A particle finite element method for analysis of industrial forming processes

==Abstract==

We present a generalized Lagrangian formulation for analysis of industrial forming processes involving thermally coupled interactions between deformable continua. The governing equations for the deformable bodies are written in a unified manner that holds both for fluids and solids. The success of the formulation lays on  a  residual-based  expression of the mass conservation equation obtained using the Finite Calculus (FIC) method that provides the necessary stability for quasi/fully incompressible situations. The  governing equations are discretized with the FEM via a mixed formulation using simplicial elements with equal linear interpolation for the velocities, the pressure and the temperature. The merits of the formulation  are demonstrated in the solution of 2D and 3D thermally-coupled forming processes using the Particle Finite Element Method (PFEM).

'''keywords''': Updated Lagrangian formulation, finite element method, incompressible fluids, consistent tangent matrix, mixed formulation, stabilized method


==1 INTRODUCTION==

There is a large number of  forming manufacturing processes in industry that involve the interaction of highly deformable continua, including viscous fluids and solids that undergo large deformations. Typically these problems also include coupled thermal effects between the interacting bodies during the forming process.

A number of numerical methods has been developed over the years for the simulation of    industrial forming processes. The different methods can be classified as those based on a pure fluid mechanics formulation, i.e. the so-called flow approach developed around  the 1980's <span id='citeF-6'></span><span id='citeF-39'></span><span id='citeF-40'></span><span id='citeF-44'></span><span id='citeF-45'></span><span id='citeF-48'></span>[[#cite-6|[6,39,40,44,45,48]]] and those based on a solid mechanics formulation <span id='citeF-11'></span><span id='citeF-12'></span><span id='citeF-19'></span><span id='citeF-31'></span><span id='citeF-47'></span>[[#cite-11|[11,12,19,31,47]]].

In this work we present a generalized continuum mechanics formulation for the analysis of industrial forming processes that involve thermally coupled interactions between  deformable continua (both fluids and solids).

The motion of the deformable bodies (either a solid or a fluid) is described using a  Lagrangian mixed formulation with the velocities, the pressure and the temperature as the unknown variables. Appropriate constitutive laws for the different constituent materials in the different parts of the analysis domain  are used. The mixed  Lagrangian formulation allows us to model the non linear interaction (including coupled thermal effects) between the different bodies involved in the forming process in a unified manner.

The continuum mechanics equations are solved with the Particle Finite Element Method (PFEM, www.cimne.com/pfem). The PFEM treats the mesh nodes in the analysis domain as virtual particles that model the behaviour of the underlying  can freely move and even separate from the  domain representing, for instance, the effect of water drops or cutting particles in drilling problems and machining. A mesh connects the nodes discretizing the continuum domain where the governing equations are solved using a stabilized FEM. Examples of application of PFEM to problems in fluid and solid mechanics including fluid-structure interaction (FSI) situations and coupled thermal flow problems can be found in <span id='citeF-1'></span><span id='citeF-2'></span><span id='citeF-4'></span><span id='citeF-5'></span><span id='citeF-9'></span><span id='citeF-13'></span><span id='citeF-14'></span><span id='citeF-15'></span><span id='citeF-16'></span><span id='citeF-17'></span><span id='citeF-18'></span><span id='citeF-20'></span><span id='citeF-26'></span><span id='citeF-27'></span><span id='citeF-34'></span><span id='citeF-35'></span><span id='citeF-37'></span><span id='citeF-38'></span><span id='citeF-43'></span>[[#cite-1|[1,2,4,5,9,13,14,15,16,17,18,20,26,27,34,35,37,38,43]]]. Early attempts of the using the PFEM for solving metal forming problems were reported in <span id='citeF-20'></span><span id='citeF-30'></span>[[#cite-20|[20,30]]].

In Lagrangian analysis procedures (such as  PFEM) the motion of the virtual particles (i.e. the nodes) and of physical particles is tracked during the transient solution. Hence, the convective terms vanish in the momentum and heat transfer equations for the continuum part of the domain and no numerical stabilization is needed for treating those terms. Another source of numerical instability however remains in Lagrangian formulation, namely the treatment of incompressibility using equal order FEM interpolation for the velocities and the pressure. In this work a stabilized form of the discretized mass conservation equation is obtaining using the Finite Calculus (FIC) method proposed by Oñate et al. <span id='citeF-21'></span><span id='citeF-22'></span><span id='citeF-23'></span><span id='citeF-24'></span><span id='citeF-25'></span><span id='citeF-28'></span><span id='citeF-29'></span><span id='citeF-30'></span><span id='citeF-36'></span>[[#cite-21|[21,22,23,24,25,28,29,30,36]]] that has excellent mass preservation features.

The generalized Lagrangian approach here described follows the ideas presented in <span id='citeF-15'></span>[[#cite-15|[15]]] where a PFEM formulation for treating fluids and solids in a unified  manner was proposed. These ideas have been extended by Franci, Oñate and Carbonell <span id='citeF-10'></span>[[#cite-10|[10]]] using a stabilized PFEM formulation based on the FIC method.

The lay-out of the paper is the following. In the next section we present the basic equations for conservation of linear momentum, mass conservation and heat transfer for a  continuum using a generalized Lagrangian framework. Next we derive the stabilized FIC form of the mass conservation equation. An incompressible continuum can be considered as the limit case of the compressible problem. Then the mixed finite element discretization using simplicial element with equal order approximation for the velocity,  the pressure and the temperature is presented and the relevant matrices and vectors of the discretized problem are given.  Details of the implicit solution of the Lagrangian FEM equations in time using an updated Lagrangian approach and a Newton-Raphson type iterative scheme are presented. The relevance of the bulk stiffness terms in the tangent matrix  for enhancing the convergence and accuracy of the iterative solution  for problems involving quasi and fully incompressible continua is discussed. The basic steps of the PFEM are described.

The efficiency and general applicability of the PFEM  are verified by solving a set of thermally coupled industrial forming problems in two (2D) and three (3D) dimensions. The excellent performance of the numerical method proposed for all the problems analyzed is highlighted.

<div id='img-1'></div>
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:draft_Samper_718004671-Figura1.png|450px]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 1:''' Initial (<math>t ={}^0 t</math>), current (<math>t={}^n t</math>) and updated (<math>t={}^{n+1} t</math>) configurations of a  domain <math>V</math> with Neumann (<math>\Gamma _t</math>) and Dirichlet (<math>\Gamma _v</math>) boundaries. Trajectory of a material point <math>i</math> and velocity  (<math>{v}_i</math>) and displacement (<math>{u}_i</math>) vectors of the point at each configuration. <math>{}^n {u}</math>, <math>{}^{n+1}{u}</math> and <math>\Delta {u}</math> denote schematically the trajectories of the overall domain between two configurations.
|}

==2 GOVERNING EQUATIONS FOR A LAGRANGIAN CONTINUUM==

We will consider a domain containing a deformable material (either a fluid or a solid) which evolves in time due to external and internal forces and prescribed velocities and thermal conditions from an ''initial configuration'' at time <math display="inline">t={}^0 t</math> (typically <math display="inline">{}^0t =0</math>) to a ''current configuration'' at time <math display="inline">t={}^n t</math>. The volume <math display="inline">V</math> and its boundary <math display="inline">\Gamma </math> at the initial and current configurations are denoted as (<math display="inline">{}^0 V, {}^0 \Gamma </math>) and (<math display="inline">{}^n V, {}^n \Gamma </math>), respectively. The goal is to find the domain that the material  occupies, as well as the velocities, strain rates and stresses (the deviatoric stresses and the pressure) and the temperature in the so-called ''updated configuration''  at time <math display="inline">{}^{n+1} t= {}^n t+\Delta t</math> (Figure 1). In the following a left superindex denotes the configuration where the variable is computed.

===2.1 Momentum equations===

The equations of conservation of linear momentum for a deformable continuum are written in the Lagrangian description as <span id='citeF-3'></span>[[#cite-3|[3]]]

<span id="eq-1"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\rho \frac{Dv_i}{Dt}-{\partial {}^{n+1}\sigma _{ij} \over \partial {}^{n+1}x_j} - {}^{n+1} b_i=0\quad , \quad i,j=1,\cdots , n_s \quad  \hbox{in } {}^{n+1}V  </math>
|}
| style="width: 5px;text-align: right;" | (1)
|}

In Eq.([[#eq-1|1]]), <math display="inline">{}^{n+1}V</math> is the analysis domain in the ''updated configuration'' at time <math display="inline">^{n+1}t</math> with boundary <math display="inline">{}^{n+1}\Gamma </math>, <math display="inline">v_i</math> and <math display="inline">b_i</math> are the velocity and body force components  along the <math display="inline">i</math>th Cartesian axis, <math display="inline">\rho </math> is the density, <math display="inline">n_s</math> is the number of space dimensions (i.e. <math display="inline">n_s=3</math> for 3D problems) and <math display="inline">{}^{n+1}\sigma _{ij}</math> are the Cauchy stresses in <math display="inline">{}^{n+1}V</math>  that are split in the deviatoric (<math display="inline">s_{ij}</math>) and pressure (<math display="inline">p</math>) components as

<span id="eq-2"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
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| style="text-align: center;" | <math>{}^{n+1}\sigma _{ij} = {}^{n+1}s_{ij}+ {}^{n+1}p \delta _{ij} </math>
|}
| style="width: 5px;text-align: right;" | (2)
|}

where <math display="inline">\delta _{ij}</math> is the Kronecker delta. The pressure is assumed to be positive for a tension state. Summation of terms with repeated indices  is assumed in Eq.([[#eq-1|1]]) and in  the following.

'''Remark 1.''' The term <math display="inline">\frac{Dv_i}{Dt}</math> in Eq.([[#eq-1|1]]) is the ''material derivative'' of the velocity. This term is typically computed in a Lagrangian framework as

<span id="eq-3"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\frac{Dv_i}{Dt} = \frac{ {}^{n+1}v_i- {}^n v_i }{\Delta t}     </math>
|}
| style="width: 5px;text-align: right;" | (3)
|}

with

<span id="eq-4"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>{}^{n+1} v_i:=v_i ({}^{n+1} {x},{}^{n+1} t)\quad ,\quad {}^n v_i:=v_i ({}^n{x},{}^n t)       </math>
|}
| style="width: 5px;text-align: right;" | (4)
|}

where <math display="inline">{}^n v_i</math> is the value of the velocity  at the material point that has the position <math display="inline">{}^n{x}</math> at time <math display="inline">t={}^nt</math> and <math display="inline">{x}</math> is the coordinates vector in a fixed Cartesian  system <span id='citeF-3'></span><span id='citeF-7'></span><span id='citeF-48'></span>[[#cite-3|[3,7,48]]].

