In this paper we consider (hierarchical, La-grange)reduced basis approximation anda posteriori error estimation for linear functional outputs of affinely parametrized elliptic coercive partial differential equa-tions. The essential ingredients are (primal-dual)Galer-kin projection onto a low-dimensional space associated with a smooth “parametric manifold” - dimension re-duction; efficient and effective greedy sampling meth-ods for identification of optimal and numerically stable approximations - rapid convergence;a posteriori er-ror estimation procedures - rigorous and sharp bounds for the linear-functional outputs of interest; and Offine-Online computational decomposition strategies - min-imummarginal cost for high performance in the real-time/embedded (e.g., parameter-estimation, control)and many-query (e.g., design optimization, multi-model/ scale)contexts. We present illustrative results for heat conduction and convection-diffusion,inviscid flow, and linear elasticity; outputs include transport rates, added mass,and stress intensity factors.

/pdf/reduced-basis-approximation-and-a-posteriori-error-1csv7ta4m6.pdf

Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations

This book provides a thorough introduction to the mathematical and algorithmic aspects of certified reduced basis methods for parametrized partial differential equations. Central aspects ranging from model construction, error estimation and computational efficiency to empirical interpolation methods are discussed in detail for coercive problems. More advanced aspects associated with time-dependent problems, non-compliant and non-coercive problems and applications with geometric variation are also discussed as examples.

Certified Reduced Basis Methods for Parametrized Partial Differential Equations

Four decades after the development of the first dynamic substructuring techniques, there is a necessity to classify the different methods in a general framework that outlines the relations between them. In this paper, a certain vision on substructuring methods is proposed, by recalling important historical milestones that allow us to understand substructuring as a domain decomposition concept. Thereafter, based on the dual and primal assembly of substructures, a general framework for the classification of the methods is presented. This framework allows us to indicate how the various classes of methods, proposed along the years, can be derived from a clear mathematical description of substructured problems. Current bottlenecks in experimental dynamic substructuring, as well as solutions found in literature, will also be briefly discussed.

/pdf/general-framework-for-dynamic-substructuring-history-review-1hai30qv08.pdf

General Framework for Dynamic Substructuring: History, Review and Classification of Techniques

We present a technique for the rapid and reliable prediction of linear-functional outputs of elliptic (and parabolic) partial differential equations with affine parameter dependence. The essential components are (i) (provably) rapidly convergent global reduced basis approximations, Galerkin projection onto a space W(sub N) spanned by solutions of the governing partial differential equation at N selected points in parameter space; (ii) a posteriori error estimation, relaxations of the error-residual equation that provide inexpensive yet sharp and rigorous bounds for the error in the outputs of interest; and (iii) off-line/on-line computational procedures, methods which decouple the generation and projection stages of the approximation process. The operation count for the on-line stage, in which, given a new parameter value, we calculate the output of interest and associated error bound, depends only on N (typically very small) and the parametric complexity of the problem; the method is thus ideally suited for the repeated and rapid evaluations required in the context of parameter estimation, design, optimization, and real-time control.

/pdf/reliable-real-time-solution-of-parametrized-partial-1wnffkep9s.pdf

Reliable Real-Time Solution of Parametrized Partial Differential Equations: Reduced-Basis Output Bound Methods

In this paper, we extend the reduced-basis approximations developed earlier for linear elliptic and parabolic partial differential equations with affine parameter dependence to problems in- volving (a) nonaffine dependence on the parameter, and (b) nonlinear dependence on the field variable. The method replaces the nonaffine and nonlinear terms with a coefficient function approximation which then permits an efficient offline-online computational decomposition. We first review the coefficient function approximation procedure: the essential ingredients are (i) a good collateral reduced-basis approximation space, and (ii) a stable and inexpensive interpolation procedure. We then apply this approach to linear nonaffine and nonlinear elliptic and parabolic equations; in each instance, we discuss the reduced-basis approximation and the associated offline-online computational procedures. Numeri- cal results are presented to assess our approach.

/pdf/efficient-reduced-basis-treatment-of-nonaffine-and-nonlinear-2pgbrpce4d.pdf

Efficient reduced-basis treatment of nonaffine and nonlinear partial differential equations

Parametric families of reduced finite element models. theory and applications

New Results on the Identification of Normal Modes from Experimental Complex Modes

The ambient temperature effect may result in limitations of vibration-based SHM approaches for civil engineering structures. This paper addresses the issue of discriminating changes in modal parame...

https://hal.inria.fr/inria-00164852/document

Merging Sensor Data from Multiple Temperature Scenarios for Vibration Monitoring of Civil Structures

A fundamental property of the Rayleigh-Ritz method is that a reduced model gives exact predictions if the solutions lie in the range of the reduction basis. For cases with and without rigid-body modes, it is shown that the strain energy norm can be used to define optimal projections of both loads and displacements onto the exactly represented subspace of a given reduced model. In practice, multiple loads or displacements are considered, and variants of the singular value decomposition based on the strain and kinetic energy norms are shown to provide ways to select important contributions. The proposed framework is used to define two new methods for optimal reduction and reduced model correction. The validity and usefulness of the proposed methods are illustrated for the example of two square plates assembled at a right angle with two very different plate thickness configurations.

Etienne Balmes

Papers

Parametric families of reduced finite element models. theory and applications

New Results on the Identification of Normal Modes from Experimental Complex Modes

Merging Sensor Data from Multiple Temperature Scenarios for Vibration Monitoring of Civil Structures

Optimal Ritz vectors for component mode synthesis using the singular value decomposition

High Modal Density, Curve Veering, Localization: A Different Perspective On The Structural Response