Diffusional evolution of precipitates in elastic media using the extended finite element and the level set methods

Duddu, Ravindra; Chopp, David L.; Voorhees, Peter W.; Moran, Brian

doi:10.1016/j.jcp.2010.11.002

Cited by 40 publications

(25 citation statements)

References 53 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…However, most of these methods typically require tools not frequently available in standard finite element and finite difference software packages. Examples of such approaches include the extended and composite finite element methods (e.g., [31,12,23,13,32,55,7,4]), immersed interface methods (e.g., [40,43,60,44,65]), virtual node methods with embedded boundary conditions (e.g., [3,73,34]), matched interface and boundary methods (e.g., [71,68,69,67,72]), modified finite volume/embedded boundary/cut-cell methods/ghost-fluid methods (e.g., [27,36,19,25,26,35,47,70,48,37,46,64,49,9,10,52,53,33,63]). In another approach, known as the fictitious domain method (e.g., [28,29,56,45]), the original system is either augmented with equations for Lagrange multipliers to enforce the boundary conditions, or the penalty method is used to enforce the boundary condi-tions weakly.…”

Section: Introductionmentioning

confidence: 99%

Analysis of the diffuse-domain method for solving PDEs in complex geometries

Lervåg

Lowengrub

2015

Communications in Mathematical Sciences

View full text Add to dashboard Cite

Abstract. In recent work, Li et al. (Comm. Math. Sci., 7:81-107, 2009) developed a diffusedomain method (DDM) for solving partial differential equations in complex, dynamic geometries with Dirichlet, Neumann, and Robin boundary conditions. The diffuse-domain method uses an implicit representation of the geometry where the sharp boundary is replaced by a diffuse layer with thickness that is typically proportional to the minimum grid size. The original equations are reformulated on a larger regular domain and the boundary conditions are incorporated via singular source terms. The resulting equations can be solved with standard finite difference and finite element software packages.Here, we present a matched asymptotic analysis of general diffuse-domain methods for Neumann and Robin boundary conditions. Our analysis shows that for certain choices of the boundary condition approximations, the DDM is second-order accurate in . However, for other choices the DDM is only first-order accurate. This helps to explain why the choice of boundary-condition approximation is important for rapid global convergence and high accuracy. Our analysis also suggests correction terms that may be added to yield more accurate diffuse-domain methods. Simple modifications of first-order boundary condition approximations are proposed to achieve asymptotically second-order accurate schemes. Our analytic results are confirmed numerically in the L 2 and L ∞ norms for selected test problems.

show abstract

Section: Introductionmentioning

confidence: 99%

Analysis of the diffuse-domain method for solving PDEs in complex geometries

Lervåg

Lowengrub

2015

Communications in Mathematical Sciences

View full text Add to dashboard Cite

show abstract

“…For instance, particle merging and splitting phenomena, which are common for particles during the phase transformations, cannot be captured easily in those BIM-based approaches. Recently, Duddu et al (Duddu et al, 2011) revisited this problem based on the same mathematical model using a combined extended finite element method (XFEM) and level set method (LSM), in which the geometry of interface was implicitly defined by the level set function so that topological changes could be easily handled.…”

Section: Introductionmentioning

confidence: 99%

Equilibrium morphology of misfit particles in elastically stressed solids under chemo-mechanical equilibrium conditions

Zhao

Bordas

2015

Journal of the Mechanics and Physics of Solids

View full text Add to dashboard Cite

This paper studies the effects of chemical, elastic and interfacial energies on the equilibrium morphology of misfit particles due to phase separation in binary alloys under chemo-mechanical equilibrium conditions. A continuum framework that governs the chemo-mechanical equilibrium of the system is first developed using a variational approach by treating the phase interface as a sharp interface endowed with interfacial excess energy. An extended finite element method (XFEM) in conjunction with the level set method is then developed to simulate the behaviors of the coupled chemo-mechanical system. The coupled chemo-mechanics model together with the 1 xzhao99@gmail.com 2 stephane.bordas@uni.lu 3 Corresponding author, j-qu@northwestern.edu (Jianmin Qu) 2 numerical techniques developed here provides an efficient simulation tool to predict the equilibrium morphologies of precipitates in phase separate alloys.

show abstract

“…We employ a sharp interface model based on a hybrid extended finite element and level set method that naturally allows complex topological changes during the morphological evolution without requiring remeshing. The level set method, originally devised for tracking moving interfaces (Osher and Sethian, 1988), is capable of describing complicated geometrical interfaces of microstructures, such as dendritic solidification (Chen et al, 1997;Zabaras et al, 2006), precipitate evolution (Duddu et al, 2011) and biofilm growth (Duddu et al, 2008;Duddu et al, 2009).. The combined method enables the handling of complex morphological transformations of multiple misfitting inhomogeneities.…”

Section: Introductionmentioning

confidence: 99%

Effects of elastic strain energy and interfacial stress on the equilibrium morphology of misfit particles in heterogeneous solids

Zhao

Duddu

Bordas

et al. 2013

Journal of the Mechanics and Physics of Solids

Self Cite

View full text Add to dashboard Cite

This paper presents an efficient sharp interface model to study the morphological transformations of misfit particles in phase separated alloys. Both the elastic anisotropy and interfacial energy are considered. The geometry of the material interface is implicitly described by the level set method so that the complex morphological transformation of microstructures can be accurately captured. A smoothed extended finite element method is adopted to evaluate the elastic field without requiring remeshing. The equilibrium morphologies of particles are shown to depend on the elastic anisotropy, interfacial energy as well as the particle size. Various morphological transformations, such as shape changes from spheres to cuboids, directional aligned platelets and particle splitting, are observed. The simulated results are in good agreement with experimental observations. The proposed model provides a useful tool in understanding the morphological transformation of precipitates, which will facilitate the analysis and design of metallic alloys.

show abstract

Diffusional evolution of precipitates in elastic media using the extended finite element and the level set methods

Cited by 40 publications

References 53 publications

Analysis of the diffuse-domain method for solving PDEs in complex geometries

Analysis of the diffuse-domain method for solving PDEs in complex geometries

Equilibrium morphology of misfit particles in elastically stressed solids under chemo-mechanical equilibrium conditions

Effects of elastic strain energy and interfacial stress on the equilibrium morphology of misfit particles in heterogeneous solids

Contact Info

Product

Resources

About