Tianyue Li

Tianyue Li

U-M AIM PhD Candidate

Posts by Collection

portfolio

publications

A scalable Ewald-free BIE framework for periodic Stokes flow via hierarchical proxy sums.

Published in arXiv, 2026

We present a BIE framework for three-dimensional particulate Stokes flow in periodic pipes with circular cross-sections, wall-bounded doubly-periodic, and triply periodic geometries that uses only the free-space Green’s function and avoids both Ewald summation and the extended linear system. Proxy sources placed on equivalent surfaces of the kernel-independent FMM (KIFMM) form the auxiliary basis, and contributions from far image boxes are captured by a hierarchical proxy sum made absolutely convergent by a net-force-zero compatibility condition.

Recommended citation: http://arxiv.org/abs/2605.30805

talks

Perturbation Approach to Stokes-Whitehead Paradox: Oseen Approximation

Published:

The Stokes equation describes low-Reynold’s number flow dynamics. Consider an uniform flow past a sphere. Although the solution to the Stokes equation satisfies both surface and far-field boundary conditions, the next order approximation to the Navier-Stokes equation does not. This is the Whitehead paradox. Instead, we consider the low Reynold’s number as an asymptotic parameter, and look at a perturbation problem to steady state, incompressible Navier Stokes equation. The Stokes equation is the unperturbed equation in this problem. Considering flow near the surface of the sphere as a boundary layer, a second order uniform approximation can be made to resolve the Whitehead paradox. This is the Oseen approximation.

Boundary integral equation analysis for spheroidal suspensions

Published:

Boundary integral equation (BIE) methods are numerical PDE methods well suited to study soft materials in nature and industry. They reduce the dimension of the problem, but are plagued by singular and near-singular integrals on and near surfaces. We use solid spheroidal harmonics to evaluate BIO on and near the surface. Coupled with smooth quadrature for far evaluation, we present a fast, spectrally accurate method for computing BIO on spheroidal suspensions.

BIE method for Stokes Flow in Axisymmetric and Periodic Geometry

Published:

Peristalsis is an important biological process in which tubes in our body contract and relax to move solid or fluid along. One important inverse problem in mimicing this behavior artificially is to find the optimal wall configuration that generates the optimal flow, or minimizes the energy loss. During the forward direction, one solves for the flow generated by a certain wall geometry. This can be done using boundary integral equations (BIEs), by imposing a surface source density on the wall. In this paper, we reformulate the periodic geometry using proxy sources and periodic boundary discrepancies. We then use axisymmetry to reduce the surface BI into a 1D integral by analytically integrating over the angle of revolution. We also discuss the special quadrature needed to address singular and near-singular integrals that appears from BIE. Finally we give convergence and simulation results.

A scalable boundary integral method for simulating particulate flows through complex periodic geometries

Published:

Understanding low Reynolds number flows and constituent particle dynamics in confined, periodic geometries is fundamental to many biophysical applications. While Boundary Integral Equation (BIE) methods are powerful for Stokes flow, they typically rely on periodic Green’s functions, which can be restrictive for complex geometries and arbitrary periodic flow conditions.

Meshing between the (slender) lines: a fast, accurate, and scalable general purpose boundary integral equation solver

Published:

Multiscale, fast, and accurate solvers are needed to model many biophysical systems in low-Re flow. Boundary Integral Equation (BIE) methods reduce dimension from the problem domain to its boundary, and can be O(N) fast with Fast Multipole Methods. We developed a general purpose BIE solver on quadrilateral mesh with accuracy, speed, and scalability, which enabled us to combine with special purpose solvers to simulate many interesting physical phenomina such as particle transport, vascular networks, and rheology.

A fast, accurate, and scalable general purpose boundary integral equation solver

Published:

Simulating fluid-structure interactions in low-Re flow is important for many bio- physical system such as cell transport, active microswimmers, and rheology of dense suspensions. However, to accurately describe large scale behavior while resolving small scale details is challenging due to the high degrees of freedom and changing domain. Boundary integral equation (BIE) methods addresses both challenges by framing the differential equation in the fluid domain as an integral equation on the domain boundary, reducing dimensionality and avoiding domain meshing and remeshing. However, addressing near-singular integrands and maintaining speed in more complex geometries are still active areas of research.

teaching

Teaching: U-M Math Department

Undergraduate course, University of Michigan, Ann Arbor, Math Department, 2023

  • Fall 2023: Math 115 (Calculus 1) Primary instructor, individual section
  • Winter 2024: Math 116 (Calculus 2) Primary instructor, group section
  • Fall 2024: Math 115 Primary instructor, individual section
  • Winter 2025: Math 116 Primary instructor, individual section
  • Fall 2026: Math 215 Lab instructor, individual lab sections.