scholarly journals Variable time step size application in implicit finite difference schemes when modelling operation of biosensors

2014 ◽  
Vol 55 ◽  
Author(s):  
Arsenij Kurbanov ◽  
Vytautas Ašeris
Keyword(s):  
Mathematical Models ◽  
Step Count ◽  
Reaction Diffusion Equations ◽  
Finite Difference Schemes ◽  
Model Parameters ◽  
Step Size ◽  
Time Step ◽  
Time Step Size ◽  
Variable Time ◽  
Variable Time Step

The operation of biosensors is described by mathematical models with reaction-diffusion equations. Due to non-linear reaction members, these models are solved using numerical methods, which often are a very time-consuming. The goal is to propose variable time step size algorithm which would reduce required calculations while preserving the accuracy of results. The proposed algorithm was applied to two biosensor models: with different diffusion and reaction. The applications of this algorithm are wide, because it is based on a few basic requirements, which are true for most biosensors models. The recommendations were made for choosing optimal algorithm parameters in general case. Algorithms efficiency was found to be dependent on mathematical models reaction part. Although the algorithm does reduce step count for all analyzed model parameters, the step count decrease ranges from 30% to tens of millions times.

A Strategy to Accelerate the Numerical Integration in the Dynamic Simulation of Multibody Systems

1997 ◽  
Author(s):  
Jesús Cardenal ◽  
Javier Cuadrado ◽  
Eduardo Bayo
Keyword(s):  
Multibody Systems ◽  
Equations Of Motion ◽  
Stiff Systems ◽  
Step Size ◽  
Step Method ◽  
Integral Of Motion ◽  
Time Step ◽  
Time Step Size ◽  
Variable Time ◽  
Numerical Integrator

Abstract This paper presents a multi-index variable time step method for the integration of the equations of motion of constrained multibody systems in descriptor form. The basis of the method is the augmented Lagrangian formulation with projections in index-3 and index-1. The method takes advantage of the better performance of the index-3 formulation for large time steps and of the stability of the index-1 for low time steps, and automatically switches from one method to the other depending on the required accuracy and values of the time step. The variable time stepping is accomplished through the use of an integral of motion, which in the case of conservative systems becomes the total energy. The error introduced by the numerical integrator in the integral of motion during consecutive time steps provides a good measure of the local integration error, and permits a simple and reliable strategy for varying the time step. Overall, the method is efficient and powerful; it is suitable for stiff and non-stiff systems, robust for all time step sizes, and it works for singular configurations, redundant constraints and topology changes. Also, the constraints in positions, velocities and accelerations are satisfied during the simulation process. The method is robust in the sense that becomes more accurate as the time step size decreases.

scholarly journals A Minimizing Movement approach to a class of scalar reaction-diffusion equations

2020 ◽  
Author(s):  
Florentine Catharina Fleißner
Keyword(s):  
General Setting ◽  
Gradient Flow ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Step Size ◽  
Time Step ◽  
Flux Boundary Condition ◽  
Time Step Size ◽  
Kantorovich Distance

The purpose of this paper is to introduce a Minimizing Movement approach to scalar reaction-diffusion equations of the form \partial_t u \ = \ \Lambda\cdot \mathrm{div}[u(\nabla F'(u) + \nabla V)] \ - \ \Sigma\cdot (F'(u) + V) u, \quad \text{ in } (0, +\infty)\times\Omega, with parameters $\Lambda, \Sigma > 0$ and no-flux boundary condition u(\nabla F'(u) + \nabla V)\cdot {\sf n} \ = \ 0, \quad \text{ on } (0, +\infty)\times\partial\Omega, which is built on their gradient-flow-like structure in the space $\mathcal{M}(\bar{\Omega})$ of finite nonnegative Radon measures on $\bar{\Omega}\subset\xR^d$, endowed with the recently introduced Hellinger-Kantorovich distance $\HK_{\Lambda, \Sigma}$. It is proved that, under natural general assumptions on $F: [0, +\infty)\to\xR$ and $V:\bar{\Omega}\to\xR$, the Minimizing Movement scheme \mu_\tau^0:=u_0\mathscr{L}^d \in\mathcal{M}(\bar{\Omega}), \quad \mu_\tau^n \text{ is a minimizer for } \mathcal{E}(\cdot)+\frac{1}{2\tau}\HK_{\Lambda, \Sigma}(\cdot, \mu_\tau^{n-1})^2, \ n\in\xN, for \mathcal{E}: \mathcal{M}(\bar{\Omega}) \to (-\infty, +\infty], \ \mathcal{E}(\mu):= \begin{cases} \int_\Omega{[F(u(x))+V(x)u(x)]\xdif x} &\text{ if } \mu=u\mathscr{L}^d, \\ +\infty &\text{ else}, \end{cases} yields weak solutions to the above equation as the discrete time step size $\tau\downarrow 0$. Moreover, a superdifferentiability property of the Hellinger-Kantorovich distance $\HK_{\Lambda, \Sigma}$, which will play an important role in this context, is established in the general setting of a separable Hilbert space.

