Using the LATIN method

The LATIN method [Ladevèze, 1998] is a general procedure for solving nonlinear problems in structural mechanics. It consists of three key features:

1. A separation of global, linear quantities from local, nonlinear quantities;
2. A two-step iterative procedure which ``builds'' the solution;
3. An ad-hoc space-time procedure.

The first two elements were adopted in [Champaney, 1996] to model assembling structures, and in Dolbow:contact in conjunction with the X-FEM to model crack growth with frictional contact acting on the crack faces.

For the present problem, knowing all quantities at time , we wish to determine the temperatures and interface fluxes (31) at the new time step such that the weak form (30) is satisfied and the temperature at the interface is equal to the melting temperature . We will refer to the latter constraint as the constitutive law on the solid-liquid interface.

We adopt an iterative strategy to resolve this problem. For notational clarity, we will drop the superscript from all quantities, and assume the iterative procedure is invoked in moving from time step to . We will also use subscripts and , etc., to refer to iteration number.

The iterative procedure adopted by the LATIN method begins by considering two sets of solution spaces. The variables of interest are the temperature , its value on the interface :

the temperature gradient and the heat fluxes on the interface (31). We use the compact notation:

to define the solution spaces. Specifically, we let:

and

satisfying (38)

The goal is then to find the element located at the intersection of and , represented geometrically in Fig. 20. We use the superscript and to denote an element in or , respectively. The iterative strategy begins with an initial element in and builds a sequence of approximate solutions , ..., , , ...until convergence. A given iteration involves two steps: and as shown in Fig. 20. The iterations stop when the ``distance'' between and (measured with an appropriate norm) is below a specified tolerance.

Two successive approximations are always tied by a given search direction. The direction used when going ``down'' from the set to the set is denoted by whereas the direction when going ``up'' from to is denoted by . The appropriate choice of the search directions in order to achieve (fast) convergence is an important aspect of the LATIN method (see [Ladevèze, 1998]).

The next two sections describe the two steps in the method:

• Going down from to in the search direction . We will see that this step is local and may be solved independently for each pair of points facing each other on the interface.
• Going up from to in the search direction . This step involves a global solve.

Figure 20: The iterative procedure in the LATIN algorithm.

1. The local step: update on the interface
   The local update involves determining a new estimate for the fluxes and temperatures on the interface given quantities obtained from a solution . The new approximation is required to satisfy the constitutive law (i.e. ). Additional equations are provided by the search direction.

   To move from an element to , the search direction is associated with a linear operator . The search equations are then
   

   
   The above can be written more compactly as

       on (40)

   
   
   with the understanding that the relationship holds on both and , separately. This compact form will be used in the remainder of this section when convenient.

   The process of determining given then involves solving the above search equations in conjunction with the constitutive law on the interface

   
   
   The resulting closed form equations for are very simple in this case. They are given by
   
   

2. The global step
   We wish to move from to . This is accomplished through a global solve in conjunction with additional equations provided by the search direction which is also associated with the operator on the interface. With a known element , the search equations are then
   

       on (43)

   
   

   This equation is used in conjunction with the weak form of the governing equations (30), by solving for and making the substitution in the surface integrals on . The problem to be solved is then: Find which satisfies

   
   
   where it is understood that
   
   

   After solving (44), the local fluxes on the interface at step are given by

   
   
   with
   
   
   Iteration over subsequent steps continues until convergence is met, measured with an appropriate norm.