Nonlinear Finite Element Methods and Computational Contact ...shortcourse2018.it.cas.cz/im/data/my/2018_Lecture_23.pdf · Nonlinear Finite Element Methods and Computational Contact

Alexander Popp

ECCOMAS Advanced Course 2018- Computational Structural Dynamics - Prague, Czech Republic, June 4-8, 2018

Nonlinear Finite Element Methods and Computational

Contact Mechanics

Institute for Mathematics and Computer-Based Simulation (IMCS)University of the Bundeswehr Munich (UniBw M), Germany

Technische Universität München

Alexander Popp, Institute for Mathematics and Computer-Based Simulation (IMCS), UniBw M

Unit 23 - CONTACT PROBLEMS 2 (Friday, 11:40-12:40)

• contact discretization techniques • mortar methods for tied contact • mortar methods for unilateral contact • semi-smooth Newton methods • parallel HPC implementation

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Agenda

ɠ METHODS 1

ɡ METHODS 2

ɢ METHODS 3

ɣ METHODS 4

ɤ APPLICATION

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Contact Discretization Techniques

Mortar Methods for Tied Contact

Mortar Methods for Unilateral Contact

Iterative Solution / Semi-Smooth Newton

Parallel HPC Implementation



• finite element method used for discretization of continuum,i.e. 2D continuum or 3D continuum

• here, we consider simple first-order and second-order Lagrangian elements2D 3-node triangle (tri3)4-node quadrilateral (quad4)6-node triangle (tri6)8-node quadrilateral (quad8)9-node quadrilateral (quad9)

• contact effects take place on theboundary, and the boundary discretizationis naturally inherited from theunderlying volume discretization

• 3D 4-node tetrahedron (tet4)8-node hexahedron (hex8)10-node tetrahedron (tet10)20-node hexahedron (hex20)27-node hexahedron (hex27)

[courtesy: P. Wriggers]

Finite element discretization

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Finite element discretization• common types of element boundaries of Lagrangian finite elements

line2 line3 quad4 tri3 quad9 quad8 tri6

2D 3D

u(2)

u(1)

λ

master side

slave side

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Finite element discretization• discretization of kinetic, internal, external and virtual work as usual • only consider contact contributions now (e.g. for unilateral contact) • for simplicity, only consider generic contributions of type

• strictly speaking, we need discretizations of the displacements (in the gap function) and of the additional Lagrange multiplier / traction field

• the product of those two then has to be integrated over the contact surface • this is the only really correct and intuitive thing to do… • this is called segment-to-segment (STS) or mortar approach (later) • however, this is NOT what is usually done…

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contact virtual work contact constraint equations



Matching / non-matching meshes• important for choice of contact discretization technique

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[cop

yrig

ht: C

OM

SOL]

MATCHING MESHES • only a very fortunate special case • impossible for large deformations • more relevant for tied contact • inflexible for mesh generation • allows for node-wise coupling

NON-MATCHING MESHES • that is what usually happens • standard for large deformations • much more challenging • gives flexibility for mesh generation • requires segment-based coupling



Overview - Contact discretization

• (1) Node-to-node (NTN) contact discretization

• (2) Node-to-segment (NTS) contact discretization

• (3) Gauss-point-to-segment (GPTS) contact discretization

• (4) Segment-to-segment (STS) contact discretization

• (5) Mortar contact discretization

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Node-to-node (NTN) contact discretization• only possible for matching meshes • evaluate node pairs point-wise

• simplification 1: no integration • simplification 2: no LM interpolation

(one Lagrange multiplier at each node)

• very easy to evaluate numerically, since not even the displacement shapefunctions enter the contact contributions, but directly the nodal displacements

• neighborhood of each node is considered with tributary area Ai

• leads to a simple point-wise coupling (also gap only evaluated at node pair)

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[courtesy: P. Wriggers]



