Organizing Committee
Abstract

This program will focus on integral equation methods, fast algorithms and their applications to fluid dynamics and materials science. Integral equation methods have been used for more than a century to establish existence and uniqueness results for a variety of elliptic, parabolic and hyperbolic partial differential equations (PDEs). From a computational perspective, they have been used most extensively in the elliptic (steady state or time harmonic) case, because of their ability to handle complex geometry, unbounded domains and radiation conditions and because of the availability of fast algorithms to reduce the cost of handling the dense matrices that arise from their discretization. These algorithms include fast multipole methods (FMM), methods based on the Fast Fourier Transform (FFT) or the non-uniform FFT (“NUFFT”), and hierarchical compression-based methods (wavelet and SVD-based schemes, H-matrices, HSS-matrices, etc.). The fundamental issue is that discretization of an elliptic boundary integral equation yields a dense N × N matrix, where N denotes the number of degrees of freedom used to describe the unknown. The straightforward application of a dense matrix to a vector requires O(N2) work, while classical Gaussian elimination techniques require O(N3) work to solve the system. The various fast algorithms listed above provide the ability to apply the discretized integral operator to a vector in O(N) or O(N log N) operations. When combined with modern iterative methods (such as GMRES), well-conditioned integral equation formulations such as second kind integral equations (SKIEs) have reduced the total work required to near optimal complexity, bringing large scale simulations within practical reach.

The program will bring a group of researchers from the US and HKUST with common interests and complementary expertise to work intensively on constructing well-conditioned integral equation formulations, developing high-order, fast, and robust algorithms with scalable implementation, and applying them to solve complex, large-scale real physical applications in multiphase flows and dislocation dynamics and to propose positive conditions on the Hadamard conjecture.

Workshop Topics

Topic 1: Efficient methods for the threshold dynamics method and interface dynamics

Interfacial phenomena play a vital role in technological applications in such diverse areas as fluids, biology and material science. Numerical simulation of interface dynamics is a difficult task. The MBO scheme is an efficient numerical method for simulating interface dynamics driven by surface energy minimization. The scheme is also generalized to wetting dynamics and droplet spreading on rough solid surfaces (X. Xu, D. Wang and X.P. Wang, 2015).

In this program, we will develop integral equation formulations and design NUFFT-based fast algorithms for these problems. As compared with existing numerical schemes, the proposed method restricts the computation to a small neighborhood of the material interfaces, provides high-order discretization for smooth or piecewise smooth cases, and achieves near optimal complexity. We will also pay particular attention to scalable implementation to take advantage of modern heterogeneous parallel computers so that real physical 3D problems can be simulated and studied within minutes and hours instead of days and weeks.

Possible Projects:

  • Develop integral equation formulations and design NUFFT-based fast algorithms for problems with complex and rough boundaries.
  • Generalize the MBO-type schemes to multiphase fluid flow (i.e. couple with the Navier-Stokes equations).
  • Develop parallel algorithms for 3D interface dynamic problems.

Topic 2: Fast Numerical Methods for Simulations of Dislocation Climb

Dislocations are line defects in crystals. The climb motion of dislocations is assisted by diffusion and emission and/or absorption of vacancies or interstitials, and plays crucial roles in the plastic deformation of crystalline materials at high temperature. Numerical simulation of the dynamics of dislocation lines, or discrete dislocation dynamics (DDD), is an important tool for the study of plasticity. However, in early DDD simulations, the formulations of dislocation climb were based only on several special cases (single, straight dislocations) for which analytical solutions are available.

Recently, Prof. Yang Xiang and collaborators derived a Green's function formulation for the climb of curved, multiple dislocations in three-dimensions (Three-dimensional formulation of dislocation climb, Y.J. Gu, Y. Xiang, S.S. Quek, and D.J. Srolovitz, J. Mech. Phys. Solids, 83, 319-337, 2015). This new formulation is able to capture the long-range contribution to the dislocation climb velocity associated with vacancy diffusion, which was missing in the previous DDD simulation models.

In this proposed collaborative research workshop, robust numerical methods will be discussed and implemented to calculate the dislocation climb velocity accurately and efficiently, including boundary integral equation methods, NUFFT method, accurate dislocation discretization method, etc. Parallelization and other high performing methods will also be discussed for dislocation climb simulations in large systems.

