JIAYIN (KAY) LU 卢嘉茵
Applied Mathematician & Computational Artist & Freelance Photographer
I. Computational Geometry, Parallel Computation
Multi-threaded Computation of the Voronoi Diagrams
Voronoi tessellation is a beautiful mathematical concept: For a set of discrete points in space, the space is associated with its closest point.
It founds many applications in different scientific domains. For example, material scientists can analyze the Voronoi diagrams of atomistic systems to study material properties. However, these systems are often large-scale, with millions of particles. As systems grow in size, the computational demands increase, necessitating efficient and scalable software solutions.
I developed a multi-threaded extension of Voro++, a software written in C++ to generate Voronoi diagrams. I tested the parallel performance on different particle distribution cases, and showed that with optimal load balancing strategies, all cases achieve near-perfect parallel efficiency.
TriMe++: Multi-threaded High-quality Meshing in 2D
Building on multi-threaded Voro++, I further extended our research to address challenges in 2D meshing. This led to the development of TriMe++, a 2D multi-threaded meshing software library in C++ using Delaunay triangulation. It is designed for large-scale quality meshing with millions of particles on complicated shapes.
TriMe++ can be useful in diverse fields ranging from computer graphics to scientific computing (e.g., in simulations using the Finite Element Method (FEM)). It is especially useful for applications that require large-scale meshing due to its computational efficiency and significant speed-up from parallelization.
Next steps
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To tackle challenges in extreme-scale computation of the Voronoi diagrams, I am currently investigating distributed-memory parallelization using OpenMPI.
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I am currently developing a 3D version of TriMe++. The goal is to incorporate the power of multi-threaded parallel computation, and efficiently compute quality tetrahedron meshes for complication shapes in 3D.
II. Continuum Mechanics, Numerical Methods, Data-driven Approach
Numerical Methods for Quasi-static Hypo-elastoplastic Materials
Many engineering materials exhibit elastoplastic behavior, including metals, granular materials, and amorphous solids such as bulk metallic glasses (BMGs). Understanding their plastic deformation is vital for predicting material failures. In many practical scenarios, such as laboratory tests, the material mechanics are in the quasi-static regime. However, few existing numerical methods can simulate this realistic physical regime.
Recent work has shown a close mathematical correspondence of quasi-static elastoplastic equations with the Incompressible Navier-Stokes equations. Therefore, a projection method analogous to fluid's projection method can be developed to solve the PDE system.
Drawing inspiration from improvements made for fluid's projection method in the past 50 years, I made several numerical improvements on the projection method for quasi-static elastoplasticity. I implemented a FEM solver for the projection step, developed a fully second-order in time numerical projection scheme, and developed an adaptive timestepping scheme for efficient computation of accurate solutions.
Multi-scale Modeling of the Bulk Metallic Glasses (BMGs)
I developed a multi-scale model to study plastic deformation behaviors of BMGs. Our collaborators developed a mesoscopic data-driven and stochastic model from molecular dynamics (MD) simulation data. I integrated this mesoscopic model into the broader macroscopic continuum framework, resolving differences in scales in physically reasonable ways.
The mesoscopic model is used to describe local plastic deformation of the material in the continuum model, allowing realistic stochastic plastic deformation behaviors not possible before. The multi-scale model enables the simulation of arbitrarily large systems infeasible for MD simulations, and enables direct comparisons of the continuum simulation and the MD simulations.
Next steps
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To understand the significantly richer and more complicated plastic deformation behaviors of BMGs in 3D, I plan to extend the numerical improvements and the multi-scale modeling to 3D.
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In a recent work, using a Spectral Deferred Correction formulation of the fluid projection method allows it to achieve arbitrarily high-order of temporal accuracy. Drawing inspiration from this, I plan to improve the numerical formulation of the projection for quasi-static elastoplasticity, to achieve arbitrarily high-order temporal accuracy.
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The numerical methods and the multi-scale modeling are not restricted to BMGs. I plan to extend the methodologies to study the plastic deformations of other elastoplastic materials.
III. Quantitative Biology, Statistics, Geometry, Machine Learning
Large-scale Study of Grasshopper Wings: Cell Geometries and Veination Networks
Insect wings represent a remarkable achievement of evolution and biological engineering. They possess qualities of being lightweight, strong, durable, and flexible, which are made possible by the presence of “wing veins”, the thickened, structure-like elements embedded within the wing’s surface.
I am digitizing the wings from a collection of over 4000 species of grasshoppers at the Field Museum Chicago. From the images, I then extract the cell segmentation and construct the venation networks. By studying the cell geometries and venation network topologies, I hope to uncover the generative rules of the pattern, and link the mechanical properties of the wings with the geometry patterns. Furthermore, with the large dataset, I will perform inter-species and inter-population comparisons of the patterns, to understand the similarities and differences from evolutionary biology perspectives.
Future goals
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Through the project, I greatly appreciate the power of ML tools in aiding the analysis of a large amount of messy biological and experimental data. Inspired by the project, I want to develop a machine learning package to automate the process of shape mapping/morphing for quantitative biology researchers.
Others: Undergraduate Research
The Nash-Shapley Model for Multi-player Games
This was my first experience in research! I studied the famous three-player mathematical poker model proposed by John Nash and Lloyd Shapley, reproduced the optimal probability strategies and the equilibrium results using Mathematica coding, and investigated several extensions of the game.
Accepted / Published
[1] Lu, Jiayin, and Lazar, Emanuel, and Rycroft, Chris. (2023). An extension to Voro++ for multithreaded computation of Voronoi cells. Computer Physics Communications. 291. 108832 (2023). [paper]
[2] Lazar, Emanuel, and Lu, Jiayin, and Rycroft, Chris. (2022). Voronoi cell analysis: The shapes of particle systems. American Journal of Physics. 90. 469-480 (2022). [paper]
Preprint / Submitted
[3] Lu, Jiayin, and Rycroft, Chris. TriMe++: Multi-threaded geometry meshing using the Delaunay Triangulation. [paper]
[4] Lu, Jiayin, and Rycroft, Chris. Numerical methods and improvements for simulating quasi- static hypo-elastoplastic materials. [paper]
[5] Lazar, Emanuel, and Lu, Jiayin, and Rycroft, Chris, and Schwartz, Deborah. Voronoi topology in two dimensions: theory, algorithms, and applications. [paper]
[6] B. Xu, Z. Wu, J. Lu, M.D. Shields, C.H. Rycroft, F. Bamer, M.L. Falk. Stochastic evolution elasto-plastic modeling of a metallic glass. 2024 (submitted).