coupled thermomechanical contact problems
TRANSCRIPT
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
Prof. Dr. Ing. Carlos Agelet de Saracibar
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
Prof. Dr. Ing. Carlos Agelet de Saracibar
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
Prof. Dr. Ing. Carlos Agelet de Saracibar
Motivation and Goals
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.
Prof. Dr. Ing. Carlos Agelet de Saracibar
Motivation and Goals
Prof. Dr. Ing. Carlos Agelet de Saracibar
Motivation and Goals
Prof. Dr. Ing. Carlos Agelet de Saracibar
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
Prof. Dr. Ing. Carlos Agelet de Saracibar
Coupled Thermomechanical Problem
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
Prof. Dr. Ing. Carlos Agelet de Saracibar
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
Prof. Dr. Ing. Carlos Agelet de Saracibar
Mechanical Problem Solid and Liquid-like Phases
Prof. Dr. Ing. Carlos Agelet de Saracibar
Mechanical Problem Solid and Liquid-like Phases
Prof. Dr. Ing. Carlos Agelet de Saracibar
Mechanical Problem Solid and Liquid-like Phases
Liquid-like phase
Solid phase
Prof. Dr. Ing. Carlos Agelet de Saracibar
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
Prof. Dr. Ing. Carlos Agelet de Saracibar
Mechanical Problem Solid Phase
T.D. Yield Stress
T.D. Viscosity T.D. Young Mod.
T.D. Hardening
20º C
500º C
700º C
Prof. Dr. Ing. Carlos Agelet de Saracibar
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
Prof. Dr. Ing. Carlos Agelet de Saracibar
Mechanical Problem
J2 Elasto-Viscoplastic model, isotropic and kinematic hardening
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
Prof. Dr. Ing. Carlos Agelet de Saracibar
Mechanical Problem
J2 Elasto-Viscoplastic model, isotropic and kinematic hardening
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
Prof. Dr. Ing. Carlos Agelet de Saracibar
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
Prof. Dr. Ing. Carlos Agelet de Saracibar
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
Prof. Dr. Ing. Carlos Agelet de Saracibar
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
Prof. Dr. Ing. Carlos Agelet de Saracibar
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
Prof. Dr. Ing. Carlos Agelet de Saracibar
Mechanical Problem
J2 Elasto-Viscoplastic model, isotropic and kinematic hardening
Mushy Zone. Liquid-Solid Continuous Transition
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
Prof. Dr. Ing. Carlos Agelet de Saracibar
Mechanical Problem
J2 Elasto-Viscoplastic model, isotropic and kinematic hardening
Mushy Zone. Liquid-Solid Continuous Transition
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
Prof. Dr. Ing. Carlos Agelet de Saracibar
Mechanical Problem
J2 Elasto-Viscoplastic model, isotropic and kinematic hardening
Mushy Zone. Liquid-Solid Continuous Transition
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
Prof. Dr. Ing. Carlos Agelet de Saracibar
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
Prof. Dr. Ing. Carlos Agelet de Saracibar
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.
Prof. Dr. Ing. Carlos Agelet de Saracibar
Mechanical Problem Incompressibility or Quasi-incompressibility Problem
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
Prof. Dr. Ing. Carlos Agelet de Saracibar
Mechanical Problem Incompressibility or Quasi-incompressibility Problem
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
Prof. Dr. Ing. Carlos Agelet de Saracibar
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
Prof. Dr. Ing. Carlos Agelet de Saracibar
Mechanical Problem Multiscale Stabilization Methods
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
Prof. Dr. Ing. Carlos Agelet de Saracibar
Mechanical Problem Multiscale Stabilization Methods
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
Prof. Dr. Ing. Carlos Agelet de Saracibar
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
Prof. Dr. Ing. Carlos Agelet de Saracibar
Mechanical Problem OSGS Stabilization Technique
Orthogonal Subgrid Scale (OSGS) technique
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
Prof. Dr. Ing. Carlos Agelet de Saracibar
Mechanical Problem OSGS Stabilization Technique
Orthogonal Subgrid Scale (OSGS) technique
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
Prof. Dr. Ing. Carlos Agelet de Saracibar
Mechanical Problem OSGS Stabilization Technique
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
Prof. Dr. Ing. Carlos Agelet de Saracibar
Mechanical Problem OSGS Stabilization Technique
Discrete OSGS stabilized variational form
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
Prof. Dr. Ing. Carlos Agelet de Saracibar
Mechanical Problem OSGS Stabilization Technique
Discrete OSGS stabilized variational form
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
Prof. Dr. Ing. Carlos Agelet de Saracibar
Mechanical Problem OSGS Stabilization Technique
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
Prof. Dr. Ing. Carlos Agelet de Saracibar
Mechanical Problem OSGS Stabilization Technique
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
Prof. Dr. Ing. Carlos Agelet de Saracibar
Mechanical Problem OSGS Stabilization Technique
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
Prof. Dr. Ing. Carlos Agelet de Saracibar
Mechanical Problem OSGS Stabilization Technique
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
Prof. Dr. Ing. Carlos Agelet de Saracibar
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
