scholarly journals Adaptive Mesh Refinement and Adaptive Time Integration for Electrical Wave Propagation on the Purkinje System

2015 ◽  
Vol 2015 ◽  
pp. 1-14 ◽  
Author(s):  
Wenjun Ying ◽  
Craig S. Henriquez
Keyword(s):  
Wave Propagation ◽  
Adaptive Algorithm ◽  
Adaptive Mesh Refinement ◽  
Method Of Lines ◽  
Adaptive Mesh ◽  
Membrane Properties ◽  
Computational Domain ◽  
Linear Diffusion ◽  
Splitting Technique ◽  
Purkinje System

A both space and time adaptive algorithm is presented for simulating electrical wave propagation in the Purkinje system of the heart. The equations governing the distribution of electric potential over the system are solved in time with the method of lines. At each timestep, by an operator splitting technique, the space-dependent but linear diffusion part and the nonlinear but space-independent reactions part in the partial differential equations are integrated separately with implicit schemes, which have better stability and allow larger timesteps than explicit ones. The linear diffusion equation on each edge of the system is spatially discretized with the continuous piecewise linear finite element method. The adaptive algorithm can automatically recognize when and where the electrical wave starts to leave or enter the computational domain due to external current/voltage stimulation, self-excitation, or local change of membrane properties. Numerical examples demonstrating efficiency and accuracy of the adaptive algorithm are presented.

scholarly journals Convergence and quasi-optimal cost of adaptive algorithms for nonlinear operators including iterative linearization and algebraic solver

2021 ◽  
Author(s):  
Alexander Haberl ◽  
Dirk Praetorius ◽  
Stefan Schimanko ◽  
Martin Vohralík
Keyword(s):  
Adaptive Algorithm ◽  
Adaptive Mesh Refinement ◽  
Elliptic Boundary ◽  
Mesh Refinement ◽  
Adaptive Mesh ◽  
Picard Iteration ◽  
Optimal Rate ◽  
Stopping Criteria ◽  
Element Discretization ◽  
Fully Adaptive

AbstractWe consider a second-order elliptic boundary value problem with strongly monotone and Lipschitz-continuous nonlinearity. We design and study its adaptive numerical approximation interconnecting a finite element discretization, the Banach–Picard linearization, and a contractive linear algebraic solver. In particular, we identify stopping criteria for the algebraic solver that on the one hand do not request an overly tight tolerance but on the other hand are sufficient for the inexact (perturbed) Banach–Picard linearization to remain contractive. Similarly, we identify suitable stopping criteria for the Banach–Picard iteration that leave an amount of linearization error that is not harmful for the residual a posteriori error estimate to steer reliably the adaptive mesh-refinement. For the resulting algorithm, we prove a contraction of the (doubly) inexact iterates after some amount of steps of mesh-refinement/linearization/algebraic solver, leading to its linear convergence. Moreover, for usual mesh-refinement rules, we also prove that the overall error decays at the optimal rate with respect to the number of elements (degrees of freedom) added with respect to the initial mesh. Finally, we prove that our fully adaptive algorithm drives the overall error down with the same optimal rate also with respect to the overall algorithmic cost expressed as the cumulated sum of the number of mesh elements over all mesh-refinement, linearization, and algebraic solver steps. Numerical experiments support these theoretical findings and illustrate the optimal overall algorithmic cost of the fully adaptive algorithm on several test cases.

scholarly journals Accelerating numerical wave propagation using wavefield adapted meshes. Part I: forward and adjoint modelling

2020 ◽  
Vol 221 (3) ◽  
pp. 1580-1590 ◽  
Author(s):  
M van Driel ◽  
C Boehm ◽  
L Krischer ◽  
M Afanasiev
Keyword(s):  
Wave Propagation ◽  
Adaptive Mesh Refinement ◽  
Waveform Inversion ◽  
Computational Cost ◽  
Model Space ◽  
Adaptive Mesh ◽  
Order Of Magnitude ◽  
Speed Up ◽  
Adjoint Modelling ◽  
The Waves

SUMMARY An order of magnitude speed-up in finite-element modelling of wave propagation can be achieved by adapting the mesh to the anticipated space-dependent complexity and smoothness of the waves. This can be achieved by designing the mesh not only to respect the local wavelengths, but also the propagation direction of the waves depending on the source location, hence by anisotropic adaptive mesh refinement. Discrete gradients with respect to material properties as needed in full waveform inversion can still be computed exactly, but at greatly reduced computational cost. In order to do this, we explicitly distinguish the discretization of the model space from the discretization of the wavefield and derive the necessary expressions to map the discrete gradient into the model space. While the idea is applicable to any wave propagation problem that retains predictable smoothness in the solution, we highlight the idea of this approach with instructive 2-D examples of forward as well as inverse elastic wave propagation. Furthermore, we apply the method to 3-D global seismic wave simulations and demonstrate how meshes can be constructed that take advantage of high-order mappings from the reference coordinates of the finite elements to physical coordinates. Error level and speed-ups are estimated based on convergence tests with 1-D and 3-D models.

