Coupled Thermomechanical Contact Problems

Prof. Dr. Ing. Carlos Agelet de Saracibar

Coupled Thermomechanical

Contact Problems

Computational Modeling of Solidification Processes

C. Agelet de Saracibar, M. Chiumenti, M. Cervera

ETS Ingenieros de Caminos, Canales y Puertos, Barcelona, UPC

International Center for Numerical Methods in Engineering (CIMNE)

COMPLAS Short Course,

Barcelona, Spain, 4 September 2007


Outline

Motivation and Goals

Coupled Thermomechanical Problem

Mechanical Problem

Solid Phase

Liquid Phase

Mushy Zone. Liquid-Solid Continuous Transition

Multigrid Scale Pressure Stabilization

Thermal Problem

Mechanical Contact Problem

Thermal Contact Problem

Computational Simulations

Concluding Remarks


Outline



Mechanical Problem

Solid Phase

Liquid Phase



Thermal Problem




Concluding Remarks



Truthful coupled thermomechanical simulation of solidification and cooling processes of industrial cast parts (usually meaning complex geometries), in particular for aluminium alloys in permanent moulds.

Thermodynamically consistent material model allowing a continuous transition between the different states of the process, from an initial fully liquid state to a final fully solid state.

Thermomechanical contact model allowing the simulation of the interaction among all casting tools as consequence of the thermal deformations generated by high temperature gradients and phase transformations during solidification and cooling processes.

Suitable stable FE formulation for linear tetrahedral elements, which allow the FE discretization of complex industrial geometries, and for incompressible or quasi-incompressible problems.






Outline



Mechanical Problem

Solid Phase

Liquid Phase



Thermal Problem




Concluding Remarks



LOOP, TIME

Fractional Step Method, Product Formula Algorithm

Solve the THERMAL PROBLEM at constant configuration

Solve the discrete variational form of the energy balance equation

, , , ,tcQ

cC L K Q Q

Obtain the temperatures map

Solve the MECHANICAL PROBLEM at constant temperature

Solve the discrete variational form of the momentum balance equation

, , , ,mc

s

u B u t u t u

Obtain the displacements map, and then the stresses map

END LOOP, TIME


Outline



Mechanical Problem

Solid Phase

Liquid Phase



Thermal Problem




Concluding Remarks


Mechanical Problem Solid and Liquid-like Phases





Liquid-like phase

Solid phase


Outline



Mechanical Problem

Solid Phase

Liquid Phase



Thermal Problem




Concluding Remarks


Mechanical Problem Solid Phase

T.D. Yield Stress

T.D. Viscosity T.D. Young Mod.

T.D. Hardening

20º C

500º C

700º C


Mechanical Problem

J2 Elasto-Viscoplastic model, isotropic and kinematic hardening

Solid Phase

The thermoelastic constitutive behaviour in solid state can be

written as,

2 dev p

p K e

G

u

s

where the is the mean pressure, is the deviatoric part of the stress tensor, is the bulk modulus, is the shear modulus and is the volumetric thermal deformation which takes also into account the thermal shrinkage due to phase-change,

( )S

G K p s

e

1 3 3Lref S S ref

S

e


Mechanical Problem


Solid Phase

The yield function can be written as,

( , , , ) : ( , )q R q s q s q

where the yield radius takes the form,

0

2( , )

3R q q


Mechanical Problem


Solid Phase

The plastic evolution equations can be written as,

where the plastic parameter takes the form,

( , , , )

( , , , ) 2 3

( , , , )

p

q

q

q

q

σ

q

ε s q n

s q

ξ s q n

1( , , , )

nq

s q

and the unit normal to the yield surface takes the form,

n s q s q


Outline



Mechanical Problem

Solid Phase

Liquid Phase



Thermal Problem




Concluding Remarks


Mechanical Problem

Viscous deviatoric model

Liquid-like Phase

In the liquid state , the thermal deformation is zero and the bulk modulus and shear modulus tends to infinity, characterizing an incompressible rigid-plastic material, such that

A purely viscous deviatoric model can be obtained as a particular case of the J2 elasto-viscoplastic model, without considering neither isotropic or kinematic hardening and setting to zero the radius of the yield surface. Then the yield function takes the form,

