Relevant bibliographies by topics / Phase-field model

Contents

  1. Journal articles
  2. Dissertations / Theses
  3. Books
  4. Book chapters
  5. Conference papers
  6. Reports

Academic literature on the topic 'Phase-field model'

Author: Grafiati
Published: 4 June 2021
Last updated: 6 September 2023

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Journal articles on the topic "Phase-field model"

1

Marconi, Umberto Marini Bettolo, Andrea Crisanti, and Giulia Iori. "Soluble phase field model." Physical Review E 56, no. 1 (July 1, 1997): 77–87. http://dx.doi.org/10.1103/physreve.56.77.

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Liu, Honghu. "Phase transitions of a phase field model." Discrete & Continuous Dynamical Systems - B 16, no. 3 (2011): 883–94. http://dx.doi.org/10.3934/dcdsb.2011.16.883.

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KATAYAMA, Yuta, Tomohiro TAKAKI, and Junji KATO. "123 Modified multi-phase-field topology optimization model." Proceedings of The Computational Mechanics Conference 2015.28 (2015): _123–1_—_123–2_. http://dx.doi.org/10.1299/jsmecmd.2015.28._123-1_.

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Fan, Ling, Walter Werner, Swen Subotić, Daniel Schneider, Manuel Hinterstein, and Britta Nestler. "Multigrain phase-field simulation in ferroelectrics with phase coexistences: An improved phase-field model." Computational Materials Science 203 (February 2022): 111056. http://dx.doi.org/10.1016/j.commatsci.2021.111056.

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Chen, Xinfu, G. Caginalp, and Christof Eck. "A rapidly converging phase field model." Discrete & Continuous Dynamical Systems - A 15, no. 4 (2006): 1017–34. http://dx.doi.org/10.3934/dcds.2006.15.1017.

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Wu, Pingping, and Yongfeng Liang. "Lattice Phase Field Model for Nanomaterials." Materials 14, no. 23 (November 29, 2021): 7317. http://dx.doi.org/10.3390/ma14237317.

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The lattice phase field model is developed to simulate microstructures of nanoscale materials. The grid spacing in simulation is rescaled and restricted to the lattice parameter of real materials. Two possible approaches are used to solve the phase field equations at the length scale of lattice parameter. Examples for lattice phase field modeling of complex nanostructures are presented to demonstrate the potential and capability of this model, including ferroelectric superlattice structure, ferromagnetic composites, and the grain growth process under stress. Advantages, disadvantages, and future directions with this phase field model are discussed briefly.
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Kim, Seong Gyoon, Won Tae Kim, and Toshio Suzuki. "Phase-field model for binary alloys." Physical Review E 60, no. 6 (December 1, 1999): 7186–97. http://dx.doi.org/10.1103/physreve.60.7186.

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Karma, Alain. "Phase-field model of eutectic growth." Physical Review E 49, no. 3 (March 1, 1994): 2245–50. http://dx.doi.org/10.1103/physreve.49.2245.

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Antanovskii, Leonid K. "A phase field model of capillarity." Physics of Fluids 7, no. 4 (April 1995): 747–53. http://dx.doi.org/10.1063/1.868598.

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Shen, C., and Y. Wang. "Phase field model of dislocation networks." Acta Materialia 51, no. 9 (May 2003): 2595–610. http://dx.doi.org/10.1016/s1359-6454(03)00058-2.

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Dissertations / Theses on the topic "Phase-field model"

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Agrawal, Vaibhav. "Multiscale Phase-field Model for Phase Transformation and Fracture." Research Showcase @ CMU, 2016. http://repository.cmu.edu/dissertations/850.

