A computational model for nucleation of solid-solid phase transformations

Chu, Yi-An; Moran, Brian

doi:10.1088/0965-0393/3/4/003

Cited by 29 publications

(22 citation statements)

References 16 publications

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“…Comparing (25) and (17), one can deduce that the present method is di erent from that proposed in Reference [19]. Although the order of (25) is identical to that of (17), the matrices K and F in (25) are di erent fromK andF in (17).…”

Section: Essential Boundary Conditionsmentioning

confidence: 76%

“…It will be shown that K is also banded. After U 1 is obtained from (25), U 2 can be obtained from (22). Because the order of matrix B 2 is very low, the computational e ort required for the calculation of K and F in (26) and (27) is not signiÿcant, so that the present scheme is e cient.…”

Section: Essential Boundary Conditionsmentioning

confidence: 99%

“…Although the order of (25) is identical to that of (17), the matrices K and F in (25) are di erent fromK andF in (17). Compared with (17), the present method is more e cient because only factorization of a n u × n u (n u N ) matrix, B 2 , is required to obtain the transformation matrix T. In the next section, a simpliÿed approach is developed so that B 2 become a diagonal matrix.…”

Section: Essential Boundary Conditionsmentioning

confidence: 99%

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Imposition of essential boundary conditions by displacement constraint equations in meshless methods

Zhang

Liu

et al. 2001

Commun. Numer. Meth. Engng.

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SUMMARYOne of major di culties in the implementation of meshless methods is the imposition of essential boundary conditions as the approximations do not pass through the nodal parameter values. As a consequence, the imposition of essential boundary conditions in meshless methods is quite awkward. In this paper, a displacement constraint equations method (DCEM) is proposed for the imposition of the essential boundary conditions, in which the essential boundary conditions is treated as a constraint to the discrete equations obtained from the Galerkin methods. Instead of using the methods of Lagrange multipliers and the penalty method, a procedure is proposed in which unknowns are partitioned into two subvectors, one consisting of unknowns on boundary u, and one consisting of the remaining unknowns. A simpliÿed displacement constraint equations method (SDCEM) is also proposed, which results in a e cient scheme with su cient accuracy for the imposition of the essential boundary conditions in meshless methods. The present method results in a symmetric, positive and banded sti ness matrix. Numerical results show that the accuracy of the present method is higher than that of the modiÿed variational principles. The present method is a exact method for imposing essential boundary conditions in meshless methods, and can be used in Galerkin-based meshless method, such as element-free Galerkin methods, reproducing kernel particle method, meshless local Petrov-Galerkin method.

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Section: Essential Boundary Conditionsmentioning

confidence: 76%

Section: Essential Boundary Conditionsmentioning

confidence: 99%

Section: Essential Boundary Conditionsmentioning

confidence: 99%

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Imposition of essential boundary conditions by displacement constraint equations in meshless methods

Zhang

Liu

et al. 2001

Commun. Numer. Meth. Engng.

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“…In a general framework, C 1 continuity on the primary variable is required when second-order derivatives of the primary variable appear in the variational statement. For example, in the nucleation of a solid-solid phase transformation based on an energy functional that is dependent on the strain and strain gradients [35] or in strain-gradient damage models [5], C 1 -continuous trial functions are required in the Galerkin implementation.…”

Section: Natural Neighbour Interpolantmentioning

confidence: 99%

Overview and recent advances in natural neighbour galerkin methods

Cueto

Sukumar

Calvo

et al. 2003

ARCO

133

119

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SummaryIn this paper, a survey of the most relevant advances in natural neighbour Galerkin methods is presented. In these methods (also known as natural element methods, NEM), the Sibson and the Laplace (non-Sibsonian) interpolation schemes are used as trial and test functions in a Galerkin procedure. Natural neighbour-based methods have certain unique features among the wide family of so-called meshless methods: a well-defined and robust approximation with no user-defined parameters on non-uniform grids, and the ability to exactly impose essential (Dirichlet) boundary conditions are particularly noteworthy. A comprehensive review of the method is conducted, including a description of the Sibson and the Laplace interpolants in two-and three-dimensions. Application of the NEM to linear and non-linear problems in solid as well as fluid mechanics is studied. Other issues that are pertinent to the vast majority of meshless methods, such as numerical quadrature, imposing essential boundary conditions, and the handling of secondary variables are also addressed. The paper is concluded with some benchmark computations that demonstrate the accuracy and the key advantages of this numerical method.

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“…[9] use a strain-based finite-element method together with nucleation theory (based on linear elasticity for evaluation a perturbed Lagrangian algorithm to study nucleation of a of the self-energy associated with nucleation and on the dilatational phase transformation for materials governed by assumption of a sharp interface with a constant interfacial a Landau-Ginzburg potential. Chu and Moran [10] use a disfree energy). The classical nucleation model [1][2][3][4] is known placement-based element-free Galerkin method, [11,12] in conto accurately represent heterogeneous nucleation behavior junction with a Landau-Ginzburg model in two dimensions, in the vicinity of equilibrium.…”

Section: Introductionmentioning

confidence: 99%

A model for nonclassical nucleation of solid-solid structural phase transformations

Chu

Moran²,

Reid³

et al. 2000

Metall Mater Trans A

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A new model for homogeneous nucleation of structural phase transformations, which can span the range of nucleation from classical to nonclassical, is presented. This model is extended from the classical nucleation theory by introducing driving-force dependencies into the interfacial free energy, the misfit strain energy, and the nucleus chemical free-energy change in order to capture the nonclassical nucleation phenomena. The driving-force dependencies are determined by matching the asymptotic solutions of the new model for the nucleus size and the nucleation energy barrier to the corresponding asymptotic solutions of the Landau-Ginzburg model for nucleation of solid-state phase transformations in the vicinity of lattice instability. Thus, no additional material parameters other than those of the classical nucleation theory and the Landau-Ginzburg model are required, and nonclassical nucleation behavior can be easily predicted based on the well-developed analytical solutions of the classical nucleation model. A comparison of the new model to the Landau-Ginzburg model for homogeneous nucleation of a dilatational transformation is presented as a benchmark example. An application to homogeneous nucleation of a cubic-to-tetragonal transformation is presented to illustrate the capability of this model. The nonclassical homogeneous nucleation behavior of the experimentally studied fcc → bcc transformation in the Fe-Co system is examined by the new model, which predicts a 20 pct reduction in the critical driving force for homogeneous nucleation.

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A computational model for nucleation of solid-solid phase transformations

Cited by 29 publications

References 16 publications

Imposition of essential boundary conditions by displacement constraint equations in meshless methods

Imposition of essential boundary conditions by displacement constraint equations in meshless methods

Overview and recent advances in natural neighbour galerkin methods

A model for nonclassical nucleation of solid-solid structural phase transformations

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