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Good morning, dear colleagues!
The title of my presentation is "Computationally efficient numerical method for heat transfer problems in the engineering of foundations construction on permafrost soils".
The reason behind temperature field computation is connected with the fact that a number of rules and regulations require determination of the following parameters:
the position of the frozen soil's upper surface beneath constructions,
the computational average of annual temperature,
the equivalent soil temperature, etc.
To define these parameters we need to know the temperature field for
the whole operational period of the construction, and
the neglect of any such computations may lead to disasters similar to
what you can see for yourselves in this picture.
To calculate the temperature field, we require specialized software, and
the criteria for effectiveness of this software is, of course, the achievement of
adequate accuracy and to also perform the computation over a minimal period of time.
In this report, I'll briefly discuss the specific problems relating to heat transfer which arise in foundations engineering. I will describe the numerical method and model that we have developed.
I will also explain the difference between our models and methods and other known methods.
I shall explain the advantages of our software application and models over similar software and models.
At the end of my presentation, I will briefly show the results of typical computations and formulate the conclusions.
It is a well-known fact that the interchanging seasons of
thawing and freezing of active soil layers
is accompanied by significant changes in soil structures
and its properties.
To take into account all of these changes
one has to consider a set of processes including
thermokarst, frost heave, cryogenic suction, formation of
frost-shattered clefts, destruction of cryopegs, etc.
The underlined processes for these phenomena are
both purely thermal and hydro-mechanical effects.
The relationship between these effects is shown schematically in this diagram.
And as one can see from the diagram, changes in temperature
lead to the migration of moisture in the pores in the soil,
and to changes in the state of solid particles that constitute the soil.
In turn, the presence of moisture leads to the necessity
to account for convective heat transfer,
and it also exerts pressure on solid particles of the soil.
Finally, the changes in the state of solid particles
lead to changes in heat content on one hand
and to the moisture redistribution in pores on the other.
The complete mathematical description of all these processes
require solution of a system consisting of:
the heat equation, Darcy's and Richard's equations, analysis of changes in thermodynamic potentials and solution of the elasto-plastic stress-strain problem.
And all of this is complicated by the fact that there is no complete reconstruction of volumetric deformation owing to the hydrostatic component of stress.
Therefore, serious problems are immediately apparent with respect to model formulation,
because it's very difficult to decide which effects are of primary importance for construction design.
At the same time, it is known that the temperature is the most actively changing
and most important parameter when designing constructions on permafrost.
This is due to the fact that all other parameters
change more slowly and over a narrower range.
That is why we have considered the heat transfer problem.
And the heat equation, shown on this slide, has been solved.
In this equation Ceff is the effective heat capacity,
while k is the thermal conductivity.
These quantities depend on temperature,
that is, the heat equation is non-linear.
Typical dependencies of these equations are shown in these figures.
As can be seen, heat capacity changes drastically
in the vicinity of the phase transition,
and this causes difficulties in the finite-difference approximation
of the original differential equation.
It is necessary to note that this heat equation can only be solved analytically in a one dimensional case with specified posed boundary conditions on a semi-infinite domain.
In all other cases we have to use numerical methods, especially when considering a 3D problem.
The problem becomes even more complicated if one attempts to factor in a set of accompanying interrelated effects.
As a rule, such detailed models require significant amounts of time to obtain the solution;
they are formulated for each specific case and should be normalized,
these are what we refer to as the ad hoc models.
If the model is very complex due to an attempt to account for a set of interrelated processes,
one encounters a problem of proving the existence of a solution
and its uniqueness for the obtained non-linear boundary-value problem.
And again, all this is complicated even further by the necessity
to design a stable algorithm,
which will lead to a convergent sequence of solutions.
So, what do we actually obtain as a result of the use of such models?
In the output, there is a plausible solution, which sometimes coincides with the measured results and sometimes doesn't.
Of course, it mainly depends on what is factored into the model and what isn't.
So, the existence of phase transition in our model was addressed with the introduction of
temperature dependencies of heat capacity and thermal conductivity,
in accordance with the presented expressions.
In these expressions Cth is the heat capacity of soil in the thawed phase,
while Cf is the heat capacity of soil in the frozen phase.
The thermal conductivity of soil in thawed and frozen phases
is similarly defined.
The omega function determines the dependence of ice content on temperature.
In this formula theta is the volumetric moisture content (water content),
and all these parameters, such as heat capacity and thermal conductivity
in thawed and frozen phases, are set separately for each soil layer.
Therefore, we can account for the
heterogeneous distribution of soil layers.
Convective heat transfer is also taken into account in the model,
and there is also the possibility of setting arbitrary shaped heat sources
and drains inside the simulation domain -
this is done in order to consider the operation of cooling devices.
For numerical solution of this problem
we used the Douglas-Rachford scheme
of the alternating direction method.
