Christian Engström - Academia.edu (original) (raw)
Papers by Christian Engström
ArXiv, 2019
In this paper we consider a sorting scheme for the removal of spurious scattering resonant pairs ... more In this paper we consider a sorting scheme for the removal of spurious scattering resonant pairs in two-dimensional electromagnetic problems and in three-dimensional acoustic problems. The novel so ...
Journal of Computational Physics, 2020
In this paper we consider scattering resonance computations in optics when the resonators consist... more In this paper we consider scattering resonance computations in optics when the resonators consist of frequency dependent and lossy materials, such as metals at optical frequencies. The proposed computational approach combines a novel hp-FEM strategy, based on dispersion analysis for complex frequencies, with a fast implementation of the nonlinear eigenvalue solver NLEIGS. Numerical computations illustrate that the pre-asymptotic phase is significantly reduced compared to standard uniform h and p strategies. Moreover, the efficiency grows with the refractive index contrast, which makes the new strategy highly attractive for metal-dielectric structures. The hp-refinement strategy together with the efficient parallel code result in highly accurate approximations and short runtimes on multi processor platforms.
Computers & Mathematics with Applications, 2016
We present a-posteriori analysis of higher order finite element approximations (hp-FEM) for quadr... more We present a-posteriori analysis of higher order finite element approximations (hp-FEM) for quadratic Fredholm-valued operator functions. Residual estimates for approximations of the algebraic eigenspaces are derived and we reduce the analysis of the estimator to the analysis of an associated boundary value problem. For the reasons of robustness we also consider approximations of the associated invariant pairs. We show that our estimator inherits the efficiency and reliability properties of the underlying boundary value estimator. As a model problem we consider spectral problems arising in analysis of photonic crystals. In particular, we present an example where a targeted family of eigenvalues cannot be guaranteed to be semisimple. Numerical experiments with hp-FEM show the predicted convergence rates. The measured effectivities of the estimator compare favorably with the performance of the same estimator on the associated boundary value problem. We also present a benchmark estimator, based on the dual weighted residual (DWR) approach, which is more expensive to compute but whose measured effectivities are close to one.
Characterization of an adaptiverenement algorithm for a meshless eigenvalue solver based on radia... more Characterization of an adaptiverenement algorithm for a meshless eigenvalue solver based on radial basis functions
Mathematical and Numerical Aspects of Wave Propagation WAVES 2003, 2003
Using Bloch waves to represent the full solution of the Maxwell equations inperiodic media, we st... more Using Bloch waves to represent the full solution of the Maxwell equations inperiodic media, we study the limit process where the material's period becomes much smaller than the wavelenght. it is se ...
Meshless methods are a promising field of numerical methods recently introduced to computational ... more Meshless methods are a promising field of numerical methods recently introduced to computational electromagnetics. The potential of conformal and multi-scale modeling and the possibility of dynamic grid refinements are very attractive features that appear more naturally in meshless methods than in classical methods. The Radial Point Interpolation Method (RPIM) uses radial basis functions for the approximation of spatial derivatives. In this publication an eigenvalue solver is introduced for RPIM in electromagnetics. Eigenmodes are calculated on the example of a cylindrical resonant cavity. It is demonstrated that the computed resonance frequencies converge to the analytical values for increasingly fine spatial discretization. The computation of eigenmodes is an important tool to support research on a timedomain implementation of RPIM. It allows a characterization of the method's accuracy and to investigate stability issues caused by the possible occurrence of non-physical solutions.
The concept of an adaptive meshless eigenvalue solver is presented and implemented for two-dimens... more The concept of an adaptive meshless eigenvalue solver is presented and implemented for two-dimensional structures. Based on radial basis functions, eigenmodes are calculated in a collocation approach for the second-order wave equation. This type of meshless method promises highly accurate results with the simplicity of a node-based collocation approach. Thus, when changing the discrete representation of a physical model, only node locations have to be adapted, hence avoiding the numerical overhead of handling an explicit mesh topology. The accuracy of the method comes at a cost of dealing with poorlyconditioned matrices. This is circumvented by applying a leaveone-out-cross-validation optimization algorithm to get stable results. A node adaptivity algorithm is presented to efficiently refine an initially coarse discretization. The convergence is evaluated in two numerical examples with analytical solutions. The most relevant parameter of the adaptation algorithm is numerically investigated and its influence on the convergence rate examined.