( ) : s s

0,

: ,p e

e K

G

u 0

0

( )L


Mechanical Problem

Viscous deviatoric model

Liquid-like Phase

The plastic evolution equation takes the form,

and the plastic strain evolution takes the simple form,

1( )

s n s s

( )p σ

ε s n

where the plastic parameter and unit normal to the yield surface take the form,

1p

ε s


Outline



Mechanical Problem

Solid Phase

Liquid Phase



Thermal Problem




Concluding Remarks


Mechanical Problem



The thermoelastic constitutive behaviour in mushy state

can be written as,

2 dev p

p K e

G

u

s

where the is the mean pressure, is the deviatoric part of the stress tensor, is the bulk modulus, is the shear modulus and is the volumetric thermal deformation due to the thermal shrinkage taking place during the liquid-solid phase-change,

( )L S

G K p s

e

0

pcL

SS

Ve f

V


Mechanical Problem



The yield function can be written as,

( , , , ) : ( , )s sq f f R q s q s q

The yield function for the solid phase is recovered for 1sf

The yield function for the liquid-like phase is recovered for 0sf

where the solid fraction satisfies,

0

1

s L

s S

f

f


Mechanical Problem



The plastic evolution equations can be written as,

where the plastic parameter takes the form,

( , , , )

( , , , ) 2 3

( , , , )

p

q s

s

q

q f

q f

σ

q

ε s q n

s q

ξ s q n

1( , , , )

nq

s q

and the unit normal to the yield surface takes the form,

s sf f n s q s q


Outline



Mechanical Problem

Solid Phase

Liquid Phase



Thermal Problem




Concluding Remarks


Mechanical Problem Incompressibility or Quasi-incompressibility Problem

Low order standard finite elements discretizations are very convenient for the numerical simulation of large scale complex industrial casting problems:

Relatively easy to generate for real life complex geometries Computational efficiency Well suited for contact problems

Low order standard finite elements lock for incompressible or quasi-incompressible problems.

Mixed u/p finite element formulations may avoid this locking behaviour, but they must satisfy a pressure stability restriction on the interpolation degree for the u/p fields given by the Babuska-Brezzi pressure stability condition.

Finite elements based on equal u/p finite element interpolation order do not satisfy the BB stability condition. Standard mixed linear/linear u/p triangles or tetrahedra elements do not satisfy the BB stability condition, exhibiting spurious pressure oscillations.



Stabilization of incompressibility or quasi-incompressibility problems using Orthogonal Subgrid Scale Method, within the framework of Variational Multiscale Methods (Hughes, 1995; Garikipati & Hughes, 1998, 2000; Codina, 2000, 2002; Chiumenti et al. 2002, 2004; Cervera et al. 2003; Agelet de Saracibar et al. 2006)

To stabilize the pressure while using mixed equal order linear u/p triangles and tetrahedra

Pressure stabilization goals



P1

Standard Galerkin linear u

element

P1/P1 OSGS

Stabilized mixed linear u/p

element

P1/P1

Standard mixed linear/linear u/p

element

Pressure map


Mechanical Problem Multiscale Stabilization Methods

Strong form of the mixed quasi-incompressible problem

Let us consider the following infinite-dimensional spaces for the displacements and pressure fields, respectively,

1: on uH u u u 2: L

Find a displacement field and pressure field such that, u p

1in

0K

p

e p

s B 0

u

Find such that, : U

in U F



Variational form of the mixed quasi-incompressible problem

Let us consider the following infinite-dimensional spaces for the displacements, displacements variations and pressure fields, respectively,

1: on uH u u u 2: L

Find a displacement field and pressure field such that, u p

1

, , , , 0in

, , 0

s

K

p

e q p q q

s v v B v t v v

u

Find such that, : U

, : inB L U V V V

1: H



Multiscale methods. Subgrid multiscale technique

Let us consider the following split of the exact solution field,

: :h U U U

where is the exact solution, is the finite

element approximated solution (belonging to the finite-dimensional FE

solution space) and is the subgrid scale solution, not

captured by the finite element mesh, belonging to the infinite-dimensional

subgrid scale space

U h h h h U

U

, ,0

h

h

hpp

uu uU U U


Mechanical Problem OSGS Stabilization Technique

Orthogonal Subgrid Scale (OSGS) technique

Within the OSGS technique, the infinite-dimensional subgrid scale space is approximated by a finite-dimensional space choosed to be orthogonal to the FE solution space (Codina, 2000)

h

U




The subgrid scale solution is then approximated, at the element level as,

1 , 1 , 1 , 1

1 0

n e n h h n h n

n

P p

p

u s B

where is a (mesh size dependent) stabilization parameter defined as, , 1e n

2

, 1 *2

ee n

ch

G

where is the secant shear modulus. *G




Using a linear interpolation for displacements (triangles/tetrahedra) and assuming that body forces belong to the FE space, the subgrid scale solution is then approximated, at the element level as,

