| 1. |
- 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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| 2. |
- 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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| 3. |
- 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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| 4. |
- 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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| 5. |
- 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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| 6. |
- Arjmand, Doghonay, 1987-, et al.
(författare)
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Analysis of heterogeneous multiscale methods for long time wave propagation problems
- 2014
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Ingår i: Multiscale Modeling & simulation. - : Society for Industrial & Applied Mathematics (SIAM). - 1540-3459 .- 1540-3467. ; 12:3, s. 1135-1166
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Tidskriftsartikel (refereegranskat)abstract
- In this paper, we analyze a multiscale method developed under the heterogeneous multiscale method (HMM) framework for numerical approximation of multiscale wave propagation problems in periodic media. In particular, we are interested in the long time O(epsilon(-2)) wave propagation, where e represents the size of the microscopic variations in the media. In large time scales, the solutions of multiscale wave equations exhibit O(1) dispersive effects which are not observed in short time scales. A typical HMM has two main components: a macromodel and a micromodel. The macromodel is incomplete and lacks a set of local data. In the setting of multiscale PDEs, one has to solve for the full oscillatory problem over local microscopic domains of size eta = O(epsilon) to upscale the parameter values which are missing in the macroscopic model. In this paper, we prove that if the microproblems are consistent with the macroscopic solutions, the HMM approximates the unknown parameter values in the macromodel up to any desired order of accuracy in terms of epsilon/eta..
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| 7. |
- Arjmand, Doghonay, et al.
(författare)
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Atomistic-continuum multiscale modelling of magnetisation dynamics at non-zero temperature
- 2018
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Ingår i: Advances in Computational Mathematics. - : Springer Science and Business Media LLC. - 1019-7168 .- 1572-9044. ; 44:4, s. 1119-1151
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Tidskriftsartikel (refereegranskat)abstract
- In this article, a few problems related to multiscale modelling of magnetic materials at finite temperatures and possible ways of solving these problems are discussed. The discussion is mainly centred around two established multiscale concepts: the partitioned domain and the upscaling-based methodologies. The major challenge for both multiscale methods is to capture the correct value of magnetisation length accurately, which is affected by a random temperature-dependent force. Moreover, general limitations of these multiscale techniques in application to spin systems are discussed.
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| 8. |
- Arjmand, Doghonay, 1987-, et al.
(författare)
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Efficient low rank approximations for parabolic control problems with unknown heat source
- 2024
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Ingår i: Journal of Computational and Applied Mathematics. - : Elsevier B.V.. - 0377-0427 .- 1879-1778. ; 450
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Tidskriftsartikel (refereegranskat)abstract
- An inverse problem of finding an unknown heat source for a class of linear parabolic equations is considered. Such problems can typically be converted to a direct problem with non-local conditions in time instead of an initial value problem. Standard ways of solving these non-local problems include direct temporal and spatial discretization as well as the shooting method, which may be computationally expensive in higher dimensions. In the present article, we present approaches based on low-rank approximation via Arnoldi algorithm to bypass the computational limitations of the mentioned classical methods. Regardless of the dimension of the problem, we prove that the Arnoldi approach can be effectively used to turn the inverse problem into a simple initial value problem at the cost of only computing one-dimensional matrix functions while still retaining the same accuracy as the classical approaches. Numerical results in dimensions d=1,2,3 are provided to validate the theoretical findings and to demonstrate the efficiency of the method for growing dimensions.
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| 9. |
- Arjmand, Doghonay, et al.
(författare)
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Estimates for the upscaling error in heterogeneous multiscale methods for wave propagation problems in locally periodic media
- 2017
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Ingår i: Multiscale Modeling & simulation. - : Society for Industrial and Applied Mathematics. - 1540-3459 .- 1540-3467. ; 15:2, s. 948-976
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Tidskriftsartikel (refereegranskat)abstract
- This paper concerns the analysis of a multiscale method for wave propagation problems in microscopically nonhomogeneous media. A direct numerical approximation of such problems is prohibitively expensive as it requires resolving the microscopic variations over a much larger physical domain of interest. The heterogeneous multiscale method (HMM) is an efficient framework to approximate the solutions of multiscale problems. In the HMM, one assumes an incomplete macroscopic model which is coupled to a known but expensive microscopic model. The micromodel is solved only locally to upscale the parameter values which are missing in the macro model. The resulting macroscopic model can then be solved at a cost independent of the small scales in the problem. In general, the accuracy of the HMM is related to how good the upscaling step approximates the right macroscopic quantities. The analysis of the method that we consider here was previously addressed only in purely periodic media, although the method itself is numerically shown to be applicable to more general settings. In the present study, we consider a more realistic setting by assuming a locally periodic medium where slow and fast variations are allowed at the same time. We then prove that the HMM captures the right macroscopic effects. The generality of the tools and ideas in the analysis allows us to establish convergence rates in a multidimensional setting. The theoretical findings here imply an improved convergence rate in one dimension, which also justifies the numerical observations from our earlier study.
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| 10. |
- Arjmand, Doghonay, et al.
(författare)
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Modelling long-range interactions in multiscale simulations of ferromagnetic materials
- 2020
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Ingår i: Advances in Computational Mathematics. - New York : Springer. - 1019-7168 .- 1572-9044. ; 46:1
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Tidskriftsartikel (refereegranskat)abstract
- Atomistic-continuum multiscale modelling is becoming an increasingly popular tool for simulating the behaviour of materials due to its computational efficiency and reliable accuracy. In the case of ferromagnetic materials, the atomistic approach handles the dynamics of spin magnetic moments of individual atoms, while the continuum approximations operate with volume-averaged quantities, such as magnetisation. One of the challenges for multiscale models in relation to physics of ferromagnets is the existence of the long-range dipole-dipole interactions between spins. The aim of the present paper is to demonstrate a way of including these interactions into existing atomistic-continuum coupling methods based on the partitioned-domain and the upscaling strategies. This is achieved by modelling the demagnetising field exclusively at the continuum level and coupling it to both scales. Such an approach relies on the atomistic expression for the magnetisation field converging to the continuum expression when the interatomic spacing approaches zero, which is demonstrated in this paper.
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