Localized orthogonal decomposition for high-order, hybrid finite elements
Duration: 03/2024 – 02/2026
Funding scheme: Funding for Mobility Cooperation with Germany
Funder: Research Council of Finland
Budget: 18,000 €
My role: Finnish principal investigator (transferred to Zhi-Song Liu in 09/2024)
Sibling project: High-order hybrid multiscale methods for rough heterogeneous structures
German principal investigator: Roland Maier
Public description
Many natural processes involve multiple scales, such as fluid flow in porous materials or wave propagation through layered soil. These systems have microscopic scales, describing material properties and textures, and macroscopic scales, where observable phenomena occur. The same principle applies in manufacturing, for example, when producing fiber-reinforced composites that alter material behavior.
Mathematically, partial differential equations with coefficients varying on a microscopic scale describe such processes. While typically only practical macroscopic information is of interest, discarding microscopic features in numerical simulations often fails to provide the desired results. However, fully resolving microscopic coefficients is computationally prohibitive.
Numerical homogenization methods offer a possible solution to this problem. This project aims to improve their performance by combining state-of-the-art hybrid finite elements with high-order multiscale approaches.
Scientific abstract
This is achieved by solving a set of auxiliary local problems in an offline phase, which then significantly lower computation times to solve the problem of interest in an online stage. Such an approach is particularly beneficial if similar problems have to be solved multiple times, as in the context of time-dependent PDEs and uncertainty quantification.
Current research aims to further improve the locality of the auxiliary problems while enhancing the accuracy, for instance, with higher-order approximations. In this project, we aim to combine the efficient multiscale method known as localized orthogonal decomposition with the advantages of hybrid finite element schemes. Doing so allows us to
- exploit the benefits (symmetric positive definite matrices, super-convergence, etc.) of hybrid discontinuous Galerkin and hybrid high-order methods,
- extend the construction and allow for higher-order convergence rates without restrictive regularity assumptions,
- investigate super-localization properties.
Keywords
high-order approximation, hybrid discontinuous Galerkin, hybrid high-order method, localized orthogonal decomposition, multiscale problems, rough coefficients, super-localized orthogonal decomposition
Related publications
Kaarnioja, Vesa; Rupp, Andreas
Quasi-Monte Carlo and discontinuous Galerkin Journal Article
In: Electronic Transactions on Numerical Analysis, vol. 60, pp. 589–617, 2024, ISSN: 1068-9613.
@article{KaarniojaR24,
title = {Quasi-Monte Carlo and discontinuous Galerkin},
author = {Vesa Kaarnioja and Andreas Rupp},
doi = {10.1553/etna_vol60s589},
issn = {1068-9613},
year = {2024},
date = {2024-12-12},
urldate = {2024-12-12},
journal = {Electronic Transactions on Numerical Analysis},
volume = {60},
pages = {589--617},
abstract = {In this study, we consider the development of tailored quasi-Monte Carlo (QMC) cubatures for non-conforming discontinuous Galerkin (DG) approximations of elliptic partial differential equations (PDEs) with random coefficients. We consider both the affine and uniform and the lognormal models for the input random field and investigate the use of QMC cubatures to approximate the expected value of the PDE response subject to input uncertainty. In particular, we prove that the resulting QMC convergence rate for DG approximations behaves in the same way as if continuous finite elements were chosen. Notably, the parametric regularity bounds for DG, which are developed in this work, are also useful for other methods such as sparse grids. Numerical results underline our analytical findings.},
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In this study, we consider the development of tailored quasi-Monte Carlo (QMC) cubatures for non-conforming discontinuous Galerkin (DG) approximations of elliptic partial differential equations (PDEs) with random coefficients. We consider both the affine and uniform and the lognormal models for the input random field and investigate the use of QMC cubatures to approximate the expected value of the PDE response subject to input uncertainty. In particular, we prove that the resulting QMC convergence rate for DG approximations behaves in the same way as if continuous finite elements were chosen. Notably, the parametric regularity bounds for DG, which are developed in this work, are also useful for other methods such as sparse grids. Numerical results underline our analytical findings.
