Computer Methods in Applied Mechanics and Engineering最新文献

{"title":"Towards quantum accelerated large-scale topology optimization","authors":"Zisheng Ye,&nbsp;Wenxiao Pan","doi":"10.1016/j.cma.2026.118819","DOIUrl":"10.1016/j.cma.2026.118819","url":null,"abstract":"<div><div>We present an efficient topology optimization (TO) method that not only enhances computational efficiency on classical computing but also provides a practical pathway for leveraging quantum computing to achieve further acceleration. The method targets large-scale, multi-material TO of three-dimensional (3D) continuum structures, beyond prior quantum TO studies limited to small-scale and single-material problems. Building on our discrete-variable TO framework (DVTO-MT), which employs multi-cut optimization and trust regions to reduce iteration counts and thereby PDE solver calls, the proposed method introduces a modified Dantzig-Wolfe (MDW) decomposition to further reduce per-iteration optimization time. The MDW method exploits the block-angular structure of the problem to decompose the mixed-integer linear program (MILP) into reduced-size global and local sub-problems. Evaluations on large-scale 3D bridge design problems demonstrate orders-of-magnitude reductions in computational time, with robust performance even for designs exceeding 50 million variables where classical MILP solvers fail to converge. Furthermore, the computationally intensive local sub-problems are transformed into equivalent quadratic unconstrained binary optimization (QUBO) formulations for quantum acceleration. The resulting QUBOs require only sparse qubit connectivity, a crucial consideration for near-term quantum hardware, and linear construction cost, offering the potential for an additional order-of-magnitude speedup. All observed and estimated speedups become more significant with increasing problem size and when extending from single-material to multi-material designs, highlighting the potential of the proposed method, coupled with quantum computing, to address the scale and complexity of real-world TO challenges.</div></div>","PeriodicalId":55222,"journal":{"name":"Computer Methods in Applied Mechanics and Engineering","volume":"453 ","pages":"Article 118819"},"PeriodicalIF":7.3,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146191973","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":"A learning-based domain decomposition method","authors":"Rui Wu ,&nbsp;Nikola Kovachki ,&nbsp;Burigede Liu","doi":"10.1016/j.cma.2026.118799","DOIUrl":"10.1016/j.cma.2026.118799","url":null,"abstract":"<div><div>Recent developments in mechanical, aerospace, and structural engineering have driven a growing need for efficient ways to model and analyze structures at much larger and more complex scales than before. While established numerical methods like the Finite Element Method remain reliable, they often struggle with computational cost and scalability when dealing with large and geometrically intricate problems. In recent years, neural network-based methods have shown promise because of their ability to efficiently approximate nonlinear mappings. However, most existing neural approaches are still largely limited to simple domains, which makes it difficult to apply to real-world partial differential equations (PDEs) involving complex geometries. In this paper, we propose a learning-based domain decomposition method (L-DDM) that addresses this gap. Our approach uses a single, pre-trained neural operator-originally trained on simple domains-as a surrogate model within a domain decomposition scheme, allowing us to tackle large and complicated domains efficiently. We provide a general theoretical result on the existence of neural operator approximations in the context of domain decomposition solution of abstract PDEs. We then demonstrate our method by accurately approximating solutions to elliptic PDEs with discontinuous microstructures in complex geometries, using a physics-pretrained neural operator (PPNO). Our results show that this approach not only outperforms current state-of-the-art methods on these challenging problems, but also offers resolution-invariance and strong generalization to microstructural patterns unseen during training.</div></div>","PeriodicalId":55222,"journal":{"name":"Computer Methods in Applied Mechanics and Engineering","volume":"453 ","pages":"Article 118799"},"PeriodicalIF":7.3,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146161065","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":"Polynomial chaos expansion for operator learning","authors":"Himanshu Sharma ,&nbsp;Lukáš Novák ,&nbsp;Michael Shields","doi":"10.1016/j.cma.2026.118796","DOIUrl":"10.1016/j.cma.2026.118796","url":null,"abstract":"<div><div>Operator learning (OL) has emerged as a powerful tool in scientific machine learning (SciML) for approximating mappings between infinite-dimensional functional spaces. One of its main applications is learning the solution operator of partial differential equations (PDEs). While