A meshless method based on a radial basis collocation approach is presented to calculate eigenval... more A meshless method based on a radial basis collocation approach is presented to calculate eigenvalues for the secondorder wave equation. Instead of an explicit mesh topology only a node distribution is required to calculate electric fields, thus facilitating dynamic alteration of the discretization of an electromagnetic problem. An algorithm is presented that automatically adapts an initially very coarse discretization by adding points where higher accuracy is required by the physics of the problem. The algorithm is applied to a cylindrical cavity resonator and the rate of convergence is compared to uniform refinements with the radial basis method and to a regular grid-based finite-difference approach.
Journal of Physics: Condensed Matter, 2007
This paper is concerned with the estimation of the volume fraction and the anisotropy of a two-co... more This paper is concerned with the estimation of the volume fraction and the anisotropy of a two-component composite from measured bulk properties. An algorithm that takes into account that measurements have errors is developed. This algorithm is used to study data from experimental measurements for a nanocomposite with an unknown nanostructure. The dependence on the nanostructure is quantified in terms of a measure in the representation formula introduced by D Bergman. We use composites with known nanostructures to illustrate the dependence on the underlying measure and show how errors in the measurements affect the estimates of the structural parameters.
International Journal for Numerical Methods in Engineering, 2011
Band structure calculations for photonic crystals require the numerical solution of eigenvalue pr... more Band structure calculations for photonic crystals require the numerical solution of eigenvalue problems. In this paper, we consider crystals composed of lossy materials with frequency-dependent permittivities. Often, these frequency dependencies are modeled by rational functions, such as the Lorentz model, in which case the eigenvalue problems are rational in the eigenvalue parameter. After spatial discretization using an interior penalty discontinuous Galerkin method, we employ a recently developed linearization technique to deal with the resulting rational matrix eigenvalue problems. In particular, the efficient implementation of Krylov subspace methods for solving the linearized eigenvalue problems is investigated in detail. Numerical experiments demonstrate that our new approach is considerably cheaper in terms of memory and computing time requirements compared to the naive approach of turning the rational eigenvalue problem into a polynomial eigenvalue problem and applying standard linearization techniques.
IEEE Transactions on Microwave Theory and Techniques, 2010
A meshless collocation method based on radial basis function (RBF) interpolation is presented for... more A meshless collocation method based on radial basis function (RBF) interpolation is presented for the numerical solution of Maxwell's equations. RBFs have attractive properties such as theoretical exponential convergence for increasingly dense node distributions. Although the primary interest resides in the time domain, an eigenvalue solver is used in this paper to investigate convergence properties of the RBF interpolation method. The eigenvalue distribution is calculated and its implications for longtime stability in time-domain simulations are established. It is found that eigenvalues with small, but nonzero, real parts are related to the instabilities observed in time-domain simulations after a large number of time steps. Investigations show that by using global basis functions, this problem can be avoided. More generally, the connection between the high matrix condition number, accuracy, and the magnitude of nonzero real parts is established.
IEEE Transactions on Microwave Theory and Techniques, 2010
For two-component composites, we address the inverse problem of estimating the structural paramet... more For two-component composites, we address the inverse problem of estimating the structural parameters and decrease measurement errors in bulk property measurements. A measurement of the effective permittivity at one frequency gives microstructural information about the composite that is used in cross-property bounds to estimate the effective permittivity at other frequencies. We use this information and inverse bounds on microstructural parameters to tighten error bars on permittivity measurements at microwave frequencies. The method can be used in the design of random and periodic composite materials for a large variety of applications. We apply the method to a composite material used in radar applications.
A new method to estimate the micro-structural parameters of anisotropic two-phase composite mater... more A new method to estimate the micro-structural parameters of anisotropic two-phase composite material is derived. The parameters are estimated using information from measurements or from numerical e ...