Prof. Dr. Ing. Carlos Agelet de Saracibar
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
Prof. Dr. Ing. Carlos Agelet de Saracibar
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
Prof. Dr. Ing. Carlos Agelet de Saracibar
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
Prof. Dr. Ing. Carlos Agelet de Saracibar
... 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
Nonlinear Contact Kinematics
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
Prof. Dr. Ing. Carlos Agelet de Saracibar
... Contact constraint is satisfied
... Contact constraint is violated
Nonlinear Contact Kinematics
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
Prof. Dr. Ing. Carlos Agelet de Saracibar
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
Prof. Dr. Ing. Carlos Agelet de Saracibar
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
Prof. Dr. Ing. Carlos Agelet de Saracibar
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
Prof. Dr. Ing. Carlos Agelet de Saracibar
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
Prof. Dr. Ing. Carlos Agelet de Saracibar
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
Prof. Dr. Ing. Carlos Agelet de Saracibar
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
Prof. Dr. Ing. Carlos Agelet de Saracibar
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
Prof. Dr. Ing. Carlos Agelet de Saracibar
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
Prof. Dr. Ing. Carlos Agelet de Saracibar
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
Prof. Dr. Ing. Carlos Agelet de Saracibar
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
Prof. Dr. Ing. Carlos Agelet de Saracibar
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
Prof. Dr. Ing. Carlos Agelet de Saracibar
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
Prof. Dr. Ing. Carlos Agelet de Saracibar
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
Prof. Dr. Ing. Carlos Agelet de Saracibar
1. Computational Simulation Aluminum (AlSi7Mg) Part in a Steel (X40CrMoV5) Mould
Mesh of cast, cooling system
and mould: 380,000 tetrahedra
Prof. Dr. Ing. Carlos Agelet de Saracibar
1. Computational Simulation Temperature
Prof. Dr. Ing. Carlos Agelet de Saracibar
1. Computational Simulation Solid Fraction
Prof. Dr. Ing. Carlos Agelet de Saracibar
1. Computational Simulation Temperature and Solid Fraction at Section 1
Prof. Dr. Ing. Carlos Agelet de Saracibar
1. Computational Simulation Pressure and J2 von Mises Effective Stress at Section 1
Prof. Dr. Ing. Carlos Agelet de Saracibar
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
Prof. Dr. Ing. Carlos Agelet de Saracibar
2. Computational Simulation Cast Iron Part in a Sand Mould
Prof. Dr. Ing. Carlos Agelet de Saracibar
2. Computational Simulation Cast Iron Part in a Sand Mould
Prof. Dr. Ing. Carlos Agelet de Saracibar
2. Computational Simulation Solid Fraction at t=500 s.
Prof. Dr. Ing. Carlos Agelet de Saracibar
2. Computational Simulation Temperature at t=1100 s.
Prof. Dr. Ing. Carlos Agelet de Saracibar
2. Computational Simulation J2 von Mises Effective Stress at Section 1
Prof. Dr. Ing. Carlos Agelet de Saracibar
2. Computational Simulation J2 von Mises Effective Stress at Section 2
Prof. Dr. Ing. Carlos Agelet de Saracibar
2. Computational Simulation J2 von Mises Effective Stress at Section 3
Prof. Dr. Ing. Carlos Agelet de Saracibar
3. Computational Simulation Geometry of the Core, Chiller and Part
Prof. Dr. Ing. Carlos Agelet de Saracibar
3. Computational Simulation Geometry of the Core and Chiller
Prof. Dr. Ing. Carlos Agelet de Saracibar
3. Computational Simulation Geometry of the Part
Prof. Dr. Ing. Carlos Agelet de Saracibar
3. Computational Simulation Filling Process
Prof. Dr. Ing. Carlos Agelet de Saracibar
3. Computational Simulation Temperature Map Evolution
Prof. Dr. Ing. Carlos Agelet de Saracibar
3. Computational Simulation Temperature Map Evolution at Section A-B
Prof. Dr. Ing. Carlos Agelet de Saracibar
3. Computational Simulation Solidification Time and Solid Fraction Map Evolution
Prof. Dr. Ing. Carlos Agelet de Saracibar
3. Computational Simulation Temperature vs Time at Different Points
Prof. Dr. Ing. Carlos Agelet de Saracibar
4. Computational Simulation Geometry of the Part
Prof. Dr. Ing. Carlos Agelet de Saracibar
4. Computational Simulation Filling of the Part
Prof. Dr. Ing. Carlos Agelet de Saracibar
4. Computational Simulation Temperature Map Evolution
Prof. Dr. Ing. Carlos Agelet de Saracibar
4. Computational Simulation Solidification Times and Solid Fraction Evolution
Prof. Dr. Ing. Carlos Agelet de Saracibar
4. Computational Simulation Temperature vs Time at Different Points
Prof. Dr. Ing. Carlos Agelet de Saracibar
4. Computational Simulation Deformation of the Part
Prof. Dr. Ing. Carlos Agelet de Saracibar
4. Computational Simulation Displacements maps
Prof. Dr. Ing. Carlos Agelet de Saracibar
4. Computational Simulation Von Mises Residual Stresses
Prof. Dr. Ing. Carlos Agelet de Saracibar
5. Computational Simulation Geometry of a Motor Block
Prof. Dr. Ing. Carlos Agelet de Saracibar
5. Computational Simulation Temperature Distribution
Prof. Dr. Ing. Carlos Agelet de Saracibar
5. Computational Simulation Equivalent Plastic Strain
Prof. Dr. Ing. Carlos Agelet de Saracibar
5. Computational Simulation J2 Effectiove Stress
Prof. Dr. Ing. Carlos Agelet de Saracibar
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
Prof. Dr. Ing. Carlos Agelet de Saracibar
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.