Altunay, Rabia; Vesterinen, Kalevi; Alander, Pasi; Immonen, Eero; Rupp, Andreas; Roininen, Lassi
Denture reinforcement via topology optimization Journal Article Forthcoming
In: Medical Engineering & Physics, Forthcoming, ISSN: 1350-4533.
@article{AltunayVAIRR24,
title = {Denture reinforcement via topology optimization},
author = {Rabia Altunay and Kalevi Vesterinen and Pasi Alander and Eero Immonen and Andreas Rupp and Lassi Roininen},
doi = {10.1016/j.medengphy.2024.104272},
issn = {1350-4533},
year = {2024},
date = {2024-12-07},
urldate = {2024-12-07},
journal = {Medical Engineering & Physics},
abstract = {We present a computational design method that optimizes the reinforcement of dentures and increases the stiffness of dentures. Our approach optimally places reinforcement in the denture, which modern multi-material three-dimensional printers could implement. The study focuses on reducing denture displacement by identifying regions that require reinforcement (E-glass material) with the help of topology optimization. Our method is applied to a three-dimensional complete lower jaw denture. We compare the displacement results of a non-reinforced denture and a reinforced denture that has two materials. The comparison results indicate that there is a decrease in the displacement in the reinforced denture. Considering node-based displacement distribution, the reinforcement reduces the displacement magnitudes in the reinforced denture compared to the non-reinforced denture. The study guides dental technicians on where to automatically place reinforcement in the fabrication process, helping them save time and reduce material usage.},
keywords = {},
pubstate = {forthcoming},
tppubtype = {article}
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We present a computational design method that optimizes the reinforcement of dentures and increases the stiffness of dentures. Our approach optimally places reinforcement in the denture, which modern multi-material three-dimensional printers could implement. The study focuses on reducing denture displacement by identifying regions that require reinforcement (E-glass material) with the help of topology optimization. Our method is applied to a three-dimensional complete lower jaw denture. We compare the displacement results of a non-reinforced denture and a reinforced denture that has two materials. The comparison results indicate that there is a decrease in the displacement in the reinforced denture. Considering node-based displacement distribution, the reinforcement reduces the displacement magnitudes in the reinforced denture compared to the non-reinforced denture. The study guides dental technicians on where to automatically place reinforcement in the fabrication process, helping them save time and reduce material usage.
Liu, Zhi-Song; Maier, Roland; Rupp, Andreas
Numerical homogenization by continuous super-resolution Online Forthcoming
Forthcoming, visited: 13.11.2024.
@online{LiuMR24,
title = {Numerical homogenization by continuous super-resolution},
author = {Zhi-Song Liu and Roland Maier and Andreas Rupp},
url = {https://arxiv.org/abs/2411.07576},
doi = {10.48550/arXiv.2411.07576},
year = {2024},
date = {2024-11-13},
urldate = {2024-11-13},
abstract = {Finite element methods typically require a high resolution to satisfactorily approximate micro and even macro patterns of an underlying physical model. This issue can be circumvented by appropriate numerical homogenization or multiscale strategies that are able to obtain reasonable approximations on under-resolved scales. In this paper, we study the implicit neural representation and propose a continuous super-resolution network as a numerical homogenization strategy. It can take coarse finite element data to learn both in-distribution and out-of-distribution high-resolution finite element predictions. Our highlight is the design of a local implicit transformer, which is able to learn multiscale features. We also propose Gabor wavelet-based coordinate encodings which can overcome the bias of neural networks learning low-frequency features. Finally, perception is often preferred over distortion so scientists can recognize the visual pattern for further investigation. However, implicit neural representation is known for its lack of local pattern supervision. We propose to use stochastic cosine similarities to compare the local feature differences between prediction and ground truth. It shows better performance on structural alignments. Our experiments show that our proposed strategy achieves superior performance as an in-distribution and out-of-distribution super-resolution strategy.},
keywords = {},
pubstate = {forthcoming},
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Finite element methods typically require a high resolution to satisfactorily approximate micro and even macro patterns of an underlying physical model. This issue can be circumvented by appropriate numerical homogenization or multiscale strategies that are able to obtain reasonable approximations on under-resolved scales. In this paper, we study the implicit neural representation and propose a continuous super-resolution network as a numerical homogenization strategy. It can take coarse finite element data to learn both in-distribution and out-of-distribution high-resolution finite element predictions. Our highlight is the design of a local implicit transformer, which is able to learn multiscale features. We also propose Gabor wavelet-based coordinate encodings which can overcome the bias of neural networks learning low-frequency features. Finally, perception is often preferred over distortion so scientists can recognize the visual pattern for further investigation. However, implicit neural representation is known for its lack of local pattern supervision. We propose to use stochastic cosine similarities to compare the local feature differences between prediction and ground truth. It shows better performance on structural alignments. Our experiments show that our proposed strategy achieves superior performance as an in-distribution and out-of-distribution super-resolution strategy.