Composite materials, i.e, mixtures of two or more materials, are commonly used in industry, becau... more Composite materials, i.e, mixtures of two or more materials, are commonly used in industry, because they often have outstanding properties in comparison with the original materials. In many cases the inhomogeneities are on a very fine scale, which makes it difficult to perform a full numerical simulation of the material. On the fine scale we have rapid oscillations in the material parameters, but we are usually interested in the behavior at a much larger scale. At the large scale the composite reacts in the same way as a homogeneous material, with some effective material properties. In this thesis, models for the effective electromagnetic properties are analyzed. Mathematically, it is the study of Maxwell's equations with rapidly oscillating coefficients. The geometry on the fine scale is unknown in the study of many man-made materials and for almost all materials in nature. When, for example, only the permittivity of the components is known but nothing about the geometry, we have bounds on the effective permittivity of the composite, that is, the permittivity of the composite cannot exceed the permittivity of the components. The effective properties of heterogeneous materials depend strongly on the microstructure. This dependence can be quantified in terms of structural parameters, such as the volume fraction and the anisotropy of the material. We discuss the possibility of bounding the structural parameters from measurements of bulk properties of a two-component composite. Moreover, we show that this method can be used in practice, not only to bound the structural parameters but the method also implies restrictions on the possible values of the components in the composite. The problem of bounding the structural parameters from known values of an effective property is called inverse homogenization. Information from measurements of one effective property can be used to improve bounds on a related property. These bounds are called cross-property bounds or coupled bounds. We use the bounds on the structural parameters to derive cross-property bounds for anisotropic materials. When the microstructure is periodic and completely known it is in principle possible to exactly determine the effective properties of the composite. Two different methods for determination of the effective properties are compared numerically and an extension from the static limit of one of the methods is given. Using Bloch waves, the extension is from the static limit (zero wave vector) to an arbitrary wave vector in the first Brillouin zone. A nonzero wave vector is necessary when the microstructure cannot be considered infinitely small compared to the wavelength, for example in the study of optically active materials (Less)
When the wavelength is much larger than the typical scale of the microstructure in a material, it... more When the wavelength is much larger than the typical scale of the microstructure in a material, it is possible to define effective or homogenized material coefficients. Two numerical methods for the ...
When the microstructure of a medium has a much smaller length scale than the typical wavelength o... more When the microstructure of a medium has a much smaller length scale than the typical wavelength of the electromagnetic fields present, it is possible to compute effective material parameters. Using ...
ArXiv, 2019
In this paper we consider a sorting scheme for the removal of spurious scattering resonant pairs ... more In this paper we consider a sorting scheme for the removal of spurious scattering resonant pairs in two-dimensional electromagnetic problems and in three-dimensional acoustic problems. The novel so ...
Journal of Computational Physics, 2020
In this paper we consider scattering resonance computations in optics when the resonators consist... more In this paper we consider scattering resonance computations in optics when the resonators consist of frequency dependent and lossy materials, such as metals at optical frequencies. The proposed computational approach combines a novel hp-FEM strategy, based on dispersion analysis for complex frequencies, with a fast implementation of the nonlinear eigenvalue solver NLEIGS. Numerical computations illustrate that the pre-asymptotic phase is significantly reduced compared to standard uniform h and p strategies. Moreover, the efficiency grows with the refractive index contrast, which makes the new strategy highly attractive for metal-dielectric structures. The hp-refinement strategy together with the efficient parallel code result in highly accurate approximations and short runtimes on multi processor platforms.
Computers & Mathematics with Applications, 2016
We present a-posteriori analysis of higher order finite element approximations (hp-FEM) for quadr... more We present a-posteriori analysis of higher order finite element approximations (hp-FEM) for quadratic Fredholm-valued operator functions. Residual estimates for approximations of the algebraic eigenspaces are derived and we reduce the analysis of the estimator to the analysis of an associated boundary value problem. For the reasons of robustness we also consider approximations of the associated invariant pairs. We show that our estimator inherits the efficiency and reliability properties of the underlying boundary value estimator. As a model problem we consider spectral problems arising in analysis of photonic crystals. In particular, we present an example where a targeted family of eigenvalues cannot be guaranteed to be semisimple. Numerical experiments with hp-FEM show the predicted convergence rates. The measured effectivities of the estimator compare favorably with the performance of the same estimator on the associated boundary value problem. We also present a benchmark estimator, based on the dual weighted residual (DWR) approach, which is more expensive to compute but whose measured effectivities are close to one.
