iMesh, iMeshP Use Cases

The use cases below describe the PDE to be solved, and any
application-imposed constraints on the parallel solution approach.
These are meant to illustrate the use of the iMesh and iMeshP
interfaces to solve these problems.

For a given use case, describe how the constructs proposed for iMesh
and iMeshP (e.g. Process, iMesh/iMeshP Instances, iMesh/iMeshP API functions,
Partition, Part) will be used to solve the problem.  While not
absolutely required, this will generally require desription of:

- the basic approach to distributing the domain and computations for
  the domain across Processes 

- what kind of communications are done in general, and how those
  interact with iMeshP constructs

- the basic steps of the computational or time step kernel, and how
  they interact with iMeshP/iMesh.


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Use case 1: Solve a discretized PDE with FEM using spatial domain
decomposition  

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Problem statement

Solve a function dF/dx = f(x) on a spatial domain with prescribed
boundary conditions.

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Use case 2: Radiation transport

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Problem Statement

In this use case, the radiation flux phi(x,E,t) is computed over a
finite element grid, with flux values stored at vertices.  In a very
simplified form, the implicit formulation of this equation can be
written as:

d/d(x,E,t) phi(x,E,t1) = f(S(x,E), phis(x,t0))

where 

phis(x,t) = sum_E{w_E phi(x,E,t)}

is the scalar flux, a weighted sum of energy-dependent fluxes phi(x,E,t).

The discretized problem is partitioned into S spatial domains {si} and
E energy domains {ej}. 

A given time step solution consists of two parts.  First, across an
energy subdomain ej, the problem is solved over the spatial subdomains
si using a domain decomposition PDE solution method; the result is an
phi(v(si),ej,t0), the radiation flux for a given energy ej at each
vertex si.  Second, the energy-based flux is converted to a scalar
phi(v(si),t0), by computing a weighted sum over energy subdomains ej of
the flux at each vertex, phi(v(si),ej,t0).  The scalar flux is used in
computing the energy-dependent flux phi(v(si),t1) on the next time step.

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Use case 3: Structural dynamics with parallel contact detection

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Problem Statement

Solve a structural dynamics problem:

d/dt^2 x(x,t) = f(x,t)

over a volumetric domain V with multiple connected sets V_k,
handling cases where connected sets come into contact and exert force
on each other when their boundaries collide.  

Solve each time step in two phases.  In the first phase, solve for new
positions x(x,t) using a standard (spatial) domain decomposition FEM,
with each process responsible for computing behavior of a spatial
subdomain.  In the second phase, for all faces bounding single volume
elements (i.e. not contiguously connected to two volume elements),
given the length of the next timestep t, compute location and time of
collision of any two faces in that set.  Solve this second phase by
distributing faces over processes according to spatial proximity,
based on some space-filling decomposition of the bounding box of all
volume elements (which will by definition enclose all bounding faces).

No restrictions should be placed on which process is assigned a volume
element and any of its connected boundary faces; that is, a volume and
its connected boundary faces are not required to be solved by the same
process.