This scheme has the advantage that the matrices
of systems of linear algebraic equations have a simple structure,
which are potrayed in this figure.
As you can see, this is a 3-diagonal matrix. SLAE with the matrices
are solved fast and effectively using the Thomas algorithm.
With this numerical scheme we managed to obtain the stability criterion,
which relates the values of space and time discretization,
and also thermophysical parameters of soil-heat capacity
and thermal conductivity.
The result has been published this year in Proceedings.
Also the problem of method convergence has been studied
and it has been determined that in order for the method
to be convergent with the exact solution,
it is necessary for the finite-difference operators
to satisfy the following conditions:
they must be densely defined, strictly monotone, and single-valued.
For the considered problem, these conditions are satisfied
and the convergence rate is determined by the expression
which relates the values of space discretization steps,
the time discretization step and thermophysical parameters.
This expression shows that convergence can be quickly achieved.
Before turning to the demonstration of the computational results
I would like to briefly discuss the issue of
the effectiveness of our method.
To do this, I propose we consider a simplified model problem.
Let there be a cubic region of soil
with linear dimensions of 10x10x10 meters.
Boundary conditions of the first type -2 degrees Celsius
is set on all the faces of the cube.
The initial temperature in the whole volume is 0 degrees,
and the phase transition temperature is assumed to be 0 degrees.
Let us require computation of the temperature field in a 1000-days' time.
Let the objective accuracy of the computation be +/- 0,25
meters in space,and +/- 2 days in time.
In this case, it appears that for the computational mesh
consisting of 68 921 nodes, our method should perform nearly
1.5e8 arithmetic operations.
To solve the same problem using implicit formulation
of the finite element method,
it is necessary to perform a number of operations exceeding 3.9e10,
which is a few orders of magnitude greater than the previous figure.
And it is important to point out that such implicit formulation of the finite element method is used in many software applications available on the market, for example, Ansys and Comsol.
So, our method clearly performs better in terms of time efficiency.
And this is our advantage: such a low computational complexity of the method
allows one to apply it in engineering computations.
It is well-known that the design engineers have to compute
a set of computations
with various configurations of cooling devices and construction objects,
and they don't have the time to wait for days and weeks to obtain the results for a single computation.
Our method has been implemented and integrated into
the Computer Aided Engineering software Frost 3D.
I will briefly mention how the computer model in this software
is constructed.
Generally, computer model construction consists of 3 steps.
In the first step design drawings of the project are used.
Boundaries of the simulation domain, locations of construction objects and cooling devices
are set according to the design drawings of the project.
In the second step the reconstruction of 2D contours
into 3D geometry is performed.
Here, one can see a part of the obtained 3D geometry.
In this figure, soil layers are made semitransparent so you can see
the location of the cooling devices - the vertical and inclined pipes.
There is a sandy embankment at the top of the modeling region,
where the oil tank with the hot oil is supposed to be located.
As I have already said, for every soil layer,
the heat capacity and thermal conductivity
in thawed and frozen phases and phase transition temperature are set.
The working conditions of the cooling devices are also set.
In the third step the computational mesh is created.
Here, an example of the obtained computational mesh is presented.
Two intermediate soil layers have been rendered invisible
in order to show how the cooling devices are located inside the soil.
In this step, boundary conditions are set.
Boundary conditions on every face can be time-dependent, that is,
we can consider seasonal temperature fluctuations of the atmosphere.
It is also possible to set various heat flows.
Boundary conditions may be of the first, second and third type.
For this particular problem, the boundary condition of the third type,
and the temperature parameter +50 degrees Celsius,
- it was set on the soil surface (which is at the interface with the oil tank).
So, we have a heat flow which should lead to the thawing of frozen soil.
It is interesting to see the results of the interplay between the cooling
devices and this heat flow from the oil tank.
Here, the computational results are represented in a 3D form.
The change in the temperature field over a time period
for December 2013 to March 2015 can be seen.
Temperature values are represented by color.
A section of the computational domain is presented so that we could
see the effect from the operation of cooling devices.
There is a period of time, when they don't operate and the effect
of their influence isn't seen, and there's a period of time when they are operational
where we can see the effect and we can observe the heat flow from above.
The same computational results can be represented clearly
in the form of isolines on a plane.
Here the section is chosen in such a way that we can see
the operation of cooling devices.
On the basis of these results, it's possible to obtain any parameter -
the estimation is based on the temperature field data,
for example, the depth of zero amplitudes, frozen bed.
All of these parameters can be determined for any moment in time
that we are interested in.
In conclusion, I would like to summarize the following: the developed method allows rapid and effective simulation of the dynamics of a temperature field which takes phase transition into account,
and allows the computation of freezing time from cooling devices.
The most important issues are that the method converges with an exact solution,
and the computational complexity of the method is lower than the computational complexity of the methods
implemented in the more well-known commercial Computer Aided Engineering software.
Thank you for your attention!