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We address two problems in this thesis. First, a phase-field model for structural phase transformations in solids and second, a model for dynamic fracture. The existing approaches for both phase transformations and fracture can be grouped into two categories. Sharp-interface models, where interfaces are singular surfaces; and regularized-interface models, such as phase-field models, where interfaces are smeared out. The former are challenging for numerical solutions because the interfaces or crack needs to be explicitly tracked, but have the advantage that the kinetics of existing interfaces or cracks and the nucleation of new interfaces can be transparently and precisely prescribed. The diffused interface models such as phasefield models do not require explicit tracking of interfaces and makes them computationally attractive. However, the specification of kinetics and nucleation is both restrictive and extremely opaque in such models. This prevents straightforward calibration of phase-field models to experiment and/or molecular simulations, and breaks the multiscale hierarchy of passing information from atomic to continuum. Consequently, phase-field models cannot be confidently used in dynamic settings. We present a model which has all the advantages of existing phase-field models but also allows us to prescribe kinetics and nucleation criteria. We present a number of examples to characterize and demonstrate the features of the model. We also extend it to the case of multiple phases where preserving kinetics of each kind of interface is more complex. We use the phase transformation model with certain changes to model dynamic fracture. We achieve the advantage of prescribing nucleation and kinetics independent of each other. We demonstrate examples of anisotropic crack propagation and crack propagation on an interface in a composite material. We also report some limitations of phase-field models for fracture which have not been mentioned in the existing literature. These limitations include dependence of effective crack width and hence the effective surface energy on the crack speed, lack of a reasonable approximation for the mechanical response of cracked region and inability to model large deformations.
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She, Minggang. "Phase field model for precipitates in crystals." Thesis, Massachusetts Institute of Technology, 2008. http://hdl.handle.net/1721.1/46020.

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Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Chemical Engineering, 2008.
Includes bibliographical references (p. 261-270).
Oxygen precipitate caused by oxygen supersaturation is the most common and important defects in Czochralski (CZ) silicon. The presence of oxygen precipitate in silicon wafer has both harmful and beneficial effects on the microelectronic device production. Oxygen precipitates are useful for gathering metallic contaminants away from the device regions and for increasing the mechanical strength of the wafer [Borghesi, 1995], but they also can destroy the electrical and mechanical characteristics of the semiconductor and microelectronic devices [Abe, 1985; Kolbesen, 1985]. The understanding of the mechanism of the formation and growth of the oxygen precipitates in CZ silicon is a key to improve the quality of silicon wafer. The goal of this thesis is to provide a full understanding of the growth of an isolated oxygen precipitate in CZ silicon and its morphological evolution by means of phase-field method, and to gain the insight of the morphological transition of the oxygen precipitate and the distribution of oxygen, vacancy, and self-interstitial around the single oxygen precipitate. The traditional approach to simulate multiphase system is the sharp interface model. Sharp interface model requires tracking the interface between phases, which make the simulation much difficult and complicate. Phase-field model offers an alternative approach for predicting mesoscale morphological and microstructure evolution of inhomogeneous multiphase system. The most significant computational advantage of a phase-field model is that explicit tracking of the interface is unnecessary. In this thesis, the phase-field model is applied to simulate the evolution of oxygen precipitates in CZ silicon. A phase-field model for a two-component inhomogeneous system was first derived to set up the framework of phase-field method and a dynamically adaptive finite element method also was built to specifically solve phase-field equations. This model was used to investigate the effects of interfacial and elastic properties on the growth of a single precipitate, coarsening of two precipitates, and competitive growth of multiple precipitates. For an isolated precipitate growth, both elastic energy and interfacial energy affect the precipitate morphological evolution.
(cont.) Numerical results show the shape of the precipitate is determined by the relative contributions of elastic energy and interfacial energy, the degree of elastic anisotropy, and the degree of interfacial anisotropy. A dimensionless length scale LS3 was defined to represent the relative contributions of the interfacial energy and elastic energy. For large LS3 (LS3 > 5), the anisotropic elasticity plays a dominant role and precipitate evolves to held the elastic anisotropy even if the interfacial anisotropy is very strong. However, if LS3 ~1 or elasticity is isotropic, the strong anisotropy ([epsilon]4 =/> 0.05 ) of the interface will be the dominant factor to determine the precipitate shape. The growth rate of an isolated precipitate follows the diffusion-controlled power law. The elasticity significantly decreases the precipitate growth rate, while the anisotropy of the interface does not. Coarsening of two precipitates was also explored with different interfacial and elastic properties. The results also show that both elasticity and interfacial anisotropy enhance the coarsening rate. For competitive growth of multiple precipitates, a gap was found to be developed between the precipitates because of the precipitate screening, but this gap could be destroyed by increasing the interfacial energy or introducing elastic energy. Based on the framework of the previous phase-field model, another phase-field model coupling CALPHAD thermodynamic assessment was developed to simulate the growth of the oxygen precipitate in CZ isilicon. An asymptotic analysis was performed to understand the phase-field model at the sharp interface limit and all physical principles of the solid precipitate growth problem were recovered. a Cristobalite and amorphous oxygen precipitates were calculated at different orientations and temperatures. Disk-like shape, square, ellipse, a slightly deformed sphere are reproduced for oxygen precipitates, which agrees with the experimental observations very well. In addition, the growth rates of amorphous precipitates and a cristobalite precipitates at different temperatures show that at high temperature 1100 °C, amorphous precipitate has the largest growth rate, while at low temperature 900 °C, a cristobalite precipitate grows faster.
(cont.) This qualitatively explained why different polymorphs and shapes of the oxygen precipitate were observed in experiments at different annealing temperatures.
by Minggang She.
Ph.D.
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Renuka, Balakrishna Ananya. "Application of a phase-field model to ferroelectrics." Thesis, University of Oxford, 2016. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.728788.

