Port reduction in parametrized component static condensation: approximation and <i>a posteriori</i> error estimation

Eftang, Jens L.; Patera, Anthony T.

doi:10.1002/nme.4543

Cited by 74 publications

(106 citation statements)

References 29 publications

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“…First, as we consider here larger global systems with a much larger number of instantiated components we introduce a new non-symmetric SCRBE approximation, which reduces both offline and online cost and memory footprint; the corresponding linear-algebraic system is subsequently symmetrized in order to (say) accommodate efficient linear solvers. We also demonstrate that our central theoretical results in particular related to a posteriori error estimation survive intact for this more efficient revision of our earlier formulations in [12]. Second, we provide a precise formulation for general geometric mappings and port space compatibility, and we demonstrate that (in the isotropic linear-elastic case) rigid-body parameters related to "docking" of component instantiations in a system do not affect the associated bilinear forms and thus do not impact offline -thanks to smaller RB space dimensions -or online -thanks to treatment of differently oriented component instantiations as effectively identical -computational cost.…”

Section: Introductionmentioning

confidence: 53%

“…In this paper, we extend our earlier work for two-dimensional scalar problems in [12] to the more demanding three-dimensional vector-field case. We focus here on applications in linear elasticity, but we note that the component synthesis and indeed RB and port approximations can be readily extended to problems in heat transfer or (frequency domain) acoustics, or any phenomenon described by a linear elliptic or parabolic [13] PDE.…”

Section: Introductionmentioning

confidence: 80%

“…For example, for the CMS approaches an eigenmode expansion (with subsequent truncation) for the port degrees of freedom is considered in [9,10], and an adaptive procedure based on a posteriori error estimators for the port reduction is considered in [11]. For the SCRBE, we introduce in [12] port reduction with empirical modes; in this case the port approximation spaces are informed by snapshots of relevant port-restricted solutions which are obtained through an offline pairwise empirical training algorithm.…”

Section: Introductionmentioning

confidence: 99%

“…This framework invokes classical residual arguments on the (RB) bubble level [6], a non-conforming approximation to the error-residual equation at the port level, and finally matrix perturbation at the system level in order to bound (under an eigenvalue proximity assumption) the error contributions from both RB approximation [5] and port reduction [12]. In actual practice, we may reduce online computational cost by consideration of a plausible and asymptotically rigorous error estimator rather than a rigorous error bound.…”

Section: Introductionmentioning

confidence: 99%

“…Second, we provide a precise formulation for general geometric mappings and port space compatibility, and we demonstrate that (in the isotropic linear-elastic case) rigid-body parameters related to "docking" of component instantiations in a system do not affect the associated bilinear forms and thus do not impact offline -thanks to smaller RB space dimensions -or online -thanks to treatment of differently oriented component instantiations as effectively identical -computational cost. Third, we introduce a http://www.amses-journal.com/content/1/1/3 new functional interpretation of our algebraic a posteriori error estimation framework in [12], which may serve to extend our approach here to larger classes of problems. And finally, we consider multi-reference parameter bound conditioners [14] for sharper error estimation.…”

Section: Introductionmentioning

confidence: 99%

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A port-reduced static condensation reduced basis element method for large component-synthesized structures: approximation and A Posteriori error estimation

Eftang

Patera

2014

Adv Model Simul Eng Sci

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Background:We consider a static condensation reduced basis element framework for efficient approximation of parameter-dependent linear elliptic partial differential equations in large three-dimensional component-based domains. The approach features an offline computational stage in which a library of interoperable parametrized components is prepared; and an online computational stage in which these component archetypes may be instantiated and connected through predefined ports to form a global synthesized system. Thanks to the component-interior reduced basis approximations, the online computation time is often relatively small compared to a classical finite element calculation. Methods: In addition to reduced basis approximation in the component interiors, we employ in this paper port reduction with empirical port modes to reduce the number of degrees of freedom on the ports and thus the size of the Schur complement system. The framework is equipped with efficiently computable a posteriori error estimators that provide asymptotically rigorous bounds on the error in the approximation with respect to the underlying finite element discretization. We extend our earlier approach for two-dimensional scalar problems to the more demanding three-dimensional vector-field case. Results and Conclusions: This paper focuses on linear elasticity analysis for large structures with tens of millions of finite element degrees of freedom. Through our procedure we effectively reduce the number of degrees of freedom to a few thousand, and we demonstrate through extensive numerical results for a microtruss structure that our approach provides an accurate, rapid, and a posteriori verifiable approximation for relevant large-scale engineering problems.

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Section: Introductionmentioning

confidence: 53%

Section: Introductionmentioning

confidence: 80%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

A port-reduced static condensation reduced basis element method for large component-synthesized structures: approximation and A Posteriori error estimation

Eftang

Patera

2014

Adv Model Simul Eng Sci

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Proper orthogonal decomposition‐based reduced basis element thermal modeling of integrated circuits

Meyer

Helenbrook

Cheng

2017

Numerical Meth Engineering

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Summary Various reduced basis element methods are compared for performing transient thermal simulations of integrated circuits. The reduced basis element method is a type of reduced order modeling that takes advantage of repeated geometrical features. It uses a reduced set of basis functions to approximate the solution of a PDE on subdomains (blocks); then these blocks are coupled together to perform a simulation on an entire domain. As the simulations are transient, a proper orthogonal decomposition basis is used, and the proper orthogonal decomposition eigenvalues from each block are used to derive error bounds for the entire simulation. These bounds are used to examine choices of block decompositions for a simplified integrated circuit structure. A decomposition that uses a single block for each transistor device is compared with a decomposition that uses one block for multiple devices. It was found that larger blocks are more computationally efficient; however, the advantage decreases if the devices within a block receive independent signals. Continuous and discontinuous methods of coupling the blocks were also compared. The coupling methods lend themselves to different solution approaches such as static condensation (continuous coupling) and block‐based inversion (discontinuous). Static condensation yielded the best convergence rate, accuracy, and operation count. Copyright © 2017 John Wiley & Sons, Ltd.

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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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Port reduction in parametrized component static condensation: approximation and a posteriori error estimation

Cited by 74 publications

References 29 publications

A port-reduced static condensation reduced basis element method for large component-synthesized structures: approximation and A Posteriori error estimation

A port-reduced static condensation reduced basis element method for large component-synthesized structures: approximation and A Posteriori error estimation

Proper orthogonal decomposition‐based reduced basis element thermal modeling of integrated circuits

Proper generalized decomposition solutions within a domain decomposition strategy

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