Two-Dimensional Simulation of Wave Propagation in a Three-Pipe Junction

2000 ◽  
Vol 122 (4) ◽  
pp. 549-555 ◽  
Author(s):  
R. J. Pearson ◽  
M. D. Bassett ◽  
P. Batten ◽  
D. E. Winterbone
Keyword(s):  
Wave Propagation ◽  
Adaptive Mesh Refinement ◽  
Adaptive Mesh ◽  
Finite Volume Scheme ◽  
Two Dimensional ◽  
Action Simulation ◽  
Pressure Time ◽  
Time Histories ◽  
Dimensional Simulation ◽  
Measured Pressure

The modelling of wave propagation in complex pipe junctions is one of the biggest challenges for simulation codes, particularly those applied to flows in engine manifolds. In the present work an inviscid two-dimensional model, using an advanced numerical scheme, has been applied to the simulation of shock-wave propagation through a three-pipe junction; the results are compared with corresponding schlieren images and measured pressure-time histories. An approximate Riemann solver is used in the shock-capturing finite volume scheme and the influence of the order of accuracy of the solver and the use of adaptive mesh refinement are investigated. The code can successfully predict the evolution and reflection of the wave fronts at the junctions whilst the run time is such as to make it feasible to include such a model as a local multi-dimensional region within a one-dimensional wave-action simulation of flow in engine manifolds. [S0742-4795(00)01304-1]

scholarly journals Numerical Simulation of Global-Scale Atmospheric Chemical Transport with High-Order Wavelet-Based Adaptive Mesh Refinement Algorithm

2016 ◽  
Vol 144 (4) ◽  
pp. 1469-1486 ◽  
Author(s):  
A. N. Semakin ◽  
Y. Rastigejev
Keyword(s):  
Numerical Simulation ◽  
Adaptive Mesh Refinement ◽  
Mesh Refinement ◽  
Computational Cost ◽  
Adaptive Mesh ◽  
Chemical Transport ◽  
Global Scale ◽  
High Order ◽  
Computational Domain ◽  
Grid Points

Abstract High computational cost associated with numerical modeling of multiscale global atmospheric chemical transport (ACT) imposes severe limitations on the spatial resolution of fixed nonadaptive grids. Recently it has been shown that the interaction of numerical diffusion caused by the crude resolution with complex velocity field of atmospheric flows leads to large numerical errors. To address the described difficulties, the authors have developed a wavelet-based adaptive mesh refinement (WAMR) method for numerical simulation of two-dimensional multiscale ACT problems. The WAMR is an adaptive method that minimizes the number of grid points by introducing a fine grid only in the locations where fine spatial scales occur and uses high-order spatial discretization throughout the computational domain. The algorithm has been tested for several challenging ACT problems. Particularly, it is shown that the method correctly simulates dynamics of a pollution plume traveling on a global scale, producing less than 1% error with a relatively low number (~105) of grid points. To achieve such accuracy, conventional nonadaptive techniques would require more than three orders of magnitude more computational resources. The method possesses good mass conservation properties; it is shown that an error in the total pollutant mass does not exceed 0.02% for this number of points. The obtained results demonstrate the WAMR’s ability to achieve high numerical accuracy for challenging ACT problems at a relatively low computational cost.

A h-Adaptive Algorithm Using Residual Error Estimates for Fluid Flows

2013 ◽  
Vol 13 (2) ◽  
pp. 461-478 ◽  
Author(s):  
N. Ganesh ◽  
N. Balakrishnan
Keyword(s):  
Adaptive Algorithm ◽  
Adaptive Mesh Refinement ◽  
Fluid Flows ◽  
Adaptive Mesh ◽  
Viscous Flows ◽  
Adaptive Strategy ◽  
Residual Error ◽  
Error Estimator ◽  
Flow Problems ◽  
Error Indicators

AbstractAlgorithms for adaptive mesh refinement using a residual error estimator are proposed for fluid flow problems in a finite volume framework. The residual error estimator, referred to as the ℜ-parameter is used to derive refinement and coarsening criteria for the adaptive algorithms. An adaptive strategy based on the ℜ-parameter is proposed for continuous flows, while a hybrid adaptive algorithm employing a combination of error indicators and the ℜ-parameter is developed for discontinuous flows. Numerical experiments for inviscid and viscous flows on different grid topologies demonstrate the effectiveness of the proposed algorithms on arbitrary polygonal grids.

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