Machine-learned force fields (MLFFs) promise to offer a computationally efficient alternative to ab initio simulations for complex molecular systems. However, ensuring their generalizability beyond training data is crucial for their wide application in studying solid materials. This work investigates the ability of a graph neural network (GNN)-based MLFF, trained on Lennard-Jones Argon, to describe solid-state phenomena not explicitly included during training. We assess the MLFF's performance in predicting phonon density of states (PDOS) for a perfect face-centered cubic (FCC) crystal structure at both zero and finite temperatures. Additionally, we evaluate vacancy migration rates and energy barriers in an imperfect crystal using direct molecular dynamics (MD) simulations and the string method. Notably, vacancy configurations were absent from the training data. Our results demonstrate the MLFF's capability to capture essential solid-state properties with good agreement to reference data, even for unseen configurations. We further discuss data engineering strategies to enhance the generalizability of MLFFs. The proposed set of benchmark tests and workflow for evaluating MLFF performance in describing perfect and imperfect crystals pave the way for reliable application of MLFFs in studying complex solid-state materials.
{"title":"Generalizability of Graph Neural Network Force Fields for Predicting Solid-State Properties","authors":"Shaswat Mohanty, Yifan Wang, Wei Cai","doi":"arxiv-2409.09931","DOIUrl":"https://doi.org/arxiv-2409.09931","url":null,"abstract":"Machine-learned force fields (MLFFs) promise to offer a computationally\u0000efficient alternative to ab initio simulations for complex molecular systems.\u0000However, ensuring their generalizability beyond training data is crucial for\u0000their wide application in studying solid materials. This work investigates the\u0000ability of a graph neural network (GNN)-based MLFF, trained on Lennard-Jones\u0000Argon, to describe solid-state phenomena not explicitly included during\u0000training. We assess the MLFF's performance in predicting phonon density of\u0000states (PDOS) for a perfect face-centered cubic (FCC) crystal structure at both\u0000zero and finite temperatures. Additionally, we evaluate vacancy migration rates\u0000and energy barriers in an imperfect crystal using direct molecular dynamics\u0000(MD) simulations and the string method. Notably, vacancy configurations were\u0000absent from the training data. Our results demonstrate the MLFF's capability to\u0000capture essential solid-state properties with good agreement to reference data,\u0000even for unseen configurations. We further discuss data engineering strategies\u0000to enhance the generalizability of MLFFs. The proposed set of benchmark tests\u0000and workflow for evaluating MLFF performance in describing perfect and\u0000imperfect crystals pave the way for reliable application of MLFFs in studying\u0000complex solid-state materials.","PeriodicalId":501162,"journal":{"name":"arXiv - MATH - Numerical Analysis","volume":"21 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142259151","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Physics-Informed Neural Networks (PINNs) represent a significant advancement in Scientific Machine Learning (SciML), which integrate physical domain knowledge into an empirical loss function as soft constraints and apply existing machine learning methods to train the model. However, recent research has noted that PINNs may fail to learn relatively complex Partial Differential Equations (PDEs). This paper addresses the failure modes of PINNs by introducing a novel, hard-constrained deep learning method -- trust-region Sequential Quadratic Programming (trSQP-PINN). In contrast to directly training the penalized soft-constrained loss as in PINNs, our method performs a linear-quadratic approximation of the hard-constrained loss, while leveraging the soft-constrained loss to adaptively adjust the trust-region radius. We only trust our model approximations and make updates within the trust region, and such an updating manner can overcome the ill-conditioning issue of PINNs. We also address the computational bottleneck of second-order SQP methods by employing quasi-Newton updates for second-order information, and importantly, we introduce a simple pretraining step to further enhance training efficiency of our method. We demonstrate the effectiveness of trSQP-PINN through extensive experiments. Compared to existing hard-constrained methods for PINNs, such as penalty methods and augmented Lagrangian methods, trSQP-PINN significantly improves the accuracy of the learned PDE solutions, achieving up to 1-3 orders of magnitude lower errors. Additionally, our pretraining step is generally effective for other hard-constrained methods, and experiments have shown the robustness of our method against both problem-specific parameters and algorithm tuning parameters.