much of the progress in this area has been driven by deep neural network-based approaches such as Deep Operator Networks (DeepONet) and Fourier Neural Operator (FNO), recent work has begun to explore traditional machine learning methods for OL. In this work, we introduce polynomial chaos expansion (PCE) as an OL method. PCE has been widely used for uncertainty quantification (UQ) and has recently gained attention in the context of SciML. For OL, we establish a mathematical framework that enables PCE to approximate operators in both purely data-driven and physics-informed settings. The proposed framework reduces the task of learning the operator to solving a system of equations for the PCE coefficients. Moreover, the framework provides UQ by simply post-processing the PCE coefficients, without any additional computational cost. We apply the proposed method to a diverse set of PDE problems to demonstrate its capabilities. Numerical results demonstrate the strong performance of the proposed method in both OL and UQ tasks, achieving excellent numerical accuracy and computational efficiency.</div></div>","PeriodicalId":55222,"journal":{"name":"Computer Methods in Applied Mechanics and Engineering","volume":"453 ","pages":"Article 118796"},"PeriodicalIF":7.3,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146161061","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":"A sparse basis for equilibrium stress fields with application for direct data-driven mechanics","authors":"Erik Prume ,&nbsp;Chenyi Ji ,&nbsp;Stefanie Reese ,&nbsp;Michael Ortiz","doi":"10.1016/j.cma.2026.118803","DOIUrl":"10.1016/j.cma.2026.118803","url":null,"abstract":"<div><div>We present a new class of solvers for direct data-driven mechanical problems based on a sparse basis representation of equilibrium stress fields. Our first contribution is an efficient algorithm for computing the required sparse null-space basis on tetrahedral meshes.</div><div>Only a single QR decomposition is needed to compute a small remaining set of dense basis vectors associated with boundary conditions and topological holes which can be handled efficiently via a partitioned Cholesky factorization. Building on this, we demonstrate how standard iterative solvers-such as the Newton-Raphson method-can be applied to direct data-driven formulations.</div><div>The proposed approach is particularly valuable for challenging problems with complex data distributions requiring systematic exploration of the space of equilibrium stress fields. To this end, we introduce an algorithm that constructs a hierarchical solution set through an eigenvalue decomposition in the joint space of equilibrium stress and compatible strain fields. We demonstrate the proposed methodology with a numerical example involving brittle fracture with probabilistic tensile strength. The resulting family of failure patterns offers valuable insights for uncertainty quantification and design decision-making.</div></div>","PeriodicalId":55222,"journal":{"name":"Computer Methods in Applied Mechanics and Engineering","volume":"453 ","pages":"Article 118803"},"PeriodicalIF":7.3,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146152936","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":"Spectral CT reconstruction based on particular and homogeneous solutions correction with feasible-domain regularization (PAHO-FEDO)","authors":"Yue Yang ,&nbsp;Yongxu Liu ,&nbsp;Ping Yang ,&nbsp;Xu Jiang ,&nbsp;Zheng Sun ,&nbsp;Xing Zhao","doi":"10.1016/j.cma.2026.118776","DOIUrl":"10.1016/j.cma.2026.118776","url":null,"abstract":"<div><div>Spectral computed tomography (SCT) employs multi-energy X-rays to scan the object and obtain polychromatic projection data, which contains rich information and thereby enables basis material decomposition. However, existing decomposition methods face two significant challenges: First, basis material decomposition is mathematically modeled as solving a highly ill-posed nonlinear system of equations, whose solution typically requires numerous inner and outer loop iterations, leading to a lengthy computation time. Moreover, due to the significant differences in the attenuation characteristics of different materials, the coefficients of the equations are imbalanced, causing some basis material images to converge slowly and thereby affecting the overall decomposition efficiency. Second, due to the limited sampling rate, both projection data and image data require interpolation, resulting in a “slope effect” in the image edge regions, which leads to erroneous decomposition of the basis materials at the edges. For the imbalanced nonlinear system of equations, a method is proposed to dynamically adjust the allocation ratio of the polychromatic projection residuals using the particular solution and homogeneous solution, which requires only one inner loop to obtain the optimal solution. We observe that the system’s weak nonlinear characteristics mean that most of the computation is concentrated in the inner loop. Therefore, our method