Reduced-order modelling numerical homogenization

Abdulle, Assyr; Bai, Yun

doi:10.1098/rsta.2013.0388

Cited by 16 publications

(19 citation statements)

References 60 publications

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“…(b) The patch U 2 (K) for a given K ∈ T H . Figure 1: Illustration of the patches U k (K) defined in (5). The dark gray part depicts the coarse element K ∈ T H .…”

Section: Localized Orthogonal Decompositionmentioning

confidence: 99%

“…The convergence analysis for a multiscale reduced basis method usually combines an existing a priori error analysis for the parameter independent multiscale approximation with error estimates for the Greedy procedure such as stated in Proposition 3.4 (see [3,5,6]). The following result guarantees convergence of the method, independent of the variations in the coefficient a ε .…”

Section: A Priori Error Analysismentioning

confidence: 99%

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A reduced basis localized orthogonal decomposition

Abdulle¹,

Henning²

2015

Journal of Computational Physics

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In this work we combine the framework of the Reduced Basis method (RB) with the framework of the Localized Orthogonal Decomposition (LOD) in order to solve parametrized elliptic multiscale problems. The idea of the LOD is to split a high dimensional Finite Element space into a low dimensional space with comparably good approximation properties and a remainder space with negligible information. The low dimensional space is spanned by locally supported basis functions associated with the node of a coarse mesh obtained by solving decoupled local problems. However, for parameter dependent multiscale problems, the local basis has to be computed repeatedly for each choice of the parameter. To overcome this issue, we propose an RB approach to compute in an "offline" stage LOD for suitable representative parameters. The online solution of the multiscale problems can then be obtained in a coarse space (thanks to the LOD decomposition) and for an arbitrary value of the parameters (thanks to a suitable "interpolation" of the selected RB). The online RB-LOD has a basis with local support and leads to sparse systems. Applications of the strategy to both linear and nonlinear problems are given.

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“…(b) The patch U 2 (K) for a given K ∈ T H . Figure 1: Illustration of the patches U k (K) defined in (5). The dark gray part depicts the coarse element K ∈ T H .…”

Section: Localized Orthogonal Decompositionmentioning

confidence: 99%

Section: A Priori Error Analysismentioning

confidence: 99%

A reduced basis localized orthogonal decomposition

Abdulle¹,

Henning²

2015

Journal of Computational Physics

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“…Furthermore, most discoveries in this paper actually exist elsewhere in various previous publications [12][13][14][15][16]; however, it is useful to provide a more systematic explanation for mechanics-based homogenization. Another important point is that theoretically the mathematical asymptotic homogenization process requires the micro-cell to be small or infinitely small to assume convergence of the process, but this is not necessary for mechanics-based homogenization.…”

Section: Introductionmentioning

confidence: 97%

Homogenization Method for Designing Novel Architectured Cellular Materials

Ma¹

2016

Proceedings of the VII European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS Congress 2016)

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Abstract. Two types of novel architectured cellular materials have been developed [1,2]. To estimate the effective material properties and to optimally design such materials, a suitable homogenization method was needed. An existing asymptotic homogenization process has provided an effective means for the multiscale modeling of continuum solids; however, that method was not easily adapted to the aforementioned materials, which are naturally discrete systems. First, we developed a strain-based homogenization method equivalent to the existing asymptotic homogenization method, but it was developed based on an engineering approach rather than on a mathematical approach such as the one used in asymptotic homogenization. The new approach separates the strain field into a homogenized strain field and a strain variation field in the local cellular domain superposed on the homogenized strain field. The Principle of Virtual Displacements for the relationship between the strain variation field and the homogenized strain field is then used to condense the strain variation field onto the homogenized strain field, and the homogenization process becomes a coordinate reduction process comparable to the Guyan Reduction used in structural dynamics analyses. The characteristic modes and the stress recovery process are also discussed. The new method is then extended to a stress-based homogenization process based on the Principle of Virtual Forces, and it is further applied to address the discrete systems of the aforementioned architectured cellular materials.

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“…While the idea of the former is to share one reduced basis on all subdomains the idea of the latter two is to generate one reduced basis for each class of subdomains which are then coupled appropriately. There also exist several approaches to use model reduction techniques for homogenization problems (see [14]) and problems with multiple scales, such as the reduced basis finite element HMM [3,4]. For the case of no scale-separation there exist the generalized MsFEM method [21], which incorporates model reduction ideas, and most recently a work combining the reduced basis method with localized orthogonal decomposition (see [6]).…”

Section: Introductionmentioning

confidence: 99%

Error Control for the Localized Reduced Basis Multiscale Method with Adaptive On-Line Enrichment

Ohlberger¹,

Schindler²

2015

SIAM J. Sci. Comput.

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Abstract. In this contribution we consider localized, robust and efficient a-posteriori error estimation of the localized reduced basis multi-scale (LRBMS) method for parametric elliptic problems with possibly heterogeneous diffusion coefficient. The numerical treatment of such parametric multi-scale problems are characterized by a high computational complexity, arising from the multi-scale character of the underlying differential equation and the additional parameter dependence. The LRBMS method can be seen as a combination of numerical multi-scale methods and model reduction using reduced basis (RB) methods to efficiently reduce the computational complexity with respect to the multi-scale as well as the parametric aspect of the problem, simultaneously. In contrast to the classical residual based error estimators currently used in RB methods, we are considering error estimators that are based on conservative flux reconstruction and provide an efficient and rigorous bound on the full error with respect to the weak solution. In addition, the resulting error estimator is localized and can thus be used in the on-line phase to adaptively enrich the solution space locally where needed. The resulting certified LRBMS method with adaptive on-line enrichment thus guarantees the quality of the reduced solution during the on-line phase, given any (possibly insufficient) reduced basis that was generated during the offline phase. Numerical experiments are given to demonstrate the applicability of the resulting algorithm with online enrichment to single phase flow in heterogeneous media.

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Reduced-order modelling numerical homogenization

Cited by 16 publications

References 60 publications

A reduced basis localized orthogonal decomposition

A reduced basis localized orthogonal decomposition

Homogenization Method for Designing Novel Architectured Cellular Materials

Error Control for the Localized Reduced Basis Multiscale Method with Adaptive On-Line Enrichment

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