Cahn--Hilliard models are central for describing the evolution of interfaces in phase separation processes and free boundary problems. In general, they have nonconstant and often degenerate mobilit...
{"title":"How do degenerate mobilities determine singularity formation in Cahn-Hilliard equations?","authors":"Catalina Pesce, A. Münch","doi":"10.1137/21m1391249","DOIUrl":"https://doi.org/10.1137/21m1391249","url":null,"abstract":"Cahn--Hilliard models are central for describing the evolution of interfaces in phase separation processes and free boundary problems. In general, they have nonconstant and often degenerate mobilit...","PeriodicalId":313703,"journal":{"name":"Multiscale Model. Simul.","volume":"2 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123779332","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study scattering by a high aspect ratio particle using boundary integral equation methods. This problem has important applications in nanophotonics problems, including sensing and plasmonic imaging. To illustrate the effect of parity and the need for adapted methods in presence of high aspect ratio particles, we consider the scattering in two dimensions by a sound-hard, high aspect ratio ellipse. This fundamental problem highlights the main challenge and provide valuable insights to tackle plasmonic problems and general high aspect ratio particles. For this problem, we find that the boundary integral operator is nearly singular due to the collapsing geometry from an ellipse to a line segment. We show that this nearly singular behavior leads to qualitatively different asymptotic behaviors for solutions with different parities. Without explicitly taking this nearly singular behavior and this parity into account, computed solutions incur a large error. To address these challenges, we introduce a new method called Quadrature by Parity Asymptotic eXpansions (QPAX) that effectively and efficiently addresses these issues. We first develop QPAX to solve the Dirichlet problem for Laplace's equation in a high aspect ratio ellipse. Then, we extend QPAX for scattering by a sound-hard, high aspect ratio ellipse. We demonstrate the effectiveness of QPAX through several numerical examples.
{"title":"Quadrature by Parity Asymptotic eXpansions (QPAX) for scattering by high aspect ratio particles","authors":"C. Carvalho, A. Kim, L. Lewis, Zois Moitier","doi":"10.1137/21M1416801","DOIUrl":"https://doi.org/10.1137/21M1416801","url":null,"abstract":"We study scattering by a high aspect ratio particle using boundary integral equation methods. This problem has important applications in nanophotonics problems, including sensing and plasmonic imaging. To illustrate the effect of parity and the need for adapted methods in presence of high aspect ratio particles, we consider the scattering in two dimensions by a sound-hard, high aspect ratio ellipse. This fundamental problem highlights the main challenge and provide valuable insights to tackle plasmonic problems and general high aspect ratio particles. For this problem, we find that the boundary integral operator is nearly singular due to the collapsing geometry from an ellipse to a line segment. We show that this nearly singular behavior leads to qualitatively different asymptotic behaviors for solutions with different parities. Without explicitly taking this nearly singular behavior and this parity into account, computed solutions incur a large error. To address these challenges, we introduce a new method called Quadrature by Parity Asymptotic eXpansions (QPAX) that effectively and efficiently addresses these issues. We first develop QPAX to solve the Dirichlet problem for Laplace's equation in a high aspect ratio ellipse. Then, we extend QPAX for scattering by a sound-hard, high aspect ratio ellipse. We demonstrate the effectiveness of QPAX through several numerical examples.","PeriodicalId":313703,"journal":{"name":"Multiscale Model. Simul.","volume":"45 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121567562","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The Fokker--Planck approximation for an elementary linear, two-dimensional kinetic model endowed with a mass-preserving integral collision process is numerically studied, along with its diffusive l...