===2.2 Constitutive equations===

===''Newtonian fluids''===

The relationship between the deviatoric Cauchy stresses <math display="inline">{}^{n+1}s_{ij}</math> and the  rates of deformation <math display="inline">\varepsilon _{ij}</math> has the standard form for a Newtonian fluid,

<span id="eq-5"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
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| style="text-align: center;" | <math>{}^{n+1}s_{ij} = 2\mu \varepsilon '_{ij} \quad \hbox{with}\quad  \varepsilon '_{ij}= \varepsilon _{ij} - {1 \over 3}\varepsilon _v \delta _{ij}\quad \hbox{and}\quad  \varepsilon _{ij} = {1 \over 2}\left({\partial {}^{n+1}v_i \over \partial {}^{n+1}x_j} +  {\partial {}^{n+1}v_j \over \partial {}^{n+1}x_i}  \right) </math>
|}
| style="width: 5px;text-align: right;" | (5)
|}

where <math display="inline">\mu </math> is the viscosity, <math display="inline">\varepsilon '_{ij}</math> are the deviatoric rates of deformation and <math display="inline">\varepsilon _v</math> is the volumetric strain rate defined as <math display="inline">\varepsilon _v= \varepsilon _{ii}</math>.

The above constitutive equation  can be written in matrix form as (for 2D problems)

<span id="eq-6"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
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| style="text-align: center;" | <math>{}^{n+1}{s} = {D} {\boldsymbol \varepsilon } \quad \hbox{with}\quad   {D}=2 \mu \left[                      \begin{array}{cccccc}2/3 & -1/3 & 0 \\                         -1/3 &2/3  &0  \\                       0 & 0 &1/2\\                      \end{array} \right]   </math>
|}
| style="width: 5px;text-align: right;" | (6)
|}

In Eq.([[#eq-6|6]]) <math display="inline">{s} = [s_{11},s_{22},s_{12}]^T</math> and <math display="inline">{\boldsymbol \varepsilon } = [\varepsilon _{11},\varepsilon _{22},\varepsilon _{12}]^T</math> are the deviatoric Cauchy stress vector and the strain rate vector, respectively.

===''Hypo-elastic solids''===

We will assume a hipo-elastic solid with a constitutive equation of the form

<span id="eq-7"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
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| style="text-align: center;" | <math>\sigma _{ij}^{\nabla ^J} = 2\bar \mu \varepsilon _{ij} + \lambda \varepsilon _v \delta _{ij}  </math>
|}
| style="width: 5px;text-align: right;" | (7)
|}

where <math display="inline">\bar \mu </math> and <math display="inline">\lambda </math> are the Lame constants and <math display="inline">\sigma _{ij}^{\nabla ^J}</math> is the Jaumann rate of the  Cauchy stresses <span id='citeF-3'></span>[[#cite-3|[3]]].

Tensor <math display="inline">{\boldsymbol \sigma }^{\nabla ^J}</math> is approximated in this work as

<span id="eq-8"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
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| 
{| style="text-align: left; margin:auto;" 
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| style="text-align: center;" | <math>{\boldsymbol \sigma }^{\nabla ^J}\simeq \frac{1}{J} L ({\boldsymbol \tau })  </math>
|}
| style="width: 5px;text-align: right;" | (8)
|}

where <math display="inline">J</math> is the determinant of the deformation tensor <math display="inline">{F}</math> <math display="inline">\left(F_{ij} ={\partial {}^{n+1}x_i \over \partial ^{n}x_j}\right)</math>, and <math display="inline">L ({\boldsymbol \tau }) </math> is the Lie derivative of the Kirchhoff stress tensor <math display="inline">\tau _{ij}</math> <span id='citeF-3'></span>[[#cite-3|[3]]].

Combining Eqs.([[#eq-7|7]]) and ([[#eq-8|8]]) gives the following approximation for <math display="inline">L ({\boldsymbol \tau })</math>

<span id="eq-9"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
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| 
{| style="text-align: left; margin:auto;" 
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| style="text-align: center;" | <math>L(\tau _{ij}) \simeq J (2\bar \mu \varepsilon _{ij} + \lambda \varepsilon _v \delta _{ij})  </math>
|}
| style="width: 5px;text-align: right;" | (9)
|}

'''Remark 2.''' Assumption ([[#eq-8|8]]) is equivalent to accepting that the Truesdell and Jaumann rates of the Cauchy stresses are identical. A more accurate assumption for solving the same problem can be found in <span id='citeF-10'></span>[[#cite-10|[10]]].

Using standard transformations of continuum mechanics gives <span id='citeF-3'></span>[[#cite-3|[3]]]

<span id="eq-10"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
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| style="text-align: center;" | <math>{L} ({\boldsymbol \tau }) =  {F}\dot {S} {F}^T = \frac{1}{\Delta t} {F} (\Delta {S}) {F}^T = \frac{J}{\Delta t}\Delta{\boldsymbol \sigma }  </math>
|}
| style="width: 5px;text-align: right;" | (10)
|}

where <math display="inline">{S}</math> are the 2nd Piola-Kirchhoff stress tensor, the upper dot denotes the time derivative and <math display="inline"> \Delta{\boldsymbol \sigma }</math> is the increment of the Cauchy stress tensor.

Substituting Eq.([[#eq-10|10]]) into ([[#eq-9|9]])  gives after grouping terms

<span id="eq-11"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
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| style="text-align: center;" | <math>{}^{n+1}\sigma _{ij}= {}^{n}\sigma _{ij} + \Delta \sigma _{ij}  </math>
|}
| style="width: 5px;text-align: right;" | (11)
|}

where

<span id="eq-12"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
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| 
{| style="text-align: left; margin:auto;" 
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| style="text-align: center;" | <math>\Delta \sigma _{ij} = \Delta s_{ij}+ \Delta p \delta _{ij}  </math>
|}
| style="width: 5px;text-align: right;" | (12)
|}

with

<span id="eq-13"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
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| style="text-align: center;" | <math>\Delta s_{ij} = 2\bar \mu \Delta t \varepsilon '_{ij}\quad \hbox{and}\quad \Delta p = \bar \kappa \Delta t \varepsilon _v  </math>
|}
| style="width: 5px;text-align: right;" | (13)
|}

where

<span id="eq-14"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
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| 
{| style="text-align: left; margin:auto;" 
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| style="text-align: center;" | <math>\bar \kappa = \frac{2}{3} \bar \mu + \lambda    </math>
|}
| style="width: 5px;text-align: right;" | (14)
|}

is the bulk modulus of the solid material.

The Cauchy stresses <math display="inline">{}^{n}{\boldsymbol \sigma }</math> in Eq.([[#eq-12|12]]) are computed on the updated configuration. These stresses can be obtained from the 2nd Piola-Kirchhoff stresses <math display="inline">{}^n{S}</math> on the current configuration  as

<span id="eq-15"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
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| style="text-align: center;" | <math>{}^{n}{\boldsymbol \sigma } = \frac{1}{J} {F}  ({}^n{S}){F}^T   </math>
|}
| style="width: 5px;text-align: right;" | (15)
|}

'''Remark 3.''' The relationship between the deviatoric Cauchy stress increments and the rates of deformation for a solid material can be written in the same matrix form as Eq.(6) substituting in matrix <math display="inline">D</math> the viscosity <math display="inline">\mu </math> by <math display="inline">\bar \mu \Delta t </math>.

===2.3 Mass conservation equation===

The  mass conservation equation can be written for both fluid and solids as <span id='citeF-3'></span><span id='citeF-7'></span><span id='citeF-48'></span>[[#cite-3|[3,7,48]]]

<span id="eq-16"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
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{| style="text-align: left; margin:auto;" 
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| style="text-align: center;" | <math>r_v =0  \quad  \hbox{in } {}^{n+1}V  </math>
|}
| style="width: 5px;text-align: right;" | (16a)
|}

with

<span id="eq-17"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
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{| style="text-align: left; margin:auto;" 
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| style="text-align: center;" | <math>r_v:=-\frac{1}{\kappa }\frac{Dp}{Dt}+\varepsilon _v </math>
|}
| style="width: 5px;text-align: right;" | (16b)
|}

For a quasi-incompressible fluid material <math display="inline">\kappa =\rho c^2</math> where <math display="inline">c</math> is the speed of sound in the fluid. For a solid material, <math display="inline">\kappa = \bar \kappa </math> with <math display="inline">\bar \kappa </math> given by Eq.([[#eq-14|14]]). Eqs.([[#2.3 Mass conservation equation|16]]) define the constitutive equation for the pressure for both solids and quasi-incompressible fluids as <math display="inline">\Delta p = \kappa  \Delta t \varepsilon _v</math>. This is consisted with the expression obtained in Eq.([[#eq-13|13]]) for a hypo-elastic solid material.

For a fully incompressible material <math display="inline">\kappa =\infty </math> and Eq.([[#eq-16|16a]]) simplifies to the standard form, <math display="inline">\varepsilon _v =0</math>. In this case the pressure increment can not be obtained from Eqs.([[#2.3 Mass conservation equation|2.3]]) and then the pressure becomes an independent variable <span id='citeF-7'></span><span id='citeF-48'></span>[[#cite-7|[7,48]]]. In our work we will retain the quasi-incompressible form of <math display="inline">r_v</math>  for convenience.

'''Remark 4.''' Note that for Newtonian fluids the deviatoric stresses are directly related to the rates of deformation by Eq.([[#eq-5|5]]), whereas for solids the ''deviatoric stress increments'' are related to the rates of deformation by Eq.([[#eq-13|13]]). On the other hand, the relationship between the pressure increment and the volumetric strain rate has the form of Eq.([[#eq-13|13]]) for both fluids and solids, with the adequate definition of the bulk modulus for each case.

===2.4 Thermal balance===

The thermal balance equation is written in a Lagrangian framework as

<span id="eq-18"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
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| style="text-align: center;" | <math>\rho c \frac{DT}{Dt} - {\partial  \over \partial {}^{n+1} x_i} \left(k {\partial T \over \partial {}^{n+1} x_i}\right)+ {}^{n+1} Q =0  \quad ,\quad  i,j=1,\cdots , n_s  \quad \hbox{in } {}^{n+1}V  </math>
|}
| style="width: 5px;text-align: right;" | (17)
|}

where <math display="inline">T</math> is the temperature, <math display="inline">c</math> is the thermal capacity, <math display="inline">k</math> is the heat conductivity and <math display="inline">Q</math> is the external heat source.

===2.5 Boundary conditions===

===''Mechanical problem''===

The boundary conditions at the Dirichlet (<math display="inline">\Gamma _v</math>) and Neumann (<math display="inline">\Gamma _t</math>) boundaries with <math display="inline">\Gamma  = \Gamma _v \cup \Gamma _t</math> are

<span id="eq-19"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
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| 
{| style="text-align: left; margin:auto;" 
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| style="text-align: center;" | <math>{}^{n+1} v_i -{}^{n+1} v_i^p =0 \qquad \hbox{on }{}^{n+1}\Gamma _v </math>
|}
| style="width: 5px;text-align: right;" | (18)
|}

<span id="eq-20"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
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| style="text-align: center;" | <math>{}^{n+1}\sigma _{ij}n_j - {}^{n+1} t_i^p =0 \quad \hbox{on }{}^{n+1}\Gamma _t \quad , \quad i,j=1,\cdots , n_s  </math>
|}
| style="width: 5px;text-align: right;" | (19)
|}

where <math display="inline">v_i^p</math> and <math display="inline">t_i^p</math> are the prescribed velocities and prescribed tractions on the respective boundaries and <math display="inline">n_j</math> are the components of the unit normal vector to the boundary <span id='citeF-3'></span><span id='citeF-7'></span><span id='citeF-48'></span>[[#cite-3|[3,7,48]]].

