Theory to Practice of Unsaturated Soil Mechanics - Unsaturated Soil Mechanics in Engineering Practice - page 19

1.6.2 Partial Differential Equation Solving
Partial differential equation solvers have been developed
in mathematics and computing science disciplines and are
increasingly being used to obtain solutions for geotechnical
engineering problems. A single PDE solver can be used to
solve several types of PDEs relevant to saturated-unsaturated
soil mechanics problems. It is also possible for more than
one physical phenomenon to be operative within a REV,
resulting in the necessity to combine the solution of multiple
PDEs in a “coupled” or “uncoupled” manner. For example,
it is necessary for both an equilibrium and a continuity PDE
to be simultaneously satisfied when solving consolidation
(or swelling) problems. In other words, it is necessary for
more than one PDE to be simultaneously solved in a cou-
pled or uncoupled mode. Commonly available PDE solvers
are generally capable of addressing these tasks.
Each PDE has a “primary” variable that can be computed,
and when there is more than one PDE to be simultaneously
solved, there will be more than one primary variable. Each
PDE may have one or more soil property to be input. If the
soil properties are a function of the primary variable being
solved, the solution becomes nonlinear. This is the general
situation when dealing with unsaturated soil problems and
as a consequence it is necessary to iterate to get a converged
solution. When the equations are highly nonlinear, it can be
a challenge to obtain a converged solution.
The ability to solve a wide range of geotechnical engi-
neering problems within a similar context gives rise to the
possibility of developing a special problem solving com-
puter platform called problem solving environment (PSE).
Gallopoulos et al., (1994) described a PSE as “ a computer
system that provides all the computational facilities needed
to solve a target class of problems. PSEs use the language of
the target class of problems, so users can run them without
specialized knowledge of the underlying computer hardware
or software. PSEs create a framework that is all things to
all people; they solve simple or complex problems, support
rapid prototyping or detailed analysis, and can be used in
introductory education or at the frontiers of science.” Par-
tial differential equation solvers form the basis for PSEs for
solving saturated-unsaturated soil mechanics problems.
All classic soil mechanics problems can be viewed in terms
of the solution of a PDE. Let us consider a few problems
commonly encountered in unsaturated soil mechanics. The
PDE for water flow through a saturated-unsaturated soil sys-
tem in either two or three dimensions is probably the most
common problem encountered. These solutions have been
extensively applied in disciplines beyond geotechnical engi-
neering such as agriculture, environmental engineering, and
water resources. Hydraulic heads are the primary variables
computed followed by solving for other variables of interest.
The water coefficient of permeability (hydraulic conductiv-
ity) is dependent upon the negative pore-water pressure and
this gives rise to a nonlinear equation with associated con-
vergence challenges.

All flow processes (i.e., water, air, and heat) have sim-
ilar PDEs that can be solved using a similar PDE solver.
Heat flow problems can readily be solved, but there are
added challenges associated with the freezing and thawing
of water. Air flow problems add the challenge of dealing
with a compressible fluid phase.
Analyses associated with slope stability, bearing capacity,
and earth pressure calculations have historically been per-
formed using plasticity and limit equilibrium methods (e.g.,
methods of slices). However, all these application areas
are increasingly being viewed as “optimization” solutions
imposed on the results of a stress analysis (Pham and Fred-
lund, 2003). Stresses computed from linear elastic analyses
with approximate elastic parameters have been shown to
be acceptable for subsequent usage in an optimization pro-
cedure (Stianson, 2008). Total stress fields are computed
in an uncoupled manner by “switching on” gravity body
forces. Techniques such as “dynamic programming” are then
used to determine the shape and location of the rupture
surface (Pham et al., 2001a, Pham and Fredlund 2003a;
Stianson, 2008). Although the finite-element-based “opti-
mization” solutions to slope stability problems have been
published, long-standing limit equilibrium analyses are still
important in geotechnical engineering practice. The limit
equilibrium analyses that have been applied for saturated
soil problems can be extended to unsaturated soil problems.
The theory associated with both limit equilibrium methods
of slices and optimization methods are presented in the topic.
The prediction of volume changes associated with expan-
sive clays and collapsible soils can also be performed using
the solution of a stress-deformation analysis (Vu and Fred-
lund, 2000). The soil properties for this problem are nonlin-
ear but can be converted to equivalent, incremental elastic
parameters. These problems can also be solved using more
elaborate elastoplastic models, but these solutions are con-
sidered to be outside the scope of this topic. In the case of
expansive or collapsible soils, the volume changes are com-
monly associated with changes in the negative pore-water
pressures (or matric suctions). Consequently, it is necessary
to combine a seepage analysis and a stress analysis in a cou-
pled or uncoupled manner in order to solve transient volume
change problems.

1.6.3 Convergence of Nonlinear Partial Differential
Equations
The nonconvergence of nonlinear PDEs is probably the sin-
gle greatest deterrent to the use of numerical methods in
engineering practice. However, there are several techniques
that have emerged that greatly assist in the solution of highly
nonlinear PDEs (Mansell et al., 2002). The most successful
technique appears to involve automatic, dynamic finite ele-
ment mesh refinement as well as mesh optimization (Oden,
1989), which are referred to as adaptive grid refinement
(ADR), methods.