october 2016 alternative implicit mpm formulations · material point methods for geohazards 17th...

ALTERNATIVE IMPLICIT MPM FORMULATIONS

Antonia Larese, Ilaria Iaconeta, Riccardo Rossi, Eugenio Oñate

Material Point Methods for Geohazards 17th October 2016

GEO-RAMP H2020 MSCA-RISE-2014 GA n.645665

OUTLINE

• MOTIVATION • IMPLICIT vs EXPLICIT FORMULATIONS • IMPLICIT MPM

• GRID BASED • MESHLESS

• VALIDATION • KRATOS open-source platform • FUTURE WORK

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MOTIVATION

CIMNE tasks (ongoing doctoral work of I. Iaconeta) Development of a NUMERICAL TOOL

for the simulation of granular flows at the MACROSCALE, focusing on DRY GRANULAR MATERIAL in STATIC and FLOWING regime

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Training in Multiscale Analysis of multi-Phase Particulate Processes FP7 PEOPLE 2013 ITN–Grant Agreement nº607453

www.t-mappp.eu

http://www.t-mappp.eu/

MOTIVATION

We look for a numerical technique • In the Continuum mechanics framework Able to • Handle large deformation and displacement (GEOMETRIC NON LINEARITY) • Handle history dependent material (MATERIAL NON-LINEARITY) and • With good conservation properties • Parallelizable • Modular (multiphase, multimaterial,…)

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MPM

IMPLICIT vs EXPLICIT FORMULATIONS

Most of MPM codes are explicit Few authors developed implicit MPM codes • Guilkey and Weiss – similarities FEM-MPM • Cummins and Brackbill – Newton-Krylov approach • Beuth – quasi static problems, higher order elements • Sanchez – quasi static problems • …

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IMPLICIT vs EXPLICIT FORMULATIONS

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IMPLICIT SCHEMES: • More complicated

• More expensive

More accurate

Stability does not depend on the wave propagation speed in the

media

Can solve from Static and Quasi static to dynamic and gravity

driven problems

More robust

More “FEM like”

IMPLICIT MPM

Two different approaches: • Grid-based MPM: it uses a fixed background grid which is

deformed and “reset” at each time step. • Meshless MPM: it employs a purely Lagrangian approach The only conceptual difference is in the calculation of the SHAPE FUNCTIONS and SHAPE FUNCTION DERIVATIVES on the material points. The similarities/differences with standard Updated Lagrangian (UL) finite elements (FE) techniques will be highlighted.

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GRID-BASED MPM

IMPLICIT MPM: GRID-BASED

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NOMENCLATURE: • i, j, k material points (MP) • I, J, K grid nodes (nodes)

Background grid: Linear Triangular elements • Unstructured grid • Easier definition of non regular

boundaries • Less accurate than quadrilateral

elements especially on the boundaries

Initial position of the MP: integration points of the FE background grid • Original mass transferred to the

MP with minimal error

IMPLICIT MPM: GRID-BASED

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Each MATERIAL POINT is defined by the material point itself and its connectivity It can be seen as a finite element with a moving integration point

Material Point i: • pi(xi, yi) • Connectivity IJK

Material Point j: • pj(xj, yj) • Connectivity IJK

IMPLICIT MPM: GRID-BASED Solution strategy Classical MPM stages are followed at each time step:

1. INITIALIZATION PHASE: definition of the initial conditions on the FE grid’s nodes

2. UL-FEM CALCULATION PHASE solution on the nodes of the background grid

3. CONVECTIVE PHASE: information is interpolated and stored on the particles which are moved on the calculated positions

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1. Initialization phase

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At the beginning of every time step, the initial conditions on the FE grid’s nodes are defined during the initialization phase. The initialization phase is composed by: a. Iterative Extrapolation on the

nodes of the material points information obtained at the previous time step tn

b. Prediction of nodal displacement, velocity and acceleration using a Newmark scheme

tn

Remark: nomenclature

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: MP displacement

: MP velocity

: MP acceleration

: superscript n or n+1 indicates the time step (tn or tn+1) on which the variable is evaluated. tn is the current (known) time step tn, while tn+1 is the next unknown time step

: subscript p refers to MP variables, while I to nodal one

: superscript it or k refer to the iteration within the iterative extrapolation and the time step respectively

1. Initialization phase a) Iterative extrapolation

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: known : identically null after grid reset

While

Loop over MP • Interpolation from nodes to MP ( )

