Summary of research paper: TheNumericalSolutionofDiffusionProblemsinStronglyHeterogeneousNon-IsotropicMaterials

The Numerical Solution of Diffusion Problems

in Strongly Heterogeneous Non-Isotropic

Materials

PRESENTED BY:

UMESH TIMALSINA(071/MSCS/669)

DEPARTMENT OF ELECTRONICS AND COMPUTER

IOE, PULCHOWK CAMPUS

Paper By:

James Hyman Mikhail Shashkov Stanly Steinberg

OUTLINE:

SLIDES 3-5: INTRODUCTION

SLIDES 6-8: THE SUPPORT OPERATORS METHOD

SLIDES 9-11: THE CONTINIUUM PROBLEM

SLIDES 12-16: THE SPACES OF DISCRETE FUNCTIONS

SLIDES 17-25: THE FINITE DIFFERENCE METHOD

2

INTRODUCTION:

Description and Investigation of new finite difference

algorithm for solving the elliptic partial difference

equation or stationary diffusion equation of the form:

The solution u=u(x,y): concentration to be solved for

temperature in heat diffusion problems and pressure flow

problems.

3

INTRODUCTION-II:

V- two dimensional space

div- divergence

grad- gradient

K=K(x,y) is a symmetric positive difference matrix

f=f(x,y) is the given forcing function

4

INTRODUCTION-III:

5

This is the robin boundary condition.

The Algorithm: Constructed using a nontrivial generalization of

the support operators method where the matrix K may be

discontinuous and the computation may not be smooth.

SUPPORT OPERATORS

METHOD- I:

Constructs discrete analogs of invariant differential

operators div and grad.

They satisfy the discrete analogs of the integral identities

responsible for the conservative property of the

continuum model.

This paper: third series of paper on support operators

method.[1]

6

SUPPORT OPERATORS

METHOD-II:

Support operators method is extended to non-diagonal

non-smooth non-diagonal tensor K and non smooth

logically rectangular grids.

Improvised accuracy for non-smooth tensor K: use the flux

operator K grad rather than gradient operator grad, as

one of the basic first order operator.

Requirement: inner product replaced by an inner

product weighted by the inverse of the materials

properties tensor k.

7

SUPPORT OPERATORS

METHOD-III:

The methods are linear, conservative and material

discontinuities are assumed to occur at the surfaces of

the grid cells.

Both heat flux and temperature are used as primary

variables.

8

THE CONTINIUM PROBLEM:

The Flux Form of the given PDE is developed using the

robin boundary conditions.

The space of scalar functions H with the inner product

(above) gives the PDE and the boundary condition to be

re-written as:

9

THE CONTINIUM PROBLEM-II:

The operator A is given by:

10

THE CONTINIUM PROBLEM-II:

The operator A can be represented in the form:

Boundary conditions: included in definitions of operators

and spaces of function in a natural way.

11

THE SPACES OF DISCRETE

FUNCTIONS:

Finite Difference: Discretizations Necessary

Types of discretizations:

Cell Centered discretization of scalar functions

Nodal and Face Centered discretizations of a vector

valued function.

12

THE SPACES OF DISCRETE

FUNCTIONS-II:

The logically

Rectangular

Grid and a

Typical cell in the

Grid[1].(assumed

To be convex unless

Otherwise stated.)

13

THE SPACES OF DISCRETE

FUNCTIONS-III:

The concept of

2D-Grids is exten-

ded to a 3D mesh

Whenever convin

ient.[1]

14

THE SPACES OF DISCRETE

FUNCTIONS-III:

The grids are interpreted as the discretizations of a map

from a curvilinear coordinate systems.

The spaces of discrete functions:

Discrete scalar functions:- straightforward

Discrete vector functions:- two possibilities for discretization

15

THE SPACES OF DISCRETE

FUNCTIONS-IV:

Vector functions are discretized as:

Using usual Cartesian coordinate components

Orthogonal Projection of Vector on the direction which is

perpendicular to the surfaces of 3-D cells at the centers of

the surfaces.

The spaces of Discrete scalar and vector function need

the inner products.

16

THE FINITE DIFFERENCE

METHOD

The support operators method is used to derive the

approximation to the divergence, flux operator, and

variable coefficient Laplacian.

First step: derive a discrete approximation to the

divergence.

Next step: use this discrete divergence to derive the

approximations of the flux operator and laplacian using

discrete analogs of integral identities.

17

THE FINITE DIFFERENCE

METHOD-II

Divergence: also called prime operator because of its

principle role.

Conservative and time invariant defination of divergence

is given by:

Now a discrete analog DIV of divergence div is derived

for both the nodal and surface discritizations.

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THE FINITE DIFFERENCE

METHOD-III

The nodal discretization for vectors give the divergence

to be,

19

THE FINITE DIFFERENCE

METHOD-IV

The surface discretization for scalar and vector functions,

give the prime operator as:

20

THE FINITE DIFFERENCE

METHOD-V

Then, the derived operator is the discrete analog of flux

operator and calculated for surface and nodal

discretization for vectors.

The discrete operators equations of the continium

operator Ω is defined as:

21

THE FINITE DIFFERENCE

METHOD-VI

Then, discrete operators on a grid for a cell node

discretization and cell surface discretization are

calculated for diagonal k, where K=kI on a rectangular

grid with cell sides hY , hX and cell volume hXhY.

The discrete analog of laplacian div grad is calculated by

choosing K=I and is given by.

22

THE FINITE DIFFERENCE

METHOD-VII

The null space for the discrete operator is a constant

function.

Cases are different for cell surface discretizations.

23

THE FINITE DIFFERENCE

METHOD-VII

The cell node and cell surface discretization give the

same form of equations as:

The fluxes can be eliminated to obtain an equation for U.

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THE FINITE DIFFERENCE

METHOD-VII

Different properties of these equations are exploited

under cell-node and cell-surface discretization to obtain

a numerical solution for the problem using methods like:

Cell-node discretization: SOR iteration, incomplete cholsky

Cell-surface discretization: Iterative methods, two-level

gradient methods, minimal residual method and minimal

correction method.

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REFERENCES:

[1] M. Shashkov and S. Steinberg, Support-operator finite-

difference algorithms for general elliptic problems, J. Comput.

Phys. 118, 131

[2] James M. Hyman, Mikhail Shashkov, Adjoint operators for thenatural discretizations of the divergence, gradient and curl on

logically rectangular grids (1997)

[3] Mikhail Shashkov and Stanly Steinberg, Solving Diffusion

Equations with Rough Coefficients in Rough Grids, 1996

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