{"title":"Physics-Informed Neural Networks with Trust-Region Sequential Quadratic Programming","authors":"Xiaoran Cheng, Sen Na","doi":"arxiv-2409.10777","DOIUrl":"https://doi.org/arxiv-2409.10777","url":null,"abstract":"Physics-Informed Neural Networks (PINNs) represent a significant advancement\u0000in Scientific Machine Learning (SciML), which integrate physical domain\u0000knowledge into an empirical loss function as soft constraints and apply\u0000existing machine learning methods to train the model. However, recent research\u0000has noted that PINNs may fail to learn relatively complex Partial Differential\u0000Equations (PDEs). This paper addresses the failure modes of PINNs by\u0000introducing a novel, hard-constrained deep learning method -- trust-region\u0000Sequential Quadratic Programming (trSQP-PINN). In contrast to directly training\u0000the penalized soft-constrained loss as in PINNs, our method performs a\u0000linear-quadratic approximation of the hard-constrained loss, while leveraging\u0000the soft-constrained loss to adaptively adjust the trust-region radius. We only\u0000trust our model approximations and make updates within the trust region, and\u0000such an updating manner can overcome the ill-conditioning issue of PINNs. We\u0000also address the computational bottleneck of second-order SQP methods by\u0000employing quasi-Newton updates for second-order information, and importantly,\u0000we introduce a simple pretraining step to further enhance training efficiency\u0000of our method. We demonstrate the effectiveness of trSQP-PINN through extensive\u0000experiments. Compared to existing hard-constrained methods for PINNs, such as\u0000penalty methods and augmented Lagrangian methods, trSQP-PINN significantly\u0000improves the accuracy of the learned PDE solutions, achieving up to 1-3 orders\u0000of magnitude lower errors. Additionally, our pretraining step is generally\u0000effective for other hard-constrained methods, and experiments have shown the\u0000robustness of our method against both problem-specific parameters and algorithm\u0000tuning parameters.","PeriodicalId":501162,"journal":{"name":"arXiv - MATH - Numerical Analysis","volume":"82 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142259093","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we propose a new approaches for low rank approximation of quaternion tensors cite{chen2019low,zhang1997quaternions,hamilton1866elements}. The first method uses quasi-norms to approximate the tensor by a low-rank tensor using the QT-product cite{miao2023quaternion}, which generalizes the known L-product to N-mode quaternions. The second method involves Non-Convex norms to approximate the Tucker and TT-rank for the completion problem. We demonstrate that the proposed methods can effectively approximate the tensor compared to the convexifying of the rank, such as the nuclear norm. We provide theoretical results and numerical experiments to show the efficiency of the proposed methods in the Inpainting and Denoising applications.
{"title":"Quaternion tensor low rank approximation","authors":"Alaeddine Zahir, Ahmed Ratnani, Khalide Jbilou","doi":"arxiv-2409.10724","DOIUrl":"https://doi.org/arxiv-2409.10724","url":null,"abstract":"In this paper, we propose a new approaches for low rank approximation of\u0000quaternion tensors\u0000cite{chen2019low,zhang1997quaternions,hamilton1866elements}. The first method\u0000uses quasi-norms to approximate the tensor by a low-rank tensor using the\u0000QT-product cite{miao2023quaternion}, which generalizes the known L-product to\u0000N-mode quaternions. The second method involves Non-Convex norms to approximate\u0000the Tucker and TT-rank for the completion problem. We demonstrate that the\u0000proposed methods can effectively approximate the tensor compared to the\u0000convexifying of the rank, such as the nuclear norm. We provide theoretical\u0000results and numerical experiments to show the efficiency of the proposed\u0000methods in the Inpainting and Denoising applications.","PeriodicalId":501162,"journal":{"name":"arXiv - MATH - Numerical Analysis","volume":"29 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142259097","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The space-time adaptive ADER finite element DG method with a posteriori correction technique of solutions on subcells by the finite-volume ADER-WENO limiter was used to simulate non-stationary compressible multicomponent reactive flows. The multicomponent composition of the reacting medium and the reactions occurring in it were described by expanding the original system of Euler equations by a system of non-stationary convection-reaction equations. The use of this method to simulate high stiff problems associated with reactions occurring in a multicomponent medium requires the use of the adaptive change in the time step. The solution of the classical problem related to the formation and propagation of a ZND detonation wave is carried out. It was shown that the space-time adaptive ADER finite element DG method with a posteriori correction technique of solutions on subcells by the finite-volume ADER-WENO limiter can be used to simulate flows without using of splitting in directions and fractional step methods.