===''Thermal problem''===

<span id="eq-21"></span>
<span id="eq-22"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
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| style="text-align: center;" | <math>{}^{n+1}T - {}^{n+1}T^p =0 \quad \hbox{on }{}^{n+1}\Gamma _T</math>
| style="width: 5px;text-align: right;" | (20)
|-
| style="text-align: center;" | <math> k {\partial T \over \partial n} + {}^{n+1} q_n^p =0 \quad \hbox{on }{}^{n+1}\Gamma _q  </math>
| style="width: 5px;text-align: right;" | (21)
|}
|}

where <math display="inline">T^p</math> and <math display="inline">q_n^p</math> are the prescribed temperature and the prescribed normal heat flux at the boundaries <math display="inline">\Gamma _T</math> and <math display="inline">\Gamma _q</math>, respectively and <math display="inline">n</math> is the direction normal to be boundary.

==3 STABILIZED MASS CONSERVATION EQUATION==

In this work we will use the following stabilized form of the mass conservation equation obtained using the Finite Calculus (FIC) approach <span id='citeF-38'></span>[[#cite-38|[38]]]

<span id="eq-23"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
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| style="text-align: center;" | <math>-\frac{1}{\kappa }\frac{Dp}{Dt}+  \varepsilon _v  +\tau {\partial {}^{n+1} \hat r_{m_i} \over \partial {}^{n+1}x_i} =0 \qquad \hbox{in }{}^{n+1}V  </math>
|}
| style="width: 5px;text-align: right;" | (22)
|}

The last term in Eq.([[#eq-23|22]]) is the so-called stabilization term introduced by the FIC technique; in this equation <math display="inline">\hat r_{m_i}</math> is a ''static momentum term'' defined as

<span id="eq-24"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\hat r_{m_i}= {\partial s_{ij} \over \partial x_j} + \frac{\partial p}{\partial x_i} + b_i   </math>
|}
| style="width: 5px;text-align: right;" | (23)
|}

and <math display="inline">\tau </math> is a ''stabilization parameter''. Eq.([[#eq-23|22]]) is a simplification of a more general FIC-based stabilized expression for quasi-incompressible fluids detailed  in <span id='citeF-38'></span>[[#cite-38|[38]]].

The stabilization parameter <math display="inline">\tau </math> is typically computed for each element  as

<span id="eq-25"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\tau = \left(\frac{8\mu }{(l^e)^2} + \frac{2\rho }{\Delta t}  \right)^{-1} </math>
|}
| style="width: 5px;text-align: right;" | (24)
|}

where <math display="inline">\Delta t</math> is the time step used for the transient solution and <math display="inline">l^e</math> is a characteristic element length computed as <math display="inline">l^e = 2(V^e)^{1/n_s}</math> where <math display="inline">V^e</math> is the element area (for 3-noded triangles) or volume (for 4-noded tetrahedra). For heterogeneous material the values of <math display="inline">\mu </math> and <math display="inline">\rho </math>  are computed at the element center <span id='citeF-38'></span>[[#cite-38|[38]]]. For  a solid material, the viscosity <math display="inline">\mu </math> is substituted by <math display="inline">\bar \mu \Delta t</math> with <math display="inline">\bar \mu </math> defined in Eq.(7).

We note that <math display="inline">\tau =0</math> for “compressible” type problems not requiring stabilization. In these cases Eq.([[#eq-23|22]]) recovers the standard form for the conservation of mass of the infinitesimal theory (see Eqs.([[#2.3 Mass conservation equation|16]])).

==4 VARIATIONAL EQUATIONS==

===4.1 Momentum equations===

Multiplying Eq.(1) by arbitrary test functions <math display="inline">w_i</math> with dimensions of velocity and integrating over the analysis domain <math display="inline">{}^{n+1}V</math> gives the weighted residual form of the momentum equations as <span id='citeF-46'></span><span id='citeF-48'></span>[[#cite-46|[46,48]]]

<span id="eq-26"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\int _{{}^{n+1}V} w_i \left(\rho \frac{Dv_i}{Dt}- {\partial {}^{n+1}\sigma _{ij} \over \partial {}^{n+1} x_j}- {}^{n+1} b_i\right)dV =0 </math>
|}
| style="width: 5px;text-align: right;" | (25)
|}

Integrating by parts  the term involving <math display="inline">\sigma _{ij}</math> and using the Neumann boundary conditions ([[#eq-20|19]]) yields the weak variational form of the momentum equations as

<span id="eq-27"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\int _{{}^{n+1}V} w_i \rho \frac{Dv_i}{Dt} dV + \int _{{}^{n+1}V} \delta \varepsilon _{ij} {}^{n+1} \sigma _{ij} dV - \int _{{}^{n+1}V} w_i {}^{n+1} b_i dV - \int _{{}^{n+1}\Gamma _t} w_i {}^{n+1} t_i^p d\Gamma =0 </math>
|}
| style="width: 5px;text-align: right;" | (26)
|}

where <math display="inline">\delta \varepsilon _{ij}= {\partial w_i \over \partial {}^{n+1}x_j}+{\partial w_j \over \partial {}^{n+1}x_i}</math> is an arbitrary (virtual) strain rate field. Eq.([[#eq-27|26]]) is the standard form of the ''principle of virtual power''  <span id='citeF-3'></span><span id='citeF-7'></span><span id='citeF-48'></span>[[#cite-3|[3,7,48]]].

Substituting  the stresses from Eq.([[#eq-2|2]]) into ([[#eq-27|26]]) gives the following expression (in matrix form)

<span id="eq-28"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\int _{{}^{n+1}V} {w}^T \rho \frac{D\hbox{v}}{Dt} dV + \!\int _{{}^{n+1}V} \delta {\boldsymbol \varepsilon }^T {}^{n+1}{s} dV  + \!\int _{{}^{n+1}V} \delta {\boldsymbol \varepsilon }^T {m} {}^{n+1}{p} dV -</math>
|-
| style="text-align: center;" | <math>  -\int _{{}^{n+1}V} {w}^T {}^{n+1}{b}dV - \!\int _{{}^{n+1}\Gamma _t} \hbox{w}^T {}^{n+1}\hbox{t}^p d\Gamma =0  </math>
|}
| style="width: 5px;text-align: right;" | (27)
|}

In Eq.([[#eq-28|27]]) <math display="inline">{w},{v}</math> and <math display="inline">\delta {\boldsymbol \varepsilon }</math> are vectors containing the test functions, the velocities and the virtual strain rates respectively; <math display="inline">{b}</math> and <math display="inline">\hbox{t}^p</math> are the body force and surface traction vectors, respectively and <math display="inline">{m}</math> is an auxiliary vector. These vectors are defined as (for 2D problems)

<span id="eq-29"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\begin{array}{l}\hbox{w} =[w_1,w_2]^T\quad ,\quad \hbox{v} =[v_1,v_2]^T \quad ,\quad \hbox{b} =[b_1,b_2]^T \quad ,\quad  \hbox{t}^p =[t_1^p,t_2^p]^T\\[.5cm]  \delta {\boldsymbol \varepsilon }= [\delta \varepsilon _{11},\delta \varepsilon _{22}, \delta \varepsilon _{12}]^T \quad ,\quad  {m}=[1,1,0]^T  \end{array} </math>
|}
| style="width: 5px;text-align: right;" | (28)
|}

===4.2 Mass conservation equation===

We multiply Eq.([[#eq-23|22]]) by arbitrary (continuous) test functions <math display="inline">q</math> (with dimensions of pressure) defined over the analysis domain. Integrating over <math display="inline">{}^{n+1}V</math> gives

<span id="eq-30"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\int _{{}^{n+1}V} -\frac{q}{\kappa } \frac{Dp}{Dt}dV +  \int _{{}^{n+1}V} q \varepsilon _v dV + \int _{{}^{n+1}V} q \tau {\partial {}^{n+1}\hat r_{m_i} \over \partial {}^{n+1}x_i} dV =0  </math>
|}
| style="width: 5px;text-align: right;" | (29)
|}

The third integral in Eq.([[#eq-30|29]]) corresponding to the stabilization term is zero in the compressible parts of the continuum.

Integrating by parts the last integral in Eq.([[#eq-30|29]]) and using ([[#eq-24|23]]) gives after some transformations <span id='citeF-38'></span>[[#cite-38|[38]]]

<span id="eq-31"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\begin{array}{r}\displaystyle \int _{{}^{n+1}V} \frac{q}{\kappa } \frac{Dp}{Dt}dV - \int _{{}^{n+1}V}q \varepsilon _v dV + \int _{{}^{n+1}V} \tau {\partial q \over \partial {}^{n+1}x_i} \left({\partial {}^{n+1}s_{ij} \over \partial {}^{n+1}x_j} + {\partial {}^{n+1}p \over \partial {}^{n+1}x_i} + {}^{n+1}b_i\right)dV\\ \displaystyle - \int _{{}^{n+1}\Gamma _t} q \tau \left[\rho \frac{Dv_n}{Dt}-\frac{2}{h_n} \left({}^{n+1}s_n +{}^{n+1} p - {}^{n+1}t_n\right)\right]d\Gamma =0 \end{array}   </math>
|}
| style="width: 5px;text-align: right;" | (30)
|}

where <math display="inline">v_n</math> and <math display="inline">t_n</math> are the velocity and the traction force normal to the boundary <math display="inline">\Gamma _t</math>, respectively and <math display="inline">s_n = s_{ij} n_j</math>. The characteristic length <math display="inline">h_n</math> in Eq.([[#eq-31|30]]) is taken as <math display="inline">h_n = \frac{l^e}{2}</math> in practice.

===4.3 Thermal balance equation===

Application of the standard weighted residual method to the thermal balance equations ([[#eq-18|17]]), ([[#eq-21|20]]) and ([[#eq-22|21]]) leads, after standard operations, to <span id='citeF-46'></span><span id='citeF-48'></span>[[#cite-46|[46,48]]]

<span id="eq-32"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\int _{{}^{n+1}V} \hat w \rho c \frac{DT}{Dt}dV + \int _{{}^{n+1}V} {\partial \hat w  \over \partial {}^{n+1}x_i}  k {\partial T \over \partial {}^{n+1}x_i}dV - \int _{{}^{n+1}V} \hat w {}^{n+1} Q dV+ \int _{{}^{n+1}\Gamma _q} \hat w {}^{n+1} q_n^p d\Gamma =0 </math>
|}
| style="width: 5px;text-align: right;" | (31)
|}

where <math display="inline">\hat w</math> are the space weighting functions for the temperature.

