| 1. |
- Arjmand, Doghonay
(författare)
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Analysis and Applications of the Heterogeneous Multiscale Methods for Multiscale Elliptic and Hyperbolic Partial Differential Equations
- 2013
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Licentiatavhandling (övrigt vetenskapligt/konstnärligt)abstract
- This thesis concerns the applications and analysis of the Heterogeneous Multiscale methods (HMM) for Multiscale Elliptic and Hyperbolic Partial Differential Equations. We have gathered the main contributions in two papers.The first paper deals with the cell-boundary error which is present in multi-scale algorithms for elliptic homogenization problems. Typical multi-scale 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. Solving the micro model requires, on the other hand, imposing boundary conditions on the boundary of the microscopic domain. Imposing a naive boundary condition leads to $O(\varepsilon/\eta)$ error in the computation, where $\varepsilon$ is the size of the microscopic variations in the media and $\eta$ is the size of the micro-domain. Until now, strategies were proposed to improve the convergence rate up to fourth-order in $\varepsilon/\eta$ at best. However, the removal of this error in multi-scale algorithms still remains an important open problem. In this paper, we present an approach with a time-dependent model which is general in terms of dimension. With this approach we are able to obtain $O((\varepsilon/\eta)^q)$ and $O((\varepsilon/\eta)^q + \eta^p)$ convergence rates in periodic and locally-periodic media respectively, where $p,q$ can be chosen arbitrarily large. In the second paper, we analyze a multi-scale method developed under the Heterogeneous Multi-Scale Methods (HMM) framework for numerical approximation of wave propagation problems in periodic media. In particular, we are interested in the long time $O(\varepsilon^{-2})$ wave propagation. In the method, the microscopic model uses the macro solutions as initial data. In short-time wave propagation problems a linear interpolant of the macro variables can be used as the initial data for the micro-model. However, in long-time multi-scale wave problems the linear data does not suffice and one has to use a third-degree interpolant of the coarse data to capture the $O(1)$ dispersive effects apperaing in the long time. In this paper, we prove that through using an initial data consistent with the current macro state, HMM captures this dispersive effects up to any desired order of accuracy in terms of $\varepsilon/\eta$. We use two new ideas, namely quasi-polynomial solutions of periodic problems and local time averages of solutions of periodic hyperbolic PDEs. As a byproduct, these ideas naturally reveal the role of consistency for high accuracy approximation of homogenized quantities.
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| 2. |
- Edelvik, Fredrik, 1972, et al.
(författare)
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Numerical homogenization of spatial network models
- 2024
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Ingår i: Computer Methods in Applied Mechanics and Engineering. - 0045-7825 .- 1879-2138. ; 418
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Tidskriftsartikel (refereegranskat)abstract
- We present and analyze a methodology for numerical homogenization of spatial networks models, e.g. heat conduction and linear deformation in large networks of slender objects, such as paper fibers. The aim is to construct a coarse model of the problem that maintains high accuracy also on the micro-scale. By solving decoupled problems on local subgraphs we construct a low dimensional subspace of the solution space with good approximation properties. The coarse model of the network is expressed by a Galerkin formulation and can be used to perform simulations with different source and boundary data, at a low computational cost. We prove optimal convergence to the micro-scale solution of the proposed method under mild assumptions on the homogeneity, connectivity, and locality of the network on the coarse scale. The theoretical findings are numerically confirmed for both scalar-valued (heat conduction) and vector-valued (linear deformation) models.
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| 3. |
- Elfverson, Daniel, et al.
(författare)
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A Multilevel Monte Carlo Method for Computing Failure Probabilities
- 2016
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Ingår i: SIAM-ASA Journal on Uncertainty Quantification. - : Society for Industrial & Applied Mathematics (SIAM). - 2166-2525. ; 4:1, s. 312-330
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Tidskriftsartikel (refereegranskat)abstract
- We propose and analyze a method for computing failure probabilities of systems modeled as numerical deterministic models (e.g., PDEs) with uncertain input data. A failure occurs when a functional of the solution to the model is below (or above) some critical value. By combining recent results on quantile estimation and the multilevel Monte Carlo method, we develop a method that reduces computational cost without loss of accuracy. We show how the computational cost of the method relates to error tolerance of the failure probability. For a wide and common class of problems, the computational cost is asymptotically proportional to solving a single accurate realization of the numerical model, i.e., independent of the number of samples. Significant reductions in computational cost are also observed in numerical experiments.
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| 4. |
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| 5. |
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| 6. |
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| 7. |
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| 8. |
- Elfverson, Daniel, 1986-
(författare)
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Multiscale Methods and Uncertainty Quantification
- 2015
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- In this thesis we consider two great challenges in computer simulations of partial differential equations: multiscale data, varying over multiple scales in space and time, and data uncertainty, due to lack of or inexact measurements.We develop a multiscale method based on a coarse scale correction, using localized fine scale computations. We prove that the error in the solution produced by the multiscale method decays independently of the fine scale variation in the data or the computational domain. We consider the following aspects of multiscale methods: continuous and discontinuous underlying numerical methods, adaptivity, convection-diffusion problems, Petrov-Galerkin formulation, and complex geometries.For uncertainty quantification problems we consider the estimation of p-quantiles and failure probability. We use spatial a posteriori error estimates to develop and improve variance reduction techniques for Monte Carlo methods. We improve standard Monte Carlo methods for computing p-quantiles and multilevel Monte Carlo methods for computing failure probability.
