| 1. |
- 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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| 2. |
- 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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| 3. |
- 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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| 4. |
- 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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| 5. |
- 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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| 6. |
- 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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| 7. |
- 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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| 8. |
- 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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| 9. |
- 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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| 10. |
- 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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