Moving Boundaries in Material Science

Transcription

1 Moving Boundaries in Material Science Michal Beneš Department of Mathematics Faculty of Nuclear Sciences and Physical Engineering Czech Technical University in Prague Mathematical Modelling Group Workshop of Center of Excellence Advanced Engineering Materials under Complex Loading Faculty of Nuclear Sciences and Physical Engineering, Prague, March 13 14, 2014

2 Outline of the presentation Department activity Collaboration Results: microstructure formation in materials dislocation dynamics surface diffusion high-performance computing Conclusion Motivation: The presentation covers the activities of the Department of Mathematics in the domain of material science. Outline 2

3 Department of Mathematics Education basic math courses for the Faculty interdisciplinary program Applications in Natural Sciences: bachelor study master study Ph.D. study major fields in: Mathematical Engineering - B.Sc., M.Sc., Ph.D. Mathematical Informatics - B.Sc., M.Sc. Applied Informatics - B.Sc. Research applications of algebra, functional analysis and geometry in mathematical physics and quantum theory scientific computing, mathematical modelling and numerical simulation of complex phenomena in high-tech design, environment protection and computer science applying algebraic number theory and discrete mathematics in symbolical dynamical systems analysis of the microscopic structure of transportation flows and modelling agent systems statistical processing of general monitoring signals with applications in acoustical defectoscopy of materials Math Department 3

4 Scientific computing and mathematical modelling Research interests The activities of the group cover mainly the development of numerical methods for solving complex problems in continuum mechanics, environment and advanced technology, and the development of cutting-edge methods in information technology: multi-phase and multi-component fluid flow transport processes accompanying the flow through porous media first-order phase transitions in condensed matter, the growth of microstructures numerical methods for free-boundary problems diffusion in turbulent flow and transition to turbulence non-linear phenomena and complex dynamics of reaction-diffusion systems edge detection and noise filtering in image processing parallelization of numerical algorithms for nonlinear evolution PDEs mathematical visualization software for mainframe computing on GPU Mathematical Modelling Group 4

5 Scientific computing and mathematical modelling Collaboration Global: Colorado School of Mines, Golden Reservoir Engineering Research Institute Palo Alto, CA Yale University University of California, Los Angeles Clarkson University, Potsdam, NY Kyushu University, Fukuoka, Japan University of Miyazaki, Japan Shibaura University, Japan Meiji University, Tokyo, Japan European: Comenius University, Bratislava, Slovakia Slovak Technical University, Bratislava, Slovakia AGH Krakow, Poland Université de Pierre et Marie Curie, Paris Technische Universität Dresden Universität Bayreuth Albert-Ludwigs Universität Freiburg, Germany Université Paris Sud, Orsay Mathematical Modelling Group 5

6 Scientific computing and mathematical modelling Collaboration Local: Institute of Thermomechanics, Czech Academy of Sciences Institute of Computer Science, Czech Academy of Sciences Faculty of Mathematics and Physics, Charles University in Prague Industrial: Honeywell Inc., U.S.A. (Honeywell Prague Laboratory) CA, Inc. U.S.A. (R and D Center Prague) NVIDIA Corp. Caterpillar Inc., Mossville, U.S.A. Mathematical Modelling Group 6

7 Mathematical Modelling and Numerical Simulation Real problem Physical problem Mathematical problem Numerical method General scheme Numerical algorithm design Advanced implementation techniques High-performance computation Comparison with physical model and real problem behaviour Mathematical Modelling 7

8 Phase-transitions in materials Phase transitions 8

9 Microstructure growth in solidification Dendritic growth - experiment RPI - NASA material: succinonitril Simulation results by the phase-field model Experiment and simulation Mathematical model is based on the system of evolutionary parabolic PDE s (phase-field equations). Numerical algorithm uses the finite-difference/volume method in space and the higher-order solver in time. Numerical simulations are performed on the high-performance systems. Phase transitions 9

10 Stefan problem with surface tension ρc u t b c (u) Ω = 0 u t=0 = u 0 λ u n s λ u n l = Lv Γ = (λ u) v Ω s a Ω l λ s, λ l heat conductivity σ = f(u ) surface tension κ Γ mean curvature of the hypersurface Γ(t) n Γ normal unit vector to Γ(t) pointing out of Ω s v Γ normal velocity of Γ(t) normal unit vector to Ω pointing out of Ω n Ω u u = σ s κ Γ α σ s v Γ Ω s (t) t=0 = Ω so Dirichlet boundary condition b c (u) = u u Ω Neumann boundary condition with the heat flux g on Ω b c (u) = (λ(u) u g) n Ω Phase transitions 10