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| 9. |
- Elfverson, Daniel, et al.
(författare)
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Multiscale methods for problems with complex geometry
- 2017
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Ingår i: Computer Methods in Applied Mechanics and Engineering. - : Elsevier BV. - 0045-7825 .- 1879-2138. ; 321, s. 103-123
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Tidskriftsartikel (refereegranskat)abstract
- We propose a multiscale method for elliptic problems on complex domains, e.g. domains with cracks or complicated boundary. For local singularities this paper also offers a discrete alternative to enrichment techniques such as XFEM. We construct corrected coarse test and trail spaces which takes the fine scale features of the computational domain into account. The corrections only need to be computed in regions surrounding fine scale geometric features. We achieve linear convergence rate in the energy norm for the multiscale solution. Moreover, the conditioning of the resulting matrices is not affected by the way the domain boundary cuts the coarse elements in the background mesh. The analytical findings are verified in a series of numerical experiments.
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| 10. |
- Elfverson, Daniel
(författare)
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On discontinuous Galerkin multiscale methods
- 2013
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Licentiatavhandling (övrigt vetenskapligt/konstnärligt)abstract
- In this thesis a new multiscale method, the discontinuous Galerkin multiscale method, is proposed. The method uses localized fine scale computations to correct a global coarse scale equation and thereby takes the fine scale features into account. We show a priori error bounds for convection dominated diffusion-convection-reaction problems with variable coefficients. We also present a posteriori error bound in the case of no convection or reaction and present an adaptive algorithm which tunes the method parameters automatically. We also present extensive numerical experiments which verify our analytical findings.
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| 11. |
- Elfverson, Daniel, et al.
(författare)
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Uncertainty Quantification for Approximate p-Quantiles for Physical Models with Stochastic Inputs
- 2014
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Ingår i: SIAM-ASA JOURNAL ON UNCERTAINTY QUANTIFICATION. - : Society for Industrial & Applied Mathematics (SIAM). - 2166-2525. ; 2:1, s. 826-850
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Tidskriftsartikel (refereegranskat)abstract
- We consider the problem of estimating the p-quantile for a given functional evaluated on solutions of a deterministic model in which model input is subject to stochastic variation. We derive upper and lower bounding estimators of the p-quantile. We perform an a posteriori error analysis for the p-quantile estimators that takes into account the effects of both the stochastic sampling error and the deterministic numerical solution error and yields a computational error bound for the estimators. We also analyze the asymptotic convergence properties of the p-quantile estimator bounds in the limit of large sample size and decreasing numerical error and describe algorithms for computing an estimator of the p-quantile with a desired accuracy in a computationally efficient fashion. One algorithm exploits the fact that the accuracy of only a subset of sample values significantly affects the accuracy of a p-quantile estimator resulting in a significant gain in computational efficiency. We conclude with a number of numerical examples, including an application to Darcy flow in porous media.
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| 12. |
- Engwer, C., et al.
(författare)
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Efficient implementation of the localized orthogonal decomposition method
- 2019
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Ingår i: Computer Methods in Applied Mechanics and Engineering. - : Elsevier B.V.. - 0045-7825 .- 1879-2138. ; 350, s. 123-153
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Tidskriftsartikel (refereegranskat)abstract
- In this paper we present algorithms for an efficient implementation of the Localized Orthogonal Decomposition method (LOD). The LOD is a multiscale method for the numerical simulation of partial differential equations with a continuum of inseparable scales. We show how the method can be implemented in a fairly standard Finite Element framework and discuss its realization for different types of problems, such as linear elliptic problems with rough coefficients and linear eigenvalue problems.
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| 13. |
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| 14. |
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| 15. |
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| 16. |
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| 17. |
- Fagerlund, Fritjof, et al.
(författare)
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Multilevel Monte Carlo methods for computing failure probability of porous media flow systems
- 2016
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Ingår i: Advances in Water Resources. - : Elsevier BV. - 0309-1708 .- 1872-9657. ; 94, s. 498-509
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Tidskriftsartikel (refereegranskat)abstract
- We study improvements of the standard and multilevel Monte Carlo method for point evaluation of the cumulative distribution function (failure probability) applied to porous media two-phase flow simulations with uncertain permeability. To illustrate the methods, we study an injection scenario where we consider sweep efficiency of the injected phase as quantity of interest and seek the probability that this quantity of interest is smaller than a critical value. In the sampling procedure, we use computable error bounds on the sweep efficiency functional to identify small subsets of realizations to solve highest accuracy by means of what we call selective refinement. We quantify the performance gains possible by using selective refinement in combination with both the standard and multilevel Monte Carlo method. We also identify issues in the process of practical implementation of the methods. We conclude that significant savings in computational cost are possible for failure probability estimation in a realistic setting using the selective refinement technique, both in combination with standard and multilevel Monte Carlo.