Liu, Zhi-Song; Büttner, Markus; Aizinger, Vadym; Rupp, Andreas
Super-resolution works for coastal simulations Online Forthcoming
Forthcoming, visited: 30.08.2024.
@online{LiuBAR24,
title = {Super-resolution works for coastal simulations},
author = {Zhi-Song Liu and Markus Büttner and Vadym Aizinger and Andreas Rupp},
url = {https://arxiv.org/abs/2408.16553},
doi = {10.48550/arXiv.2408.16553},
year = {2024},
date = {2024-08-30},
urldate = {2024-08-30},
abstract = {Learning fine-scale details of a coastal ocean simulation from a coarse representation is a challenging task. For real-world applications, high-resolution simulations are necessary to advance understanding of many coastal processes, specifically, to predict flooding resulting from tsunamis and storm surges. We propose a Deep Network for Coastal Super-Resolution (DNCSR) for spatiotemporal enhancement to efficiently learn the high-resolution numerical solution. Given images of coastal simulations produced on low-resolution computational meshes using low polynomial order discontinuous Galerkin discretizations and a coarse temporal resolution, the proposed DNCSR learns to produce high-resolution free surface elevation and velocity visualizations in both time and space. To efficiently model the dynamic changes over time and space, we propose grid-aware spatiotemporal attention to project the temporal features to the spatial domain for non-local feature matching. The coordinate information is also utilized via positional encoding. For the final reconstruction, we use the spatiotemporal bilinear operation to interpolate the missing frames and then expand the feature maps to the frequency domain for residual mapping. Besides data-driven losses, the proposed physics-informed loss guarantees gradient consistency and momentum changes. Their combination contributes to the overall 24% improvements in RMSE. To train the proposed model, we propose a large-scale coastal simulation dataset and use it for model optimization and evaluation. Our method shows superior super-resolution quality and fast computation compared to the state-of-the-art methods.},
keywords = {},
pubstate = {forthcoming},
tppubtype = {online}
}
Learning fine-scale details of a coastal ocean simulation from a coarse representation is a challenging task. For real-world applications, high-resolution simulations are necessary to advance understanding of many coastal processes, specifically, to predict flooding resulting from tsunamis and storm surges. We propose a Deep Network for Coastal Super-Resolution (DNCSR) for spatiotemporal enhancement to efficiently learn the high-resolution numerical solution. Given images of coastal simulations produced on low-resolution computational meshes using low polynomial order discontinuous Galerkin discretizations and a coarse temporal resolution, the proposed DNCSR learns to produce high-resolution free surface elevation and velocity visualizations in both time and space. To efficiently model the dynamic changes over time and space, we propose grid-aware spatiotemporal attention to project the temporal features to the spatial domain for non-local feature matching. The coordinate information is also utilized via positional encoding. For the final reconstruction, we use the spatiotemporal bilinear operation to interpolate the missing frames and then expand the feature maps to the frequency domain for residual mapping. Besides data-driven losses, the proposed physics-informed loss guarantees gradient consistency and momentum changes. Their combination contributes to the overall 24% improvements in RMSE. To train the proposed model, we propose a large-scale coastal simulation dataset and use it for model optimization and evaluation. Our method shows superior super-resolution quality and fast computation compared to the state-of-the-art methods.