11 Mean curvature flow Evolution law v Γ = κ Γ + F Notations: Γ closed curve in R 2 hypersurface in R n n Γ normal vector v Γ normal velocity κ Γ mean curvature F forcing term or closed Phase transitions 11

12 Three-dimensional growth Three-dimensional pattern growth Parameters are u = 1.0, L = 2.0, β = 300, a = 2.0, α = 3, L 1 = L 2 = L 3 = 4.0, initial radius = 0.05, N 1 = N 2 = 400, ξ = Anisotropy: g(n) = 1 0.4(n n4 2 + n4 3 ) 3D growth 12

13 Three-dimensional growth Three-dimensional pattern growth Parameters are u = 1.0, L = 2.0, β = 300, a = 2.0, α = 3, L 1 = L 2 = L 3 = 4.0, initial radius = 0.05, N 1 = N 2 = 400, ξ = Anisotropy: g(n) = 1+0.4(n 4 1 +n4 2 +n )+ 0.4(3(n 4 1 +n4 2 +n4 3 )+66n2 1 n2 2 n ) 3D growth 13

14 Three-dimensional growth Three-dimensional pattern growth 3D growth 14

15 Dendritic growth Parameters: u = 1.0, L = 2.0, β = 300, a = 2.0, α = 3, L 1 = L 2 = L 3 = 4.0, initial radius = 0.05, N 1 = N 2 = 400, ξ = ) Anisotropy: Φ 0 (ρn) = ρ (1 0.4(n n4 2 + n4 3 ) 3D growth 15

16 Dynamics of Dislocations in Crystalline Materials

17 Dislocation dynamics microstructural line defects in atomic lattice of crystalline material along which the regular crystallographic arrangement of atoms is disturbed (dislocation lines): closed curves - dipolar loops line ending at the surface of the crystal - dislocation curve line defects slip along slip planes; a slip displacement carried by a single dislocation is called the Burgers vector dipolar loops are rectangles of the size several tens by single units of nanometers (in average 120nm by 8nm for Nickel crystal at room temperature) allowed to slip only along the Burgers vector; size of dislocation curve varies in scale of µm F y b F x 2l F Force interaction curve and dipolar loop z 2h x dipolar-loop geometry Dislocation dynamics 17

18 Dislocation dynamics - references Pauš P., Kratochvl J., Beneš M. A dislocation dynamics analysis of the critical cross-slip annihilation distance and the cyclic saturation stress in fcc single crystals at different temperatures, Acta Materialia, Volume 61, Issue 20, December 2013, Pages , Impact factor Pauš P., Kratochvl J., Beneš M. Mechanisms controlling the cyclic saturation stress and the critical cross-slip annihilation distance in copper single crystals, Philosophical Magazine Letters, DOI: / , Published online: 27 Nov 2013, Impact factor Pauš P., Beneš M., Kratochvl J., Simulation of dislocation annihilation by cross-slip, Acta Physica Polonica A, 122, No. 3 (2012) , Impact factor Minrik V., Beneš M. and Kratochvl J., Simulation of Dynamical Interaction between Dislocations and Dipolar Loops, Journal of Applied Physics 107, (2010), Impact factor: Minárik V., Kratochvíl J., Mikula K. and Beneš M., Numerical simulation of dislocation dynamics, in Numerical Mathematics and Advanced Applications, ENUMATH 2003 (peer reviewed proceedings), pp , eds. Feistauer M., Dolejší V., Knobloch P., Najzar K., Springer Verlag, 2004, ISBN Kratochvíl J., Saxlová M., Sweeping mechanism of dislocation pattern formation, Scripta Metallurgica et Materialia, 26 (1): (1992). Hirth J.P. and Lothe J., Theory of Dislocations, John Wiley Dislocation dynamics 18

19 Dipolar loops 2 types of dipolar loop: vacancy and intersticial, each in 2 stable configurations 4 types of stress fields interaction force per unit-length is given by the Peach-Koehler equation: f i = ε ijk σ jm b m s k, f i σ jm b m s k i-th component of the interaction force per unit length of the dislocation line components of the stress field tensor at the dislocation position components of the Burgers vector components of unit vector s which has direction of the dislocation line Dislocation dynamics 19

