Meshfree Methods for Partial Differential Equations II: 43 (Lecture Notes in Computational Science a

Editorial Reviews. From the Back Cover. Meshfree methods for the solution of partial differential Meshfree Methods for Partial Differential Equations II: 43 ( Lecture Notes in Computational Science and Engineering) - Kindle edition by Michael Griebel, Marc Alexander Kindle Store; ›; Kindle eBooks; ›; Science & Math.
Table of contents

• Refine your editions:
• Prof. Dr. Thomas-Peter Fries
• Meshfree Methods for Partial Differential Equations II : Marc Alexander Schweitzer :

Thus, choosing shape parameters has been an active topic in approximation theory. It has been showed that the scaling technique allows a large range of acceptable shape parameters, while greater shape parameters perform more accurately than smaller ones [ 23 ].

We extend the scaling technique in [ 23 ] to three-dimensional case. It can be done in each of the x , y , and z directions in the local domain. After re-scaling the interpolating RBFs, a relatively large shape parameter c can be used even for extremely small grid distances. In this section, the solution procedure to approximate spatial-temporal calcium concentrations is presented.

The coupled system described in Equations 1 — 4 is solved by the following steps:. For a large number of collocation points or a high-dimensional space, the time consumption can be very high if a brute-force searching algorithm is used. There are a number of established methods of finding the nearest neighbors of a given point [ 42 , 43 , 44 , 45 , 46 ]. In our work, the k -dimensional tree or kd-tree for short data structure is used for its high efficiency [ 42 , 43 ].

The most time-consuming part of the procedure is Step 4 and 5 because the solutions of the diffusion-reaction equations have to be iteratively updated. To find the steady-state solution to the system of PDEs, the following convergence criterion may be used:. Thus, the computational cost will depend on the time-step size.

In explicit time stepping methods, a small sampling distance in space generally implies the need for a small time-step, which apparently results in slow convergence. However, large time-steps often lead to diverging or oscillating numerical results.

• 2 editions of this work;
• Meshfree Methods for Partial Differential Equations II.
• Top Authors.
• Bitch on Wheels: The True Story of Black Widow Killer Sharon Nelson.
• Publikationen - AG Technomathematik - TU Kaiserslautern?
• Career Planning and Development--In Reverse?
• Delayed Flight, An Inspirational Romance.

A good choice of time-step size can improve the rate of convergence and the stability as well. On the other hand, an explicit time discretization form is only conditionally stable and the stability analysis for diffusion equations yields the following Courant—Friedrichs—Lewy condition [ 47 ]:. The above requirement is equivalent to:. The diffusion coefficients are fixed as shown in Table I. Thus, the time-step size approximately satisfies:. This is much smaller than 4 ms as in the FEM in [ 30 ]. While it is necessary to have a small time-step size to ensure a stable solution, the meshless method still significantly outperforms the FEM in computational efficiency.

By contrast, our meshless scheme only took about 5 minutes to simulate the same time course on a single processor of the same type, despite a much smaller time-step size being used. A very similar model has been used in [ 30 ] except that the t-tubule in our work is modelled as a cube-shaped box instead of a cylinder. As in [ 30 ], the LCC current density is uniformly distributed on the cell surface and the surface of t-tubule.

The surface areas and volumes are listed in Table II. The results shown in Figure 3 are very similar to those in [ 30 ]. In this numerical experiment, the t-tubule is modeled as a cube with a dimension of 0. In both h and i , the second picture shows a cross-section view along the t-tubule. Also, we select three feature spots along each scanned line at 0. The time course concentrations of the mobile and stationary buffers are shown in d — g in Figure 4.

In the presence of LCC current densities, the concentrations of all buffers increase. To investigate how the thickness of the t-tubule may affect calcium signaling, we consider the t-tubule as a line zero-thickness instead of a cube while keeping the length unchanged. However, we assume the same cell volume, cell surface areas and t-tubule surface area as used in Table II.

This experiment shows that, while real t-tubules have certain thickness, treating them as line or curve structures can simplify mathematical simulations with comparable numerical accuracy. In this numerical experiment, the t-tubule is modeled as a single line. Consider a t-tubule-free whole-cell model described in Equation As shown in Figure 6 , our model is still able to predict a high gradient near boundary regions. In this subsection, we discuss two issues regarding the sensitivity of the described meshless method to the shape parameter c and the number of nodes used.

We first test the sensitivity to the shape parameter using the single t-tubule model on the left of Figure 1. However, when c becomes larger, the calcium concentration remains the same. In all three cases, there are 24, interior points and 6, boundary points in the domain. We then test the sensitivity of the meshless method to the number of nodes by using the whole-cell model on the right of Figure 1.

The shape parameter c is chosen as The reason is that the total fluxes of the model are independent of the number of nodes chosen. When the grid distance becomes two times smaller, the number of boundary nodes is expected to increase by four times, meaning that each boundary node is carrying four times fewer calcium fluxes.