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| 18. |
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| 19. |
- Görtz, Morgan, 1994, et al.
(författare)
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A numerical multiscale method for fiber networks
- 2021
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Ingår i: World Congress in Computational Mechanics and ECCOMAS Congress. - : CIMNE. - 2696-6999. ; 300
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Konferensbidrag (refereegranskat)abstract
- Fiber network modeling can be used for studying mechanical properties of paper [1]. The individual fibers and the bonds in-between constitute a detailed representation of the material. However, detailed microscale fiber network models must be resolved with efficient numerical methods. In this work, a numerical multiscale method for discrete network models is proposed that is based on the localized orthogonal decomposition method [4]. The method is ideal for these network problems, because it reduces the maximum size of the problem, it is suitable for parallelization, and it can effectively solve fracture propagation. The problem analyzed in this work is the nodal displacement of a fiber network given an applied load. This problem is formulated as a linear system that is solved by using the aforementioned multiscale method. To solve the linear system, the multiscale method constructs a low-dimensional solution space with good approximation properties [5, 2]. The method is observed to work well for unstructured fiber networks, with optimal rates of convergence obtainable for highly localized configurations of the method.
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| 20. |
- Görtz, Morgan, 1994, et al.
(författare)
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Iterative method for large-scale Timoshenko beam models assessed on commercial-grade paperboard
- 2024
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Ingår i: Computational Mechanics. - 1432-0924 .- 0178-7675. ; In Press
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Tidskriftsartikel (refereegranskat)abstract
- Large-scale structural simulations based on micro-mechanical models of paper products require extensive numerical resources and time. In such models, the fibrous material is often represented by connected beams. Whereas previous micro-mechanical simulations have been restricted to smaller sample problems, large-scale micro-mechanical models are considered here. These large-scale simulations are possible on a non-specialized desktop computer with 128GB of RAM using an iterative method developed for network models and based on domain decomposition. Moreover, this method is parallelizable and is also well-suited for computational clusters. In this work, the proposed memory-efficient iterative method is numerically validated for linear systems resulting from large networks of Timoshenko beams. Tensile stiffness and out-of-plane bending stiffness are simulated and validated for various commercial-grade three-ply paperboards consisting of layers composed of two different types of paper fibers. The results of these simulations show that a linear network model produces results consistent with theory and published experimental data
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| 21. |
- Görtz, Morgan, 1994, et al.
(författare)
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Iterative solution of spatial network models by subspace decomposition
- 2023
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Ingår i: Mathematics of Computation. - 0025-5718 .- 1088-6842. ; 93, s. 233-58
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Tidskriftsartikel (refereegranskat)abstract
- We present and analyze a preconditioned conjugate gradient method (PCG) for solving spatial network problems. Primarily, we consider diffusion and structural mechanics simulations for fiber based materials, but the methodology can be applied to a wide range of models, fulfilling a set of abstract assumptions. The proposed method builds on a classical subspace decomposition into a coarse subspace, realized as the restriction of a finite element space to the nodes of the spatial network, and localized subspaces with support on mesh stars. The main contribution of this work is the convergence analysis of the proposed method. The analysis translates results from finite element theory, including interpolation bounds, to the spatial network setting. A convergence rate of the PCG algorithm, only depending on global bounds of the operator and homogeneity, connectivity and locality constants of the network, is established. The theoretical results are confirmed by several numerical experiments.
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| 22. |
- Görtz, Morgan, 1994, et al.
(författare)
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Multiscale methods for solving wave equations on spatial networks
- 2023
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Ingår i: Computer Methods in Applied Mechanics and Engineering. - : Elsevier BV. - 0045-7825. ; 410
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Tidskriftsartikel (refereegranskat)abstract
- We present and analyze a multiscale method for wave propagation problems, posed on spatial networks. By introducing a coarse scale, using a finite element space interpolated onto the network, we construct a discrete multiscale space using the localized orthogonal decomposition (LOD) methodology. The spatial discretization is then combined with an energy conserving temporal scheme to form the proposed method. Under the assumption of well-prepared initial data, we derive an a priori error bound of optimal order with respect to the space and time discretization. In the analysis, we combine the theory derived for stationary elliptic problems on spatial networks with classical finite element results for hyperbolic problems. Finally, we present numerical experiments that confirm our theoretical findings. (c) 2023 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).
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| 23. |
- Görtz, Morgan, 1994, et al.