: unkown

While

Loop over MP • Interpolation from nodes to MP ( , ) • Extrapolation from MP to nodes ( , , )


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: known : identically null after grid reset : unkown

Nodal momentum :

Nodal inertia :

Nodal mass :


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: known : identically null after grid reset : unkown

While

Loop over MP • Interpolation from nodes to MP ( , ) • Extrapolation from MP to nodes ( ) • Evaluation and Update of nodal


Once

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=

=

= identically zero

To be used in the Newmark prediction

Result of the iterative extrapolation

1. Initialization phase b) Prediction (Newmark)

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• Prediction of nodal displacement, velocity and acceleration using a Newmark scheme

: known : known (from iterative extrapolation)

2. Updated Lagrangian phase

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: current configuration

: initial (undeformed) configuration

tn

• CALCULATION PHASE

• From tn to tn+1

• Implicit formulation: solution of an

algebraic system of equations

• FE traditional steps

tn+1

2. Updated Lagrangian phase

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FE PROCEDURE: a. ELEMENTAL SYSTEM.

The local left-hand-side (lhs) and right-handside (rhs) are evaluated in the current configuration

b. ASSEMBLING. The global LHS and RHS are obtained by assembling the local contributions

c. SOLVING. The system is iteratively solved. is calculated

• During the iterative procedure the nodes are allowed to move, accordingly to the nodal solution

• The material points do not change their local position within the geometrical element until the solution has reached convergence

: current configuration

: initial (undeformed) configuration

tn

tn+1

2. Updated Lagrangian phase Algebraic solution system

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• NON LINEAR PROBLEMS • Geometrical non linearity • Material non linearity

• STATIC and DYNAMIC PROBLEMS

FE PROCEDURE: a. ELEMENTAL SYSTEM. b. ASSEMBLING. c. SOLVING.

The system is iteratively solved is calculated

d. CHECK CONVERGENCE • NEWMARK CORRECTION

2. Updated Lagrangian phase Check convergence

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NO

2. Updated Lagrangian phase Newmark correction

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FE PROCEDURE: a. ELEMENTAL SYSTEM. b. ASSEMBLING. c. SOLVING.

The system is iteratively solved is calculated

d. CHECK CONVERGENCE • NEWMARK CORRECTION

2. Updated Lagrangian phase Check convergence

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NO YES

END Go to next step:

Convective Phase

3. Convective phase

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1. Nodal information at time tn+1 are interpolated back onto the material points.

MP displacement:

MP velocity:

MP acceleration:

MP position (update):

2. MP position is updated

3. Convective phase

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1. Nodal information at time tn+1 are interpolated back onto the material points.

2. MP position is updated 3. The undeformed FE grid is

recovered. 4. The material points

connectivities are updated (identify the element in which each MP falls)

UPDATED LIST OF MATERIAL POINTS WITHIN EACH ELEMENT

UNDEFORMED GRID RECOVERED

MP POSITION UPDATED

MESHLESS MPM

IMPLICIT MPM: MESHLESS

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• The Lagrangian nodes are moving during the process

• They are used for the computation of the shape functions in a meshless fashion

• No mesh, no connectivity is present

• Nodes will carry their history

• No need of extrapolating information from MP to the nodes

IMPLICIT MPM: MESHLESS How to compute the MP shape functions?

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MOVING LEAST SQUARES technique to approximate a given function u(x) using polynomial functions of the type

unknown coefficients

Nodal value of the function uI = u(xI)

IMPLICIT MPM: MESHLESS How to compute the MP shape functions?

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MOVING LEAST SQUARES technique


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MLS shape functions: • Comply the PARTITION OF UNITY property at the material points positions: • Do NOT comply the DELTA KRONECKER property at the nodes

• Are NOT INTERPOLANTS


The MLS shape functions do NOT comply the DELTA KRONECKER property at the nodes

Problem on the imposition of Dirichlet boundary conditions

These should be treated using a MULTIPOINT CONSTRAINT technique. We choose Lagrangian Multipliers

(check http://www.colorado.edu/engineering/CAS/courses.d/IFEM.d/IFEM.Ch08.d/IFEM.Ch08.pdf and http://www.colorado.edu/engineering/CAS/courses.d/IFEM.d/IFEM.Ch09.d/IFEM.Ch09.pdf )

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http://www.colorado.edu/engineering/CAS/courses.d/IFEM.d/IFEM.Ch08.d/IFEM.Ch08.pdf






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Look for all nodes I such that: Compute and such that being u a given function and uI its value on node I

