Reduced basis technique for nonlinear analysis of structures

Noor, Ahmed K.; Peters, Jeanne M.

doi:10.2514/6.1979-747

Cited by 136 publications

(191 citation statements)

References 14 publications

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“…In this simple interpretation, this is certainly not a recent idea and initial work grew out of two related lines of inquiry: one focusing on the need for effective, many-query design evaluation [44], and one from the desire for efficient parameter continuation methods for nonlinear problems [4,114,115,117].…”

Section: Historical Background and Perspectivesmentioning

confidence: 99%

Certified Reduced Basis Methods for Parametrized Partial Differential Equations

Hesthaven¹,

Rozza²,

Stamm³

2016

SpringerBriefs in Mathematics

865

1,221

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The final publication is available at Springer via http://dx.doi.org/10.1007/978-3-319-22470-1International audienceThis 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

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Section: Historical Background and Perspectivesmentioning

confidence: 99%

Certified Reduced Basis Methods for Parametrized Partial Differential Equations

Hesthaven¹,

Rozza²,

Stamm³

2016

SpringerBriefs in Mathematics

865

1,221

View full text Add to dashboard Cite

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“…The essential acceleration ingredients are Galerkin projection onto a low dimensional space associated with an optimally sampled smooth parametric manifold [1,5,12,14] -rapid convergence; rigorous a posteriori error bounds for the field variable and associated functional outputs of interest -reliability and control; and Offline-Online computational decomposition strategies -rapid response in the real-time and many-query contexts. In the Online stage, given a new parameter value, we rapidly calculate the reduced basis (output) approximation and associated reduced basis error bound: the operation count is independent of N and depends only on N N and Q; here Q is the number of terms in the affine parameter expansion of the operator.…”

Section: Introductionmentioning

confidence: 99%

A natural-norm Successive Constraint Method for inf-sup lower bounds

Huynh

Knezevic

Chen

et al. 2010

Computer Methods in Applied Mechanics and Engineering

112

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We present a new approach for the construction of lower bounds for the inf-sup stability constants required in a posteriori error analysis of reduced basis approximations to affinely parametrized partial differential equations. We combine the "linearized" inf-sup statement of the naturalnorm approach with the approximation procedure of the Successive Constraint Method (SCM): the former (natural-norm) provides an economical parameter expansion and local concavity in parameter -a small(er) optimization problem which enjoys intrinsic lower bound properties; the latter (SCM) provides a systematic optimization framework -a Linear Program (LP) relaxation which readily incorporates continuity and stability constraints. The natural-norm SCM requires a parameter domain decomposition: we propose a greedy algorithm for selection of the SCM control points as well as adaptive construction of the optimal subdomains. The efficacy of the natural-norm SCM is illustrated through numerical results for two types of non-coercive problems: the Helmholtz equation (for acoustics, elasticity, and electromagnetics), and the convection-diffusion equation.

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“…The method takes its roots in domain decomposition methods and reduced basis discretizations (Fink & Rheinboldt (1983), Noor & Peters (1980), Prud'homme et al (2002)), and its applications extend to, for example, control and optimization problems. The basic idea is to first decompose the computational domain into a series of subdomains that are similar to a few reference domains (or generic computational parts).…”

mentioning

confidence: 99%

A reduced basis element method for the steady Stokes problem: Application to hierarchical flow systems

Løvgren¹,

Maday²,

Rønquist³

2006

MIC

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The reduced basis element method is a new approach for approximating the solution of problems described by partial differential equations. The method takes its roots in domain decomposition methods and reduced basis discretizations (Fink & Rheinboldt (1983), Noor & Peters (1980), Prud'homme et al. (2002)), and its applications extend to, for example, control and optimization problems. The basic idea is to first decompose the computational domain into a series of subdomains that are similar to a few reference domains (or generic computational parts). Associated with each reference domain are precomputed solutions corresponding to the same governing partial differential equation, but solved for different choices of some underlying parameter. In this work, the parameters are representing the geometric shape associated with a computational part. The approximation corresponding to a new shape is then taken to be a linear combination of the precomputed solutions, mapped from the reference domain for the part to the actual domain. We extend earlier work (Maday & Rønquist (2002), ) in this direction to solve incompressible fluid flow problems governed by the steady Stokes equations. Particular focus is given to constructing the basis functions, to the mapping of the velocity fields, to satisfying the inf-sup condition, and to "gluing" the local solutions together in the multidomain case (Belgacem et al. (2000)). We also demonstrate an algorithm for choosing the most efficient precomputed solutions. Two-dimensional examples are presented for pipes, bifurcations, and couplings of pipes and bifurcations in order to simulate hierarchical flow systems.

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Reduced basis technique for nonlinear analysis of structures

Cited by 136 publications

References 14 publications

Certified Reduced Basis Methods for Parametrized Partial Differential Equations

Certified Reduced Basis Methods for Parametrized Partial Differential Equations

A natural-norm Successive Constraint Method for inf-sup lower bounds

A reduced basis element method for the steady Stokes problem: Application to hierarchical flow systems

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