Computer Methods in Applied Mechanics and Engineering最新文献

{"title":"Differentiability in unrolled training of neural physics simulators on transient dynamics","authors":"Bjoern List,&nbsp;Li-Wei Chen,&nbsp;Kartik Bali,&nbsp;Nils Thuerey","doi":"10.1016/j.cma.2024.117441","DOIUrl":"10.1016/j.cma.2024.117441","url":null,"abstract":"<div><div>Unrolling training trajectories over time strongly influences the inference accuracy of neural network-augmented physics simulators. We analyze these effects by studying three variants of training neural networks on discrete ground truth trajectories. In addition to commonly used one-step setups and fully differentiable unrolling, we include a third, less widely used variant: unrolling without temporal gradients. Comparing networks trained with these three modalities makes it possible to disentangle the two dominant effects of unrolling, training distribution shift and long-term gradients. We present a detailed study across physical systems, network sizes, network architectures, training setups, and test scenarios. It also encompasses two modes of computing the simulation trajectories. In <em>prediction</em> setups, we rely solely on neural networks to compute a trajectory. In contrast, <em>correction</em> setups include a numerical solver that is supported by a neural network. Spanning all these variations, our study provides the empirical basis for our main findings: A non-differentiable but unrolled training setup supported by a numerical solver in a correction setup can yield substantial improvements over a fully differentiable prediction setup not utilizing this solver. We also quantify a difference in the accuracy of models trained in a fully differentiable setup compared to their non-differentiable counterparts. Differentiable setups perform best in a direct comparison of correction networks, and the same is observed when comparing prediction setups among each other. In both cases, the accuracy of unrolling without temporal gradients comes relatively close. Furthermore, we empirically show that these behaviors are invariant to changes in the underlying physical system, the network architecture and size, and the numerical scheme. These results motivate integrating non-differentiable numerical simulators into training setups even if full differentiability is unavailable. We also observe that the convergence rate of common neural architectures is low compared to numerical algorithms. This encourages the use of <em>correction</em> approaches combining neural and numerical algorithms to utilize the benefits of both.</div></div>","PeriodicalId":55222,"journal":{"name":"Computer Methods in Applied Mechanics and Engineering","volume":"433 ","pages":"Article 117441"},"PeriodicalIF":6.9,"publicationDate":"2024-10-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142529267","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A quasi-meshfree method for nonlinear solid mechanics: Separating domain discretization from solution discretization","authors":"Joseph Bishop ,&nbsp;Mike Tupek ,&nbsp;Jacob Koester","doi":"10.1016/j.cma.2024.117459","DOIUrl":"10.1016/j.cma.2024.117459","url":null,"abstract":"<div><div>In many applications, domains of interest are geometrically complex containing numerous small features. These features are typically removed in a manual process to facilitate a conventional element-based meshing process. This manual defeaturing process is dependent upon the goals of the simulation and typically involves subjective heuristics. To provide a flexible and easily adaptable discretization process of the governing equations that is independent of the domain discretization, an element-free Galerkin method is proposed in which a fine-scale triangulation is used to first discretize the fully featured domain, but then a coarse-scale element-free discretization is used to approximate the solution of the governing equations. The fine-scale triangulation can be of poor quality and extremely refined since it is not used directly to approximate the solution of the governing equations. The coarse-scale element-free basis has local support and can be adapted through refinement or coarsening without the need to alter the fine-scale triangulation or other geometric considerations. The element-free basis functions are constructed using a conventional moving-least-squares procedure, but the initial weight functions are constructed using manifold geodesics for general applicability to non-convex domains. The weak form of the governing equations is integrated using a secondary coarse-scale element-free basis and a gradient projection technique. The projected-gradient methodology ensures the necessary consistency properties to pass the patch test and obtain optimal rates of convergence. The overall method is termed quasi-meshfree since both meshfree and mesh-based concepts are used. Several verification problems and nonlinear application examples are presented to demonstrate the overall method.