Abstract : Inhomogeneity, as the source of various multiscale effects in systems such as materials and data, play essential roles in the material properties and the data structure of these defected systems with multiple scales. The complexity of modeling defects and their impact to the properties of the systems present new challenges for mathematical modeling and analysis. Multi-scale, multi-physics and multi-fidelity models are required to accurately describe the complicated phenomena associated with the inhomogeneity. Speakers in this minisymposium will discuss recent advances in modeling approaches and simulation methods, and new findings obtained in analysis and simulations.
[04025] Recent progress on multiscale coupling for crystalline defects
Format : Talk at Waseda University
Author(s) :
Lei Zhang (Shanghai Jiao Tong University)
Yangshuai Wang (University of British Columbia)
Abstract : In this talk, we present some recent progress on the multiscale coupling methods for crystalline defects, which include: 1) MeshAC, a three-dimensional mesh package designed for atomistic-to-continuum; 2) (a/c) coupling with higher order far-field boundary conditions; 3) Adaptive QM/MM coupling with machine-learned interatomic potentials (MLIP).
[02228] Mathematical perspectives in modeling microstructures in metallic materials
Format : Online Talk on Zoom
Author(s) :
Yejun Gu (Agency for Science Technology and Research)
Abstract : The mechanical properties of metallic materials are strongly dependent on their microstructural features (e.g. morphology and distribution of dislocations, grains, and other defects). Thus it necessitates a comprehensive understanding of the microstructure evolutions in materails during deformation. I will present some attempts of applying mathematical methods to address this problem. The presentation consists of two parts: one is the fast algorithms for solving partial differential equations, in order to precisely describe the microstructure evolutions; the other is the probabilistic description and statistical analysis, which are employed to quantify the relationship between the microstructures and mechanical properties.
[04547] A Three-Dimensional Continuum Simulation Method for Grain Boundary Motion Incorporating Dislocation Structure
Format : Talk at Waseda University
Author(s) :
Xiaoxue Qin (Shanghai University)
Abstract : We develop a continuum model for the dynamics of grain boundaries in three dimensions that incorporates the motion and reaction of the constituent dislocations. The continuum model is based on a simple representation of densities of curved dislocations on the grain boundary. Illposedness due to nonconvexity of the total energy is fixed by a numerical treatment based on a projection method that maintains the connectivity of the constituent dislocations. An efficient simulation method is developed, in which the critical but computationally expensive long-range interaction of dislocations is replaced by another projection formulation that maintains the constraint of equilibrium of the dislocation structure described by the Frank’s formula. This continuum model is able to describe the grain boundary motion and grain rotation due to both coupling and sliding effects, to which the classical motion by mean curvature model does not apply. Comparisons with atomistic simulation results show that our continuum model is able to give excellent predictions of evolutions of low angle grain boundaries and their dislocation structures.
[04727] A nonlocal elasiticity model for simulating the static and dynamic problems of crystalline defects in materials
Format : Talk at Waseda University
Author(s) :
Xiaoyin Wang (Wuhan University)
Abstract : In this work, we present a nonlocal elasticity model in order to solve the static and dynamic problems related to crystalline defects such as dislocations, interfaces, etc. In our model, a superposition framework based on nonlocal description is used to solve the stress and displacement field due to defects. The interaction of dislocation with other types of defects such as cracks can be modeled with higher accuracy due to the consideration of nonlocal effects. The model is solved by optimization based numerical techniques in order to accelerate the simulations. We also apply this model to problems related to crystal interfaces.
[05384] An IBVP of a model for motion of grain boundaries
Format : Talk at Waseda University
Author(s) :
Peicheng Zhu (Shanghai University)
Abstract : We shall prove global existence of weak solutions to an initial-boundary value problem for a novel phase-field model which is proposed as an attempt to describe the motion of grain boundaries, a type of interface motion by interface diffusion driven by bulk free energy in elastically deformable solids. Its applications include important processes arising in Materials science, e.g., Sintering. In this model the evolution equation for an order parameter is a non-uniformly, degenerate parabolic equation of fourth order, which differs from the Cahn-Hilliard equation by a non-smooth term of the gradient of the unknown.
