Subject: SITeam Proceedings 4/14
Date: Wed, 14 Apr 1999 16:55:16 -0500
From: "Robert A. Fiedler" 

Amit described the ALE formulation for solid mechanics and how
this would fit into our iterative time stepping scheme for the
coupled problem (including regression).

The basic idea of the ALE (Arbitrary Lagrangian Eulerian) approach
is to write the evolution equations in a reference frame that is
moving at some grid velocity (with respect to the fixed lab frame).

For the fluid equations, this change of reference frames shows
up only in the computation of the fluxes of the mass, momentum, and
energy through the surfaces of the mesh zones.  For example, the mass
flux through a surface in a fixed (Eulerian) coordinate system is

  rho_gas * v_gas * area

in a fixed frame.  In a frame moving at velocity v_grid, the mass
flux is

  rho_gas * (v_gas - v_grid) * area

If v_grid = 0, we recover the Eulerian form.  If v_grid = v_gas,
the mass flux vanishes.  This is the same as the Lagrangian
formulation, in which the coordinate system follows the motion of
the fluid.

In order for the fluid equations to retain their simple form
despite being expressed in a moving frame, we solve them in
integral form, commonly known as the finite volume method.
All this means is that we solve for the mass, momentum, and
energy contained in a set of finite volumes (the mesh zones).
Note that these volumes are not constant -- they change as
the grid moves.  See the integral form at

  http://www.csar.uiuc.edu/F_bcs/sld011.htm

The familiar Eulerian differential form of the fluid equations
is derived by considering a finite volume and then taking the
limit as the volume shrinks to infinitesimal size.
One may obtain a differential form for the fluid equations
for a moving reference frame, but the finite volume formulation
is simpler and guarantees global mass, momentum, and energy
conservation.

In solid mechanics, a finite element method is to be used rather
than a finite volume one.  Amit showed us the procedure for
transforming the equations from Lagrangian coordinates to a
frame in which the physical extent of the solid material is
mapped to a domain of fixed size.

The inertial term, rho * displacement_double_dot, when transformed
expands to 5 terms containing products of various first and second
order time and space derivatives of the coordinate transformation
and the displacement -- just from the chain rule.

For both the fluids and solids, the coordinate transformation
(or, equivalently, the grid velocity) needs to be specified.
The boundary condition is that the transformed surface maps to
the (moving) physical surface.  We are free to define the mapping
(or motion) of the interior points in any way that makes sense.

Prasad efficiently generates a new mesh given the new location
of the surface and the previous mesh.  He then calculates a
grid velocity from

  v_grid = (x_new - x_old)/delta_t

He plugs this grid velocity into the ALE finite volume fluid
equations and updates the mass, momentum and energy in each
mesh zone.  Note that no interpolation has to be done to
get values on the new mesh -- the effect of the motion of the
mesh is taken into account by including the grid velocity in
the ALE finite volume equations.  There is no loss of accuracy no
matter how the grid velocity is defined, as long as the
mesh retains its nice geometrical properties (not too skewed).

Amit presented one standard way of defining the transformation
in which Laplace's equation is solved to obtain the
mapping at interior points, again with the boundary
condition that the transformed surface maps to the physical
one.  In one dimension, this is equivalent to making all the
elements the same physical size (a linear stretching).
Solving Laplace's equation for the transformation would not add
too much to the computational cost, since it adds just another
linear system to solve which is simpler and has the same number
of degrees of freedom as the equation of motion.  We may also
consider an approach similar to Prasad's if we can come up with
a quick way to generate a new mesh.

The coupled GEN1 code will include the ALE fluids and solids
modules plus the jump conditions complete with regression.
Time stepping will likely proceed in a manner similar to GENH:
we will first take ~10 steps in the fluids under the assumption
that we know the velocity of the solid surface
at the old and new time levels.  We plug the assumed solid
velocity into the jump conditions (as well as some fluid
variables) to get the regression rate (with respect to the
fixed lab frame; in the reference frame of the moving solid
surface, the regression rate is just a*P**n).  The location of
the surface is determined by integrating this regression rate,
by whatever method (particle tracking, level sets, ... ).

Once the fluids calculation finishes those ~10 steps, the
pressure, other fluid surface quantities, and the location of
the surface at the new time level are sent to the solids
module.  The solids module solves the transformed equation
of motion and updates the coordinate transformation so
that the transformed surface maps to the new physical surface
location.  If needed, the fluids and solids calculations
are repeated (with increasingly better estimates of what the
other module will get at the new time level) until the
convergence criteria are satisfied.

Eric pointed out that we may be able to decrease the number of
elements in the solid as the propellant burns away without
losing much accuracy because we can still represent the solution
quite well with fewer points.