Characterization of an adaptiverenement algorithm for a meshless eigenvalue solver based on radia... more Characterization of an adaptiverenement algorithm for a meshless eigenvalue solver based on radial basis functions
Mathematical and Numerical Aspects of Wave Propagation WAVES 2003, 2003
Using Bloch waves to represent the full solution of the Maxwell equations inperiodic media, we st... more Using Bloch waves to represent the full solution of the Maxwell equations inperiodic media, we study the limit process where the material's period becomes much smaller than the wavelenght. it is se ...
Meshless methods are a promising field of numerical methods recently introduced to computational ... more Meshless methods are a promising field of numerical methods recently introduced to computational electromagnetics. The potential of conformal and multi-scale modeling and the possibility of dynamic grid refinements are very attractive features that appear more naturally in meshless methods than in classical methods. The Radial Point Interpolation Method (RPIM) uses radial basis functions for the approximation of spatial derivatives. In this publication an eigenvalue solver is introduced for RPIM in electromagnetics. Eigenmodes are calculated on the example of a cylindrical resonant cavity. It is demonstrated that the computed resonance frequencies converge to the analytical values for increasingly fine spatial discretization. The computation of eigenmodes is an important tool to support research on a timedomain implementation of RPIM. It allows a characterization of the method's accuracy and to investigate stability issues caused by the possible occurrence of non-physical solutions.
The concept of an adaptive meshless eigenvalue solver is presented and implemented for two-dimens... more The concept of an adaptive meshless eigenvalue solver is presented and implemented for two-dimensional structures. Based on radial basis functions, eigenmodes are calculated in a collocation approach for the second-order wave equation. This type of meshless method promises highly accurate results with the simplicity of a node-based collocation approach. Thus, when changing the discrete representation of a physical model, only node locations have to be adapted, hence avoiding the numerical overhead of handling an explicit mesh topology. The accuracy of the method comes at a cost of dealing with poorlyconditioned matrices. This is circumvented by applying a leaveone-out-cross-validation optimization algorithm to get stable results. A node adaptivity algorithm is presented to efficiently refine an initially coarse discretization. The convergence is evaluated in two numerical examples with analytical solutions. The most relevant parameter of the adaptation algorithm is numerically investigated and its influence on the convergence rate examined.
A meshless method based on a radial basis collocation approach is presented to calculate eigenval... more A meshless method based on a radial basis collocation approach is presented to calculate eigenvalues for the secondorder wave equation. Instead of an explicit mesh topology only a node distribution is required to calculate electric fields, thus facilitating dynamic alteration of the discretization of an electromagnetic problem. An algorithm is presented that automatically adapts an initially very coarse discretization by adding points where higher accuracy is required by the physics of the problem. The algorithm is applied to a cylindrical cavity resonator and the rate of convergence is compared to uniform refinements with the radial basis method and to a regular grid-based finite-difference approach.
Journal of Physics: Condensed Matter, 2007
This paper is concerned with the estimation of the volume fraction and the anisotropy of a two-co... more This paper is concerned with the estimation of the volume fraction and the anisotropy of a two-component composite from measured bulk properties. An algorithm that takes into account that measurements have errors is developed. This algorithm is used to study data from experimental measurements for a nanocomposite with an unknown nanostructure. The dependence on the nanostructure is quantified in terms of a measure in the representation formula introduced by D Bergman. We use composites with known nanostructures to illustrate the dependence on the underlying measure and show how errors in the measurements affect the estimates of the structural parameters.
International Journal for Numerical Methods in Engineering, 2011
Band structure calculations for photonic crystals require the numerical solution of eigenvalue pr... more Band structure calculations for photonic crystals require the numerical solution of eigenvalue problems. In this paper, we consider crystals composed of lossy materials with frequency-dependent permittivities. Often, these frequency dependencies are modeled by rational functions, such as the Lorentz model, in which case the eigenvalue problems are rational in the eigenvalue parameter. After spatial discretization using an interior penalty discontinuous Galerkin method, we employ a recently developed linearization technique to deal with the resulting rational matrix eigenvalue problems. In particular, the efficient implementation of Krylov subspace methods for solving the linearized eigenvalue problems is investigated in detail. Numerical experiments demonstrate that our new approach is considerably cheaper in terms of memory and computing time requirements compared to the naive approach of turning the rational eigenvalue problem into a polynomial eigenvalue problem and applying standard linearization techniques.