Hauck, Moritz; Målqvist, Axel; Rupp, Andreas
Arbitrary order approximations at constant cost for Timoshenko beam network models Online Forthcoming
Forthcoming, visited: 19.07.2024.
@online{HauckMR24,
title = {Arbitrary order approximations at constant cost for Timoshenko beam network models},
author = {Moritz Hauck and Axel Målqvist and Andreas Rupp},
url = {https://arxiv.org/abs/2407.14388},
doi = {10.48550/arXiv.2407.14388},
year = {2024},
date = {2024-07-19},
urldate = {2024-07-19},
abstract = {This paper considers the numerical solution of Timoshenko beam network models, comprised of Timoshenko beam equations on each edge of the network, which are coupled at the nodes of the network using rigid joint conditions. Through hybridization, we can equivalently reformulate the problem as a symmetric positive definite system of linear equations posed on the network nodes. This is possible since the nodes, where the beam equations are coupled, are zero-dimensional objects. To discretize the beam network model, we propose a hybridizable discontinuous Galerkin method that can achieve arbitrary orders of convergence under mesh refinement without increasing the size of the global system matrix. As a preconditioner for the typically very poorly conditioned global system matrix, we employ a two-level overlapping additive Schwarz method. We prove uniform convergence of the corresponding preconditioned conjugate gradient method under appropriate connectivity assumptions on the network. Numerical experiments support the theoretical findings of this work.},
keywords = {},
pubstate = {forthcoming},
tppubtype = {online}
}
This paper considers the numerical solution of Timoshenko beam network models, comprised of Timoshenko beam equations on each edge of the network, which are coupled at the nodes of the network using rigid joint conditions. Through hybridization, we can equivalently reformulate the problem as a symmetric positive definite system of linear equations posed on the network nodes. This is possible since the nodes, where the beam equations are coupled, are zero-dimensional objects. To discretize the beam network model, we propose a hybridizable discontinuous Galerkin method that can achieve arbitrary orders of convergence under mesh refinement without increasing the size of the global system matrix. As a preconditioner for the typically very poorly conditioned global system matrix, we employ a two-level overlapping additive Schwarz method. We prove uniform convergence of the corresponding preconditioned conjugate gradient method under appropriate connectivity assumptions on the network. Numerical experiments support the theoretical findings of this work.
Vedral, Joshua; Rupp, Andreas; Kuzmin, Dmitri
Strongly consistent low-dissipation WENO schemes for finite elements Online Forthcoming
Forthcoming, visited: 08.07.2024.
@online{VedralRK24,
title = {Strongly consistent low-dissipation WENO schemes for finite elements},
author = {Joshua Vedral and Andreas Rupp and Dmitri Kuzmin},
url = {https://arxiv.org/abs/2407.04646},
doi = {10.48550/arXiv.2407.04646},
year = {2024},
date = {2024-07-08},
urldate = {2024-07-08},
abstract = {We propose a way to maintain strong consistency and facilitate error analysis in the context of dissipation-based WENO stabilization for continuous and discontinuous Galerkin discretizations of conservation laws. Following Kuzmin and Vedral (J. Comput. Phys. 487:112153, 2023) and Vedral (arXiv preprint arXiv:2309.12019), we use WENO shock detectors to determine appropriate amounts of low-order artificial viscosity. In contrast to existing WENO methods, our approach blends candidate polynomials using residual-based nonlinear weights. The shock-capturing terms of our stabilized Galerkin methods vanish if residuals do. This enables us to achieve improved accuracy compared to weakly consistent alternatives. As we show in the context of steady convection-diffusion-reaction (CDR) equations, nonlinear local projection stabilization terms can be included in a way that preserves the coercivity of local bilinear forms. For the corresponding Galerkin-WENO discretization of a CDR problem, we rigorously derive a priori error estimates. Additionally, we demonstrate the stability and accuracy of the proposed method through one- and two-dimensional numerical experiments for hyperbolic conservation laws and systems thereof. The numerical results for representative test problems are superior to those obtained with traditional WENO schemes, particularly in scenarios involving shocks and steep gradients.},
