Reduced basis method and domain decomposition for elliptic problems in networks and complex parametrized geometries

Iapichino, Laura; Quarteroni, Alfio; Rozza, Gianluigi

doi:10.1016/j.camwa.2015.12.001

Cited by 56 publications

(48 citation statements)

References 26 publications

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“…Combinations of the RB method with DD methods have been considered in [58,59,48,5,47,29,79,49,62,64,16]. Here, intra-element RB approximations are for instance coupled by either polynomial Lagrange multipliers [58,59], generalized Legendre polynomials [47], FE basis functions [49], or empirical modes generated from local solutions of the PDE [29,64,16] on the interface. In order to address parameterized multiscale problems the local approximation spaces are for instance spanned by eigenfunctions of an eigenvalue problem on the space of harmonic functions in [28], generated by solving the global parameterized PDE and restricting the solution to the respective subdomain in [70,3], or enriched in the online stage by local solutions of the PDE, prescribing the insufficient RB solution as Dirichlet boundary conditions in [70,3].…”

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confidence: 99%

Randomized Local Model Order Reduction

Buhr¹,

Smetana²

2018

SIAM J. Sci. Comput.

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In this paper we propose local approximation spaces for localized model order reduction procedures such as domain decomposition and multiscale methods. Those spaces are constructed from local solutions of the partial differential equation (PDE) with random boundary conditions, yield an approximation that converges provably at a nearly optimal rate, and can be generated at close to optimal computational complexity. In many localized model order reduction approaches like the generalized finite element method, static condensation procedures, and the multiscale finite element method local approximation spaces can be constructed by approximating the range of a suitably defined transfer operator that acts on the space of local solutions of the PDE. Optimal local approximation spaces that yield in general an exponentially convergent approximation are given by the left singular vectors of this transfer operator [I. Babuška and R. Lipton 2011, K. Smetana and A. T. Patera 2016]. However, the direct calculation of these singular vectors is computationally very expensive. In this paper, we propose an adaptive randomized algorithm based on methods from randomized linear algebra [N. Halko et al. 2011], which constructs a local reduced space approximating the range of the transfer operator and thus the optimal local approximation spaces. Moreover, the adaptive algorithm relies on a probabilistic a posteriori error estimator for which we prove that it is both efficient and reliable with high probability. Several numerical experiments confirm the theoretical findings.Key words. localized model order reduction, randomized linear algebra, domain decomposition methods, multiscale methods, a priori error bound, a posteriori error estimation AMS subject classifications. 65N15, 65N12, 65N55, 65N30, 65C20, 65N25

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confidence: 99%

Randomized Local Model Order Reduction

Buhr¹,

Smetana²

2018

SIAM J. Sci. Comput.

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“…In [9] deformation patterns from an analysis of the assembled structure are employed. Moreover, local reduced models are generated from parametrized Lagrange or Fourier modes and coupled via FE basis functions in [14]. Finally, empirical modes generated from local solutions of the PDE are suggested in [5,7,20].…”

Section: Introductionmentioning

confidence: 99%

Static Condensation Optimal Port/Interface Reduction and Error Estimation for Structural Health Monitoring

Smetana

2019

IUTAM Symposium on Model Order Reduction of Coupled Systems, Stuttgart, Germany, May 22–25, 2018

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Having the application in structural health monitoring in mind, we propose reduced port spaces that exhibit an exponential convergence for static condensation procedures on structures with changing geometries for instance induced by newly detected defects. Those reduced port spaces generalize the port spaces introduced in [K. Smetana and A.T. Patera, SIAM J. Sci. Comput., 2016] to geometry changes and are optimal in the sense that they minimize the approximation error among all port spaces of the same dimension. Moreover, we show numerically that we can reuse port spaces that are constructed on a certain geometry also for the static condensation approximation on a significantly different geometry, making the optimal port spaces well suited for use in structural health monitoring.

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“…The aim being is problems in which the domain can be built as a nonoverlapping union of a small set of subdomains that are geometrically similar. () In this sense, Iapichino et al proposed the reduced basis hybrid method to solve Stokes flow in a domain composed by different nonoverlapping subdomains. These subdomains are grouped into geometrically similar subdomains, that is, subdomains with the same number of sides in 2D.…”

Section: Introductionmentioning

confidence: 99%

“…This approach requires at the online phase to solve a global problem that includes the reduced basis (standard in this technique) plus a coarse finite element mesh (to capture the normal fluxes on the interfaces) and Lagrange multipliers (to impose the continuity of the primal function on the interface). This was further improved in the work of Iapichino et al by introducing (apart from the geometry) a set of parameters that characterize the profile of the function on the internal interfaces of the computational domain. This implies to solve, at the online phase, a system with many unknowns as parameters characterizing the solution on the interfaces plus the entire reduced basis for each subdomain of the DD.…”

Section: Introductionmentioning

confidence: 99%

Proper generalized decomposition solutions within a domain decomposition strategy

Huerta

Nadal

Chinesta

2018

Numerical Meth Engineering

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Domain decomposition strategies and proper generalized decomposition are efficiently combined to obtain a fast evaluation of the solution approximation in parameterized elliptic problems with complex geometries. The classical difficulties associated to the combination of layered domains with arbitrarily oriented midsurfaces, which may require in-plane-out-of-plane techniques, are now dismissed. More generally, solutions on large domains can now be confronted within a domain decomposition approach. This is done with a reduced cost in the offline phase because the proper generalized decomposition gives an explicit description of the solution in each subdomain in terms of the solution at the interface. Thus, the evaluation of the approximation in each subdomain is a simple function evaluation given the interface values (and the other problem parameters). The interface solution can be characterized by any a priori user-defined approximation. Here, for illustration purposes, hierarchical polynomials are used. The repetitiveness of the subdomains is exploited to reduce drastically the offline computational effort. The online phase requires solving a nonlinear problem to determine all the interface solutions. However, this problem only has degrees of freedom on the interfaces and the Jacobian matrix is explicitly determined. Obviously, other parameters characterizing the solution (material constants, external loads, and geometry) can also be incorporated in the explicit description of the solution. KEYWORDS domain decomposition, parameterized solutions, proper generalized decomposition, reduced-order models INTRODUCTIONProper generalized decomposition (PGD 1-4 ) has proven its advantages and applicability in many parameterized problems. PGD reduces the computational complexity induced by a large number of dimensions (the sum of the number of spatial dimensions plus the number of parameters) to the iterative resolution of low dimensional problems, usually one dimensional (1D) or two dimensional (2D). Moreover, the PGD approach provides an explicit expression of the approximated solution. Thus, the online phase, that is, the evaluation of the approximation given the parameters, is very fast and does not require any extra interpolation or another solve (typical, for instance, in Proper Orthogonal Decomposition strategies).However, PGD has never been combined successfully with domain decomposition (DD) strategies. The methodology proposed here makes it possible. The resolution is divided into 2 phases. The first one builds (offline) a PGD model (ie, an

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Reduced basis method and domain decomposition for elliptic problems in networks and complex parametrized geometries

Cited by 56 publications

References 26 publications

Randomized Local Model Order Reduction

Randomized Local Model Order Reduction

Static Condensation Optimal Port/Interface Reduction and Error Estimation for Structural Health Monitoring

Proper generalized decomposition solutions within a domain decomposition strategy

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