| 1. |
- Abdulle, Assyr, et al.
(författare)
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A parabolic local problem with exponential decay of the resonance error for numerical homogenization
- 2021
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Ingår i: Mathematical Models and Methods in Applied Sciences. - 0218-2025. ; 31:13
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Tidskriftsartikel (refereegranskat)abstract
- This paper aims at an accurate and efficient computation of effective quantities, e.g., the homogenized coefficients for approximating the solutions to partial differential equations with oscillatory coefficients. Typical multiscale methods are based on a micro-macro coupling, where the macromodel describes the coarse scale behaviour, and the micro model is solved only locally to upscale the effective quantities, which are missing in the macro model. The fact that the micro problems are solved over small domains within the entire macroscopic domain, implies imposing artificial boundary conditions on the boundary of the microscopic domains. A naive treatment of these artificial boundary conditions leads to a first order error in ε/δ, where ε < δ represents the characteristic length ofthe small scale oscillations and δ^d is the size of micro domain. This error dominates all other errors originating from the discretization of the macro and the micro problems, and its reduction is a main issue in today’s engineering multiscale computations. The objective of the present work is to analyse a parabolic approach, first announced in [A. Abdulle,D. Arjmand, E. Paganoni, C. R. Acad. Sci. Paris, Ser. I, 2019], for computing the homogenized coefficients with arbitrarily high convergence rates in ε/δ. The analysis covers the setting of periodic microstructure,and numerical simulations are provided to verify the theoretical findings for more general settings, e.g. non-periodic micro structures.
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| 2. |
- Abdulle, Assyr, et al.
(författare)
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A parabolic local problem with exponential decay of the resonance error for numerical homogenization
- 2021
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Ingår i: Mathematical Models and Methods in Applied Sciences. - : World Scientific Pub Co Pte Ltd. - 0218-2025 .- 1793-6314. ; 31:13, s. 2733-2772
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Tidskriftsartikel (refereegranskat)abstract
- This paper aims at an accurate and efficient computation of effective quantities, e.g. the homogenized coefficients for approximating the solutions to partial differential equations with oscillatory coefficients. Typical multiscale methods are based on a micro–macro-coupling, where the macromodel describes the coarse scale behavior, and the micromodel is solved only locally to upscale the effective quantities, which are missing in the macromodel. The fact that the microproblems are solved over small domains within the entire macroscopic domain, implies imposing artificial boundary conditions on the boundary of the microscopic domains. A naive treatment of these artificial boundary conditions leads to a first-order error in ?/??/δ, where ?<??<δ represents the characteristic length of the small scale oscillations and ??δd is the size of microdomain. This error dominates all other errors originating from the discretization of the macro and the microproblems, and its reduction is a main issue in today’s engineering multiscale computations. The objective of this work is to analyze a parabolic approach, first announced in A. Abdulle, D. Arjmand, E. Paganoni, C. R. Acad. Sci. Paris, Ser. I, 2019, for computing the homogenized coefficients with arbitrarily high convergence rates in ?/??/δ. The analysis covers the setting of periodic microstructure, and numerical simulations are provided to verify the theoretical findings for more general settings, e.g. non-periodic microstructures.
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| 3. |
- Abdulle, Assyr, et al.
(författare)
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AN ELLIPTIC LOCAL PROBLEM WITH EXPONENTIAL DECAY OF THE RESONANCE ERROR FOR NUMERICAL HOMOGENIZATION
- 2023
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Ingår i: Multiscale Modeling & simulation. - : Society for Industrial & Applied Mathematics (SIAM). - 1540-3459 .- 1540-3467. ; 21:2, s. 513-541
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Tidskriftsartikel (refereegranskat)abstract
- Numerical multiscale methods usually rely on some coupling between a macroscopic and a microscopic model. The macroscopic model is incomplete as effective quantities, such as the homogenized material coefficients or fluxes, are missing in the model. These effective data need to be computed by running local microscale simulations followed by a local averaging of the microscopic information. Motivated by the classical homogenization theory, it is a common practice to use local elliptic cell problems for computing the missing homogenized coefficients in the macro model. Such a consideration results in a first order error O(E/8), where E represents the wavelength of the microscale variations and 8 is the size of the microscopic simulation boxes. This error, called ``resonance error,"" originates from the boundary conditions used in the microproblem and typically dominates all other errors in a multiscale numerical method. Optimal decay of the resonance error remains an open problem, although several interesting approaches reducing the effect of the boundary have been proposed over the last two decades. In this paper, as an attempt to resolve this problem, we propose a computationally efficient, fully elliptic approach with exponential decay of the resonance error.