significantly accelerates convergence. For the “slope effect”, we propose a feasible-domain regularization method based on the dilation operator to constrain the reconstructed material densities within plausible ranges and compensate for edge information, thereby reducing decomposition errors. The above method, based on particular and homogeneous solutions correction with feasible-domain regularization, is referred to as PAHO-FEDO, which perfectly solves the basis material decomposition problem. Numerical simulations and real data experiments demonstrate that the proposed method outperforms existing model-based state-of-the-art methods in terms of decomposition accuracy, edge preservation, and convergence speed, with better RMSE/PSNR/SSIM values, minor mean Euclidean edge errors, and fewer iterations to reach convergence, thereby providing an efficient solution to the challenges encountered in current SCT material decomposition.</div></div>","PeriodicalId":55222,"journal":{"name":"Computer Methods in Applied Mechanics and Engineering","volume":"453 ","pages":"Article 118776"},"PeriodicalIF":7.3,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146191980","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":"Concurrent topology optimization of two-scale structures considering high-cycle fatigue damage","authors":"Xiaopeng Zhang ,&nbsp;Zheng Ni ,&nbsp;Yaguang Wang ,&nbsp;Junling Fan","doi":"10.1016/j.cma.2026.118820","DOIUrl":"10.1016/j.cma.2026.118820","url":null,"abstract":"<div><div>Two-scale structures demonstrate great potential in engineering due to their superior mechanical performance. However, under variable-amplitude loading, the analysis of structural fatigue response is complex, which makes the fatigue design of two-scale structures a challenge. In this study, we propose a concurrent topology optimization method considering high-cycle fatigue damage under variable-amplitude loading, which controls the maximum fatigue damage by designing the microstructure and its distribution at the macro scale under given volume constraints at both scales.​ To facilitate fatigue analysis under complex loading conditions, the rainflow counting method is employed to convert load history into analyzable cyclic loads. By incorporating the Palmgren-Miner linear cumulative damage rule into the microscale homogenization method, the fatigue damage at the microscale can be effectively analyzed. In fatigue damage analysis, three damage models signed von Mises, Brown-Miller, and Dang Van are considered. To address the challenge of microscale fatigue localization caused by highly nonlinear damage distribution, penalized fatigue damage constraints are defined by scaling the fatigue damage values. Based on the adjoint variable method, sensitivity analysis for the fatigue damage constraints is performed to update the design variables through the Method of Moving Asymptotes (MMA). Numerical examples demonstrate that the optimized design can effectively control fatigue damage. The results confirm that fatigue damage is more severe under tensile than compressive loading, a fact that directly leads to differing optimal designs.</div></div>","PeriodicalId":55222,"journal":{"name":"Computer Methods in Applied Mechanics and Engineering","volume":"453 ","pages":"Article 118820"},"PeriodicalIF":7.3,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146191979","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":"Temporal convergence analysis of the generalized finite element method for multi-scale heat transfer","authors":"T.J. Miller ,&nbsp;Patrick J. O’Hara ,&nbsp;Jack J. McNamara","doi":"10.1016/j.cma.2026.118808","DOIUrl":"10.1016/j.cma.2026.118808","url":null,"abstract":"<div><div>The temporal accuracy, convergence, and efficiency of the generalized finite element method (GFEM) with time-dependent enrichment functions are investigated for applications to multi-scale, transient heat transfer problems involving localized, non-stationary thermal loads. The GFEM enables the construction of solution-tailored enrichments from either analytical or numerical considerations, enabling the simultaneous resolution of spatial and temporal scales on coarse, fixed FEM meshes. The global-local GFEM (GFEM^<em>gl</em>) is used to build numerical enrichments. The temporal behavior of the GFEM is investigated using the generalized trapezoidal method as the time integrator to assess the viability of time-stepping within this framework. Numerical experiments performed on problems exhibiting sharp temporal and highly localized spatial gradients show that the GFEM^<em>gl</em> only obtains robust convergence when employing transient local problems, representing an advancement over previous studies. Performance analyses demonstrate that the GFEM with time-dependent shape functions efficiently delivers high-fidelity solutions compared to standard FEM formulations. The results demonstrate strong potential of GFEM for significant reductions in computational costs.