==5 FEM DISCRETIZATION==

We discretize  the analysis domain into finite elements with <math display="inline">n</math> nodes in the standard manner leading to a mesh with a total number of <math display="inline">N_e</math> elements and <math display="inline">N</math> nodes <span id='citeF-33'></span><span id='citeF-46'></span>[[#cite-33|[33,46]]]. In our work we will choose simple 3-noded linear triangles (<math display="inline">n=3</math>) for 2D problems and 4-noded tetrahedra (<math display="inline">n=4</math>) for 3D problems with local linear shape functions <math display="inline">N_i^e</math> defined for each node <math display="inline">i</math> (<math display="inline">i=1,\cdots , n</math>) of element <math display="inline">e</math> <span id='citeF-33'></span><span id='citeF-47'></span>[[#cite-33|[33,47]]]. The velocity components, the  pressure and the temperature are interpolated over the mesh in terms of their nodal values in the same manner using the  ''global linear shape functions'' <math display="inline">N_j</math> spanning over the elements sharing node <math display="inline">j</math> (<math display="inline">j=1,\cdots , N</math>) <span id='citeF-33'></span><span id='citeF-46'></span>[[#cite-33|[33,46]]]. In matrix form we have (for 2D problems)

<span id="eq-33"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>{v} = {N}_v\bar {v}  \quad ,\quad p = {N}_p\bar {p}\quad  , \quad T = {N}_T \bar {T}  </math>
|}
| style="width: 5px;text-align: right;" | (32)
|}

where

<span id="eq-34"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\begin{array}{c}\displaystyle \bar {v} = \left\{\begin{matrix}\bar{v}^1\\\bar{v}^2\\\vdots \\ \bar{v}^N\end{matrix}  \right\}\quad \hbox{with}~ \bar{v}^i = \left\{\begin{matrix}\bar{v}^i_1\\\bar{v}^i_2\end{matrix}  \right\}~, ~ \bar {p} =\left\{\begin{matrix}\bar{p}^1\\\bar{p}^2\\\vdots \\ \bar{p}^{N}\end{matrix}  \right\}~, ~ \bar {T}= \left\{\begin{matrix}\bar{T}_1\\\bar{T}_2\\\vdots \\ \bar{T}^N\end{matrix}  \right\}\\[.5cm] \displaystyle {N}_v = [{N}_1, {N}_2,\cdots , {N}_N ]^T ~,~ {N}_p = {N}_T = [{N}_1, { N}_2,\cdots , {N}_N ] \end{array} </math>
|}
| style="width: 5px;text-align: right;" | (33)
|}

with <math display="inline">{N}_j = N_j {I}_2</math> where <math display="inline">{I}_2</math> is the <math display="inline">2\times 2</math> unit matrix.

In Eq.([[#eq-34|33]]) vectors <math display="inline">\bar{v}</math>, <math display="inline">\bar {p}</math> and <math display="inline">\bar {T}</math>  contain the nodal velocities, the nodal pressures and the nodal temperatures for the whole mesh, respectively and the upperindex denotes the nodal value for each vector or scalar magnitude.

Substituting Eqs.([[#eq-33|32]]) into Eqs.([[#eq-27|26]]), ([[#eq-31|30]]) and ([[#eq-32|31]]) and choosing a Galerkin formulation with <math display="inline">w_i = q =  \hat w_i =N_i</math> leads to the following system of algebraic equations

<span id="eq-35"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>{M}_0 {\dot{\bar{v}}} + {g} +{Q}{}^{n+1}\bar {p}- {f}_v={0} </math>
|}
| style="width: 5px;text-align: right;" | (34a)
|}

<span id="eq-36"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\hbox{M}_1 {\dot{\bar{p}}}-{Q}^T {}^{n+1}\bar{v} + ({L}+{M}_b){}^{n+1} \bar {p}- {f}_p={0} </math>
|}
| style="width: 5px;text-align: right;" | (34b)
|}

<span id="eq-37"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>{C} {\dot{\bar{T}}}+\hat {L} {}^{n+1}\bar{T} -  {f}_T={0}  </math>
|}
| style="width: 5px;text-align: right;" | (34c)
|}

where <math display="inline">{\dot{\bar{a}}}</math>  denotes the material time derivative of the components of  a vector <math display="inline">{a}</math>. The element form of the matrices and vectors in Eqs.([[#5 FEM DISCRETIZATION|5]]) is given in Box 1.

<div class="center" style="font-size: 75%;">
[[File:Draft_Samper_718004671_9250_BoxI.png|550px]]

'''Box 1'''. Element form of the matrices and vectors in Eqs.(34) (for 2D problems).
</div>



We note that Eqs.([[#5 FEM DISCRETIZATION|34]]) involve the geometry at the updated configuration (<math display="inline">{}^{n+1}V</math>). This geometry is unknown; hence the solution of Eqs.([[#5 FEM DISCRETIZATION|34]]) has to be found iteratively. The iterative solution scheme chosen in this work is presented in the next section.

<br/>

'''Remark 5.''' The boundary terms of vector <math display="inline">{f}_p</math> can be incorporated into the matrices of Eq.([[#eq-36|36]]). This leads to a non symmetrical set of equations. These boundary terms are computed here iteratively within the incremental solution scheme.

'''Remark 6.''' The presence of matrix <math display="inline">{M}_b</math> in Eq.([[#eq-36|36]]) allows us to compute the pressure without the need of prescribing its value at the free surface <span id='citeF-38'></span>[[#cite-38|[38]]].

==6 TRANSIENT SOLUTION OF THE DISCRETIZED EQUATIONS==

Eqs.([[#5 FEM DISCRETIZATION|34]]) are solved in time with an implicit Newton-Raphson type iterative scheme <span id='citeF-3'></span><span id='citeF-7'></span><span id='citeF-47'></span>[[#cite-3|[3,7,47]]]. The basic steps within a time increment <math display="inline">[n,n+1]</math> are:<br/>

* Initialize variables: <math display="inline">({}^{n+1}\hbox{x}^1,{}^{n+1}\bar{v}^1,{}^{n+1}{s}^1,{}^{n+1}\bar{p}^1,  {}^{n+1}\bar{T}^1)\equiv  \left\{{}^{n}{x},{}^{n}\bar{v},{}^{n}{s},{}^{n}\bar{p},{}^{n}\bar{T}\right\}</math>.<p> In the following <math display="inline">(\cdot )^i</math> denotes a value computed at the <math display="inline">i</math>th iteration. </p>
* Iteration loop: <math display="inline">i=1,\cdots , N_{iter}</math>.<p> For each iteration. 

</p>

===''Step 1. Compute the initial Cauchy stresses <math>{}^{n}{\boldsymbol \sigma }^i</math> for solid elements''===

<span id="eq-38"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>{}^{n} {\boldsymbol \sigma }^i =  \frac{1}{J^i}{F}^i [{}^{n}{S}] {F}^{iT} \quad \hbox{with } F_{kl}^i = {\partial {}^{n+1}x_k^i \over \partial {}^{n} x_l} \quad \hbox{and } {J^i} = \vert {F}^i\vert   </math>
|}
| style="width: 5px;text-align: right;" | (35a)
|}

with

<span id="eq-39"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>{}^{n}{S} = {}^{n} {\boldsymbol \sigma }\quad \hbox{and}\quad {}^{n}{\sigma }_{ij} = {}^{n}s_{ij} + {}^{n}p \delta _{ij}  </math>
|}
| style="width: 5px;text-align: right;" | (35b)
|}

===''Step 2. Compute the nodal velocity increments <math>\Delta \bar {v}</math>''===

From Eq.([[#5 FEM DISCRETIZATION|34]]a)

<span id="eq-40"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>{H}_v^i \Delta \bar {v} = - {}^{n+1}\bar{\hbox{r}}_m^i \rightarrow  \Delta \bar{v} </math>
|}
| style="width: 5px;text-align: right;" | (36a)
|}

with the momentum residual <math display="inline">\bar{r}_m</math> and the iteration matrix <math display="inline">{H}_v</math>  given by

<span id="eq-41"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\bar{\hbox{r}}_m = \hbox{M}_0 {\dot{\bar{v}}} + \hbox{g}+\hbox{Q}\bar {p}-\hbox{f}_v\quad , \quad \hbox{H}_v = \frac{1}{\Delta t} \hbox{M}_0 + \hbox{K} + \hbox{K}_v  </math>
|}
| style="width: 5px;text-align: right;" | (36b)
|}

where <math display="inline"> \hbox{K}_{v}</math> is the bulk tangent matrix (see Remark 9) and

<span id="eq-42"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\hbox{K}_{ij}^e = \int _{{}^nV^e} \left(\hbox{B}_i^{eT} {\boldsymbol{\cal C}}_T \hbox{B}_j^{e} dV + {G}^T_i [{S} ]  {G}_j \Delta t \right)dV   </math>
|}
| style="width: 5px;text-align: right;" | (36c)
|}

where <math display="inline">{\boldsymbol{\cal C}}_T</math> is the tangent constitutive matrix emanating from the linearization of Eq.([[#eq-35|34a]]) (expressed in the current configuration) with respect to the nodal velocities <span id='citeF-10'></span><span id='citeF-37'></span>[[#cite-10|[10,37]]]. The second term in matrix <math display="inline">\hbox{K}^e</math> is explained in  Remark 7.

===''Step 3. Update the nodal velocities''===

<span id="eq-43"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>{}^{n+1}\bar{v}^{i+1}= \bar{v}^i + \Delta \bar{v} </math>
|}
| style="width: 5px;text-align: right;" | (37)
|}

===''Step 4. Compute the nodal pressures''===

From Eq.([[#eq-36|34b]]) we obtain

<span id="eq-44"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\hbox{H}_p^i  {}^{n+1}\bar{p}^{i+1}= \frac{1}{\Delta t} \hbox{M}_1\bar{p}^i  + {Q}^T \bar{v}^{i+1}+ \bar{f}^{i}_p  \rightarrow {}^{n+1}\bar{p}^{i+1}   </math>
|}
| style="width: 5px;text-align: right;" | (38a)
|}

with

<span id="eq-45"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\hbox{H}_p = \frac{1}{\Delta t} \hbox{M}_1 + \hbox{L} + \hbox{M}_b </math>
|}
| style="width: 5px;text-align: right;" | (38b)
|}

===''Step 5. Update the nodal coordinates and the deformation gradient''===

<span id="eq-46"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>{}^{n+1}{x}^{i+1}= {x}^i + \frac{1}{2} (\bar{v}^{i+1} + {}^{n}\bar{v})\Delta t </math>
|}
| style="width: 5px;text-align: right;" | (39)
|}

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>{}^{n+1}F^{i+1}_{ij} = {\partial {}^{n+1}{x}^{i+1}_i \over \partial {}^{n}{x_j}}</math>
|}
|}

A more accurate expression for computing <math display="inline">{}^{n+1}{x}^{i+1}</math> can be used involving the  nodal accelerations <span id='citeF-37'></span>[[#cite-37|[37]]].

===''Step 6.Compute the deviatoric Cauchy stresses''===

''Fluids:''

<span id="eq-47"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>{}^{n+1}{s}^{i+1} = {D}_T\hbox{B}\bar {v}^{i+1}  </math>
|}
| style="width: 5px;text-align: right;" | (40a)
|}

''Solids:''

<span id="eq-48"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\Delta  {s}= {D}_T  \hbox{B} \Delta \bar {v}  </math>
|}
| style="width: 5px;text-align: right;" | (40b)
|}

where <math display="inline">{D}_T</math> is the tangent constitutive matrix for the deviatoric Cauchy stresses. For elastic solids and Newtonian fluids <math display="inline">{D}_T = {D}</math> where <math display="inline">{D}</math> is given by Eq.([[#eq-6|6]]) and <math display="inline">\mu =\bar \mu \Delta t</math> for solids.

For non-linear continua, the adequate expression for <math display="inline">{D}_T</math> has to be used <span id='citeF-3'></span><span id='citeF-10'></span>[[#cite-3|[3,10]]].