1 , 1 , 1

1 0

n e n h h n

n

P p

p

u

where is a (mesh size dependent) stabilization parameter defined as, , 1e n

2

, 1 *2

ee n

ch

G

where is the secant shear modulus. *G



Discrete OSGS stabilized variational form

Find such that, for any h hU

,,stab h h stab h h hB L U V V V

0,h hV

where the stabilized variational forms can be written as,

, 1 , 11, , ,

elem

e

n

stab h h h h e n h h n heB B P p q

U V U V

stab h hL LV V




Let the projection of the discrete pressure gradient onto the FE solution space and the pressure gradient projection and FE associated spaces, respectively.

Then the orthogonal projection of the discrete pressure gradient onto the FE soluctiuon space takes the form,

1, hH , 1 , 1:h n h h nP p

, 1 , 1 , 1h h n h n h nP p p




Find such that, , 1 , 1 , 1, ,h n h n h n h h hp u

,0 ,0, ,h h h h h hq v

, 1 1 , 1 1

1

1 , 1 , 1 , 1 1 , 1 , 1 11

1 , 1 , 1

, , , , 0

, , , 0

, 0

elem

e

s

h n n h h n n h h h

n

n h n h h n h e n n h n h n n hK e

n h n h n h

p

q p q p q

p

s v v B v t v

u

for any



Strong form of the stabilized mixed quasi-incompressible problem

Let us consider the following infinite-dimensional spaces for the displacements, pressure and pressure gradient projection fields, respectively,

1: on uH u u u 2: L

Find a displacement field , a pressure field and a pressure gradient projection such that,

u p

1 2 0 inK

p

p p

p

s B 0

u

0

1H



Staggered solution procedure. Problem 1

Given find such that, , 1 , 1,h n h n h hp u

,0,h h h hq v

, 1 1 , 1 1

1

1 , 1 , 1 , , 1 ,1

, , , , 0

, , , 0elem

e

s

h n n h h n n h h h

n

n h n h h n h e n n h n h n n hK e

p

q p q p q

s v v B v t v

u

for any

,h n h



Staggered solution procedure. Problem 2

Given find such that, , 1 , 1,h n h n h hp u , 1h n h

, 1 1 , 1 , 11, 0

elem

e

n

e n n h n h n hep

,0h h for any



Matrix form of the staggered solution procedure

Problem 1. Solve the algebraic system of equations to compute the incremental discrete displacements and pressures unknowns

Problem 2. Compute the discrete continuous pressure gradient projections unknowns

1

( ) ( ) ( ) ( ) ( )

1 1 1 , 1

( ) ( ) ( ) ( )

1 1 , 1 , 1

n

i i i i i

T n n n u n

i T i i i

n n p n n p n

1

1 , 1 , 1 1ΠIn n n n


Outline



Mechanical Problem

Solid Phase

Liquid Phase



Thermal Problem




Concluding Remarks


Thermal Problem

Variational enthalpy form of energy balance

, , , ,tcQ

cH K Q Q

where the enthalpy rate is given by,

H C L

where the latent heat release is given by,

s ldf dfL L L

d d

where the solid fraction satisfies,

0

1

s L

s S

f

f

Energy Balance


Outline



Mechanical Problem

Solid Phase

Liquid Phase



Thermal Problem




Concluding Remarks


Nonlinear Contact Kinematics

(1)X

(2)X

(1)x

(2)x

(1)

(2)

(1)

t

(2)

t

(1) (1)

(2) (2)

(1)

(2)

(1)

(2)

(2)N

Reference Configurations

0t

(1)N

(1)n

(2)n

Current Configurations

t

Slave Body

Master Body

(2) (2)

(2) (1) (1) (1) (2) (2), ARGMIN t tt

X

X X X X (2) (1) (2) (2), tt x X X

45 16/12/2013


... Coordinates of the closest-point-projection of the slave particle onto the master surface at current configuration

... Coordinates of the master particle at reference configuration, defined as the closest-point-projection of the slave particle onto the master surface at the current configuration