A Multi-Index Variable Time Step Method for the Dynamic Simulation of Multibody Systems

1996 ◽  
Author(s):  
Jesús Cardenal ◽  
Javier Cuadrado ◽  
Eduardo Bayo
Keyword(s):  
Multibody Systems ◽  
New Method ◽  
Step Method ◽  
Time Step ◽  
Multi Index ◽  
Time Step Size ◽  
Variable Time ◽  
Simulation Process ◽  
Variable Time Step ◽  
Time Step Method

Abstract This paper presents a new multi-index variable time step method for the integration of the equations of motion of constrained multibody systems in descriptor form. The basis of the method is the augmented Lagrangian formulation with projections in index-3 and index-1. The new method takes advantage of the better performance of the index-3 formulation for large time steps and of the stability of the index-1 for low time steps, and automatically switches from one method to the other depending on the required accuracy and values of the time step. The modification of time steps is accomplished through the use of a global system invariant such as the kinetic energy stored in the penalty system. This energy provides a good measure of the global error introduced by the numerical integration during the simulation process, and permits a simple and reliable strategy for varying the time step. Overall, the new method is quite efficient and robust: it is suitable for stiff and non-stiff systems, it is robust for all time step sizes, it works for singular configurations, redundant constraints and topology changes. Also, the constraints in positions, velocities and accelerations are satisfied to machine precision during the simulation process. The method is always more accurate as the time step size decreases.

scholarly journals A Nonoscillatory Second-Order Time-Stepping Procedure for Reaction-Diffusion Equations

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-15
Author(s):  
Philku Lee ◽  
George V. Popescu ◽  
Seongjai Kim
Keyword(s):  
Numerical Solution ◽  
Reaction Diffusion ◽  
Second Order ◽  
Reaction Diffusion Equations ◽  
Step Size ◽  
Time Step ◽  
Potential Oscillations ◽  
Time Step Size ◽  
Time Stepping ◽  
Spurious Oscillations

After a theory of morphogenesis in chemical cells was introduced in the 1950s, much attention had been devoted to the numerical solution of reaction-diffusion (RD) partial differential equations (PDEs). The Crank–Nicolson (CN) method has been a common second-order time-stepping procedure. However, the CN method may introduce spurious oscillations for nonsmooth data unless the time step size is sufficiently small. This article studies a nonoscillatory second-order time-stepping procedure for RD equations, called a variable-θmethod, as a perturbation of the CN method. In each time level, the new method detects points of potential oscillations to implicitly resolve the solution there. The proposed time-stepping procedure is nonoscillatory and of a second-order temporal accuracy. Various examples are given to show effectiveness of the method. The article also performs a sensitivity analysis for the numerical solution of biological pattern forming models to conclude that the numerical solution is much more sensitive to the spatial mesh resolution than the temporal one. As the spatial resolution becomes higher for an improved accuracy, the CN method may produce spurious oscillations, while the proposed method results in stable solutions.

scholarly journals Analysis of a Decoupled Time-Stepping Scheme for Evolutionary Micropolar Fluid Flows

2016 ◽  
Vol 2016 ◽  
pp. 1-13 ◽  
Author(s):  
S. S. Ravindran
Keyword(s):  
Micropolar Fluid ◽  
Stokes Equations ◽  
Fluid Model ◽  
Navier Stokes ◽  
Optimal Order ◽  
Step Size ◽  
Time Step ◽  
Order Error ◽  
Time Step Size ◽  
Time Stepping

Micropolar fluid model consists of Navier-Stokes equations and microrotational velocity equations describing the dynamics of flows in which microstructure of fluid is important. In this paper, we propose and analyze a decoupled time-stepping algorithm for the evolutionary micropolar flow. The proposed method requires solving only one uncoupled Navier-Stokes and one microrotation subphysics problem per time step. We derive optimal order error estimates in suitable norms without assuming any stability condition or time step size restriction.

On the Influence of Inflow Model Selection for Time-Domain Tiltrotor Aeroelastic Analysis

2021 ◽  
Author(s):  
Ethan Corle ◽  
Matthew Floros ◽  
Sven Schmitz
Keyword(s):  
Experimental Data ◽  
Particle Method ◽  
Comprehensive Analysis ◽  
Solution Stability ◽  
Step Size ◽  
Time Step ◽  
Time Step Size ◽  
Vortex Particle Method ◽  
Whirl Flutter ◽  
Vortex Particle

The methods of using the viscous vortex particle method, dynamic inflow, and uniform inflow to conduct whirl-flutter stability analysis are evaluated on a four-bladed, soft-inplane tiltrotor model using the Rotorcraft Comprehensive Analysis System. For the first time, coupled transient simulations between comprehensive analysis and a vortex particle method inflow model are used to predict whirl-flutter stability. Resolution studies are performed for both spatial and temporal resolution in the transient solution. Stability in transient analysis is noted to be influenced by both. As the particle resolution is refined, a reduction in simulation time-step size must also be performed. An azimuthal time step size of 0.3 deg is used to consider a range of particle resolutions to understand the influence on whirl-flutter stability predictions. Comparisons are made between uniform inflow, dynamic inflow, and the vortex particle method with respect to prediction capabilities when compared to wing beam-bending frequency and damping experimental data. Challenges in assessing the most accurate inflow model are noted due to uncertainty in experimental data; however, a consistent trend of increasing damping with additional levels of fidelity in the inflow model is observed. Excellent correlation is observed between the dynamic inflow predictions and the vortex particle method predictions in which the wing is not part of the inflow model, indicating that the dynamic inflow model is adequate for capturing damping due to the induced velocity on the rotor disk. Additional damping is noted in the full vortex particle method model, with the wing included, which is attributed to either an interactional aerodynamic effect between the rotor and the wing or a more accurate representation of the unsteady loading on the wing due to induced velocities.

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