(författare)
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Network model for predicting structural properties of paper
- 2022
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Ingår i: Nordic Pulp & Paper Research Journal. - : Walter de Gruyter GmbH. - 0283-2631 .- 2000-0669. ; 37:4, s. 712-24
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Tidskriftsartikel (refereegranskat)abstract
- Paper simulations that resolve the entire microscopic fiber structure are typically time-consuming and require extensive resources. Several such modeling approaches have been proposed to analyze different properties in paper. However, most use non-linear and time-dependent models resulting in high computational complexity. Resolving these computational issues would increase its usefulness in industrial applications. The model proposed in this work was developed in collaboration with companies in the papermaking industry within the Innovative Simulation of Paper (ISOP) project. A linear network model is used for efficiency, where 1-D beams represent the fibers. Similar models have been proposed in the past. However, in this work, the paper models are three-dimensional, a new dynamic bonding technique is used, and more extensive simulations are evaluated. The model is used to simulate tensile stiffness, tensile strength, and bending resistance. These simulated results are compared to experimental and theoretical counterparts and produce representable results for realistic parameters. Moreover, an off-the-shelf computer accessible to a paper developer can evaluate these models structural properties efficiently.
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| 24. |
- Hauck, Moritz, 1997, et al.
(författare)
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Super-localization of spatial network models
- 2024
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Ingår i: Numerische Mathematik. - 0029-599X .- 0945-3245. ; 156:3, s. 901-926
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Tidskriftsartikel (refereegranskat)abstract
- Spatial network models are used as a simplified discrete representation in a wide range of applications, e.g., flow in blood vessels, elasticity of fiber based materials, and pore network models of porous materials. Nevertheless, the resulting linear systems are typically large and poorly conditioned and their numerical solution is challenging. This paper proposes a numerical homogenization technique for spatial network models which is based on the super-localized orthogonal decomposition (SLOD), recently introduced for elliptic multiscale partial differential equations. It provides accurate coarse solution spaces with approximation properties independent of the smoothness of the material data. A unique selling point of the SLOD is that it constructs an almost local basis of these coarse spaces, requiring less computations on the fine scale and achieving improved sparsity on the coarse scale compared to other state-of-the-art methods. We provide an a posteriori analysis of the proposed method and numerically confirm the method’s unique localization properties. In addition, we show its applicability also for high-contrast channeled material data.
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| 25. |
- Hellman, F., et al.
(författare)
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Contrast Independent Localization of Multiscale Problems
- 2017
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Ingår i: Multiscale Modeling & Simulation. - : Society for Industrial & Applied Mathematics (SIAM). - 1540-3459 .- 1540-3467. ; 15:4, s. 1325-1355
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Tidskriftsartikel (refereegranskat)abstract
- The accuracy of many multiscale methods based on localized computations suffers from high contrast coefficients since the localization error generally depends on the contrast. We study a class of methods based on the variational multiscale method, where the range and kernel of a quasi-interpolation operator de fines the method. We present a novel interpolation operator for two-valued coefficients and prove that it yields contrast independent localization error under physically justified assumptions on the geometry of inclusions and channel structures in the coefficient. The idea developed in the paper can be transferred to more general operators and our numerical experiments show that the contrast independent localization property follows.
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| 26. |
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| 27. |
- Hellman, F., et al.
(författare)
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Contrast Independent Localization of Multiscale Problems
- 2017
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Ingår i: Multiscale Modeling and Simulation. - : Society for Industrial & Applied Mathematics (SIAM). - 1540-3467 .- 1540-3459. ; 15:4, s. 1325-1355
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Tidskriftsartikel (refereegranskat)abstract
- The accuracy of many multiscale methods based on localized computations suffers from high contrast coefficients since the localization error generally depends on the contrast. We study a class of methods based on the variational multiscale method, where the range and kernel of a quasi-interpolation operator de fines the method. We present a novel interpolation operator for two-valued coefficients and prove that it yields contrast independent localization error under physically justified assumptions on the geometry of inclusions and channel structures in the coefficient. The idea developed in the paper can be transferred to more general operators and our numerical experiments show that the contrast independent localization property follows.
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| 28. |
- Hellman, Fredrik
(författare)
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Multiscale and multilevel methods for porous media flow problems
- 2015
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Licentiatavhandling (övrigt vetenskapligt/konstnärligt)abstract
- We consider two problems encountered in simulation of fluid flow through porous media. In macroscopic models based on Darcy's law, the permeability field appears as data.The first problem is that the permeability field generally is not entirely known. We consider forward propagation of uncertainty from the permeability field to a quantity of interest. We focus on computing p-quantiles and failure probabilities of the quantity of interest. We propose and analyze improved standard and multilevel Monte Carlo methods that use computable error bounds for the quantity of interest. We show that substantial reductions in computational costs are possible by the proposed approaches.The second problem is fine scale variations of the permeability field. The permeability often varies on a scale much smaller than that of the computational domain. For standard discretization methods, these fine scale variations need to be resolved by the mesh for the methods to yield accurate solutions. We analyze and prove convergence of a multiscale method based on the Raviart–Thomas finite element. In this approach, a low-dimensional multiscale space based on a coarse mesh is constructed from a set of independent fine scale patch problems. The low-dimensional space can be used to yield accurate solutions without resolving the fine scale.
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| 29. |
- Hellman, Filip, 1984, et al.
