Introduction
Many software packages for numerical computations allow users to use a static adaptive (hereby referred to as adaptive) step in the construction of an orthogonal hexahedral structured computational mesh. This means that informed users can employ their experience to get a more accurate computation without significantly increasing the computation time by specifying the areas of the computational domain in which, in their opinion, it is necessary to apply more detailed partitioning (use a smaller spatial step) as compared to the rest of the computational domain.
When properly used, adaptive partitioning of the computational domain is a powerful tool in numerical computations to increase accuracy. However, when the above option is overused, the computational time can increase dramatically without necessarily altering the accuracy of the computation to any significant degree. In this article, we describe the theoretical advantages and disadvantages of using adaptive partitioning of the computational domain, and also give two examples for numerical computations of thermal fields in soils. In the first example, application of the adaptive step is appropriate; this is not the case, however, in the second.
Disadvantages of using the adaptive mesh
ПIn the numerical solution of non-stationary problems, the time step is selected according to the numerical scheme stability criterion. For example, for the numerical solution of a heat conduction problem via the finite difference approximation method, the selection criterion of the time step _{} has the following form:
_{}
where _{} are the minimum spatial steps along the axes _{} respectively, and the constant C is selected based on the physical parameters of the problem.
When applying adaptive partitioning of the computational domain, strong reduction of the spatial step along any direction in the vicinity of a single node should be avoided. This practice does not produce a tangible increase in the accuracy of computations in general but, according to equation (1), it significantly reduces the time step and hence increases the computational time.
It is also worth separately mentioning the problem of the adaptive mesh application to finite approximation of differential operators. The problem is that the order of approximation of differential operators in the general case decreases. Let’s consider, for example, the finite-difference approximation of the operator _{}. Applying the approximation to a standard template, we obtain the following formula:
_{}
where _{}. Assuming sufficient smoothness of the function u, on the basis of the Taylor formula, we obtain the following equalities:
_{}
_{}
Inserting equalities (3) and (4) into (2), we obtain the following formula:
_{}
It follows from formula (5) that, when _{}, the operator _{} is approximated with first-order accuracy. At the same time, on the regular mesh _{}, and the operator _{} is approximated with second-order accuracy.
Advantages of using of the adaptive partitioning
Despite the fact that the difference operator is approximated with first-order accuracy in the general case, if, in practice, _{} and _{} do not differ significantly _{} the summand _{} in (5) will be quite small and will not have much impact on the accuracy of the approximation.
It is also worth mentioning that if the user wants to achieve a more accurate solution on a small area, it makes sense to perform the refinement of the mesh in this area only, instead of refining the mesh throughout the computational domain. In this case, the use of the adaptive mesh can significantly reduce the number of nodes of the computational mesh, and thus significantly reduce the computational time and the amount of memory used.
The need for using an adaptive mesh arises in cases when there is a need for a detailed discretization of small objects with complex shapes. The use of a uniform mesh in such cases leads to an excessive increase in the number of nodes, which makes it difficult to construct the model and to perform further computations on it.
Comparison of computation results
Example 1
With the help of the Frost 3D software package, we modeled the problem of heat conduction with phase transitions. The top view of the created model is demonstrated in Figure 1:
Figure 1 – Top view of the computational domain
The computational domain contains of four parallelepipeds and a particularity in the form of an asterisk in the center. The width of the domain is 60 m and the height is 20 m. The physical parameters of the materials are represented in Table 1.
Table 1 – Physical parameters of materials
Parameter |
Upper left and lower right parallelepiped |
Upper right and lower left box |
Asterisk |
Temperature at the initial time moment, ^{o}C |
20 |
10 |
-5 |
Volumetric heat capacity at the temperature higher than that of the phase transition, MJ/(m^{3}К) |
4.14 |
1.14 |
4.14 |
Volumetric heat capacity at the temperature lower than that of the phase transition, MJ/(m^{3}K) |
2.14 |
0.54 |
2.14 |
Thermal conductivity at the temperature higher than that of the phase transition, W/(m K) |
1.8 |
1.8 |
1.8 |
Thermal conductivity at the temperature lower than that of the phase transition, W/(m K) |
2.1 |
2.1 |
2.1 |
The phase transition temperature in all the materials is 0 ^{o}C. Zero heat flux is specified as a boundary condition at the boundaries of the computational domain.
For the above model, we construct a computational mesh using a uniform step along all three coordinate axes. The mesh step along each coordinate axis is 1.0 m. Consequently, the computational domain contains 78,141 nodes. As seen in Figure 2, the particularity in the center when using the uniform mesh is discretized in a very approximate fashion.
Figure 2 – Domain partitioning with uniform step
We now construct a computational mesh with approximately the same number of nodes using the adaptive step, with refinement in the center. The minimum step along the coordinate axes is 0.155 m and the maximum is 1.5 m. The entire domain contains 84,375 nodes. As seen in Figure 3, the complex object in the center is discretized much more accurately than in the case of the uniform mesh.
Figure 3 – Fragment of domain partitioning with irregular step
In order to verify the impact of rough discretization on the obtained result, we perform numerical computation and consider the dependence of temperature on time in the center of the computational domain for both the uniform and adaptive meshes (Figure 1).
The graphs below demonstrates the dependence of temperature (Kelvin) on time (days).
Analyzing the graphs, we note that the results of the computations differ significantly. We can see that the phase transition occurred on the sixtieth day for the uniform spatial step, while the same phenomenon for the adaptive step occurs on the one-hundred-and-thirtieth day.
Example 2
In example 2 we construct a similar model to that for Example 1, with the only difference being that there is no particularity in the center in the form of an asterisk. Similar to Example 1, we carry out two computations: one on the uniform mesh, and the second on the adaptive mesh. The steps of the meshes along the coordinate axes are equal to those used in Example 1.
Now we can examine the behavior of the temperature in the central point of the computational domain. Graph 2 demonstrates the dependencies of the temperature on time when using the uniform and adaptive meshes.
Comparing the results of the computations, we find that the thermal fields do not differ so significantly as in Example 1. However, in some numerical schemes using the time step selection criterion described above, the application of the adaptive mesh significantly increases the computational time, which indicates the potential inexpediency of using the adaptive step.
Conclusion
Summing up the advantages and disadvantages of applying adaptive partitioning of the computational domain, we note that the application of this option makes sense when the domain contains elements whose geometric features are much smaller than the dimensions of the computational domain . In addition, the adaptive mesh can be used if the applied numerical method does not lose orders of approximation with irregular partitioning of the domain and also possesses unconditional stability when selecting the time step. In other cases, application of adaptive partitioning does not significantly improve the accuracy of computations, but still considerably increases computational time.