: search radius

: MP coordinates

: nodal coordinates

At every time step a SEARCH of the neighbor nodes to each MP should be performed

IMPLICIT MPM: MESHLESS Solution strategy

1. INITIALIZATION PHASE • NO extrapolation from particle (the nodes keep their history) • Newmark prediction

identical to Grid based MPM but

Prediction made on the nodes

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1. INITIALIZATION PHASE • NO extrapolation from particle • Newmark prediction

2. UL-FEM CALCULATION PHASE • (identical to Grid based MPM)

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3. CONVECTIVE PHASE • Update MP position interpolating nodal information

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3. CONVECTIVE PHASE • Update MP position • Update the list of nodes such that

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CONSTITUTIVE LAWS

• HYPERELASTIC LAW: NEO-HOOKEAN MATERIAL

Ψ 𝐽𝐽,𝑪𝑪𝒆𝒆 =12𝐾𝐾𝑙𝑙𝑙𝑙2 𝐽𝐽 +

12𝜇𝜇 𝑡𝑡𝑡𝑡 𝑪𝑪𝒆𝒆 − 3

𝑺𝑺 = 2𝜕𝜕Ψ𝜕𝜕𝑪𝑪𝒆𝒆

𝑪𝑪𝒆𝒆 ℂ = 4𝜕𝜕2Ψ

𝜕𝜕𝑪𝑪𝒆𝒆𝜕𝜕𝑪𝑪𝒆𝒆

Ψ : Free energy function

S : II Piola-Kirchhoff stress tensor ℂ: IV order constitutive tensor

CONSTITUTIVE LAWS Elastic regime

• HYPOELASTIC LAW – future work

• HYPERELASTIC-PLASTICITY: J2 PLASTIC THEORY “Computational Inelasticity”, Simo & Hughes (1998)

𝑭𝑭 = 𝑭𝑭𝑒𝑒𝑭𝑭𝑝𝑝 𝑭𝑭 = 𝐽𝐽1/3𝑭𝑭�

𝝉𝝉 = 𝐾𝐾 𝑙𝑙𝑙𝑙𝐽𝐽 𝟏𝟏 +12𝐺𝐺 𝑑𝑑𝑑𝑑𝑑𝑑𝒃𝒃�𝑒𝑒

𝑑𝑑𝝉𝝉𝑛𝑛+1 = ℂ�𝑛𝑛+1: 𝑑𝑑𝜺𝜺𝑛𝑛+1 − 𝜺𝜺𝑛𝑛+1𝑝𝑝

Multiplicative decomposition of the total deformation gradient F

KIRCHHOFF STRESS TENSOR 𝝉𝝉

𝑭𝑭 : Total deformation gradient

𝑭𝑭�: Volume-preserving part of 𝑭𝑭

𝒃𝒃�𝑒𝑒: Elastic part of Left Cauchy-Green tensor

𝐾𝐾 : Bulk modulus 𝐺𝐺 : Shear modulus

CONSTITUTIVE LAWS Plastic regime

ELASTIC PLASTIC


• HYPERELASTIC-PLASTICITY: J2 PLASTIC THEORY “Computational Inelasticity”, Simo & Hughes (1998)


• Associative rate-independent Von Mises

• Visco plastic law

• Non-associative Drucker-Prager

• …future work

1. ASSOCIATIVE RATE-INDEPENDENT VON MISES PLASTIC LAW

Yield surface

• Kuhn-Tucker restriction

• Plastic flow rule

Return mapping

State function at iteration i

H : Hardening contribution

∆γ: Plastic multiplier

�̅�𝐺 = 𝐺𝐺3𝑡𝑡𝑡𝑡 𝒃𝒃�𝑒𝑒 : Modified shear modulus

𝝈𝝈𝑌𝑌: Yield stress

𝜶𝜶: Equivalent plastic strain


P

• Metal plasticity

2. VISCO PLASTIC LAW Duvaut-Lions’ model

η: Viscosity Case of constant viscosity


Yield surface

State function at iteration i

• NON NEWTONIAN plastics


Return mapping

3. NON-ASSOCIATIVE DRUCKER-PRAGER YIELD CRITERION (Hofstetter & Taylor, 1990)

Plastic potential

𝜇𝜇 : Coefficient of friction 𝑘𝑘: Cohesion 𝜉𝜉: Dilatancy coefficient


Yield surface

• Kuhn-Tucker restriction


Return mapping

NUMERICAL EXAMPLES

STATIC ANALYSIS OF 2D CANTILEVER BEAM SUBJECTED TO SELF-WEIGHT

NUMERICAL EXAMPLES

DYNAMIC ANALYSIS OF 2D CANTILEVER BEAM SUBJECTED TO SELF-WEIGHT

NUMERICAL EXAMPLES


Displacement FEM CODE

MPM CODE

NUMERICAL EXAMPLES


Velocity along y-direction FEM CODE

MPM CODE

VALIDATION PLANAR GRANULAR COLUMN COLLAPSE

NON-ASSOCIATIVE DRUCKER-PRAGER YIELD CRITERION (Hofstetter & Taylor, 1990)