(författare)
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Multiscale Mixed Finite Elements
- 2016
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Ingår i: Discrete and Continuous Dynamical Systems - Series S. - : American Institute of Mathematical Sciences (AIMS). - 1937-1632 .- 1937-1179. ; 9:5, s. 1269-1298
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Tidskriftsartikel (refereegranskat)abstract
- In this work, we propose a mixed finite element method for solving elliptic multiscale problems based on a localized orthogonal decomposition (LOD) of Raviart-Thomas finite element spaces. It requires to solve local problems in small patches around the elements of a coarse grid. These computations can be perfectly parallelized and are cheap to perform. Using the results of these patch problems, we construct a low dimensional multiscale mixed finite element space with very high approximation properties. This space can be used for solving the original saddle point problem in an efficient way. We prove convergence of our approach, independent of structural assumptions or scale separation. Finally, we demonstrate the applicability of our method by presenting a variety of numerical experiments, including a comparison with an MsFEM approach.
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| 30. |
- Hellman, Fredrik, et al.
(författare)
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Multiscale mixed finite elements, Discrete and Continuous Dynamical Systems
- 2016
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Ingår i: Discrete and Continuous Dynamical Systems. Series S. - : American Institute of Mathematical Sciences (AIMS). - 1937-1632 .- 1937-1179. ; 9, s. 1269-1298
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Tidskriftsartikel (refereegranskat)abstract
- In this work, we propose a mixed finite element method for solving elliptic multiscale problems based on a localized orthogonal decomposition (LOD) of Raviart--Thomas finite element spaces. It requires to solve local problems in small patches around the elements of a coarse grid. These computations can be perfectly parallelized and are cheap to perform. Using the results of these patch problems, we construct a low dimensional multiscale mixed finite element space with very high approximation properties. This space can be used for solving the original saddle point problem in an efficient way. We prove convergence of our approach, independent of structural assumptions or scale separation. Finally, we demonstrate the applicability of our method by presenting a variety of numerical experiments, including a comparison with an MsFEM approach.
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| 31. |
- Hellman, Fredrik, 1985, et al.
(författare)
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Numerical Homogenization of Elliptic PDEs with Similar Coefficients
- 2019
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Ingår i: Multiscale Modeling and Simulation. - : SIAM PUBLICATIONS. - 1540-3467 .- 1540-3459. ; 17:2, s. 650-674
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Tidskriftsartikel (refereegranskat)abstract
- We consider a sequence of elliptic partial differential equations (PDEs) with different but similar rapidly varying coefficients. Such sequences appear, for example, in splitting schemes for time-dependent problems (with one coefficient per time step) and in sample based stochastic integration of outputs from an elliptic PDE (with one coefficient per sample member). We propose a parallelizable algorithm based on Petrov-Galerkin localized orthogonal decomposition that adaptively (using computable and theoretically derived error indicators) recomputes the local corrector problems only where it improves accuracy. The method is illustrated in detail by an example of a time-dependent two-pase Darcy flow problem in three dimensions.
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| 32. |
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| 33. |
- Hellman, Fredrik
(författare)
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Numerical Methods for Darcy Flow Problems with Rough and Uncertain Data
- 2017
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- We address two computational challenges for numerical simulations of Darcy flow problems: rough and uncertain data. The rapidly varying and possibly high contrast permeability coefficient for the pressure equation in Darcy flow problems generally leads to irregular solutions, which in turn make standard solution techniques perform poorly. We study methods for numerical homogenization based on localized computations. Regarding the challenge of uncertain data, we consider the problem of forward propagation of uncertainty through a numerical model. More specifically, we consider methods for estimating the failure probability, or a point estimate of the cumulative distribution function (cdf) of a scalar output from the model.The issue of rough coefficients is discussed in Papers I–III by analyzing three aspects of the localized orthogonal decomposition (LOD) method. In Paper I, we define an interpolation operator that makes the localization error independent of the contrast of the coefficient. The conditions for its applicability are studied. In Paper II, we consider time-dependent coefficients and derive computable error indicators that are used to adaptively update the multiscale space. In Paper III, we derive a priori error bounds for the LOD method based on the Raviart–Thomas finite element.The topic of uncertain data is discussed in Papers IV–VI. The main contribution is the selective refinement algorithm, proposed in Paper IV for estimating quantiles, and further developed in Paper V for point evaluation of the cdf. Selective refinement makes use of a hierarchy of numerical approximations of the model and exploits computable error bounds for the random model output to reduce the cost complexity. It is applied in combination with Monte Carlo and multilevel Monte Carlo methods to reduce the overall cost. In Paper VI we quantify the gains from applying selective refinement to a two-phase Darcy flow problem.
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| 34. |
- Hellman, Fredrik, et al.
