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Development of 1st-order Particle Discretization Scheme (PDS) for analysis of cracking phenomena

Mahendra Kumar Pala , Lalith Wijerathneb, Muneo Horic

a Doctorate Student, Department of Civil Engineering, The University of Tokyo, Tokyo, Japanb Associate Professor, Earthquake Research Institute, The University of Tokyo , Tokyo, Japanc Professor, Earthquake Research Institute, The University of Tokyo , Tokyo, Japan

About meHello, I am Mahendra Kumar Pal,

And I am pursuing Doctorate course

in Department of Civil Engineering,

the University of Tokyo.

I have received my B.Tech degree from B.I.E.T.

Jhansi affiliated to UP technical University Lucknow

in 2010, and M.Tech from Indian Institute of

Technology, Hyderabad in 2012.

My research interests are computational solid

mechanics, functional analysis, structural

mechanics, earthquake engineering, mathematical

modelling and numerical simulation.

Travelling, event planning and management are my

favorite hobbies. I have explored the beautiful island

called, Japan from its north-most Hokkaido, to

Okinawa, the south most part of the land of rising

sun.

Currently, I am the president of the University of

Tokyo Indian Students’Association (UTISA).

Introduction

PDS approximate the function and derivative as

union of local polynomial expansion about the

mother point of conjugate domain tessellation, Φ𝛼,

and Ψ𝛽.

Function approximation:

𝑓𝑑 𝒙 =

𝛼

𝑛

𝑓𝛼𝑛 𝑃𝑛 𝒙 − 𝒙𝛼 𝜑𝛼(𝒙)

Derivative approximation: over conjugate domain

𝑔𝑑 𝒙 =

𝛽

𝑛

𝑔𝛽𝑛 𝑃𝑛 𝒙 − 𝒙𝛽 ψ𝛽(𝒙)

For example: 𝑃4 𝒙 − 𝒙𝛽 = {1, 𝑥 − 𝑥𝛽 , 𝑦 − 𝑦𝛽 , (𝑧 − 𝑧𝛽)}

1st order PDS

Convergence rate 1st order PDS

1st order PDS-FEM

Numerical example and results

Summary

• Efficient treatment of boundary

conditions

• Implementing techniques for

modeling propagating cracks

• Implementation of 3D PDS-FEM

Material properties and boundary conditions

• Young’s Modulus = 1000MPa, Poisson’s ratio = 0.33

• Stress far field =10 MPa

Future work

• 1st order PDS-FEM is proposed and

implemented for 2D problems

• 2nd order convergence rate is

obtained by approximating the

unknown variables with first order

polynomial series expansion

𝑛

𝐼𝛼𝑚𝑛 𝑓𝛼𝑛 = 𝑃𝑚 𝒙 − 𝒙𝛼 𝜑𝛼 𝒙 𝑓(𝒙)𝑑𝑉

𝑛

𝐼𝛽𝑚𝑛 𝑔𝑖𝛽𝑛= 𝑃𝑚 𝒙 − 𝒙𝛼 ψ𝛽 𝒙 𝑓,𝑖

𝑑(𝒙)𝑑𝑉

Where , 𝐼𝛽𝑚𝑛 = 𝑃𝑚 𝒙 − 𝒙𝛽 𝑃𝑛(𝒙 − 𝒙𝛽)ψ𝛽(𝒙)𝑑𝑉

6 unit

3 un

it

1 unit

Symmetric Axis

1

Delaunay tessellation ψ𝛽

Voronoi tessellation 𝜑𝛼

Mother-point

Center of gravity

Function approximation

Derivative approximation

1

m

Elevated displacement field

m

Crude MeshFiner Mesh

1

MPa

Crude Mesh

Finer Mesh

Elevated stress field

𝑢 =𝑢𝛼0

𝑢𝛼1

𝑢𝛼2

=𝑢𝛼0

𝑢𝛼1

0

𝑣 =𝑣𝛼0

𝑣𝛼1

𝑣𝛼2

=𝑣𝛼0

𝑣𝛼1

0In order to investigate the

efficiency of the formulation:

𝑢,𝑥 , 𝑢,𝑦 𝑎𝑛𝑑 𝑣,𝑥 , 𝑣,𝑦 has been

calculated from the analytical solution

0th order PDS

Analytical

Approximation

Function

Derivative

1st order PDS

Comparison between 0th order and 1st order PDS

Convergence rate for derivative approximation

Governing equation

𝒖𝑑(𝒙) =

𝛼

𝑛

𝒖𝛼𝑛𝑃𝑛 𝒙 − 𝒙𝛼 𝜑𝛼(𝒙)

𝝐𝑑 𝒙 =

𝛽

𝑛

𝝐𝛽𝑛𝑃𝑛 𝒙 − 𝒙𝛽 ψ𝛽(𝒙)

𝝈𝑑 𝒙 =

𝛽

𝑛

𝝈𝛽𝑛𝑃𝑛 𝒙 − 𝒙𝛽 ψ𝛽(𝒙)

Displacement vector and body force approximation:

Stress & strain field approximation:

{𝒖𝛼𝑛}, {𝝐𝛽𝑛} and {𝝈𝛽𝑛} are the set of unknowns

For a static equilibrium problem, displacement vector,

body force vector, stress and strain field are the

variable

Contact address: mahendra@eri.u-tokyo.ac.jp

0.001

0.01

20

Log 1

0(L

2no

rm)

Log10 (dof)

Convergence rate for displacement

1

2

60

0.01

0.1

20

Log 10

(L2

-nor

m)

Log10 (dof)

Convergence rate for stress

1

2

60

Numerical Example

F

Body in static equilibrium

uu

Consider the infinitesimal deformation of linear elastic body

Lagrange for above governing equation

The displacement vector, stress and strain fields are

unknowns variable.

𝐿 𝒖, 𝝈, 𝝐 = 1

2𝝐: 𝒄: 𝝐 − 𝝈: 𝝐 − 𝛻𝒖 𝑑𝑉

𝛻. 𝝈 + 𝒃 = 𝒇 in Ω

𝒖 = 𝒖 on 𝜕Ω

0.01

0.1

1

10 100 1000

Log 1

0(L

2-

norm

)

Log10 (Dof)

Sin(x)

Arbitrary function

𝑓(𝑥)=𝑥/√(10𝑥+𝑥^3 )

Minimization of error norm

𝐸 = 𝑓 − 𝑓𝑑(𝑥)2𝑑𝑉 & 𝐸 = 𝑓,𝑖

𝑑 − 𝑔𝑑(𝑥)2𝑑𝑉

leads to