[04942] Stochastic Continuum Models for High–Entropy Alloys with Short-range Order
Format : Talk at Waseda University
Author(s) :
Luchan Zhang (Shenzhen University)
Yahong Yang (Hong Kong University of Science and Technology)
Yang Xiang (Hong Kong University of Science and Technology)
Abstract : High entropy alloys (HEAs) are a class of novel materials that exhibit superb engineering properties. It has been demonstrated by extensive experiments and first principles/atomistic simulations that short-range order in the atomic level randomness strongly influences the properties of HEAs. In this talk, we present stochastic continuum models for HEAs with short-range order from atomistic models. A proper continuum limit is obtained such that the mean and variance of the atomic level randomness together with the short-range order described by a characteristic length are kept in the process from the atomistic interaction model to the continuum equation. The obtained continuum model with short range order is in the form of an Ornstein–Uhlenbeck (OU) process, which validates our previous continuum model adopting the OU process phenomenologically for HEAs with short range order. We derive such stochastic continuum models with short-range order for both elasticity in HEAs without defects and HEAs with dislocations (line defects). The obtained stochastic continuum models are based on the energy formulations, whose variations lead to stochastic partial differential equations.
[03918] GAS: A Gaussian Mixture Distribution-Based Adaptive Sampling Method for PINNs
Format : Talk at Waseda University
Author(s) :
Cheng Yuan (Wuhan University)
Abstract : With the recent study of deep learning in scientific computation, the Physics-Informed Neural Networks (PINNs) method has drawn widespread attention for solving Partial Differential Equations (PDEs). Compared to traditional methods, PINNs can efficiently handle high-dimensional problems, but the accuracy is relatively low, especially for highly irregular problems. Inspired by the idea of adaptive finite element methods and incremental learning, we propose GAS, a Gaussian mixture distribution-based adaptive sampling method for PINNs. During the training procedure, GAS uses the current residual information to generate a Gaussian mixture distribution for the sampling of additional points, which are then trained together with historical data to speed up the convergence of the loss and achieve higher accuracy. Several numerical simulations on 2D and 10D problems show that GAS is a promising method that achieves state-of-the-art accuracy among deep solvers, while being comparable with traditional numerical solvers.
[05016] An Elastic Interaction-Based Loss Function in Image Segmentation and Detection
Format : Talk at Waseda University
Author(s) :
Yaxin FENG (Hong Kong University of Science and Technology)
Yuan Lan (Huawei Theory Lab)
Yang Xiang (Hong Kong University of Science and Technology)
Luchan Zhang (Shenzhen University)
Abstract : Deep learning techniques have shown their success in image processing since they are easy to manipulate and robust to various types of datasets. The commonly used pixel-wise loss functions result in a bottleneck to achieve high precision for complicated structures in biomedical and autonomous driving science images. For example, the predicted small blood vessels in retinal images are often disconnected or even missed under the supervision of the pixel-wise losses, and the existence of lanes needed to be inferred even when they are occluded by cars or human. This long-range elastic interaction-based training strategy addresses these problem. In this strategy, convolutional neural network (CNN) learns the target region under the guidance of the elastic interaction energy between the boundary of the predicted region and that of the actual object. Under the supervision of the proposed loss, the boundary of the predicted region is attracted strongly by the object boundary and tends to stay connected.
[03366] Phase field model for self-climb of prismatic dislocation loops by vacancy pipe diffusion
Format : Online Talk on Zoom
Author(s) :
Xiaohua NIU (Xiamen University of Technology)
Abstract : In this talk, we present a phase field model for the self-climb motion of prismatic dislocation loops via vacancy pipe diffusion driven by elastic interactions. This conserved dynamics model is developed under the framework of the Cahn-Hilliard equation with incorporation of the climb force on dislocations, and is based on the dislocation self-climb velocity formulation established in Ref. (Niu et al., 2017). Asymptotic analysis shows that the proposed phase field model gives the dislocation self-climb velocity accurately in the sharp interface limit. Numerical simulations of evolution, translation, coalescence and repelling of prismatic loops by self-climb show excellent agreement with discrete dislocation dynamics simulation results and the experimental observation. Also a phase field model for the motion of prismatic dislocation loops by both conservative climb and non-conservative climb is also developed. The simulations will be shown to illustrate the influence of the self-climb in the dislocation climb process.
[05129] Global weak solutions to an initial-boundary value problem of a phase-field model for motion of grain boundaries
Author(s) :
Luchan Zhang (Shenzhen University)
Peicheng Zhu (Shanghai University)
Abstract : We shall prove global existence of weak solutions to an initial-boundary value problem for a novel phase-field model which is proposed as an attempt to describe the motion of grain boundaries, a type of interface motion by interface diffusion driven by bulk free energy in elastically deformable solids. Its applications include important processes arising in Materials science, e.g., Sintering. In this model the evolution equation for an order parameter is a non-uniformly, degenerate parabolic equation of fourth order, which differs from the Cahn-Hilliard equation by a non-smooth term of the gradient of the unknown.