(författare)
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Numerical upscaling for heterogeneous materials in fractured domains
- 2021
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Ingår i: Esaim - Mathematical Modelling and Numerical Analysis - Modelisation Mathematique Et Analyse Numerique. - : EDP Sciences. - 0764-583X .- 2822-7840 .- 1290-3841. ; 55
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Tidskriftsartikel (refereegranskat)abstract
- We consider numerical solution of elliptic problems with heterogeneous diffusion coefficients containing thin highly conductive structures. Such problems arise e.g. in fractured porous media, reinforced materials, and electric circuits. The main computational challenge is the high resolution needed to resolve the data variation. We propose a multiscale method that models the thin structures as interfaces and incorporate heterogeneities in corrected shape functions. The construction results in an accurate upscaled representation of the system that can be used to solve for several forcing functions or to simulate evolution problems in an efficient way. By introducing a novel interpolation operator, defining the fine scale of the problem, we prove exponential decay of the shape functions which allows for a sparse approximation of the upscaled representation. An a priori error bound is also derived for the proposed method together with numerical examples that verify the theoretical findings. Finally we present a numerical example to show how the technique can be applied to evolution problems.
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| 35. |
- Hellman, Fredrik, et al.
(författare)
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NUMERICAL UPSCALING OF PERTURBED DIFFUSION PROBLEMS
- 2020
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Ingår i: Siam Journal on Scientific Computing. - : Society for Industrial & Applied Mathematics (SIAM). - 1064-8275 .- 1095-7197. ; 42:4
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Tidskriftsartikel (refereegranskat)abstract
- In this paper we study elliptic partial differential equations with rapidly varying diffusion coefficient that can be represented as a perturbation of a reference coefficient. We develop a numerical method for efficiently solving multiple perturbed problems by reusing local computations performed with the reference coefficient. The proposed method is based on the Petrov-Galerkin localized orthogonal decomposition (PG-LOD), which allows for straightforward parallelization with low communication overhead and memory consumption. We focus on two types of perturbations: local defects, which we treat by recomputation of multiscale shape functions, and global mappings of a reference coefficient for which we apply the domain mapping method. We analyze the proposed method for these problem classes and present several numerical examples.
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| 36. |
- Hellman, Fredrik, 1985, et al.
(författare)
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Numerical upscaling of perturbed diffusion problems
- 2020
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Ingår i: SIAM Journal of Scientific Computing. - 1064-8275 .- 1095-7197. ; 42:4, s. A2014-A2036
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Tidskriftsartikel (refereegranskat)abstract
- In this paper we study elliptic partial differential equations with rapidly varying diffusion coefficient that can be represented as a perturbation of a reference coefficient. We develop a numerical method for efficiently solving multiple perturbed problems by reusing local computations performed with the reference coefficient. The proposed method is based on the Petrov-Galerkin localized orthogonal decomposition (PG-LOD), which allows for straightforward parallelization with low communication overhead and memory consumption. We focus on two types of perturbations: local defects, which we treat by recomputation of multiscale shape functions, and global mappings of a reference coefficient for which we apply the domain mapping method. We analyze the proposed method for these problem classes and present several numerical examples.
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| 37. |
- Hellman, Fredrik, et al.
(författare)
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Well-posedness and finite element approximation of mixed dimensional partial differential equations
- 2024
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Ingår i: BIT NUMERICAL MATHEMATICS. - 0006-3835 .- 1572-9125. ; 64:1
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Tidskriftsartikel (refereegranskat)abstract
- In this article, a mixed dimensional elliptic partial differential equation is considered, posed in a bulk domain with a large number of embedded interfaces. In particular, well-posedness of the problem and regularity of the solution are studied. A fitted finite element approximation is also proposed and an a priori error bound is proved. For the solution of the arising linear system, an iterative method based on subspace decomposition is proposed and analyzed. Finally, numerical experiments are presented and rapid convergence using the proposed preconditioner is achieved, confirming the theoretical findings.
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| 38. |
- Henning, Patrick, et al.
(författare)
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A localized orthogonal decomposition method for semi-linear elliptic problems
- 2014
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Ingår i: Mathematical Modelling and Numerical Analysis. - : EDP Sciences. - 2822-7840 .- 0764-583X .- 1290-3841. ; 48:5, s. 1331-1349
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Tidskriftsartikel (refereegranskat)abstract
- In this paper we propose and analyze a localized orthogonal decomposition (LOD) method for solving semi-linear elliptic problems with heterogeneous and highly variable coefficient functions. This Galerkin-type method is based on a generalized finite element basis that spans a low dimensional multiscale space. The basis is assembled by performing localized linear fine-scale computations on small patches that have a diameter of order H | log (H) | where H is the coarse mesh size. Without any assumptions on the type of the oscillations in the coefficients, we give a rigorous proof for a linear convergence of the H1-error with respect to the coarse mesh size even for rough coefficients. To solve the corresponding system of algebraic equations, we propose an algorithm that is based on a damped Newton scheme in the multiscale space.
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| 39. |
- Henning, Patrick, et al.
