A discretize-then-map approach for the treatment of parameterized geometries in model order reduction

Taddei, Tommaso; Zhang, Lei

doi:10.1016/j.cma.2021.113956

Cited by 13 publications

(5 citation statements)

References 39 publications

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“…j=1 , connectivity matrix T and elements {D k, } N e k=1 , we denote by U a generic element of X and we denote by U ∈ R DN v the corresponding FE vector associated with the Lagrangian basis of T a , for all ∈ L. Following [14], we pursue a discretize-then-map treatment of parameterized geometries: given the mesh T a L i , we state the local variational problems in the deformed mesh…”

Section: Instantiated Systemmentioning

confidence: 99%

Component-based model order reduction procedure for large scales Thermo-Hydro-Mechanical systems

Iollo,

Sambataro,

Taddei

2023

10th Edition of the International Conference on Computational Methods for Coupled Problems in Science and Engineering

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In this work we develop a component-based model order reduction (CB-pMOR) procedure for a class of problems in nonlinear mechanics with internal variables. The work is motivated by applications to thermo-hydro-mechanical (THM) systems for radioactive waste disposal. The THM system is coupled, time-dependent, and highly nonlinear; furthermore, the solution to the problem depends on several parameters, which might be related to the geometric configuration (e.g. the number of repositories, their distance or their size) or the material properties of the medium. We investigate the effectiveness of the proposed method in terms of accuracy and computational costs for a two-dimensional THM system in the case of overlapping partitions.

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Section: Instantiated Systemmentioning

confidence: 99%

Component-based model order reduction procedure for large scales Thermo-Hydro-Mechanical systems

Iollo,

Sambataro,

Taddei

2023

10th Edition of the International Conference on Computational Methods for Coupled Problems in Science and Engineering

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“…To speed up computations, we should thus resort to hyper-reduction techniques [7,18,27,62,78]. The choice of the hyper-reduction procedure strongly depends on the PDE model of interest, on the underlying high-fidelity numerical scheme, and on the geometrical parameterization: we refer to [72] for a discussion on the treatment of geometry parameterizations. We further observe that evaluation of (17a) involves evaluation of u j,µ in the mapped quadrature points of the mesh Ω Li : this evaluation is extremely expensive for unstructured meshes and thus requires a specialized treatment.…”

Section: Residual Assembly and Algebraic Formulation Of The Reduced-o...mentioning

confidence: 99%

Localized model reduction for nonlinear elliptic partial differential equations: localized training, partition of unity, and adaptive enrichment

Smetana¹,

Taddei²

2022

Preprint

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We propose a component-based (CB) parametric model order reduction (pMOR) formulation for parameterized nonlinear elliptic partial differential equations (PDEs). CB-pMOR is designed to deal with large-scale problems for which full-order solves are not affordable in a reasonable time frame or parameters' variations induce topology changes that prevent the application of monolithic pMOR techniques. We rely on the partition-of-unity method (PUM) to devise global approximation spaces from local reduced spaces, and on Galerkin projection to compute the global state estimate. We propose a randomized data compression algorithm based on oversampling for the construction of the components' reduced spaces: the approach exploits random boundary conditions of controlled smoothness on the oversampling boundary. We further propose an adaptive residual-based enrichment algorithm that exploits global reduced-order solves on representative systems to update the local reduced spaces. We prove exponential convergence of the enrichment procedure for linear coercive problems; we further present numerical results for a two-dimensional nonlinear diffusion problem to illustrate the many features of our proposal and demonstrate its effectiveness.

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“…EQ procedures also dubbed mesh sampling and weighting have been first proposed in References 4‐6 and further developed in several other works including Reference 7: the key feature of EQ is to recast the problem of hyper‐reduction as a sparse representation problem and then resort to state‐of‐the‐art techniques in machine learning and signal processing to estimate the solution to the resulting optimization problem. Here, we rely on the approach employed in Reference 8, which combines the methods in References 4 and 5 and relies on non‐negative least‐squares to estimate the solution to the sparse representation problem.…”

Section: Introductionmentioning

confidence: 99%

“…As discussed in Section 3, the presence of internal variables requires several changes to the EQ approach in Reference 8. Our approach relies on a different treatment of primary and internal variables compared to the works in Reference 9 and 10, as explained in Section 3.2.…”

Section: Introductionmentioning

confidence: 99%

“…In addition, in the present work we aim at solving coupled systems that model the interaction between mechanic, thermal and hydraulic response, as described in Section 4. In Section 3, we clarify to what extent the management of internal variables requires a careful adaptation of the MOR technique illustrated in Reference 8 and we briefly compare our treatment of internal variables with References 9 and 10.…”

Section: Introductionmentioning

confidence: 99%

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An adaptive projection‐based model reduction method for nonlinear mechanics with internal variables: Application to thermo‐hydro‐mechanical systems

Iollo

Sambataro

Taddei

2022

Numerical Meth Engineering

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We propose a projection‐based monolithic model order reduction procedure for a class of problems in nonlinear mechanics with internal variables. The work is motivated by applications to thermo‐hydro‐mechanical (THM) systems for radioactive waste disposal. THM equations model the behavior of temperature, pore water pressure, and solid displacement in the neighborhood of geological repositories, which contain radioactive waste and are responsible for a significant thermal flux toward the Earth's surface. We develop an adaptive sampling strategy based on the POD‐Greedy method, and we develop an element‐wise empirical quadrature hyper‐reduction procedure to reduce assembling costs. We present numerical results for a two‐dimensional THM system to illustrate and validate the proposed methodology.

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A discretize-then-map approach for the treatment of parameterized geometries in model order reduction

Cited by 13 publications

References 39 publications

Component-based model order reduction procedure for large scales Thermo-Hydro-Mechanical systems

Component-based model order reduction procedure for large scales Thermo-Hydro-Mechanical systems

Localized model reduction for nonlinear elliptic partial differential equations: localized training, partition of unity, and adaptive enrichment

An adaptive projection‐based model reduction method for nonlinear mechanics with internal variables: Application to thermo‐hydro‐mechanical systems

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