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| 4. |
- Abdulle, Assyr, et al.
(författare)
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Exponential decay of the resonance error in numerical homogenization via parabolic and elliptic cell problems
- 2019
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Ingår i: Comptes rendus. Mathematique. - : Elsevier BV. - 1631-073X .- 1778-3569. ; 357:6, s. 545-551
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Tidskriftsartikel (refereegranskat)abstract
- This paper presents two new approaches for finding the homogenized coefficients of multiscale elliptic PDEs. Standard approaches for computing the homogenized coefficients suffer from the so-called resonance error, originating from a mismatch between the true and the computational boundary conditions. Our new methods, based on solutions of parabolic and elliptic cell problems, result in an exponential decay of the resonance error.
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| 5. |
- Arjmand, Doghonay, et al.
(författare)
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A FINITE ELEMENT HETEROGENEOUS MULTISCALE METHOD WITH IMPROVED CONTROL OVER THE MODELING ERROR
- 2016
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Ingår i: Communications in Mathematical Sciences. - : International Press of Boston. - 1539-6746 .- 1945-0796. ; 14:2, s. 463-487
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Tidskriftsartikel (refereegranskat)abstract
- Multiscale partial differential equations (PDEs) are difficult to solve by traditional numerical methods due to the need to resolve the small wavelengths in the media over the entire computational domain. We develop and analyze a Finite Element Heterogeneous Multiscale Method (FE-HMM) for approximating the homogenized solutions of multiscale PDEs of elliptic, parabolic, and hyperbolic type. Typical multiscale methods require a coupling between a micro and a macro model. Inspired from the homogenization theory, traditional FE-HMM schemes use elliptic PDEs as the micro model. We use, however, the second order wave equation as our micro model independent of the type of the problem on the macro level. This allows us to control the modeling error originating from the coupling between the different scales. In a spatially fully discrete a priori error analysis we prove that the modeling error can be made arbitrarily small for periodic media, even if we do not know the exact period of the oscillations in the media. We provide numerical examples in one and two dimensions confirming the theoretical results. Further examples show that the method captures the effective solutions in general non-periodic settings as well.
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| 6. |
- Arjmand, Doghonay, 1987-, et al.
(författare)
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A Finite Element Heterogenous Multiscale Method with Improved Control Over the Modeling Error
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Annan publikation (övrigt vetenskapligt/konstnärligt)abstract
- Multiscale partial dierential equations (PDEs) are difficult to solve by traditional numerical methods due to the need to resolve the small wavelengths in the media over the entire computational domain. We develop and analyze a Finite Element Heterogeneous Multiscale Method (FE-HMM) for approximating the homogenized solutions of multiscale PDEs of elliptic, parabolic,and hyperbolic type. Typical multiscale methods require a coupling between a micro and a macromodel. Inspired from the homogenization theory, traditional FE-HMM schemes use elliptic PDEs as the micro model. We use, however, the second order wave equation as our micro model independent of the type of the problem on the macro level. This allows us to control the modeling error originating by the coupling between the dierent scales. In a spatially fully discrete a priori error analysis we prove that the modeling error can be made arbitrarily small for periodic media, even if we do not know the exact period of the oscillations in the media. We provide numerical examples in one and two dimensions confirming the theoretical results. Further examples show that the method captures the effective solutions in general non-periodic settings as well
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| 7. |
- Arjmand, Doghonay, 1987-, et al.
(författare)
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A note on the Taylor s decomposition on four points for a third-order differential equation
- 2007
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Ingår i: Applied Mathematics and Computation. - : Elsevier. - 0096-3003 .- 1873-5649. ; 188:2, s. 1483-1490
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Tidskriftsartikel (refereegranskat)abstract
- Taylor's decomposition on four points is presented. three-step difference schemes generated by the Taylor's decomposition on fourpoints for the numerical solutions of an initial-value problem, a boundary-value problem, and a nonlocal boundary-value problem for a third-order ordinary differential equation are constructed. Numerical examples are given.
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| 8. |
- Arjmand, Doghonay, et al.
