Nonconforming finite element approximation of crystalline microstructure

Li, Bo; Luskin, Mitchell

doi:10.1090/s0025-5718-98-00941-7

Cited by 31 publications

(36 citation statements)

References 37 publications

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“…As is well-known, nonconforming elements (e.g., Crouzeix-Raviart (CR) linear elements [11] and the rotated Q 1 -element [12], [18], [10]) are very useful to seek numerical solutions of many physical problems (see [11], [12], [13], [15], [16], [17], [18], [27], [10]). A quadrilateral version of the rotated Q 1 -element was studied in [18], but it is only suitable for uniform asymptotic rectangles.…”

Section: Introductionmentioning

confidence: 99%

Untitled

2004

Math. Comp.

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Abstract.In this paper we analyze the finite element discretization for the first-order system least squares mixed model for the second-order elliptic problem by means of using nonconforming and conforming elements to approximate displacement and stress, respectively. Moreover, on arbitrary regular quadrilaterals, we propose new variants of both the rotated Q 1 nonconforming element and the lowest-order Raviart-Thomas element.

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Section: Introductionmentioning

confidence: 99%

Untitled

2004

Math. Comp.

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“…The fact that the energy density for the cubic to orthorhombic transformation has six wells makes this transformation significantly more difficult to analyze than the two and three-well transformations since the additional wells give the crystal more freedom to deform without the cost of additional energy. The numerical analysis of both conforming and nonconforming finite element approximations for the two-well and three-well transformations has been presented by Li and Luskin [30,31,32,33]. The stability theory was also used by Luskin and Ma to analyze the microstructure in ferromagnetic crystals [35].…”

Section: Introductionmentioning

confidence: 99%

The Simply Laminated Microstructure in Martensitic Crystals that Undergo a Cubic-to-Orthorhombic Phase Transformation

Bhattacharya

Luskin

1999

Archive for Rational Mechanics and Analysis

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Abstract. We study simply laminated microstructures of a martensitic crystal capable of undergoing a cubic to orthorhombic transformation of type P (432) → P (222) . The free energy density modeling such a crystal is minimized on six energy wells that are pairwise rank-one connected. We consider the energy minimization problem with Dirichlet boundary data compatible with an arbitrary but fixed simple laminate. We first show that for all but a few isolated values of transformation strains, this problem has a unique Young measure solution solely characterized by the boundary data that represents the simply laminated microstructure. We then present a theory of stability for such a microstructure, and apply it to the conforming finite element approximation to obtain the corresponding error estimates for the finite element energy minimizers.

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“…This analysis was extended to the cubic to tetragonal transformation in [22]. For constant volume fractions, an analysis for a nonconforming finite element approximation was given in [23].…”

Section: Introductionmentioning

confidence: 99%

Approximation of a Martensitic Laminate with Varying Volume Fractions

Ли

Luskin

1999

ESAIM: M2AN

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Abstract. We give results for the approximation of a laminate with varying volume fractions for multi-well energy minimization problems modeling martensitic crystals that can undergo either an orthorhombic to monoclinic or a cubic to tetragonal transformation. We construct energy minimizing sequences of deformations which satisfy the corresponding boundary condition, and we establish a series of error bounds in terms of the elastic energy for the approximation of the limiting macroscopic deformation and the simply laminated microstructure. Finally, we give results for the corresponding finite element approximation of the laminate with varying volume fractions.Résumé. Nous considérons des problèmes de minimisation d'énergie avec de multiples puits de potentiel. De tels problèmes modélisent, pour des cristaux martensitiques, des transitions de phase d'un réseau orthorhombiqueà monoclinique, ou d'un réseau cubiqueà tétragonal, par exemple. Des résultats d'approximation des structures laminaires correspondantes, avec fractions volumiques variables, sont donnés. Des suites minimisantes, avec déformations compatibles avec les conditions au bord, sont construites et permettent l'obtention de plusieurs estimations d'erreur concernant l'approximation de la déformation macroscopique limite en fonction de l'énergieélastique. Finalement, nous décrivons des résultats concernant l'approximation paréléments finis de la structure laminaire avec fractions volumiques variables.

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Nonconforming finite element approximation of crystalline microstructure

Cited by 31 publications

References 37 publications

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The Simply Laminated Microstructure in Martensitic Crystals that Undergo a Cubic-to-Orthorhombic Phase Transformation

Approximation of a Martensitic Laminate with Varying Volume Fractions

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