===''Step 7.Compute the  stresses''===

''Solids:''<br/>

Cauchy stresses:

<span id="eq-49"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>{}^{n+1}\sigma _{ij}^{i+1} = {}^{n}\sigma _{ij}^{i} + \Delta  {s}_{ij} + \Delta \bar{p}\delta _{ij}  </math>
|}
| style="width: 5px;text-align: right;" | (41a)
|}

with

<span id="eq-50"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\Delta \bar{p}= {}^{n+1}\bar{p}^{i+1} - {}^{n}\bar {p}  </math>
|}
| style="width: 5px;text-align: right;" | (41b)
|}

2d Piola Kirchhoff stresses:

<span id="eq-51"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>{}^{n+1}{S}^{i+1} = J {F}^{-1}[{}^{n+1}{\boldsymbol \sigma }^{i+1}] {F}^{-T} ~~\hbox{with } ~~{F} \equiv {}^{n+1}{F}^{i+1}  </math>
|}
| style="width: 5px;text-align: right;" | (41c)
|}

''Fluids:''

<span id="eq-52"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>{}^{n+1}\sigma _{ij}^{i+1} = {}^{n+1}{s}^{i+1}_{ij} + {}^{n+1}{p}^{i+1}\delta _{ij}  </math>
|}
| style="width: 5px;text-align: right;" | (41d)
|}

===''Step 8.Compute the nodal temperatures''===

<span id="eq-53"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\left[\frac{1}{\Delta t}{C} + \hat{L}   \right]\Delta \bar{T}=- {}^{n+1} \bar{r}^i_T \quad ,\quad  {}^{n+1}\bar{T}^{i+1}= {}^{n}\bar{T}^i + \Delta \bar{T} </math>
|}
| style="width: 5px;text-align: right;" | (42)
|}

with

<span id="eq-54"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\bar {r}_T= {C} {\dot{\bar{T}}} + \hat{L} \bar{T} - {f}_T </math>
|}
| style="width: 5px;text-align: right;" | (43)
|}

===''Step 9. Check convergence''===

Verify the following conditions:

<span id="eq-55"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\begin{array}{c}\displaystyle \Vert {}^{n+1}\bar {v}^{i+1}- {}^{n+1}\bar{v}^{i}\Vert \le e_v \Vert {}^{n}\bar {v}\Vert \\ \displaystyle \Vert {}^{n+1}\bar {p}^{i+1}-{}^{n+1}\bar {p}^i\Vert \le e_p  \Vert {}^{n}\bar{p}\Vert \\ \displaystyle \Vert {}^{n+1}\bar {T}^{i+1} - {}^{n+1}\bar {T}^i\Vert \le e_T  \Vert {}^{n}\bar{T}\Vert  \end{array}   </math>
|}
| style="width: 5px;text-align: right;" | (44)
|}

where <math display="inline">e_v</math>, <math display="inline">e_p</math> and <math display="inline">e_T</math> are prescribed error norms. In the examples presented in the paper we have set <math display="inline">e_v = e_p=e_T = 10^{-3}</math>.

If  conditions ([[#eq-55|44]]) are satisfied then make <math display="inline">n \leftarrow n+1 </math> and proceed to the next  time step. Otherwise, make the iteration counter <math display="inline">i \leftarrow i+1 </math> and repeat Steps 1&#8211;9.

The derivatives and integrals in all the matrices of the iteration matrix <math display="inline">{H}_v</math>  are computed on the ''updated discretized configuration at time'' <math display="inline">n</math> (<math display="inline">{}^nV</math>) while the residual vectors <math display="inline">\bar{r}_T</math> and <math display="inline">\bar{r}_m</math> are computed in the current configuration. This is a particular version of the ''updated Lagrangian formulation'' <span id='citeF-3'></span><span id='citeF-10'></span><span id='citeF-37'></span><span id='citeF-47'></span><span id='citeF-48'></span>[[#cite-3|[3,10,37,47,48]]].

'''Remark 7.''' The  tangent deviatoric constitutive matrix <math display="inline">{\boldsymbol{\cal C}}_T</math> in Eq.([[#eq-42|36c]]) depends on the constitutive model chosen. For Newtonian fluids <math display="inline">{\boldsymbol{\cal C}}_T = {\boldsymbol{\cal C}}</math> where <math display="inline">{\boldsymbol{\cal C}}</math> is obtained by transforming the deviatoric constitutive tensor <math display="inline">{c}_{ijkl}=\mu (\delta _{ik}\delta _{jl} + \delta _{il}\delta _{jk})</math> from the updated to the current configuration as <math display="inline"> {\cal C}_{ijkl}=F^{-1}_{A_i}F^{-1}_{B_j} F^{-1}_{C_k} F^{-1}_{D_l}     {c}_{ABCD}</math> <span id='citeF-3'></span><span id='citeF-37'></span>[[#cite-3|[3,37]]]. Matrix <math display="inline">\boldsymbol{\cal C}</math> is formed by applying Voigt rule to the terms of tensor <math display="inline">{\cal C}_{ijkl}</math>. A similar expression is obtained for elastic solids changing <math display="inline">\mu </math> by <math display="inline">\bar \mu \Delta t</math> <span id='citeF-10'></span>[[#cite-10|[10]]].

The second term in the integral of Eq.([[#eq-42|36c]]) represents the initial stress contribution to matrix <math display="inline">{K}</math> <span id='citeF-3'></span><span id='citeF-10'></span><span id='citeF-37'></span>[[#cite-3|[3,10,37]]]. Matrices <math display="inline">{G}</math> and <math display="inline">[ S]</math> in Eq.([[#eq-42|36c]]) are given by (for 2D problems)

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math> {G} = \begin{bmatrix}     \bar {G} & \bar {0}\\ \bar {0}& \bar {G}     \end{bmatrix}~~,~~  \bar {G} = \begin{bmatrix}   {}_nN_{1,1} & 0 &|& {}_nN_{2,1} & 0 &|& \cdots & {}_nN_{n,1}\\     {}_nN_{1,2} & 0 &|& {}_nN_{2,2} & 0 &|& \cdots & {}_nN_{n,1}     \end{bmatrix} ~~,~~  \bar {0} = \left\{\begin{matrix} 0\\0\end{matrix} \right\}     </math>
|}
|}

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>[{ S}] =  \begin{bmatrix}{S} &  {0}\\ {0}& {S}     \end{bmatrix}~~,~~ {0} = \begin{bmatrix}0&0\\0&0\end{bmatrix} ~~,~~     {}_nN_{i,j}={\partial N_i \over \partial ^{n}x_j}     </math>
|}
| style="width: 5px;text-align: right;" | (45)
|}

where <math display="inline">{S}</math> is the 2d Piola-Kirchhoff stress tensor.

'''Remark 8.''' The iteration matrix <math display="inline">\hbox{H}_v</math> in Eq.([[#eq-40|36a]]) is an ''approximation of the exact tangent matrix for the solid/fluid problem'' in the updated Lagrangian formulation <span id='citeF-3'></span><span id='citeF-10'></span><span id='citeF-37'></span>[[#cite-3|[3,10,37]]]. The form of <math display="inline">\hbox{H}_v</math>  used in this work has yielded good results  with convergence achieved for the nodal values for the velocities,  the pressure and the temperature in 3&#8211;4 iterations for all the problems analyzed.

'''Remark 9.'''  Including the  bulk stiffness matrix <math display="inline">\hbox{K}_v</math> in <math display="inline">\hbox{H}_v</math> is important for the fast convergence, mass preservation and overall accuracy of the iterative solution for quasi and fully incompressible materials <span id='citeF-9'></span><span id='citeF-38'></span>[[#cite-9|[9,38]]].     The element expression of <math display="inline">\hbox{K}_v</math> can be obtained as <span id='citeF-38'></span>[[#cite-38|[38]]]

<span id="eq-57"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\hbox{K}_v^e = \int _{{}^nV^e} \hbox{B}^T \hbox{m} \theta \Delta t\kappa \hbox{m}^T \hbox{B}dV </math>
|}
| style="width: 5px;text-align: right;" | (46)
|}

where <math display="inline">\theta </math> is a positive number such that <math display="inline">0<   \theta \le 1</math> that has the role of preventing the ill-conditioning of the iteration matrix <math display="inline">{H}_v</math> for highly incompressible materials. For a fully incompressible material (<math display="inline">\kappa =\infty </math>), a finite  value of <math display="inline">\kappa </math> is used in practice within <math display="inline">{K}_v</math> as this helps to obtaining an accurate solution with reduced mass loss in few iterations per time step <span id='citeF-9'></span>[[#cite-9|[9]]]. These considerations, however, do not affect the value of <math display="inline">\kappa </math> within matrix <math display="inline">{M}_1</math> in Eq.([[#eq-36|34b]]) that vanishes for a fully incompressible material. A similar approach for improving mass conservation in incompressible fluids was proposed  in <span id='citeF-41'></span>[[#cite-41|[41]]].

'''Remark 10.''' The time step within a time interval <math display="inline">[n,n+1]</math> is chosen as <math display="inline">\Delta t =\min \left(\frac{^n l_{\min }^e}{\vert {}^n{v}\vert _{\max }},\Delta t_b\right)</math> where <math display="inline">^n l_{\min }^e</math> is the minimum characteristic distance  of all elements in the mesh, with <math display="inline">l^e</math> computed as explained in Section 3, <math display="inline">\vert ^n{v}\vert _{\max }</math> is the maximum value of the modulus of the velocity of all nodes in the mesh  and <math display="inline">\Delta t_b</math> is the critical time step of all nodes approaching a solid boundary defined as <math display="inline">\Delta t_b = \min \left(\frac{^n l_b}{\vert ^n{v}_b\vert _{\max }}\right)</math> where <math display="inline">^n l_b</math> is the distance from the node to the boundary and <math display="inline">^n{v}_b</math> is the velocity of the node. This definition of <math display="inline">\Delta t</math> intends that no node crosses a solid boundary during a time step <span id='citeF-38'></span>[[#cite-38|[38]]].

'''Remark 11.''' The material properties for the fluid and the solid may be dependent on the temperature. This effect is accounted for by updating the material properties in terms of the temperature within the iteration loop.

==7 ABOUT THE PARTICLE FINITE ELEMENT METHOD (PFEM)==

===7.1 The basis of the PFEM===

Let us consider a domain <math display="inline">V</math> containing  fluid and solid subdomains. Each subdomain is characterized by a set of points, hereafter termed ''virtual particles''. The virtual particles contain all the information  for defining the geometry and the material and mechanical properties of the underlying subdomain. In the PFEM both subdomains are modelled using an ''updated'' ''Lagrangian formulation'' <span id='citeF-3'></span><span id='citeF-47'></span>[[#cite-3|[3,47]]].