</div></div>","PeriodicalId":55222,"journal":{"name":"Computer Methods in Applied Mechanics and Engineering","volume":"432 ","pages":"Article 117459"},"PeriodicalIF":6.9,"publicationDate":"2024-10-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142530367","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Fading regularization method for the stationary Stokes data assimilation problem","authors":"Hatem Zayeni ,&nbsp;Amel Ben Abda ,&nbsp;Franck Delvare","doi":"10.1016/j.cma.2024.117450","DOIUrl":"10.1016/j.cma.2024.117450","url":null,"abstract":"<div><div>In this study, we address the ill-posed stationary Stokes data assimilation (DA) problem using the fading regularization method (FRM). It involves reconstructing the fluid velocity field throughout the study domain <span><math><mi>Ω</mi></math></span> in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span>, where <span><math><mi>n</mi></math></span> is the dimension of the space, as well as the boundary conditions, using knowledge of some observations of the fluid velocity field <span><math><msup><mrow><mi>u</mi></mrow><mrow><mi>o</mi><mi>b</mi><mi>s</mi></mrow></msup></math></span> measured within a limited domain <span><math><mi>ω</mi></math></span> included in <span><math><mi>Ω</mi></math></span>. Using the FRM, the main ill-posed problem is transformed into a sequence of well-posed constraint optimization problems and simplifies the resolution of DA problem. Additionally, we prove the convergence of both the continuous and the discrete formulations. This method is implemented numerically using the method of fundamental solutions (MFS) and several numerical simulations are shown to illustrate the performance of the algorithm in terms of efficiency, accuracy, convergence, stability, and robustness to noisy data, as well as its ability to deblur the data in <span><math><mi>ω</mi></math></span>.</div></div>","PeriodicalId":55222,"journal":{"name":"Computer Methods in Applied Mechanics and Engineering","volume":"433 ","pages":"Article 117450"},"PeriodicalIF":6.9,"publicationDate":"2024-10-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142529320","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Reliability-based composite pressure vessel design optimization with cure-induced stresses and spatial material variability","authors":"B. Van Bavel ,&nbsp;O. Shishkina ,&nbsp;D. Vandepitte ,&nbsp;D. Moens","doi":"10.1016/j.cma.2024.117463","DOIUrl":"10.1016/j.cma.2024.117463","url":null,"abstract":"<div><div>The future green hydrogen economy requires reliable and affordable composite pressure vessels. These vessels are expensive to manufacture, for a large part due to the high cost of carbon fibers in the composite layup. This study minimizes the thickness (and cost) of a composite pressure vessel layup without a reduction of its reliability. The study applies a multiphysics and multiscale uncertainty quantification framework that predicts the nondeterministic vessel burst pressure. A thermomechanical curing simulation accounts for cure-induced stress. It shows a good qualitative agreement with experimental measurements. A previously validated nondeterministic mechanical burst simulation accounts for experimentally measured spatial material variability of fiber misalignment, fiber volume fraction, and fiber strength. The workflow is coupled with a global optimization strategy that minimizes the layup thickness and retains the same 1% probability of failure pressure as a baseline pressure vessel. The optimization varies the number of layers in the layup, and their winding angle. A 27.3% thickness reduction is achieved.</div></div>","PeriodicalId":55222,"journal":{"name":"Computer Methods in Applied Mechanics and Engineering","volume":"432 ","pages":"Article 117463"},"PeriodicalIF":6.9,"publicationDate":"2024-10-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142530365","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"GFN: A graph feedforward network for resolution-invariant reduced operator learning in multifidelity applications","authors":"Oisín M. Morrison ,&nbsp;Federico Pichi ,&nbsp;Jan S. Hesthaven","doi":"10.1016/j.cma.2024.117458","DOIUrl":"10.1016/j.cma.2024.117458","url":null,"abstract":"<div><div>This work presents a novel resolution-invariant model order reduction strategy for multifidelity applications. We base our architecture on a novel neural network layer developed in this work, the graph feedforward network, which extends the concept of feedforward networks to graph-structured data by creating a direct link between the weights of a neural network and the nodes of a mesh, enhancing the interpretability of the network. We exploit the method’s capability of training and testing on different mesh sizes in an autoencoder-based reduction strategy for parameterised partial differential equations. We show that this extension comes with provable guarantees on the performance via error bounds. The capabilities of the proposed methodology are tested on three challenging benchmarks, including advection-dominated phenomena and problems with a high-dimensional parameter space. The method results in a more lightweight and highly flexible strategy when compared to state-of-the-art models, while showing excellent generalisation performance in both single fidelity and multifidelity scenarios.