An adaptive moving mesh method for forced curve shortening flow

We propose a novel adaptive moving mesh method for the numerical solution of a forced curve shortening geometric evolution equation. Control of the mesh quality is obtained using a tangential mesh velocity derived from a mesh equidistribution principle, where a positive adaptivity measure or monitor function is approximately equidistributed along the evolving curve. Central finite differences are used to discretise in space the governing evolution equation for the position vector and a second-order implicit scheme is used for the temporal integration. Simulations are presented indicating the generation of meshes which resolve areas of high curvature and are of second-order accuracy. Furthermore, the new method delivers improved solution accuracy compared to the use of uniform arc-length meshes

A second-order in time, BGN-based parametric finite element method for geometric flows of curves

Over the last two decades, the field of geometric curve evolutions has attracted significant attention from scientific computing. One of the most popular numerical methods for solving geometric flows is the so-called BGN scheme, which was proposed by Barrett, Garcke, and Nurnberg (J. Comput. Phys., 222 (2007), pp. 441{467), due to its favorable properties (e.g., its computational efficiency and the good mesh property). However, the BGN scheme is limited to first-order accuracy in time, and how to develop a higher-order numerical scheme is challenging. In this paper, we propose a fully discrete, temporal second-order parametric finite element method, which incorporates a mesh regularization technique when necessary, for solving geometric flows of curves. The scheme is constructed based on the BGN formulation and a semi-implicit Crank-Nicolson leap-frog time stepping discretization as well as a linear finite element approximation in space. More importantly, we point out that the shape metrics, such as manifold distance and Hausdorff distance, instead of function norms, should be employed to measure numerical errors. Extensive numerical experiments demonstrate that the proposed BGN-based scheme is second-order accurate in time in terms of shape metrics. Moreover, by employing the classical BGN scheme as a mesh regularization technique when necessary, our proposed second-order scheme exhibits good properties with respect to the mesh distribution.Comment: 35 pages, 9 figure

Numerical simulation of fully nonlinear interaction between steep waves and 2D floating bodies using the QALE-FEM method

This paper extends the QALE-FEM (quasi arbitrary Lagrangian–Eulerian finite element method) based on a fully nonlinear potential theory, which was recently developed by the authors [Q.W. Ma, S. Yan, Quasi ALE finite element method for nonlinear water waves, J. Comput. Phys, 212 (2006) 52–72; S. Yan, Q.W. Ma, Application of QALE-FEM to the interaction between nonlinear water waves and periodic bars on the bottom, in: 20th International Workshop on Water Waves and Floating Bodies, Norway, 2005], to deal with the fully nonlinear interaction between steep waves and 2D floating bodies. In the QALE-FEM method, complex unstructured mesh is generated only once at the beginning of calculation and is moved to conform to the motion of boundaries at other time steps, avoiding the necessity of high cost remeshing. In order to tackle challenges associated with floating bodies, several new numerical techniques are developed in this paper. These include the technique for moving mesh near and on body surfaces, the scheme for estimating the velocities and accelerations of bodies as well as the forces on them, the method for evaluating the fluid velocity on the surface of bodies and the technique for shortening the transient period. Using the developed techniques and methods, various cases associated with the nonlinear interaction between waves and floating bodies are numerically simulated. For some cases, the numerical results are compared with experimental data available in the public domain and good agreement is achieved

Large-scale structural analysis: The structural analyst, the CSM Testbed and the NAS System

The Computational Structural Mechanics (CSM) activity is developing advanced structural analysis and computational methods that exploit high-performance computers. Methods are developed in the framework of the CSM testbed software system and applied to representative complex structural analysis problems from the aerospace industry. An overview of the CSM testbed methods development environment is presented and some numerical methods developed on a CRAY-2 are described. Selected application studies performed on the NAS CRAY-2 are also summarized

Geometric partial differential equations: Theory, numerics and applications

This workshop concentrated on partial differential equations involving stationary and evolving surfaces in which geometric quantities play a major role. Mutual interest in this emerging field stimulated the interaction between analysis, numerical solution, and applications

Simulation of action potential propagation based on the ghost structure method

In this paper, a ghost structure (GS) method is proposed to simulate the monodomain model in irregular computational domains using finite difference without regenerating body-fitted grids. In order to verify the validity of the GS method, it is first used to solve the Fitzhugh-Nagumo monodomain model in rectangular and circular regions at different states (the stationary and moving states). Then, the GS method is used to simulate the propagation of the action potential (AP) in transverse and longitudinal sections of a healthy human heart, and with left bundle branch block (LBBB). Finally, we analyze the AP and calcium concentration under healthy and LBBB conditions. Our numerical results show that the GS method can accurately simulate AP propagation with different computational domains either stationary or moving, and we also find that LBBB will cause the left ventricle to contract later than the right ventricle, which in turn affects synchronized contraction of the two ventricles

Integrated Heart - Coupling multiscale and multiphysics models for the simulation of the cardiac function

Mathematical modelling of the human heart and its function can expand our understanding of various cardiac diseases, which remain the most common cause of death in the developed world. Like other physiological systems, the heart can be understood as a complex multiscale system involving interacting phenomena at the molecular, cellular, tissue, and organ levels. This article addresses the numerical modelling of many aspects of heart function, including the interaction of the cardiac electrophysiology system with contractile muscle tissue, the sub-cellular activation-contraction mechanisms, as well as the hemodynamics inside the heart chambers. Resolution of each of these sub-systems requires separate mathematical analysis and specially developed numerical algorithms, which we review in detail. By using specific sub-systems as examples, we also look at systemic stability, and explain for example how physiological concepts such as microscopic force generation in cardiac muscle cells, translate to coupled systems of differential equations, and how their stability properties influence the choice of numerical coupling algorithms. Several numerical examples illustrate three fundamental challenges of developing multiphysics and multiscale numerical models for simulating heart function, namely: (i) the correct upscaling from single-cell models to the entire cardiac muscle, (ii) the proper coupling of electrophysiology and tissue mechanics to simulate electromechanical feedback, and (iii) the stable simulation of ventricular hemodynamics during rapid valve opening and closure