Node-to-segment (NTS) contact discretization• possible also for non-matching meshes • the standard approach for many years • still today in many commercial codes

• simplification 1: no integration • simplification 2: no LM interpolation

(one Lagrange multiplier at each node)

• still quite easy to evaluate numerically, but now at least the displacement shape functions enter the contact contributions, due to NTS projection

• same format as for NTN but gap now requires (nonlinear) NTS projection

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Node-to-segment (NTS) - Typical problems• inconsistent (only nodes are checked for contact),

failure to satisfy so-called patch tests exactly—> possible solution: tailored modifications, but none reallyunphysical oscillations in so-called dropping-edge problems —-> possible solution: none really

• introduces slave-master-concept (contact only checked on 1 side),results may depend massively on choice of slave side —> possible solution: so-called two-pass algorithms

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Gauss-point-to-segment (GPTS) approach• possible for non-matching meshes • extension of the NTS approach • overcomes most major NTS problems

• simplification 1: no LM interpolation(one Lagrange multiplier at each Gauss point)

• now, the integration is really carried out (different from NTN / NTS) • more involved to evaluate numerically, since not only a point-wise model any

more, but including numerical integration (easily factor 10 in CPU time)

• kind of an intermediate step between NTS and STS methods

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Z

�c

�N�gNd� !nsegX

s=1

Z

�c,seg


s=1

ngpX

g=1

wi�Ni�gNiJi



Gauss-point-to-segment (GPTS) - Problems• still, patch tests are not satisfied exactly, but only within integration error • unbiased treatment only possible with two-half-pass (2hp) version

• big new problem: inf-sup-stability (or LBB-stability)

• important criterion for mixed finite element formulations(here: displacements and Lagrange multipliers)

• GPTS overly simplifies Lagrange multiplier interpolation —> not LBB-stable • effects of LBB-stability/-instability difficult to assess

(1) when Lagrange multipliers are used, the formulation must be LBB-stable(2) when a penalty formulation is used, an LBB-unstable approach can be ok, however, usually robustness problems for high penalty parameters then

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Segment-to-segment (STS) approach• the ideal approach for non-matching meshes • intuitive handling of contact integrals • no simplifications / no collocation

• LM interpolation: with a set of FEM shape functions(real Lagrange multiplier field on the entire surface)

• full integral evaluation with FEM-discretized fields u and λ

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Z

�c


s=1

Z

�c,seg


s=1

ngpX

g=1

wi�Ni�gNiJi

same formulation as for GPTS,but now λN is not defined pointwise

at the Gauss points (collocation), but rather as an interpolated fieldwith shape functions and discrete

nodal values

�h =mX

j=1

�j�j



Segment-to-segment (STS) approach• early STS formulation [e.g. Simo and Wriggers 1985] had some problems • how to adequately define the Lagrange multiplier interpolation,

i.e. how to define it such that the LBB-stability condition is fulfilled? • first attempt: piecewise constant Lagrange multiplier —> LBB-unstable • other attempts: independent intermediate surface —> difficult • general framework to deal with these problems: mortar methods

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graphical interpretation of (integral) STS approach[courtesy: P. Wriggers]



Mortar approach• special type of STS approach • (1) No independent intermediate surface is introduced for LM

interpolation, but one of the two surface discretization is re-used • (2) The LM interpolation (shape functions) is chosen identical to the

displacement interpolation on this side of the contact surface • (3) The patch test is passed exactly and LBB-stability is guaranteed,

which is not possible in this combination for any other method

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however, we will later see that there is also analternative (so-called dual mortar) approach,where this second criterion is dropped, but

the beneficial properties are still all kept

�j

N (1)j

N (2)k

�(2)c

�(1)c⇠(1) !