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Choudhury, Abhik Narayan [Verfasser]. "Quantitative phase-field model for phase transformations in multi-component alloys / Abhik Narayan Choudhury." Karlsruhe : KIT Scientific Publishing, 2013. http://www.ksp.kit.edu.

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Eiken, Janin [Verfasser]. "A Phase-Field Model for Technical Alloy Solidification / Janin Eiken." Aachen : Shaker, 2010. http://d-nb.info/1124364226/34.

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Ahmad, Noor Atinah. "Phase-field model of rapid solidification of a binary alloy." Thesis, University of Southampton, 1997. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.242477.

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Koslowski, Marisol Ortiz Michael. "A phase-field model of dislocations in ductile single crystals /." Diss., Pasadena, Calif. : California Institute of Technology, 2003. http://resolver.caltech.edu/CaltechETD:etd-05302003-094155.

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Xu, Ying. "TWO-DIMENSIONAL SIMULATION OF SOLIDIFICATION IN FLOW FIELD USING PHASE-FIELD MODEL|MULTISCALE METHOD IMPLEMENTATION." Lexington, Ky. : [University of Kentucky Libraries], 2006. http://lib.uky.edu/ETD/ukymeen2006d00524/YingXu_Dissertation_2006.pdf.

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Thesis (Ph. D.)--University of Kentucky, 2006.
Title from document title page (viewed on January 25, 2007). Document formatted into pages; contains: xiii, 162 p. : ill. (some col.). Includes abstract and vita. Includes bibliographical references (p. 151-157).
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Baba, Karim Sidi. "Adaptive finite element computations of a double obstacle phase field model." Thesis, University of Sussex, 1998. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.244349.

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Ronquillo, David Carlos. "Magnetic-Field-Driven Quantum Phase Transitions of the Kitaev Honeycomb Model." The Ohio State University, 2020. http://rave.ohiolink.edu/etdc/view?acc_num=osu1587035230123328.

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Books on the topic "Phase-field model"

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B, McFadden Geoffrey, Wheeler A. A, and National Institute of Standards and Technology (U.S.), eds. A phase-field model with convection: Numerical simulations. [Gaithersburg, MD]: U.S. Dept. of Commerce, Technology Administration, National Institute of Standards and Technology, 2000.

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B, McFadden Geoffrey, Wheeler A. A, and National Institute of Standards and Technology (U.S.), eds. A phase-field model of solidification with convection. Gaithersburg, MD: U.S. Dept. of Commerce, Technology Administration, National Institute of Standards and Technology, 1998.

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B, McFadden Geoffrey, Wheeler A. A, and National Institute of Standards and Technology (U.S.), eds. A phase-field model of solidification with convection. Gaithersburg, MD: U.S. Dept. of Commerce, Technology Administration, National Institute of Standards and Technology, 1998.

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B, McFadden Geoffrey, Wheeler A. A, and National Institute of Standards and Technology (U.S.), eds. A phase-field model of solidification with convection. Gaithersburg, MD: U.S. Dept. of Commerce, Technology Administration, National Institute of Standards and Technology, 1998.

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B, McFadden Geoffrey, Wheeler A. A, and National Institute of Standards and Technology (U.S.), eds. A phase-field model with convection: Numerical simulations. [Gaithersburg, MD]: U.S. Dept. of Commerce, Technology Administration, National Institute of Standards and Technology, 2000.

