literature study: - high order material point...

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Literature study:High Order Material Point Method

Roel Tielen

February 11 2016

Outline

• Introduction master project

• Description MPM

• Numerical challenges with MPM

• Benchmark problems

• Prelimenary numerical results

• Outlook

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Introduction master project

Objective:Development of a class of high-order material point methods that:

• solve/reduce some numerical challenges within classical MPM

• reduce computing times

• enable a more accurate prediction of physical quantities

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Material point method

Combined particle-mesh approach:

• Material points represent acontinuum

• Equation of motion is solved onfixed background grid

• Material points move throughdomain over time

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Description MPM

A time step of MPM consists of three steps:

1 Project particle properties onto nodes

2 Solve equation of motion on background grid

3 Update particle properties from solution at the nodes

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Projection from particles to nodes

Construction of mass matrix Mt and force vector Fint,t

Mt(i ,j) =

∫Ωφi (x)φj(x)ρ(x)dx ≈

np∑p=1

φi (xtp)φj(x

tp)mp

Fint,t(i) =

∫Ωσ(x)∇φi (x)dx ≈

np∑p=1

σtp∇φi (x tp)V tp

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Solving equation on background grid

• At each time step we obtain

Mtat = Ft .

• Determine lumped mass matrix

Mt →MLt .

• Solve equation of motion for i ∈ 1, . . . , nn

ati =Fti

MLt(i ,i)

.

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Update particle properties

Update particle properties (velocity, stress, etc.) from nodal quantitiesby evaluating basis functions at particle positions.

Example:

v t+∆tp = v tp + ∆t

nn∑i=1

φi (xtp)ati

∆εt+∆tp =

nn∑i=1

∇φi (x tp)∆ut+∆ti

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Numerical challenges with MPM

The quality of the MPM solution is affected by:

• Grid crossing errors

• Quadrature rule used in MPM

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Grid crossing error

Grid crossing:The movement of a particle from one element to another element

Effect of grid crossing:

• Non-physical increase/decrease of internal force at node

• Influences quality of MPM solution

xi−1 xi+1xi

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Grid crossing error

Internal force at node xi is determined by:

Fint,t(i) =

np∑p=1

σtp∇φi (x tp)V tp .

In standard MPM, ∇φi is discontinuous at element boundary:

φi

xi−1 xi xi+1

1

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Quadrature rule MPM

Within MPM particles serve as integration points:

∫Ωφi (x)φj(x)ρ(x)dx ≈

np∑p=1

φi (xtp)φj(x

tp)mp

This numerical integration rule is in general not exact.

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Benchmark problems

Three benchmark problems are considered:

• Vibrating string with initial velocity

• Soil column under self weight

• Vibrating bar with dynamic traction 1

1M. Steffen, R. M. Kirby, M. Berzins. Analysis and reduction of quadratureerrors in the material point method (MPM). Int. J. Numer. Methods. Engrg, 76(2008), pp. 922-948.

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Vibrating string

v0(x)

x0 L

∂2u

∂t2=

E

ρ

∂2u

∂x2

Boundary conditions:

u(0, t) = 0

u(L, t) = 0

Initial conditions:

u(x , 0) = 0

∂u

∂t(x , 0) = v0 sin

(πxL

)

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Soil column under self weight

soil

y

H

0

∂2u

∂t2=

E

ρ

∂2u

∂y2− g

Boundary conditions:

u(0, t) = 0

∂u

∂y(H, t) = 0

Initial conditions:

u(y , 0) = 0

∂u

∂t(y , 0) = 0

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Implementation MPM

• Modified Lagrangian algorithm 2

• Euler-Cromer time integration scheme

• Piecewise linear basis functions

• 1D implementation in Matlab

2(AL)-Kafaji, I, Formulation of a dynamic material point method (MPM) forgeomechanical problems, Institut fur Geotechnik der Universitat Stuttgart, (2013)

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Results

In this presentation focus on:

• Investigation of grid crossing error

• Spatial convergence

Numerical Approximation:

uex = unum +O(∆xn) +O(∆t)

RMS Error:

eRMS =√

1np

∑npp=1 (unum(xp, t)− uex(xp, t))2

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Vibrating string

100 101 10210−6

10−5

10−4

10−3

10−2

Number of elements [−]

RM

S E

rror

[m]

4 PPC6 PPC8 PPC

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Vibrating string

For the vibrating string problem, when particles (almost) stay equallydistributed:

• MPM shows 2nd order convergence in space.

• Increasing number of PPC, decreases the RMS error.

Order of convergence consistent with literature. 3

3M. Gong. Improving the Material Point Method. The University of NewMexico (2015)

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101 10210−4

10−3

10−2

Number of elements [−]

RM

S E

rror

[m]

4 PPC6 PPC8 PPC

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40 elements, no grid crossing

0 0.5 1 1.5 2 2.5−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

time [s]

velo

city

[m/s

]

Velocity of particle 40

MPMExact

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80 elements, grid crossing

0 0.5 1 1.5 2 2.5−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

time [s]

velo

city

[m/s

]


MPMExact

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• Convergence only for a low number of elements.

• Grid crossings seriously effect quality of the solution.

Similar observations by Bardenhagen et al.4 and Steffen et al.5

4S. G. Bardenhagen, E. M. Kober. The generalized interpolation material pointmethod. Comput Model Engrg Sci, 5 (2004), pp. 477-495.


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High order MPM

Quadratic B-spline basis functions:

• are positive functions

• have continuous derivatives

• lead to lower quadrature errors 6

• possess partition of unity property

B-spline basis functions (seem) well suited for MPM!


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B-splines

0 1 2 3 4 5 60

0.2

0.4

0.6

0.8

1

1.2

1.4

x

y

B spline basis functions

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Outlook

B-spline MPM already implemented, next steps:

• Comparison with standard MPM (Benchmark problems)

• Apply an MLS approach to decrease quadrature error

• Decrease quadrature error with B-spline approximation

• Investigate 3rd order B-spline basis functions

• Extension of B-spline MPM to 2D

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Linear vs. B-spline MPM

0 0.5 1 1.5 2 2.5−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

time [s]

velo

city

[m/s

]


standard MPMExact solution

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Linear vs. B-spline MPM

0 0.5 1 1.5 2 2.5−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

time [s]

velo

city

[m/s

]


B−spline MPMExact solution

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