... Coordinates of the slave particle at reference configuration


Closest-Point-Projection onto the Master Surface

(2) (2)

(2) (1) (1) (1) (2) (2), argmin t tt

X

X X X X

(2) (1) (2) (2) (1), ,tt tx X X X

(1)X

(2)X

(2)x

46 16/12/2013


... Contact constraint is satisfied

... Contact constraint is violated


Normal Gap Function

(1) (1) (1) (2) (2) (1), ,N t tg t t X X X X

(2) (2) (1) , ,t t n x X

... Outward unit normal at the closest-point-projection on the master surface at current configuration

... Normal gap on current configuration Ng

0Ng

0Ng

47 16/12/2013


Nominal Contact Traction,

Nominal Contact Pressure

(1) (1) (1) (1) (1) (1), , t tT X P X N X

(1) (1) (1) (1) (1), , ,Nt t t t T X X n X

(1) (1) (1) (1) (1), , , Nt t t tX T X n X

(1) (1) (1) (2) (2) (1) (1) (1), , , , Nt t t t tX T X n X X T X

Setting (1) (1) (2) (2) (1), , ,t t t n X n X X

(1) (1) (1), ,Nt t t T X X

The nominal contact traction vector is defined as,

where the nominal contact pressure is defined as,

48 16/12/2013


If then

Contact Kinematic/Pressure Constraints

(1) (1) (1) (1), 0, , 0, , , 0N N N Nt t g t t t g t X X X X

(1) , 0Ng t X

Kühn-Tucker contact/separation optimality conditions

Contact consistency or persistency condition

(1) (1) (1) (1), 0, , 0, , , 0N N N Nt t g t t t g t X X X X

49 16/12/2013


Penalty Regularized Contact Constraints

(1) (1), ,N N Nt t g tX X

The penalty regularization of the Kühn-Tucker contact/separation optimality conditions and contact persistency condition takes the form

where is the contact penalty parameter. N

The penalty regularized contact constraints provides a constitutive-like equation for the nominal contact pressure as a function of the normal gap, allowing a displacement-driven formulation of the contact problem.

50 16/12/2013


Augmented Lagrangian Regularized Contact Constraints

(1) (1), , N N N Nt t g tX X

The augmented Lagrangian regularization of the Kühn-Tucker contact/ separation optimality conditions and contact persistency condition take the form

where is the contact penalty parameter and is the lagrange multiplier subjected to the following Kühn-Tucker optimality constraints

N N

0, 0, 0N N N Ng g

51 16/12/2013


The contact variational contribution to the variational form of the momentum balance equation takes the form,

Variational Contact Form

Variational Contact Form

(1)0,c

c t N NG t g d

52 16/12/2013


Mechanical Contact Problem Block-iterative Solution Scheme

Arrow-shape block system. Block-iterative solution scheme

Within the framework of penalty/augmented lagrangian methods, in order to get a better conditioned numerical problem, the resulting system of equations is arranged collecting cast, mould and interface variables, leading to a block arrow-shape system of equations of the form,

,

,

, ,

0

0

cast c cast cast cast

mold c mold mold mold

c cast c mold c c c


Mechanical Contact Problem Block-iterative Solution Scheme

Arrow-shape block system. Block-iterative solution scheme

The arrow-shape block system of equations can be solved using a block-iterative solution scheme given by,

( 1) ( )

,

( 1) ( )

,

( 1) ( 1) ( 1)

