coefficients. These formulations have received considerable attention recently as a promising domain-decomposition approach to boundary element methods.

Boundary value problems for the Poisson equation in the exterior of an open bounded Lipschitz curve C can be recast as first-kind boundary integral equations featuring weakly singular or hypersingular boundary integral operators (BIOs). Based on the recent discovery in [C. Jerez-Hanckes and J. Nedelec, SIAM J. Math. Anal., 44 (2012), pp. 2666-2694]...

We consider the tensorized operator for the Maxwell cavity source problem in frequency domain. Such formulations occur when computing statistical moments of the fields under a stochastic volume excitation. We establish a discrete inf-sup condition for its Ritz-Galerkin discretization on sparse tensor product edge element spaces built on nested sequ...

We study the electrical influence of an electrode over an axon of nonzero thickness in time using a two-dimensional finite element formulation. Although our inspiration comes from the practice of peripheral nerve stimulation, other types of neural tissue excitation can benefit from the model. Our formulation combines a Hodgkin-Huxley model to accou...

We present a two-dimensional boundary integral formulation of nerve impulse propagation. A nerve impulse is a potential difference across the cellular membrane that propagates along the nerve fiber. The traveling transmembrane potential is produced by the transfer of ionic species between the intra- and extra-cellular mediums. This current flux acr...

The Atacama Cosmology Telescope (ACT) is a 6m telescope designed to map the Cosmic Microwave Background (CMB) simultaneously at 145GHz, 220 GHz and 280 GHz. Its off-axis Gregorian design is intended to minimize and control the off-axis sidelobe response, which is critical for scientific purposes. The expected sidelobe level for this kind of design...

The Atacama Cosmology Telescope (ACT) is a 6 m telescope designed to map the Cosmic Microwave Background (CMB) simultaneously at 145 GHz, 220 GHz and 280GHz. The receiver in ACT, the Millimeter Bolometer Array Camera, features 1000 TES bolometers in each band. The detector performance depends critically on the total optical loading, requiring the s...

We introduce explicit and exact variational formulations for the weakly singular and hypersingular operators over an open interval as well as for their corresponding inverses. Contrary to the case of a closed curve, these operators no longer map fractional Sobolev spaces in a dual fashion but degenerate into different subspaces depending on their e...

We extend classic Sommerfeld and Silver-Muller radiation conditions for bounded scatterers to acoustic and electromagnetic fields propagating over three isotropic homogeneous layers in three dimensions. If x = (x(1), x(2), x(3)) is an element of R-3, with x(3) denoting the direction orthogonal to the layers, standard conditions only hold for the ou...

We propose a 2-D finite element/boundary element hybrid method for calculating the spatial distribution and frequency response of electromagnetic waves coming from a semiconductor laser when interacting with a finite-sized photonic crystal. We thus provide a flexible tool for the design of novel optical and microwave devices, among other applicatio...

Consider a flat smooth manifold \(\Gamma_m \subset \mathbb{R}^3\) of codimension one with Lipschitz boundary ∂Γm and large aspect ratios such as the one depicted in Figure 19.1. Let the associated unbounded domain \(\Omega := \mathbb{R}^3 \backslash \bar{\Gamma}_m\) be isotropic and homogeneous for the moment. We seek solutions \(u \in H^1_{loc}(\O...

We present a novel boundary integral formulation of the Helmholtz transmission problem for bounded composite scatterers (that is, piecewise constant material parameters in “subdomains”) that directly lends itself to operator preconditioning via Calderón projectors. The method relies on local traces on subdomains and weak enforcement of transmission...

We present a complete computation of the surface x1-periodic piezoelectric Green's function based on the asymptotic decomposition method and Poisson's summation formula. Spectral poles associated to surface acoustic waves render plane waves as expected. Behavior at small speed - large slownesses - portrays an oscillatory decay along the transversal...

In this work, the singular behavior of charges at corners and edges on the metallized areas in SAW transducers are investigated. In particular, it is demonstrated that a tensor product of the commonly used Tchebychev bases overestimates the singularities at corners, and, hence, it cannot be used in a proper boundary element method formulation. On t...

We propose and demonstrate experimentally the concept of the annular interdigital transducer that focuses acoustic waves on the surface of a piezoelectric material to a single, diffraction-limited spot. The shape of the transducing fingers follows the wave surface. Experiments conducted on lithium niobate substrates evidence that the generated surf...

Interdigital transducers (IDT) are widely used to generate surface acoustic waves directly on piezoelectric materials. However, in most applications, the generating fingers are straight, giving rise to the emission of plane waves. One notable exception is the circular IDT proposed by Day and Koerber for isotropic substrates [IEEE Trans. Sonics and...

We present a boundary element model for calculating the mechanical displacements and surface charge distribution for SAW IDTs assuming perfect flat conductors with null mass. For this, we take into account the full 3D piezoelectric surface Green's dyad and solve the integral equation for the electric surface charge when a potential over the metalli...

The computation of the two-dimensional harmonic spatial-domain Green's function at the surface of a piezoelectric half-space is discussed. Starting from the known form of the Green's function expressed in the spectral domain, the singular contributions are isolated and treated separately. It is found that the surface acoustic wave contributions (i....

In this work, the singular behavior of the charges at corners and edges of the 3D electrodes-bus system are investigated. These considerations are then used for the definition of a hybrid element model of SAW transducers

We consider the computation of the harmonic 3D spatial-domain Green's function of the surface of a piezoelectric substrate. Starting from the Green's function expressed in the spectral domain, the singular contributions are isolated and treated separately. It is found that the surface acoustic wave contributions give rise to an anisotropic generali...

The effects of capillary bonding of Si wafers in methanol have been investigated. Wafers brought into contact in alcohol generally leads to less particle trapping at the interface, and that liquid surface tension is useful in pulling bonding wafer together. The intrusion of methanol at the interface appears to affect the bond strength of hydrophili...

Incluye resumen en español. Tesis (Magíster en Ciencias de la Ingeniería)--Pontificia Universidad Católica de Chile, 2005.