{"title":"Diffusive Limit of a Two-Dimensional Well-Balanced Scheme for the Free Klein-Kramers Equation","authors":"L. Gosse","doi":"10.1137/20M1337077","DOIUrl":"https://doi.org/10.1137/20M1337077","url":null,"abstract":"The Fokker--Planck approximation for an elementary linear, two-dimensional kinetic model endowed with a mass-preserving integral collision process is numerically studied, along with its diffusive l...","PeriodicalId":313703,"journal":{"name":"Multiscale Model. Simul.","volume":"201 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132416428","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Nonlinear multi-scale problems are ubiquitous in materials science and biology. Complicated interactions between nonlinearities and (nonseparable) multiple scales pose a major challenge for analysis and simulation. In this paper, we study the numerical homogenization for multi-scale elliptic PDEs with monotone nonlinearity, in particular the Leray-Lions problem (a prototypical example is the p-Laplacian equation), where the nonlinearity cannot be parameterized with low dimensional parameters, and the linearization error is non-negligible. We develop the iterated numerical homogenization scheme by combining numerical homogenization methods for linear equations, and the so-called"quasi-norm"based iterative approach for monotone nonlinear equation. We propose a residual regularized nonlinear iterative method, and in addition, develop the sparse updating method for the efficient update of coarse spaces. A number of numerical results are presented to complement the analysis and valid the numerical method.
{"title":"Iterated numerical homogenization for multi-scale elliptic equations with monotone nonlinearity","authors":"Xinliang Liu, Eric T. Chung, Lei Zhang","doi":"10.1137/21m1389900","DOIUrl":"https://doi.org/10.1137/21m1389900","url":null,"abstract":"Nonlinear multi-scale problems are ubiquitous in materials science and biology. Complicated interactions between nonlinearities and (nonseparable) multiple scales pose a major challenge for analysis and simulation. In this paper, we study the numerical homogenization for multi-scale elliptic PDEs with monotone nonlinearity, in particular the Leray-Lions problem (a prototypical example is the p-Laplacian equation), where the nonlinearity cannot be parameterized with low dimensional parameters, and the linearization error is non-negligible. We develop the iterated numerical homogenization scheme by combining numerical homogenization methods for linear equations, and the so-called\"quasi-norm\"based iterative approach for monotone nonlinear equation. We propose a residual regularized nonlinear iterative method, and in addition, develop the sparse updating method for the efficient update of coarse spaces. A number of numerical results are presented to complement the analysis and valid the numerical method.","PeriodicalId":313703,"journal":{"name":"Multiscale Model. Simul.","volume":"83 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125935816","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
E. Autry, Daniel Carter, G. Herschlag, Zach Hunter, J. Mattingly
{"title":"Metropolized Multiscale Forest Recombination for Redistricting","authors":"E. Autry, Daniel Carter, G. Herschlag, Zach Hunter, J. Mattingly","doi":"10.1137/21m1406854","DOIUrl":"https://doi.org/10.1137/21m1406854","url":null,"abstract":"","PeriodicalId":313703,"journal":{"name":"Multiscale Model. Simul.","volume":"19 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131380388","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Electromagnetic heating is the process where a composite material absorbs applied electromagnetic radiation and converts this energy to internal energy in the material. While homogenization models ...
电磁加热是复合材料吸收施加的电磁辐射并将该能量转换为材料内部能量的过程。而同质化模型……
{"title":"High-Frequency Homogenization for Electromagnetic Heating of Periodic Media","authors":"J. M. Gaone, B. Tilley, V. Yakovlev","doi":"10.1137/20m1369415","DOIUrl":"https://doi.org/10.1137/20m1369415","url":null,"abstract":"Electromagnetic heating is the process where a composite material absorbs applied electromagnetic radiation and converts this energy to internal energy in the material. While homogenization models ...","PeriodicalId":313703,"journal":{"name":"Multiscale Model. Simul.","volume":"140 ","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"113983007","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study the band structure of self-adjoint elliptic operators $mathbb{A}_g= -nabla cdot sigma_{g} nabla$, where $sigma_g$ has the symmetries of a honeycomb tiling of $mathbb{R}^2$. We focus on the case where $sigma_{g}$ is a real-valued scalar: $sigma_{g}=1$ within identical, disjoint"inclusions", centered at vertices of a honeycomb lattice, and $sigma_{g}=g gg1 $ (high contrast) in the complement of the inclusion set (bulk). Such operators govern, e.g. transverse electric (TE) modes in photonic crystal media consisting of high dielectric constant inclusions (semi-conductor pillars) within a homogeneous lower contrast bulk (air), a configuration used in many physical studies. Our approach, which is based on monotonicity properties of the associated energy form, extends to a class of high contrast elliptic operators that model heterogeneous and anisotropic honeycomb media. Our results concern the global behavior of dispersion surfaces, and the existence of conical crossings (Dirac points) occurring in the lowest two energy bands as well as in bands arbitrarily high in the spectrum. Dirac points are the source of important phenomena in fundamental and applied physics, e.g. graphene and its artificial analogues, and topological insulators. The key hypotheses are the non-vanishing of the Dirac (Fermi) velocity $v_D(g)$, verified numerically, and a spectral isolation condition, verified analytically in many configurations. Asymptotic expansions, to any order in $g^{-1}$, of Dirac point eigenpairs and $v_D(g)$ are derived with error bounds. Our study illuminates differences between the high contrast behavior of $mathbb{A}_g$ and the corresponding strong binding regime for Schroedinger operators.
