Search

Meshfree methods for in silico modelling and simulation of cardiac electrophysiology are gaining more and more popularity. These methods do not require a mesh and are more suitable than the Finite Element Method (FEM) to simulate the activity of complex geometrical structures like the human heart. However, challenges such as numerical integration accuracy and time efficiency remain and limit their applicability. Recently, the Fragile Points Method (FPM) has been introduced in the meshfree methods family. It uses local, simple, polynomial, discontinuous functions to construct trial and test functions in the Galerkin weak form. This allows for accurate integration and improved efficiency while enabling the imposition of essential and natural boundary conditions as in the FEM. In this work, we consider the application of FPM for cardiac electrophysiology simulation. We derive the cardiac monodomain model using the FPM formulation and we solve several benchmark problems in 2D and 3D. We show that FPM leads to solutions of similar accuracy and efficiency with FEM while alleviating the need for a mesh. Additionally, FPM demonstrates better convergence than FEM in the considered benchmarks., Comment: 14 pages, 7 figures, 3 tables, research article

We present the meshfree Mixed Collocation Method (MCM) to solve the monodomain model for numerical simulation of cardiac electrophysiology. We apply MCM to simulate cardiac electrical propagation in 2D tissue sheets and 3D tissue slabs as well as in realistic large-scale biventricular anatomies. Capitalizing on the meshfree property of MCM, we introduce an immersed grid approach for automated generation of nodes in the modeled cardiac domains. We demonstrate that MCM solutions are in agreement with FEM solutions, thus confirming their suitability for simulation of cardiac electrophysiology both in healthy and disease conditions, such as left-bundle-branch block (LBBB) and myocardial infarction. Despite the fact that the computational time for MCM calculations is longer than for FEM, its efficiency in dealing with domains presenting irregularity, nonlinearity and discontinuity make MCM a promising alternative for heart's electrical investigations., Comment: 13 pages, 6 figures, Under review version

The monodomain model is widely used in in-silico cardiology to describe excitation propagation in the myocardium. Frequently, operator splitting is used to decouple the stiff reaction term and the diffusion term in the monodomain model so that they can be solved separately. Commonly, the diffusion term is solved implicitly with a large time step while the reaction term is solved by using an explicit method with adaptive time stepping. In this work, we propose a fully explicit method for the solution of the decoupled monodomain model. In contrast to semi-implicit methods, fully explicit methods present lower memory footprint and higher scalability. However, such methods are only conditionally stable. We overcome the conditional stability limitation by proposing a dual adaptive explicit method in which adaptive time integration is applied for the solution of both the reaction and diffusion terms. In a set of numerical examples where cardiac propagation is simulated under physiological and pathophysiological conditions, results show that our proposed method presents preserved accuracy and improved computational efficiency as compared to standard operator splitting-based methods., Comment: 11 pages, 5 figures, 1 table. Paper under review in International Journal for Numerical Methods in Engineering

The Radial Point Interpolation Mixed Collocation (RPIMC) method is proposed in this paper for transient analysis of diffusion problems. RPIMC is an efficient purely meshless method where the solution of the field variable is obtained through collocation. The field function and its gradient are both interpolated (mixed collocation approach) leading to reduced $C$-continuity requirement compared to strong-form collocation schemes. The method's accuracy is evaluated in heat conduction benchmark problems. The RPIMC convergence is compared against the Meshless Local Petrov-Galerkin Mixed Collocation (MLPG-MC) method and the Finite Element Method (FEM). Due to the delta Kronecker property of RPIMC, improved accuracy can be achieved as compared to MLPG-MC. RPIMC is proven to be a promising meshless alternative to FEM for transient diffusion problems., Comment: Accepted version for publication in Engineering Analysis with Boundary Elements

We present the Cell-based Maximum Entropy (CME) approximants in E3 space by constructing the smooth approximation distance function to polyhedral surfaces. CME is a meshfree approximation method combining the properties of the Maximum Entropy approximants and the compact support of element-based interpolants. The method is evaluated in problems of large strain elastodynamics for three-dimensional (3D) continua using the well-established Meshless Total Lagrangian Explicit Dynamics (MTLED) method. The accuracy and efficiency of the method is assessed in several numerical examples in terms of computational time, accuracy in boundary conditions imposition, and strain energy density error. Due to the smoothness of CME basis functions, the numerical stability in explicit time integration is preserved for large time step. The challenging task of essential boundary conditions imposition in non-interpolating meshless methods (e.g., Moving Least Squares) is eliminated in CME due to the weak Kronecker-delta property. The essential boundary conditions are imposed directly, similar to the Finite Element Method. CME is proven a valuable alternative to other meshless and element-based methods for large-scale elastodynamics in 3D., Comment: Added link to the repository where the CME approximants code can be found