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B, McFadden Geoffrey, Wheeler A. A, and National Institute of Standards and Technology (U.S.), eds. A phase-field model of solidification with convection. Gaithersburg, MD: U.S. Dept. of Commerce, Technology Administration, National Institute of Standards and Technology, 1998.

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B, McFadden Geoffrey, Wheeler A. A, and National Institute of Standards and Technology (U.S.), eds. A phase-field model with convection: Numerical simulations. [Gaithersburg, MD]: U.S. Dept. of Commerce, Technology Administration, National Institute of Standards and Technology, 2000.

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B, McFadden Geoffrey, Wheeler A. A, and National Institute of Standards and Technology (U.S.), eds. A phase-field model with convection: Numerical simulations. [Gaithersburg, MD]: U.S. Dept. of Commerce, Technology Administration, National Institute of Standards and Technology, 2000.

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B, McFadden Geoffrey, Wheeler A. A, and National Institute of Standards and Technology (U.S.), eds. A phase-field model of solidification with convection. Gaithersburg, MD: U.S. Dept. of Commerce, Technology Administration, National Institute of Standards and Technology, 1998.

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B, McFadden Geoffrey, Wheeler A. A, and National Institute of Standards and Technology (U.S.), eds. A phase-field model of solidification with convection. Gaithersburg, MD: U.S. Dept. of Commerce, Technology Administration, National Institute of Standards and Technology, 1998.

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Book chapters on the topic "Phase-field model"

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Plapp, Mathis. "Phase-Field Models." In Multiphase Microfluidics: The Diffuse Interface Model, 129–75. Vienna: Springer Vienna, 2012. http://dx.doi.org/10.1007/978-3-7091-1227-4_4.

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Steinbach, Ingo, and Hesham Salama. "Quantum Phase Field." In Lectures on Phase Field, 79–90. Cham: Springer Nature Switzerland, 2023. http://dx.doi.org/10.1007/978-3-031-21171-3_8.

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AbstractIn this chapter a new paradigm for the understanding of the physical world is developed based on phase-field type non-linear wave solutions. Two 1-dimensional wave fronts, right- and left-moving are combined to define the doubloon, the basic element of mass and space. The interface energy of a standard phase field defines the mass of elementary particles. The non-zero values of the phase field, the core of the doubloon, defines space in an introverted view. The sum constraint of the multi-phase-field model defines a doubloon network which is closed in itself. The model can be applied to investigate scale formation in the universe.
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Steinbach, Ingo, and Hesham Salama. "Analytics." In Lectures on Phase Field, 17–29. Cham: Springer Nature Switzerland, 2023. http://dx.doi.org/10.1007/978-3-031-21171-3_2.

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AbstractThe chapter reviews the basic analytic solutions of phase-field models at the mesoscopic scale. First the phase-field equation is derived by the Clausius-Duhem relation applied to a simple phase-field functional. The concept of variational derivative, related to the gradient contribution in the phase-field functional, is introduced. The traveling wave solution, or solitonian wave solution, for the phase-field equation in 1-dimension is derived for two relevant potential functions. Finally, the relations between model parameters and physically defined material parameters are derived.
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Steinbach, Ingo, and Hesham Salama. "Stress–Strain and Fluid Flow." In Lectures on Phase Field, 69–77. Cham: Springer Nature Switzerland, 2023. http://dx.doi.org/10.1007/978-3-031-21171-3_7.

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AbstractIn this chapter first a multi-phase-field model considering transformation strain and elastic energy is developed. It utilizes the expansion into multiple phases of the multi-phase-field model. In particular, the treatment of the diffuse interface region as an effective medium is discussed in the context of homogenization theory. The model is applied to martensitic transformation within finite strain framework. In the second part of the lecture coupling for solute transport by melt flow is discussed. The model is applied to equiaxed dendritic solidification of MgAl in a shear flow.
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Steinbach, Ingo, and Hesham Salama. "Concentration." In Lectures on Phase Field, 49–59. Cham: Springer Nature Switzerland, 2023. http://dx.doi.org/10.1007/978-3-031-21171-3_5.