However, the corresponding volume of each boundary node is reduced by eight times, yielding a doubled local calcium concentration around each boundary node. The numerical results are compared with those in [ 30 ] to validate the proposed approaches. Additionally, our method is flexible enough to handle complex geometric domains with competitive accuracy and low time consumption.

However, the explicit time-stepping method has a very strict constraint on time-step. To avoid using impractically small time-step to obtain the stable result, an implicit scheme can be considered.

Refine your editions:

Implicit time-stepping methods transform time-dependent problems into a series of Helmholtz or modified Helmholtz equations. Although these methods are numerically harder to implement, we could save significant computational time using these methods due to much larger time steps. Our ongoing research is focused on the utilization of implicit methods to model spatial-temporal calcium dynamics in ventricular myocytes.

We shall also integrate realistic ultra-structures of T-tubules and SR from 3D imaging data into our models and investigate how SR releasing and up-taking channels affect calcium signaling. The content is solely the responsibility of the authors and does not necessarily represent the official views of the sponsors. National Center for Biotechnology Information , U. Int j numer method biomed eng. Author manuscript; available in PMC Feb 1.

The publisher's final edited version of this article is available at Int j numer method biomed eng. See other articles in PMC that cite the published article. SUMMARY Spatial-temporal calcium dynamics due to calcium release, buffering and re-uptaking plays a central role in studying excitation-contraction E-C coupling in both normal and diseased cardiac myocytes.

Meshless methods, local radial basis function collocation method LRBFCM , numerical simulation, calcium dynamics, ventricular myocytes. Table I The parameters used in the paper. Name Symbol Value Ref. Open in a separate window. According to [ 37 , 34 ], a ventricular myocytes may be simplified to repeated structural units consisting of a single t-tubule and its surrounding half sarcomeres.

The t-tubule is assumed to be a tiny cube located vertically in the center of the domain, as shown on the left of Figure 1. This simplified geometric model had been studied in [ 30 ], where the FEM was utilized to find the numerical solutions of the equations. The same geometry is considered in our work so that we can validate our meshless numerical approach. We shall also consider a simplified whole-cell model, as shown on the right of Figure 1.

The cell surface is described by the following parametric equation: Definition 1 Let R d be a d -dimensional Euclidean space. The coupled system described in Equations 1 — 4 is solved by the following steps: Set the initial conditions and boundary conditions at the domain nodes;. Solve the diffusion-reaction equations for each time-step e. Approximate the concentration of the stationary buffer, [ CaB s ], by Equation Table II The cell geometry used in case 1.

Compartment volume V mc Table III The cell geometry used in case 2. Discussions In this subsection, we discuss two issues regarding the sensitivity of the described meshless method to the shape parameter c and the number of nodes used. The corresponding numbers of nodes used are , , and , , respectively. Calcium cycling and signaling in cardiac myocytes. Annual Review of Physiology. A 3D monte carlo analysis of the role of dyadic space geometry in spark generation.

Interplay of ryanodine receptor distribution and calcium dynamics. Wu L, Kwok Y. A front-fixing finite difference method for the valuation of American options. Cambridge University Press; The partition of unity finite element method: Basic theory and applications. Methods in Applied Mechanics and Engineering. Techniques of Scientific Computing.

Convergence analysis of methods for solving general equations. Schaback R, Wendland H. Using compactly supported radial basis functions to solve partial differential equations. Improved multiquadric approximation for partial differential equations. Atluri SN, Shen S.

Moving Beyond the Finite Element Method. Mesh-free radial basis function network methods with domain decomposition for approximation of functions and numerical solution of poisson's equation. Engineering Analysis with Boundary Elements. World Scientific Press; Singapore: We use cookies to give you the best possible experience.

By using our website you agree to our use of cookies. Dispatched from the UK in 3 business days When will my order arrive? Home Contact Us Help Free delivery worldwide. Description Over the past years meshfree methods for the solution of partial di? One of the reasons for this development is the fact that meshfree d- cretizationsandparticlemodels areoftenbetter suitedto copewithgeometric changes of the domain of interest than mesh-based discretization techniques such as?

Furthermore, the computational costs associated with mesh generation are eliminated in me- free approaches, since they are based only on a set of independent points. From the modelling point of view, meshfree methods gained much interest in recent years since they may provide an e?

In light of these developments the Sonderforschungsbereich and the Gesellschaft fur Mathematik und Mechanik sponsored the second interna Itwas hostedby the Institut fur.. The organizers Ivo Babu? The objective of the workshop was not only to strengthen the mathematical - derstanding and analysis of meshfree discretizations but also to promote the exchange of ideas on their implementation and application. Product details Format Paperback pages Dimensions Illustrations note IX, p.

Math Hacks Richard Cochrane. Partial Differential Equations Lawrence C. The Magic of Math Arthur Benjamin. Oscillations in Nonlinear Systems J.