Possible Projects:

  • Construct second kind integral equation formulation for the dislocation climb in the equilibrium state of vacancy diffusion.
  • Develop and implementing FMM/NUFFT based fast algorithms for solving the integral equation formulation for the vacancy diffusion equation in dislocation climb.
  • Apply the above algorithm to study large-scale dislocation climb problems.

Topic 3: Experimentation on the Hadamard Conjecture on Clamped Plate Problem

In 1908, Hadamard conjectured that the Green's function for the clamped plate problem, or mathematically, the first Dirichlet problem of the biharmonic equation on a convex domain is nonnegative. However, after 1949 numerous counterexamples disproved the positivity conjecture of Hadamard. The first result in this direction came by Duffin (1949), who showed that the Green function changes sign on a long rectangle. Garabedian then showed change of sign of the Green function in ellipses with ratio of half axes ≈ 1.6 (1951). Hedenmalm, Jakobsson and Shimorin (2002) mention that sign change occurs already in ellipses with ratio of half axes ≈ 1.2. Sign change is also proven by Coffman-Duffin (1980) in any bounded domain containing a corner, the angle of which is not too large. Their arguments are based on previous results by Osher and Seif (1973) and cover, in particular, squares. This means that neither in arbitrarily smooth uniformly convex nor in rather symmetric domains the Green function needs to be positive. In fact, convexity is neither sufficient nor necessary for a positive Green function. And the question of under what geometry the Green function will be nonnegative remains open.

In this program, we will study this problem numerically, and we will aim for our numerical results to provide new insights and lead to some positive conjectures on the geometry of the domain for which the Green function for the clamped plate problem is nonnegative. Our numerical study is based on a second kind integral equation formulation for the clamped plate problem. As the negative value of the Green function often first appears on the points close to the boundary, one needs high-order quadrature scheme in order to be able to compute the involved nearly singular integrals accurately. Finally, since one needs to scan many different geometries, fast algorithms are also in urgent need. We plan to develop and apply the FMM-accelerated QBX (quadrature by expansion) scheme for the discretization and evaluation of associated integral operators.

Possible Projects:

  • Implement parallel FMM-accelerated QBX scheme with arbitrary precision.
  • Study the Hadamard conjecture in the case of ellipses and identifying the ellipse with smallest ratio of half axes where the Green function becomes negative.
  • Study the Hadamard conjecture in the case of an arbitrary convex smooth curve to provide some conjecture and insight about this case.

Topic 4: Efficient Simulation of Photonic Metamaterials

Metamaterials are materials whose properties at the scale of a propagating wave with which they are interacting (that is, above the atomic scale, but well below the macroscopic bulk scale) determine key properties of the interaction. Metamaterials often consist of periodic arrays of structures, with purposefully placed ‘defects’ in the periodic lattice to achieve desired properties. For example, photonic crystals typically consist of a unit cell designed such that the bulk materials has one or multiple band gaps, that is, a range of wavelengths at which EM waves will not propagate in the bulk material. Defects then allow the creation of carefully designed propagation pathways within the non-propagating bulk materials.

Because of the mismatch in scale between the wavelength of light and the scale of the bulk structure, simulation of metamaterials is a formidable task that could stand to benefit tremendously from better computational tools. Material interfaces in metamaterials are typically sharp, leading to PDE BVPs with piecewise constant coefficients, which makes these problems very amenable to being studied with the help of boundary integral equation methods. High-order accuracy, as provided by the QBX quadrature scheme, is a key ingredient, since the propagation properties of EM waves are sensitive to small perturbations, making simulation at low orders of accuracy unaffordable.

The quasi-periodization method of Barnett and Greengard (2010) is another recent advance that has the (thus far unexploited) potential to aid in the study of metamaterials. In this program, we propose to create a computational toolkit for the study of metamaterials. Envisioned contributions extending beyond the creation of the toolkit include enabling the study of materials with bulk-scale defects as well as efficient and accurate numerical methods for the truncation of such calculations in the integral equation setting.

Possible Projects:

  • Design and implement a scheme to couple the QBX quadrature scheme embedded in a high-frequency FMM in three dimensions with a variant of the Barnett/Greengard periodization method.
  • Investigate truncation and acceleration schemes for bulk calculations.
  • Develop distributed-memory parallel algorithms for approaching large-scale metamaterial simulations.

Completed Projects

Organizer Talks
Student Talks