</div></div>","PeriodicalId":55222,"journal":{"name":"Computer Methods in Applied Mechanics and Engineering","volume":"432 ","pages":"Article 117458"},"PeriodicalIF":6.9,"publicationDate":"2024-10-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142530366","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"An ensemble score filter for tracking high-dimensional nonlinear dynamical systems","authors":"Feng Bao ,&nbsp;Zezhong Zhang ,&nbsp;Guannan Zhang","doi":"10.1016/j.cma.2024.117447","DOIUrl":"10.1016/j.cma.2024.117447","url":null,"abstract":"<div><div>We propose an ensemble score filter (EnSF) for solving high-dimensional nonlinear filtering problems with superior accuracy. A major drawback of existing filtering methods, e.g., particle filters or ensemble Kalman filters, is the low accuracy in handling high-dimensional and highly nonlinear problems. EnSF addresses this challenge by exploiting the score-based diffusion model, defined in a pseudo-temporal domain, to characterize the evolution of the filtering density. EnSF stores the information of the recursively updated filtering density function in the score function, instead of storing the information in a set of finite Monte Carlo samples (used in particle filters and ensemble Kalman filters). Unlike existing diffusion models that train neural networks to approximate the score function, we develop a training-free score estimation method that uses a mini-batch-based Monte Carlo estimator to directly approximate the score function at any pseudo-spatial–temporal location, which provides sufficient accuracy in solving high-dimensional nonlinear problems while also saving a tremendous amount of time spent on training neural networks. High-dimensional Lorenz-96 systems are used to demonstrate the performance of our method. EnSF provides superior performance, compared with the state-of-the-art Local Ensemble Transform Kalman Filter, in reliably and efficiently tracking extremely high-dimensional Lorenz systems (up to 1,000,000 dimensions) with highly nonlinear observation processes.</div></div>","PeriodicalId":55222,"journal":{"name":"Computer Methods in Applied Mechanics and Engineering","volume":"432 ","pages":"Article 117447"},"PeriodicalIF":6.9,"publicationDate":"2024-10-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142530363","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Mesh-driven resampling and regularization for robust point cloud-based flow analysis directly on scanned objects","authors":"Monu Jaiswal,&nbsp;Ashton M. Corpuz,&nbsp;Ming-Chen Hsu","doi":"10.1016/j.cma.2024.117426","DOIUrl":"10.1016/j.cma.2024.117426","url":null,"abstract":"<div><div>Point cloud representations of three-dimensional objects have remained indispensable across a diverse array of applications, given their ability to represent complex real-world geometry with just a set of points. The high fidelity and versatility of point clouds have been utilized in directly performing numerical analysis for engineering applications, bypassing the labor-intensive and time-consuming tasks of creating analysis-suitable CAD models. However, point clouds exhibit various levels of quality, often containing defects such as holes, noise, and sparse regions, leading to sub-optimal geometry representation that can impact the stability and accuracy of any analysis study. This paper aims to overcome such challenges by proposing a novel method that expands upon our recently developed direct point cloud-to-CFD approach based on immersogeometric analysis. The proposed method features a mesh-driven resampling technique to fill any unintended gaps and regularize the point cloud, making it suitable for CFD analysis. Additionally, ghost penalty stabilization is employed for incompressible flow to improve the conditioning and stability compromised by the small cut elements in immersed methods. The developed method is validated against standard benchmark geometries and real-world point clouds obtained in-house with photogrammetry. Results demonstrate the proposed framework’s robustness in facilitating CFD simulations directly on point clouds of varying quality, underscoring its potential for practical applications in analyzing real-world structures.</div></div>","PeriodicalId":55222,"journal":{"name":"Computer Methods in Applied Mechanics and Engineering","volume":"432 ","pages":"Article 117426"},"PeriodicalIF":6.9,"publicationDate":"2024-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142530364","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Transient anisotropic kernel for probabilistic learning on manifolds","authors":"Christian Soize ,&nbsp;Roger Ghanem","doi":"10.1016/j.cma.2024.117453","DOIUrl":"10.1016/j.cma.2024.117453","url":null,"abstract":"<div><div>PLoM (Probabilistic Learning on Manifolds) is a method introduced in 2016 for handling small training datasets by projecting an Itô equation from a stochastic dissipative Hamiltonian dynamical system, acting as the MCMC generator, for which the KDE-estimated probability measure with the training dataset is the invariant measure. PLoM performs a projection on a reduced-order vector basis related to the training dataset, using the diffusion maps (DMAPS) basis constructed with a time-independent isotropic kernel. In this paper, we propose a new ISDE projection vector basis built from a transient anisotropic kernel, providing an alternative to the DMAPS basis to improve statistical surrogates for stochastic manifolds with heterogeneous data. The construction ensures that for times near the initial time, the DMAPS basis coincides with the transient basis. For larger times, the differences between the two bases are characterized by the angle of their spanned vector subspaces. The optimal instant yielding the optimal transient basis is determined using an estimation of mutual information from Information Theory, which is normalized by the entropy estimation to account for the effects of the number of realizations used in the estimations. Consequently, this new vector basis better represents statistical dependencies in the learned probability measure for any dimension. Three applications with varying levels of statistical complexity and data heterogeneity validate the proposed theory, showing that the transient anisotropic kernel improves the learned probability measure.