The Schwarz alternating method in solid mechanics

Mota, Alejandro; Tezaur, Irina Kalashnikova; Alleman, Coleman

doi:10.1016/j.cma.2017.02.006

Cited by 27 publications

(43 citation statements)

References 54 publications

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“…This procedure is repeated until the stopping criterion is satisfied. These coupling algorithms have their origin to the classical Schwarz coupling methods [25,48,57] and an iterative patching algorithms [25].…”

Section: Coupling Methodsmentioning

confidence: 99%

Interfacing Finite Elements with Deep Neural Operators for Fast Multiscale Modeling of Mechanics Problems

Yin¹,

Zhang²,

Yu³

et al. 2022

Preprint

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Multiscale modeling is an effective approach for investigating multiphysics systems with largely disparate size features, where models with different resolutions or heterogeneous descriptions are coupled together for predicting the system's response. The solver with lower fidelity (coarse) is responsible for simulating domains with homogeneous features, whereas the expensive high-fidelity (fine) model describes microscopic features with refined discretization, often making the overall cost prohibitively high, especially for timedependent problems. In this work, we explore the idea of multiscale modeling with machine learning and employ DeepONet, a neural operator, as an efficient surrogate of the expensive solver. DeepONet is trained offline using data acquired from the fine solver for learning the underlying and possibly unknown finescale dynamics. It is then coupled with standard PDE solvers for predicting the multiscale systems with new boundary/initial conditions in the coupling stage. The proposed framework significantly reduces the computational cost of multiscale simulations since the DeepONet inference cost is negligible, facilitating readily the incorporation of a plurality of interface conditions and coupling schemes. We present various benchmarks to assess accuracy and speedup, and in particular we develop a coupling algorithm for a timedependent problem, and we also demonstrate coupling of a continuum model (finite element methods, FEM) with a neural operator representation of a particle system (Smoothed Particle Hydrodynamics, SPH) for a uniaxial tension problem with hyperelastic material. What makes this approach unique is that a well-trained over-parametrized DeepONet can generalize well and make predictions at a negligible cost.

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Section: Coupling Methodsmentioning

confidence: 99%

Interfacing Finite Elements with Deep Neural Operators for Fast Multiscale Modeling of Mechanics Problems

Yin¹,

Zhang²,

Yu³

et al. 2022

Preprint

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“…Albany is an open-source 4 C++ object-oriented, parallel, unstructured-grid, implicit finite element code for solving general PDEs, developed using the "Agile Components" code development strategy [66] with mature modular libraries from the Trilinos [39] project 5 . Over the years, Albany has hosted a number of science and engineering applications, including the Aeras global atmosphere code [73], the Albany Land-Ice (ALI) [78] ice sheet model solver, the Quantum Computer Aided Design (QCAD) [36] simulator, the ACE thermo-mechanical terrestrial model of Arctic coastal erosion [34], and the Laboratory for Computational Mechanics (LCM) [74,52,51] research code. This last project comprises Albany LCM and is specifically targeted at solid mechanics applications, such as the ones considered in this paper.…”

Section: Implementation In the Albany Multi-physics Finite Element Codementioning

confidence: 99%

Preconditioned Least-Squares Petrov-Galerkin Reduced Order Models

Lindsay¹,

Fike²,

Tezaur³

et al. 2022

Preprint

Self Cite

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In this paper, we introduce a methodology for improving the accuracy and efficiency of reduced-order models (ROMs) constructed using the least-squares Petrov-Galerkin (LSPG) projection method through the introduction of preconditioning. Unlike prior related work, which focuses on preconditioning the linear systems arising within the ROM numerical solution procedure to improve linear solver performance, our approach leverages a preconditioning matrix directly within the minimization problem underlying the LSPG formulation. Applying preconditioning in this way has the potential to improve ROM accuracy for several reasons. First, preconditioning the LSPG formulation changes the norm defining the residual minimization, which can improve the residual-based stability constant bounding the ROM solution's error. The incorporation of a preconditioner into the LSPG formulation can have the additional effect of scaling the components of the residual being minimized to make them roughly of the same magnitude, which can be beneficial when applying the LSPG method to problems with disparate scales (e.g., dimensional equations, multi-physics problems). Importantly, we demonstrate that an 'ideal preconditioned' LSPG ROM (a ROM in which the preconditioner is the inverse of

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“…This is in line with [34], which leads to the Global-Local formulation through the concept of non-intrusiveness. Here the global and local level are solved in a multiplicative manner according to the idea of Schwarz' alternating method [75].…”

Section: Dirichlet-neumann Type Boundary Conditionsmentioning

confidence: 99%

An adaptive global–local approach for phase-field modeling of anisotropic brittle fracture

Noii

Aldakheel

Wick

et al. 2020

Computer Methods in Applied Mechanics and Engineering

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This work addresses an efficient Global-Local approach supplemented with predictor-corrector adaptivity applied to anisotropic phase-field brittle fracture. The phase-field formulation is used to resolve the sharp crack surface topology on the anisotropic/non-uniform local state in the regularized concept. To resolve the crack phase-field by a given single preferred direction, second-order structural tensors are imposed to both the bulk and crack surface density functions. Accordingly, a split in tension and compression modes in anisotropic materials is considered. A Global-Local formulation is proposed, in which the full displacement/phase-field problem is solved on a lower (local) scale, while dealing with a purely linear elastic problem on an upper (global) scale. Robin-type boundary conditions are introduced to relax the stiff local response at the global scale and enhancing its stabilization. Another important aspect of this contribution is the development of an adaptive Global-Local approach, where a predictor-corrector scheme is designed in which the local domains are dynamically updated during the computation. To cope with different finite element discretizations at the interface between the two nested scales, a non-matching dual mortar method is formulated. Hence, more regularity is achieved on the interface. Several numerical results substantiate our developments.

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The Schwarz alternating method in solid mechanics

Cited by 27 publications

References 54 publications

Interfacing Finite Elements with Deep Neural Operators for Fast Multiscale Modeling of Mechanics Problems

Interfacing Finite Elements with Deep Neural Operators for Fast Multiscale Modeling of Mechanics Problems

Preconditioned Least-Squares Petrov-Galerkin Reduced Order Models

An adaptive global–local approach for phase-field modeling of anisotropic brittle fracture

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