Agenda

ɠ METHODS 1

ɡ METHODS 2

ɢ METHODS 3

ɣ METHODS 4

ɤ APPLICATION

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• non-conforming discretization methods for partial differential equations

• originating from domain decomposition

• coupling of separate, non-matching

discretizations (or even different physical fields) on non-overlapping subdomains

• potential for mathematical optimality (inf-sup stability, a priori error estimates)

➢ mortar methods “glue“ together the subdomain solutions in a variationally consistent manner

➢ corresponding continuity conditions are

enforced using Lagrange multipliers

Mortar finite element methods – Motivation

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submodels in industrial application (e.g. due to different analysts)

Why mortar methods?• coupled problems with node-matching interface meshes are “straightforward“ • however, many real-world scenarios forbid matching grids (e.g. contact dynamics, complex fluid-structure interaction, …)

different resolution requirements in cumbersome mesh generation for physical domains (e.g. aeroelasticity) complex geometries (e.g. biomechanics)

➢mortar methods guarantee a consistent load / motion transfer at non-conforming interfaces (where collocation methods fail)

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λ

Γc

u(1)

u(2)

Model problem – Solid mechanics mesh tying• IBVP of nonlinear 3D elastodynamics (here for two subdomains: i=1,2)

• weak saddle point formulation with interface Lagrange multipliers λ

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Model problem – Solid mechanics mesh tying• kinetic, internal and external virtual work contributions

• virtual work of Lagrange multipliers (i.e. interface tractions) and weak form of kinematic continuity constraints

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line2 line3 quad4 tri3 quad9 quad8 tri6

2D 3D

u(2)

u(1)

λ

master side

slave side (+ Lagrange multipliers)

Mortar finite element discretization• common types of Lagrangian mortar finite elements

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• discrete Lagrange multipliers

➢ most natural choice: ➢ trace space relation with underlying FE ➢ discrete inf-sup stability guaranteed ➢ optimal a priori error estimates can be derived ➢ (rather) straightforward for higher-order FE ➢ (rather) straightforward for IGA, etc.

➢ but: global support of interface coupling /projected basis functions [WOHLMUTH 2000]

e.g. quadratic interpolation 2D (line3)

how to define these shape functions?

e.g. global vs. local support

Standard mortar methods

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• biorthogonality

➢ stable local support of basis functions [WOHLMUTH 2000] ➢ constraints at the interface can be locally satisfied

• element-wise construction [FLEMISCH 2007] and linearization possible [POPP ET AL. 2009/2010]

• extension to higher-order elements [POPP ET AL. 2012] and isogeometric analysis [SEITZ, POPP ET AL. 2016]

1

-1

2

0

1

-1

2

0

N1 N2

Φ1 Φ2

e.g. linear 2D (line2)

e.g. bilinear 3D (quad4)

how to define these shape functions? 1

4

1

-2

Dual mortar methods – Biorthogonality

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Mortar methods – Crosspoints• crosspoint = vertex where more than two subdomains meet

• requires modification of the Lagrange multiplier shape functions • otherwise, problems due to over-constraining • most popular approach: remove discrete Lagrange multiplier from crosspoint

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standard mortar approach [Puso 2003] dual mortar approach



Mortar coupling – Discrete form• remember mesh tying virtual work contribution to weak form

• insert discretization of displacements and Lagrange multipliers

• with a suitable discrete mapping from slave to master side • similar approach also for weak form of constraints (due to symmetry)

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Mortar coupling – Mesh imprinting• discrete coupling matrices

➢ dual LM make D diagonal ➢ mixed shape functions in M

require accurate projection,and segmentation (in 3D)

➢ method inspired by [PUSO 2004]

➢ fast element-based schemespossible [FARAH, POPP ET AL. 2015]

➢ consistent boundary handlingimportant [POPP ET AL. 2013]

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▪ Continuous field of normals

▪ Projection (e.g. master à slave)

▪ Mapping (e.g. linear interpolation)

Contact segments

Mortar projection and integration (2D)

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Mortar projection and integration (3D)

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Mortar coupling – Mesh imprinting• discrete coupling matrices