(Mast et al., 2015)


𝐿𝐿� =𝑙𝑙 𝑡𝑡 − 𝑙𝑙0𝑙𝑙𝑓𝑓 − 𝑙𝑙0

𝐻𝐻� =𝑙𝑙0 − ℎ 𝑡𝑡ℎ0 − ℎ𝑓𝑓

𝐻𝐻𝑓𝑓 =ℎ𝑓𝑓𝑙𝑙0

𝐿𝐿𝑓𝑓 =𝑙𝑙𝑓𝑓 − 𝑙𝑙0𝑙𝑙0

a Hf Lf Hf* Lf*

1 1 0.85-1.17 1 0.847

1.5 0.853-1.032 1.458-2.2 1.14 1.54

2 0.9432-1.175 1.804-2.574 1.23 2.13


IRREDUCIBLE FORMULATION (displacement)

MIXED FORMULATION (displacement-pressure)


Arrangement of softer and stiffer photoelastic disks

PISTON 1: initial pre-compression on the disks

PISTON 2: sinusoidal impulse imposed at the bottom of the sample

VALIDATION PHOTOELASTIC DISKS

In collaboration with the MULTI SCALE MECHANICS group of the University of Twente (NL) MSc G. Oliveri, Prof. V. Magnanimo , Prof. S. Luding

PISTON 1

PISTON 2

OBJECTIVE: study of wave propagation in heterogeneous media

Pre-compression: 200g Material composition: 70% stiffer - 30% softer particles



EVOLUTION OF THE CHAIN OF STRESS


EXPERIMENT MPM APPLICATION

EVOLUTION OF THE CHAIN OF STRESS

FUTURE WORK

• 3D • Constitutive laws (T-MAPPP) • Quadrilateral elements • Frictional boundary conditions

• Multi-phase

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Kratos is a framework for building multi-disciplinary (MULTI-PHYSICS) finite element programs. It provides several tools for fast implementation of finite element applications. CFD, CSD, Thermally Coupled Problems, Particles, ...

OPEN SOURCE

The dynamic nature of KRATOS itself is the principal reason of the continued evolution.

FLEXIBILITY

Kratos can be used with research purposes or by engineers looking for a solution to complex industrial problems

PARALLEL HPC

High performance computing in an OpenMP/MPI - based software.

www.cimne.com/kratos

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∫dΩ Ind. R&D Test

KRATOS www.cimne.com/kratos

KRATOS – Core-Application approach www.cimne.com/kratos

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CORE KRATOS CORE: Contains the basic/common tools for a computational code - Data structure - Solvers - Spatial containers - ….

KRATOS APPLICATIONS: Contains the physics

Fluid Dynamic Application

Solid Mechanics Application

DEM Application

etc…

KRATOS – Core-Application approach www.cimne.com/kratos

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CORE

Fluid Dynamic Application

Solid Mechanics Application

DEM Application

Thermo-Mechanics Applications

Particle Mechanics Application (MPM)

FSI Application

COUPLED or DERIVED APPLICATION • High reusability • Flexibility • Reduced conflicts

etc…

Thank you!

Antonia Larese [email protected]

www.cimne.com/kratos Kratos FORUM: kratos-wiki.cimne.upc.edu/forum

REFERENCES: • Iaconeta, I., Guo, Z. Larese, A., Rossi, R., An implicit grid-based and a meshless

MPM formulation for problems in solid mechanics. Submitted to International Journal for Numerical Methods in Engineering (2016)

• Iaconeta, I., Larese, A., Rossi, R., Oñate, E., An implicit material point method applied to granular flows. Proceeding of the 1st International Conference on the Material Point Method, MPM 2017, Delft, Netherlands (2017)

http://www.cimne.com/kratos

october 2016 alternative implicit mpm formulations · material point methods for geohazards 17th...

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