scholarly journals Reduced order method for finite difference modeling of cardiac propagation

2020 ◽  
Vol 6 (3) ◽  
pp. 107-110
Author(s):  
Riasat Khan ◽  
Shahebul Hasan Shahebul Hasan ◽  
Kwong Ng
Keyword(s):  
Finite Difference ◽  
Basis Functions ◽  
Galerkin Projection ◽  
Finite Difference Approximation ◽  
Difference Approximation ◽  
Dimensional System ◽  
Bidomain Model ◽  
Reduced Order ◽  
Bidomain Equations ◽  
Low Dimensional

AbstractEfficient numerical simulation of cardiac electrophysiology is crucial for studying the electrical properties of the heart tissue. The cardiac bidomain model is the most widely accepted representation of the electrical behaviour of the heart muscle. The bidomain model offers fast cardiac potential variation, which can lead to high computational cost due to the required large grid sizes and small time steps. In this paper, the complexity of the finite difference approximation of the bidomain equations is reduced with the model order reduction technique. Proper orthogonal decomposition, a projection-based algorithm, is used to efficiently approximate the original high fidelity cardiac bidomain model with a low-dimensional system of equations. The low-dimensional basis functions are computed first from the ‘snapshots,’ which contain the solutions of the full-order system for different temporal and spatial parameters. Galerkin projection of the original cardiac bidomain system onto the subspace of the reduced order basis functions reduces the size of the linear system. Numerical results confirm the efficiency of the proposed reduced order modeling technique, reducing the simulation time by a factor of 9.54, while maintaining an RMS error of 0.769 mV between the original full order solution and the reduced order POD solution.

scholarly journals Shape Sensitivity Analysis in Flow Models Using a Finite-Difference Approach

2010 ◽  
Vol 2010 ◽  
pp. 1-22 ◽  
Author(s):  
Imran Akhtar ◽  
Jeff Borggaard ◽  
Alexander Hay
Keyword(s):  
Finite Difference ◽  
Standard Method ◽  
Elliptic Cylinder ◽  
Galerkin Projection ◽  
Reduced Order Models ◽  
Shape Sensitivity ◽  
Reduced Order ◽  
Flow Models ◽  
Low Dimensional ◽  
Parameter Values

Reduced-order models have a number of practical engineering applications for unsteady flows that require either low-dimensional approximations for analysis and control or repeated simulation over a range of parameter values. The standard method for building reduced-order models uses the proper orthogonal decomposition (POD) and Galerkin projection. However, this standard method may be inaccurate when used “off-design” (at parameter values not used to generate the POD). This phenomena is exaggerated when parameter values describe the shape of the flow domain since slight changes in shape can have a significant influence on the flow field. In this paper, we investigate the use of POD sensitivity vectors to improve the accuracy and dynamical system properties of the reduced-order models to problems with shape parameters. To carry out this study, we consider flows past an elliptic cylinder with varying thickness ratios. Shape sensitivities (derivatives of flow variables with respect to thickness ratio) computed by finite-difference approximations are used to compute the POD sensitivity vectors. Numerical studies test the accuracy of the new bases to represent flow solutions over a range of parameter values.

scholarly journals Stable and Efficient Modeling of Anelastic Attenuation in Seismic Wave Propagation

2012 ◽  
Vol 12 (1) ◽  
pp. 193-225 ◽  
Author(s):  
N. Anders Petersson ◽  
Björn Sjögreen
Keyword(s):  
Wave Equation ◽  
Finite Difference ◽  
Half Space ◽  
Material Model ◽  
Second Order ◽  
Finite Difference Approximation ◽  
Difference Approximation ◽  
Space Problem ◽  
Standard Linear Solid ◽  
Standard Linear

AbstractWe develop a stable finite difference approximation of the three-dimensional viscoelastic wave equation. The material model is a super-imposition of N standard linear solid mechanisms, which commonly is used in seismology to model a material with constant quality factor Q. The proposed scheme discretizes the governing equations in second order displacement formulation using 3N memory variables, making it significantly more memory efficient than the commonly used first order velocity-stress formulation. The new scheme is a generalization of our energy conserving finite difference scheme for the elastic wave equation in second order formulation [SIAM J. Numer. Anal., 45 (2007), pp. 1902-1936]. Our main result is a proof that the proposed discretization is energy stable, even in the case of variable material properties. The proof relies on the summation-by-parts property of the discretization. The new scheme is implemented with grid refinement with hanging nodes on the interface. Numerical experiments verify the accuracy and stability of the new scheme. Semi-analytical solutions for a half-space problem and the LOH.3 layer over half-space problem are used to demonstrate how the number of viscoelastic mechanisms and the grid resolution influence the accuracy. We find that three standard linear solid mechanisms usually are sufficient to make the modeling error smaller than the discretization error.

On Low-Dimensional Galerkin Modeling of Confined Twin-Jet Flow and Heat Transfer

2001 ◽  
Author(s):  
H. Gunes ◽  
K. Gocmen ◽  
L. Kavurmacioglu
Keyword(s):  
Heat Transfer ◽  
Jet Flow ◽  
Basis Functions ◽  
Galerkin Projection ◽  
Temperature Fields ◽  
Transitional Flow ◽  
Flow And Heat Transfer ◽  
Galerkin Models ◽  
Velocity And Temperature Fields ◽  
Low Dimensional

Abstract The two-dimensional incompressible non-isothermal confined twin-jet flow has been numerically studied in the transitional flow regime by a finite volume technique. Results have been obtained for the velocity and temperature distributions close to the onset of temporal oscillations. Next, the proper orthogonal decomposition (POD) is applied to the instantaneous flow and temperature data to obtain POD-based basis functions for both velocity and temperature fields. These basis functions are capable to identify the coherent structures in the velocity and temperature fields. The low-dimensional Galerkin models of the full Navier-Stokes and energy equations are constructed by the Galerkin projection onto basis functions. Since the low-dimensional Galerkin models are much easier to analyze than the full governing equations, basic insights into important mechanisms of dynamically complex flow and heat transfer (e.g. flow instabilities) can be easily studied by these models. The numerical implications, the validity of the models and their performance characteristics are discussed.

A parameter-uniform finite difference scheme for singularly perturbed parabolic problem with two small parameters

2021 ◽  
Author(s):  
Tesfaye Aga Bullo ◽  
Guy Aymard Degla ◽  
Gemechis File Duressa
Keyword(s):  
Finite Difference ◽  
Difference Scheme ◽  
Finite Difference Scheme ◽  
Variable Parameter ◽  
Accurate Solution ◽  
Singularly Perturbed ◽  
Finite Difference Approximation ◽  
Parabolic Problems ◽  
Difference Approximation ◽  
Uniform Mesh

A parameter-uniform finite difference scheme is constructed and analyzed for solving singularly perturbed parabolic problems with two parameters. The solution involves boundary layers at both the left and right ends of the solution domain. A numerical algorithm is formulated based on uniform mesh finite difference approximation for time variable and appropriate piecewise uniform mesh for the spatial variable. Parameter-uniform error bounds are established for both theoretical and experimental results and observed that the scheme is second-order convergent. Furthermore, the present method produces a more accurate solution than some methods existing in the literature.   

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