➢ dual LM make D diagonal ➢ mixed shape functions in M

require accurate projection,and segmentation (in 3D)

➢ method inspired by [PUSO 2004]

➢ fast element-based schemespossible [FARAH, POPP ET AL. 2015]

➢ consistent boundary handlingimportant [POPP ET AL. 2013]

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Solution algorithm – Implicit time integration• spatially discrete virtual work of Lagrange multipliers

and spatially discrete form of kinematic continuity constraints

• semi-discrete (spatially discrete / time continuous) system

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Solution algorithm – Implicit time integration• just using standard implicit time integration schemes here,

e.g. generalized-alpha method [CHUNG 1993], e.g. generalized energy-momentum method [KUHL 1999]

• when to evaluate mesh tying forces fmt and constraints gmt? • forces evaluated at generalized midpoint (typical of alpha methods)

• constraints evaluated at endpoint (will be important for unilateral contact) • irrelevant here due to linearity of mesh tying constraints

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Solution algorithm – Linearized system

•

➢ increased system size due to Lagrange multiplier DOFs ➢ classical saddle point system (symmetric but indefinite) ➢ however, conforming discretization yields a linear system that is both symmetric and positive definite

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u(2)

u(1)

λ



• extract slave quantities from rows 3 and 4

➢ negligible computational costs due to dual LM approach ➢ leads to condensed linearized system that is again symmetric and positive definite

projection operator P

Condensation of dual Lagrange multipliers

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Mesh tying – Patch tests

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displacement and stress solutionare exact up to machine precision

if the interface is not curved(no other contact discretization

approach can do this for arbitrarynon-matching meshes)



Mesh tying – Patch tests

treatment of crosspoints in 3D

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treatment of crosspoints in 2D



➢ optimal rates of spatial convergence are preserved despite non-conformity

Mesh tying – Convergence

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Alexander Popp, Institute for Mathematics and Computer-Based Simulation (IMCS), UniBw M!38

Mesh tying – Dynamics



➢ exact conservation of linear momentum, angular momentum and energy

energy-momentum method (EMM) used as time integration scheme

Mesh tying – Dynamics

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Agenda

ɠ METHODS 1

ɡ METHODS 2

ɢ METHODS 3

ɣ METHODS 4

ɤ APPLICATION

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Problem setup• IBVP of nonlinear elastodynamics

• HSM conditions and Coulomb friction law

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Weak formulation• convex cone of Lagrange multipliers

• final weak problem statement (saddle point formulation)

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Finite deformation contact and frictionAdditional complexities

- active set strategy / semi-smooth Newton method for inequality constraints

- separate treatment of normal and tangential directions (e.g. frictionless sliding, friction)

- large deformations and sliding require permanent re-evaluation and also a full linearization of coupling terms (e.g. D/M)

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Semi-discrete problem formulation• mortar discretization similar to mesh tying case

(i.e. interface projection and segmentation, coupling matrices D and M)

• discretization of contact virtual work contribution

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Semi-discrete problem formulation• compact notation using mortar matrices

• discretization of weak form of contact constraints leads to weighted nodal expressions (e.g. here nodal HSM conditions, frictionless contact)- no proof given here - (see Hüeber 2008, dissertation Univ. Stuttgart)

• with the so-called weighted gap:

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Problem formulation• full semi-discrete problem formulation for frictionless contact

• time discretization with generalized-alpha method yields- but several issues have to be addressed in dynamics (skipped here) -

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Conservation laws• semi-discrete linear and angular momentum conservation?

• linear momentum conservation no problem(if D and M are carefully integrated using segments / cells for both)

• angular momentum conservation not exactly possible(since this would require either gap vector be zero or collinear to LM)

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Agenda

ɠ METHODS 1

ɡ METHODS 2

ɢ METHODS 3

ɣ METHODS 4

ɤ APPLICATION

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▪ Discrete KKT conditions stated as inequalities

▪ First idea for active set strategy

→ active set

→ inactive set

→ all slave nodes

à after each Newton step, active as well as inactive condition will be violated (in general) à how to update the active contact set (in the nonlinear realm)?