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AbstractIn this chapter the coupling to solute concentration in an alloy is reviewed. During transformation the concentration between the phases must be redistributed, causing solute diffusion. First different approached to handle this problem are discussed comparatively. Special emphasis is given to the problem of scale in a mesoscopic phase field model. In detail the currently most advanced model, the so-called finite interface dissipation model is explained in detail. The chapter concludes with the discussion of the effect of multi-component diffusion and its handling in a phase-field model. As an example, solidification under additive manufacturing condition is presented with special emphasis of the microsegregation after solidification in a multi-component alloy.
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Steinbach, Ingo, and Hesham Salama. "Multi-Phase-Field Approach." In Lectures on Phase Field, 61–68. Cham: Springer Nature Switzerland, 2023. http://dx.doi.org/10.1007/978-3-031-21171-3_6.

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AbstractIn this chapter the extension of a phase-field model for two phases to multiple phases is presented. This relates to the treatment of triple lines and junctions between several phases, or grains in a multicrystalline structure. The conservation constraint of the sum of all fields in one material point is realized using a Lagrange formalism. The free energy functional is expanded in pairs of phases, as well as the equation of motion of individual phase fields in dependence on all other fields. As example coarsening and texture evolution in a multi grain structure with anisotropic interface energy is presented.
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Kränkel, Mirko, and Dietmar Kröner. "A Phase-Field Model for Flows with Phase Transition." In Theory, Numerics and Applications of Hyperbolic Problems II, 243–54. Cham: Springer International Publishing, 2018. http://dx.doi.org/10.1007/978-3-319-91548-7_19.

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Steinbach, Ingo, and Hesham Salama. "Capillarity." In Lectures on Phase Field, 31–39. Cham: Springer Nature Switzerland, 2023. http://dx.doi.org/10.1007/978-3-031-21171-3_3.

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AbstractThe chapter reviews the basics of the effect of capillarity, i.e. the influence of interface energy on microstructure evolution in materials. This will be done from a phenomenological aspect on the one hand and related to the representation of capillarity in a phase-field model on the other hand. The expression for curvature related to the gradient of the phase field is derived as well as the expression for Herring torque at an interface with anisotropic interface energy
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Provatas, Nikolas, Tatu Pinomaa, and Nana Ofori-Opoku. "Special Cases of the Grand Potential Phase Field Model." In Quantitative Phase Field Modelling of Solidification, 51–80. Boca Raton: CRC Press, 2021. http://dx.doi.org/10.1201/9781003204312-9.

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Steinbach, Ingo, and Hesham Salama. "Temperature." In Lectures on Phase Field, 41–47. Cham: Springer Nature Switzerland, 2023. http://dx.doi.org/10.1007/978-3-031-21171-3_4.

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AbstractIn this chapter the coupling of the phase field to a second, external field is discussed for the case of non-isothermal systems. A thermodynamic consistent derivation of the set of equations for the phase-field and temperature field is presented, starting from an entropy functional. In the second part of the chapter the systematic error, which arises in a phase-field model at the mesoscopic scale, is discussed. Finally, the so-called thin interface limit is presented, which is an elegant remedy of this systematic error by an asymptotic matching condition. The example in this section relates to solidification under additive manufacturing conditions.
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Conference papers on the topic "Phase-field model"

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Anderson, D., G. McFadden, and A. Wheeler. "A phase-field model of convection with solidification." In 40th AIAA Aerospace Sciences Meeting & Exhibit. Reston, Virigina: American Institute of Aeronautics and Astronautics, 2002. http://dx.doi.org/10.2514/6.2002-891.

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Yan, S., and R. Müller. "An Efficient Phase Field Model for Fatigue Fracture." In 15th World Congress on Computational Mechanics (WCCM-XV) and 8th Asian Pacific Congress on Computational Mechanics (APCOM-VIII). CIMNE, 2022. http://dx.doi.org/10.23967/wccm-apcom.2022.018.

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Talamini, Brandon, Andrew Stershic, and Michael Tupek. "A variational phase-field model of ductile fracture." In Proposed for presentation at the 16th U.S. National Congress on Computational Mechanics held July 25-29, 2021 in virtual,. US DOE, 2021. http://dx.doi.org/10.2172/1884174.

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Petrini, Ana Luísa, José Luiz Boldrini, Carlos Lamarca Carvalho Sousa Esteves, and Marco Bittencourt. "Phase field tensor model for damage induced anisotropy." In 8th International Symposium on Solid Mechanics. ABCM, 2022. http://dx.doi.org/10.26678/abcm.mecsol2022.msl22-0094.