20 Dislocation dynamics Γ j (t) : v Γj = T κ Γj + F (t, Γ 1 (t),..., Γ M (t), Λ 1,..., Λ N ) for j = 1,..., M Λ i (t) : dx (j) dt = 1 BP F (j) x, total(t, x, Γ 1 (t),..., Γ M (t), Λ 1,..., Λ N ) for i = 1,..., N 10 dipolar loops of V1 and V2 type clustering together. The graph on the left shows the spatial positions of interacting dislocation and dipolar loops, the graph on the right shows the projection of positions onto the glide plane. Dislocation dynamics 20

21 Dislocation - dipolar loop interaction 10 dipolar loops of V1 and V2 type clustering together. The graph on the left shows the spatial positions of interacting dislocation and dipolar loops, the graph on the right shows the projection of positions onto the glide plane. Dislocation dynamics 21

22 Dislocation - cross-slip Double cross slip, τ app = 30MPa, t (0, 0.324), plane distance D = 48nm, dislocation curve discretized by M = 150 nodes. Dislocation dynamics 22

23 Surface diffusion

24 Matematical model of elastic flow Evolution law of the hypersurface in R n V = Γ (H + F ) on Γ Notation: Γ curve in R 2 or hypersurface in R n n Γ normal vector V is normal velocity of Γ Γ is Laplace-Beltrami operator with respect to Γ H is mean curvature of Γ F forcing term Surface diffusion 24

25 Motivation body-shape dynamics as a result of surfacial processes surface destruction as a result of external stress and vibration computer data processing Terminology mismatch in physical and mathematical theories: R.F. Holub, M. Beneš and B.D. Honeyman. Interfacial Phenomena in Fluid Dynamics: Linking atomistic and macroscopic properties: Can They Explain the Transport Anomalies? in Proceedings of Czech-Japanese Seminar in Applied Mathematics 2004, editors M. Beneš, J. Mikyška, T. Oberhuber, Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague, 2005, ISBN pp References: Hoang Dieu H., Beneš M. and Oberhuber T. Numerical Simulation of Anisotropic Mean Curvature of Graphs in Relative Geometry, Acta Polytechnica Hungarica Vol. 10, No. 7, 2013, Impact factor Beneš M. Numerical Solution for Surface Diffusion on Graphs, In: Bene M., Kimura M. and Nakaki T., Eds. Proceedings of Czech Japanese Seminar in Applied Mathematics 2005, COE Lecture Note Vol.3, Faculty of Mathematics, Kyushu University Fukuoka, October 2006, pp E. Bänsch, P. Morin, R.H. Nochetto. Surface diffusion of graphs: Variational formulation, error analysis, and simulation. SIAM J. Numer. Anal. 42 (2): W.W. Mullins, Theory of Thermal Grooving, J. Appl. Phys. 28(3), , 1957 R.J. Asaro and W.A. Tiller, Interface Morphology Development During Stress Corrosion Cracking: Part I. Via Surface Diffusion, Met. Trans. 3, , 1972 Surface diffusion 25

26 Qualitative computational results by I - part 2 F Ω t N T tol mesh CPU (0,2)x(0,2) x Table of the experiment parameters (2DCir/E1). Surface diffusion 26

27 Qualitative computational results by II - part 1 F Ω t N T tol mesh CPU 0.0 (0,0.5)x(0,0.5) Table of the finest experiment parameters (ELAST/E8.2) made on geraldine. Surface diffusion 27

28 Anisotropic surface diffusion - part 1 Surface diffusion 28

29 Anisotropic surface diffusion - part 2 Surface diffusion 29

30 Anisotropic surface diffusion - part 3 Surface diffusion 30

31 High-performance computing HPC 31

32 Parallel computing Issues: 2D and 3D spatial domains complex temporal dynamics shared-memory computing (OpenMP) distributed-memory computing (MPI) Facilities: AMD Opteron Linux cluster CINECA Bologna supercomputing center in Italy HPC 32

33 GPU computing CUDA Research Center at the Deparment assigned by the NVIDIA Corp. The CUDA Research Center at the Czech Technical University in Prague concentrates mainly on the research and development of new numerical methods for solving partial differential equations in many domains like image processing, medical data processing, material science, combustion and models of atmosphere. Our aim is not only to solve the mentioned problems but also to systematically develop library for easy and efficient numerical computing on the GPU based on CUDA. For this purpose we intensively work on more fundamental problems like efficient sparse matrix formats for GPUs, matrix solvers, multigrid and domain decomposition methods. GPUs also allow us to do more complex computations in higher precision arithmetic and to study its effect on numerical simulations of highly non-linear or chaotic physical phenomena. HPC 33

34 Outlook Connecting issues: Mathematical and numerical models of new challenging problems High-performance computing and advanced implementation techniques Outlook 34