à fixed-point active set strategy (2 nested loops)

do Active set loop (index i)do Newton iteration (index k)

solve for increment Δd k

while unconverged (e.g. )while unconverged (e.g. )

Primal-dual active set strategy (PDASS)

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Semi-smooth Newton method• reformulate each set of discrete nodal KKT conditions (inequality constraints)

within single nonlinear complementarity (NCP) function

➢ equivalent to KKT conditions ➢ NCP functions only C0-continuous ➢ residual notation of both inactive and active contact constraints ➢ semi-smooth Newton method

[Alart and Curnier 1991], [Christensen et al. 1998], [Hüeber and Wohlmuth 2005]

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one

sem

i-sm

ooth

New

ton

loop

Full linearization of all deformation-dependent quantities guarantees quadratic convergence in the limit!

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Extension to Coulomb friction• how to derive an objective relative tangential velocity? • intuitive guess from simple discretization

• apply rigid body rotation Q to check for frame indifference

• alternative based on mortar time derivatives [PUSO/LAURSEN 2004]

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not frame indifferent !

Backward Euler time integration



NCP function for 3D Coulomb friction

Extension to Coulomb friction• same framework as for frictionless case • semi-smooth Newton method for stick / slip inequality conditions • fully consistent linearization • condensation of dual Lagrange multipliers

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➢ high accuracy of Lagrange multiplier solution even for coarse meshes

Examples – 2D/3D Hertzian contact

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numerical results for contact tractions

convergence behavior of nonlinear solver

Numerical examples – Hertzian contact

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➢ quadratic convergence of semi-smooth Newton due to full linearization

Deformation and Newton convergence for a representative time step

hex8 tet4 hex27

A. Popp, M. Gitterle, M.W. Gee, W.A. Wall, A dual mortar approach for 3D finite deformation contact with consistent linearization, IJNME, 83 (2010), pp. 1428-1465

Examples – 3D ironing

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Further examples

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Agenda

ɠ METHODS 1

ɡ METHODS 2

ɢ METHODS 3

ɣ METHODS 4

ɤ APPLICATION

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HPC code framework• multiphysics research code BACI • based on overlapping domain

decomposition (Trilinos) and MPI

• mortar toolbox ➢ mesh tying, contact, friction ➢ penalty, Uzawa, standard and dual LM

• efficient (self) contact search algorithm ➢ bounding volumes based on k-DOPs ➢ parallel hierarchical structure (binary tree)

How to achieve optimal parallel scalability over a wide range w.r.t. number of cores?

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finite element evaluation

contact segment evaluation

iterative linear solvers

partitioning based on overlapping domain decomposition (e.g. ParMETIS, Trilinos Zoltan) [Heroux et al. 2004]

efficient mortar integration[Farah et al. 2015]

re-partitioning and load balancing

efficient precondi-tioners (AMG) [Vanek et al. 1996]

applicable to contact problems

✔ ?

✔

?

✔

Code components

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• parallel distribution of finite element mesh based on over-lapping domain decomposition

• non-optimal scalability if contact interfaces are involved

• parallel distribution “optimized“ for element evaluation, but not efficient for contact evaluation (very localized, variable)

à parallel scalability for large-scale simulations is limited by contact interfaces

Parallel implementation and scalability

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number of processors

com

puta

tion

time

[s]

1/x

• model problem: mortar mesh tying • 2,136,177 DOFs (15k mortar nodes) • CPU time includes everything related to mortar coupling (search, projection, • integration, assembly of mortar terms)

➢why does this happen?

e.g. 32 procs, but only 5 procs do all mortar evaluation!