{"title":"High Contrast Elliptic Operators in Honeycomb Structures","authors":"M. Cassier, M. Weinstein","doi":"10.1137/21m1408968","DOIUrl":"https://doi.org/10.1137/21m1408968","url":null,"abstract":"We study the band structure of self-adjoint elliptic operators $mathbb{A}_g= -nabla cdot sigma_{g} nabla$, where $sigma_g$ has the symmetries of a honeycomb tiling of $mathbb{R}^2$. We focus on the case where $sigma_{g}$ is a real-valued scalar: $sigma_{g}=1$ within identical, disjoint\"inclusions\", centered at vertices of a honeycomb lattice, and $sigma_{g}=g gg1 $ (high contrast) in the complement of the inclusion set (bulk). Such operators govern, e.g. transverse electric (TE) modes in photonic crystal media consisting of high dielectric constant inclusions (semi-conductor pillars) within a homogeneous lower contrast bulk (air), a configuration used in many physical studies. Our approach, which is based on monotonicity properties of the associated energy form, extends to a class of high contrast elliptic operators that model heterogeneous and anisotropic honeycomb media. Our results concern the global behavior of dispersion surfaces, and the existence of conical crossings (Dirac points) occurring in the lowest two energy bands as well as in bands arbitrarily high in the spectrum. Dirac points are the source of important phenomena in fundamental and applied physics, e.g. graphene and its artificial analogues, and topological insulators. The key hypotheses are the non-vanishing of the Dirac (Fermi) velocity $v_D(g)$, verified numerically, and a spectral isolation condition, verified analytically in many configurations. Asymptotic expansions, to any order in $g^{-1}$, of Dirac point eigenpairs and $v_D(g)$ are derived with error bounds. Our study illuminates differences between the high contrast behavior of $mathbb{A}_g$ and the corresponding strong binding regime for Schroedinger operators.","PeriodicalId":313703,"journal":{"name":"Multiscale Model. Simul.","volume":"35 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121685276","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This article addresses the problem of estimating the Koopman generator of a Markov process. The direct computation of the infinitesimal generator is not easy because of the discretization of the st...
{"title":"Estimation of the Koopman Generator by Newton's Extrapolation","authors":"R. Sechi, A. Sikorski, Marcus Weber","doi":"10.1137/20M1333006","DOIUrl":"https://doi.org/10.1137/20M1333006","url":null,"abstract":"This article addresses the problem of estimating the Koopman generator of a Markov process. The direct computation of the infinitesimal generator is not easy because of the discretization of the st...","PeriodicalId":313703,"journal":{"name":"Multiscale Model. Simul.","volume":"839 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131017447","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Finite Temperature Cauchy-Born Rule and Stability of Crystalline Solids with Point Defects","authors":"T. Luo, Yang Xiang, J. Yang","doi":"10.1137/20m1341520","DOIUrl":"https://doi.org/10.1137/20m1341520","url":null,"abstract":"","PeriodicalId":313703,"journal":{"name":"Multiscale Model. Simul.","volume":"69 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114650748","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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