(författare)
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Localized orthogonal decomposition techniques for boundary value problems
- 2014
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Ingår i: SIAM Journal on Scientific Computing. - : Society for Industrial & Applied Mathematics (SIAM). - 1064-8275 .- 1095-7197. ; 36:4
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Tidskriftsartikel (refereegranskat)abstract
- In this paper we propose a local orthogonal decomposition method (LOD) for elliptic partial differential equations with inhomogeneous Dirichlet and Neumann boundary conditions. For this purpose, we present new boundary correctors which preserve the common convergence rates of the LOD, even if the boundary condition has a rapidly oscillating fine scale structure. We prove a corresponding a priori error estimate and present numerical experiments. We also demonstrate numerically that the method is reliable with respect to thin conductivity channels in the diffusion matrix. Accurate results are obtained without resolving these channels by the coarse grid and without using patches that contain the channels.
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| 40. |
- Henning, Patrick, et al.
(författare)
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The finite element method for the time-dependent gross-pitaevskii equation with angular momentum rotation
- 2017
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Ingår i: SIAM Journal on Numerical Analysis. - : Society for Industrial and Applied Mathematics. - 0036-1429 .- 1095-7170. ; 55:2, s. 923-952
-
Tidskriftsartikel (refereegranskat)abstract
- We consider the time-dependent Gross Pitaevskii equation describing the dynamics of rotating Bose Einstein condensates and its discretization with the finite element method. We analyze a mass conserving Crank-Nicolson-type discretization and prove corresponding a priori error estimates with respect to the maximum norm in time and the L-2- and energy-norm in space. The estimates show that we obtain optimal convergence rates under the assumption of additional regularity for the solution to the Gross Pitaevskii equation. We demonstrate the performance of the method in numerical experiments.
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| 41. |
- Henning, P., et al.
(författare)
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THE FINITE ELEMENT METHOD FOR THE TIME-DEPENDENT GROSS-PITAEVSKII EQUATION WITH ANGULAR MOMENTUM ROTATION
- 2017
-
Ingår i: Siam Journal on Numerical Analysis. - : Society for Industrial & Applied Mathematics (SIAM). - 0036-1429 .- 1095-7170. ; 55:2, s. 923-952
-
Tidskriftsartikel (refereegranskat)abstract
- We consider the time-dependent Gross Pitaevskii equation describing the dynamics of rotating Bose Einstein condensates and its discretization with the finite element method. We analyze a mass conserving Crank-Nicolson-type discretization and prove corresponding a priori error estimates with respect to the maximum norm in time and the L-2- and energy-norm in space. The estimates show that we obtain optimal convergence rates under the assumption of additional regularity for the solution to the Gross Pitaevskii equation. We demonstrate the performance of the method in numerical experiments.
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| 42. |
- Henning, Patrick, et al.
(författare)
-
Two-level discretization techniques for ground state computations of Bose-Einstein condensates
- 2014
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Ingår i: SIAM Journal on Numerical Analysis. - : Society for Industrial & Applied Mathematics (SIAM). - 0036-1429 .- 1095-7170. ; 52:4, s. 1525-1550
-
Tidskriftsartikel (refereegranskat)abstract
- This work presents a new methodology for computing ground states of Bose--Einstein condensates based on finite element discretizations on two different scales of numerical resolution. In a preprocessing step, a low-dimensional (coarse) generalized finite element space is constructed. It is based on a local orthogonal decomposition of the solution space and exhibits high approximation properties. The nonlinear eigenvalue problem that characterizes the ground state is solved by some suitable iterative solver exclusively in this low-dimensional space, without significant loss of accuracy when compared with the solution of the full fine scale problem. The preprocessing step is independent of the types and numbers of bosons. A postprocessing step further improves the accuracy of the method. We present rigorous a priori error estimates that predict convergence rates $H^3$ for the ground state eigenfunction and $H^4$ for the corresponding eigenvalue without pre-asymptotic effects; $H$ being the coarse scale discretization parameter. Numerical experiments indicate that these high rates may still be pessimistic.
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| 43. |
- Holst, Michael J., et al.
(författare)
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Convergence analysis of finite element approximations of the Joule heating problem in three spatial dimensions
- 2010
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Ingår i: BIT Numerical Mathematics. - : Springer. - 0006-3835 .- 1572-9125. ; 50:4, s. 781-795
-
Tidskriftsartikel (refereegranskat)abstract
- In this paper we present a finite element discretization of the Joule-heating problem. We prove existence of solution to the discrete formulation and strong convergence of the finite element solution to the weak solution, up to a sub-sequence. We also present numerical examples in three spatial dimensions. The first example demonstrates the convergence of the method in the second example we consider an engineering application.
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| 44. |
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| 45. |
- Jensen, M., et al.