{"title":"Space-time adaptive ADER-DG finite element method with LST-DG predictor and a posteriori sub-cell WENO finite-volume limiting for simulation of non-stationary compressible multicomponent reactive flows","authors":"I. S Popov","doi":"arxiv-2409.09932","DOIUrl":"https://doi.org/arxiv-2409.09932","url":null,"abstract":"The space-time adaptive ADER finite element DG method with a posteriori\u0000correction technique of solutions on subcells by the finite-volume ADER-WENO\u0000limiter was used to simulate non-stationary compressible multicomponent\u0000reactive flows. The multicomponent composition of the reacting medium and the\u0000reactions occurring in it were described by expanding the original system of\u0000Euler equations by a system of non-stationary convection-reaction equations.\u0000The use of this method to simulate high stiff problems associated with\u0000reactions occurring in a multicomponent medium requires the use of the adaptive\u0000change in the time step. The solution of the classical problem related to the\u0000formation and propagation of a ZND detonation wave is carried out. It was shown\u0000that the space-time adaptive ADER finite element DG method with a posteriori\u0000correction technique of solutions on subcells by the finite-volume ADER-WENO\u0000limiter can be used to simulate flows without using of splitting in directions\u0000and fractional step methods.","PeriodicalId":501162,"journal":{"name":"arXiv - MATH - Numerical Analysis","volume":"16 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142259150","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The space-time adaptive ADER-DG finite element method with LST-DG predictor and a posteriori sub-cell ADER-WENO finite-volume limiting was used for simulation of multidimensional reacting flows with detonation waves. The presented numerical method does not use any ideas of splitting or fractional time steps methods. The modification of the LST-DG predictor has been developed, based on a local partition of the time step in cells in which strong reactivity of the medium is observed. This approach made it possible to obtain solutions to classical problems of flows with detonation waves and strong stiffness, without significantly decreasing the time step. The results obtained show the very high applicability and efficiency of using the ADER-DG-PN method with a posteriori sub-cell limiting for simulating reactive flows with detonation waves. The numerical solution shows the correct formation and propagation of ZND detonation waves. The structure of detonation waves is resolved by this numerical method with subcell resolution even on coarse spatial meshes. The smooth components of the numerical solution are correctly and very accurately reproduced by the numerical method. Non-physical artifacts of the numerical solution, typical for problems with detonation waves, such as the propagation of non-physical shock waves and weak detonation fronts ahead of the main detonation front, did not arise in the results obtained. The results of simulating rather complex problems associated with the propagation of detonation waves in significantly inhomogeneous domains are presented, which show that all the main features of detonation flows are correctly reproduced by this numerical method. It can be concluded that the space-time adaptive ADER-DG-PN method with a posteriori sub-cell ADER-WENO finite-volume limiting is perfectly applicable to simulating complex reacting flows with detonation waves.
