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Maximum entropy-based variational multiscale element-free Galerkin methods for scalar advection–diffusion problems

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Abstract

This paper presents a new parameter-free stability term via Galerkin mesh-free scheme to solve the multiscale behavior of steady-state advection-dominated advection–diffusion problems where the classical Galerkin method is shown to be unstable. A variational multiscale element-free Galerkin method (VMEFG) is proposed based on first-order local maximum entropy (LME) basis functions, which are owning weak Kronecker delta property that enables the imposition of essential boundary conditions with ease. In the present work, the finite space of the problem is enlarged by the infinite bubble space in the background integration cell. These bubbles constitute the fine scale solve locally (captures) the thin layers associated with the flow. Then, this bubble space solution is substituted into the coarse scales to recover the VMEFG method globally, which determines the parameter-free stability term naturally. The developed LME-based VMEFG is studied against three standard benchmark problems exhibiting boundary/internal layers and observed to be stable and robust in resolving all the layers. The effect of various values of locality parameter in defining the LME approximants is carried out between the values 1.2 and 2. It is found that the suboptimal solution is arrived at the locality parameter value, 1.5 and above 1.8, is not recommended. Convergence test is also carried out with two different mass functions and validated with the analytical solution and found that the computational efficiency of the present method is in good agreement with analytical solution.

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Abbreviations

a:

Unknown constants in the approximation

b :

Constant convective velocity

G :

Hessian matrix

h :

Characteristic nodal spacing

H :

Shannon’s entropy

\(k\) :

Diffusivity (m2 s−1)

K1, K2, K3 :

Sub-cells of Delaunay triangular cell

L 2 :

Error measure norm

n :

Outward unit normal

N :

Number of nodes

N c :

Number of Delaunay triangular cells

Pe:

Peclet number

\(q_{{\text{N}}}\) :

Neumann boundary heat flux (W m−2)

r i :

Radius of the support for each node

s :

Prescribed source

T :

Unknown scalar in the problem

T h :

Approximation of the trail solution

\(\bar{T}\) :

Coarse scale of the trail solution

\(T^{\prime}\) :

Fine scale of the trail solution

T D :

Scalar value on Dirichlet boundary

w :

Prior mass function

\(\bar{w}\) :

Coarse scale of the mass function

\(w'\) :

Fine scale of the mass function

x :

Coordinates of the nodes

x i :

Scattered node set

\(\tilde{{x}}_{{\text{i}}}\) :

Shifted coordinates

\(H_{0}^{1}\) :

Hilbert space

L 2 :

Lebesgue space

V :

Subspace of Hilbert space

V h :

Subspace of V

\(\varOmega\) :

Domain of the problem

\({{\text{d}}\varOmega }\), \(\varGamma\) :

Boundary of the domain

\({\varGamma}_{{\text{N}}}\) :

Neumann boundary conditions imposed boundary

\(\xi\) :

Bubble function

\(\beta\) :

Pareto optimal parameter

\(\gamma\) :

Locality parameter

λ :

Lagrange Multiplier vector

\(\emptyset_{{\text{i}}}\) :

Basis functions

\(\tau\) :

Stabilization parameter

a:

Local Point of interest within the Delaunay triangular cell

cell:

Delaunay triangular cell of interest

i:

Node of interest

j:

Neighbor Node

d:

Spatial dimension

num:

Numerical solution

exact:

Exact solution

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  1. School of Mechanical Engineering, SASTRA Deemed to be University, Thanjavur, 613401, India

    Sreehari Peddavarapu & Srinivasan Raghuraman

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  1. Sreehari Peddavarapu
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Peddavarapu, S., Raghuraman, S. Maximum entropy-based variational multiscale element-free Galerkin methods for scalar advection–diffusion problems. J Therm Anal Calorim 141, 2527–2540 (2020). https://doi.org/10.1007/s10973-020-09845-y

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