The solution steps within a time step in the PFEM are as follows:

<div id='img-2'></div>
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:draft_Samper_718004671-Figure2a.png|600px|Sequence of steps to update a “cloud” of virtual particles (nodes) representing a domain containing a fluid and a solid  from ⁿt  to ⁿ⁺²t. u, v,a,ɛ , ̇ɛ and σ  denote the displacement, the velocity, the acceleration, the strain, the strain rates and the Cauchy stresses, respectively.]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 2:''' Sequence of steps to update a “cloud” of virtual particles (nodes) representing a domain containing a fluid and a solid  from <math>^n t</math>  to <math>{}^{n+2}t</math>. <math>{u}, {v},{a},{\boldsymbol \varepsilon } , \dot {\boldsymbol \varepsilon }</math> and <math>{\boldsymbol \sigma }</math>  denote the displacement, the velocity, the acceleration, the strain, the strain rates and the Cauchy stresses, respectively.
|}

<ol>

<li>The starting point at each time step is the cloud of points <math display="inline">C</math> in the fluid and solid domains. For instance <math display="inline">^{n} C</math> denotes the cloud at time <math display="inline">t={}^n t </math> (Figure [[#img-1|1]]). The virtual particles are assumed to be coincident with the nodes in the finite element mesh generated in Step 3. </li>

<li>Identify the boundaries defining the analysis domain <math display="inline">^{n} V</math>, as well as the subdomains in the fluid and the solid. This is an essential step as some boundaries (such as the free surface in fluids) may be severely distorted during the solution, including separation and re-entering of nodes. The Alpha Shape method <span id='citeF-8'></span>[[#cite-8|[8]]] is used for the boundary definition. Clearly, the accuracy in the reconstruction of the boundaries depends on the number of points in the vicinity of each boundary and on the Alpha Shape parameter. In the problems solved in this work the Alpha Shape method has been implementation as described in <span id='citeF-14'></span><span id='citeF-26'></span>[[#cite-14|[14,26]]]. </li>

<li> Discretize  the analysis domain <math display="inline">^{n} V</math> with a finite element mesh <math display="inline">{}^{n} M</math> using the virtual particles as the mesh nodes. We use an efficient mesh generation scheme based on an enhanced Delaunay tesselation <span id='citeF-13'></span><span id='citeF-14'></span>[[#cite-13|[13,14]]]. </li>

<li> Solve the Lagrangian equations of motion for the overall continuum using the standard FEM. Compute the state variables in  the next (updated) configuration for <math display="inline">^n t+\Delta t</math>: velocities, pressure, strain rate and viscous stresses in the fluid and displacements, stresses and strains in the solid. </li>

<li>  Move the mesh nodes to a new position <math display="inline">^{n+1} C</math> where ''n''+1 denotes the time <math display="inline">{}^n t +\Delta t</math>, in terms of the time increment size. </li>

<li> Go back to step 1 and repeat the solution for the next time step to obtain <math display="inline">{}^{n+2} C</math>. </li>

</ol>

Note that the key differences between the PFEM and the classical FEM are the remeshing technique and the identification of the domain boundary at each time step.

The CPU time required for meshing grows linearly with the number of nodes. As a general rule,  meshing consumes for 3D problems around 15% of the total CPU time per time step, while the solution of the equations (with typically 3 iterations per time step) and the system assembly consume approximately 70% and 15% of the CPU time per time step, respectively. These figures refer to analyses in a single processor Pentium IV PC <span id='citeF-35'></span>[[#cite-35|[35]]]. Considerable speed can be gained using parallel computing techniques.

Application of the PFEM in fluid and solid mechanics and in fluid-structure interaction problems can be found in <span id='citeF-1'></span><span id='citeF-2'></span><span id='citeF-4'></span><span id='citeF-5'></span><span id='citeF-8'></span><span id='citeF-9'></span><span id='citeF-13'></span><span id='citeF-14'></span><span id='citeF-15'></span><span id='citeF-16'></span><span id='citeF-17'></span><span id='citeF-18'></span><span id='citeF-20'></span><span id='citeF-26'></span><span id='citeF-27'></span><span id='citeF-34'></span><span id='citeF-35'></span><span id='citeF-37'></span><span id='citeF-38'></span><span id='citeF-41'></span><span id='citeF-43'></span>[[#cite-1|[1,2,4,5,8,9,13,14,15,16,17,18,20,26,27,34,35,37,38,41,43]]] as well in www.cimne.com/pfem.

===7.2 Treatment of contact conditions in the PFEM===

Known velocities at boundaries in the PFEM are prescribed in strong form to the boundary nodes. These nodes might belong to fixed external boundaries or to moving boundaries linked to the interacting solids. Surface tractions are applied to the Neumann part of the boundary, as it is usual in the FEM.

Contact between fluid particles and fixed boundaries is accounted for by the incompressibility condition which  naturally prevents fluid nodes to penetrate into the solid boundaries <span id='citeF-14'></span><span id='citeF-26'></span><span id='citeF-32'></span><span id='citeF-35'></span>[[#cite-14|[14,26,32,35]]].

The contact between two solid interfaces is treated by introducing a layer of ''contact elements'' between the two interacting solid interfaces (Figure [[#img-3|3]]). This contact layer is ''automatically created during the mesh generation step'' by prescribing a minimum distance <math display="inline">\left(h_{c} \right)</math> between two solid boundaries. If the distance exceeds the minimum value <math display="inline">\left(h_{c} \right)</math> then the generated elements are treated as fluid elements. Otherwise the elements are treated as contact elements where a relationship between the tangential and normal forces and the corresponding displacement is introduced  in <span id='citeF-26'></span><span id='citeF-32'></span><span id='citeF-35'></span>[[#cite-26|[26,32,35]]].

This algorithm allows us to model complex frictional contact conditions between two or more interacting bodies moving in water in  a simple manner. The algorithm has  been used to model frictional contact situations between rigid or elastic solids in structural mechanics applications, such as soil/rock excavation problems <span id='citeF-4'></span><span id='citeF-5'></span>[[#cite-4|[4,5]]]. The frictional contact algorithm described above has been extended by Oliver ''et al.'' <span id='citeF-20'></span>[[#cite-20|[20]]] for analysis of metal cutting and machining problems.

<div id='img-3'></div>
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 65%;max-width: 100%;"
|-
|[[Image:draft_Samper_718004671-contact-solid-boundaries.png|390px|Layer of contact elements at a soil-solid interface modelled with the PFEM]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 3:''' Layer of contact elements at a soil-solid interface modelled with the PFEM
|}

==8 EXAMPLES==

===8.1 Numerical examples of manufacturing problems===

Some examples are presented next to show the capabilities of the method for the simulation of industrial forming processes. All the problems presented involve large strains and big changes in the geometry boundaries. The “particle” description of the continuum assumed in the PFEM introduces the capability of adapting the geometry to the computed displacement solution automatically.

The numerical solution is computed taking in account thermal and mechanical effects with the solution strategy described in Section 6.

The changes in the geometry are recognized by the PFEM features. As explained in Section 7.1, the finite element mesh is obtained by  a reconnection of the particles and a remeshing of the domain. A refinement of the critical areas at each time step has been performed. The criterion to detect the critical areas for refining the mesh is based on the evaluation of the plastic energy dissipation, but other mechanical variables can be used for this purpose. Details of the application of the PFEM to solid mechanics problems can be found in <span id='citeF-4'></span><span id='citeF-5'></span>[[#cite-4|[4,5]]].

===8.2 Extrusion of a steel plate===

Extrusion is one of the techniques used to reduce the section of pieces of metal and form a reduced piece of the same object. This example considers a piece of steel with the dimensions shown in  Figure [[#img-4|4]] pushed towards a rigid wall die. Two different shapes of the wall die have been analysed. Both models are depicted in Figure [[#img-4|4]]. The symmetry of the problem allows us to consider only the upper part of the piece and a 2D simulation to solve the problem. The material is modelled with a thermo-elastoplastic constitutive law with a Huber-Von Misses yield surface. The constitutive model can be found in <span id='citeF-42'></span>[[#cite-42|[42]]]. The physical properties of the material (steel) considered in the example are shown in Box 2.


{| class="wikitable" style="text-align: left; margin: 1em auto;"

|-
| colspan='2' style="border-left: 2px solid;border-right: 2px solid;border-left: 2px solid;" | ''MATERIAL PROPERTIES'':

|-
| style="border-left: 2px solid;" | Young  Modulus 
| style="border-right: 2px solid;" | <math> 206.9 \times 10^9  Pa</math>
|-
| style="border-left: 2px solid;" | Poisson Coefficient 
| style="border-right: 2px solid;" | 0.29 
|-
| style="border-left: 2px solid;" | Yield Stress 
| style="border-right: 2px solid;" | <math>450\times 10^6  Pa</math>
|-
| style="border-left: 2px solid;" | Linear Hardening Modulus &nbsp;~&nbsp;
| style="border-right: 2px solid;" | <math>129.24\times 10^6  Pa</math>
|-
| style="border-left: 2px solid;" | Reference Hardening 
| style="border-right: 2px solid;" | <math>450\times 10^6  Pa</math>
|-
| style="border-left: 2px solid;" | Saturation Hardening 
| style="border-right: 2px solid;" | <math>715\times 10^6  Pa</math>
|-
| style="border-left: 2px solid;" | Hardening Exponent 
| style="border-right: 2px solid;" | 16.93 
|-
| style="border-left: 2px solid;" | Conductivity 
| style="border-right: 2px solid;" | <math>45 N/s K</math>
|-
| style="border-left: 2px solid;" | Density 
| style="border-right: 2px solid;" | <math>7800 kg/m^3 </math>
|-
| style="border-left: 2px solid;" | Specific Capacity 
| style="border-right: 2px solid;" | <math>460  m^2/s^2 K</math>
|-
| style="border-left: 2px solid;" | Flow Stress Softening 
| style="border-right: 2px solid;" | <math>0.002 K^{-1}</math>
|-
| style="border-left: 2px solid;" | Hardening Softening 
| style="border-right: 2px solid;" | <math>0.002 K^{-1}</math>
|-
| style="border-left: 2px solid;" | Dissipation Factor 
| style="border-right: 2px solid;" | 0.9 
|- style="border-bottom: 2px solid;"
| style="border-left: 2px solid;" | Expansion Coefficient 
| style="border-right: 2px solid;" | <math>10^{-5} K^{-1} </math>

|}<br/>

'''Box 2.''' Material properties for the steel extrusion problem

An imposed velocity is set on the left side of the models: <math display="inline">10 mm/s</math> for the vertical die wall and <math display="inline">2.5 mm/s</math> for the inclined wall. In both models the vertical displacement is imposed to a zero value at the lower side of the steel plate. The initial temperature is <math display="inline">^\circ </math>T=293.15 K and the left boundary is fixed to this temperature during the analysis.

<div id='img-4'></div>
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 95%;max-width: 100%;"
|-
|[[Image:draft_Samper_718004671-dimensions_wall_step.png|570px|Extrusion of a steel plate with two wall dies. Dimensions expressed in mm. ]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 4:''' Extrusion of a steel plate with two wall dies. Dimensions expressed in mm. 
|}

<div id='img-5'></div>
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 95%;max-width: 100%;"
|-
|[[Image:draft_Samper_718004671-extrusion_step_mesh.png|570px|Extrusion of a steel plate at three  time instants: t = 0.09s, t = 2.59s, t = 5.19s. Vertical die wall]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 5:''' Extrusion of a steel plate at three  time instants: t = 0.09s, t = 2.59s, t = 5.19s. Vertical die wall
|}

The results of the extrusion analysis with the vertical step are shown in Figures [[#img-5|5]] and [[#img-6|6]]. Different time instants are considered, from the beginning to the end of the process. The contact forces with the wall, the temperature of the model, the plastic strain and the plastic strain rate are measures that can be obtained from the numerical results. With the distribution and the values of these quantities the extrusion process can be studied and optimized.