{"title":"Space-time adaptive ADER-DG finite element method with LST-DG predictor and a posteriori sub-cell ADER-WENO finite-volume limiting for multidimensional detonation waves simulation","authors":"I. S. Popov","doi":"arxiv-2409.09911","DOIUrl":"https://doi.org/arxiv-2409.09911","url":null,"abstract":"The space-time adaptive ADER-DG finite element method with LST-DG predictor\u0000and a posteriori sub-cell ADER-WENO finite-volume limiting was used for\u0000simulation of multidimensional reacting flows with detonation waves. The\u0000presented numerical method does not use any ideas of splitting or fractional\u0000time steps methods. The modification of the LST-DG predictor has been\u0000developed, based on a local partition of the time step in cells in which strong\u0000reactivity of the medium is observed. This approach made it possible to obtain\u0000solutions to classical problems of flows with detonation waves and strong\u0000stiffness, without significantly decreasing the time step. The results obtained\u0000show the very high applicability and efficiency of using the ADER-DG-PN method\u0000with a posteriori sub-cell limiting for simulating reactive flows with\u0000detonation waves. The numerical solution shows the correct formation and\u0000propagation of ZND detonation waves. The structure of detonation waves is\u0000resolved by this numerical method with subcell resolution even on coarse\u0000spatial meshes. The smooth components of the numerical solution are correctly\u0000and very accurately reproduced by the numerical method. Non-physical artifacts\u0000of the numerical solution, typical for problems with detonation waves, such as\u0000the propagation of non-physical shock waves and weak detonation fronts ahead of\u0000the main detonation front, did not arise in the results obtained. The results\u0000of simulating rather complex problems associated with the propagation of\u0000detonation waves in significantly inhomogeneous domains are presented, which\u0000show that all the main features of detonation flows are correctly reproduced by\u0000this numerical method. It can be concluded that the space-time adaptive\u0000ADER-DG-PN method with a posteriori sub-cell ADER-WENO finite-volume limiting\u0000is perfectly applicable to simulating complex reacting flows with detonation\u0000waves.","PeriodicalId":501162,"journal":{"name":"arXiv - MATH - Numerical Analysis","volume":"3 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142259152","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In recent years, variational quantum algorithms have garnered significant attention as a candidate approach for near-term quantum advantage using noisy intermediate-scale quantum (NISQ) devices. In this article we introduce kernel descent, a novel algorithm for minimizing the functions underlying variational quantum algorithms. We compare kernel descent to existing methods and carry out extensive experiments to demonstrate its effectiveness. In particular, we showcase scenarios in which kernel descent outperforms gradient descent and quantum analytic descent. The algorithm follows the well-established scheme of iteratively computing classical local approximations to the objective function and subsequently executing several classical optimization steps with respect to the former. Kernel descent sets itself apart with its employment of reproducing kernel Hilbert space techniques in the construction of the local approximations -- which leads to the observed advantages.
{"title":"Kernel Descent -- a Novel Optimizer for Variational Quantum Algorithms","authors":"Lars Simon, Holger Eble, Manuel Radons","doi":"arxiv-2409.10257","DOIUrl":"https://doi.org/arxiv-2409.10257","url":null,"abstract":"In recent years, variational quantum algorithms have garnered significant\u0000attention as a candidate approach for near-term quantum advantage using noisy\u0000intermediate-scale quantum (NISQ) devices. In this article we introduce kernel\u0000descent, a novel algorithm for minimizing the functions underlying variational\u0000quantum algorithms. We compare kernel descent to existing methods and carry out\u0000extensive experiments to demonstrate its effectiveness. In particular, we\u0000showcase scenarios in which kernel descent outperforms gradient descent and\u0000quantum analytic descent. The algorithm follows the well-established scheme of\u0000iteratively computing classical local approximations to the objective function\u0000and subsequently executing several classical optimization steps with respect to\u0000the former. Kernel descent sets itself apart with its employment of reproducing\u0000kernel Hilbert space techniques in the construction of the local approximations\u0000-- which leads to the observed advantages.","PeriodicalId":501162,"journal":{"name":"arXiv - MATH - Numerical Analysis","volume":"40 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142259147","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper we study the dynamics of metamaterials composed of high-contrast subwavelength resonators and show the existence of localised modes in such a setting. A crucial assumption in this paper is time-modulated material parameters. We prove a so-called capacitance matrix approximation of the wave equation in the form of an ordinary differential equation. These formulas set the ground for the derivation of a first-principles characterisation of localised modes in terms of the generalised capacitance matrix. Furthermore, we provide numerical results supporting our analytical results showing for the first time the phenomenon of space-time localised waves in a perturbed time-modulated metamaterial. Such spatio-temporal localisation is only possible in the presence of subwavelength resonances in the unperturbed structure. We introduce the time-dependent degree of localisation to quantitatively determine the localised modes and provide a variety of numerical experiments to illustrate our formulations and results.