IEEE Transactions on Microwave Theory and Techniques, 2010
A meshless collocation method based on radial basis function (RBF) interpolation is presented for... more A meshless collocation method based on radial basis function (RBF) interpolation is presented for the numerical solution of Maxwell's equations. RBFs have attractive properties such as theoretical exponential convergence for increasingly dense node distributions. Although the primary interest resides in the time domain, an eigenvalue solver is used in this paper to investigate convergence properties of the RBF interpolation method. The eigenvalue distribution is calculated and its implications for longtime stability in time-domain simulations are established. It is found that eigenvalues with small, but nonzero, real parts are related to the instabilities observed in time-domain simulations after a large number of time steps. Investigations show that by using global basis functions, this problem can be avoided. More generally, the connection between the high matrix condition number, accuracy, and the magnitude of nonzero real parts is established.
IEEE Transactions on Microwave Theory and Techniques, 2010
For two-component composites, we address the inverse problem of estimating the structural paramet... more For two-component composites, we address the inverse problem of estimating the structural parameters and decrease measurement errors in bulk property measurements. A measurement of the effective permittivity at one frequency gives microstructural information about the composite that is used in cross-property bounds to estimate the effective permittivity at other frequencies. We use this information and inverse bounds on microstructural parameters to tighten error bars on permittivity measurements at microwave frequencies. The method can be used in the design of random and periodic composite materials for a large variety of applications. We apply the method to a composite material used in radar applications.
A new method to estimate the micro-structural parameters of anisotropic two-phase composite mater... more A new method to estimate the micro-structural parameters of anisotropic two-phase composite material is derived. The parameters are estimated using information from measurements or from numerical e ...
Composite materials, i.e, mixtures of two or more materials, are commonly used in industry, becau... more Composite materials, i.e, mixtures of two or more materials, are commonly used in industry, because they often have outstanding properties in comparison with the original materials. In many cases the inhomogeneities are on a very fine scale, which makes it difficult to perform a full numerical simulation of the material. On the fine scale we have rapid oscillations in the material parameters, but we are usually interested in the behavior at a much larger scale. At the large scale the composite reacts in the same way as a homogeneous material, with some effective material properties. In this thesis, models for the effective electromagnetic properties are analyzed. Mathematically, it is the study of Maxwell's equations with rapidly oscillating coefficients. The geometry on the fine scale is unknown in the study of many man-made materials and for almost all materials in nature. When, for example, only the permittivity of the components is known but nothing about the geometry, we have bounds on the effective permittivity of the composite, that is, the permittivity of the composite cannot exceed the permittivity of the components. The effective properties of heterogeneous materials depend strongly on the microstructure. This dependence can be quantified in terms of structural parameters, such as the volume fraction and the anisotropy of the material. We discuss the possibility of bounding the structural parameters from measurements of bulk properties of a two-component composite. Moreover, we show that this method can be used in practice, not only to bound the structural parameters but the method also implies restrictions on the possible values of the components in the composite. The problem of bounding the structural parameters from known values of an effective property is called inverse homogenization. Information from measurements of one effective property can be used to improve bounds on a related property. These bounds are called cross-property bounds or coupled bounds. We use the bounds on the structural parameters to derive cross-property bounds for anisotropic materials. When the microstructure is periodic and completely known it is in principle possible to exactly determine the effective properties of the composite. Two different methods for determination of the effective properties are compared numerically and an extension from the static limit of one of the methods is given. Using Bloch waves, the extension is from the static limit (zero wave vector) to an arbitrary wave vector in the first Brillouin zone. A nonzero wave vector is necessary when the microstructure cannot be considered infinitely small compared to the wavelength, for example in the study of optically active materials (Less)
When the wavelength is much larger than the typical scale of the microstructure in a material, it... more When the wavelength is much larger than the typical scale of the microstructure in a material, it is possible to define effective or homogenized material coefficients. Two numerical methods for the ...
When the microstructure of a medium has a much smaller length scale than the typical wavelength o... more When the microstructure of a medium has a much smaller length scale than the typical wavelength of the electromagnetic fields present, it is possible to compute effective material parameters. Using ...