Parallel scalability – first results

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body1 (slave)

body2 (master)

slave interface

redistribution

redistribute the mortar elements in this zone equally among all processors

Dynamic load balancing strategy• Step 1 (mesh tying and contact) ➢ redistribution of mortar interface ➢ (independent of the underlying distribution) ➢ non-local assembly of mortar terms

• Step 2 (contact only) ➢ restrict redistribution to current contact ➢ proximity (i.e. where work needs to be done) ➢ dynamic redistribution based on some ➢ balance measure (i.e. processor workload)

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• model problem: mortar mesh tying • 2,136,177 DOFs (15k mortar nodes)

➢ reduction in CPU time up to factor 10 ➢ especially important for contact (re-evaluation in every Newton step) (expensive contact stiffness terms)

Parallel scalability – try again

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➢ stability of semi-smooth Newton also for multiple interfaces / self contact

Self contact and multibody contact

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➢ simulation of particle contacts based on nonlinear elasticity and friction

Examples – Self contact / multibody contact

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➢ dynamic load balancing algorithm assures optimal parallel scalability

Examples – Three-dimensional particles

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• impact of two thin-walled structures (13,994,880 DOFs)

• 500 time steps computed on LRZ SuperMUC cluster, three Intel Xeon nodes (= 120 cores)

• total computation time approximately 48 hours

• iterative linear solver with multigrid (AMG) preconditioner

➢ dynamic load balancing assures parallel scalability of mortar coupling

Numerical examples – Large scale model

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Numerical examples – Large scale model

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Numerical examples(Parallel redistribution and dynamic load balancing)

• 3D rolling tire contact simulation • approximately 500,000 DOFs

distributed on 16 processors • scalability due to load balancing • robustness of semi-smooth Newton

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Numerical examples(Parallel redistribution and dynamic load balancing)

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Contact search(just one slide - no comprehensive introduction)

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• essential for efficient large-scale algorithms • prohibitive complexity of brute force approach • see [WILLIAMS AND O’CONNOR 1995], [WRIGGERS 2006] • here parallel version of binary tree from [YANG AND LAURSEN 2008] —> hierarchical (binary) tree structure —> discrete orientation polytopes (k-DOPs) as bounding volumes —> e.g. typically 8-DOPs in 2D / 18-DOPs in 3D



Self contact search(just one slide - no comprehensive introduction)

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• again, essential for efficient large-scale algorithms • see [BENSON AND HALLQUIST 1990], [WRIGGERS 2006] • here self contact binary tree from [YANG AND LAURSEN 2008] —> hierarchical (binary) tree structure with k-DOPs —> single self contact surface more demanding —> no a priori assignment of slave / master surface —> dual graph for bottom-up construction (not top-down)



THE END!



Summary and outlook• text 1 • text 2 • text 3 • text 4

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Thank you very much for your attention!

Some recent papers / preprints: Popp, A., Wriggers, P. (Eds.) (2018): Contact Modeling for Solids and Particles, CISM International Centre for Mechanical Sciences 585, Springer International Publishing Hiermeier, M., Wall, W.A., Popp, A. (2018): A truly variationally consistent and symmetric mortar-based contact formulation for finite deformations, Computer Methods in Applied Mechanics and Engineering, under review Seitz, A., Wall, W.A., Popp, A. (2018): A computational approach for thermo-elasto-plastic frictional contact based on a monolithic formulation employing non-smooth nonlinear complementarity functions, Advanced Modeling and Simulation in Engineering Sciences, 5:5 (open access) Farah, P., Wall, W.A., Popp, A. (2018): A mortar finite element approach for point, line and surface contact, International Journal for Numerical Methods in Engineering, published online

Please contact me for any questions: [email protected]

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Nonlinear Finite Element Methods and Computational Contact ...shortcourse2018.it.cas.cz/im/data/my/2018_Lecture_23.pdf · Nonlinear Finite Element Methods and Computational Contact