(författare)
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A time dependent approach for removing the cell boundary error in elliptic homogenization problems
- 2016
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Ingår i: Journal of Computational Physics. - : Elsevier. - 0021-9991 .- 1090-2716. ; 314, s. 206-227
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Tidskriftsartikel (refereegranskat)abstract
- This paper concerns the cell-boundary error present in multiscale algorithms for elliptic homogenization problems. Typical multiscale methods have two essential components: a macro and a micro model. The micro model is used to upscale parameter values which are missing in the macro model. To solve the micro model, boundary conditions are required on the boundary of the microscopic domain. Imposing a naive boundary condition leads to O(epsilon/eta) error in the computation, where epsilon is the size of the microscopic variations in the media and eta is the size of the micro-domain. The removal of this error in modern multiscale algorithms still remains an important open problem. In this paper, we present a time-dependent approach which is general in terms of dimension. We provide a theorem which shows that we have arbitrarily high order convergence rates in terms of epsilon/eta in the periodic setting. Additionally, we present numerical evidence showing that the method improves the O(epsilon/eta) error to O(epsilon) in general non-periodic media.
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| 9. |
- Arjmand, Doghonay, et al.
(författare)
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An equation-free approach for second order multiscale hyperbolic problems in non-divergence form
- 2018
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Ingår i: Communications in Mathematical Sciences. - 1539-6746 .- 1945-0796. ; 16:8, s. 2317-2343
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Tidskriftsartikel (refereegranskat)abstract
- The present study concerns the numerical homogenization of second order hyperbolic equations in non-divergence form, where the model problem includes a rapidly oscillating coefficient function. These small scales influence the large scale behavior, hence their effects should be accurately modelled in a numerical simulation. A direct numerical simulation is prohibitively expensive since a minimum of two points per wavelength are needed to resolve the small scales. A multiscale method, under the equation-free methodology, is proposed to approximate the coarse scale behaviour of the exact solution at a cost independent of the small scales in the problem. We prove convergence rates for the upscaled quantities in one as well as in multi-dimensional periodic settings. Moreover, numerical results in one and two dimensions are provided to support the theory.
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| 10. |
- Arjmand, Doghonay, 1987-
(författare)
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Analysis and Applications of Heterogeneous Multiscale Methods for Multiscale Partial Differential Equations
- 2015
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- This thesis centers on the development and analysis of numerical multiscale methods for multiscale problems arising in steady heat conduction, heat transfer and wave propagation in heterogeneous media. In a multiscale problem several scales interact with each other to form a system which has variations over a wide range of scales. A direct numerical simulation of such problems requires resolving the small scales over a computational domain, typically much larger than the microscopic scales. This demands a tremendous computational cost. We develop and analyse multiscale methods based on the heterogeneous multiscale methods (HMM) framework, which captures the macroscopic variations in the solution at a cost much lower than traditional numerical recipes. HMM assumes that there is a macro and a micro model which describes the problem. The micro model is accurate but computationally expensive to solve. The macro model is inexpensive but incomplete as it lacks certain parameter values. These are upscaled by solving the micro model locally in small parts of the domain. The accuracy of the method is then linked to how accurately this upscaling procedure captures the right macroscopic effects. In this thesis we analyse the upscaling error of existing multiscale methods and also propose a micro model which significantly reduces the upscaling error invarious settings. In papers I and IV we give an analysis of a finite difference HMM (FD-HMM) for approximating the effective solutions of multiscale wave equations over long time scales. In particular, we consider time scales T^ε = O(ε−k ), k =1, 2, where ε represents the size of the microstructures in the medium. In this setting, waves exhibit non-trivial behaviour which do not appear over short time scales. We use new analytical tools to prove that the FD-HMM accurately captures the long time effects. We first, in Paper I, consider T^ε =O(ε−2 ) and analyze the accuracy of FD-HMM in a one-dimensional periodicsetting. The core analytical ideas are quasi-polynomial solutions of periodic problems and local time averages of solutions of periodic wave equations.The analysis naturally reveals the role of consistency in HMM for high order approximation of effective quantities over long time scales. Next, in paperIV, we consider T^ε = O(ε−1 ) and use the tools in a multi-dimensional settingto analyze the accuracy of the FD-HMM in locally-periodic media where fast and slow variations are allowed at the same time. Moreover, in papers II and III we propose new multiscale methods which substantially improve the upscaling error in multiscale elliptic, parabolic and hyperbolic partial differential equations. In paper II we first propose a FD-HMM for solving elliptic homogenization problems. The strategy is to use the wave equation as the micro model even if the macro problem is of elliptic type. Next in paper III, we use this idea in a finite element HMM setting and generalize the approach to parabolic and hyperbolic problems. In a spatially fully discrete a priori error analysis we prove that the upscaling error can be made arbitrarily small for periodic media, even if we do not know the exact period of the oscillations in the media.
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