</div></div>","PeriodicalId":55222,"journal":{"name":"Computer Methods in Applied Mechanics and Engineering","volume":"432 ","pages":"Article 117453"},"PeriodicalIF":6.9,"publicationDate":"2024-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142530369","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"High-efficient sample point transform algorithm for large-scale complex optimization","authors":"Caihua Zhou ,&nbsp;Haixin Zhao ,&nbsp;Shengli Xu","doi":"10.1016/j.cma.2024.117451","DOIUrl":"10.1016/j.cma.2024.117451","url":null,"abstract":"<div><div>Decomposition algorithms and surrogate model methods are frequently employed to address large-scale, intricate optimization challenges. However, the iterative resolution phase inherent to decomposition algorithms can potentially alter the background vector, leading to the repetitive evaluation of samples across disparate iteration cycles. This phenomenon significantly diminishes the computational efficiency of optimization. Accordingly, a novel approach, designated the Sample Point Transformation Algorithm (SPTA), is put forth in this paper as a means of enhancing efficiency through a process of mathematical deduction. The mathematical deduction reveals that the difference between sample points in each iteration loop is a simple function related to the inter-group dependent variables. Consequently, the SPTA method achieves the comprehensive transformation of the sample set by establishing a surrogate model of the difference between the sample sets of two cycles with a limited number of sample points, as opposed to conducting a substantial number of repeated samplings. This SPTA is employed to substitute the most time-consuming step of direct calculation in the classical optimization process. To validate the calculation efficiency, a series of numerical examples were conducted, demonstrating an improvement of approximately 75 % while maintaining optimal accuracy. This illustrates the advantage of the SPTA in addressing large-scale and complex optimization problems.</div></div>","PeriodicalId":55222,"journal":{"name":"Computer Methods in Applied Mechanics and Engineering","volume":"432 ","pages":"Article 117451"},"PeriodicalIF":6.9,"publicationDate":"2024-10-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142530368","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Tackling the curse of dimensionality in fractional and tempered fractional PDEs with physics-informed neural networks","authors":"Zheyuan Hu ,&nbsp;Kenji Kawaguchi ,&nbsp;Zhongqiang Zhang ,&nbsp;George Em Karniadakis","doi":"10.1016/j.cma.2024.117448","DOIUrl":"10.1016/j.cma.2024.117448","url":null,"abstract":"<div><div>Fractional and tempered fractional partial differential equations (PDEs) are effective models of long-range interactions, anomalous diffusion, and non-local effects. Traditional numerical methods for these problems are mesh-based, thus struggling with the curse of dimensionality (CoD). Physics-informed neural networks (PINNs) offer a promising solution due to their universal approximation, generalization ability, and mesh-free training. In principle, Monte Carlo fractional PINN (MC-fPINN) estimates fractional derivatives using Monte Carlo methods and thus could lift CoD. However, this may cause significant variance and errors, hence affecting convergence; in addition, MC-fPINN is sensitive to hyperparameters. In general, numerical methods and specifically PINNs for tempered fractional PDEs are under-developed. Herein, we extend MC-fPINN to tempered fractional PDEs to address these issues, resulting in the Monte Carlo tempered fractional PINN (MC-tfPINN). To reduce possible high variance and errors from Monte Carlo sampling, we replace the one-dimensional (1D) Monte Carlo with 1D Gaussian quadrature, applicable to both MC-fPINN and MC-tfPINN. We validate our methods on various forward and inverse problems of fractional and tempered fractional PDEs, scaling up to 100,000 dimensions. Our improved MC-fPINN/MC-tfPINN using quadrature consistently outperforms the original versions in accuracy and convergence speed in very high dimensions. Code is available at <span><span>https://github.com/zheyuanhu01/Tempered_Fractional_PINN</span><svg><path></path></svg></span>.</div></div>","PeriodicalId":55222,"journal":{"name":"Computer Methods in Applied Mechanics and Engineering","volume":"432 ","pages":"Article 117448"},"PeriodicalIF":6.9,"publicationDate":"2024-10-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142446240","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}