Singularly perturbed diffusion-advection-reaction processes on extremely large three-dimensional curvilinear networks with a periodic microstructure -- efficient solution strategies based on homogenization theory

2016-8
Kropat, Erik
Meyer-Nieberg, Silja
Weber, Gerhard-Wilhelm
Boundary value problems on large periodic networks arise in many applications such as soil mechanics in geophysics or the analysis of photonic crystals in nanotechnology. As a model example, singularly perturbed elliptic differential equations of second order are addressed. Typically, the length of periodicity is very small compared to the size of the covered region. The overall complexity of the networks raises serious problems on the computational side. The high density of the graph, the huge number of edges and vertices and highly oscillating coefficients necessitate solution schemes, where even a numerical approximation is no longer feasible. Realizing that such a system depends on two spatial scales - global scale (full domain) and local scale (microstructure) - a two-scale asymptotic analysis for network differential equations is applied. The limit process leads to a homogenized model on the full domain. The homogenized coefficients cover the micro-oscillations and the topology of the periodic network and characterize the effective behaviour. The approximate model's quality is guaranteed by error estimates. Furthermore, singularly perturbed microscopic models with a decreasing diffusion part and transport-dominant problems are discussed. The effectiveness of the two-scale limit analysis is demonstrated by numerical examples of diffusion-advection-reaction problems on large periodic grids.
Numerical Algebra, Control and Optimization

Suggestions

Bridging the gap between variational homogenization results and two-scale asymptotic averaging techniques on periodic network structures
Kropat, Erik; Meyer-Nieberg, Silja; Weber, Gerhard Wilhelm (American Institute of Mathematical Sciences (AIMS), 2017)
In modern material sciences and multi-scale physics homogenization approaches provide a global characterization of physical systems that depend on the topology of the underlying microgeometry. Purely formal approaches such as averaging techniques can be applied for an identification of the averaged system. For models in variational form, two-scale convergence for network functions can be used to derive the homogenized model. The sequence of solutions of the variational microcsopic models and the correspondi...
Numerical solutions of boundary value problems; applications in ferrohydrodynamics and magnetohydrodynamics
Şenel, Pelin; Tezer, Münevver; Department of Mathematics (2017)
In this thesis, steady, laminar, fully developed flows in pipes subjected to a point magnetic source or uniform magnetic field are simulated by the dual reciprocity boundary element method (DRBEM). The Navier-Stokes and energy equations are solved in terms of the velocity, pressure and the temperature of the fluid which are all of the original variables of the problem. The missing pressure equation is derived and pressure boundary conditions are generated by a finite difference approximation and the DRBEM c...
Discontinuous Galerkin Methods for Convection Diffusion Equations with Random Coefficients
Çiloğlu, Pelin; Yücel, Hamdullah (null; 2019-09-11)
Partial differential equations (PDEs) with random input data is one of the most powerful tools to model oil and gas production as well as groundwater pollution control. However, the information available on the input data is very limited, which causes high level of uncertainty in approximating the solution to these problems. To identify the random coefficients, the well–known technique Karhunen Loéve (K–L) expansion has some limitations. K–L expansion approach leads to extremely high dimensional system...
Numerical method for conform reflection
Kushnarov, Andriy; Öktem, Hakan; Department of Scientific Computing (2010)
Conformal map has application in a lot of areas of science, e.g., fluid flow, heat conduction, solidification, electromagnetic, etc. Especially conformal map applied to elasticity theory can provide most simple and useful solution. But finding of conformal map for custom domain is not trivial problem. We used a numerical method for building a conformal map to solve torsion problem. In addition it was considered an infinite system method to solve the same problem. Results are compared.
Multiscale Modeling of Thin-Wire Coupling Problems Using Hybridization of Finite Element and Dipole Moment Methods and GPU Acceleration
ÖZGÜN, ÖZLEM; Mittra, Raj; Kuzuoğlu, Mustafa (2020-01-01)
In this article, a hybrid numerical method, called finite element method (FEM) + dipole moment (DM), is presented for efficient solution of multiscale electromagnetic radiation and scattering problems that involve structures with fine features, such as thin-wire antennas or objects. In this method, the FEM is hybridized with the DM approach to help ease certain computational burdens, such as mesh refinement, ill-conditioning, memory overload, and long computation times, when solving multiscale problems with...
Citation Formats
E. Kropat, S. Meyer-Nieberg, and G.-W. Weber, “Singularly perturbed diffusion-advection-reaction processes on extremely large three-dimensional curvilinear networks with a periodic microstructure -- efficient solution strategies based on homogenization theory,” Numerical Algebra, Control and Optimization, pp. 183–219, 2016, Accessed: 00, 2020. [Online]. Available: https://hdl.handle.net/11511/52276.