Residual-free bubbles for the Helmholtz equation

Franca, Leopoldo P.; Farhat, Charbel; Macedo, Antonini; Lesoinne, Michel

doi:10.1002/(sici)1097-0207(19971115)40:21<4003::aid-nme199>3.0.co;2-z

Cited by 144 publications

(82 citation statements)

References 10 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The fact that bubble functions disappear on element boundaries [17][18][19][20][21][22][23][24] makes it possible to remove the equations that correspond to these functions from the set of elemental equations. This procedure is called static condensation [27].…”

Section: Variational Multiscale Methods Using Bubble Functionsmentioning

confidence: 99%

“…Multiscale variational approach is generally used to take into account the variations of field unknown ranging over different physical scales without using excessively refined computational girds [16]. Normally, in this approach the field unknown (T) is divided into two parts as are, generally, high order polynomials which are zero on the element boundaries [17][18][19][20][21][22][23][24].…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

A multiscale finite element space–time discretization method for transient transport phenomena using bubble functions

Nassehi

Parvazinia

2009

Finite Elements in Analysis and Design

View full text Add to dashboard Cite

show abstract

Section: Variational Multiscale Methods Using Bubble Functionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A multiscale finite element space–time discretization method for transient transport phenomena using bubble functions

Nassehi

Parvazinia

2009

Finite Elements in Analysis and Design

View full text Add to dashboard Cite

show abstract

“…The preceding conclusion is not at odds with the notable improvements provided by local enrichments in problems where lack of lower semicontinuity is not an issue, for instance advection-diffusion problems [32,10,33,11]-which can also be rewritten as the minimization of a residual in H −1 (see [51])-or recovering the inf-sup condition for kinematically constrained systems (e.g., incompressibility [3], mortar methods [6,36]). …”

Section: Finite-dimensional Space Of Local Enrichments and E T W (Fmentioning

confidence: 99%

Concurrent Multiscale Computing of Deformation Microstructure by Relaxation and Local Enrichment with Application to Single‐Crystal Plasticity

Conti¹,

Hauret²,

Ortiz³

2007

Multiscale Model. Simul.

View full text Add to dashboard Cite

Abstract. This paper is concerned with the effective modeling of deformation microstructures within a concurrent multiscale computing framework. We present a rigorous formulation of concurrent multiscale computing based on relaxation; we establish the connection between concurrent multiscale computing and enhanced-strain elements; and we illustrate the approach in an important area of application, namely, single-crystal plasticity, for which the explicit relaxation of the problem is derived analytically. This example demonstrates the vast effect of microstructure formation on the macroscopic behavior of the sample, e.g., on the force/travel curve of a rigid indentor. Thus, whereas the unrelaxed model results in an overly stiff response, the relaxed model exhibits a proper limit load, as expected. Our numerical examples additionally illustrate that ad hoc element enhancements, e.g., based on polynomial, trigonometric, or similar representations, are unlikely to result in any significant relaxation in general.Key words. multiscale computing, relaxation, microstructure, finite elements, enhanced strain, single-crystal plasticity AMS subject classifications. 74G65, 74C05DOI. 10.1137/0606623321. Introduction. The problem addressed in this paper concerns the effective modeling of deformation microstructures within a concurrent multiscale computing framework. In many applications of interest, materials develop fine microstructure on multiple length and time scales in response to loading [5,53,58,49]. Examples of such microstructures include martensite; subgrain dislocation structures; dislocation walls and networks; ferroelectric domains; shear bands; spall planes; and others. In addition, materials such as polycrystalline metals may exhibit processing microstructure from the outset, prior to the onset of deformation. The macroscopic behavior of such materials is too complex to be amenable to modeling based on simple representational schemes, such as afforded by continuum thermodynamics, symmetry groups, linearization, polynomial approximations, empirical fitting and calibration, and other similar schemes. Indeed, empirical models are a major source of error and uncertainty in engineering applications, and the empirical paradigm does not offer a systematic means of reducing such error and uncertainty.Multiscale modeling aims to eliminate empiricism and uncertainty from material models by systematically identifying the rate-controlling mechanisms at all scales and the fundamental laws that govern those mechanisms, and by bridging the relevant

show abstract

“…To systematically treat various singularly perturbed problems, residual-free bubbles were introduced in [6,10,11,12,13,14]. These bubbles are produced by solving, exactly or not, differential equations at the element level, involving the differential operator of the problem.…”

Section: Introductionmentioning

confidence: 99%

Towards multiscale functions: enriching finite element spaces with local but not bubble-like functions

Franca

Madureira

Valentin

2005

Computer Methods in Applied Mechanics and Engineering

Self Cite

View full text Add to dashboard Cite

Abstract. In this paper we propose a novel way, via finite elements to treat problems that can be singular perturbed, a reaction-diffusion equation in our case. We enrich the usual piecewise linear or bilinear finite element trial spaces with local solutions of the original problem, as in the Residual Free Bubble (RFB) setting, but do not require these functions to vanish on each element edge, a departure from the RFB paradigm. Such multiscale functions have an analytic expression, for triangles and rectangles. Bubbles are the choice for the test functions allowing static condensation, thus our method is of Petrov-Galerkin type. We perform several numerical validations which confirm the good performance of the method.

show abstract

Residual-free bubbles for the Helmholtz equation

Abstract: The Galerkin method enriched with residual-free bubbles is considered for approximating the solution of the Helmholtz equation. Two-dimensional tests demonstrate the improvement over the standard Galerkin method and the Galerkin-least-squares method using piecewise bilinear interpolations.

Cited by 144 publications

References 10 publications

A multiscale finite element space–time discretization method for transient transport phenomena using bubble functions

A multiscale finite element space–time discretization method for transient transport phenomena using bubble functions

Concurrent Multiscale Computing of Deformation Microstructure by Relaxation and Local Enrichment with Application to Single‐Crystal Plasticity

Towards multiscale functions: enriching finite element spaces with local but not bubble-like functions

Contact Info

Product

Resources

About