{"title":"Space-Time Wave Localisation in Systems of Subwavelength Resonators","authors":"Habib Ammari, Erik Orvehed Hiltunen, Liora Rueff","doi":"arxiv-2409.10100","DOIUrl":"https://doi.org/arxiv-2409.10100","url":null,"abstract":"In this paper we study the dynamics of metamaterials composed of\u0000high-contrast subwavelength resonators and show the existence of localised\u0000modes in such a setting. A crucial assumption in this paper is time-modulated\u0000material parameters. We prove a so-called capacitance matrix approximation of\u0000the wave equation in the form of an ordinary differential equation. These\u0000formulas set the ground for the derivation of a first-principles\u0000characterisation of localised modes in terms of the generalised capacitance\u0000matrix. Furthermore, we provide numerical results supporting our analytical\u0000results showing for the first time the phenomenon of space-time localised waves\u0000in a perturbed time-modulated metamaterial. Such spatio-temporal localisation\u0000is only possible in the presence of subwavelength resonances in the unperturbed\u0000structure. We introduce the time-dependent degree of localisation to\u0000quantitatively determine the localised modes and provide a variety of numerical\u0000experiments to illustrate our formulations and results.","PeriodicalId":501162,"journal":{"name":"arXiv - MATH - Numerical Analysis","volume":"17 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142259148","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
An adaptation of the arbitrary high order ADER-DG numerical method with local DG predictor for solving the IVP for a first-order non-linear ODE system is proposed. The proposed numerical method is a completely one-step ODE solver with uniform steps, and is simple in algorithmic and software implementations. It was shown that the proposed version of the ADER-DG numerical method is A-stable and L-stable. The ADER-DG numerical method demonstrates superconvergence with convergence order 2N+1 for the solution at grid nodes, while the local solution obtained using the local DG predictor has convergence order N+1. It was demonstrated that an important applied feature of this implementation of the numerical method is the possibility of using the local solution as a solution with a subgrid resolution, which makes it possible to obtain a detailed solution even on very coarse coordinate grids. The scale of the error of the local solution, when calculating using standard representations of single or double precision floating point numbers, using large values of the degree N, practically does not differ from the error of the solution at the grid nodes. The capabilities of the ADER-DG method for solving stiff ODE systems characterized by extreme stiffness are demonstrated. Estimates of the computational costs of the ADER-DG numerical method are obtained.
提出了一种带有局部DG预测器的任意高阶ADER-DG数值方法，用于求解一阶非线性ODE系统的IVP。结果表明，所提出的 ADER-DG 数值方法具有 A 稳定性和 L 稳定性。ADER-DG 数值方法对网格节点上的解具有超收敛性，收敛阶数为 2N+1，而使用局部 DG 预测器得到的局部解的收敛阶数为 N+1。结果表明，这种数值方法的一个重要应用特征是可以将局部解用作具有子网格分辨率的解，这使得即使在非常粗糙的坐标网格上也能获得详细的解。当使用单精度或双精度浮点数的标准表示，并使用较大的阶数 N 值进行计算时，局部解的误差范围实际上与网格节点上的解的误差并无差别。演示了 ADER-DG 方法求解以极端刚度为特征的刚性 ODE 系统的能力，并估算了 ADER-DG 数值方法的计算成本。
{"title":"Arbitrary high order ADER-DG method with local DG predictor for solutions of initial value problems for systems of first-order ordinary differential equations","authors":"I. S. Popov","doi":"arxiv-2409.09933","DOIUrl":"https://doi.org/arxiv-2409.09933","url":null,"abstract":"An adaptation of the arbitrary high order ADER-DG numerical method with local\u0000DG predictor for solving the IVP for a first-order non-linear ODE system is\u0000proposed. The proposed numerical method is a completely one-step ODE solver\u0000with uniform steps, and is simple in algorithmic and software implementations.