scholarly journals Accelerating the Finite-Element Method for Reaction-Diffusion Simulations on GPUs with CUDA

Micromachines ◽  
2020 ◽  
Vol 11 (9) ◽  
pp. 881 ◽  
Author(s):  
Hedi Sellami ◽  
Leo Cazenille ◽  
Teruo Fujii ◽  
Masami Hagiya ◽  
Nathanael Aubert-Kato ◽  
...  
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Temporal Dynamics ◽  
Dna Nanotechnology ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
The Finite Element Method ◽  
Highly Nonlinear ◽  
Spatio Temporal ◽  
Element Method

DNA nanotechnology offers a fine control over biochemistry by programming chemical reactions in DNA templates. Coupled to microfluidics, it has enabled DNA-based reaction-diffusion microsystems with advanced spatio-temporal dynamics such as traveling waves. The Finite Element Method (FEM) is a standard tool to simulate the physics of such systems where boundary conditions play a crucial role. However, a fine discretization in time and space is required for complex geometries (like sharp corners) and highly nonlinear chemistry. Graphical Processing Units (GPUs) are increasingly used to speed up scientific computing, but their application to accelerate simulations of reaction-diffusion in DNA nanotechnology has been little investigated. Here we study reaction-diffusion equations (a DNA-based predator-prey system) in a tortuous geometry (a maze), which was shown experimentally to generate subtle geometric effects. We solve the partial differential equations on a GPU, demonstrating a speedup of ∼100 over the same resolution on a 20 cores CPU.

scholarly journals Projected Finite Elements for Systems of Reaction-Diffusion Equations on Closed Evolving Spheroidal Surfaces

2017 ◽  
Vol 21 (3) ◽  
pp. 718-747 ◽  
Author(s):  
Necibe Tuncer ◽  
Anotida Madzvamuse
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Projection Operator ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Time Dependent ◽  
Diffusion Equations ◽  
Surface Evolution ◽  
Element Discretization ◽  
Element Method

AbstractThe focus of this article is to present the projected finite element method for solving systems of reaction-diffusion equations on evolving closed spheroidal surfaces with applications to pattern formation. The advantages of the projected finite element method are that it is easy to implement and that it provides a conforming finite element discretization which is “logically” rectangular. Furthermore, the surface is not approximated but described exactly through the projection. The surface evolution law is incorporated into the projection operator resulting in a time-dependent operator. The time-dependent projection operator is composed of the radial projection with a Lipschitz continuous mapping. The projection operator is used to generate the surface mesh whose connectivity remains constant during the evolution of the surface. To illustrate the methodology several numerical experiments are exhibited for different surface evolution laws such as uniform isotropic (linear, logistic and exponential), anisotropic, and concentration-driven. This numerical methodology allows us to study new reaction-kinetics that only give rise to patterning in the presence of surface evolution such as theactivator-activatorandshort-range inhibition; long-range activation.

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