, ,

i i

cast cast cast c cast c

i i

mold mold mold c mold c

i i i

c c c c cast cast c mold mold


Outline



Mechanical Problem

Solid Phase

Liquid Phase



Thermal Problem




Concluding Remarks


Thermal Contact Problem Heat Conduction

Heat conduction flux is given by,

cond cond c mq h

where the heat conduction coefficient is considered to be a function

of the contact pressure , thermal resistance by conduction ,

Vickers hardness and an exponent coefficient , and is given by,

1

n

Ncond N

cond e

th t

R H

nNt

eHcondR


Thermal Contact Problem Heat Conduction

Thermal resistance by conduction depends on the air trapped

between cast and mould due to the surfaces roughness and on the

mould coating, and is given by,

2

z ccond

a c

RR

k k

condR

where is the mean peak-to-valley heights of rough surfaces, given

by,

2 2

, ,z z cast z mouldR R R

zR

and is the thermal conductivity of the air trapped between the cast

and mould surfaces roughness, is the coating thickness and is

the thermal conductivity of the coating

ak

c ck


Thermal Contact Problem Heat Convection

Heat convection flux is given by,

conv conv c mq h

where the heat convection coefficient is considered to be a function of

the contact gap , the mean peak-to-valley heights of rough surfaces

, the thermal conductivity of the air trapped between the cast and

mould surfaces roughness , the coating thickness and the

thermal conductivity of the coating , and is given by,

1

max ,conv N

N z a c c

h gg R k k

Ng

zR

ak cck


Thermal Contact Problem Heat Radiation

Heat radiation flux is given by,

4 4

1 1 1

a c m

rad

c m

q

where,

Boltzman constant a

cast temperature c

mould temperature m

cast emissivity c

mould emissivity m


Outline



Mechanical Problem

Solid Phase

Liquid Phase



Thermal Problem




Concluding Remarks


1. Computational Simulation Aluminum (AlSi7Mg) Part in a Steel (X40CrMoV5) Mould

Mesh of cast, cooling system

and mould: 380,000 tetrahedra


1. Computational Simulation Temperature


1. Computational Simulation Solid Fraction


1. Computational Simulation Temperature and Solid Fraction at Section 1


1. Computational Simulation Pressure and J2 von Mises Effective Stress at Section 1


1. Computational Simulation Convergence Behaviour at a Typical Step

Penalty

1 1.000000E+3

2 2.245836E+2

3 2.093789E+2

4 7.473996E+1

5 5.873453E+1

6 9.986438E+0

7 3.762686E-2

Augmented

Lagrangian

1 1.000000E+3

2 8.957635E+2

3 7.846474E+0

4 6.735237E-3

5A 5.734238E+1

6 6.723579E-3

7A 3.946447E-3

Block-iterative

Penalty

1 1.000000E+3

2 6.84654E+1

3 8.57626E-2

4 2.97468E+1

5 4.845342E-2

6 4.734127E-3


2. Computational Simulation Cast Iron Part in a Sand Mould


2. Computational Simulation Cast Iron Part in a Sand Mould


2. Computational Simulation Solid Fraction at t=500 s.


2. Computational Simulation Temperature at t=1100 s.


2. Computational Simulation J2 von Mises Effective Stress at Section 1






3. Computational Simulation Geometry of the Core, Chiller and Part


3. Computational Simulation Geometry of the Core and Chiller


3. Computational Simulation Geometry of the Part


3. Computational Simulation Filling Process


3. Computational Simulation Temperature Map Evolution


3. Computational Simulation Temperature Map Evolution at Section A-B


3. Computational Simulation Solidification Time and Solid Fraction Map Evolution


3. Computational Simulation Temperature vs Time at Different Points


4. Computational Simulation Geometry of the Part


4. Computational Simulation Filling of the Part


4. Computational Simulation Temperature Map Evolution


4. Computational Simulation Solidification Times and Solid Fraction Evolution


4. Computational Simulation Temperature vs Time at Different Points


4. Computational Simulation Deformation of the Part


4. Computational Simulation Displacements maps


4. Computational Simulation Von Mises Residual Stresses


5. Computational Simulation Geometry of a Motor Block


5. Computational Simulation Temperature Distribution


5. Computational Simulation Equivalent Plastic Strain


5. Computational Simulation J2 Effectiove Stress


Outline



Mechanical Problem

Solid Phase

Liquid Phase



Thermal Problem




Concluding Remarks


Concluding Remarks

A suitable coupled thermomechanical model allowing the simulation of complex industrial casting processes has been introduced.

A thermomechanical J2 viscoplastic material model allowing a continuous transition from an initial liquid-like state to a final solid state has been introduced. The thermomechanical J2 viscoplastic model reduces to a purely deviatoric viscous model for the liquid-like phase.

A thermomechanical contact model has been introduced. Ill conditioning induced by the penalty method used in the mechanical contact formulation has been smoothed introducing a block-iterative algorithm.

An OSGS stabilized mixed formulation for tetrahedral elements has been used to avoid locking and spurious pressure oscillations due to the quasi- incompressibility constraints arising in J2 viscoplasticity models.

The formulation developed has been implemented in the FE software VULCAN and a number of computational simulations have been shown.

Documents

Coupled Thermomechanical Contact Problems