keywords = {},
pubstate = {forthcoming},
tppubtype = {online}
}
We propose a way to maintain strong consistency and facilitate error analysis in the context of dissipation-based WENO stabilization for continuous and discontinuous Galerkin discretizations of conservation laws. Following Kuzmin and Vedral (J. Comput. Phys. 487:112153, 2023) and Vedral (arXiv preprint arXiv:2309.12019), we use WENO shock detectors to determine appropriate amounts of low-order artificial viscosity. In contrast to existing WENO methods, our approach blends candidate polynomials using residual-based nonlinear weights. The shock-capturing terms of our stabilized Galerkin methods vanish if residuals do. This enables us to achieve improved accuracy compared to weakly consistent alternatives. As we show in the context of steady convection-diffusion-reaction (CDR) equations, nonlinear local projection stabilization terms can be included in a way that preserves the coercivity of local bilinear forms. For the corresponding Galerkin-WENO discretization of a CDR problem, we rigorously derive a priori error estimates. Additionally, we demonstrate the stability and accuracy of the proposed method through one- and two-dimensional numerical experiments for hyperbolic conservation laws and systems thereof. The numerical results for representative test problems are superior to those obtained with traditional WENO schemes, particularly in scenarios involving shocks and steep gradients.
Cheng, Hanz Martin; Helin, Tapio; Manninen, Ville-Petteri; Holopainen, Timo; Jokinen, Juha; Sorvari, Samu; Rupp, Andreas
Recovery of transversely-isotropic elastic material parameters in induction motor rotors Online Forthcoming
Forthcoming, visited: 10.05.2024.
@online{ChengHMHJSR24,
title = {Recovery of transversely-isotropic elastic material parameters in induction motor rotors},
author = {Hanz Martin Cheng and Tapio Helin and Ville-Petteri Manninen and Timo Holopainen and Juha Jokinen and Samu Sorvari and Andreas Rupp},
url = {https://arxiv.org/abs/2405.06388},
doi = {10.48550/arXiv.2405.06388},
year = {2024},
date = {2024-05-10},
urldate = {2024-05-10},
abstract = {We propose numerical algorithms for recovering parameters in eigenvalue problems for linear elasticity of transversely isotropic materials. Specifically, the algorithms are used to recover the elastic constants of a rotor core. Numerical tests show that in the noiseless setup, two pairs of bending modes are sufficient for recovering one to four parameters accurately. To recover all five parameters that govern the elastic properties of electric engines accurately, we require three pairs of bending modes and one torsional mode. Moreover, we study the stability of the inversion method against multiplicative noise; for tests in which the data contained multiplicative noise of at most 1%, we find that all parameters can be recovered with an error less than 10%. },
keywords = {},
pubstate = {forthcoming},
tppubtype = {online}
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We propose numerical algorithms for recovering parameters in eigenvalue problems for linear elasticity of transversely isotropic materials. Specifically, the algorithms are used to recover the elastic constants of a rotor core. Numerical tests show that in the noiseless setup, two pairs of bending modes are sufficient for recovering one to four parameters accurately. To recover all five parameters that govern the elastic properties of electric engines accurately, we require three pairs of bending modes and one torsional mode. Moreover, we study the stability of the inversion method against multiplicative noise; for tests in which the data contained multiplicative noise of at most 1%, we find that all parameters can be recovered with an error less than 10%.
Di Pietro, Daniele Antonio; Dong, Zhaonan; Kanschat, Guido; Matalon, Pierre; Rupp, Andreas
Homogeneous multigrid for hybrid discretizations: application to HHO methods Online Forthcoming
Forthcoming, visited: 25.03.2024.
