Lecture 5. Fdm - Flac

Lecture #5. Introduction to Finite Difference

Method

NUMERICAL METHODS AND GEOMECHANICS

5.1 Fundamental background of FDM5.2 Numerical formulation of FLAC5.3 Analysis example

NUMERICAL METHODS AND GEOMECHANICSNUMERICAL METHODS AND GEOMECHANICS

A SIMPLE MECHANICAL ANALOG

m

F(t)Newtons Law of Motion duF m a mdt

= =

For a continuous body, this can be generalized as iji ij

du gdt x

= +

where = mass density,xj = coordinate vector (x,y)

ij = components of the stress tensor, andgi = gravitation

, ,u u u

5.1 Fundamental background of FDM Lecture #5. Introduction to Finite Difference Method


STRESS-STRAIN EQUATIONS

In addition to the law of motion, a continuous material must obey a constitutive relation - that is, a relation between stresses and strains.For an elastic material this is:

In general, the form is as follows:

where



5.2 Numerical Formulation of FLAC Lecture #5. Introduction to Finite Difference Method

FLAC & FLAC3D solves the full dynamic equations of motion even for quasi-static problems. This has advantages for problems that involve physical instability, such as collapse, as will be explained later.

To model the static response of a system, a relaxation scheme is used in which damping absorbs kinetic energy. This approach can model collapse problems in a more realistic and efficient manner than other schemes, e.g., matrix-solution methods.

What is FLAC?



Basic Explicit Calculation Cycle

Equilibrium Equation(Equation of Motion)

Stress - Strain Relation(Constitutive Equation)

For all gridpoints (nodes)

For all zones (elements)i ij jF n L=

new stresses

nodal forces

Gauss theorem

strain rates

velocities

ijii

j

du gdt x

= +



In the finite difference method, each derivative in the previous equations (motion & stress-strain) is replaced by an algebraic expression relating variables at specific locations in the grid.

The algebraic expressions are fully explicit; all quantities on the right-hand side of the expressions are known. Consequently each element (zone or gridpoint) in a FLAC grid appears to be physically isolated from its neighbors during one calculational timestep. This is the basis of the calculation cycle:

A GENERAL FINITE-DIFFERENCE FORMULA


FLACs grid is internally composed of triangles. These are combined into quadrilaterals. The scheme for deriving difference equations for a polygon is described as follows:

Overlaid Triangular element Nodal force vectorElements with velocity vectors



FLAC: For all elements...

Gauss theorem,iS A

i

fn fdS dAx=

is used to derived a finite difference formula for elements of arbitrary shape.

( )biu nodal velocity

b

a( )aiu nodal velocity

S

For a polygon the formula becomes1

iSi

f f n Sx A

This formula is applied to calculating the strain increments, eij, for a zone:

( )( ) ( )1212

a bii i j

Sj

jiij

j i

u u u n Sx A

uue tx x

+ = +


The outward flux of a vector field through a closed surface is equal to the volume integral of the divergence of the region inside the surface


FLAC: For all gridpoints...

Once all stresses have been calculated, gridpoint forces

are derived from the resulting tractions acting on the

sides of each triangle. For example,

Then a classical central finite-difference formula is used

to obtain new velocities and displacements:

( in large strain mode)



Overlay & Mixed-Discretization Formulation of FLAC:

+ /2 =

Each is constant-stress/constant-strain:

Volume strain averaged over . Deviatoric strain evaluated for

and separately (Mixed discretization procedure)

Solution is Updated Lagrangian (grid moves with the material), and explicit (local changes do not affect neighbours in one timestep )



FLAC3Ds grid is internally composed of tetrahedral elements. These arecombined into hexahedral elements. The scheme for deriving difference equations for a tetrahedron is described as follows:

8 node zone with two overlays of 5 tetrahedra in each overlay



Various degenerate forms of elements are also available in FLAC3D to enable the grid to conform to required geometry of boundaries:



FLAC3D: For all elements... Gauss theorem,

is used to derived a finite difference formula for elements of arbitrary shape -

This formula is applied to calculating the strain rates, , for a zone:ij

Face area ( )fS

( )fin

Face unit normal vector

VVolume( )fiv

Average face

velocity

Hence,

( )liv

where,

ii j

j

vvx

,i

i jj

vvx



FLAC3D: For all gridpoints...

Once all stresses have been calculated, gridpoint forces are derived from the resulting tractions acting on the sides of each tetrahedron. For example,

for each triangular face. Then, contributions from each face and each tetrahedron (within a hexahedron) are summed -

Sum of contribution at global node of all tetrahedra meeting at that node

Then a classical central finite-difference formula is used to obtain new velocities and displacements:

( in large strain mode)



Overlay & Mixed-Discretization Formulation of FLAC3D :

+ /2 =

Forces from 2 overlays are averaged at gridpoints gives symmetry

Each

. Deviatoric strain evaluated for:

(Mixed discretization procedure) gives accurate plastic flow.Solution is Updated Lagrangian (grid moves with the material), and explicit (local changes do not affect neighbours in one timestep )

is constant-stress/constant-strain.

Volume strain averaged over:



Methods of solution in time domain

displacement u

force F

x

F

stress

u

numerical grid

EXPLICITAll elements:

{ } { }( ),F f u = (nonlinear law)

All nodes:

{ } Fu tm =

Repeat for n time-steps

No iterationswithin stepsInformation cannot physically propagate between elements during one time step

Assume (u)are fixed

Assume (F)are fixed

Correct ifmin

p

xtC

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Lecture 5. Fdm - Flac