</div></div>","PeriodicalId":55222,"journal":{"name":"Computer Methods in Applied Mechanics and Engineering","volume":"453 ","pages":"Article 118808"},"PeriodicalIF":7.3,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146191978","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":"A rotation-based approach to third medium contact regularization","authors":"Vilmer Dahlberg,&nbsp;Filip Sjövall,&nbsp;Anna Dalklint,&nbsp;Mathias Wallin","doi":"10.1016/j.cma.2026.118801","DOIUrl":"10.1016/j.cma.2026.118801","url":null,"abstract":"<div><div>The third medium contact method utilizes a fictitious “third” medium to implicitly model contact interactions. When contact occurs, the fictitious medium is severely deformed which necessitates numerical regularization to ensure numerical stability. Ultimately, this regularization should promote good element quality but otherwise not interfere with the modeling of the contact mechanics. One approach penalizes both stretch and rotational deformation modes using the displacement Hessian which requires higher order elements. Another approach introduces additional degrees of freedom to penalize an approximation of the rotation gradient which drastically increases the system size. We propose a new regularization based on an approximation of the rotation gradient in the fictitious medium, which does not penalize stretch deformation modes and can be used with first-order elements. The efficacy of our method is exemplified using several numerical examples including benchmark tests, an investigation of parasitic forces in the third medium and a novel application to general loading condition.</div></div>","PeriodicalId":55222,"journal":{"name":"Computer Methods in Applied Mechanics and Engineering","volume":"453 ","pages":"Article 118801"},"PeriodicalIF":7.3,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146152926","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":"Gap-SBM: A new conceptualization of the shifted boundary method with optimal convergence for the Neumann and Dirichlet problems","authors":"J. Haydel Collins ,&nbsp;Kangan Li ,&nbsp;Alexei Lozinski ,&nbsp;Guglielmo Scovazzi","doi":"10.1016/j.cma.2026.118793","DOIUrl":"10.1016/j.cma.2026.118793","url":null,"abstract":"<div><div>We propose and mathematically analyze a new Shifted Boundary Method for the treatment of Dirichlet and Neumann boundary conditions, with provable optimal accuracy in the <em>L</em>²- and <em>H</em>¹-norms of the error. The proposed method is built on three stages. First, the distance map between the SBM surrogate boundary and the true boundary is used to construct an approximation to the geometry of the gap between the two. Then, the representations of the numerical solution and test functions are extended from the surrogate domain to the such gap. Finally, approximate quadrature formulas and specific shift operators are applied to integrate a variational formulation that also involves the fields extended in the gap. An extensive set of two- and three-dimensional tests demonstrates the theoretical findings and the overall optimal performance of the proposed method.</div></div>","PeriodicalId":55222,"journal":{"name":"Computer Methods in Applied Mechanics and Engineering","volume":"453 ","pages":"Article 118793"},"PeriodicalIF":7.3,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146152929","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":"Multi-Level Monte Carlo training of neural operators","authors":"James Rowbottom ,&nbsp;Stefania Fresca ,&nbsp;Pietro Lio ,&nbsp;Carola-Bibiane Schönlieb ,&nbsp;Nicolas Boullé","doi":"10.1016/j.cma.2026.118800","DOIUrl":"10.1016/j.cma.2026.118800","url":null,"abstract":"<div><div>Operator learning is a rapidly growing field that aims to approximate nonlinear operators related to partial differential equations (PDEs) using neural operators. These rely on discretization of input and output functions and are, usually, expensive to train for large-scale problems at high-resolution. Motivated by this, we present a Multi-Level Monte Carlo (MLMC) approach to train neural operators by leveraging a hierarchy of resolutions of function discretization. Our framework relies on using gradient corrections from fewer samples of fine-resolution data to decrease the computational cost of training while maintaining a high level accuracy. The proposed MLMC training procedure can be applied to any architecture accepting multi-resolution data. Our numerical experiments on a range of state-of-the-art models and test-cases demonstrate improved computational efficiency compared to traditional single-resolution training approaches, and highlight the existence of a Pareto curve between accuracy and computational time, related to the number of samples per resolution.</div></div>","PeriodicalId":55222,"journal":{"name":"Computer Methods in Applied Mechanics and Engineering","volume":"453 ","pages":"Article 118800"},"PeriodicalIF":7.3,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146152931","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}