@online{DiPietroDKMR24,
title = {Homogeneous multigrid for hybrid discretizations: application to HHO methods},
author = {Di Pietro, Daniele Antonio and Zhaonan Dong and Guido Kanschat and Pierre Matalon and Andreas Rupp},
url = {https://arxiv.org/abs/2403.15858
https://inria.hal.science/hal-04518103},
doi = {10.48550/arXiv.2403.15858},
year = {2024},
date = {2024-03-25},
urldate = {2024-03-25},
abstract = {We prove the uniform convergence of the geometric multigrid V-cycle for hybrid high-order (HHO) and other discontinuous skeletal methods. Our results generalize previously established results for HDG methods, and our multigrid method uses standard smoothers and local solvers that are bounded, convergent, and consistent. We use a weak version of elliptic regularity in our proofs. Numerical experiments confirm our theoretical results.},
keywords = {},
pubstate = {forthcoming},
tppubtype = {online}
}
We prove the uniform convergence of the geometric multigrid V-cycle for hybrid high-order (HHO) and other discontinuous skeletal methods. Our results generalize previously established results for HDG methods, and our multigrid method uses standard smoothers and local solvers that are bounded, convergent, and consistent. We use a weak version of elliptic regularity in our proofs. Numerical experiments confirm our theoretical results.
Lu, Peipei; Maier, Roland; Rupp, Andreas
A localized orthogonal decomposition strategy for hybrid discontinuous Galerkin methods Online Forthcoming
Forthcoming, visited: 28.07.2023.
@online{LuMR23,
title = {A localized orthogonal decomposition strategy for hybrid discontinuous Galerkin methods},
author = {Peipei Lu and Roland Maier and Andreas Rupp},
url = {https://arxiv.org/abs/2307.14961},
doi = {10.48550/arXiv.2307.14961},
year = {2023},
date = {2023-07-28},
urldate = {2023-07-28},
abstract = {We formulate and analyze a multiscale method for an elliptic problem with an oscillatory coefficient based on a skeletal (hybrid) formulation. More precisely, we employ hybrid discontinuous Galerkin approaches and combine them with the localized orthogonal decomposition methodology to obtain a coarse-scale skeletal method that effectively includes fine-scale information. This work is a first step to reliably merge hybrid skeletal formulations and localized orthogonal decomposition and unite the advantages of both strategies. Numerical experiments are presented to illustrate the theoretical findings. },
keywords = {},
pubstate = {forthcoming},
tppubtype = {online}
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We formulate and analyze a multiscale method for an elliptic problem with an oscillatory coefficient based on a skeletal (hybrid) formulation. More precisely, we employ hybrid discontinuous Galerkin approaches and combine them with the localized orthogonal decomposition methodology to obtain a coarse-scale skeletal method that effectively includes fine-scale information. This work is a first step to reliably merge hybrid skeletal formulations and localized orthogonal decomposition and unite the advantages of both strategies. Numerical experiments are presented to illustrate the theoretical findings.
Kazarnikov, Alexey; Ray, Nadja; Haario, Heikki; Lappalainen, Joona; Rupp, Andreas
Parameter estimation for cellular automata Online Forthcoming
Forthcoming, visited: 01.02.2023.
@online{KazarnikovRHLR23,
title = {Parameter estimation for cellular automata},
author = {Alexey Kazarnikov and Nadja Ray and Heikki Haario and Joona Lappalainen and Andreas Rupp},
url = {https://arxiv.org/abs/2301.13320},
doi = {10.48550/arXiv.2301.13320},
year = {2023},
date = {2023-02-01},
urldate = {2023-02-01},
abstract = {Self organizing complex systems can be modeled using cellular automaton models. However, the parametrization of these models is crucial and significantly determines the resulting structural pattern. In this research, we introduce and successfully apply a sound statistical method to estimate these parameters. The method is based on constructing Gaussian likelihoods using characteristics of the structures such as the mean particle size. We show that our approach is robust with respect to the method parameters, domain size of patterns, or CA iterations. },
keywords = {},
pubstate = {forthcoming},
tppubtype = {online}
}
Self organizing complex systems can be modeled using cellular automaton models. However, the parametrization of these models is crucial and significantly determines the resulting structural pattern. In this research, we introduce and successfully apply a sound statistical method to estimate these parameters. The method is based on constructing Gaussian likelihoods using characteristics of the structures such as the mean particle size. We show that our approach is robust with respect to the method parameters, domain size of patterns, or CA iterations.