\u0000It was shown that the proposed version of the ADER-DG numerical method is\u0000A-stable and L-stable. The ADER-DG numerical method demonstrates\u0000superconvergence with convergence order 2N+1 for the solution at grid nodes,\u0000while the local solution obtained using the local DG predictor has convergence\u0000order N+1. It was demonstrated that an important applied feature of this\u0000implementation of the numerical method is the possibility of using the local\u0000solution as a solution with a subgrid resolution, which makes it possible to\u0000obtain a detailed solution even on very coarse coordinate grids. The scale of\u0000the error of the local solution, when calculating using standard\u0000representations of single or double precision floating point numbers, using\u0000large values of the degree N, practically does not differ from the error of the\u0000solution at the grid nodes. The capabilities of the ADER-DG method for solving\u0000stiff ODE systems characterized by extreme stiffness are demonstrated.\u0000Estimates of the computational costs of the ADER-DG numerical method are\u0000obtained.","PeriodicalId":501162,"journal":{"name":"arXiv - MATH - Numerical Analysis","volume":"40 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142259100","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Mingwu Li, Thomas Thurnher, Zhenwei Xu, Shobhit Jain
The theory of spectral submanifolds (SSMs) has emerged as a powerful tool for constructing rigorous, low-dimensional reduced-order models (ROMs) of high-dimensional nonlinear mechanical systems. A direct computation of SSMs requires explicit knowledge of nonlinear coefficients in the equations of motion, which limits their applicability to generic finite-element (FE) solvers. Here, we propose a non-intrusive algorithm for the computation of the SSMs and the associated ROMs up to arbitrary polynomial orders. This non-intrusive algorithm only requires system nonlinearity as a black box and hence, enables SSM-based model reduction via generic finite-element software. Our expressions and algorithms are valid for systems with up to cubic-order nonlinearities, including velocity-dependent nonlinear terms, asymmetric damping, and stiffness matrices, and hence work for a large class of mechanics problems. We demonstrate the effectiveness of the proposed non-intrusive approach over a variety of FE examples of increasing complexity, including a micro-resonator FE model containing more than a million degrees of freedom.
谱子芒福德（SSM）理论已成为构建高维非线性机械系统的严格、低维降阶模型（ROM）的有力工具。直接计算 SSMs 需要明确了解运动方程中的非线性系数，这限制了其对通用有限元求解器的适用性。在这里，我们提出了一种非侵入式算法，用于计算运动方程中的非线性系数和相关的 ROM，最高可达任意多项式阶。我们的表达式和算法适用于具有高达三次阶非线性的系统，包括速度相关非线性项、非对称阻尼和刚度矩阵，因此适用于大量力学问题。我们通过各种复杂度不断增加的 FE 例子，包括包含超过一百万个自由度的微谐振器 FE 模型，证明了所提出的非侵入式方法的有效性。
{"title":"Data-free Non-intrusive Model Reduction for Nonlinear Finite Element Models via Spectral Submanifolds","authors":"Mingwu Li, Thomas Thurnher, Zhenwei Xu, Shobhit Jain","doi":"arxiv-2409.10126","DOIUrl":"https://doi.org/arxiv-2409.10126","url":null,"abstract":"The theory of spectral submanifolds (SSMs) has emerged as a powerful tool for\u0000constructing rigorous, low-dimensional reduced-order models (ROMs) of\u0000high-dimensional nonlinear mechanical systems. A direct computation of SSMs\u0000requires explicit knowledge of nonlinear coefficients in the equations of\u0000motion, which limits their applicability to generic finite-element (FE)\u0000solvers. Here, we propose a non-intrusive algorithm for the computation of the\u0000SSMs and the associated ROMs up to arbitrary polynomial orders. This\u0000non-intrusive algorithm only requires system nonlinearity as a black box and\u0000hence, enables SSM-based model reduction via generic finite-element software.\u0000Our expressions and algorithms are valid for systems with up to cubic-order\u0000nonlinearities, including velocity-dependent nonlinear terms, asymmetric\u0000damping, and stiffness matrices, and hence work for a large class of mechanics\u0000problems. We demonstrate the effectiveness of the proposed non-intrusive\u0000approach over a variety of FE examples of increasing complexity, including a\u0000micro-resonator FE model containing more than a million degrees of freedom.","PeriodicalId":501162,"journal":{"name":"arXiv - MATH - Numerical Analysis","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142259099","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Lorenzo Panzeri, Alessio Fumagalli, Laura Longoni, Monica Papini, Diego Arosio