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Witterstein, G., Kamel Ariffin Mohd Atan, and Isthrinayagy S. Krishnarajah. "A Phase Field Model for Stem Cell Differentiation." In INTERNATIONAL CONFERENCE ON MATHEMATICAL BIOLOGY 2007: ICMB07. AIP, 2008. http://dx.doi.org/10.1063/1.2883870.

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Sondershaus, R., and R. Müller. "Phase field model for simulating fracture of ice." In 8th European Congress on Computational Methods in Applied Sciences and Engineering. CIMNE, 2022. http://dx.doi.org/10.23967/eccomas.2022.219.

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Popov, Dmitri I. "Dynamics of eutectic microstructure during phase nucleation and phase termination: phase-field model computer simulations." In Fifth International Workshop on Nondestructive Testing and Computer Simulations in Science and Engineering, edited by Alexander I. Melker. SPIE, 2002. http://dx.doi.org/10.1117/12.456253.

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Tapia Armenta, Juan, and Gilberto Mariscal. "Equidistribution algorithm for a two-dimensional phase field model." In 2006 Seventh Mexican International Conference on Computer Science. IEEE, 2006. http://dx.doi.org/10.1109/enc.2006.13.

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Ying Dong and Jim Ji. "Phase unwrapping using region-based Markov Random Field model." In 2010 32nd Annual International Conference of the IEEE Engineering in Medicine and Biology Society (EMBC 2010). IEEE, 2010. http://dx.doi.org/10.1109/iembs.2010.5627494.

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Vodička, Roman. "A computational model of interface and phase-field fracture." In FRACTURE AND DAMAGE MECHANICS: Theory, Simulation and Experiment. AIP Publishing, 2020. http://dx.doi.org/10.1063/5.0033946.

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Reports on the topic "Phase-field model"

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Li, Yulan, Shenyang Y. Hu, Ke Xu, Jonathan D. Suter, John S. McCloy, Bradley R. Johnson, and Pradeep Ramuhalli. Preliminary Phase Field Computational Model Development. Office of Scientific and Technical Information (OSTI), December 2014. http://dx.doi.org/10.2172/1177715.

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Anderson, D. M., G. B. McFadden, and A. A. Wheeler. A phase-field model with convection:. Gaithersburg, MD: National Institute of Standards and Technology, 2000. http://dx.doi.org/10.6028/nist.ir.6442.

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Anderson, D. M., G. B. McFadden, and A. A. Wheeler. A phase-field model with convection:. Gaithersburg, MD: National Institute of Standards and Technology, 2000. http://dx.doi.org/10.6028/nist.ir.6568.

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Wheeler, A. A., W. J. Boettinger, and G. B. McFadden. A phase-field model for isothermal phase transitions in binary alloys. Gaithersburg, MD: National Institute of Standards and Technology, 1991. http://dx.doi.org/10.6028/nist.ir.4662.

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Wheeler, A. A., B. T. Murray, and R. J. Schaefer. Computation of dendrites using a phase field model. Gaithersburg, MD: National Institute of Standards and Technology, 1992. http://dx.doi.org/10.6028/nist.ir.4894.

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Anderson, D. M., G. B. McFadden, and A. A. Wheeler. A phase-field model of solidification with convection. Gaithersburg, MD: National Institute of Standards and Technology, 1998. http://dx.doi.org/10.6028/nist.ir.6237.

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Aagesen, Larry Kenneth, and Daniel Schwen. MARMOT Phase-Field Model for the U-Si System. Office of Scientific and Technical Information (OSTI), September 2016. http://dx.doi.org/10.2172/1389724.

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McFadden, G. B., J. J. Eggleston, and P. W. Voorhees. A phase-field model for high anisotropic interfacial energy. Gaithersburg, MD: National Institute of Standards and Technology, 2001. http://dx.doi.org/10.6028/nist.ir.6706.

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Tikare, V., D. Fan, S. J. Plimpton, and R. M. Fye. Massively Parallel Methods for Simulating the Phase-Field Model. Office of Scientific and Technical Information (OSTI), December 2000. http://dx.doi.org/10.2172/773880.

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10

Wheeler, A. A., W. J. Boettinger, and G. B. McFadden. A phase-field model of solute trapping during solidification. Gaithersburg, MD: National Institute of Standards and Technology, 1992. http://dx.doi.org/10.6028/nist.ir.4922.

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