High Resolution Multi-organ Hemodynamic Simulation with High Performance Computing

Rongliang Chen
(Shenzhen Institutes of Advanced Technology, Chinese Academy of Sciences)

Abstract: Patient-specific blood flow simulations have the potential to provide quantitative predictive tools for virtual surgery, treatment planning, and risk stratification. To accurately resolve the blood flows based on the patient-specific geometry and parameters is still a big challenge because of the complex geometry and the turbulence, and it is also important to obtain the results in a short amount of computing time so that the simulation can be used in surgery planning. In this talk, we will precent some recent results of the multi-organ blood flow simulations with patient-specific geometry and parameters on a large-scale supercomputer. Several mathematical, biomechanical, and supercomputing issues will be discussed in detail. We will also report the parallel performance of the methods on a supercomputer with a large number of processors.

ICTM: A fast algorithm for interface related optimization problems

Dong Wang
(The Chinese University of Hong Kong)

Abstract: In this talk, we will present an efficient iterative convolution thresholding method (ICTM) for solving interface related optimization problems. The method is showed to be unconditionally stable, efficient, simple, easy to code and applicable to a wide range of problems. Applications in image segmentation and surface reconstruction from point clouds will be presented.

A Petrov-Galerkin finite element-meshfree formulation for multi-dimensional fractional diffusion equations

Zeng Lin
(Shenzhen Institutes of Advanced Technology, Chinese Academy of Sciences)

Abstract: Meshfree methods with arbitrary order smooth approximation are very attractive for accurate numerical modeling of fractional differential equations, especially for multi-dimensional problems. However, the non-local property of fractional derivatives poses considerable difficulty and complexity for the numerical simulations of fractional differential equations and this issue becomes much more severe for meshfree methods due to the rational nature of their shape functions. In order to resolve this issue, a new weak formulation regarding multi-dimensional Riemann-Liouville fractional diffusion equations is introduced through unequally splitting the original fractional derivative of the governing equation into a fractional derivative for the weight function and an integer derivative for the trial function. Accordingly, a Petrov-Galerkin finite element-meshfree method is developed, where smooth reproducing kernel meshfree shape functions are adopted for the trial function approximation to enhance the solution accuracy, and the discretization of weight function is realized by the explicit finite element shape functions with an analytical fractional derivative evaluation to further reduce the computational complexity and improve efficiency. The proposed method enables a direct and efficient employment of meshfree approximation, and also eliminates the undesirable singular integration problem arising in the fractional derivative computation of meshfree shape functions. A nonlinear extension of the proposed method to the fractional Allen-Cahn equation is presented as well. The effectiveness of the proposed methodology is consistently demonstrated by numerical results.

On discontinuous and continuous approximations to Volterra integral equations of the second kind

Hui Liang
(Harbin Institute of Technology)

Abstract: Collocation and Galerkin methods in the discontinuous and globally continuous piecewise polynomial spaces, in short, denoted as DC, CC, DG and CG methods respectively, are employed to solve Volterra integral equations (VIEs) of the second kind. It is proved that the quadrature DG and CG (QDG and QCG) methods obtained from the DG and CG methods by approximating the inner products by suitable numerical quadrature formulas, are equivalent to the DC and CC methods, respectively. In addition, the fully discretised DG and CG (FDG and FCG) methods are equivalent to the corresponding fully discretised DC and CC (FDC and FCC) methods. The convergence theories are established for DG and CG methods, and their semi-discretised (QDG and QCG) and fully discretised (FDG and FCG) versions. In particular, it is proved that the CG method for second-kind VIEs possess a similar convergence to the DG method for first-kind VIEs. Numerical examples illustrate the theoretical results.