Electrical resistivity tomography is a suitable technique for non-invasive monitoring of municipal solid waste landfills, but accurate sensitivity analysis is necessary to evaluate the effectiveness and reliability of geoelectrical investigations and to properly design data acquisition. Commonly, a thin high-resistivity membrane in placed underneath the waste to prevent leachate leakage. In the construction of a numerical framework for sensitivity computation, taking into account the actual dimensions of the electrodes and, in particular, of the membrane, can lead to extremely high computational costs. In this work, we present a novel approach for numerically computing sensitivity effectively by adopting a mixed-dimensional framework, where the membrane is approximated as a 2D object and the electrodes as 1D objects. The code is first validated against analytical expressions for simple 4-electrode arrays and a homogeneous medium. It is then tested in simplified landfill models, where a 2D box-shaped liner separates the landfill body from the surrounding media, and 48 electrodes are used. The results show that electrodes arranged linearly along both sides of the perimeter edges of the box-shaped liner are promising for detecting liner damage, with sensitivity increasing by 2-3 orders of magnitude, even for damage as small as one-sixth of the electrode spacing in diameter. Good results are also obtained when simulating an electrical connection between the landfill and the surrounding media that is not due to liner damage. The next steps involve evaluating the minimum number of configurations needed to achieve suitable sensitivity with a manageable field effort and validating the modeling results with downscaled laboratory tests.
{"title":"Sensitivity analysis with a 3D mixed-dimensional code for DC geoelectrical investigations of landfills: synthetic tests","authors":"Lorenzo Panzeri, Alessio Fumagalli, Laura Longoni, Monica Papini, Diego Arosio","doi":"arxiv-2409.10326","DOIUrl":"https://doi.org/arxiv-2409.10326","url":null,"abstract":"Electrical resistivity tomography is a suitable technique for non-invasive\u0000monitoring of municipal solid waste landfills, but accurate sensitivity\u0000analysis is necessary to evaluate the effectiveness and reliability of\u0000geoelectrical investigations and to properly design data acquisition. Commonly,\u0000a thin high-resistivity membrane in placed underneath the waste to prevent\u0000leachate leakage. In the construction of a numerical framework for sensitivity\u0000computation, taking into account the actual dimensions of the electrodes and,\u0000in particular, of the membrane, can lead to extremely high computational costs.\u0000In this work, we present a novel approach for numerically computing sensitivity\u0000effectively by adopting a mixed-dimensional framework, where the membrane is\u0000approximated as a 2D object and the electrodes as 1D objects. The code is first\u0000validated against analytical expressions for simple 4-electrode arrays and a\u0000homogeneous medium. It is then tested in simplified landfill models, where a 2D\u0000box-shaped liner separates the landfill body from the surrounding media, and 48\u0000electrodes are used. The results show that electrodes arranged linearly along\u0000both sides of the perimeter edges of the box-shaped liner are promising for\u0000detecting liner damage, with sensitivity increasing by 2-3 orders of magnitude,\u0000even for damage as small as one-sixth of the electrode spacing in diameter.\u0000Good results are also obtained when simulating an electrical connection between\u0000the landfill and the surrounding media that is not due to liner damage. The\u0000next steps involve evaluating the minimum number of configurations needed to\u0000achieve suitable sensitivity with a manageable field effort and validating the\u0000modeling results with downscaled laboratory tests.","PeriodicalId":501162,"journal":{"name":"arXiv - MATH - Numerical Analysis","volume":"2 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142259096","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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