A similar set of results for a different extrusion problems using a die wall with a progressive reduction of the section are depicted in Figures [[#img-7|7]] and [[#img-8|8]].

<div id='img-6'></div>
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 95%;max-width: 100%;"
|-
|[[Image:draft_Samper_718004671-extrusion_step_results.png|570px|Results of the extrusion of a steel plate  (vertical die wall) at t = 2.59s. Nodal forces in Newtons, pressure in N/mm², temperature in Kelvins and plastic strain rate in 1/s. ]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 6:''' Results of the extrusion of a steel plate  (vertical die wall) at t = 2.59s. Nodal forces in Newtons, pressure in <math>N/mm^2</math>, temperature in Kelvins and plastic strain rate in <math>1/s</math>. 
|}

<div id='img-7'></div>
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 95%;max-width: 100%;"
|-
|[[Image:draft_Samper_718004671-extrusion_slope_mesh.png|570px|Extrusion of a steel plate at three  time instants: t = 0.9s, t = 11.9s, t = 24.9s. Inclined die wall]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 7:''' Extrusion of a steel plate at three  time instants: t = 0.9s, t = 11.9s, t = 24.9s. Inclined die wall
|}

<div id='img-8'></div>
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 95%;max-width: 100%;"
|-
|[[Image:draft_Samper_718004671-extrusion_slope_results.png|570px|Results of the extrusion of a steel plate at t = 11.9s (inclined die wall). Nodal forces in Newtons, pressure in N/mm², temperature in Kelvins and plastic strain rate in 1/s. ]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 8:''' Results of the extrusion of a steel plate at t = 11.9s (inclined die wall). Nodal forces in Newtons, pressure in <math>N/mm^2</math>, temperature in Kelvins and plastic strain rate in <math>1/s</math>. 
|}

<div id='img-9'></div>
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 95%;max-width: 100%;"
|-
|[[Image:draft_Samper_718004671-single_shear_cut.png|570px|Cutting of a steel plate with a single cutter. Dimensions in mm.]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 9:''' Cutting of a steel plate with a single cutter. Dimensions in mm.
|}

<div id='img-10'></div>
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 95%;max-width: 100%;"
|-
|[[Image:draft_Samper_718004671-double_shear_cut.png|570px|Cutting of a steel plate with a two symmetric cutters. Dimensions in mm.]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 10:''' Cutting of a steel plate with a two symmetric cutters. Dimensions in mm.
|}

===8.3 Shear cutting===

The next example shows the cutting of solid plate in 2D. It is a thermo-mechanical problem which involves large deformations and the process of cutting by means of a rigid tool. The initial conditions of the problem are depicted in Figures [[#img-9|9]] and [[#img-10|10]]. A thermo-elastoplastic constitutive law with the same  material properties as described in the previous example has been used. As previously mentioned, the PFEM combines the Lagrangian updating of the configuration with a redefinition of the domain geometry. These features allow to model naturally the cutting region by  refining  the solid model in the tool tip and by  inserting  new particles (nodes) when large flat boundaries appear at the tip contact zone.

The cutting tool moves downwards with a velocity of <math display="inline">5 mm/s</math> in the single cutter model and with a velocity of <math display="inline">200mm/s</math> in the problem with two cutters. In both models the horizontal displacement is imposed to zero in the outer sides of the plate. The initial temperature in the domain is set to 293.15 K and is kept fixed to this value in the outer boundary sides during  the analysis.

The PFEM results of the material deformation and the temperature distribution for various time instants are shown in Figures [[#img-11|11]] and [[#img-12|12]].

<div id='img-11'></div>
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 95%;max-width: 100%;"
|-
|[[Image:draft_Samper_718004671-single_cut_results.png|570px|Cutting process. Results at the onset,  the middle and  the end of the process using a single cutter. Nodal forces are expressed in Newtons and the temperature in Kelvins.]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 11:''' Cutting process. Results at the onset,  the middle and  the end of the process using a single cutter. Nodal forces are expressed in Newtons and the temperature in Kelvins.
|}

<div id='img-12'></div>
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 95%;max-width: 100%;"
|-
|[[Image:draft_Samper_718004671-double_cut_results.png|570px|Results at the onset, the middle and  the end of a plate cutting process with two symmetric cutters. Nodal forces are expressed in Newtons and the temperature in Kelvins.]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 12:''' Results at the onset, the middle and  the end of a plate cutting process with two symmetric cutters. Nodal forces are expressed in Newtons and the temperature in Kelvins.
|}

<div id='img-13'></div>
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 95%;max-width: 100%;"
|-
|[[Image:draft_Samper_718004671-dimensions_forge.png|570px|Forging of a steel cylinder against to curved rigid walls. Dimensions in meters.]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 13:''' Forging of a steel cylinder against to curved rigid walls. Dimensions in meters.
|}

===8.4 Forging of a cylindrical steel piece===

The next example considers the forging of a  steel cylinder. The material properties are the same as in Box 2. The mould is defined by two rigid walls that move  towards each other at the same speed, (<math display="inline">1 m/s</math>)  and enclose the steel piece. The dimensions of the model are depicted in Figure [[#img-13|13]]. An axisymmetric solution is considered. The initial temperature of the steel material is <math display="inline">^\circ </math>T =293.15 K.

The results of the material deformation, the pressure contours and the temperature distribution for various time instants are shown in Figures [[#img-14|14]] and [[#img-15|15]].

<div id='img-14'></div>
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 95%;max-width: 100%;"
|-
|[[Image:draft_Samper_718004671-forge_mesh_force.png|570px|Meshes and wall nodal forces at the beginning,  the middle and  the end of the forging process. Nodal forces are expressed in Newtons.]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 14:''' Meshes and wall nodal forces at the beginning,  the middle and  the end of the forging process. Nodal forces are expressed in Newtons.
|}

<div id='img-15'></div>
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 95%;max-width: 100%;"
|-
|[[Image:draft_Samper_718004671-forge_pressure_temperature.png|570px|Pressure and temperatures at the beginning,  the middle and  the end of the forging process. Pressure is expressed in Pascals and the temperature in Kelvins.]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 15:''' Pressure and temperatures at the beginning,  the middle and  the end of the forging process. Pressure is expressed in Pascals and the temperature in Kelvins.
|}

===8.5 Hot forging of a metal piece===

Metal hot forging consists on deforming a work piece  at high temperature under compressive forces in order to obtain the desired shape of the final product. It is a largely used process in industrial metal manufacturing as it allows an improvement of the material properties of the final part via a controlled deformation process. For example, via  hot forging it is possible to reduce the impurities and  defects in the material, thereby avoiding stress concentrations in the final product.

The metal forging can be performed using an open die or a closed die. In this work, the latter case is considered. The geometry and the problem data for the 2D simulation are shown in Figure [[#img-16|16]]. The forging process is performed by controlling the displacement of the upper rectangular die which is pushed downwards with a velocity v=0.01<math display="inline">m/s</math> for a duration of 5.65<math display="inline">s</math>, while the material rigid walls are kept at the same position during the analysis. At the beginning of the process the metal piece and the lower walls have temperatures <math display="inline">^\circ </math>T=2000 K and <math display="inline">^\circ </math>T =1000 K, respectively. The forging is carried out through two steps: during the first one the piece is compressed by the descending rectangular  die; then the die is stopped and from t=6.0<math display="inline">s</math> to t=50.0<math display="inline">s</math> the piece is cooled down.  The cooling process is carried out by applying the temperature given by the following step function to both the upper rectangular die and the internal walls.

<span id="eq-58"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\displaystyle  \begin{array}{l}\displaystyle  T=850K   6.0s < t \leq 8.5s\\ \displaystyle T=600K   8.5s < t \leq 10.5s\\ \displaystyle T=300K   10.5s < t \leq 50s\\ \end{array}   </math>
|}
| style="width: 5px;text-align: right;" | (58)
|}

It is assumed that the rectangular die has the same temperature of the metal piece during the compression phase. The normal heat flux through the surfaces in contact with the air has not been taken into account in the analysis.

The metal is modelled as an isotropic incompressible non-Newtonian fluid with a non-linear viscosity  given by the following relation <span id='citeF-44'></span><span id='citeF-45'></span>[[#cite-44|[44,45]]]

<span id="eq-59"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\mu = \frac{\sigma _{y}}{\sqrt{3}\bar \varepsilon }  </math>
|}
| style="width: 5px;text-align: right;" | (59)
|}

where <math display="inline">{\sigma _{y}}</math> is the uniaxial yield stress of the material and <math display="inline">\bar{{\varepsilon }}</math> is the deviatoric strain rate invariant defined as:

<span id="eq-60"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\bar \varepsilon = \sqrt{\frac{2}{3}{\varepsilon }_{ij} {\varepsilon }_{ij}}  </math>
|}
| style="width: 5px;text-align: right;" | (60)
|}

The uniaxial yield stress <math display="inline">{\sigma _{y}}</math> depends on the temperature as follows:

<span id="eq-61"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\displaystyle  \begin{array}{l}\displaystyle {\sigma _{y}}=157.7                                                            T \leq 1500\\ \displaystyle {\sigma _{y}}=157.7+3.6\cdot (T-1500)^2-3250\cdot (T-1500)     1500< T \leq 1930 \\ \displaystyle {\sigma _{y}}=79.4                                                               T >1930 \\ \end{array}   </math>
|}
| style="width: 5px;text-align: right;" | (61)
|}

The constitutive equation ([[#eq-59|59]]) is typical in the study of industrial forming processes by the plastic/viscoplastic flow approach <span id='citeF-44'></span><span id='citeF-45'></span><span id='citeF-48'></span>[[#cite-44|[44,45,48]]].

The metal piece has been discretized using 6849 3-noded triangular finite elements. The 2D coupled thermal-mechanical simulation has been run for 50<math display="inline">s</math> with <math display="inline">\Delta t</math>=<math display="inline">0.0005s</math>.

The snapshots of Figure [[#img-17|17]] refer to the initial compression phase that has a duration of 5.65<math display="inline">s</math>. The upper rectangular die moves downwards forcing the metal piece to take the shape of the lower rigid walls.

Figure [[#img-18|18]] shows four representative snapshots of the cooling period.

