# 武汉大学厦门大学计算数学研讨会

 09:20-09:30 开幕式，合影（主持人：杜魁） 学术报告（主持人：吕锡亮） 09:30-10:10 杨志坚：Cauchy-Born Approximation at Finite Temperature for System with Dilute Defects 10:10-10:30 休息 10:30-11:10 陈竑焘：A Recovery Based Linear Finite Element Method for 4th Order Problems 11:15-11:55 黄雪莹：Patient-specific CT-based 3D FSI model for left ventricle in hypertrophic obstructive cardiomyopathy 学术报告（主持人：杜魁） 14:00-14:40 陈国贤：Subcell Hydrostatic Reconstruction Scheme of Gravitational Flows with Vacuum 14:45-15:25 吕锡亮：Regularization Methods for Linear Inverse Problems with Sparse Constraint 15:25-15:55 休息 15:55-16:35 邱建贤：An Modified WENO Schemes for Hyperbolic Conservation Laws 16:35-18:00 自由讨论

Cauchy-Born approximation at finite temperature for System with Dilute Defects

Constitutive relations play important roles in the study of particular materials. Either experiments or massive simulations were needed to get such material properties. In this talk, I will introduce our recent work on the efforts of effectively evaluating material constitutive relations. It can be considered as generalization of traditional Cauchy-Born approximation, which only works for perfect system at zero-temperature.

A Recovery Based Linear Finite Element Method for 4th Order Problems

We analyze a gradient recovery based linear finite element method to solve string equations and the corresponding eigenvalue problems. Our method uses only $C^0$ element, which avoids complicated construction of $C^1$ elements and nonconforming elements. Optimal error bounds under various Sobolev norms are established. Moreover, after a post-processing the recovered gradient is superconvergent to the exact one. Finally, some numerical experiments are presented to validate our theoretical findings.

Patient-specific CT-based 3D FSI model for left ventricle in hypertrophic obstructive cardiomyopathy

Left ventricular outflow tract obstruction is observed in 70% of patients with hypertrophic cardiomyopathy, which occurs in about 1 of every 500 adults in the general population. It has been widely believed that the motion of the mitral valve, in particular, its systolic anterior motion (SAM), attributes significantly to such obstruction. For a better understanding of the mitral valve motion, a 3D patient-specific fluid-structure interaction model of the left ventricle from a patient with hypertrophic obstructive cardiomyopathy based on computed tomography (CT) scan images was proposed in this study. The entire 3D left ventricle, including the mitral valve, was reconstructed from contrast enhanced CT images and the computational analysis was performed in ADINA. Displacement, structural stress, pressure, flow velocity and shear stress within the left ventricle and mitral valve were extracted to characterize their behavior. The maximum shear stress on mitral valve was 9.68 . It was found that the pressure on its posterior leaflet was higher than that on the anterior leaflet and the peak pressure on the mitral valve was 93.5 mmHg which occurred at pre-SAM time. High angles of attack (54.3 ±22.4o) were found in this patient. The methodology established in this study may have the potential to clarify the mechanisms of SAM and ultimately optimize surgical planning by comparing the mechanical results obtained from preoperative and postoperative models.

Subcell Hydrostatic Reconstruction Scheme of Gravitational Flows with Vacuum

We design a scheme for the Euler equations with gravity by our new designed subcell hydrostatic reconstruction method which is originally for the shallow water equations [SIAM Journal on Numerical Analysis, 55(2):758-784, 2017.]. The key difficulty for such problem is to give a proper definition of the non-conservative product of measures due to the gravitational potential, such that two properties are preserved: one is the well-balancing which can preserve the steady state exactly (hydrostatic isothermal steady state are considered in this paper); another one is the non-negativity of both the gas density and pressure. The numerical experiments demonstrate the scheme’s robustness.

Regularization Methods for Linear Inverse Problems with Sparse Constraint

In this talk, we consider the linear inverse problems of recovering a sparse vector from noisy measurement data. Two different class of regularization methods are proposed: iterative regularization method and variational regularization method. For iterative regularization method, we showed that ADMM method is a regularization method, which explained why ADMM works well for the image debluring problem. For the variational regularization method, we provided an algorithm of primal-dual active set type for a class of convex/nonconvex sparsity-promoting penalties. A novel necessary optimality condition for the global minimizer using the associated thresholding operator is derived. Upon introducing the dual variable, the active set can be determined from the primal and dual variables. This relation lends itself to an iterative algorithm of active set type which at each step involves updating the primal variable only on the active set and then updating the dual variable explicitly. This approach can also extend to the group sparse model. Numerical examples are given to validate the theoretical results.

An Modified WENO Schemes for Hyperbolic Conservation Laws

In this presentation, a class of modified weighted essentially non-oscillatory (MWENO) schemes is presented in the finite difference framework for solving the hyperbolic conservation laws. These schemes adapt between the linear upwind scheme and the WENO scheme automatically by the usage of a new simple switching principle. The methodology to reconstruct numerical fluxes for the MWENO schemes is split into two parts: if all extreme points of the reconstruction polynomial for numerical flux in the big spatial stencil are located outside of the stencil, the the numerical flux is approximated directly by the reconstruction polynomial, and the approximation is a linear and high order accuracy; otherwise the WENO procedure in {\em \{G.-S. Jiang and C.-W. Shu, J. Comput. Phys., 126 (1996), 202-228\}} is applied to reconstruct the numerical flux. The main advantage of these new MWENO schemes is their robustness and efficiency comparing with the classical WENO schemes specified in {\em \{G.-S. Jiang and C.-W. Shu, J. Comput. Phys., 126 (1996), 202-228\}}. The MWENO schemes can be applied to compute some extreme test cases such as the Sedov blast wave, the Leblanc and the high Mach number astrophysical jet problems et al. by using a normal CFL number without any further positivity preserving procedure for the purpose of controlling the concurrence of the negative density and pressure. Extensive numerical results are provided to illustrate the good performance of the MWENO schemes.