(författare)
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Finite element convergence for the time-dependent Joule heating problem with mixed boundary conditions
- 2022
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Ingår i: Ima Journal of Numerical Analysis. - : Oxford University Press (OUP). - 0272-4979 .- 1464-3642. ; 42:1, s. 199-228
-
Tidskriftsartikel (refereegranskat)abstract
- We prove strong convergence for a large class of finite element methods for the time-dependent Joule heating problem in three spatial dimensions with mixed boundary conditions on Lipschitz domains. We consider conforming subspaces for the spatial discretization and the backward Euler scheme for the temporal discretization. Furthermore, we prove uniqueness and higher regularity of the solution on creased domains and additional regularity in the interior of the domain. Due to a variational formulation with a cut-off functional, the convergence analysis does not require a discrete maximum principle, permitting approximation spaces suitable for adaptive mesh refinement, responding to the difference in regularity within the domain.
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| 46. |
- Kettil, Gustav, 1990, et al.
(författare)
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A Multiscale Method for Discrete Fiber Network Models
- 2018
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Ingår i: 6th European Conference on Computational Mechanics (Solids, Structures and Coupled Problems). 7th European Conference on Computational Fluid Dynamics, 11-15 June 2018, Glasgow, UK.
-
Konferensbidrag (övrigt vetenskapligt/konstnärligt)abstract
- The mechanics of paper depends on the properties of its fibers and bonds. Modeling paper as a network [1] will include effects of single fibers and bonds, capturing heterogeneous properties. In the ISOP (Innovative Simulation of Paper) project at Fraunhofer-Chalmers Centre, the forming process is simulated [2, 3]. To investigate the mechanical properties of the resulting simulated paper sheets a network approach is utilized. Numerical investigation of fiber networks is demanding due to the large number of fibers and bonds, fluctuation of their properties, and the non-regular network structure. Multiscale methods are useful tools to circumvent such problems. In this work a multi-scale approach for fiber networks is developed, based on a FEM-method for continua [4]. Consider a fiber network governed by a model resulting in an equation Kx = F, where K describes the network properties, x are node displacements, and F are applied forces. The idea of the multi-scale method is to consider a subset of all nodes, denoted coarse nodes, which in turn represents a coarse grid. At each coarse node a basis function is defined similarly as in the finite element method. By solving a system including the coarse nodes an approximation would be attained, however this approximation would leave out the fine scale effects of the heterogeneous network. Instead the coarse basis functions are modified by solving a local system at each coarse node, including surrounding fine nodes. These modified basis functions are thereafter used when solving the global system, resulting in an approximation of the network displacements now including effects from the fine scale.
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| 47. |
- Kettil, Gustav, 1990, et al.
(författare)
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A Multiscale Methodology for Simulation of Mechanical Properties of Paper
- 2020
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Ingår i: Proceedings of the 6th European Conference on Computational Mechanics: Solids, Structures and Coupled Problems, ECCM 2018 and 7th European Conference on Computational Fluid Dynamics, ECFD 2018. - 9788494731167 ; 2020, s. 2795-2806
-
Konferensbidrag (refereegranskat)abstract
- In this work a multiscale framework developed for simulation of mechanical properties of paper is presented. The framework consists of two major parts. In the first part the forming process of a paper machine is simulated using the fiber suspension model developed in [8]. Fluid dynamics together with an advanced contact calculation method enables detailed simulation of the lay down process. The resulting paper sheet is used as input to the second part of the framework. In the second part the fiber configuration attained from the unique forming simulations is transformed into a network representation, enabling simulation of mechanical properties. The paper mechanics is governed by a fiber network model. To study macroscale properties a novel numerical upscaling method for networks has been developed. In this paper the complete simulation methodology is outlined and discussed.
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| 48. |
- Kettil, Gustav, et al.
(författare)
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Numerical upscaling of discrete network models
- 2020
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Ingår i: BIT (Copenhagen). - : Springer Science and Business Media LLC. - 0006-3835 .- 1572-9125. ; 60:1, s. 67-92
-
Tidskriftsartikel (refereegranskat)abstract
- In this paper a numerical multiscale method for discrete networks is presented. The method gives an accurate coarse scale representation of the full network by solving sub-network problems. The method is used to solve problems with highly varying connectivity or random network structure, showing optimal order convergence rates with respect to the mesh size of the coarse representation. Moreover, a network model for paper-based materials is presented. The numerical multiscale method is applied to solve problems governed by the presented network model.
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| 49. |
- Lang, Annika, 1980, et al.
(författare)
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LOCALIZED ORTHOGONAL DECOMPOSITION FOR A MULTISCALE PARABOLIC STOCHASTIC PARTIAL DIFFERENTIAL EQUATION
- 2024
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Ingår i: MULTISCALE MODELING & SIMULATION. - 1540-3459 .- 1540-3467. ; 22:1, s. 204-229
-
Tidskriftsartikel (refereegranskat)abstract
- A multiscale method is proposed for a parabolic stochastic partial differential equation with additive noise and highly oscillatory diffusion. The framework is based on the localized orthogonal decomposition (LOD) method and computes a coarse -scale representation of the elliptic operator, enriched by fine -scale information on the diffusion. Optimal order strong convergence is derived. The LOD technique is combined with a (multilevel) Monte Carlo estimator and the weak error is analyzed. Numerical examples that confirm the theoretical findings are provided, and the computational efficiency of the method is highlighted.
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| 50. |
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