<div id='img-16'></div>
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:draft_Samper_718004671-forgingInput2.png|400px|]]
|[[Image:draft_Samper_718004671-forgingData.png|200px|]]
|-
| colspan="2"|[[Image:draft_Samper_718004671-metalData.png|600px|Hot forging of a metal piece. Initial geometry, coordinates of corner points and material properties.]]
|- style="text-align: center; font-size: 75%;"
| colspan="2" | '''Figure 16:''' Hot forging of a metal piece. Initial geometry, coordinates of corner points and material properties.
|}

<div id='img-17'></div>
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:draft_Samper_718004671-forging800.png|400px|]]
|[[Image:draft_Samper_718004671-forging1500.png|400px|]]
|-
|[[Image:draft_Samper_718004671-forging3610.png|400px|]]
|[[Image:draft_Samper_718004671-tempComp.png|400px|]]
|-
|[[Image:draft_Samper_718004671-forging5320.png|400px|]]
|[[Image:draft_Samper_718004671-forging5560.png|400px|]]
|-
| colspan="2"|[[Image:draft_Samper_718004671-forging5650.png|300px|Hot forging of a metal piece. Compression phase. Snapshots of the deformation with temperature contours at different time instants.]]
|- style="text-align: center; font-size: 75%;"
| colspan="2" | '''Figure 17:''' Hot forging of a metal piece. Compression phase. Snapshots of the deformation with temperature contours at different time instants.
|}

<div id='img-18'></div>
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:draft_Samper_718004671-forgingCooling8450.png|400px|]]
|[[Image:draft_Samper_718004671-forgingCooling10445.png|400px|]]
|-
|[[Image:draft_Samper_718004671-forgingCooling25000.png|400px|]]
|[[Image:draft_Samper_718004671-forgingCooling50000.png|400px|]]
|-
| colspan="2"|[[Image:draft_Samper_718004671-tempCooling.png|300px|Hot forging of a metal piece. Cooling period. Snapshots of the  temperature contours on the deformed geometry at different time steps.]]
|- style="text-align: center; font-size: 75%;"
| colspan="2" | '''Figure 18:''' Hot forging of a metal piece. Cooling period. Snapshots of the  temperature contours on the deformed geometry at different time steps.
|}

<div id='img-19'></div>
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:draft_Samper_718004671-ballsInput2.png|400px|]]
|[[Image:draft_Samper_718004671-ballDataGeo.png|400px|]]
|-
| colspan="2"|[[Image:draft_Samper_718004671-ballDataMat.png|600px|Falling of three solid objects in a heated tank filled with a fluid. Initial geometry, thermal conditions and material properties.]]
|- style="text-align: center; font-size: 75%;"
| colspan="2" | '''Figure 19:''' Falling of three solid objects in a heated tank filled with a fluid. Initial geometry, thermal conditions and material properties.
|}

===8.6 Falling of three solid objects in a heated tank filled with fluid===

Three solid objects with the same shape fall from the same height into a tank containing a fluid at rest. The geometry, the problem data, and the initial thermal conditions are shown in Figure [[#img-19|19]].

The fluid in the tank has an initial temperature of T=340<math display="inline">K</math>, while the solid bodies from left to right have initial temperatures of T=180 K, T=200 K, and T=220 K, respectively. The solid and the fluid domains have been discretized with a finite element mesh of 9394 3-noded triangular elements. The simulation has been run for eight seconds with <math display="inline">\Delta t</math>=<math display="inline">0.0001 s</math>. The heat flux in the normal direction is assumed to be zero for the boundaries in contact with the air and the walls. Figure [[#img-20|20]] shows six representative snapshots of the numerical simulation with the temperature results plotted over the fluid domain and the objects domains.

<div id='img-20'></div>
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:draft_Samper_718004671-Balls664.png|400px|]]
|[[Image:draft_Samper_718004671-Balls1816.png|400px|]]
|-
|[[Image:draft_Samper_718004671-Balls2656.png|400px|]]
|[[Image:draft_Samper_718004671-tempIce.png|400px|]]
|-
|[[Image:draft_Samper_718004671-Balls4500.png|400px|]]
|[[Image:draft_Samper_718004671-Balls7000.png|400px|]]
|-
| colspan="2"|[[Image:draft_Samper_718004671-Balls8000.png|300px|Falling of three solid objects in a heated tank filled with a fluid. Snapshots with temperature contours at different time steps.]]
|- style="text-align: center; font-size: 75%;"
| colspan="2" | '''Figure 20:''' Falling of three solid objects in a heated tank filled with a fluid. Snapshots with temperature contours at different time steps.
|}

<div id='img-21'></div>
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 85%;max-width: 100%;"
|-
|[[Image:draft_Samper_718004671-REPBalls6hGraph.png|510px|Falling of three solid objects in a heated tank filled with a fluid. Evolution of the temperature at the center of the three objects.]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 21:''' Falling of three solid objects in a heated tank filled with a fluid. Evolution of the temperature at the center of the three objects.
|}

In the graph of Figure [[#img-21|21]] the evolution of the temperature at the central point of the three solid objects is plotted.

===8.7 Melting of an ice block===

A cylindrical ice block at initial temperature <math display="inline">^\circ </math>T=270 K is dropped into a tank containing water at rest at an initial temperature <math display="inline">^\circ </math>T=340 K. The initial geometry, the problem data and the initial thermal conditions for the 2D simulation are shown in Figure [[#img-22|22]].

<div id='img-22'></div>
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:draft_Samper_718004671-iceInput2.png|400px|]]
|[[Image:draft_Samper_718004671-iceDataGeo.png|400px|]]
|-
| colspan="2"|[[Image:draft_Samper_718004671-iceDataMat.png|600px|Melting of an ice cylinder. Geometry, material data and initial thermal conditions.]]
|- style="text-align: center; font-size: 75%;"
| colspan="2" | '''Figure 22:''' Melting of an ice cylinder. Geometry, material data and initial thermal conditions.
|}

Ice is treated as a hypoelastic solid until  some of its elements reach the fusion temperature (T=273.15 K ). These elements are transferred to the fluid domain and take the physical properties of the fluid. For this analysis the following assumptions were made: the mechanical and thermal properties of water and  ice do not change with  temperature and the heat normal flux along the boundaries in contact with the air or the walls have been considered to be zero.

In Figure [[#img-23|23]] snapshots of some representative instants of the analysis are shown. The temperature contours are plotted over the water domain and the ice block.

In Figure [[#img-24|24]] the detail of the melting of the ice block is illustrated. The finite element mesh is drawn over the solid and the fluid domains.

<div id='img-23'></div>
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:draft_Samper_718004671-Ice44.png|400px|]]
|[[Image:draft_Samper_718004671-Ice110.png|400px|]]
|-
|[[Image:draft_Samper_718004671-Ice263.png|400px|]]
|[[Image:draft_Samper_718004671-tempIce.png|400px|]]
|-
|[[Image:draft_Samper_718004671-Ice455.png|400px|]]
|[[Image:draft_Samper_718004671-Ice1001.png|400px|]]
|-
| colspan="2"|[[Image:draft_Samper_718004671-Ice1600.png|300px|Melting of an ice cylinder. Snapshots of the melting process with temperature contours at different time steps.]]
|- style="text-align: center; font-size: 75%;"
| colspan="2" | '''Figure 23:''' Melting of an ice cylinder. Snapshots of the melting process with temperature contours at different time steps.
|}

<div id='img-24'></div>
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:draft_Samper_718004671-iceZoom281.png|400px|]]
|[[Image:draft_Samper_718004671-iceZoom470.png|400px|]]
|-
|[[Image:draft_Samper_718004671-iceZoom977.png|400px|]]
|[[Image:draft_Samper_718004671-iceZoom1274.png|400px|]]
|-
|[[Image:draft_Samper_718004671-iceZoom1469.png|400px|]]
|[[Image:draft_Samper_718004671-iceZoom1547.png|400px|Melting of an ice cylinder. Zoom of the ice domain at different time instants.]]
|- style="text-align: center; font-size: 75%;"
| colspan="2" | '''Figure 24:''' Melting of an ice cylinder. Zoom of the ice domain at different time instants.
|}

===8.8 Falling and warming of a solid sphere in a cilindrical tank containing hot water===

A solid sphere at <math display="inline">^\circ </math>T=270 K is dropped into a cilndrical tank containing still water  at <math display="inline">^\circ </math>T=340 K. In  Figure [[#img-22|22]] the geometry and the problem data are given. The 3D simulation is run for a duration of 3<math display="inline">s</math>  with <math display="inline">\Delta t</math>=<math display="inline">0.001 s</math>. A finite element mesh of 144663 4-noded tetrahedral elements has been used to discretize both the fluid domain and the sphere. Figure [[#img-26|26]] shows six representative snapshots of the analysis. The temperature contours have been plotted over the fluid domain and the sphere.

Figure [[#img-27|27]] shows the  evolution of the temperature at the center of the sphere versus time. The solid material has high conductivity and this explains its fast warming. After three seconds, it practically reaches the equilibrium temperature. We note that the  normal heat flux along the boundaries in contact with the air have been considered to be zero.

<div id='img-25'></div>
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:draft_Samper_718004671-ball3DInput.png|400px|]]
|[[Image:draft_Samper_718004671-ball3DDataGeo.png|400px|]]
|-
| colspan="2"|[[Image:draft_Samper_718004671-ball3DDataMat.png|600px|Warming of a solid sphere with initial temperature ^∘T=270 K falling into a cylindrical tank containing a fluid at ^∘T=340 K. Initial geometry and material properties.]]
|- style="text-align: center; font-size: 75%;"
| colspan="2" | '''Figure 25:''' Warming of a solid sphere with initial temperature <math>^\circ </math>T=270 K falling into a cylindrical tank containing a fluid at <math>^\circ </math>T=340 K. Initial geometry and material properties.
|}

<div id='img-26'></div>
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:draft_Samper_718004671-ball3D8.png|400px|]]
|[[Image:draft_Samper_718004671-ball3D17.png|400px|]]
|-
|[[Image:draft_Samper_718004671-ball3D38.png|400px|]]
|[[Image:draft_Samper_718004671-tempIce.png|400px|]]
|-
|[[Image:draft_Samper_718004671-ball3D53.png|400px|]]
|[[Image:draft_Samper_718004671-ball3D71.png|400px|]]
|-
| colspan="2"|[[Image:draft_Samper_718004671-ball3D300.png|300px|Warming of a solid sphere (^∘T=270 K) falling into a cylindrical tank containing a fluid at ^∘T=340 K. Snapshots of the  motion of the sphere and the temperature contours at different time steps.]]
|- style="text-align: center; font-size: 75%;"
| colspan="2" | '''Figure 26:''' Warming of a solid sphere (<math>^\circ </math>T=270 K) falling into a cylindrical tank containing a fluid at <math>^\circ </math>T=340 K. Snapshots of the  motion of the sphere and the temperature contours at different time steps.
|}

<div id='img-27'></div>
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 80%;max-width: 100%;"
|-
|[[Image:draft_Samper_718004671-ball3DEvolution.png|480px|Warming of a solid sphere (^∘T=270 K) falling into a cylindrical tank containing a fluid at ^∘T=340 K. Evolution of the temperature at the center of the sphere.]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 27:''' Warming of a solid sphere (<math>^\circ </math>T=270 K) falling into a cylindrical tank containing a fluid at <math>^\circ </math>T=340 K. Evolution of the temperature at the center of the sphere.
|}

==9 CONCLUDING REMARKS==

We have presented a unified Lagrangian formulation for analysis of industrial forming processes involving thermally coupled interactions between deformable continua containing fluids and solids. A  residual-based  expression of the mass conservation equation obtained using the FIC method  provides the necessary stability for quasi/fully incompressible situations. The  governing equations for the generalized continuum are discretized with the FEM using simplicial elements with equal linear interpolation for the velocities, the pressure and the temperature. The merits of the formulation in terms of its general applicability have been demonstrated in the solution of a variety of thermally-coupled industrial forming processes using the Particle Finite Element Method (PFEM).

==ACKNOWLEDGEMENTS==

This research was partially supported by the Advanced Grant  project SAFECON of the European Research Council and the HFLUIDS project of the National Research Programme of Spain.

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